Some generalizations of Darbo’s theorem and applications to fractional integral equations

In this paper, some generalizations of Darbo’s fixed point theorem are presented. An existence result for a class of fractional integral equations is given as an application of the obtained results.


Introduction and preliminaries
Let E be a Banach space over R (or C) with respect to a certain norm · . For any subsets X and Y of E, we have the following notations: X denotes the closure of X; conv(X) denotes the convex hull of X; P(X) denotes the set of nonempty subsets of X; X + Y and λX (λ ∈ R) stand for algebraic operations on sets X and Y . We denote by B E the family of all nonempty bounded subsets of E. Finally, if X is a nonempty subset of E and T : X → X is a given operator, we denote by Fix(T) the set of fixed points of T, that is, Fix(T) = {x ∈ X : Tx = x}. Banaś and Goebel [] introduced the following axiomatic definition of the concept of a measure of noncompactness.
Definition . Let σ : B E → [, ∞) be a given mapping. We say that σ is a BG-measure of noncompactness (in the sense of Banaś and Gobel) on E if the following conditions are satisfied: (i) For every X ∈ B E , σ (X) =  iff X is precompact.
(v) If {X n } ⊆ B E is a decreasing sequence (w.r.t. ⊆) of closed sets such that σ (X n ) →  as n → ∞, then X ∞ := ∞ n= X n is nonempty.
Let X be a nonempty, bounded, closed, and convex subset of the Banach space E. We denote by D X the set of self-mappings D : X → X satisfying the following conditions: (i) D is a continuous mapping.
(ii) There exist σ : B E → [, ∞), a BG-measure of noncompactness on E, and a constant k ∈ (, ) such that The following result is known as Darbo ) Let D : X → X be a mapping that belongs to F X . Then D has at least one fixed point.
Observe that D X ⊆ F X . In fact, let D : X → X be a given mapping that belongs to D X . Let ε > . From the definition of D X , there is some k ∈ (, ) such that for any nonempty subset W of X . Let δ ε = (  k -)ε. Then for any nonempty subset W of X , we have In [], Dhage introduced the following axiomatic definition of the measure of noncompactness.
Definition . Let σ : B E → [, ∞) be a given mapping. We say that σ is a D-measure of noncompactness (in the sense of Dhage) on E if the following conditions are satisfied: (iii) For every X ∈ B E , we have is a BG-measure of noncompactess on E, then σ is a D-measure of noncompactess on E.
In this paper, using the axiomatic definition of the measure of noncompactness given by Dhage, we obtain new generalizations of Theorem .. Finally, an existence result for a certain class of fractional integral equations will be given as an application.

Main results
Let X be a nonempty, bounded, closed, and convex subset of a Banach space E. We continue to use the same notations presented in the previous section of this paper.
Let F X be the set of self-mappings D : X → X satisfying the following conditions: (i) D is a continuous mapping.
(ii) There exists σ : B E → [, ∞), a D-measure of noncompactness on E, such that for all ε > , there exists some δ ε >  for which We have the following result.
Theorem . Let D : X → X be a mapping that belongs to F X . Then D has at least one fixed point.
The result of Theorem . can be obtained using the same arguments of the proof of Theorem . in []. By Theorem ., we want just to mention that Theorem . is still valid for any D-measure of noncompactness.
Let G X be the set of mappings D : (ω  ) ω is nondecreasing and right continuous; (ω  ) for every ε > , there exists γ ε >  such that The following lemma can be proved using a similar argument as in the proof of Theorem . in [].

Lemma . We have
Using Theorem . and Lemma ., we obtain the following result.
Corollary . Let D : X → X be a mapping that belongs to G X . Then D has at least one fixed point.
Let be the set of functions ϕ : [, ∞) → [, ∞) satisfying the conditions: Let H X be the set of mappings D : X → X such that where ϕ ∈ and σ : B E → [, ∞) is a D-measure of noncompactness.

Proof
Take we obtain the desired result.
Using Corollary . and Lemma ., we obtain the following result.
Corollary . Let D : X → X be a mapping that belongs to H X . Then D has at least one fixed point.
Let I X be the set of mappings D : X → X such that

Lemma . We have
Using Corollary . and Lemma ., we obtain the following result.
Corollary . Let D : X → X be a mapping that belongs to I X . Then D has at least one fixed point.
Let J X be the set of mappings D : X → X such that where σ : Theorem . Let D : X → X be a mapping that belongs to J X . Then D has at least one fixed point.
Proof Consider the sequence {X n } of subsets of E defined by By induction, we observe easily that If for some N , we have σ (X N ) = , then by the property (i) of the D-measure of noncompactness, X N is compact. Since D(X N ) ⊆ X N (from (.)), Schauder's fixed point theorem applied to the self-mapping D : X N → X N gives the desired result. So, without loss of the generality, we may assume that σ (X n ) > , n = , , , . . . .
Let K X be the set of mappings D : X → X such that (θ  ) there exist k ∈ (, ) and a D-measure of noncompactness σ : We have the following result.
Theorem . Let D : X → X be a mapping that belongs to K X . Then D has at least one fixed point.
Proof Consider the sequence {X n } of subsets of E defined by (.). As in the proof of Theorem ., without loss of the generality, we may assume that σ (X n ) > , n = , , , . . . .
Passing to the limit as n → ∞, we obtain The rest of the proof is similar to that in the proof of Theorem ..
Corollary . Let D : X → X be a continuous mapping. Suppose that there exist a constant k ∈ (, ) and a D-measure of noncompactness σ : for any W ∈ P(X ) with σ (W )σ (DW ) > . Then D has at least one fixed point.

Proof
Taking in Theorem ., we obtain the desired result.
Remark . Observe that D X ⊆ K X . In fact, if D : X → X belongs to D X , that is, Let L X be the set of mappings D : X → X such that Then there is some r ≥  such that lim n→∞ σ (X n ) = r.
If r > , then from the property (ζ  ), we have which contradicts (.). As consequence, we have The rest of the proof is similar to the proof of Theorem ..

Remark . Taking
where k ∈ (, ) is a constant, we obtain Theorem ..

An existence result for a fractional integral equation
The measure of noncompactness argument is a useful tool in Nonlinear Analysis. In particular, such argument can be used to obtain existence results for various classes of integral equations. For more details on the applications of the measure of noncompactness concept, we refer the reader to [, , , , -] and the references therein.
In this section, we discuss the existence of solutions to the fractional integral equation We suppose that the following conditions are satisfied.
(iv) The function g : [, T] → R is C  and nondecreasing.
(v) There exists r  >  such that and is a Banach space. Let W be a nonempty and bounded subset of E. Let us define the map- Let B E be the set of all nonempty bounded subsets of E. Then the mapping is a BG-measure of noncompactness (then it is a D-measure of noncompactness) on the space E (see []). We have the following existence result.
Theorem . Under the assumptions (i)-(v), equation (.) has at least one solution y * ∈ E. Moreover, we have y * ≤ r  .
Proof Let us consider the operator D defined on E by At first, we show that the operator D maps E into itself. Set From the assumption (i), we have just to show that H maps E into itself. In order to prove this fact, let us fix some y ∈ E. Observe that Hy : [, T] → R is a well-defined function. In fact, using the assumptions (iii) and (iv), for all t ∈ [, T] we have (Hy)(t) ≤ t  g (s)|u(s, y(s))| (g(t)g(s)) -α ds that is, Let us prove the continuity of Hy at . To do this, let {t n } be a sequence in [, T] such that t n →  + as n → ∞. From (.), for all n we have Passing to the limit as n → ∞ and using the continuity of g at , we obtain lim n→∞ (Hy)(t n ) =  = (Hy)().
Then Hy is continuous at .
where B(, r  ) = z ∈ E : z ≤ r  . Now, we claim that the operator D : B(, r  ) → B(, r  ) is continuous. From (.), we can write D in the form and Hy is defined by (.). In order to prove our claim, it is sufficient to show that the operators G and H are continuous on B(, r  ). First of all, we show that G is a continuous operator on B(, r  ). To do this, we take a sequence {y n } ⊂ B(, r  ) and y ∈ B(, r  ) such that y ny →  as n → ∞, and we have to prove that Gy n -Gy →  as n → ∞. In fact, for all t ∈ [, T], using the condition (ii), we have ≤ ψ y ny ≤ y ny .

Thus we have
Gy n -Gy ≤ y ny , for all n.
Passing to the limit as n → ∞ in the above inequality, we obtain lim n→∞ Gy n -Gy = .
Passing to the limit superior as ρ →  + and using the fact that ψ is upper semi-continuous, we obtain Then, from the assumption (v), we obtain σ (DW ) ≤ ψ σ (W ) .
As a consequence, for any nonempty subsets W of B(, r  ), we have Under the assumptions on the function ψ, the operator D : B(, r  ) → B(, r  ) belongs to the family of operators L X , where X = B(, r  ). Then by Theorem ., we deduce that D has at least one fixed point y * ∈ B(, r  ), which is a solution to equation (.).