Jleli and Samet Advances in Difference Equations (2017) 2017:21 DOI 10.1186/s13662-017-1076-7
R E S E A R C H Open Access Solvability of a q-fractional integral equation arising in the study of an epidemic model Mohamed Jleli and Bessem Samet*
*Correspondence: [email protected] Abstract Department of Mathematics, College of Science, King Saud We study the solvability of a functional equation involving q-fractional integrals. Such University, P.O. Box 2455, Riyadh, an equation arises in the study of the spread of an infectious disease that does not 11451, Saudi Arabia induce permanent immunity. Our method is based on the noncompactness measure argument in a Banach algebra and an extension of Darbo’s fixed point theorem. MSC: 62P10; 26A33; 45G10 Keywords: epidemic model; q-calculus; fractional order; measure of noncompactness
1 Introduction In this paper, we deal with the solvability of the q-fractional integral equation
N t gi(t, x(t)) x(t)= f (t)+ (t – qs)(αi–)u s, x(s) d s , t ∈ I, (.) i (α ) i q i= q i
where I = [, ], q ∈ (, ), αi >,fi : I → R,andgi, ui : I × R → R (i =,...,N).
For N =,q =,andα =,equation(.) arises in the study of the spread of an infectious disease that does not induce permanent immunity (see [–]). Equation (.)canbewrittenas
N αi · · ∈ x(t)= fi(t)+gi t, x(t) Iq ui , x( ) (t) , t I, i=
αi where Iq is the q-fractional integral of order αi defined by [] t αi (αi–) ∈ Iq h(t)= (t – qs) h(s) dqs, t I. q(αi)
Via noncompactness measure argument in a Banach algebra, we provide sufficient con- ditions for the existence of at least one solution to equation (.). We also give an example in order to illustrate our existence result.
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 2 of 15
The concept of noncompactness measure was used by many authors in order to study the solvability of various classes of integral equations (see [–] and references therein). Very recently, Jleli, Mursaleen, and Samet [] studied the solvability of a functional equa- tion involving the q-fractional integral equation t f (t, x(t)) (α–) x(t)=F t, x a(t) , (t – qs) u s, x(s) dqs , t ∈ [, ]. q(α)
Using the noncompactness measure tool, the authors obtained an existence result for such an equation. To the best of our knowledge, [] is the only work dealing with the solvability of a q-fractional integral equation with noncompactness measure. In this paper, we use a different approach in order to establish our existence result. We first establish a fixed point theorem in a Banach algebra via a measure of noncompactness satisfying condition (m). Next, via the obtained fixed point result, we prove that, under certain conditions, equation (.) has at least one solution.
2 Preliminaries on quantum calculus The concept of quantum calculus was introduced by Jackson [, ]. This subject is rich in history and has many applications (see [–]). In this section, we recall some basic facts on quantum calculus and present additional properties that will be used later. For more details, we refer to [].
Let q ∈ [, ∞)\{}.Forx ∈ R,wedefinetheq-real number [x]q by
–qx [x] = . q –q
The q-factorial of x is defined by
k– i (x, q) =, (x, q)k = –xq , k =,,...,∞. i=
For (a, b) ∈ R,theq-analog of (a – b)k is defined by
k– (a – b)() =, (a – b)(k) = a – qib , k =,,.... i=
For β ∈ R,(a, b) ∈ R,anda ≥ ,
∞ a – bqi (a – b)(β) = aβ . a – bqβ+i i=
Note that, for b =,wehave
a(β) = aβ .
Lemma . ([]) If β >and ≤ a ≤ b ≤ t, then
(t – b)(β) ≤ (t – a)(β). Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 3 of 15
The q-gamma function is given by
( – q)(x–) (x)= , x ∈{/ ,–,–,...}. q ( – q)x–
We have the following property:
q(x +)=[x]q q(x).
Let f :[,T] → R be a given function (T > ). We define the q-integral of the function f by
t ∞ n n Iqf (t)= f (s) dqs = t( – q) f tq q , t ∈ [, T]. n=
If η ∈ [, T], then T T η f (s) dqs = f (s) dqs – f (s) dqs. η
Lemma . Let f :[,]→ R be a continuous function. Then t t
f (s) dqs ≤ f (s) dqs, t ∈ [, ].
Remark . If ≤ t ≤ t ≤ andf :[,]→ R is a continuous function, then the in- equality t t f (s) dqs ≤ f (s) dqs t t
is not satisfied in general. As a counterexample, we refer to [], p..
Let f :[,]→ R be a given function. The fractional q-integral of order α ≥ ofthe function f is given by Iq f (t)=f (t)and t α (α–) ∈ Iq f (t)= (t – qs) f (s) dqs, t [, ], α >. q(α)
Note that, for α =,wehave
∈ Iqf (t)=Iqf (t), t [, ].
For f ≡ ,
α α ∈ Iq (t)= t , t [, ]. q(α +) Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 4 of 15
3 A fixed point theorem in a Banach algebra via a measure of noncompactness satisfying condition (m) In this section, we recall the axiomatic approach of noncompactness measure introduced by Banas and Goebel []. Next, we establish a fixed point theorem in a Banach algebra via a measure of noncompactness satisfying condition (m). This fixed point result plays an important rule in the proof of our existence result.
Let (E, ·) be a Banach Algebra over R withrespecttoacertainnorm·E.Wedenote
by E the zero vector of E.Forx ∈ E and r >,wedenotebyB(x, r)theopenballinE of center x and radius r,thatis,
B(x, r)= y ∈ E : x – yE < r .
We denote by P(E) the set of all nonempty subsets of E.IfM ∈ P(E), then the symbol M denotes the closure of M.ThesymbolConv(M) stands for the convex hull of M.For (M, N) ∈ P(E) × P(E)andα ∈ R,wedenote
M + N = x + y :(x, y) ∈ M × N
and
αM = {αx : x ∈ M}.
We denote by Pb(E) the set of all nonempty bounded subsets of E.ForM ∈ Pb(E), we denote by M the norm of M,thatis,
M = sup xE : x ∈ M .
We denote by Prc(E) the set of all relatively compact subsets of E.ForM, N ∈ P(E), we denote by MN the product set
MN = x · y :(x, y) ∈ M × N .
In what follows, we recall the axiomatic approach of a measure of noncompactness in- troduced by Banas and Goebel [].
Definition . Let μ : Pb(E) → [, ∞)beagivenmapping.Wesaythatμ is a measure of noncompactness in E if it satisfies the following axioms: – (A) The family ker μ = μ ({}) is a subset of Prc(E).
(A) (M, N) ∈ Pb(E) × Pb(E), M ⊂ N ⇒ μ(M) ≤ μ(N).
(A) μ(M)=μ(M), M ∈ Pb(E).
(A) μ(Conv(M)) = μ(M), M ∈ Pb(E). (A) μ(λM +(–λ)N) ≤ λμ(M)+(–λ)μ(N) for λ ∈ [, ] and
(M, N) ∈ Pb(E) × Pb(E).
(A) If {Mn} is a sequence of closed sets from Pb(E) such that Mn+ ⊂ Mn for n =,,... ∞ and if limn→∞ μ(Mn)=, then the intersection set M∞ = n= Mn is nonempty.
Now, let us recall the following important result, which is called Drabo’s fixed point theorem [, ]. Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 5 of 15
Lemma . Let D be a nonempty, bounded, closed, and convex subset of E, and let T : D → D be a continuous mapping. Suppose that there exists a constant k ∈ (, ) such that
μ(TM) ≤ kμ(M)
for any nonempty subset M of D, where μ is a measure of noncompactness in E. Then T has at least one fixed point in D.
The following concept was introduced by Banas and Olszowy [].
Definition . Let μ be a measure of noncompactness in E.Wesaythatμ satisfies con- dition (m)if
μ(MN) ≤Mμ(N)+Nμ(M), (M, N) ∈ Pb(E) × Pb(E).
Lemma . Let μ be a measure of noncompactness in E satisfying condition (m). Let
{Mi}i=,...,q be a finite sequence in Pb(E), q ≥ . Then q q q μ Mi ≤ Mjμ(Mi). (.) i= i= j=,j= i
Proof We shall use the induction principle. For q =,(.) follows immediately from Def- inition ..Supposenowthat(.)issatisfiedforsomeq ≥ . We have to prove that q+ q+ q+ μ Mi ≤ Mjμ(Mi). (.) i= i= j=,j= i
Using (.) and Definition .,wehave q+ q μ Mi = μ Mi Mq+ i= i= q q ≤ μ Mi Mq+ + μ(Mq+) Mi i= i= q q ≤ μ Mi Mq+ + μ(Mq+) Mj i= j= q q q ≤ Mjμ(Mi)Mq+ + μ(Mq+) Mj i= j=,j= i j= q q+ q = Mjμ(Mi)Mq+ + μ(Mq+) Mj i= j=,j= i j= q+ q+ = Mjμ(Mi). i= j=,j= i
Thus, we proved (.). Finally, (.) follows by the induction principle. Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 6 of 15
Now, we deal with the fixed point problem ∈ Findx D such that N (.) x = i= Tix,
where D ∈ P(E)andTi : D → E, i =,...,N, N ≥ , are given operators. We have the following fixed point result.
Theorem . Assume that D is nonempty, bounded, closed, and convex subset of the Ba- nach algebra E. Assume also that the following conditions are satisfied:
(i) Ti is continuous, i =,...,N. (ii) TiD is bounded, i =,..., N. ⊂ N (iii) TD D, where Tx = i= Tix. { }q ⊂ ∞ (iv) There exists a finite sequence ki i= (, ) such that
μ(TiM) ≤ kiμ(M), i =,...,N,
for any nonempty subset M of D, where μ is a measure of noncompactness in E satisfying condition (m). q q (v) i= ki j=,j= i TjD <. Then problem (.) has at least one solution in D.
Proof Let M be nonempty subset of D. Using Lemma . andtheconsideredassumptions, we obtain N N N μ(TM)=μ TiM ≤ TjMμ(TiM) i= i= j=,j= i N N N N ≤ ki TjMμ(M) ≤ ki TjD μ(M). i= j=,j= i i= j=,j= i
Thus, we have proved that
μ(TM) ≤ kμ(M)
for every nonempty subset M of D,where
N N k = ki TjD <. i= j=,j= i
The result follows by Lemma ..
Remark . For N =,Theorem. reduces to a fixed point theorem established in [].
Remark . For N =,Theorem. reduces to Lemma .. Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 7 of 15
4 Main result In this section, we state and prove our main result concerning the existence of solutions to equation (.). Let E = C(I; R) be the Banach space of all real-valued continuous functions in I equipped with the norm
uE = max u(t) : t ∈ I , u ∈ E.
Clearly, E is a Banach algebra with respect to the operation · defined by
(u · v)(t)=u(t)v(t), t ∈ I,(u, v) ∈ E × E.
Let M ∈ Pb(E). For x ∈ M and ε >,set
ω(x, ε)=sup x(t)–x(s) : t, s ∈ I, |t – s|≤ε ,
ω(M, ε)=sup ω(x, ε):x ∈ M .
It was proved in [] that the mapping μ : Pb(E) → [, ∞)definedby
μ(M)= lim ω(M, ε), M ∈ Pb(E), (.) ε→+
is a measure of noncompactness in E.Moreover,μ satisfies condition (m)(see[]). Equation (.)canbewrittenas
N x = Tx = Tix,(.) i=
where fo,r i =,...,N, g (t, x(t)) t i (αi–) (Tix)(t)=fi(t)+ (t – qs) ui s, x(s) dqs, t ∈ I. q(αi)
We consider the following assumption:
(A) For i =,...,N, the functions fi : I → R and gi, ui : I × R → R are continuous.
Lemma . For every i =,...,N, the operator Ti : E → E is well defined.
Proof Fix i ∈{,...,N}.Wehavejusttoprovethattheoperator t (αi–) (Six)(t)= (t – qs) ui s, x(s) dqs, t ∈ I,(.)
maps E into it self. To do this, let us fix x ∈ E. For all t ∈ I,wehave
∞ n n+ (αi–) n n (Six)(t)=t( – q) q t – q t ui tq , x tq n= ∞ α αi n n+ ( i–) n n = t ( – q) q –q ui tq , x tq . n= Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 8 of 15
Since qn+ ∈ (, ), by Lemma . we have (α –) –qn+ i ≤ ( – )(αi–) =.
Therefore, the continuity of ui and the Weierstrass convergence theorem give us the de- sired result.
We suppose also that the following assumptions are satisfied:
(A) For all i =,...,N,thereexistconstantsCi >and di >such that
di gi(t, x)–gi(t, y) ≤ Ci|x – y| ,(t, x, y) ∈ I × I × R.
(A) For all i =,...,N,thereexistaconstantei >and a nondecreasing continuous
function ϕi :[,∞) → [, ∞) such that ei ui(t, x)–ui(t, y) ≤ ϕi |x – y| ,(t, x, y) ∈ I × I × R.
(A) For all i =,...,N,wehave
ui(t,)=, t ∈ I.
(A) There exists r >such that
di ∗ ei (Cir + gi )ϕi(r ) /N fiE + ≤ r , q(αi +)
where
∗ ∈ gi = max gi(t,) : t I .
Lemma . Under Assumptions (A)-(A), for all i =,...,N, we have /N TiB(E, r) ⊂ B E, r .
Proof Let i ∈{,...,N} be fixed, and let x ∈ B(E, r). For all t ∈ I,wehave |g (t, x(t))| t i (αi–) (Tix)(t) ≤ fi(t) + (t – qs) ui s, x(s) dqs q(αi) |g (t, x(t))| t i (αi–) ≤fiE + (t – qs) ui s, x(s) dqs q(αi) |g (t, x(t)) – g (t,)| + |g (t,)| t i i i (αi–) ≤fiE + (t – qs) ui s, x(s) dqs. q(αi)
Using Assumption (A), we obtain ∗ ≤ di gi t, x(t) – gi(t,) + gi(t,) Cir + gi , Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 9 of 15
By Assumptions (A) and (A) we have ei ui s, x(s) = ui s, x(s) – ui(s,) ≤ ϕi r .
Therefore, we obtain
∗ (C rdi + g )ϕ (rei ) t i i i (αi–) (Tix)(t) ≤fiE + (t – qs) dqs q(αi) ∗ (C rdi + g )ϕ (rei ) i i i αi = fiE + t q(αi +)
di ∗ ei (Cir + gi )ϕi(r ) ≤fiE + . q(αi +)
By Assumption (A) we get
/N TixE ≤ r ,
which proves the desired result.
Lemma . Under Assumptions (A)-(A), for all i =,...,N, /N Ti : B(E, r) → B E, r
is continuous.
Proof Under the considered assumptions, we have just to prove that, for all i =,...,N,the
operator Si : B(E, r) → E defined by (.) is continuous. To do this, let us fix i ∈{,...,N}
and ε >suchthatx – yE ≤ ε,(x, y) ∈ B(E, r) × B(E, r). For all t ∈ I,wehave t (αi–) (Six)(t)–(Siy)(t) = (t – qs) ui s, x(s) – ui s, y(s) dqs t (αi–) ≤ (t – qs) ui s, x(s) – ui s, y(s) dqs i ε αi ≤ ur( )t [αi]q ui (ε) ≤ r , [αi]q
where
i ∈ ∈ | |≤ ur(ε)=sup ui(t, x)–ui(t, y) : t I, x, y [–r, r], x – y ε .
Therefore,
i ur(ε) Six – SiyE ≤ . [αi]q Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 10 of 15
Passing to the limit as ε → and using the uniform continuity of ui on the compact set I × [–r, r], we obtain
ui (ε) lim r =, → ε [αi]q
which yields the desired result.
Next, we need the following additional assumption:
(A) For all i =,...,N, there exists a constant λi >such that di ei t + ϕi t ≤ λit, t ≥ .
Lemma . Under assumptions (A)-(A), for all i =,...,N, we have
μ(TiM) ≤ kiμ(M)
for every nonempty subset M of B(E, r), where ei di ∗ Ciϕi(r ) (Cir + gi ) ki = λi max , . q(αi +) q(αi)
Proof Fix i ∈{,...,N}.LetM be a nonempty subset of B(E, r), ε >,andx ∈ M.Let
(t, t) ∈ I × I be such that |t – t|≤ε. Without loss of the generality, we may assume that
t ≥ t. Therefore, we have
(Tix)(t)–(Tix)(t) g (t , x(t )) t i (αi–) ≤ fi(t)–fi(t) + (t – qs) ui s, x(s) dqs q(αi) g (t , x(t )) t i (αi–) – (t – qs) ui s, x(s) dqs q(αi) g (t , x(t )) t i (αi–) ≤ ω(fi, ε)+ (t – qs) ui s, x(s) dqs q(αi) g (t , x(t )) t i (αi–) – (t – qs) ui s, x(s) dqs q(αi) g (t , x(t )) t i (αi–) + (t – qs) ui s, x(s) dqs q(αi) g (t , x(t )) t i (αi–) – (t – qs) ui s, x(s) dqs q(αi) |g (t , x(t )) – g (t , x(t ))| t i i (αi–) ≤ ω(fi, ε)+ (t – qs) ui s, x(s) dqs q(αi) |g (t , x(t ))| t t i (αi–) (αi–) + (t – qs) ui s, x(s) dqs – (t – qs) ui s, x(s) dqs q(αi)
:= ω(fi, ε)+(I)+(II), Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 11 of 15
where
ω(fi, ε)=sup fi(t)–fi(s) : t, s ∈ I, |t – s|≤ε .
Now, let us estimate (I)and(II). • Estimate of (I). We have t ϕ (rei ) (αi–) i (t – qs) ui s, x(s) dqs ≤ . q(αi) q(αi +)
On the other hand,
gi t, x(t) – gi t, x(t) ≤ gi t, x(t) – gi t, x(t) + gi t, x(t) – gi t, x(t)
di ≤ Ci x(t)–x(t) + ω(gi, ε)
di ≤ Ciω(x, ε) + ω(gi, ε),
where
ω(gi, ε)=sup gi(t, x)–gi(s, x) : t, s ∈ I, |t – s|≤ε, x ∈ [–r, r] .
Therefore,
(C ω(x, ε)di + ω(g , ε))ϕ (rei ) (I) ≤ i i i . (.) q(αi +)
• Estimate of (II). First, observe that ≤ ∗ gi t, x(t) gi t, x(t) – gi(t,) + gi .
Therefore, by Assumption (A), ∗ ≤ di gi t, x(t) Cir + gi .
Next, we have t t (αi–) (αi–) (t – qs) ui s, x(s) dqs – (t – qs) ui s, x(s) dqs ∞ n n+ (αi–) αi n n αi n n =(–q) q –q t ui q t, x q t – t ui q t, x q t . n=
On the other hand, αi n n αi n n t ui q t, x q t – t ui q t, x q t ≤ αi n n n n t ui q t, x q t – ui q t, x q t αi n n αi n n + t ui q t, x q t – t ui q t, x q t Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 12 of 15
n n ei ≤ ϕi x q t – x q t + Ai(ε) ei ≤ ϕi ω(x, ε) + Ai(ε),
where Ai(ε)=sup H(τ, s, x)–H τ , s , x : τ, s, τ , s ∈ I, τ – τ ≤ ε, s – s ≤ ε, x ∈ [–r, r]
and
αi H(τ, s, x)=τ ui(s, x), (τ, s, x) ∈ I × I × R.
As a consequence, we get t t (αi–) (αi–) (t – qs) ui s, x(s) dqs – (t – qs) ui s, x(s) dqs ei ≤ ϕi ω(x, ε) + Ai(ε).
Then
∗ (C rdi + g )(ϕ (ω(x, ε)ei )+A (ε)) (II) ≤ i i i i .(.) q(αi)
Now, combining (.)with(.), we get
di ei di ∗ ei (Ciω(x, ε) + ω(gi, ε))ϕi(r ) (Cir + gi )(ϕi(ω(x, ε) )+Ai(ε)) ω(Tix, ε) ≤ ω(fi, ε)+ + . q(αi +) q(αi)
Therefore,
d e (Ciω(M, ε) i + ω(gi, ε))ϕi(r i ) ω(TiM, ε) ≤ ω(fi, ε)+ q(αi +) ∗ (C rdi + g )(ϕ (ω(M, ε)ei )+A (ε)) + i i i i . q(αi)
Passing to the limit as ε → , we obtain
di ei di ∗ ei Ciμ(M) ϕi(r ) (Cir + gi )ϕi(μ(M) ) μ(TiM) ≤ + . q(αi +) q(αi)
Then
∗ C ϕ (rei ) (C rdi + g ) i i i i di ei μ(TiM) ≤ max , μ(M) + ϕi μ(M) . q(αi +) q(αi)
By Assumption (A) we obtain ei di ∗ Ciϕi(r ) (Cir + gi ) μ(TiM) ≤ λi max , μ(M), q(αi +) q(αi)
which proves the desired result. Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 13 of 15
Now, we are able to state and prove our main result.
Theorem . Suppose that Assumptions (A)-(A) are satisfied. If
N ei di ∗ N– Ciϕi(r ) (Cir + gi ) r N λ max , < , (.) i (α +) (α ) i= q i q i
∗ then equation (.) has at least one solution x ∈ B(E, r).
Proof Observe that x ∈ B(E, r) is a solution to equation (.)ifandonlyifx is a solution
to (.), where D = B(E, r), and T is given by (.).Inordertoproveourexistenceresult, we have to check that all the assumptions of Theorem . are satisfied. By Lemma .,for
all i =,...,N, Ti : B(E, r) → E is a continuous operator. By Lemma ., for all i =,...,N,
TiB(E, r) is bounded. Moreover, for all x ∈ B(E, r), we have N N N ≤ ≤ /N Tx E = Tix Tix E r = r. i= E i= i=
Therefore, TB(E, r) ⊂ B(E, r). By Lemma . condition (iv) of Theorem . is satisfied with ei di ∗ Ciϕi(r ) (Cir + gi ) ki = λi max , , i =,...,N. q(αi +) q(αi)
On the other hand, from Lemma . and (.)wehave
q q N ei di ∗ N– Ciϕi(r ) (Cir + gi ) k T B( , r) ≤ r N λ max , <. i j E i (α +) (α ) i= j=,j= i i= q i q i
Therefore, all conditions of Theorem . are satisfied, and the desired result follows.
We end the paper with the following illustrative example.
Example Consider the functional equation t t x(t) t x(s) x t t s (/) d s t ∈ ( )= + + ( – /) q , [, ]. (.) ( + s )
Equation (.)isaaparticularcaseofequation(.)with
t N =, q = , α = , f (t)= , t x x g(t, x)= + , u(t, x)= . +t
Obviously, the functions f :[,]→ R and g, u :[,]× R → R are continuous. Then Assumption (A) of Theorem . is satisfied. For all (t, x, y) ∈ [, ] × R × R,wehave
( ) g (t, x)–g (t, y) ≤ |x – y|. Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 14 of 15
Therefore, Assumption (A) is satisfied with
( ) C = , d =.
For all (t, x, y) ∈ [, ] × R × R,wehave
u (t, x)–u (t, y) ≤ |x – y|.
Moreover, u(t, ) = for all t ∈ [, ]. Therefore, Assumptions (A) and (A) are satisfied with
t ϕ (t)= , e =.
Note that, in this case, we have
∗ ( ) f = , g = . E
≥ It is not difficult to check that Assumption (A) is satisfied for every r .Forevery t ≥ , we have
td + ϕ te = t.
Then Assumption (A) is satisfied with λ = .Moreover,forr = ,wehave e d ∗ Cϕ(r ) (Cr + g ) λ max , =.<. q(α +) q(α)
∗ ∈ By Theorem . we deduce that equation (.) admits at least one solution x B(E, ).
Competing interests The authors declare to have no competing interests.
Authors’ contributions Both authors contributed equally in writing this paper. Both authors read and approved the final manuscript.
Acknowledgements The second author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
Received: 8 September 2016 Accepted: 4 January 2017
References 1. Brestovanska, E: Qualitative behaviour of an integral equation related to some epidemic model. Demonstr. Math. 36(3), 603-609 (2003) 2. Gripenberg, G: On some epidemic models. Q. Appl. Math. 39, 317-327 (1981) 3. Olaru, IM: Generalization of an integral equation related to some epidemic models. Carpath. J. Math. 26, 92-96 (2010) 4. Agarwal, RP: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365-370 (1969) 5. Aghajani, A, Allahyari, R, Mursaleen, M: A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68-77 (2014) 6. Aghajani, A, Banas, J, Jalilian, Y: Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62, 1215-1227 (2011) 7. Aghajani, A, Mursaleen, M, Shole Haghighi, A: Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness. Acta Math. Sci. 35(3), 552-566 (2015) Jleli and Samet Advances in Difference Equations (2017) 2017:21 Page 15 of 15
8. Aghajani, A, Pourhadi, E, Trujillo, JJ: Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 16(4), 962-977 (2013) 9. Banas, J, Caballero, J, Rocha, J, Sadarangani, K: Monotonic solutions of a class of quadratic integral equations of Volterra type. Comput. Math. Appl. 49, 943-952 (2005) 10. Banas, J, Goebel, K: Measures of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math., vol. 60. Dekker, New York (1980) 11. Banas, J, Martinon, A: Monotonic solutions of a quadratic integral equation of Volterra type. Comput. Math. Appl. 47, 271-279 (2004) 12. Banas, J, Mursaleen, M: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Dehli (2014) 13. Banas, J, Olszowy, L: On a class of measures of non-compactness in Banach algebras and their application to nonlinear integral equations. Z. Anal. Anwend. 28, 475-498 (2009) 14. Banas,´ J, O’Regan, D: On existence and local attractivity of solutions of a quadratic integral equation of fractional order. J. Math. Anal. Appl. 345, 573-582 (2008) 15. Darwish, MA, Sadarangani, K: On a quadratic integral equation with supremum involving Erdélyi-Kober fractional order. Math. Nachr. 288(5-6), 566-576 (2015) 16. Jleli, M, Mursaleen, M, Samet, B: Q-Integral equations of fractional orders. Electron. J. Differ. Equ. 2016, 17 (2016) 17. Jackson, FH: On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253-281 (1908) 18. Jackson, FH: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193-203 (1910) 19. Abdeljawad, T, Baleanu, D: Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 16, 4682-4688 (2011) 20. Ahmad, B, Nieto, J: Basic theory of nonlinear third-order q-difference equations and inclusions. Math. Model. Anal. 18(1), 122-135 (2013) 21. Annaby, MH, Mansour, ZS: q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012) 22. Atici, FM, Eloe, PW: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14(3), 333-344 (2007) 23. Ernst, T: The History of q-Calculus and a New Method. U. U. D. M. Report 2000:16, Department of Mathematics, Uppsala University (2000) 24. Jarad, F, Abdeljawad, T, Baleanu, D: Stability of q-fractional non-autonomous systems. Nonlinear Anal., Real World Appl. 14(1), 780-784 (2013) 25. Kac, V, Cheung, P: Quantum Calculus. Springer Science & Business Media (2002) 26. Wu, GC, Baleanu, D: New applications of the variational iteration method-from differential equations to q-fractional difference equations. Adv. Differ. Equ. 2013, 21 (2013) 27. Ferreira, R: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010) 28. Darbo, G: Punti uniti in transformazioni a condomino non compatto. Rend. Semin. Mat. Univ. Padova 24, 84-92 (1955)