Stabilities of Mixed Type Quintic-Sextic Functional Equations in Various Normed Spaces

Malaya Journal of Matematik, Vol. 9, No. 1, 217-243, 2021

https://doi.org/10.26637/MJM0901/0038

Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces

John Micheal Rassias1, Elumalai Sathya2, Mohan Arunkumar 3*

Abstract In this paper, we introduce ”Mixed Type Quintic - Sextic functional equations” and then provide their general solution, and prove generalized Ulam - Hyers stabilities in Banach spaces and Fuzzy normed spaces, by using both the direct Hyers - Ulam method and the alternative fixed point method. Keywords Quintic functional equation, sextic functional equation, mixed type quintic - sextic functional equation, generalized Ulam - Hyers stability, Banach space, Fuzzy Banach space, Hyers - Ulam method, alternative fixed point method. AMS Subject Classification 39B52, 32B72, 32B82.

1Pedagogical Department - Mathematics and Informatics, The National and Kapodistrian University of Athens,4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece. 2Department of Mathematics, Shanmuga Industries Arts and Science College, Tiruvannamalai - 606 603, TamilNadu, India. 3Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India. *Corresponding author: 1 [email protected]; 2 [email protected]; 3 [email protected] Article History: Received 11 December 2020; Accepted 24 January 2021 c 2021 MJM.

Contents such problems the interested readers can refer the monographs of [1,4,5,8, 18, 22, 24–26, 33, 36, 37, 41, 43, 48]. 1 Introduction...... 217 The general solution of Quintic and Sextic functional 2 General Solution...... 218 equations 3 Stability Results In Banach Space ...... 219 f (x + 3y) − 5 f (x + 2y) + 10 f (x + y) − 10 f (x) 3.1 Hyers - Ulam Method...... 219 + 5 f (x − y) − f (x − 2y) = 120 f (y) (1.1) 3.2 Alternative Fixed Point Method...... 225 and 4 Stability Results In Fuzzy Banach Space ...... 229 f (x + 3y) − 6 f (x + 2y) + 15 f (x + y) − 20 f (x) + 15 f (x − y) 4.1 Deﬁnitions on Fuzzy Banach Spaces...... 229 − 6 f (x − 2y) + f (x − 3y) = 720 f (y) (1.2) 4.2 Hyers - Ulam Method...... 229 4.3 Alternative Fixed Point Method...... 237 was introduced and investigated by T.Z. Xu et. al., [47] and establish the generalized Ulam - Hyers stability in quasi References...... 241 β−normed spaces via ﬁxed point method . In this paper, we introduce the Mixed Type Quintic- Sex- 1. Introduction tic functional equation of the form The stability problem for functional equation is originated 1 E(w + 4v) − 5E(w + 3v) − Eq(w + 3v) + 10E(w + 2v) from a question of S.M. Ulam [45] under group homomor- 2 s phisms and positively answered for an additive functional 5 + Eq(w + 2v) − 10E(w + v) − 5 Eq(w + v) equation on Banach spaces by D.H. Hyers [23] and T. Aoki [2]. 2 s s It was further generalized and marvelous outcome has been q obtained by number of authors one can refer [21, 35, 38, 42]. + 5E(w) + 5 Es (w) − E(w − v) Over the last seven decades, the above problem was tack- 5 − Eq(w − v) + Eq(w − 2v) = 120E(v) + 300 Eq(v) led by numerous authors and its solutions via various forms of 2 s s s functional equations were discussed. For more information on (1.3) Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 218/243

q where Es (w) = E(w) + E(−w) which is different from and using oddness of E, we arrive the subsequent equations 5 6 (1.1) and (1.2). It is easy to verify that E(w) = c1w + c2w E(0) = 0 is the solution of the functional equation (1.3) for any positive E(8w) − 5E(6w) + 10E(4w) − 129E(2w) = 0 (2.2) constants c1,c2. E(8w) − 5E(7w) + 10E(6w) − 10E(5w) + 5E(4w) The main aim of this paper is to provide the general so- − E(3w) − 120E(w) = 0 (2.3) lution and generalized Ulam - Hyers stabilities of (1.3) in Banach spaces and fuzzy normed spaces, by using both the E(7w) − 5E(6w) + 10E(5w) − 10E(4w) + 5E(3w) direct Hyers - Ulam method and the alternative ﬁxed point − E(2w) − 120E(w) = 0 (2.4) method. E(6w) − 5E(5w) + 10E(4w) − 10E(3w) Now, we present the result due to Margolis, Diaz [28] and + 5E(2w) − 121E(w) = 0 (2.5) Radu [34] for ﬁxed point theory. E(5w) − 5E(4w) + 10E(3w) − 10E(2w) − 115E(w) = 0 (2.6)

Theorem 1.1. [28, 34] Suppose that for a complete general- E(4w) − 5E(3w) + 10E(2w) − 129E(w) = 0 (2.7) ized metric space (Ω,δ) and a strictly contractive mapping E(3w) − 4E(2w) − 115E(w) = 0 (2.8) T : Ω −→ Ω with Lipschitz constant L. Then, for each given x ∈ Ω , either for all w ∈ U1. Subtracting (2.3) from (2.2), one can see that 5E(7w) − 15E(6w) + 10E(5w) + 5E(4w) + E(3w) d(T nx,T n+1x) = ∞ ∀ n ≥ 0, − 129E(2w) + 120E(w) = 0 (2.9)

for all w ∈ U1. Multiplying (2.4) by 5 and subtracting from or there exists a natural number n0 such that (2.9), one can observe that n n+1 (FPC1) d(T x,T x) < ∞ for all n ≥ n0 ; (FPC2) The sequence (T nx) is convergent to a fixed point y∗ E(6w) − 40E(5w) + 55E(4w) − 24E(3w) of T − 1245E(2w) − 720E(w) = 0 (2.10) (FPC3) y∗ is the unique fixed point of T in the set ∆ = {y ∈ Ω : d(T n0 x,y) < ∞}; for all w ∈ U1. Multiplying (2.5) by 10 and subtracting from ∗ 1 (2.10), one can find that (FPC4) d(y ,y) ≤ 1−L d(y,Ty) for all y ∈ ∆. 10E(5w) − 45E(4w) + 76E(3w) − 174E(2w) + 1930E(w) = 0 (2.11)

2. General Solution for all w ∈ U1. Multiplying (2.6) by 10 and subtracting from (2.11), one can verify that In this section, we test the general solution of the functional 5E(4w) − 24E(3w) − 74E(2w) + 3080E(w) = 0 equation (1.3). To prove the solution, we deﬁne U and U 1 2 (2.12) be real vector spaces.

for all w ∈ U1. Multiplying (2.7) by 5 and subtracting from (2.12), one can see that Theorem 2.1. For an odd mapping E : U1 −→ U2 fulﬁll- ing the functional equation (1.3) for all w,v ∈ U1, then E is E(3w) − 124E(2w) + 3725E(w) = 0 (2.13) quintic. for all w ∈ U1. Subtracting (2.8) from (2.13), one can arrive

120E(2w) − 3840E(w) = 0 (2.14) Proof. Given E : U1 −→ U2 is an odd function. Using oddness of E in (1.3), one can obtain that for all w ∈ U1. Thus it follows from (2.14), we achieve

120E(2w) = 3840E(w) =⇒ E(2w) = 32E(w) E(w + 4v) − 5E(w + 3v) + 10E(w + 2v) − 10E(w + v) =⇒ E(2w) = 25E(w) (2.15) + 5E(w) − E(w − v) = 120E(v) (2.1) for all w ∈ U1. Hence E is quintic. for all w,v ∈ U1. Now, interchanging (w,v) by (0,0), (0,2w), Theorem 2.2. For an even mapping E : U1 −→ U2 fulﬁlling (4w,w), (3w,w), (2w,w), (w,w), (0,w) and (−w,w) in (2.1) the functional equation (1.3) for all w,v ∈ U1, then E is sextic.

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Proof. Given E : U1 −→ U2 is an even function. Using even- for all w ∈ U1. Thus it follows from (2.29), we achieve ness of E in (1.3), one can obtain that 720E(2w) = 46080E(w) =⇒ E(2w) = 26E(w) E(w + 4v) − 6E(w + 3v) + 15E(w + 2v) − 20E(w + v) (2.30) + 15E(w) − 6E(w − v) + E(w − 2v) = 720E(v) for all w ∈ U . Hence E is sextic. (2.16) 1 Hereafter, through this article, we use the following nota- for all w,v ∈ . Now, interchanging (w,v) by (0,0), (0,2w), U1 tions: (4w,w), (3w,w), (2w,w), (w,w), (0,w) and (−w,w) in (2.16) and using evenness of E, we arrive the subsequent equations • The functional equation can be taken as

E(0) = 0 1 q E (w,v) =E(w + 4v) − 5E(w + 3v) − Es (w + 3v) E(8w) − 6E(6w) + 16E(4w) − 746E(2w) = 0 (2.17) 2 5 E(8w) − 6E(7w) + 15E(6w) − 20E(5w) + 15E(4w) + 10E(w + 2v) + Eq(w + 2v) 2 s − 6E(3w) + E(2w) − 720E(w) = 0 (2.18) q − 10E(w + v) − 5 Es (w + v) E(7w) − 6E(6w) + 15E(5w) − 20E(4w) + 15E(3w) q − 6E(2w) − 719E(w) = 0 (2.19) + 5E(w) + 5 Es (w) − E(w − v) E(6w) − 6E(5w) + 15E(4w) − 20E(3w) 5 − Eq(w − v) + Eq(w − 2v) + 15E(2w) − 726E(w) = 0 (2.20) 2 s s q E(5w) − 6E(4w) + 15E(3w) − 20E(2w) − 704E(w) = 0 − 120E(v) − 300 Es (v) , (2.21) q E(4w) − 6E(3w) + 16E(2w) − 746E(w) = 0 (2.22) where Es (w) = E(w) + E(−w) . 2E(3w) − 12E(2w) − 690E(w) = 0 (2.23) • Let α = {−1,+1}. for all w ∈ U . Subtracting (2.18) from (2.17), one can see 1 • Define a constant ξ as that 2, i f ψ = 0; 6E(7w) − 21E(6w) + 20E(5w) + E(4w) ξ = ψ 1 , i f ψ = 1. + 6E(3w) − 747E(2w) + 720E(w) = 0 (2.24) 2 for all w ∈ U1. Multiplying (2.19) by 6 and subtracting from 3. Stability Results In Banach Space (2.24), one can observe that In this section, we confirm the generalized Ulam - Hyers 15E(6w) − 70E(5w) + 121E(4w) − 84E(3w) stability in Banach space using Hyers - Ulam method and − 711E(2w) + 5034E(w) = 0 (2.25) the alternative fixed point method. In order to establish the stability results, let us take W1 be a normed space and W2 be for all w ∈ U1. Multiplying (2.20) by 15 and subtracting from a Banach space. (2.25), one can find that 3.1 Hyers - Ulam Method 20E(5w) − 104E(4w) + 216E(3w) Theorem 3.1. For an odd mapping Eq : W1 −→ W2 fulfilling − 936E(2w) + 15924E(w) = 0 (2.26) the functional inequality for all w ∈ U1. Multiplying (2.21) by 20 and subtracting from Eq(w,v) ≤ S (w,v) (3.1) (2.26), one can verify that for all w,v ∈ W1. Then there exists one and only quintic func- 16E(4w) − 84E(3w) − 536E(2w) + 30004E(w) = 0 tion Q5 : W1 −→ W2 which satisfying the functional equation (2.27) (1.3) and the functional inequality for all w ∈ U1. Multiplying (2.22) by 16 and subtracting from 1 ∞ 1 Q (w) − E (w) ≤ × γα w, γα w (2.27), one can see that 5 q 5 ∑ 5γα S5 2 2 2 · 5! 1−α 2 γ= 2 12E(3w) − 792E(2w) + 41940E(w) = 0 (2.28) (3.2) for all w ∈ U . Multiplying (2.23) by 6 and subtracting from 1 for all w ∈ W . The mapping Q is defined as (2.28), one can arrive 1 5 1 αβ 720E(2w) − 46080E(w) = 0 (2.29) Q5(w) = lim Eq 2 w (3.3) β→∞ 25αβ

219 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 220/243

2 for all w ∈ W1, where S : W1 −→ [0,∞) is a function fulﬁll- for all w ∈ W1. Multiplying (3.9) by 5, we see that ing the condition 5Eq(7w) − 25Eq(6w) + 50Eq(5w) − 50Eq(4w) 1 αβ αβ lim 2 w,2 v = 0 (3.4) +25Eq(3w) − 5Eq(2w) − 600Eq(w) ≤ 5S (3w,w) 5αβ S β→∞ 2 (3.15)

γα γα for all w,v ∈ W1. The function S5 2 w,2 w is deﬁned by for all w ∈ W1. It follows from (3.14) and (3.15), we arrive

10Eq(6w) − 40Eq(5w) + 55Eq(4w) − 24Eq(3w) γα γα γα γα γα S5 2 w,2 w = S (0,2 · 2w) + S (2 · 4w,2 w) −124Eq(2w) − 720Eq(w) γα γα + 5S (2 · 3w,2 w) ≤ S (0,2w) + S (4w,w) + 5S (3w,w) (3.16) + 10S (2γα · 2w,2γα w) for all w ∈ W1. Multiplying (3.10) by 10, we ﬁnd that + 10S (2γα w,2γα w)

+ 5S (0,2γα w) 10Eq(6w) − 50Eq(5w) + 100Eq(4w) − 100Eq(3w) γα γα + S (−2 w,2 w) (3.5) +50Eq(2w) − 1210Eq(w) ≤ 10S (2w,w) (3.17) for all w ∈ W1. for all w ∈ W1. It follows from (3.17) and (3.16), we obtain

Proof. Using oddness of E in (3.1), one can obtain that 10Eq(5w) − 45Eq(4w) + 76Eq(3w)

−174Eq(2w) + 1930Eq(w) Eq(w + 4v) − 5Eq(w + 3v) + 10Eq(w + 2v) ≤ S (0,2w) + S (4w,w) + 5S (3w,w) + 10S (2w,w) − 10E (w + v) + 5E (w) − E (w − v) q q q (3.18)

−120Eq(v) ≤ S (w,v) (3.6) for all w ∈ W1. Multiplying (3.11) by 10, we see that for all w,v ∈ W1. Now, interchanging (w,v) by (0,2w), (4w,w), (3w,w), (2w,w), (w,w), (0,w) and (−w,w) in (3.6) and using 10Eq(5w) − 50Eq(4w) + 100Eq(3w) oddness of E, we arrive the subsequent inequalities −100Eq(2w) − 1150Eq(w) ≤ 10S (w,w) (3.19) Eq(8w) − 5Eq(6w) + 10Eq(4w) − 129Eq(2w) for all w ∈ . From (3.18) and (3.19), we get ≤ S (0,2w) (3.7) W1

Eq(8w) − 5Eq(7w) + 10Eq(6w) − 10Eq(5w) 5Eq(4w) − 24Eq(3w) − 74Eq(2w) + 3080Eq(w)

+5Eq(4w) − Eq(3w) − 120Eq(w) ≤ S (4w,w) (3.8) ≤ S (0,2w) + S (4w,w) + 5S (3w,w) + 10 (2w,w) + 10 (w,w) (3.20) Eq(7w) − 5Eq(6w) + 10Eq(5w) − 10Eq(4w) S S

+5Eq(3w) − Eq(2w) − 120Eq(w) ≤ S (3w,w) (3.9) for all w ∈ W1. Multiplying (3.12) by 5, we have

Eq(6w) − 5Eq(5w) + 10Eq(4w) − 10Eq(3w) 5Eq(4w) − 25Eq(3w) + 50Eq(2w) − 645Eq(w)

+5Eq(2w) − 121Eq(w) ≤ S (2w,w) (3.10) ≤ 5S (0,w) (3.21)

Eq(5w) − 5Eq(4w) + 10Eq(3w) for all w ∈ W1. Combining (3.20) and (3.21), we arrive −10Eq(2w) − 115Eq(w) ≤ S (w,w) (3.11)

Eq(3w) − 124Eq(2w) + 3725Eq(w) Eq(4w) − 5Eq(3w) + 10Eq(2w) − 129Eq(w) ≤ S (0,w) (3.12) ≤ S (0,2w) + S (4w,w) + 5S (3w,w) + 10 (2w,w) + 10 (w,w) + 5 (0,w) (3.22) Eq(3w) − 4Eq(2w) − 115Eq(w) ≤ S (−w,w) S S S (3.13) for all w ∈ W1. It follows from (3.13) and (3.22), we achieve for all w ∈ W1. From (3.7) and (3.8), we have 120Eq(2w) − 3840Eq(w)

5Eq(7w) − 15Eq(6w) + 10Eq(5w) + 5Eq(4w) ≤ S (0,2w) + S (4w,w) + 5S (3w,w) + 10S (2w,w) + 10 (w,w) + 5 (0,w) + (−w,w) (3.23) +Eq(3w) − 129Eq(2w) + 120Eq(w) S S S

≤ Eq(8w) − 5Eq(6w) + 10Eq(4w) − 129Eq(2w) for all w ∈ W1. Let us take + E (8w) − 5E (7w) + 10E (6w) − 10E (5w) q q q q S (w,w) = S (0,2w) + S (4w,w) + 5S (3w,w) 5 +5Eq(4w) − Eq(3w) − 120Eq(w) + 10S (2w,w) + 10S (w,w) + 5S (0,w) ≤ S (0,2w) + S (4w,w) (3.14) + S (−w,w) (3.24)

220 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 221/243

for all w ∈ W1. Using (3.24) in (3.24), we reach Thus,

120Eq(2w) − 3840Eq(w) ≤ S5(w,w) (3.25) kQ5(w) − R5(w)k

1 n δ δ o for all w ∈ W1. It follows from (3.25) that = Q (2 w) − R (2 w) 25δ 5 5 1 n o Eq(2w) 1 δ δ δ δ − Eq(w) ≤ × S5(w,w) (3.26) ≤ Q5(2 w) − Eq(2 w) + R5(2 w) − Eq(2 w) 25 25 · 5! 25δ ∞ 1 1 γ+δ γ+δ 1 ≤ × S5 2 w,2 w (3.32) for all w ∈ W1. Changing w by 2w and multiply by in 4 ∑ 5(γ+δ) 25 2 · 5! γ=0 2 (3.26) and adding the resultant inequality to (3.26), one can obtain for all w ∈ W1. Letting δ tends to ∞ in (3.32), we have 2 Q5(w) − R5(w) = 0 which implies Q5(w) = R5(w) for all Eq(2 w) − Eq(w) w ∈ W . Hence the theorem holds for α = 1. 210 1 w On changing w by in (3.25), we get 1 1 2 ≤ 5 × S5(w,w) + 5 S5(2w,2w) (3.27) 2 · 5! 2 w 1 w w 5 Eq(w) − 2 Eq ≤ × S5 , (3.33) for all w ∈ W1. Generalized for a positive integer β, we have 2 5! 2 2 w β β−1 γ γ for all w ∈ W . Replacing w by in (3.33) and multiplying Eq(2 w) 1 S (2 w,2 w) 1 − E (w) ≤ × 5 2 5β q 5 ∑ 5γ by 25 and adding the resultant inequality to (3.33), we arrive 2 2 · 5! γ=0 2 (3.28) w E (w) − 210E q q 22 δ 5δ for all w ∈ W1. By deﬁning w by 2 w and dividing by 2 in 1 h w w w w i ≤ × S , + 25S , (3.34) (3.28) and letting β → ∞, it shows that the sequence 5! 5 2 2 5 22 22

( β ) Eq(2 w) for all w ∈ W1. Generalizing for a positive integer β, we see 25β that

5β w Eq(w) − 2 Eq is a Cauchy sequence. Since W2 is complete, this sequence 2β converges to a point Q5(w) in W2. Thus, we define this func- 1 β w w tion by ≤ × 25β−1 , ∑ S5 β β 5! γ=1 2 2 E (2β w) q β Q5(w) = lim (3.29) 1 w w β→∞ 25β = × 25β , (3.35) 5 ∑ S5 β β 2 · 5! γ=1 2 2 for all w ∈ W1. Taking limit as β tends to ∞ in (3.28) and using (3.29), we arrive (3.2) for the case α = 1. for all w ∈ W1. The rest of the proof is similar clues that of To prove that the existing Q5(w) satisfies the functional case α = 1. Hence the proof is complete. equation (1.3), changing (w,v) by (2β w,2β v) and dividing by 25β in (3.1), we get The following corollary is an immediate consequence of Theorem 3.1 regarding the Ulam - Hyers stability [23] of the 1 β β 1 β β functional equation (1.3). Eq(2 w,2 v) ≤ × S (2 w,2 v) (3.30) 25β 25β Corollary 3.2. For an odd mapping Eq : W1 −→ W2 fulfilling for all w,v ∈ W1. Letting β tends to ∞ in (3.30) using (3.4), the functional inequality (3.29), we obtain

Eq(w,v) ≤ Φ (3.36) Q5(w,v) = 0 (3.31) for all w,v ∈ W1 where Φ > 0 is a constant. Then there exists for all w,v ∈ . Thus Q satisﬁes the functional equation W1 5 one and only quintic function Q5 : W1 −→ W2 which satisfying (1.3). the functional equation (1.3) and the functional inequality It is easy to prove that the existence of Q5 is unique. In- deed, let R5 be an another quintic mapping satisfying (1.3) 33Φ Q5(w) − Eq(w) ≤ (3.37) and (3.2). Now 5!|31|

δ 5δ δ 5δ Q5(2 w) = 2 Q5(w) and R5(2 w) = 2 R5(w). for all w ∈ W1.

221 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 222/243

The following corollary is an immediate consequence of Theorem 3.5. For an even mapping Es : W1 −→ W2 fulfilling Theorem 3.1 regarding the Ulam - Hyers - THRassias stability the functional inequality [38] of the functional equation (1.3). kEs(w,v)k ≤ S (w,v) (3.44) Corollary 3.3. For an odd mapping Eq : W1 −→ W2 fulfilling the functional inequality for all w,v ∈ W1. Then there exists one and only sextic function Q6 : W1 −→ W2 which satisfying the functional equation (1.3) Φ ||w||φ + ||v||φ ; Eq(w,v) ≤ (3.38) and the functional inequality Φ||w||φ1 + ||v||φ2 ; ∞ 1 1 γα γα for all w,v ∈ W1 where Φ > 0 is a constant and φ,φ1,φ2 6= 5. kQ (w) − Es(w)k ≤ × S 2 w,2 w 6 26 · 6! ∑ 26γα 6 Then there exists one and only quintic function Q5 : W1 −→ 1−α γ= 2 which satisfying the functional equation (1.3) and the W2 (3.45) functional inequality for all w ∈ . The mapping Q is defined as  φ W1 6 Γ5T Φ||w||  ;  5!|25 − 2φ | 1 αβ Q5(w) − Eq(w) ≤ Q6(w) = lim Es 2 w (3.46) Γ Φ||w||φ1 Γ Φ||w||φ2 β→∞ 26αβ  5T1 + 5T2  5 φ 5 φ ; 5!|2 − 2 1 | 5!|2 − 2 2 | 2 for all w ∈ W1, where S : W1 −→ [0,∞) is a function fulfill- (3.39) ing the condition where 1 αβ αβ lim S 2 w,2 v = 0 (3.47) φ φ φ β→∞ 6αβ Γ5T = 11 · 2 + 5 · 3 + 4 + 43 ; 2 φ φ φ Γ = 10 · 2 1 + 5 · 3 1 + 4 1 + 32 ; (3.40) γα γα 5T1 for all w,v ∈ W1. The function S6 2 w,2 w is defined by φ2 Γ5T2 = 2 + 11 ; γα γα S6 2 w,2 w for all w ∈ W1. = S (0,2γα · 2w) + S (2γα · 4w,2γα w) The following corollary is an immediate consequence of + ( γα · w, γα w) + ( γα · w, γα w) Theorem 3.1 regarding the Ulam - Hyers - JMRassias stability 5S 2 3 2 10S 2 2 2 γα γα γα [42] of the functional equation (1.3). + 10S (2 w,2 w) + 5S (0,2 w) + S (−2γα w,2γα w) (3.48) Corollary 3.4. For an odd mapping Eq : W1 −→ W2 fulfilling the functional inequality for all w ∈ W1. 2 2 Φ ||w||φ ||v||φ + ||w|| φ + ||v|| φ ; Proof. Using evenness of E in (3.44), one can obtain that Eq(w,v) ≤ Φ||w||φ1 ||v||φ2 + ||w||φ1+φ2 + ||v||φ1+φ2 ; kE (w + 4v) − 6E (w + 3v) + 15E (w + 2v) (3.41) s s s − 20Es(w + v) + 15Es(w) − 6Es(w − v) for all w,v ∈ W1 where Φ > 0 is a constant and 2φ,φ1 +φ2 6= 5. +Es(w − 2v) − 720Es(v)k ≤ S (w,v) (3.49) Then there exists one and only quintic function Q5 : W1 −→ W2 which satisfying the functional equation (1.3) and the for all w,v ∈ W1. Now, interchanging (w,v) by (0,2w), (4w,w), functional inequality (3w,w), (2w,w), (w,w), (0,w) and (−w,w) in (3.49) and using evenness of E, we arrive the subsequent inequalities  Γ Φ||w||φ  5J ;  5 2φ 5!|2 − 2 | kEs(8w) − 6Es(6w) + 16Es(4w) − 746Es(2w)k Q5(w) − Eq(w) ≤ φ +φ (3.42) Γ J Φ||w|| 1 2  5 1 ; ≤ S (0,2w) (3.50)  5!|25 − 2φ1+φ2 | kEs(8w) − 6Es(7w) + 15Es(6w) − 20Es(5w) where +15Es(4w) − 6Es(3w) + Es(2w) − 720Es(w)k φ 2φ φ 2φ φ 2φ ≤ S (4w,w) (3.51) Γ5J = 10 · 2 + 11 · 2 + 5(3 + 3 ) + 4 + 4 + 54 ; kE (7w) − 6E (6w) + 15E (5w) − 20E (4w) φ1+φ2 φ1+φ2 φ1+φ2 s s s s Γ5J1 = 11 · 2 + 5 · 3 + 4 +15Es(3w) − 6Es(2w) − 719Es(w)k ≤ S (3w,w) φ φ φ +4 1 + 5 · 3 1 + 10 · 2 1 + 54 ; (3.52)

(3.43) kEs(6w) − 6Es(5w) + 15Es(4w) − 20Es(3w)

+15Es(2w) − 726Es(w)k ≤ S (2w,w) (3.53) for all w ∈ W1.

222 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 223/243

kEs(5w) − 6Es(4w) + 15Es(3w) for all w ∈ W1. From (3.63) and (3.64), we arrive −20Es(2w) − 704Es(w)k ≤ S (w,w) (3.54) k12Es(3w) − 792Es(2w) + 41940Es(w)k kE ( w) − E ( w) + E ( w) − E (w)k ≤ ( ,w) s 4 6 s 3 16 s 2 746 s S 0 ≤ S (0,2w) + S (4w,w) + 6S (3w,w) (3.55) + 15S (2w,w) + 20S (w,w) + 16S (0,w) (3.65) k2Es(3w) − 12Es(2w) − 690Es(w)k ≤ S (−w,w) (3.56) for all w ∈ W1. Multiplying (3.56) by 6 and it follows from (3.65), we achieve for all w ∈ W1. From (3.50) and (3.51), we have k720Es(2w) − 46080Es(w)k k6Es(7w) − 21Es(6w) + 20Es(5w) + Es(4w) ≤ S (0,2w) + S (4w,w) + 6S (3w,w) + 15S (2w,w) +6Es(3w) − 747Es(2w) + 720Es(w)k + 20S (w,w) + 16S (0,w) + 6S (−w,w) (3.66) ≤ kEs(8w) − 6Es(6w) + 16Es(4w) − 746Es(2w)k for all w ∈ W1. Let us take + kEs(8w) − 6Es(7w) + 15Es(6w) − 20Es(5w) (w,w) = (0,2w) + (4w,w) + 6 (3w,w) +15Es(4w) − 6Es(3w) + Es(2w) − 720Es(w)k S6 S S S ≤ S (0,2w) + S (4w,w) (3.57) + 15S (2w,w) + 20S (w,w) + 16S (0,w) + 6S (−w,w) (3.67) for all w ∈ W1. Multiplying (3.52) by 6, we see that for all w ∈ W1. Using (3.67) in (3.67), we reach k6Es(7w) − 36Es(6w) + 90Es(5w) − 120Es(4w) k720Es(2w) − 46080Es(w)k ≤ S6(w,w) (3.68) +90Es(3w) − 36Es(2w) − 5514Es(w)k ≤ 6S (3w,w) (3.58) for all w ∈ W1. It follows from (3.68) that for all w ∈ W1. It follows from (3.57) and (3.58), we arrive Es(2w) 1 − Es(w) ≤ × S6(w,w) (3.69) 26 26 · 6! k15Es(6w) − 70Es(5w) + 121Es(4w) − 84Es(3w) 1 for all w ∈ W1. Changing w by 2w and multiply by in −711Es(2w) + 5034Es(w)k 26 (3.69) and adding the resultant inequality to (3.69), one can ≤ S (0,2w) + S (4w,w) + 6S (3w,w) (3.59) obtain 2 for all w ∈ W1. Multiplying (3.53) by 15, we ﬁnd that Es(2 w) 1 1 − Es(w) ≤ S6(w,w) + S6(2w,2w) 212 26 · 6! 26 k15E (6w) − 90E (5w) + 225E (4w) − 300E (3w) s s s s (3.70) +225Es(2w) − 10890Es(w)k ≤ 15S (2w,w) (3.60) for all w ∈ W1. Generalized for a positive integer β, we have for all w ∈ W1. Combining (3.60) and (3.59), we obtin β β−1 γ γ Es(2 w) 1 S (2 w,2 w) − E (w) ≤ × 6 k20Es(5w) − 104Es(4w) + 216Es(3w) 6β s 6 ∑ 6γ 2 2 · 6! γ=0 2 −936Es(2w) + 15924Es(w)k (3.71) ≤ S (0,2w) + S (4w,w) for all w ∈ W1. The rest of the proof is similar lines to that of + 6S (3w,w) + 15S (2w,w) (3.61) Theorem 3.1. Hence the proof is complete. for all w ∈ W1. Multiplying (3.54) by 20, we get The following corollary is an immediate consequence of Theorem 3.5 regarding the Ulam - Hyers stability [23] of the k20E (5w) − 120E (4w) + 300E (3w) s s s functional equation (1.3). −400Es(2w) − 14080Es(w)k ≤ 20S (w,w) (3.62) Corollary 3.6. For an even mapping Es : W1 −→ W2 fulﬁlling for all w ∈ W1. It follows from (3.61) and (3.62), we have the functional inequality

k16Es(4w) − 84Es(3w) − 536Es(2w) + 30004Es(w)k kEs(w,v)k ≤ Φ (3.72) ≤ (0,2w) + (4w,w) + 6 (3w,w) S S S for all w,v ∈ W1 where Φ > 0 is a constant. Then there exists + 15S (2w,w) + 20S (w,w) (3.63) one and only sextic function Q6 : W1 −→ W2 which satisfying the functional equation (1.3) and the functional inequality for all w ∈ W1. Multiplying (3.55) by 16, we see that 65Φ kQ6(w) − Es(w)k ≤ (3.73) k16Es(4w) − 96Es(3w) + 256Es(2w) − 11936Es(w)k 6!|63| ≤ 16S (0,w) (3.64) for all w ∈ W1.

223 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 224/243

The following corollary is an immediate consequence of Theorem 3.9. For a mapping E : W1 −→ W2 fulfilling the Theorem 3.5 regarding the Ulam - Hyers - THRassias stability functional inequality [38] of the functional equation (1.3). kE (w,v)k ≤ S (w,v) (3.80) Corollary 3.7. For an even mapping Es : W1 −→ W2 fulfilling the functional inequality for all w,v ∈ W1. Then there exists one and only quintic function Q : W −→ W and one and only sextic function Φ||w||φ + ||v||φ ; 5 1 2 kEs(w,v)k ≤ (3.74) Q6 : W1 −→ W2 which satisfying the functional equation (1.3) ||w||φ1 + ||v||φ2 Φ ; and the functional inequality for all w,v ∈ W1 where Φ > 0 is a constant and φ,φ1,φ2 6= 6. kE(w) − Q5(w) − Q6(w)k Then there exists one and only sextic function Q6 : W1 −→ which satisfying the functional equation (1.3) and the 1 ∞ 1 n W2 ≤ γα w, γα w 6 ∑ 5γα S5 2 2 functional inequality 2 · 5! 1−α 2 γ= 2  φ o Γ6T Φ||w|| + − γα w,− γα w  ; S5 2 2  6!|26 − 2φ | kQ6(w) − Es(w)k ≤ φ φ 1 ∞ 1 n Γ6T1 Φ||w|| 1 Γ6T2 Φ||w|| 2 + γα w, γα w  + ; 7 ∑ 6γα S6 2 2  6!|26 − 2φ1 | 6!|26 − 2φ2 | 2 · 6! 1−α 2 γ= 2 (3.75) γα γα o + S6 − 2 w,−2 w (3.81) where for all w ∈ W1. The mappings Q5 and Q6 are defined in (3.3) = · φ + · φ + φ + Γ6T 16 2 6 3 4 91 ; and (3.46) for all w ∈ W , where S : W 2 −→ [0,∞) is a 1 1 Γ = 15 · 2φ1 + 6 · 3φ1 + 4φ1 + 26 ; (3.76) function fulfilling the conditions (3.4) and (3.47) for all w,v ∈ 6T1 γα γα γα γα φ2 W1. The functions S5 2 w,2 w and S6 2 w,2 w Γ6T2 = 2 + 65 ; are given in (3.5) and (3.48) for all w ∈ W1. for all w ∈ W . 1 Proof. We know that by definition of odd function, we have The following corollary is an immediate consequence of Theorem 3.5 regarding the Ulam - Hyers - JMRassias stability Eq(w) − Eq(−w) EO(w) = (3.82) [42] of the functional equation (1.3). 2

Corollary 3.8. For an even mapping Es : W1 −→ W2 fulﬁlling for all w ∈ W1. It follows from (3.82) that the functional inequality 1n o kEO(w,v)k ≤ Eq(w,v) − Eq(−w,−v) Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; 2 kEs(w,v)k ≤ + + 1n o Φ ||w||φ1 ||v||φ2 + ||w||φ1 φ2 + ||v||φ1 φ2 ; ≤ S (w,v) + S (−w,−v) (3.83) 2 (3.77) for all w ∈ W1. Thus by Theorem 3.1, we arrive for all w,v ∈ W1 where Φ > 0 is a constant and 2φ,φ1 +φ2 6= 6. Then there exists one and only sextic function Q6 : W1 −→ kQ5(w) − EO(w)k W2 which satisfying the functional equation (1.3) and the ∞ 1 1 n γα γα functional inequality ≤ S5 2 w,2 w 26 · 5! ∑ 25γα γ= 1−α  2φ 2 Γ6J Φ||w||  ; γα γα o  6!|26 − 22φ | + S5 − 2 w,−2 w (3.84) kQ6(w) − Es(w)k ≤ φ +φ (3.78) Γ J Φ||w|| 1 2  6 1 ;  6!|26 − 2φ1+φ2 | for all w ∈ W1. We know that by deﬁnition of even function, we have where Eq(w) + Eq(−w) φ 2φ φ 2φ φ 2φ EE (w) = (3.85) Γ6J = 16 · 2 + 15 · 2 + 6(3 + 3 ) + 4 + 4 + 117 ; 2 φ1+φ2 φ1+φ2 φ1 Γ6J1 = 16 · 2 + 5(3 + 3 ) for all w ∈ W1. It follows from (3.85) that +4φ1+φ2 + 4φ1 + 15 · 2φ1 + 90 ; 1n o kE (w,v)k ≤ E (w,v) − E (−w,−v) E 2 s s (3.79) 1n o ≤ S (w,v) + S (−w,−v) (3.86) for all w ∈ W1. 2

224 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 225/243

for all w ∈ W1. Thus by Theorem 3.5, we arrive The following corollary is an immediate consequence of Theorem 3.9 regarding the Ulam - Hyers - JMRassias stability kQ6(w) − EE (w)k [42] of the functional equation (1.3). ∞ 1 1 n γα γα ≤ w, w Corollary 3.12. For a mapping E : W1 −→ W2 fulﬁlling the 7 ∑ 6γα S6 2 2 2 · 6! 1−α 2 functional inequality γ= 2 o γα γα + S6 − 2 w,−2 w (3.87) Φ ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; kE (w,v)k ≤ Φ||w||φ1 ||v||φ2 + ||w||φ1+φ2 + ||v||φ1+φ2 ; w ∈ for all W1. Now, deﬁne a function (3.93)

E(w) = EO(w) + EE (w) (3.88) for all w,v ∈ W1 where Φ > 0 is a constant and 2φ,φ1 + φ 6= 5,6. Then there exists one and only quintic function for all w ∈ W . Combining (3.88), (3.84) and (3.87), we 2 1 Q : −→ and one and only sextic function Q : −→ derived our result. 5 W1 W2 6 W1 W2 which satisfying the functional equation (1.3) and the The following corollary is an immediate consequence of functional inequality Theorem 3.9 regarding the Ulam - Hyers stability [23] of the kE(w) − Q (w) − Q (w)k functional equation (1.3). 5 6  n o Γ5J + Γ6J Φ||w||2φ ; For a mapping fulfilling the  5!|25−22φ | 6!|26−22φ | Corollary 3.10. E : W1 −→ W2 ≤ n o functional inequality Γ5J1 + Γ6J1 Φ||w||φ1+φ2 ;  5!|25−2φ1+φ2 | 6!|26−2φ1+φ2 | (3.94) kE (w,v)k ≤ Φ (3.89) 0 where Γgs are defined in (3.43) and (3.79) respectively for all for all w,v ∈ W1 where Φ > 0 is a constant. Then there w ∈ W1. exists one and only quintic function Q5 : W1 −→ W2 and one and only sextic function Q6 : W1 −→ W2 which satisfying the 3.2 Alternative Fixed Point Method functional equation (1.3) and the functional inequality Theorem 3.13. For an odd mapping Eq : W1 −→ W2 fulfilling 33 65 the functional inequality (3.1) for all w,v ∈ W1, where S : 2 kE(w) − Q5(w) − Q6(w)k ≤ + Φ (3.90) −→ [0,∞) is a function fulfilling the condition 5!|31| 6!|63| W1 1 β β for all w ∈ W1. lim S ξψ w,ξψ v = 0 (3.95) β→∞ 5β ξψ The following corollary is an immediate consequence of Theorem 3.9 regarding the Ulam - Hyers - THRassias stability for all w,v ∈ W1. Then there exists one and only quintic func- [38] of the functional equation (1.3). tion Q5 : W1 −→ W2 which satisfying the functional equation (1.3) and the functional inequality Corollary 3.11. For a mapping E : W1 −→ W2 fulfilling the 1− functional inequality L ψ Q5(w) − Eq(w) ≤ S w,w (3.96) 1 − L Φ||w||φ + ||v||φ ; kE (w,v)k ≤ (3.91) for all w ∈ . If = ( ), with the property Φ||w||φ1 + ||v||φ2 ; W1 L L ψ 1 for all w,v ∈ W1 where Φ > 0 is a constant and φ,φ1,φ2 6= 5,6. 5 S5 ξψ w,ξψ w = LS5 w,w (3.97) ξψ Then there exists one and only quintic function Q5 : W1 −→ W2 and one and only sextic function Q6 : W1 −→ W2 which with the condition that satisfying the functional equation (1.3) and the functional 1 w w inequality S w,w = S , , (3.98) 5 5! 5 2 2 kE(w) − Q5(w) − Q6(w)k  where S5 w,w is defined in (3.24) for all w ∈ W1. Γ5T Γ6T  + Φ||w||φ ;  5!|25 − 2φ | 6!|26 − 2φ | Proof. Consider the set  n ≤ Γ5T1 + Γ6T1 Φ||w||φ1 (3.92) n o 5!|25−2φ1 | 6!|26−2φ1 |  B = E1|E1 : W1 −→ W2,E1(0) = 0 (3.99)  Γ Γ o  + 6T2 + 5T2 Φ||w||φ2 ;  6!|26−2φ2 | 5!|25−2φ2 | Let us introduce the generalized metric on (3.99) by where Γgs are defined in (3.40) and (3.76) respectively for all n o w ∈ W1. inf ζ : kE1(w) − E2(w)k ≤ ζS5 w,w (3.100)

225 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 226/243

for all w ∈ W1. One can easy to see verify that (3.100) is for all w ∈ W1. Finally, by condition (FPC4) of Theorem 1.1, complete with respect to the defined metric. Define a mapping we get ϒ : → by B B 1 Eq(w) − Q5(w) ≤ Eq(w) − ϒEq(w) 1 1 − L ϒEq(w) = 5 Eq(ξψ ) (3.101) 1−ψ ξψ L =⇒ Eq(w) − Q5(w) ≤ S5(w,w) 1 − L for all w ∈ W1. Now, for any E1,E2 ∈ B, we arrive for all w ∈ W1. This completes the proof of the theorem. kE1(w) − E2(w)k ≤ ζS5 w,w The following corollary is an immediate consequence of Theorem 3.13 regarding the Ulam - Hyers stability [23], 1 1 =⇒ E ( ) − E ( ) ≤ w,w Ulam - Hyers - THRassias stability [38] and Ulam - Hyers - 5 1 ξψ 5 2 ξψ ζS5 ξψ ξψ JMRassias stability [42] of the functional equation (1.3). Corollary 3.14. For an odd mapping : −→ fulfill- =⇒ kϒE1(w) − ϒE2(w)k ≤ L ζS5 w,w Eq W1 W2 ing the functional inequality for all w ∈ . This implies that ϒ is a strictly contractive W1  Φ; mapping on B with Lipschitz constant L . It follows from  φ φ Eq(w,v) ≤ Φ ||w|| + ||v|| ; (3.26) that  Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ;

Eq(2w) 1 (3.109) − Eq(w) ≤ S5(w,w) (3.102) 25 25 · 5! for all w,v ∈ W1 where Φ > 0 and φ is a constant. Then there exists one and only quintic function Q which for all w ∈ W1. 5 : W1 −→ W2 For the case ψ = 0, it follows from (3.97), (3.98), (3.101) satisfying the functional equation (1.3) and the functional and (3.102), inequality  33Φ 1−0  ; ϒEq(w) − Eq(w) ≤ LS5(w,w) = L S5(w,w)   5!|31| (3.103)  Γ Φ||w||φ 5T 6= Q5(w) − Eq(w) ≤ 5 φ ; φ 5 (3.110) for all w ∈ W1. 5!|2 − 2 |  2φ Replacing w by w in (3.102), we obtain  Γ5JΦ||w|| 2  ; 2φ 6= 5  5!|25 − 22φ | 5 w 1 w w Eq(w) − 2 Eq ≤ S , (3.104) where Γgs are defined in (3.40) and (3.43) respectively for all 2 5! 5 2 2 w ∈ W1. for all w ∈ W1. Proof. If we take For the case ψ = 1, it follows from (3.97), (3.98), (3.101)  and (3.104),  Φ; (w,v) = ||w||φ + ||v||φ ; 1−1 S Φ Eq(w) − ϒEq(w) ≤ S5(w,w) = L S5(w,w)  Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; (3.105) (3.111) for all w ∈ W1. From (3.103) and (3.105), we see that β β for all w,v ∈ W1. Changing (w,v) by (ξψ w,ξψ v) and dividing 1−ψ 5β Eq(w) − ϒEq(w) ≤ L S5(w,w) (3.106) by ξψ in (3.111) and letting β tends to ∞, we see that (3.95) holds for all w,v ∈ W1. for all w ∈ W1. Thus, condition (FPC1) of Theorem 1.1 holds. It follows from (3.98), (3.111) and (3.24), one can find It follows from condition (FPC2 ) of Theorem 1.1 , that there that exists a fixed point Q5 of ϒ in L such that S w,w 1 Q5(w) = lim Eq ξψ w (3.107) 1 w w β→∞ ξ 5 = S , ψ 5! 2 2  33Φ for all w ∈ W . To prove the existing Q satisfies (1.3), the ; 1 5  5!  φ φ φ φ proof is similar to that of Theorem 3.1.  11·2 +5·3 +4 +43 Φ||w|| Again by condition (FPC3) of Theorem 1.1, Q5 is the = φ ; 5!2 unique fixed point of ϒ in the set  φ 2φ φ 2φ φ 2φ 2φ  10·2 +11·2 +5(3 +3 )+4 +4 +54 Φ||w||  ; n o 5!22φ ∆ = Q5 : Eq(w) − Q5(w) ≤ ∞ (3.108) (3.112)

226 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 227/243

5−φ 5−φ for all w ∈ W1. For ψ = 1 : L = ξψ = 2 : φ > 5 Again, it follows from (3.97), (3.111) and (3.24), one can 1− observe that L ψ Q5(w) − Eq(w) ≤ S5 w,w 1 − L 1 1−1 25−φ 5 S5 ξψ w,ξψ w ξψ = S w,w − 5−φ 5  33Φ 1 2 5 ; φ  5! ξψ 2  = S w,w  φ φ φ φ φ 5 5  11·2 +5·3 +4 +43 Φ||w|| 2 − 2 − ; = 5! ξ 5 φ ψ 1 1 2φ−5  For ψ = 0 : L = = = 2 : 2φ < 5  10·2φ +11·22φ +5(3φ +32φ )+4φ +42φ +54 Φ||w||2φ 5−2φ 25−2φ  ξψ  5−2φ ; 5! ξψ  1−ψ L  LS5 w,w ; Q (w) − E (w) ≤ w,w  5 q S5  1 − L  1−0  2φ−5 = LS5 w,w ; (3.113) 2  = S5 w,w  1 − 22φ−5   w,w ; 22φ  LS5 = S w,w 25 − 22φ 5 for all w ∈ . Thus, the functional inequality (3.96) holds W1 5−2φ 5−2φ for the following cases. For ψ = 1 : L = ξψ = 2 : 2φ > 5

1 1 1−ψ For ψ = 0 : L = = = 2−5 L 5 25 Q5(w) − Eq(w) ≤ S5 w,w ξψ 1 − L 1−1 25−2φ 1−ψ L = S5 w,w Q5(w) − Eq(w) ≤ S5 w,w 5−2φ 1 − L 1 − 2 2φ 1−0 2 −5 = S w,w 2 22φ − 25 5 = S w,w 1 − 2−5 5 1 = S5 w,w 31 Theorem 3.15. For an even mapping Es : W1 −→ W2 fulﬁlling 5 5 the functional inequality (3.44) for all w,v ∈ W1, where S : For ψ = 1 : L = ξψ = 2 2 W1 −→ [0,∞) is a function fulﬁlling the condition

1−ψ 1 β β L lim S ξψ w,ξψ v = 0 (3.114) Q5(w) − Eq(w) ≤ S5 w,w β→∞ 6β 1 − L ξψ 1−1 5 2 for all w,v ∈ W1. Then there exists one and only sextic function = S5 w,w Q : W −→ W which satisfying the functional equation (1.3) 1 − 25 6 1 2 and the functional inequality 1 = S5 w,w −31 L 1−ψ kQ6(w) − Es(w)k ≤ S6 w,w (3.115) 1 1 1 − L For = = = = φ−5 < ψ 0 : L 5−φ 5−φ 2 : φ 5 ξψ 2 for all w ∈ W1. If L = L (ψ), with the property 1 1−ψ w, w = w,w (3.116) L 6 S6 ξψ ξψ LS6 Q5(w) − Eq(w) ≤ S5 w,w ξψ 1 − L 1−0 φ−5 with the condition that 2 = S5 w,w 1 w w 1 − 2φ−5 S6 w,w = S6 , , (3.117) 6! 2 2 2φ = 5 S5 w,w 2 − 2φ where S6 w,w is deﬁned in (3.67) for all w ∈ W1.

227 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 228/243

Proof. Consider the set Theorem 3.17. For a mapping E : W1 −→ W2 fulfilling the n o functional inequality (3.80) for all w,v ∈ W1, where S : 2 B = E2|E2 : W1 −→ W2,E2(0) = 0 (3.118) W1 −→ [0,∞) is a function fulfilling the conditions (3.95) and (3.114) for all w,v ∈ W1. Then there exists one and only Let us introduce the generalized metric on (3.118) by quintic function Q5 : W1 −→ W2 and one and only sextic func- n o tion Q6 : W1 −→ W2 which satisfying the functional equation inf ζ : kE1(w) − E2(w)k ≤ ζS6 w,w (3.119) (1.3) and the functional inequality for all w ∈ W1. One can easy to see verify that (3.119) is kE(w) − Q5(w) − Q6(w)k complete with respect to the defined metric. Define a mapping L 1−ψ n o ϒ : → by ≤ S w,w + S − w,−w (3.123) B B 1 − L 1 for all w ∈ W1. If L = L (ψ), with the properties and condi- ϒEs(w) = 6 Es(ξψ ) (3.120) ξψ tions (3.97), (3.116)(3.98), (3.117) for all w ∈ W1. for all w ∈ W1. Now, for any E1,E2 ∈ B, we arrive Proof. From (3.83) of Theorem 3.9 and by Theorem 3.13, we obtain kE1(w) − E2(w)k ≤ ζS6 w,w 1 L 1−ψ n o kQ5(w) − EO(w)k ≤ × S w,w +S −w,−w 1 1 2 1 − L =⇒ E ( ) − E ( ) ≤ w,w 6 1 ξψ 6 2 ξψ ζS6 (3.124) ξψ ξψ for all w ∈ W1. Also, from (3.85) of Theorem 3.9 and by =⇒ kϒE1(w) − ϒE2(w)k ≤ L ζS6 w,w Theorem 3.15, we obtain for all w ∈ W . This implies that ϒ is a strictly contractive 1 1 L 1−ψ n o mapping on B with Lipschitz constant L . The rest of the kQ6(w) − EsE(w)k ≤ × S w,w +S −w,−w proof is similar lines to that of Theorem 3.13. This completes 2 1 − L the proof of the theorem. (3.125) for all w ∈ . Finally, from (3.88) of of Theorem 3.9 and The following corollary is an immediate consequence W1 (3.124), (3.125), we prove our desired result. of Theorem 3.15 regarding the Ulam - Hyers stability [23], Ulam - Hyers - THRassias stability [38] and Ulam - Hyers - The following corollary is an immediate consequence JMRassias stability [42] of the functional equation (1.3). of Theorem 3.17 regarding the Ulam - Hyers stability [23], Ulam - Hyers - THRassias stability [38] and Ulam - Hyers - Corollary 3.16. For an even mapping Es : W1 −→ W2 fulfilling the functional inequality JMRassias stability [42] of the functional equation (1.3). Corollary 3.18. For an mapping : −→ fulfilling the  E W1 W2 Φ; functional inequality  φ φ kEs(w,v)k ≤ Φ ||w|| + ||v|| ;   Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; Φ;  φ φ (3.121) kE (w,v)k ≤ Φ ||w|| + ||v|| ;  Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; for all w,v ∈ W1 where Φ > 0 and φ is a constant. Then there (3.126) exists one and only sextic function Q6 : W1 −→ W2 which satisfying the functional equation (1.3) and the functional for all w,v ∈ W1 where Φ > 0 and φ is a constant. Then there inequality exists one and only quintic function Q5 : W1 −→ W2 and one and only sextic function Q : −→ which satisfying the  6 W1 W2 65Φ functional equation (1.3) and the functional inequality  ;  6!|63|  φ  Γ6T Φ||w|| kE(w) − Q5(w) − Q6(w)k kQ6(w) − Es(w)k ≤ ; φ 6= 6 (3.122) 6!|26 − 2φ |  33 65   + 2Φ;  ||w||2φ  5!|31| 6!|63|  Γ6J Φ  φ  ; 2 6= 6 Γ T Γ6T 2Φ||w||  6 2φ φ ≤ 5 + ; 6= 5,6 6!|2 − 2 | 5!|25−2φ | 6!|26−2φ | φ   Γ5J Γ6J 2φ 0  5 2φ + 6 2φ 2Φ||w|| ; 2φ 6= 5,6 where Γgs are defined in (3.76) and (3.79) respectively for all 5!|2 −2 | 6!|2 −2 | w ∈ W1. (3.127) 0 Proof. The proof of the corollary is similar clues and ideas of where Γgs are defined in (3.40), (3.43), (3.76) and (3.79) Corollary 3.14. Hence the details of the proof are omitted. respectively for all w ∈ W1.

228 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 229/243

4. Stability Results In Fuzzy Banach 4.2 Hyers - Ulam Method Space Theorem 4.7. For an odd mapping Eq : W1 −→ W2 fulfilling the functional inequality In this section, we confirm the generalized Ulam - Hyers stability in Fuzzy Banach space using Hyers and Radus methods. 0 Fuzzy theory was initiated by Zadeh [49] in 1965. Cur- N (Eq(w,v),z) ≥ N (S (w,v),z) (4.1) rently, this theory is a powerful tool for modeling uncertainty and vagueness in miscellaneous problems arising in the field for all w,v ∈ W1 and all z ∈ W1. Then there exists one and of science and engineering. We use the definition of fuzzy only quintic function Q5 : W1 −→ W2 which satisfying the normed spaces given in [11] and [29–32]. functional equation (1.3) and the functional inequality

4.1 Definitions on Fuzzy Banach Spaces 5 0 5!|2 − ε| z Definition 4.1. Let X be a real linear space. A function N (Q (w) − Eq(w),z) ≥ N S (w,w), 5 5 33 N : X × R −→ [0,1] is said to be a fuzzy norm on X if for all x,y ∈ X and all s,t ∈ R, (4.2) (FNS1) N(x,c) = 0 for c ≤ 0; (FNS2) x = 0 if and only if N(x,c) = 1 for all c > 0; for all w ∈ W1 and all z ∈ W1. The mapping Q5 is defined as t (FNS3) N(cx,t) = N x, |c| if c 6= 0; 1 (FNS4) N(x + y,s +t) ≥ min{N(x,s),N(y,t)}; lim Q (w) − E 2αβ w ,z = 1 (4.3) N 5 5αβ q (FNS5) N(x,·) is a non-decreasing function on R and β→∞ 2 limt→∞N(x,t) = 1; (FNS6) for x 6= 0,N(x,·) is (upper semi) continuous on . 2 R for all w ∈ W1 and all z ∈ W1, where S : W1 −→ W3 is a The pair (X,N) is called a fuzzy normed linear space. function fulfilling the condition One may regard N(X,t) as the truth-value of the statement the norm of x is less than or equal to the real number t’. lim N 0 S 2αβ w,2αβ v ,25αβ z = 1 (4.4) Example 4.2. Let (X,|| · ||) be a normed linear space. Then β→∞  t  , t > 0, x ∈ X, for all w,v ∈ W and all z ∈ W with the condition that N (x,t) = t + kxk 1 1  0, t ≤ 0, x ∈ X N 0 S 2αβ w,2αβ w ,z = N 0 εαβ S w,w ,z (4.5) is a fuzzy norm on X. Definition 4.3. Let (X,N) be a fuzzy normed linear space. for all w ∈ W1 and all z ∈ W1 for some ε > 0 with 0 < Let xn be a sequence in X. Then xn is said to be convergent if ε α there exists x ∈ X such that lim N(xn − x,t) = 1 for all t > 0. < 1. The function S5 w,w is defined by n→∞ 25 In that case, x is called the limit of the sequence xn and we denote it by N − lim xn = x. n 0 0 n→∞ S5(w,w) = min N (S (0,2w),z) + N (S (4w,w),z) Definition 4.4. A sequence x in X is called Cauchy if for n + N 0 (S (3w,w),z) + N 0 (S (2w,w),z) each ε > 0 and each t > 0 there exists n0 such that for all + 0 ( (w,w),z) + 0 ( (0,w),z) n ≥ n0 and all p > 0, we have N(xn+p − xn,t) > 1 − ε. N S N S o Definition 4.5. Every convergent sequence in a fuzzy normed + N 0 (S (−w,w),z) (4.6) space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed for all w ∈ W and all z ∈ W . space is called a fuzzy Banach space. 1 1

Deﬁnition 4.6. A mapping f : X −→ Y between fuzzy normed Proof. Using oddness of E in (4.1), one can obtain that spaces X and Y is continuous at a point x0 if for each sequence {xn} covering to x0 in X, the sequence f {xn} converges to N (Eq(w + 4v) − 5Eq(w + 3v) + 10Eq(w + 2v) f (x0) . If f is continuous at each point of x0 ∈ X then f is said to be continuous on X. −10Eq(w + v) + 5Eq(w) − Eq(w − v) − 120Eq(v),z) 0 The stability of a quiet number of functional equations in ≥ N (S (w,v),z) (4.7) Fuzzy Banach space were inspected in [3, 6, 7, 9, 10, 29–32]. In order to establish results, we need the following assump- for all w,v ∈ W1 and all z ∈ W1. Now, interchanging (w,v) by tions. Let W1 be a linear space, W2 be a Fuzzy Banach space (0,2w), (4w,w), (3w,w), (2w,w), (w,w), (0,w) and (−w,w) and W3 be a Fuzzy normed space. in (4.7) and using oddness of Eq, we arrive the subsequent

229 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 230/243 inequalities have

N (Eq(8w) − 5Eq(6w) + 10Eq(4w) − 129Eq(2w),z) N (10Eq(6w) − 50Eq(5w) + 100Eq(4w) − 100Eq(3w) 0 ≥ N (S (0,2w),z) (4.8) 0 +50Eq(2w) − 1210Eq(w),10z) ≥ N (S (2w,w),z) N (Eq(8w) − 5Eq(7w) + 10Eq(6w) − 10Eq(5w) (4.18) +5Eq(4w) − Eq(3w) − 120Eq(w),z) ≥ N 0 (S (4w,w),z) (4.9) for all w ∈ W1 and all z ∈ W1. Combining (4.18) and (4.17), we obtain N (Eq(7w) − 5Eq(6w) + 10Eq(5w) − 10Eq(4w)

+5Eq(3w) − Eq(2w) − 120Eq(w),z) N (10Eq(5w) − 45Eq(4w) + 76Eq(3w) − 174Eq(2w) ≥ N 0 (S (3w,w),z) (4.10) +1930Eq(w),z + z + 5z + 10z) N (Eq(6w) − 5Eq(5w) + 10Eq(4w) − 10Eq(3w) n ≥ min N 0 (S (0,2w),z) + N 0 (S (4w,w),z) +5Eq(2w) − 121Eq(w),z) 0 o ≥ N (S (2w,w),z) (4.11) + N 0 (S (3w,w),z) + N 0 (S (2w,w),z) (4.19) N (Eq(5w) − 5Eq(4w) + 10Eq(3w) − 10Eq(2w) 0 −115Eq(w),z) ≥ N (S (w,w),z) (4.12) for all w ∈ W1 and all z ∈ W1. Multiplying (4.12) by 10, we N (Eq(4w) − 5Eq(3w) + 10Eq(2w) ﬁnd that 0 −129Eq(w),z) ≥ N (S (0,w),z) (4.13) N (10Eq(5w) − 50Eq(4w) + 100Eq(3w) − 100Eq(2w) N (Eq(3w) − 4Eq(2w) − 115Eq(w),z) 0 0 −1150Eq(w),10z) ≥ N (S (w,w),z) (4.20) ≥ N (S (−w,w),z) (4.14) for all w ∈ W1 and all z ∈ W1. From (4.8) and (4.9), we get for all w ∈ W1 and all z ∈ W1. It follows from (4.19) and (4.20), we arrive N (5Eq(7w) − 15Eq(6w) + 10Eq(5w) + 5Eq(4w)

+Eq(3w) − 129Eq(2w) + 120Eq(w),z + z) N (5Eq(4w) − 24Eq(3w) − 74Eq(2w) + 3080Eq(w) n ,z + z + 5z + 10z + 10z) ≥ min N (Eq(8w) − 5Eq(6w) + 10Eq(4w) n ≥ min 0 ( (0,2w),z) + 0 ( (4w,w),z) −129Eq(2w),z) N S N S 0 0 + N (Eq(8w) − 5Eq(7w) + 10Eq(6w) − 10Eq(5w) + N (S (3w,w),z) + N (S (2w,w),z) o 0 o +5Eq(4w) − Eq(3w) − 120Eq(w),z) + N (S (w,w),z) (4.21) n o ≥ min N 0 (S (0,2w),z) + N 0 (S (4w,w),z) for all w ∈ W1 and all z ∈ W1. Multiplying (4.13) by 5, we see (4.15) that for all w ∈ W1 and all z ∈ W1. Multiplying (4.10) by 5, we see that N (5Eq(4w) − 25Eq(3w) + 50Eq(2w) − 645Eq(w),5z) ≥ N 0 (S (0,w),z) (4.22) N (5Eq(7w) − 25Eq(6w) + 50Eq(5w) − 50Eq(4w)

+25Eq(3w) − 5Eq(2w) − 600Eq(w),5z) for all w ∈ W1 and all z ∈ W1. From (4.21) and (4.22), we ≥ N 0 (S (3w,w),z) (4.16) have for all w ∈ W1 and all z ∈ W1. It follows from (4.15) and N (Eq(3w) − 124Eq(2w) + 3725Eq(w) (4.16), we arrive ,z + z + 5z + 10z + 10z + 5z) n 0 0 N (10Eq(6w) − 40Eq(5w) + 55Eq(4w) − 24Eq(3w) ≥ min N (S (0,2w),z) + N (S (4w,w),z) −124E (2w)720E (w),z + z + 5z) 0 0 q q + N (S (3w,w),z) + N (S (2w,w),z) n 0 0 ≥ min ( (0,2w),z) + ( (4w,w),z) 0 0 o N S N S + N (S (w,w),z) + N (S (0,w),z) o + N 0 (S (3w,w),z) (4.17) (4.23) for all w ∈ W1 and all z ∈ W1. Multiplying (4.11) by 10, we for all w ∈ W1 and all z ∈ W1. It follows from (4.14) and

230 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 231/243

δ (4.23), we arrive for all w ∈ W1 and all z ∈ W1. Interchanging w by 2 w in (4.30) and using (4.5), (FNS3) and then switching z by εδ z, N (120Eq(2w) − 3840Eq(w), we achieve z + z + 5z + 10z + 10z + 5z + z) n 0 0 β+δ δ β−1 ! ≥ min N (S (0,2w),z) + N (S (4w,w),z) Eq(2 w) Eq(2 w) ε γ+δ 33z N − , 25β+δ 25δ ∑ 25 25 · 5! + N 0 (S (3w,w),z) + N 0 (S (2w,w),z) γ=0 0 0 0 + N (S (w,w),z) + N (S (0,w),z) ≥ N (S5(w,w),z) (4.31) o + N 0 (S (−w,w),z) for all w ∈ W1 and all z ∈ W1 for β > δ ≥ 0. With the help of 0 = N (S5(w,w),z) (4.24) (FNS3), (4.31) can be remodiﬁed as for all w ∈ W1 and all z ∈ W1. The above equation can be ! E (2β+δ w) E (2δ w) written as (4.24) that q − q ,z N 5β+δ 5δ 0 2 2 N (120Eq(2w) − 3840Eq(w),33z) ≥ N (S5(w,w),z) (4.25)     for all w ∈ W1 and all z ∈ W1. Using (FNS3) in (4.25), one 0  z  can get that ≥ N S5(w,w),  (4.32)  δ+β−1   ε γ 33  Eq(2w) 33z 0  ∑ 5 5  N − Eq(w), ≥ N (S (w,w),z) 2 2 · 5! 25 25 · 5! 5 γ=δ (4.26) for all w ∈ W and all z ∈ W . β 1 1 for all w ∈ W1 and all z ∈ W1. Changing w by 2 w and using ε α β−1 ε γ (4.5), (FNS3) in (4.26), we arrive By data < 1 and since < ∞, by the 25 ∑ 25 β+1 β ! γ=0 Eq(2 w) Eq(2 w) 33z N − , Cauchy criterion for convergence and (FNS5) implies that 25(β+1) 25β 2525β · 5! ( ) 0 β β β ≥ N S5(2 w,2 w),z Eq(2 w) 25β 0 β = N ε S5(w,w),z z = 0 (w,w), (4.27) is a Cauchy sequence in W2. Since W2 is a fuzzy Banach N S5 β ε space, this sequence converges to some point Q5 ∈ W2. Thus, β Q −→ for all w ∈ W1 and all z ∈ W1. Switching z by ε z in (4.27) deﬁne the mapping 5 : W1 W2 by β+1 β β ! Eq(2 w) Eq(2 w) ε 33z N − , 1 β 5(β+1) 5β 5β 25 · 5! lim N Q5(w) − Eq 2 w ,z = 1 (4.33) 2 2 2 β→∞ 25β 0 ≥ N (S5(w,w),z) (4.28) for all w ∈ W and all z ∈ W . Setting δ = 0 and approaching for all w ∈ W and all z ∈ W . One can easy to verify that 1 1 1 1 β tends to ∞ in (4.32), we reach E (2β w) β−1 E (2γ+1w) E (2γ w) q − E (w) = q − q . 5β q ∑ 5(γ+1) 5γ 5 2 γ=0 2 2 0 z(2 − ε)5! N (Q5(w) − Eq(w),z) ≥ N S5(w,w), (4.29) 33 (4.34) From (4.28) and (4.29), we reach

β β−1 γ ! Eq(2 w) ε 33z for all w ∈ and all z ∈ . − E (w), W1 W1 N 5β q ∑ 5γ 5 2 γ=0 2 2 · 5! To prove that the existing Q5(w) satisﬁes the functional equation (1.3), changing (w,v) by (2β w,2β v) and dividing by β−1 γ+1 γ γ [ Eq(2 w) Eq(2 w) ε z 5β ≥ min N − , 2 in (4.1), we get 5(γ+1) 25γ 25γ 25 · 5! γ=0 2 1 β−1 ( (w,v),z) = (2β w,2β v),z [ 0 0 N Eq N 5β Eq ≥ min N (S5(w,w),z) = N (S5(w,w),z) 2 γ=0 ≥ N 0 S (2β w,2β v),25β z (4.35) (4.30)

231 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 232/243

for all w,v ∈ W1 and all z ∈ W1. Now, for all w ∈ W1 and all z ∈ W1. With the help of (4.33) and 1 (4.35) in (4.36) N Q (w + 4v) − 5Q (w + 3v) − Q q (w + 3v) 5 5 2 5s 5 1 + 10Q (w + 2v) + Q q (w + 2v) − 10Q (w + v) N Q5(w + 4v) − 5Q5(w + 3v) − Q q (w + 3v) 5 2 5s 5 2 5s − 5 Q q (w + v) + 5Q (w) + 5 Q q (w) 5 5s 5 5s + 10Q (w + 2v) + Q q (w + 2v) − 10Q (w + v) 5 2 5s 5 5 − Q5(w − v) − Q5q (w − v) + Q5q (w − 2v) − 5 Q q (w + v) + 5Q (w) + 5 Q q (w) 2 s s 5s 5 5s −120Q (v) − 300 Q q (v) ,z 5 5 5s − Q (w − v) − Q q (w − v) + Q q (w − 2v) 5 5s 5s 2 1 β z ≥ min N Q5(w + 4v) − Eq(2 (w + 4v)), , −120Q (v) − 300 Q q (v) ,z 25β 15 5 5s 1 β z ≥ min{1,1,1,1,1,1,1,1,1,1,1,1,1,1, N −5Q5(w + 3v) + 5 Eq(2 (w + 3v)), , 25β 15 0 β β 5β o N S (2 w,2 v),2 z (4.37) 1 1 1 β z N − Q q (w + 3v) + Eq(2 (w + 3v)), , 2 5s 2 25β 15 for all w ∈ W1 and all z ∈ W1. Approaching β tends to ∞ 1 β z N 10Q (w + 2v) − 10 Eq(2 (w + 2v)), , 5 25β 15 (4.37) and applying (4.3), we obtain 5 5 1 β z N Q q (w + 2v) − Eq(2 (w + 2v)), , 2 5s 2 25β 15 1 N Q (w + 4v) − 5Q (w + 3v) − Q q (w + 3v) 5 5 2 5s 1 β z N −10Q (w + v) + 10 Eq(2 (w + v)), , 5 5β 15 5 2 + 10Q (w + 2v) + Q q (w + 2v) − 10Q (w + v) 5 2 5s 5 1 β z N −5Q q (w + v) + 5 Eq(2 (w + v)), , 5s 5β − 5 Q q (w + v) + 5Q (w) + 5 Q q (w) 2 15 5s 5 5s 1 β z 5 N +5Q5(w) − 5 Eq(2 (w)), , − Q (w − v) − Q q (w − v) + Q q (w − 2v) 25β 15 5 2 5s 5s 1 z β −120Q5(v) − 300 Q q (v) ,z = 1 (4.38) N 5Q q (w) − 5 Eq(2 (w)), , 5s 5s 25β 15 1 β z N −Q5(w − v) + Eq(2 (w − v)), , w ∈ z ∈ 25β 15 for all W1 and all W1. Using (FNS2) in (4.38) we see that 5 5 1 β z N − Q q (w − v) + Eq(2 (w − v)), , 2 5s 2 25β 15 1 q 1 β z Q5(w + 4v) − 5Q5(w + 3v) − Q5 (w + 3v) Q q (w − 2v) − E (2 (w − 2v)), , 2 s N 5s 5β q 2 15 5 + 10Q5(w + 2v) + Q5q (w + 2v) − 10Q5(w + v) 1 β z 2 s N −120Q (v) + 120 Eq(2 (v)), , 5 25β 15 − 5 Q q (w + v) + 5Q (w) + 5 Q q (w) 5s 5 5s 1 β z −300Q q (v) + 300 E (2 (v)), , N 5s 5β q 5 2 15 − Q5(w − v) − Q5q (w − v) + Q5q (w − 2v) 2 s s 1 β 1 β N Eq(2 (w + 4v)) − 5 Eq(2 (w + 3v)) = 120Q (v) + 300 Q q (v) 25β 25β 5 5s

1 1 β 1 β − Eq(2 (w + 3v)) + 10 Eq(2 (w + 2v)) 2 25β 25β for all w ∈ W1 and all z ∈ W1 which shows that Q5(w) satisﬁes 5 1 β 1 β + Eq(2 (w + 2v)) − 10 Eq(2 (w + v)) the functional equation (1.3). 2 25β 25β It is easy to prove that the existence of Q5 is unique. In- 1 β 1 β − 5 Eq(2 (w + v)) + 5 Eq(2 (w)) deed, let R be an another quintic mapping satisfying (1.3) 25β 25β 5 and (4.2). Now 1 β 1 β + 5 Eq(2 (w)) − Eq(2 (w − v)) 25β 25β δ 5δ δ 5δ 5 1 β 1 β Q5(2 w) = 2 Q5(w) and R5(2 w) = 2 R5(w). − Eq(2 (w − v)) + Eq(2 (w − 2v)) 2 25β 25β 1 β 1 β z −120 Eq(2 (v)) − 300 Eq(2 (v)) , 232 25β 25β 15 (4.36) Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 233/243

Thus, for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant and φ,φ1,φ2 6= 5. Then there exists one and only quintic function N (Q5(w) − R5(w),z) Q5 : W1 −→ W2 which satisfying the functional equation (1.3) δ δ 5δ and the functional inequality = N Q5(2 w) − R5(2 w),2 z ! N (Q (w) − Eq(w),z) 25δ z 5 0 δ δ 5 5 ≥ N S5(2 w,2 w), 5!(2 − ε)  0 φ 5!|2 −2φ | 66  N Γ5T Φ|w| , 33 ; ≥ 5 φ ! 0 φ φ 5!|2 −2 | 5δ  N Γ5T1 Φ|w| 1 + Γ5T2Φ|w| 2 , ; 0 5 2 z 33 = N S5(w,w), 5!(2 − ε) (4.39) εδ 66 (4.45) 0 where Γgs are defined in (3.40) respectively for all w ∈ W1 for all w ∈ W1 and all z ∈ W1. Since and all z ∈ W1. 5δ ! 0 5 2 z The following corollary is an immediate consequence of lim N S5(w,w), 5!(2 − ε) = 1 (4.40) δ→∞ εδ 66 Theorem 4.7 regarding the Ulam - Hyers - JMRassias stability [42] of the functional equation (1.3). because 25δ z Corollary 4.10. For an odd mapping Eq : W1 −→ W2 fulfill- lim 5!(25 − ε) = ∞ ing the functional inequality δ→∞ εδ 66 0 φ φ 2φ 2φ for all z ∈ W1. Letting δ tends to ∞ in (4.39), using (4.40) and N Φ |w| |v| + |w| + |v| ,z ; N (Eq(w,v),z) ≥ 0 φ φ φ +φ φ +φ (FNS2), we have Q5(w) − R5(w) = 0 which implies Q5(w) = N Φ |w| 1 |v| 2 + |w| 1 2 + |v| 1 2 ,z ; R5(w) for all w ∈ W1. Hence the theorem holds for α = 1. w (4.46) On changing w by in (4.25), we get 2 for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant and 2φ,φ1 + φ2 6= 5. Then there exists one and only quintic func- 5 w z w w N Eq(w) − 2 E , ≥ S , (4.41) 2 5! 5 2 2 tion Q5 : W1 −→ W2 which satisfying the functional equation (1.3) and the functional inequality for all w ∈ W1 and all z ∈ W1. The rest of the proof is similar clues that of case α = 1. Hence the proof is complete. N (Q5(w) − Eq(w),z) 5 2  0 φ 5!|2 −2 φ |  N Γ5J Φ|w| , ; The following corollary is an immediate consequence of ≥ 33 (4.47) | 5− φ1+φ2 | z Theorem 4.7 regarding the Ulam - Hyers stability [23] of the 0 φ1+φ2 5! 2 2  N Γ5J1 Φ|w| , 33 ; functional equation (1.3). 0 where Γ s are defined in (3.43) respectively for all w ∈ W1 Corollary 4.8. For an odd mapping : −→ fulfilling g Eq W1 W2 and all z ∈ W . the functional inequality 1 Theorem 4.11. For an even mapping Es : W1 −→ W2 fulfilling 0 N (Eq(w,v),z) ≥ N (Φ,z) (4.42) the functional inequality 0 for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant. N (Es(w,v),z) ≥ N (S (w,v),z) (4.48) Then there exists one and only quintic function Q5 : W1 −→ for all w,v ∈ W1 and all z ∈ W1. Then there exists one and only W2 which satisfying the functional equation (1.3) and the sextic function Q6 : W1 −→ W2 which satisfying the functional functional inequality equation (1.3) and the functional inequality 0 5!|31|z 6 N (Q5(w) − Eq(w),z) ≥ N Φ, (4.43) 0 6!|2 − ε| z 33 N (Q (w) − Es(w),z) ≥ N S (w,w), 6 6 65 for all w ∈ W1 and all z ∈ W1. (4.49)

The following corollary is an immediate consequence of for all w ∈ W1 and all z ∈ W1. The mapping Q6 is defined as Theorem 4.7 regarding the Ulam - Hyers - THRassias stability 1 αβ [38] of the functional equation (1.3). lim N Q6(w) − Es 2 w ,z = 1 (4.50) β→∞ 26αβ Corollary 4.9. For an odd mapping Eq : W −→ W fulfilling 1 2 for all w ∈ W and all z ∈ W , where S : W 2 −→ W is a the functional inequality 1 1 1 3 function fulfilling the condition 0 φ φ N Φ |w| + |v| ,z ; 0 αβ αβ 6αβ N (Eq(w,v),z) ≥ 0 φ φ (4.44) lim N S 2 w,2 v ,2 z = 1 (4.51) N Φ |w| 1 + |v| 2 ,z ; β→∞

233 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 234/243

for all w,v ∈ W1 and all z ∈ W1 with the condition that

N (6Es(7w) − 21Es(6w) + 20Es(5w) + Es(4w) N 0 S 2αβ w,2αβ w ,z = N 0 εαβ S w,w ,z (4.52) +6Es(3w) − 747Es(2w) + 720Es(w),z + z) n ≥ min N (Es(8w) − 6Es(6w) + 16Es(4w) for all w ∈ W1 and all z ∈ W1 for some ε > 0 with 0 < ε α −746E (2w),z) < 1. The function S w,w is deﬁned by s 26 6 + N (Es(8w) − 6Es(7w) + 15Es(6w) − 20Es(5w) o 0 0 +15Es(4w) − 6Es(3w) + Es(2w) − 720Es(w),z) S6(w,w) = min N (S (0,2w),z) + N (S (4w,w),z) 0 0 n o + N (S (3w,w),z) + N (S (2w,w),z) ≥ min N 0 (S (0,2w),z) + N 0 (S (4w,w),z) 0 0 + N (S (w,w),z) + N (S (0,w),z) (4.62) 0 +N (S (−w,w),z) (4.53) for all w ∈ W1 and all z ∈ W1. Multiplying (4.57) by 6, we see that for all w ∈ W1 and all z ∈ W1. N (6Es(7w) − 36Es(6w) + 90Es(5w) − 120Es(4w) Proof. Using evenness of E in (4.48), one can obtain that +90Es(3w) − 36Es(2w) − 5514Es(w),6z) ≥ N 0 (S (3w,w),z) (4.63)

N (Es(w + 4v) − 6Es(w + 3v) + 15Es(w + 2v) − 20Es(w + v) for all w ∈ W1 and all z ∈ W1. It follows from (4.62) and +15Es(w) − 6Es(w − v) + Es(w − 2v) − 720Es(v),z) (4.63), we arrive ≥ N 0 (S (w,v),z) (4.54) N (15Es(6w) − 70Es(5w) + 121Es(4w) − 84Es(3w)

−711Es(2w) + 5034Es(w),z + z + 6z) for all w,v ∈ W1 and all z ∈ W1. Now, interchanging (w,v) by n (0,2w), (4w,w), (3w,w), (2w,w), (w,w), (0,w) and (−w,w) ≥ min N 0 (S (0,2w),z) + N 0 (S (4w,w),z) in (4.54) and using evenness of E , we arrive the subsequent s o inequalities + N 0 (S (3w,w),z) (4.64)

for all w ∈ W1 and all z ∈ W1. Multiplying (4.58) by 15, we N (Es(8w) − 6Es(6w) + 16Es(4w) − 746Es(2w),z) get ≥ N 0 (S (0,2w),z) (4.55)

N (Es(8w) − 6Es(7w) + 15Es(6w) − 20Es(5w) N (15Es(6w) − 90Es(5w) + 225Es(4w) − 300Es(3w)

+15Es(4w) − 6Es(3w) + Es(2w) − 720Es(w),z) +225Es(2w) − 10890Es(w),15z) ≥ N 0 (S (4w,w),z) (4.56) ≥ N 0 (S (2w,w),z) (4.65) N (Es(7w) − 6Es(6w) + 15Es(5w) − 20Es(4w) for all w ∈ W1 and all z ∈ W1. Combining (4.65) and (4.64), +15Es(3w) − 6Es(2w) − 719Es(w),z) we obtain ≥ N 0 (S (3w,w),z) (4.57) (20E (5w) − 104E (4w) + 216E (3w) − 936E (2w) N (Es(6w) − 6Es(5w) + 15Es(4w) − 20Es(3w) N s s s s +15924E (w),z + z + 6z + 15z) +15Es(2w) − 726Es(w),z) s 0 n ≥ N (S (2w,w),z) (4.58) ≥ min N 0 (S (0,2w),z) + N 0 (S (4w,w),z)

N (Es(5w) − 6Es(4w) + 15Es(3w) − 20Es(2w) 0 0 o 0 + N (S (3w,w),z) + N (S (2w,w),z) (4.66) −704Es(w),z) ≥ N (S (w,w),z) (4.59) (E (4w) − 6E (3w) + 16E (2w) − 746E (w),z) N s s s s for all w ∈ W1 and all z ∈ W1. Multiplying (4.59) by 20, we ≥ N 0 (S (0,w),z) (4.60) ﬁnd that

N (2Es(3w) − 12Es(2w) − 690Es(w),z) 0 N (20Es(5w) − 120Es(4w) + 300Es(3w) − 400Es(2w) ≥ N (S (−w,w),z) (4.61) 0 −14080Es(w),20z) ≥ N (S (w,w),z) (4.67) for all w ∈ W1 and all z ∈ W1. From (4.55) and (4.56), we for all w ∈ W1 and all z ∈ W1. From (4.66) and (4.67), we have arrive

234 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 235/243

β+1 β ! Es(2 w) Es(2 w) 65z N (16Es(4w) − 84Es(3w) − 536Es(2w) N − , 26(β+1) 26β 2626β · 6! +30004Es(w),z + z + 6z + 15z + 20z) 0 β β n 0 0 ≥ ( w, w),z ≥ min N (S (0,2w),z) + N (S (4w,w),z) N S6 2 2 0 0 0 β + N (S (3w,w),z) + N (S (2w,w),z) = N ε S6(w,w),z o 0 0 z + N (S (w,w),z) (4.68) = N S6(w,w), (4.74) εβ β for all w ∈ W1 and all z ∈ W1. Multiplying (4.60) by 16, we for all w ∈ W1 and all z ∈ W1. Switching z by ε z in (4.74) see that ! E (2β+1w) E (2β w) εβ 65z s − s , N 6(β+1) 6β 6β 6 N (16Es(4w) − 96Es(3w) + 256Es(2w) 2 2 2 2 · 6! −11936E (w),16z) ≥ 0 ( (0,w),z) (4.69) 0 s N S ≥ N (S6(w,w),z) (4.75) for all w ∈ W1 and all z ∈ W1. It follows from (4.68) and for all w ∈ W1 and all z ∈ W1. The rest of the proof is similar (4.69), we arrive clues that of Theorem 4.7. Hence the proof is complete. The following corollary is an immediate consequence of N (12Es(3w) − 792Es(2w) + 41940Es(w) Theorem 4.11 regarding the Ulam - Hyers stability [23] of the ,z + z + 6z + 15z + 20z + 16z) functional equation (1.3). n 0 0 ≥ min N (S (0,2w),z) + N (S (4w,w),z) Corollary 4.12. For an even mapping Es : W1 −→ W2 fulﬁll- ing the functional inequality + N 0 (S (3w,w),z) + N 0 (S (2w,w),z) 0 0 0 o N (Es(w,v),z) ≥ N (Φ,z) (4.76) + N (S (w,w),z) + N (S (0,w),z) (4.70) for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant. for all w ∈ W1 and all z ∈ W1. Multiplying (4.61) by 6 and it Then there exists one and only sextic function Q6 : W1 −→ follows from and (4.70), we arrive W2 which satisfying the functional equation (1.3) and the functional inequality N (720Es(2w) − 46080Es(w), 0 6!|63|z N (Q (w) − Es(w),z) ≥ N Φ, (4.77) z + z + 6z + 15z + 20z + 16z + 6z) 6 65 n ≥ 0 ( ( , w),z) + 0 ( ( w,w),z) min N S 0 2 N S 4 for all w ∈ W1 and all z ∈ W1. + N 0 (S (3w,w),z) + N 0 (S (2w,w),z) The following corollary is an immediate consequence + N 0 (S (w,w),z) + N 0 (S (0,w),z) of Theorem 4.11 regarding the Ulam - Hyers - THRassias o stability [38] of the functional equation (1.3). + N 0 (S (−w,w),z) Corollary 4.13. For an even mapping Es : W1 −→ W2 fulﬁll- 0 = N (S6(w,w),z) (4.71) ing the functional inequality 0 Φ|w|φ + |v|φ ,z; for all w ∈ W1 and all z ∈ W1. The above equation can be N N (Es(w,v),z) ≥ 0 φ φ (4.78) written as N Φ |w| 1 + |v| 2 ,z ;

0 for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant and N (720Es(2w) − 46080Es(w),65z) ≥ N (S6(w,w),z) φ,φ1,φ2 6= 6. Then there exists one and only sextic function (4.72) Q6 : W1 −→ W2 which satisfying the functional equation (1.3) and the functional inequality for all w ∈ W1 and all z ∈ W1. Using (FNS3) in (4.72), one can get that N (Q6(w) − Es(w),z) 6  0 φ 6!|2 −2φ | E (2w) 65z  N Γ6T Φ|w| , ; s 0 ≥ 65 N − Es(w), ≥ N (S6(w,w),z) | 6− φ | 26 26 · 6! 0 φ1 φ2 6! 2 2  N Γ6T1 Φ|w| + Γ6T2 Φ|w| , 65 ; (4.73) (4.79)

β 0 for all w ∈ W1 and all z ∈ W1. Changing w by 2 w and using where Γgs are deﬁned in (3.76) respectively for all w ∈ W1 (4.52), (FNS3) in (4.73), we arrive and all z ∈ W1.

235 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 236/243

The following corollary is an immediate consequence Proof. We know that by deﬁnition of odd function, we have of Theorem 4.11 regarding the Ulam - Hyers - JMRassias E (w) − E (−w) stability [42] of the functional equation (1.3). E (w) = q q (4.84) O 2 Corollary 4.14. For an even mapping Es : W1 −→ W2 fulﬁll- ing the functional inequality for all w ∈ W1. It follows from (4.84) that

N (EO(u,v),z) N (Es(w,v),z) 0 φ φ 2φ 2φ 1n o N Φ |w| |v| + |w| + |v| ,z ; = N Eq(w,v) − Eq(−w,−v) ,z ≥ 2 0 Φ|w|φ1 |v|φ2 + |w|φ1+φ2 + |v|φ1+φ2 ,z; N n o (4.80) = N Eq(w,v) − Eq(−w,−v) ,2z n o for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant and ≥ min N (Eq(w,v),z),N (Eq(−w,−v),z) (4.85) 2φ,φ1 +φ2 6= 6. Then there exists one and only sextic function Q6 : W1 −→ W2 which satisfying the functional equation (1.3) for all w ∈ W1. Thus by Theorem 4.7, we arrive and the functional inequality N (Es(w) − Q5(w),z) N (Q6(w) − Es(w),z) 5 0 5!|2 − ε| z  6 2φ ≥ min N S5(w,w), , 0 φ 6!|2 −2 | 33  N Γ6J Φ|w| , 65 ; ≥ 6 φ +φ (4.81) 5 0 φ +φ 6!|2 −2 1 2 | z 0 5!|2 − ε| z  N Γ6J1 Φ|w| 1 2 , ; N S (−w,−w), (4.86) 65 5 33 where 0 s are deﬁned in (3.79) respectively for all w ∈ Γg W1 for all w ∈ W1 and all z ∈ W1. and all z ∈ W1. We know that by deﬁnition of even function, we have

Theorem 4.15. For a mapping E : W1 −→ W2 fulﬁlling the E (w) + E (−w) E (w) = q q (4.87) functional inequality E 2

0 N (E (w,v),z) ≥ N (S (w,v),z) (4.82) for all w ∈ W1. It follows from (4.87) that for all w,v ∈ W1 and all z ∈ W1. Then there exists one and N (EE (u,v),z) only quintic function Q5 : W1 −→ W2 and a one and only 1n o = N Es(w,v) + Es(−w,−v) ,z sextic function Q6 : W1 −→ W2 which satisfying the functional 2 equation (1.3) and the functional inequality n o = N Es(w,v) + Es(−w,−v) ,2z

N (E(w) − Q5(w) − Q6(w),z) n o ≥ min N (Es(w,v),z),N (Es(−w,−v),z) (4.88) 5 0 5!|2 − ε| z ≥ min N S5(w,w), , 33 for all w ∈ W1. Thus by Theorem 4.11, we arrive 5!|25 − ε| z N 0 S (−w,−w), , 5 33 N (Es(w) − Q6(w),z) 6 6!|26 − ε| z 0 6!|2 − ε| z N 0 S (w,w), , ≥ min N S6(w,w), , 6 65 65 6 6!|26 − ε| z 0 6!|2 − ε| z N 0 S (−w,−w), (4.83) N S6(−w,−w), (4.89) 6 65 65

for all w ∈ W1 and all z ∈ W1. Now, define a function for all w ∈ W1 and all z ∈ W1. The mappings Q5 , Q6 are defined in (4.3), (4.50) for all w ∈ W1 and all z ∈ W1, where 2 E(w) = EO(w) + EE (w) (4.90) S : W1 −→ W3 is a function fulfilling the conditions (4.4), (4.51) for all w,v ∈ W1 and all z ∈ W1 with the conditions for all w ∈ W1. Combining (4.86), (4.89) and (4.90), we reach that (4.5), (4.52) for all w,v ∈ W1 and all z ∈ W1 for some ε α ε α our result. ε > 0 with 0 < < 1, 0 < < 1. The functions 25 26 The following corollary is an immediate consequence of S5 w,w , S6 w,w are defined in (4.6), (4.53) for all w ∈ Theorem 4.15 regarding the Ulam - Hyers stability [23] of the W1 and all z ∈ W1. functional equation (1.3).

236 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 237/243

Corollary 4.16. For a mapping E : W1 −→ W2 fulfilling the for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant functional inequality and 2φ,φ1 + φ2 6= 6. Then there exists one and only quintic function Q5 : W1 −→ W2 and a one and only sextic function 0 N (E (w,v),z) ≥ N (Φ,z) (4.91) Q6 : W1 −→ W2 which satisfying the functional equation (1.3) and the functional inequality for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant. Then there exists one and only quintic function Q5 : W1 −→ W2 N (E(w) − Q5(w) − Q6(w),z) and a one and only sextic function Q6 : W1 −→ W2 which  n 5 2φ min 0 |w|φ , 5!|2 −2 | , satisfying the functional equation (1.3) and the functional  N Γ5J Φ 33  6 2φ inequality  0 φ 6!|2 −2 | o  N Γ6J Φ|w| , ; ≥ 65 n 5!|25−2φ1+φ2 | z 0 |w|φ1+φ2 , , N (E(w) − Q5(w) − Q6(w),z)  min N Γ5J1 Φ 33  6 φ +φ  0 + 6!|2 −2 1 2 | z o 0 5!|31|z 0 6!|63|z  Γ Φ|w|φ1 φ2 , ; ≥ min N Φ, ,N Φ,  N 6J1 65 33 65 (4.96) (4.92) 0 where Γgs are defined in (3.43) and (3.79) respectively for all for all w ∈ W1 and all z ∈ W1. w ∈ W1 and all z ∈ W1. The following corollary is an immediate consequence of Theorem 4.15 regarding the Ulam - Hyers - THRassias 4.3 Alternative Fixed Point Method stability [38] of the functional equation (1.3). Theorem 4.19. For an odd mapping Eq : W1 −→ W2 fulfilling the functional inequality (4.1) for all w,v ∈ W1 and all z ∈ W1, 2 Corollary 4.17. For a mapping E : W1 −→ W2 fulfilling the where S : W1 −→ [0,∞) is a function fulfilling the condition functional inequality

0 φ φ 0 β β 5β N Φ |w| + |v| ,z ; lim N S ξψ w,ξψ v ,ξψ z = 1 (4.97) N (E (w,v),z) ≥ 0 (4.93) → N Φ |w|φ1 + |v|φ2 ,z ; β ∞

for all w,v ∈ W1 and all z ∈ W1. Then there exists one and for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 is a constant only quintic function Q5 : W1 −→ W2 which satisfying the and φ,φ1,φ2 6= 6. Then there exists one and only quintic functional equation (1.3) and the functional inequality function Q5 : W1 −→ W2 and a one and only sextic function Q6 : W1 −→ W2 which satisfying the functional equation (1.3) 1−ψ 0 L and the functional inequality N (Q5(w) − Eq(w),z) ≥ N S5(w,w), z (4.98) 1 − L

N (E(w) − Q5(w) − Q6(w),z) for all w ∈ W1 and all z ∈ W1. If L = L (ψ), with the prop-  n 5 φ min 0 |w|φ , 5!|2 −2 | , erty  N Γ5T Φ 33  6 φ o  0 φ 6!|2 −2 | !  N Γ6T Φ|w| , 65 ; 0 1 0 ≥ n 5 φ 0 φ φ 5!|2 −2 | N 5 S5 ξψ w,ξψ w ,z = N LS5 w,w ,z (4.99)  min N Γ5T1 Φ|w| 1 + Γ5T2 Φ|w| 2 , , ξ  33 ψ  6!|26−2φ | o  0 Γ Φ|w|φ1 + Γ Φ|w|φ2 , ;  N 6T1 6T2 65 with the condition that (4.94) 33 w w 0 S5 w,w = S5 , , (4.100) where Γgs are deﬁned in (3.40) and (3.76) respectively for all 5! 2 2 w ∈ W1 and all z ∈ W1. where S5 w,w is deﬁned in (4.24) for all w ∈ W1 and all The following corollary is an immediate consequence z ∈ W1. of Theorem 4.15 regarding the Ulam - Hyers - JMRassias stability [42] of the functional equation (1.3). Proof. Consider the set

Corollary 4.18. For a mapping E : W −→ W fulﬁlling the n o 1 2 B = E |E : W −→ W ,E (0) = 0 (4.101) functional inequality 1 1 1 2 1

N (E (w,v),z) Let us introduce the generalized metric on (4.101) by 0 φ φ 2φ 2φ N Φ |w| |v| + |w| + |v| ,z ; n 0 o ≥ 0 + + inf ζ : (E (w) − E (w),z) ≥ w,w ,ζ z N Φ |w|φ1 |v|φ2 + |w|φ1 φ2 + |v|φ1 φ2 ,z ; N 1 2 N S5 (4.95) (4.102)

237 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 238/243

for all w ∈ W1 and all z ∈ W1. One can easy to see verify that Theorem 1.1 , that there exists a fixed point Q5 of ϒ in L (4.102) is complete with respect to the defined metric. Define such that a mapping ϒ : B → B by ! 1 lim Q (w) − E ξ w ,z = 1 (4.109) 1 N 5 5 q ψ β→∞ ξψ ϒEq(w) = 5 Eq(ξψ ) (4.103) ξψ for all w ∈ W1 and all z ∈ W1. To prove the existing Q5 satisfies for all w ∈ W1 and all z ∈ W1. Now, for any E1,E2 ∈ B, we (1.3), the proof is similar to that of Theorem 4.7. arrive Again by condition (FPC3) of Theorem 1.1, Q5 is the 0 unique fixed point of ϒ in the set N (E1(w) − E2(w),z) ≥ N S w,w ,ζ z 5 n o ! ∆ = Q : N (Eq(w) − Q (w),z) ≥ ∞ (4.110) 1 1 5 5 N 5 E1(ξψ ) − 5 E2(ξψ ),z ξψ ξψ for all w ∈ W1 and all z ∈ W1. Finally, by condition (FPC4) of 0 5 Theorem 1.1 , we get ≥ N S5 w,w ,ζ ξψ z N (Eq(w) − Q5(w),z) N (ϒE (w) − ϒE (w),z) ≥ N 0 S w,w ,L ζ z 1 2 5 1 ≥ N Eq(w) − ϒEq(w), z 1 − L for all w ∈ W1 and all z ∈ W1. This implies that ϒ is a strictly 1−ψ contractive mapping on B with Lipschitz constant L . It 0 L = N S5(w,w), z follows from (4.26) that 1 − L Eq(2w) 33z 0 for all w ∈ W1 and all z ∈ W1. This completes the proof of the N − Eq(w), ≥ N (S (w,w),z) 25 25 · 5! 5 theorem. (4.104) The following corollary is an immediate consequence for all w ∈ W1 and all z ∈ W1. of Theorem 4.19 regarding the Ulam - Hyers stability [23], For the case ψ = 0, it follows from (4.99), (4.100), (4.103), Ulam - Hyers - THRassias stability [38] and Ulam - Hyers - (FNS3) and (4.104), JMRassias stability [42] of the functional equation (1.3).

Corollary 4.20. For an odd mapping Eq : W1 −→ W2 fulﬁll- N (ϒEq(w) − Eq(w),z) ing the functional inequality 0 0 1−0 ≥ N (S5(w,w),L z) = N S5(w,w),L z (4.105)  Φ;  φ φ N (Eq(w,v),z) ≥ Φ ||w|| + ||v|| ; for all w ∈ W1 and all z ∈ W1.  Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; Replacing w by w in (4.104), we obtain 2 (4.111) 5 w 33z N Eq(w) − 2 Eq , for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 and φ is a 2 5! constant. Then there exists one and only quintic function 0 w w Q : −→ which satisfying the functional equation (1.3) ≥ N S5 , ,z (4.106) 5 W1 W2 2 2 and the functional inequality for all w ∈ W1 and all z ∈ W1. N (Q (w) − Eq(w),z) For the case = 1, it follows from (4.99), (4.100), (4.103), 5 ψ  (FNS3) and (4.106), 0 , 33 z  N Φ 5!|31| ;  0 ≥ 0 ||w||φ , 33 z ; 6= 5 N (Eq(w) − ϒEq(w),z) ≥ N (S5(w,w),1 · z) N Γ5T Φ 5!|2φ −25| φ  0 1−1  0 ||w||2φ , 33 z ; 2 6= 5 = N S5(w,w),L z  N Γ5J Φ 5!|22φ −25| φ (4.107) (4.112) for all w ∈ and all z ∈ . From (4.105) and (4.107), we 0 W1 W1 where Γgs are deﬁned in (3.40) and (3.43) respectively for all see that w ∈ W1 and all z ∈ W1. 0 1−ψ N (Eq(w) − ϒEq(w),z) ≥ N S5(w,w),L z Proof. If we take (4.108)  N 0 (Φ,z); 0  0 φ φ for all w ∈ W1 and all z ∈ W1. Thus, condition (FPC1) of N (S (w,v),z) = N Φ ||w|| + ||v|| ,z ; 0 Theorem 1.1 holds. It follows from condition (FPC2 ) of  N Φ ||w||φ ||v||φ + ||w||2φ + ||v||2φ ,z ;

238 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 239/243

5−φ 5−φ (4.113) For ψ = 1 : L = ξψ = 2 : φ > 5 for all w,v ∈ . Changing (w,v) by ( β w, β v) and dividing W1 ξψ ξψ 1−ψ 5β 0 L by in (4.113) and letting tends to , we see that (4.97) N (Q5(w) − Eq(w),z) ≥ N S5 w,w , z ξψ β ∞ 1 − L holds for all w,v ∈ and all z ∈ . W1 W1  1−1  It follows from (4.100), (4.113) and (4.24), one can ﬁnd 25−φ 0 that = N S5 w,w , z  1 − 25−φ  S w,w φ 0 2 33 w w = N S5 w,w , z = S , 2φ − 25 5! 2 2  33Φ ; 1 1 5! For = = = = φ−5 <  ψ 0 : L 5−φ 5−φ 2 : φ 5  φ φ φ φ 2  33 11·2 +5·3 +4 +43 Φ||w|| ξψ = φ ; 5!2  φ 2φ φ 2φ φ 2φ 2φ 1−ψ  33 10·2 +11·2 +5(3 +3 )+4 +4 +54 Φ||w|| 0 L  ; N (Q5(w) − Eq(w),z) ≥ N S5 w,w , z 5!22φ 1 − L (4.114)  1−0  2φ−5 0 for all w ∈ W1 and all z ∈ W1. =  w,w , z N S5 φ−5  Again, it follows from (4.99), (4.113) and (4.24), one can 1 − 2 observe that φ  0 2 0 = N S5 w,w , z  N S5 w,w ,L z ; 25 − 2φ   !  5−2φ 5−2φ 0 1  0 For ψ = 1 : L = ξψ = 2 : 2φ > 5 N S5 ξψ w,ξψ w ,z = N S5 w,w ,L z ; ξ 5 ψ   1−ψ  0 L  N 0 S w,w ,L z ; N (Q5(w) − Eq(w),z) ≥ N S5 w,w , z  5 1 − L (4.115)  1−1  25−2φ 0 for all w ∈ W1 and all z ∈ W1. Thus, the functional inequality = N S w,w , z  5 − 5−2φ  (4.98) holds for the following cases. 1 2

5 5 2φ For ψ = 1 : L = ξ = 2 2 ψ = N 0 S w,w , z 5 22φ − 25 1−ψ 0 L 1 1 2φ−5 N (Q5(w) − Eq(w),z) ≥ N S5 w,w , z For ψ = 0 : L = = = 2 : 2φ < 5 1 − L 5−2φ 5−2φ ξψ 2  1−1  5 2 0   1−ψ = N S5 w,w , 5 z 0 L  1 − 2  N (Q5(w) − Eq(w),z) ≥ N S5 w,w , z 1 − L  1−0  0 1 22φ−5 = N S5 w,w , z −31 = N 0 S w,w , z  5 1 − 22φ−5  1 1 For ψ = 0 : L = = = 2−5 ξ 5 25 22φ ψ = N 0 S w,w , z 5 25 − 22φ 1−ψ 0 L N (Q5(w) − Eq(w),z) ≥ N S5 w,w , z 1 − L  1−0  Theorem 4.21. For an even mapping : −→ fulﬁlling −5 Es W1 W2 2 the functional inequality (4.1) for all w,v ∈ and all z ∈ , 0   W1 W1 = N S5 w,w , −5 z 2  1 − 2  where S : W1 −→ [0,∞) is a function fulﬁlling the condition 0 1 0 β β 6β = N S5 w,w , z lim N S ξψ w,ξψ v ,ξψ z = 1 (4.116) 31 β→∞

239 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 240/243

for all w,v ∈ W1 and all z ∈ W1. Then there exists one and only For the case ψ = 0, it follows from (4.118), (4.119), sextic function Q6 : W1 −→ W2 which satisfying the functional (4.122), (FNS3) and (4.123), equation (1.3) and the functional inequality 0 N (ϒEs(w) − Es(w),z) ≥ N (S6(w,w),L z) 1−ψ 0 L = 0 (w,w), 1−0z N (Q6(w) − Es(w),z) ≥ N S6(w,w), z (4.117) N S6 L 1 − L (4.124) for all w ∈ W1 and all z ∈ W1. If L = L (ψ), with the prop- for all w ∈ W1 and all z ∈ W1. w erty Replacing w by 2 in (4.123), we obtain ! 1 6 w 65z 0 w w 0 w, w ,z = N0 L w,w ,z (4.118) N Es(w) − 2 Es , ≥ N S6 , ,z N 6 S6 ξψ ξψ S6 2 6! 2 2 ξψ (4.125) with the condition that for all w ∈ W1 and all z ∈ W1. 65 w w For the case ψ = 1, it follows from (4.118), (4.119), S w,w = S , , (4.119) 6 6! 6 2 2 (4.122), (FNS3) and (4.125), 0 N (Es(w) − ϒEs(w),z) ≥ N (S6(w,w),1 · z) where S6 w,w is deﬁned in (4.24) for all w ∈ W1 and all 0 1−1 z ∈ W1. = N S6(w,w),L z (4.126) Proof. Consider the set for all w ∈ W and all z ∈ W . From (4.124) and (4.126), we n o 1 1 B = E2|E2 : W1 −→ W2,E2(0) = 0 (4.120) see that 0 1−ψ N (Es(w) − ϒEs(w),z) ≥ N S (w,w),L z Let us introduce the generalized metric on (4.120) by 6 (4.127)

n 0 o inf ζ : N (E1(w) − E2(w),z) ≥ N S6 w,w ,ζ z for all w ∈ W1 and all z ∈ W1. Thus, condition (FPC1) of Theorem 1.1 holds. The rest of the proof is similar lines (4.121) to that of Theorem 4.19. This completes the proof of the theorem. for all w ∈ W1 and all z ∈ W1. One can easy to see verify that (4.121) is complete with respect to the defined metric. Define The following corollary is an immediate consequence a mapping ϒ : B → B by of Theorem 4.21 regarding the Ulam - Hyers stability [23], 1 Ulam - Hyers - THRassias stability [38] and Ulam - Hyers - ϒEs(w) = 6 Es(ξψ ) (4.122) JMRassias stability [42] of the functional equation (1.3). ξψ Corollary 4.22. For an even mapping Es : W1 −→ W2 fulfill- for all w ∈ W1 and all z ∈ W1. Now, for any E1,E2 ∈ B, we ing the functional inequality arrive  Φ; 0  N (E1(w) − E2(w),z) ≥ N S w,w ,ζ z φ φ 6 N (Es(w,v),z) ≥ Φ ||w|| + ||v|| ; !  Φ||w||φ ||v||φ + ||w||2φ + ||v||2φ ; 1 1 N 6 E1(ξψ ) − 6 E2(ξψ ),z (4.128) ξψ ξψ 0 5 for all w,v ∈ W1 and all z ∈ W1 where Φ > 0 and φ is a ≥ N S6 w,w ,ζ ξψ z constant. Then there exists one and only sextic function Q6 : 0 W1 −→ W2 which satisfying the functional equation (1.3) and N (ϒE1(w) − ϒE2(w),z) ≥ N S6 w,w ,L ζ z the functional inequality for all w ∈ W1 and all z ∈ W1. This implies that ϒ is a strictly contractive mapping on B with Lipschitz constant L . It N (Q6(w) − Es(w),z) follows from (4.26) that  0 Φ, 65 z ;  N 6!|63| Es(2w) 65z 0  N − Es(w), ≥ N (S6(w,w),z) ≥ 0 Γ Φ||w||φ , 65 z ; φ 6= 6 26 26 · 6! N 6T 6!|2φ −26|  (4.123)  0 ||w||2φ , 65 z ; 2 6= 6  N Γ6J Φ 6!|22φ −26| φ (4.129) for all w ∈ W1 and all z ∈ W1.

240 Stabilities of mixed type Quintic-Sextic functional equations in various normed spaces — 241/243

0 where Γgs are defined in (3.76) and (3.79) respectively for all w ∈ W1 and all z ∈ W1. N (E(w) − Q5(w) − Q6(w),z)  n o Proof. The proof of the corollary is similar clues and ideas of 0 , 33 z , 0 , 65 z  min N Φ 5!|31| N Φ 6!|63| ; Corollary 4.20. Hence the details of the proof are omitted.  n  min 0 Γ Φ||w||φ , 33 z ,  N 5T 5!|2φ −25|  o ≥ 0 ||w||φ , 65 z ; 6= 5,6 N Γ6T Φ 6!|2φ −26| φ Theorem 4.23. For a mapping E : W −→ W fulfilling the  n 1 2  min 0 Γ Φ||w||2φ , 33 z , functional inequality (4.82) for all w,v ∈ and all z ∈ ,  N 5J 5!|22φ −25| W1 W1  o where : 2 −→ [0,∞) is a function fulfilling the condition  0 ||w||2φ , 65 z ; 2 6= 6 S W1  N Γ6J Φ 6!|22φ −26| φ (4.97), (4.116) for all w,v ∈ W1 and all z ∈ W1. Then there (4.132) exists one and only quintic function Q5 : W1 −→ W2 and a 0 one and only sextic function Q6 : W1 −→ W2 which satisfying where Γgs are defined in (3.40), (3.76), (3.43) and (3.79) the functional equation (1.3) and the functional inequality respectively for all w ∈ W1 and all z ∈ W1.

N (E(w) − Q5(w) − Q6(w),z) Acknowledgment 1−ψ 0 L This work is dedicated to all the mathematicians and research ≥ min N S5(w,w), z , 1 − L scholars working in this ﬁeld. 1−ψ 0 L N S5(−w,−w), z , 1 − L References 1−ψ 0 L [1] J. Aczel and J. Dhombres, Functional Equations in Sev- N S6(w,w), z , 1 − L eral Variables, Cambridge Univ, Press, 1989. 1−ψ [2] T. Aoki, On the stability of the linear transformation in 0 L N S6(−w,−w), z (4.130) 1 − L Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. [3] M. Arunkumar, Three Dimensional Quartic Functional Equation In Fuzzy Normed Spaces, Far East Journal of for all w ∈ W1 and all z ∈ W1. If L = L (ψ), with the properties and conditions (4.99), (4.118)(4.100), (4.119) Applied Mathematics, 41(2), (2010), 88-94. [4] M. Arunkumar, John M. Rassias, On the generalized where S5 w,w , S6 w,w are deﬁned in (4.24), (4.71)for Ulam-Hyers stability of an AQ-mixed type functional all w ∈ W1 and all z ∈ W1. equation with counter examples, Far East Journal of Ap- plied Mathematics, Volume 71, No. 2, (2012), 279-305. Proof. The proof of the the proof is similar lines to that of [5] M. Arunkumar, Perturbation of n Dimensional AQ - Theorem 4.15. mixed type Functional Equation via Banach Spaces and Banach Algebra: Hyers Direct and Alternative Fixed Point Methods, International Journal of Advanced Mathe- The following corollary is an immediate consequence matical Sciences (IJAMS), Vol. 2 (1), (2014), 34-56. of Theorem 4.23 regarding the Ulam - Hyers stability [23], [6] M. Arunkumar and T. Namachivayam, Stability of n- Ulam - Hyers - THRassias stability [38] and Ulam - Hyers - dimensional quadratic functional Equation in Fuzzy JMRassias stability [42] of the functional equation (1.3). normed spaces: Direct And Fixed Point Methods, Pro- ceedings of the International Conference on Mathemati-

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