Positive solutions of fractional integral equations by the technique of measure of noncompactness

In the present study, we work on the problem of the existence of positive solutions of fractional integral equations by means of measures of noncompactness in association with Darbo’s fixed point theorem. To achieve the goal, we first establish new fixed point theorems using a new contractive condition of the measure of noncompactness in Banach spaces. By doing this we generalize Darbo’s fixed point theorem along with some recent results of (Aghajani et al. (J. Comput. Appl. Math. 260:67-77, 2014)), (Aghajani et al. (Bull. Belg. Math. Soc. Simon Stevin 20(2):345-358, 2013)), (Arab (Mediterr. J. Math. 13(2):759-773, 2016)), (Banaś et al. (Dyn. Syst. Appl. 18:251-264, 2009)), and (Samadi et al. (Abstr. Appl. Anal. 2014:852324, 2014)). We also derive corresponding coupled fixed point results. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.


Introduction
Fractional calculus is the study of integrals and derivatives of an arbitrary order. Fractional calculus seeks to find the integrals and derivatives of a real or even complex order using the Gamma function, Euler's generalization of the factorials. In modern times differential/integral equations with nonintegral order have drawn the attention of numerous researchers due to their wide applications in several fields of science and engineering. The need for fractional order differential/integral equations stems in part from the fact that many phenomena cannot be modeled by differential/integral equations with integer derivatives. Analytical and numerical techniques have been implemented to study such equations.
Due to the importance of fractional calculus, it is necessary to discuss the related problems and work on them. In this work, we study the problem of the existence of positive solutions for integral equations of the form where (·) is the (Euler) Gamma function defined by (γ ) = ∞  t γ - e -t dt. Let us recall that the function h(t, x) involved in equation (.) generates the superposition operator H defined by the formula (Hx)(t) = h(t, x(t)), where x = x(t) is an arbitrary function defined on I (cf. [, ]). We are going to show that equation (.) has a positive solution that belongs to space C + (I) = C([, ]; R + ). The obtained results extend several papers (see [-, ], for example). Finally, an example is presented to show the efficiency of our results.

Preliminaries
Throughout this paper, we assume that (E, · ) is a real Banach space with zero element . Let R = (-∞, +∞), R + = [, +∞), and N = {, , , . . .}. Let B(x, r) denote the closed ball centered at x with radius r. The symbol B r stands for the ball B(, r). For X, a nonempty subset of E, we denote by X and Conv X the closure and the closed convex hull of X, respectively. Moreover, let us denote by M E the family of nonempty bounded subsets of E and by N E its subfamily consisting of all relatively compact subsets of E. We use the following definition of measure of noncompactness (MNC, for short) given in [].
is a sequence of closed sets from M E such that X n+ ⊂ X n for n = , , . . . and if lim n→∞ β(X n ) = , then the intersection set X ∞ = ∞ n= X n is nonempty.
The subfamily ker β, defined by ( • ), represents the kernel of the MNC β and since Therefore, X ∞ ∈ ker β. From now on we denote by β an MNC and C to be a nonempty, bounded, closed and convex subset of a Banach space E.
Darbo's fixed point theorem (DFPT) is a very important generalization of Schauder's fixed point theorem and Banach's fixed point theorem.

New fixed point theorem for shifting distance functions
To complete the proof, we need the following notions.
We are now in a position to state and prove a new DFPT theorem.
Theorem . Let T be a self-continuous operator on C such that for any subset X of C, where F ∈ F, ( , ) ∈ ϒ, and ϕ : R + − → R + is a continuous function. Then T has at least one fixed point in C.
Proof We start with C  = C and construct a sequence {C n } such that C n+ = Conv(TC n ), for n ≥ . TC  = TC ⊆ C = C  , C  = Conv(TC  ) ⊆ C = C  . Therefore by continuing this process we have If ∃ a natural number N such that β(C N ) = , then C N is compact and concludes the result through Schauder's fixed point theorem. So we consider β(C n ) >  for n ≥ . Also, by (.), This implies that {F(β(C n ), ϕ(β(C n )))} is a nonincreasing sequence of positive real numbers by () of Definition .. Hence, there is an r ≥  such that Then, in view of (.) and () of Definition ., we get r =  and hence Now since C n is a nested sequence, in view of ( • ) of Definition ., we conclude that C ∞ = ∞ n= C n is a nonempty, closed, and convex subset of C. Besides we know that C ∞ belongs to ker β. So C ∞ is compact and invariant under the mapping T. Consequently, Schauder's fixed point theorem implies this result in C ∞ , but as C ∞ ⊂ C, the result is true in C.
Taking F(a, b) = a + b in Theorem ., we obtain the following.

Theorem . Let T be a self-continuous operator on C such that
for any subset X of C, where ( , ) ∈ ϒ and ϕ : R + − → R + is a continuous function. Then T has at least one fixed point in C.
Remark . Set ϕ ≡ , (t) = t, t ≥  and let the function satisfy lim n→∞ n (t) =  for any t ≥  in Theorem .. Then we get Theorem . of [].
Taking = I in Theorem ., we have the following result.

Corollary . Let T be a self-continuous operator on C such that
for any subset X of C, where F ∈ F and ϕ : if u n ≤ (v n ) for all n, then w = . Then T has at least one fixed point in C.
Following Proposition  (see []) and Theorem ., we conclude the following.

Theorem . Let T be a self-continuous operator on C such that
for any subset X of C, where F ∈ F and χ ∈ . Then T has at least one fixed point in C.

Coupled fixed point theorems
Here we derive some new coupled fixed point (CFP) results by means of the MNC.
The first CFP result is the following. Proof Consider the map G : X  → X  defined by the formula G is continuous due to the continuity of G. Following [], we define a new MNC in the space X × X as where X i , i = ,  denote the natural projections of X. Without loss of generality, we assume M is a nonempty subset of X  . Hence, by condition (.) and using ( • ) of Definition . we conclude that Following Theorem ., G has at least one fixed point in X  , and hence G has a CFP.
The second outcome of this section is the following.
Proof Consider the map G : X × X → X × X, defined by the formula G is continuous due to the continuity of G. Following [], we express β as a new MNC in the space X  as where X i , i = ,  denote the natural projections of M. Without loss of generality, we take M, a nonempty subset of X  . Following the previous theorem, we have Hence, by condition (.) and using ( • ) of Definition . we obtain Hence, using Theorem ., we conclude that G has at least one fixed point in X  , and thus G has a CFP.
Remark . We can derive some new CFP results from Theorems .-., if we take F(a, b) = a + b with various settings for , and ϕ. Let X be the set of all nonempty and bounded subsets of C + (I). Then the mapping ω  :

Solvability of an implicit fractional integral equation
for all t ∈ I and all x, y ∈ R. Additionally we assume that ϕ is superadditive, i.e., ϕ(t) + ϕ(s) ≤ ϕ(t + s) for all t, s ∈ R + , (B  ) the superposition operator H generated by the function h(t, x) satisfies for any nonnegative function x the condition where : R + → R + with (t) = ϕ(t  ) is the same function as in (B  ) and Theorem ., (B  ) the function g : I × I × R → R is continuous such that g(I × I × R + ) ⊆ R + and G  = sup g t, s, x(s) : t, s ∈ I, x ∈ C + (I) < ∞, (B  ) the function f : I → R + is C  + and nondecreasing, (B  ) the inequality has a positive solution r  such that λ = Proof For x ∈ C + (I), consider the operators F and T defined on the space C + (I) by the formulas Firstly, we prove that F is self-mapping on C + (I). To this end, it suffices to verity that if x ∈ C + (I), then Fx ∈ C + (I). Fix > , let x ∈ C + (I), and let t  , t  ∈ I (without loss of generality assume that t  ≥ t  ) and |t t  | ≤ . Then we get Therefore, if we denote Using the notion of uniform continuity of the function g on the set I  × [-r  , r  ] and f on the set I, we have ω g ( , ·) − →  and ω(f , ) − →  as − → . Consequently Fx ∈ C + (I) and thus Tx ∈ C + (I). Also, we have Hence, Thus, if x ≤ r  we obtain from assumption (B  ) the estimate Consequently, the operator T maps the ball what we want to do is to show that the operator T is continuous on B r  . For this purpose, let {x n } be a sequence in B r  such that x n → x. We have to show that Tx n → Tx. In fact, for each t ∈ I, we have where G = sup g(t, s, x)g(t, s, y) : t, s ∈ I, |x|, |y| ≤ r  , |x -y| ≤ . Indeed, Thus, taking the supremum on X, we obtain From the uniform continuity of the function g on the set I × I × R + and h on the set I × [, r  ] and the continuity of the function a on I, we have ω g ( , ·) → , ζ r  (h, ) →  and ω(a, ) →  as → . So we let →  to obtain • Assumption (B  ). For t ∈ I and x, y ∈ R, we have where ϕ(t) = t, t ≥ . So assumption (B  ) is satisfied with M  = . • Assumption (B  ). It is trivial. Indeed, taking an arbitrary nonnegative function x ∈ C + (I) and t  , t  ∈ I such that t  ≤ t  , we obtain (Hx)(t  ) -(Hx)(t  ) -(Hx)(t  ) -(Hx)(t  ) • Assumption (B  ). It is trivial with G  ≤   . • Assumption (B  ). It is trivial. • Assumption (B  ). In this case inequality (.) has the form