International Journal of Statistics and Applied Mathematics 2016; 1(4): 06-12

ISSN: 2456-1452 Maths 2016; 1(4):06-12 © 2016 Stats & Maths Ulam stability of radical functional equation in the www.mathsjournal.com Received: 02-09-2016 sense of Hyers, Rassias and Gavruta Accepted: 03-10-2016

BV Senthil Kumar BV Senthil Kumar, Ashish Kumar and P Narasimman Department of Mathematics, C. Abdul Hakeem College of Engg. & Tech., Melvisharam, Abstract Tamil Nadu, India The aim of this paper is to obtain the general solution of a radical functional equation of the form + + 2 = ( ) + ( ) Ashish Kumar and investigate its generalized Hyers-Ulam-Rassias stability. Further, we extend the results pertinent to Department of Mathematics, D.H. Hyers, Th.M. Rassias and J.M.𝑠𝑠�𝑥𝑥 Rassias.𝑦𝑦 �We𝑥𝑥𝑥𝑥 �also𝑠𝑠 provide𝑥𝑥 𝑠𝑠 𝑦𝑦counter-examples for critical values RPS Degree College, relevant to Hyers-Ulam-Rassias stability, Ulam-Gavruta-Rassias stability and J.M. Rassias stability Mahendergarh, Haryana, India controlled by mixed product-sum of powers of norms.

P Narasimman Department of Mathematics, Keywords: Radical functional equation, Generalized Hyers-Ulam stability. Thiruvalluvar University College 2000 Mathematics Subject Classification: 39B52, 39B72 of Arts and Science, Gajalnaickanpatti, Tirupattur, 1. Introduction Tamil Nadu, India The stability of functional equations is an interesting topic that has been dealt for the last six decades. In mathematics, a stipulation in which a slight disturbance in a system does not create

a considerable disturbing consequence on that system. An equation is said to be stable if a slightly different solution is close to the exact solution of that equation. In mathematical models of physical problems, the deviations in measurements will result with errors and these deviations can be dealt with the stability of equations. Hence the stability of equations is essential in mathematical models. A stable solution will be sensible in spite of such deviations. [28] An interesting and eminent talk given by S.M. Ulam in 1940, inspired to study the investigation of stability of functional equations. He posed the following question pertaining to the stability of homomorphisms in groups: Let be a group and be a metric group with metric ( , ). Given 0 does there exist a 0 such that if a function : fulfills ( ( ), ( ) ( ) for all , , then 𝒳𝒳 𝒴𝒴 𝑑𝑑 ∙ ∙ 𝜖𝜖 ≥ there exists a homomorphism : with ( ( ), ( )) for all ? If𝛿𝛿 ≥the answer is confirmative,𝑓𝑓 then𝒳𝒳 → the𝒴𝒴 functional𝑑𝑑 𝑓𝑓 equation𝑥𝑥𝑥𝑥 𝑓𝑓 𝑥𝑥 for𝑓𝑓 𝑦𝑦homomorphism≤ 𝛿𝛿 𝑥𝑥 is𝑦𝑦 said∈ 𝒳𝒳 to be stable. The foremost answer to𝑔𝑔 the𝒳𝒳 question→ 𝒴𝒴 of𝑑𝑑 S.M.𝑓𝑓 𝑥𝑥 Ulam𝑔𝑔 𝑥𝑥 was≤ provided𝜖𝜖 𝑥𝑥 by∈ D.H.𝒳𝒳 Hyers [12]. He brilliantly answered the question of Ulam by considering and as Banach spaces. The

result of Hyers is stated in the following celebrated theorem. The stability resulted provided by Hyers in the following theorem is referred as Hyers-Ulam stability.𝒳𝒳 𝒴𝒴

[12] Theorem 1.1. (D.H. Hyers )Let : be a mapping between Banach spaces such that

( 𝑓𝑓+ 𝒳𝒳) → 𝒴𝒴( ) ( ) (1.1)

for all , and for some ‖𝑓𝑓>𝑥𝑥0, then𝑦𝑦 − the𝑓𝑓 𝑥𝑥limit− 𝑓𝑓 𝑦𝑦 ‖ ≤ 𝛿𝛿

𝑥𝑥 𝑦𝑦 ∈ 𝒳𝒳 𝛿𝛿 ( ) = lim 2 (2 ) (1.2) Correspondence −𝑛𝑛 −𝑛𝑛 BV Senthil Kumar exists for each . Then there𝐴𝐴 𝑥𝑥 exist 𝑛𝑛a→ unique∞ 𝑓𝑓 additive𝑥𝑥 mapping : such that Department of Mathematics, C. Abdul Hakeem College of 𝑥𝑥 ∈ 𝒳𝒳 𝐴𝐴 𝒳𝒳 → 𝒴𝒴 Engg. & Tech., Melvisharam, ( ) ( ) (1.3) Tamil Nadu, India for all . In addition if ( ) is‖ 𝑓𝑓continuous𝑥𝑥 − 𝐴𝐴 𝑥𝑥 in‖ ≤ for𝛿𝛿 each fixed , then is linear. ~6~ 𝑥𝑥 ∈ 𝒳𝒳 𝑓𝑓 𝑡𝑡𝑡𝑡 𝑡𝑡 𝑥𝑥 ∈ 𝒳𝒳 𝐴𝐴 International Journal of Statistics and Applied Mathematics

After Hyers gave a positive answer to Ulam’s question, a of norms. We also prove that the functional equation (1.6) is huge number of papers have been published in association not stable for critical values associated with the results of with various simplifications of Ulam’s problem and Hyers Th.M. Rassias and J.M. Rassias via counter-examples. theorem. Hyers theorem was generalized by T. Aoki [1] in Throughout this paper, let us assume that be the space of 1950 for additive mappings. non-negative real numbers. Since there is no elucidation for the boundedness of Cauchy 𝑋𝑋 difference ( + ) ( ) ( )in the expression of (1.1), 2. Preliminaries in the year 1978, Th.M. Rassias [26] tried to weaken the In this section, we introduce the following definition and stipulation 𝑓𝑓for𝑥𝑥 the𝑦𝑦 Cauchy− 𝑓𝑓 𝑥𝑥 difference− 𝑓𝑓 𝑦𝑦 and thrived in proving theorem in order to proceed further to our main results. what is now known to be the Hyers-Ulam-Rassias stability for the Additive Cauchy Equation. This jargon is reasonable Definition 2.1. A mapping : is called square root because the theorem of Th.M. Rassias has strongly persuaded mapping, if the functional equation (1.6) holds for all , many mathematicians studying stability problems of . Note that the mapping 𝑠𝑠is 𝑋𝑋called→ ℝ square root mapping functional equation. because the following algebraic identity 𝑥𝑥 𝑦𝑦 ∈ During 1982-1989, J.M. Rassias ([18-20]) provided a further 𝑋𝑋 𝑠𝑠 generalization of the result of D.H. Hyers and established a + + 2 = + theorem using weaker conditions controlled by a product of

different powers of norms and this type of stability is termed �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥 √𝑥𝑥 �𝑦𝑦 as Ulam-Gavruta-Rassias stability. holds for all , and satisfies the functional equations A generalized and modified form of the theorem evolved by Th.M. Rassias was promoted by P. Gavruta [10] who replaced 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 (2 𝑠𝑠 ) = 2 ( ) (2.1) the unbounded Cauchy difference by motivating into study a and 2𝑛𝑛 𝑛𝑛 general control function within the viable approach designed 𝑠𝑠(2 𝑥𝑥 ) = 2 𝑠𝑠 𝑥𝑥( ) (2.2) by Th.M. Rassias and this type of stability is celebrated as for all and .− 2𝑛𝑛 −𝑛𝑛 Generalized Hyers-Ulam stability. 𝑠𝑠 𝑥𝑥 𝑠𝑠 𝑥𝑥 The investigation of stability of functional equations Theorem𝑥𝑥 ∈ 2.2.𝑋𝑋 Let𝑛𝑛 :∈ ℕ be a square root mapping involving with the mixed type product-sum of powers of satisfying (1.6). Then satisfies the general functional norms is introduced by K. Ravi et al.[23] is called J.M. Rassias equations (2.1) and𝑠𝑠 (2.2)𝑋𝑋 → forℝ all and . stability controlled by mixed product-sum of powers of 𝑠𝑠 norms. Proof. Letting = 0 and = 0 𝑥𝑥in ∈(1.6),𝑋𝑋 we𝑛𝑛 obtain∈ ℕ (0) = 0. For the last four decades, many mathematicians have Plugging ( , ) in ( , ) in (1.6), one finds investigated various stability results of several functional 𝑥𝑥 𝑦𝑦 𝑠𝑠 equations like additive, quadratic, cubic, quartic, quintic, 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥 (2 ) = 2 ( ) (2.3) sextic, septic, octic and nonic, functional equations (refer [2-4, 2 [7, 8, 14, 16, 17, 21, 22, 29, 30]). There are many interesting results for all . Now, replacing𝑠𝑠 𝑥𝑥 by 4𝑠𝑠 𝑥𝑥in (2.3), one obtains concerning this problem and many research monographs are also available in functional equations, one can see ([5, 6, 13, 15, 𝑥𝑥 ∈ 𝑋𝑋 (2 𝑥𝑥) = 2𝑥𝑥 ( ) (2.4) 27] ). 4 2 K. Ravi and P. Narasimman [24] obtained the generalized for all . Again replacing𝑠𝑠 𝑥𝑥 by𝑠𝑠 𝑥𝑥4 in (2.3), we get Hyers-Ulam stability of a radical reciprocal-quadratic (2 ) = 2 ( ), for all . Proceeding further and using functional equation induction6 𝑥𝑥 ∈on3𝑋𝑋 a positive integer 𝑥𝑥 , we 𝑥𝑥achive (2.1). Next, 𝑠𝑠 𝑥𝑥 (𝑠𝑠 ,𝑥𝑥 ) , 𝑥𝑥 ∈ 𝑋𝑋 ( ) ( ) considering as in (1.6), one obtains + = (1.4) 𝑥𝑥 𝑥𝑥 𝑛𝑛 ( ) ( ) 𝑓𝑓 𝑥𝑥 𝑓𝑓 𝑦𝑦 4 4 2 2 𝑥𝑥 𝑦𝑦 � (�2 ) = 2 ( ) (2.5) 𝑓𝑓��𝑥𝑥 𝑦𝑦 � 𝑓𝑓 𝑥𝑥 +𝑓𝑓 𝑦𝑦 in Banach spaces. The reciprocal-quadratic function ( ) = −2 −1 is a solution of the functional equation (1.4). for all . Now, 𝑠𝑠 letting𝑥𝑥 as𝑠𝑠 𝑥𝑥 in (2.6), we 𝑐𝑐 [25] 𝑓𝑓 𝑥𝑥 𝑥𝑥 K.2 Ravi et al. investigated the generalized Hyers-Ulam obtain (2 ) = 2 ( ), for all . Proceeding further 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 𝑥𝑥 4 stability of a generalized radical-quadrtic functional equation and using− induction6 − 3on a positive integer , we obtain (2.2), which 𝑠𝑠completes𝑥𝑥 the proof.𝑠𝑠 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 ( ) ( ) 𝑛𝑛 ± + = ( ± ) (1.5) 2 2 ( ) ( ) �𝑥𝑥 +𝑦𝑦 𝑓𝑓 𝑥𝑥 𝑓𝑓 𝑦𝑦 3. General Solution of Functional Equation (1.6) 2 2 2 𝑓𝑓 � 𝑛𝑛 � 𝑚𝑚𝑚𝑚��𝑥𝑥 𝑦𝑦 � 𝑛𝑛 𝑚𝑚 𝑓𝑓 𝑥𝑥 +𝑓𝑓 𝑦𝑦 In this section, we obtain the general solution of functional in Felbin’s type fuzzy normed linear spaces. equation (1.6) in the setting of space of real numbers. In this paper, we introduce a new radical type functional equation of the form Theorem 3.1. A mapping : satisfies (1.6) if and only if there exists an identity mapping : such that + + 2 = ( ) + ( ). (1.6) 𝑓𝑓 𝑋𝑋 → ℝ ( ) = ( ), 𝐼𝐼for𝑋𝑋 all→ ℝ . The𝑠𝑠�𝑥𝑥 radical𝑦𝑦 function�𝑥𝑥𝑥𝑥� 𝑠𝑠( 𝑥𝑥) = 𝑠𝑠 𝑦𝑦 is a solution of the functional equation (1.6). We achieve the general solution of the Proof. Let satisfy𝑓𝑓 𝑥𝑥 equation�𝐼𝐼 𝑥𝑥 (1.6). Then 𝑥𝑥 is∈ a𝑋𝑋 square root functional equation 𝑠𝑠(1.6)𝑥𝑥 and√𝑥𝑥 investigate its generalized mapping and ( ) = , for all . If is an identity Hyers-Ulam stability, Hyers-Ulam stability, Hyers-Ulam- mapping, then𝑓𝑓 𝑓𝑓 Rassias stability, Ulam-Gavruta-Rassias stability and J.M. 𝑓𝑓 𝑥𝑥 √𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 𝐼𝐼 Rassias stability controlled by mixed product-sum of powers ( ) = = ( ), for all . ~7~ �𝐼𝐼 𝑥𝑥 √𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 International Journal of Statistics and Applied Mathematics

Conversely, assume that there exists an identity mapping (4 , 4 ) : such that ( ) = ( ), for all . Hence 1 ∞ 1 𝑚𝑚+𝑖𝑖 𝑚𝑚+𝑖𝑖 0 as 𝑚𝑚+𝑖𝑖 . 2 𝑖𝑖=0 2 ≤ ∑ 𝜙𝜙 𝑥𝑥 𝑥𝑥 + + 2 = + + 2 𝐼𝐼 𝑋𝑋 → ℝ 𝑓𝑓 𝑥𝑥 �𝐼𝐼 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 This shows that →the sequence𝑚𝑚 → ∞ {2 (4 )} is a Cauchy sequence. Allow in (4.6), we− 𝑛𝑛arrive𝑛𝑛 at (4.3). To show 𝑓𝑓� 𝑥𝑥 𝑦𝑦 �𝑥𝑥 𝑥𝑥� = �𝐼𝐼�𝑥𝑥+ 𝑦𝑦+ 2 �𝑥𝑥 𝑥𝑥� that satisfies (1.6), replacing ( , ) 𝑓𝑓by (4𝑥𝑥 , 4 ) in (4.2) and multiplying by𝑛𝑛 2→ ∞, we obtain 𝑛𝑛 𝑛𝑛 �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥 = + 𝑠𝑠 −𝑛𝑛 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 2 = + , 2 4 + + 2 (4 ) (4 ) ��√𝑥𝑥 �, for𝑦𝑦� all , − 𝑛𝑛 𝑛𝑛 2 (4 , 4 𝑛𝑛) 𝑛𝑛 (4.7) √𝑥𝑥 �𝑦𝑦 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 �𝑓𝑓 � �𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥�� − 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑦𝑦 � which completes the proof of Theorem 3.1. for all , . Allowing�−𝑛𝑛 𝑛𝑛 𝑛𝑛 in (4.7), we see that satisfies (1.6) for ≤all , 𝜙𝜙 𝑥𝑥. To 𝑦𝑦prove is unique square 4. Generalized Hyers-Ulam Stability of Equation (1.6) root mapping𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋satisfying (1.6),𝑛𝑛 → let∞ us assume : be𝑠𝑠 In this section, we investigate the generalized Hyers-Ulam another square root mapping𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 which satisfies𝑠𝑠 (1.6)′ and the stability of the functional equation (1.6) in the setting of space inequality (4.3). Clearly and satisfy (2.1) and𝑠𝑠 using𝑋𝑋 → ℝ(4.3), of real numbers. We also extend the stability results in the we have ′ spirit of Hyers, Th.M. Rassias and J.M. Rassias. 𝑠𝑠 𝑠𝑠 | ( ) ( )| = 2 | (4 ) (4 )| : × Theorem 4.1. Let be a mapping satisfying ′ 2 (| (4 )−𝑛𝑛 ′(4 𝑛𝑛 )| + | (4𝑛𝑛 ) (4 )|)

𝑠𝑠 𝑥𝑥 −𝑛𝑛𝑠𝑠 𝑥𝑥′ 𝑛𝑛(4 𝑠𝑠, 4 𝑛𝑛 𝑥𝑥 )− 𝑠𝑠 𝑛𝑛𝑥𝑥 𝑛𝑛 (4.8) ( , ) =𝜙𝜙 𝑋𝑋 𝑋𝑋 → ℝ(4 , 4 ) < (4.1) ≤ ∞ 𝑠𝑠1 𝑥𝑥 𝑛𝑛−+𝑖𝑖 𝑓𝑓 𝑛𝑛+𝑥𝑥𝑖𝑖 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 𝑛𝑛+𝑖𝑖 ∞ 1 𝑖𝑖 𝑖𝑖 𝑖𝑖=0 2 𝑖𝑖+1 ≤ ∑ 𝜙𝜙 𝑥𝑥 𝑥𝑥 𝛷𝛷 𝑥𝑥 𝑦𝑦 ∑𝑖𝑖=0 2 𝜙𝜙 𝑥𝑥 𝑦𝑦 ∞ for all . Allowing in (4.8) and using (4.1), we for all , . Let : be a mapping such that find that is unique. This completes the proof of Theorem 4.1. 𝑥𝑥 ∈ 𝑋𝑋 𝑛𝑛 → ∞ 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 𝑓𝑓 𝑋𝑋 → ℝ + + 2 ( ) ( ) ( , ) (4.2) 𝑠𝑠 Theorem 4.2. Let : × be a mapping satisfying for all�𝑓𝑓� 𝑥𝑥 , 𝑦𝑦 . �Then𝑥𝑥𝑥𝑥� −there𝑓𝑓 𝑥𝑥 exists− 𝑓𝑓 𝑦𝑦 a� ≤uniqu𝜙𝜙 𝑥𝑥e 𝑦𝑦square root mapping : which satisfies (1.6) and the inequality 𝜙𝜙 𝑋𝑋 𝑋𝑋 → ℝ 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 ( , ) = ∞ 2 , < 4 4 𝑠𝑠 𝑋𝑋 → | ℝ( ) ( )| ( , ) (4.3) 𝑖𝑖 𝑥𝑥 𝑦𝑦 Φ 𝑥𝑥 𝑦𝑦 � 𝜙𝜙 � 𝑖𝑖+1 𝑖𝑖+1� ∞ 𝑖𝑖=0 for all . The𝑓𝑓 mapping𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ( ≤) Φis defined𝑥𝑥 𝑥𝑥 by for all , . Let : be a mapping such that (4.2) holds for all , . Then there exists a unique square root 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 𝑓𝑓 𝑋𝑋 → ℝ 𝑥𝑥 ∈ 𝑋𝑋 ( ) = lim 𝑠𝑠 𝑥𝑥 (4 ) (4.4) mapping : which satisfies (1.6) and the inequality 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 1 𝑛𝑛 𝑛𝑛 𝑠𝑠 𝑥𝑥 𝑛𝑛→∞ 2 𝑓𝑓 𝑥𝑥 𝑠𝑠 𝑋𝑋 → ℝ| ( ) ( )| ( , ) for all and . ( , ) ( , ) 2 Proof. Plugging into in (4.2) and dividing by , for all . The mapping𝑓𝑓 𝑥𝑥 − 𝑠𝑠 (𝑥𝑥 ) is≤ definedΦ 𝑥𝑥 𝑥𝑥 by 𝑥𝑥 ∈ 𝑋𝑋 𝑛𝑛 ∈ ℕ we obtain 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 ( ) = lim𝑠𝑠 𝑥𝑥 2 (4 ) (4 ) ( ) ( , ) (4.5) 𝑛𝑛 −𝑛𝑛 1 1 𝑠𝑠 𝑥𝑥 𝑛𝑛→∞ 𝑓𝑓 𝑥𝑥 for all and . �2 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥 � ≤ 2 𝜙𝜙 𝑥𝑥 𝑥𝑥 for all . Now, replacing by 4 in (4.5), dividing by 2 and summing the resulting inequality with (4.5), one finds 𝑥𝑥 ∈ 𝑋𝑋 𝑛𝑛 ∈ ℕ Proof. Plugging ( , ) into , in (4.2), one finds 𝑥𝑥 ∈ 𝑋𝑋 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 1 1 𝑥𝑥 𝑦𝑦 �4 4� (4 ) ( ) 1 (4 , 4 ) ( ) 2 , (4.9) 2 2 2 𝑖𝑖 𝑖𝑖 𝑥𝑥 𝑥𝑥 𝑥𝑥 � 2 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥 � ≤ � 𝑖𝑖+1 𝜙𝜙 𝑥𝑥 𝑥𝑥 �𝑓𝑓 𝑥𝑥 − 𝑓𝑓 �4�� ≤ 𝜙𝜙 �4 4� 𝑖𝑖=0 for all . Now, replacing by in (4.9), multiplying by 2 for all . Using induction arguments, we conclude that 𝑥𝑥 and summing the resulting inequality with (4.9), one gets 𝑥𝑥 ∈ 𝑋𝑋 𝑥𝑥 4 1 𝑥𝑥 ∈ 𝑋𝑋 1 (4 ) ( ) 𝑛𝑛−1 (4 , 4 ) 2 2 𝑛𝑛 𝑖𝑖 𝑖𝑖 ( ) 2 1 2 , 𝑛𝑛 𝑖𝑖+1 4 4 4 � 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥 � ≤ � 𝜙𝜙(4 𝑥𝑥 , 4 𝑥𝑥 ) (4.6) 2 𝑥𝑥 𝑖𝑖 𝑥𝑥 𝑥𝑥 𝑖𝑖=0 �𝑓𝑓 𝑥𝑥 − 𝑓𝑓 � 2�� ≤ � 𝜙𝜙 � 𝑖𝑖+1 𝑖𝑖+1� ∞ 1 𝑖𝑖 𝑖𝑖 𝑖𝑖=0 𝑖𝑖+1 ≤ ∑𝑖𝑖=0 2 𝜙𝜙 𝑥𝑥 𝑥𝑥 for all . Proceeding further and using induction on a for all . In order to prove the convergence of the positive integer , we conclude that sequence {2 (4 )}, replace by 4 in (4.6) and 𝑥𝑥 ∈ 𝑋𝑋 multiply𝑥𝑥 by∈ 2𝑋𝑋−𝑛𝑛 , we𝑛𝑛 find that for > > 𝑚𝑚0 𝑛𝑛 𝑓𝑓 𝑥𝑥 𝑥𝑥 𝑥𝑥 −𝑚𝑚 | ( ) 2 (4 )| 𝑛𝑛−1 2 , |2 (4 ) 2 (4 )|𝑛𝑛 𝑚𝑚 4 4 𝑛𝑛 −𝑛𝑛 𝑖𝑖 𝑥𝑥 𝑥𝑥 −𝑚𝑚 𝑚𝑚 −=𝑛𝑛−2𝑚𝑚 | 𝑛𝑛(+4𝑚𝑚 ) 2 (4 )| 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥 ≤ � 𝜙𝜙 � 𝑖𝑖+1 𝑖𝑖+1� 𝑖𝑖=0 2 , 𝑓𝑓 𝑥𝑥 − −𝑓𝑓𝑚𝑚 𝑚𝑚𝑥𝑥 −𝑛𝑛 𝑛𝑛+𝑚𝑚 ∞ 𝑖𝑖 𝑥𝑥 𝑥𝑥 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑥𝑥 𝑖𝑖=0 𝑖𝑖+1 𝑖𝑖+1 ~8~ ≤ ∑ 𝜙𝜙 �4 4 � International Journal of Statistics and Applied Mathematics

for all . The rest of the proof is obtained by similar for all , , then there exists a unique square root arguments as in Theorem 4.1. mapping : satisfying the functional equation (1.6) and 𝑥𝑥 ∈ 𝑋𝑋 the inequality𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 Corollary 4.3. Let : be a mapping for which there 𝑠𝑠 𝑋𝑋 → ℝ exists a constant 0, independent of , such that the | ( ) ( )| | | | | ,for all . functional inequality𝑓𝑓 𝑋𝑋 → ℝ 3𝑐𝑐3 2𝛼𝛼 2𝛼𝛼 𝑐𝑐 ≥ 𝑥𝑥 𝑦𝑦 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ≤ 2−4 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 Proof. By choosing ( , ) = | | | | + (| | + + + 2 ( ) ( ) | | ) , for all , in Theorems 4.1 and𝛼𝛼 4.2,𝛼𝛼 the proof2𝛼𝛼 is 𝜙𝜙 𝑥𝑥 𝑦𝑦 𝑐𝑐3� 𝑥𝑥 𝑦𝑦 𝑥𝑥 ,�𝑓𝑓�𝑥𝑥 𝑦𝑦 �:𝑥𝑥𝑥𝑥� − 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑦𝑦 � ≤ 𝑐𝑐 obtained.2𝛼𝛼 holds for . Then defined by 𝑦𝑦 � 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋

𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 𝑠𝑠 𝑋𝑋 → 1ℝ 5. Counter-Examples for Singular Cases ( ) = lim (4 ) In this section, we present counter-examples to prove that the 2 −𝑛𝑛 functional equation (1.6) isnot stable for singular cases in 𝑠𝑠 𝑥𝑥 𝑛𝑛→∞ 𝑛𝑛 𝑓𝑓 𝑥𝑥 Corollaries 4.4, 4.5 and 4.6. is a square root mapping satisfying the functional equation [9] (1.6) and Using the idea of Z. Gajda , we construct the following counter-example to prove that the Corollary 4.4 is false for | ( ) ( )| = . 1 2 for all and 𝑓𝑓. 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ≤ 𝑐𝑐 Define𝑝𝑝 a mapping : by

𝑥𝑥 ∈ 𝑋𝑋 (𝑛𝑛,∈ )ℕ= , 𝑓𝑓 𝑋𝑋 → ℝ ( ) Proof. Taking , for all in Theorem 4.1, ( ) = (5.1) ( , ) = 𝑛𝑛 we get . From (4.3), one finds ∞ 𝜑𝜑 4 𝑥𝑥 𝑛𝑛 𝜙𝜙 𝑥𝑥 𝑦𝑦 𝑐𝑐 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 𝑛𝑛=0 2 for all , where𝑓𝑓 the𝑥𝑥 function∑ : is given by 𝜙𝜙 𝑥𝑥 𝑥𝑥 𝑐𝑐 1 | ( ) ( )| ∞ 2 2 𝑥𝑥 ∈ 𝑋𝑋 , for all 𝜑𝜑, 𝑋𝑋 →(0ℝ,1) 𝑐𝑐 ( ) = (5.2) 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ≤ � 𝑖𝑖 ≤ 𝑐𝑐 , otherwise. 𝑖𝑖=0 𝜇𝜇 𝑥𝑥 𝑥𝑥 𝑦𝑦 ∈ for all . 𝜑𝜑 𝑥𝑥 � √ Then the function serves𝜇𝜇 as a counter-example for = as 𝑥𝑥 ∈ 𝑋𝑋 Corollary 4.4. Let 0 be fixed and . If a mapping presented in the following theorem. 1 2 : satisfies the inequality 1 𝑓𝑓 𝑝𝑝 1 2 𝑐𝑐 ≥ 𝑝𝑝 ≠ Theorem 5.1. The mapping defined above satisfies 𝑓𝑓 𝑋𝑋 → ℝ+ + 2 ( ) ( ) (| | + | | ) 𝑓𝑓 𝑝𝑝 𝑝𝑝 + + 2 ( ) ( ) 12 | | + | | (5.3) �𝑓𝑓�𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥� − 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑦𝑦 � ≤ 𝑐𝑐1 𝑥𝑥 𝑦𝑦 1 1 for all , , then there exists a unique square root 2 2 mapping : satisfying the functional equation (1.6) and for�𝑓𝑓� 𝑥𝑥all 𝑦𝑦, �𝑥𝑥. 𝑥𝑥Therefore� − 𝑓𝑓 𝑥𝑥 −there𝑓𝑓 𝑦𝑦 do� ≤ not 𝜇𝜇exist� 𝑥𝑥 a square𝑦𝑦 � root 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 mapping : and a constant > 0 such that 𝑠𝑠| 𝑋𝑋( →) ℝ ( )| | | 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 | | , for all . 2𝑐𝑐1 𝑝𝑝 𝑠𝑠 𝑋𝑋 → ℝ 𝛽𝛽 𝑝𝑝 | ( ) ( )| | | (5.4) 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ≤ 2−4 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 1 Proof. Considering ( , ) = (| | + | | ), for all , 2 in Theorems 4.1 and 4.2, we arrive 𝑝𝑝at the required𝑝𝑝 results. 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ≤ 𝛽𝛽 𝑥𝑥 1 for all . 𝜙𝜙 𝑥𝑥 𝑦𝑦 𝑐𝑐 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 ∈ Corollary𝑋𝑋 4.5. Let : be a mapping. If there exist 𝑥𝑥 ∈ 𝑋𝑋 | ( )| Proof. | ( )| = 1 = , : = + and 0 such that | 𝑛𝑛| −1 ∞ 𝜑𝜑 4 𝑥𝑥 ∞ 𝜇𝜇 1 1 𝑓𝑓 𝑋𝑋 → ℝ 𝑛𝑛 𝑛𝑛 2 . Hence 𝑓𝑓 𝑥𝑥is bounded≤ ∑𝑛𝑛=0 by2 2 . ≤If ∑| 𝑛𝑛|=0+2 | | 𝜇𝜇 � ,− then2� the 𝑎𝑎 𝑏𝑏 𝜌𝜌 𝑎𝑎 𝑏𝑏 ≠ 2 𝑐𝑐2 ≥ 1 1 + + 2 ( ) ( ) | | | | left-hand side of (5.3) is less than 6 . Now,2 suppose2 1 that0 < 𝜇𝜇 𝑓𝑓 𝜇𝜇 𝑥𝑥 𝑦𝑦 ≥ 2 𝑎𝑎 𝑏𝑏 2 | | + | | < . Then there exists a positive integer such for all�𝑓𝑓 �𝑥𝑥, 𝑦𝑦 , then�𝑥𝑥𝑥𝑥 �there− 𝑓𝑓 𝑥𝑥exists− 𝑓𝑓 a𝑦𝑦 �unique≤ 𝑐𝑐 𝑥𝑥square𝑦𝑦 root 1 1 𝜇𝜇 that2 2 1 mapping : 2 𝑥𝑥 𝑦𝑦 𝑘𝑘 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 𝑠𝑠 𝑋𝑋 → ℝ | | + | | < . | ( ) ( )| | | , for all . 1 1 (5.5) | | 1 1 𝑐𝑐2 𝜌𝜌 2 2 𝜌𝜌 𝑘𝑘+1 𝑘𝑘 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 ≤ 2−4 𝑥𝑥 𝑥𝑥 ∈ 𝑋𝑋 2 ≤ 𝑥𝑥 𝑦𝑦 2 Proof. The proof is obtained by letting ( , ) = | | | | , | | + | | < Hence 1 1 implies for all , in Theorems 4.1 and 4.2. 𝑎𝑎 𝑏𝑏 2 2 1 2 𝑘𝑘 42 | | + 4 | | < 1 𝜙𝜙 𝑥𝑥 𝑦𝑦 𝑐𝑐 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑘𝑘 1 𝑘𝑘 1 Corollary𝑥𝑥 𝑦𝑦 4.6.∈ 𝑋𝑋 Let : be a mapping. If there exist or 24 2 + 42 2< 1 𝑘𝑘𝑥𝑥 1 𝑘𝑘 𝑦𝑦1 0 > 0 2 2 2 2 and with such that 3 or 4 < , 4 < 𝑓𝑓 𝑋𝑋1 → ℝ 𝑐𝑐 ≥ 𝑘𝑘−1 𝑥𝑥1 𝑦𝑦𝑘𝑘−1 1 + + 2 ( ) ( ) 2 2 1 2 2 1 𝛼𝛼 𝛼𝛼 ≠ 4 | | | | + (| | + | | ) and consequently 𝑥𝑥 2 𝑦𝑦 2 �𝑓𝑓�𝑥𝑥 𝑦𝑦 𝑥𝑥𝑥𝑥� − 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑦𝑦 � � 𝛼𝛼 𝛼𝛼 2𝛼𝛼 2𝛼𝛼 ≤ 𝑐𝑐3� 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 � ~9~ International Journal of Statistics and Applied Mathematics

| ( ) ( )| 4 ( ), 4 ( ), 4 + + 2 (0,1). = + (5.9) 𝑓𝑓 𝑥𝑥 −𝑠𝑠 𝑥𝑥 𝑘𝑘−1 𝑘𝑘−1 𝑘𝑘−1 𝑥𝑥 𝑦𝑦 �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥� ∈ 𝑠𝑠𝑢𝑢𝑢𝑢𝑥𝑥≠0 √𝑥𝑥 ∞ Therefore, for each value of = 0,1,2, … , 1, we obtain for every square root mapping : .

𝑛𝑛 𝑘𝑘 − 4 ( ), 4 ( ), 4 + + 2 (0,1) Proof. From the relation (5.9), it𝑠𝑠 follows𝑋𝑋 → ℝ that 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑥𝑥 𝑦𝑦 �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥� ∈ and 4 + + 2 (4 ) (4 ) = 0 | ( ) ( )| | ( ) ( )| , 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝜑𝜑 � �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥�� − 𝜑𝜑 𝑥𝑥 − 𝜑𝜑 𝑦𝑦 𝑥𝑥≠0 𝑓𝑓 𝑥𝑥 − 𝑠𝑠 𝑥𝑥 𝑛𝑛∈ℕ 𝑛𝑛≠0 𝑓𝑓 𝑛𝑛| |− 𝑠𝑠 (𝑛𝑛) for = 0,1,2, … , 1. Using (5.5) and the definition of , 𝑠𝑠 𝑠𝑠𝑠𝑠 =≥ 𝑠𝑠𝑠𝑠𝑠𝑠 , we obtain √𝑥𝑥 √𝑛𝑛 ln 𝑛𝑛√−𝑛𝑛√𝑛𝑛𝑠𝑠 1 = 𝑛𝑛∈ℕ,𝑛𝑛≠0|ln| | (1)| = . 𝑛𝑛 𝑘𝑘 − 𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠 √𝑛𝑛 We have prove (5.8) is true. ( ) 𝑛𝑛∈ℕ 𝑛𝑛≠0 + + 2 ( ) 𝑠𝑠𝑠𝑠𝑠𝑠 𝑛𝑛 − 𝑠𝑠 ∞ 1 Case 1. If = = 0 in (5.8), then �𝑓𝑓�𝑥𝑥∞ 𝑦𝑦 �4 𝑥𝑥𝑥𝑥�+− 𝑓𝑓+𝑥𝑥2− 𝑓𝑓 𝑦𝑦 � (4 ) (4 ) 2 | (0) (0) (0)| = | (0)| = 0. 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑥𝑥 𝑦𝑦 ≤ � 𝑛𝑛 �𝜑𝜑 � �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥�� − 𝜑𝜑 𝑥𝑥 − 𝜑𝜑 𝑦𝑦 � 𝑛𝑛=0 1 Case 2. If , 𝑓𝑓> 0−, then𝑓𝑓 the− left𝑓𝑓 -hand side𝑓𝑓 of (5.8) becomes ∞ 4 + + 2 (4 ) (4 ) 2 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑥𝑥 𝑦𝑦 ≤ � 𝑛𝑛 �𝜑𝜑 � �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥�� − 𝜑𝜑 𝑥𝑥 − 𝜑𝜑 𝑦𝑦 � + + 2 ( ) ( ) 𝑛𝑛=𝑘𝑘 3 ∞ 2 � =𝑓𝑓�𝑥𝑥 𝑦𝑦+ + 2𝑥𝑥𝑥𝑥� − 𝑓𝑓ln𝑥𝑥 −+𝑓𝑓 +𝑦𝑦 2� 𝜇𝜇 � 𝑛𝑛 ≤ � 12 | | + | | 𝑛𝑛=𝑘𝑘 1 1 . ��𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥� ln�(𝑥𝑥 ) 𝑦𝑦 ln�(𝑥𝑥𝑥𝑥)� 12𝜇𝜇 2 2 𝑘𝑘+1 = + ln + + 2 ln( ) ln( ) ≤ 2 ≤ 𝜇𝜇 � 𝑥𝑥 𝑦𝑦 � − √𝑥𝑥 𝑥𝑥 − �𝑦𝑦 𝑦𝑦 Therefore, the inequality (5.3) holds true. We claim that the + + 2 + + 2 square root functional equation (1.6) is not stable for = in = �√𝑥𝑥ln �𝑦𝑦� �𝑥𝑥 𝑦𝑦 +�𝑥𝑥𝑥𝑥�ln− √𝑥𝑥 𝑥𝑥 − �𝑦𝑦 𝑦𝑦 Corollary 4.4. 1 𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥 𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥 𝑝𝑝 2 √𝑥𝑥 � � �𝑦𝑦 � � Assume that there exists a square root mapping : = ln 1 + +𝑥𝑥2 + ln 1 + + 2𝑦𝑦 . satisfying (5.4). Therefore, we have 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑥𝑥

𝑠𝑠 𝑋𝑋 → ℝ √𝑥𝑥 � 𝑥𝑥 �𝑥𝑥� �𝑦𝑦 � 𝑦𝑦 �𝑦𝑦� We have to prove that there exists a constant 0 such that | ( )| ( + 1)| | . (5.6) 1 2 2 𝑐𝑐 ≥ However,𝑓𝑓 𝑥𝑥 ≤ we𝛽𝛽 can choose𝑥𝑥 a positive integer with > + ln 1 + + 2 + ln 1 + + 2 1. If (0, 4 ), then 4 (0,1) for all = 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑥𝑥 0,1,2, … , 1 and1− 𝑚𝑚therefore 𝑛𝑛 𝑚𝑚 𝑚𝑚𝑚𝑚 𝛽𝛽 √𝑥𝑥 � � � �𝑦𝑦 � � � 𝑥𝑥 𝑥𝑥 | | | | 𝑦𝑦 𝑦𝑦 𝑥𝑥 ∈ 𝑥𝑥 ∈ 𝑛𝑛 1 (5.10) 𝑚𝑚 − 𝑎𝑎 �2−𝑎𝑎� (4 ) 2 2 | ( )| = ∞ 𝑚𝑚−1 = > ( + 1) , > 0 ≤=𝑐𝑐 𝑥𝑥 𝑦𝑦 𝑛𝑛 𝑛𝑛 for all . With , the inequality (5.10) is equivalent 2 2 𝑦𝑦 𝜑𝜑 𝑥𝑥 𝜇𝜇√𝑥𝑥 to the inequality 𝑓𝑓 𝑥𝑥 � 𝑛𝑛 ≥ � 𝑛𝑛 𝑚𝑚𝑚𝑚√𝑥𝑥 𝛽𝛽 √𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑡𝑡 𝑛𝑛=0 𝑛𝑛=0 which contradicts (5.6). Therefore, the square root functional 1 1 2 ln 1 + + 2 + ln 1 + + equation (1.6) is not stable for = in Corollary 4.4. 𝑎𝑎 1 1 The following example illustrates that the functional equation � − 𝑎𝑎 � > 0. (5.11) 𝑝𝑝 2 2 � 𝑡𝑡 √𝑡𝑡� 𝑡𝑡 � , � (1.6) is not stable when = + = in Corollary 4.5. 𝑡𝑡 𝑡𝑡 √𝑡𝑡 2 Inspired by the counter-example provided1 by P. Gavruta in By using L’ Hospital rule, ≤ 𝑐𝑐 𝑡𝑡 2 [11], we present the following𝜌𝜌 𝑎𝑎 example𝑏𝑏 which establishes that = + = 1 2 1 2 + the Corollary 4.5 is false for . lim ln 1 + + = lim = 0 1 𝑎𝑎 𝑎𝑎 + 2 + 1 𝜌𝜌 𝑎𝑎 𝑏𝑏 2 𝑡𝑡 � √𝑡𝑡� Define a mapping : by and lim𝑡𝑡→0 𝑡𝑡 ln �1 + + �= lim𝑡𝑡→0 = 0 𝑡𝑡 𝑡𝑡 𝑎𝑎 𝑎𝑎𝑡𝑡 𝑡𝑡 | | 0 𝑎𝑎 1 2√ 1 𝑡𝑡 �2+√𝑡𝑡�√ ( ) = (5.7) 𝑓𝑓 𝑋𝑋 → ℝ 𝑡𝑡→∞ 𝑡𝑡 � 𝑡𝑡 √𝑡𝑡� 𝑎𝑎 𝑡𝑡→∞ 𝑡𝑡+2√𝑡𝑡+1 0 = 0. and since the function 𝑥𝑥 𝑙𝑙𝑙𝑙 𝑥𝑥 𝑖𝑖𝑖𝑖 𝑥𝑥 ≠ 𝑓𝑓 𝑥𝑥 �√ 1 2 Then the function𝑖𝑖𝑖𝑖 𝑥𝑥 defined in (5.7) turns out to be a counter- ln 1 + + 𝑎𝑎 example for = as proved in the following theorem. 𝑓𝑓 𝑡𝑡 ⟶ 𝑡𝑡 � � 1 is continuous on (0, ), it follows𝑡𝑡 that√𝑡𝑡 there is a constant 𝜌𝜌 2 Theorem 5.2. Let be such that 0 < < . Then there is a = ( ) > 0 such that 1 ∞ function : and a constant 0 satisfying 2 2 𝑎𝑎 𝑎𝑎 2 𝑐𝑐 𝑐𝑐 𝑎𝑎 ln 1 + + , > 0. (5.12) + + 2 ( ) ( ) 2 | | | | (5.8) 𝑓𝑓 𝑋𝑋 → ℝ 𝑐𝑐 ≥ 1 𝑎𝑎 1 2 2 , 𝑎𝑎 �2−𝑎𝑎� If we replace by and 𝑡𝑡by √𝑡𝑡 in (5.12), it follows that for all and 2 𝑡𝑡 � � ≤ 𝑐𝑐 𝑡𝑡 � 𝑓𝑓�𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥� − 𝑓𝑓 𝑥𝑥 − 𝑓𝑓 𝑦𝑦 � ≤ 𝑐𝑐 𝑥𝑥 𝑦𝑦 1 1 𝑥𝑥 𝑦𝑦 ∈ 𝑋𝑋 𝑡𝑡 𝑡𝑡 𝑎𝑎 �2 − 𝑎𝑎� ~10~ International Journal of Statistics and Applied Mathematics

ln 1 + + 2 , > 0. (5.13) 1 1 1 ∞ 4 + + 2 (4 ) (4 ) 1 2 � −𝑎𝑎� 2 2 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑡𝑡 2 � 𝑡𝑡 √𝑡𝑡� ≤ 𝑐𝑐 � − 𝑎𝑎� 𝑡𝑡 From (5.12) and (5.13), we conclude that ≤ � 𝑛𝑛 �𝜑𝜑 � �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥�� − 𝜑𝜑 𝑥𝑥 − 𝜑𝜑 𝑦𝑦 � 𝑛𝑛=0 1 ∞ 4 + + 2 (4 ) (4 ) ln 1 + + 2 + ln 1 + + , > 0. 2 𝑛𝑛 𝑛𝑛 𝑛𝑛 1 𝑎𝑎 1 2 𝑐𝑐2 1 𝑛𝑛 � −𝑎𝑎� 2 ≤ � �𝜑𝜑 � �𝑥𝑥 𝑦𝑦 �𝑥𝑥𝑥𝑥�� − 𝜑𝜑 𝑥𝑥 − 𝜑𝜑 𝑦𝑦 � 2 � 𝑡𝑡 √𝑡𝑡� 𝑡𝑡 � 𝑡𝑡 √𝑡𝑡� ≤ 2 ≤ 𝑐𝑐 𝑡𝑡 𝑛𝑛=𝑘𝑘 3 𝑡𝑡 ∞ That is, (5.11) holds and hence (5.8) holds. 2 In order to show that the functional equation (1.6) is not stable 𝜇𝜇 ≤ � 𝑛𝑛 if = in Corollary 4.6, consider the functions and 𝑛𝑛=𝑘𝑘 12 | | | | + | | + | | . 1 1 1 1 1 12𝜇𝜇 4 4 2 2 defined in (5.1) and (5.2) respectively. Then the function 𝑘𝑘+1 𝛼𝛼 4 𝑓𝑓 𝜑𝜑 ≤ 2 ≤ 𝜇𝜇 � 𝑥𝑥 𝑦𝑦 � 𝑥𝑥 𝑦𝑦 �� defined in (5.1) becomes a counter-example for = as Therefore, the inequality (5.14) holds true. The rest of the 1 𝑓𝑓 presented in the following theorem. proof is similar to that of Theorem 5.1. 𝛼𝛼 4

Theorem 5.3. The mapping defined in (5.1) satisfies the 6. Conclusion inequality Thus we have obtained the general solution and proved the 𝑓𝑓 Ulam stability of the functional equation (1.6) pertinent to + + 2 ( ) ( ) D.H. Hyers, Th.M. Rassias, J.M. Rassias and P. Gavruta. Further, we have shown that the functional equation (1.6) is 12 | | | | + | | + | | � 𝑓𝑓 � 𝑥𝑥 𝑦𝑦 � 𝑥𝑥𝑥𝑥� − 𝑓𝑓 𝑥𝑥 1− 𝑓𝑓1 𝑦𝑦 � 1 1 (5.14) not stable for singular cases by illustrating counter-examples. 4 4 2 2

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