On an integral equation under Henstock–Kurzweil–Pettis integrability

In this paper, we investigate the set of solutions for nonlinear Volterra type integral equations in Banach spaces in the weak sense and under Henstock–Kurzweil–Pettis integrability. Moreover, a fixed point result is presented for weakly sequentially continuous mappings defined on the function space C(K, X), where K is compact Hausdorff and X is a Banach space. The main condition is expressed in terms of axiomatic measure of weak noncompactness.

In this paper, motivated by these examinations we focus on the existence of solutions in the weak sense for the nonlinear Volterra type integral equation in Banach spaces involving the Henstock-Kurzweil-Pettis integral. The main tools used in our study are associated with the techniques of measure of weak noncompactness, properties and convergence theorems mainly of Vitali type for Henstock-Kurzweil-Pettis integrals based on the notion of equi-integrability (see [11]). By using these tools, we are able to prove not only the existence of solutions of the considered integral equation, but also we can obtain a topological structure of the set of these solutions.
In the next section, we give some preliminary facts and present a fixed point result for function spaces. Our ideas were motivated originally by a theorem of Dobrakov [13] which has the interest to characterize weakly convergent sequences in C(K , X ) with weakly convergent sequences in X without equicontinuity conditions. In the last section, we use our fixed point result and the techniques of the theory of measure of weak noncompactness presented in Section 2 to establish existence principle for the nonlinear Volterra integral equation (1.1) under Henstock-Kurzweil-Pettis integrability. By imposing some conditions expressed in terms of the measure of weak noncompactness on f and k, we define an operator over the Banach space of continuous functions from a compact interval to a Banach space, whose fixed points are solutions of (1.1).

Preliminaries
Let X be a Banach space with the norm ||.|| and let K be a compact and Hausdorff space. In what follows, we denote by C(K , X ) the Banach space of all continuous functions from K to X , endowed with the sup-norm ||.|| ∞ defined by ||x|| ∞ = sup{ x(t) , t ∈ K } for each x ∈ C(K , X ). Definition 2.1 Let X be a Banach space and C a lattice with a least element, which is denoted by 0. By a measure of weak noncompactness on X , we mean a function defined on the set of all bounded subsets of X with value in C satisfying: (1) (conv( )) = ( ), for all bounded subsets ⊆ X , where conv denotes the closed convex hull of , (2) for any bounded subsets 1 , 2 of X we have If ( ) = 0, then is relatively weakly compact in X .
The above notion is a generalization of the well-known De Blasi measure of weak noncompactness β (see [10]) defined on each bounded set of X by Note for all bounded subsets , 1 , 2 of X , Note that β is the counterpart for the weak topology of the classical Hausdorff measure of noncompactness.
For more examples and properties of measures of weak noncompactness, we refer the reader to [2,4,5,21,22]. Definition 2.2 A function f : X 1 −→ X 2 , where X 1 and X 2 are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent (x n ) n ⊂ X 1 with x n x, we have f x n f x. Here, denotes weak convergence.
The following fixed point result due to Arino et al. [3] will be used throughout this section. Theorem 2.3 Let X be a metrizable locally convex linear topological space and let C be a weakly compact convex subset of X . Then, any weakly sequentially continuous map F : C −→ C has a fixed point.
To characterize weak convergence in C(K , X ), we use Dobrakov's theorem Theorem 2.4 ([13, Theorem 9]) Let K be a compact Hausdorff space and X a Banach space. Let (x n ) be a bounded sequence in C(K , X ), and x ∈ C(K , X ). Then, (x n ) is weakly convergent to x if and only if (x n (t)) is weakly convergent to x(t) for each t ∈ K . Now, we state the following Ambrosetti's type lemma which will be useful in the sequel. Lemma 2.5 Let V ⊆ C(K , X ) be a family of strongly equicontinuous functions. Then (a) The next fixed point result has the advantage to omit any equicontinuity conditions. Theorem 2.6 Let K be a compact Hausdorff space, X is a Banach space with Q a non-empty closed convex subset of C(K , X ) and is a measure of weak noncompactness on X . Suppose F : Q −→ Q satisfying: Then, the set of fixed points of F is non-empty and weakly compact in C(K , X ).
Proof Let S be the set of fixed points of F in Q. We claim that S is non-empty. Indeed, let x 0 ∈ F(Q) and G the family of all closed bounded convex subsets D of Then, H is closed and convex, and which is a contradiction, so (H ) = 0. Since, H is a weakly closed subset of C(K , X ) (notice a convex subset of a Banach space is closed iff it is weakly closed), then H is a weakly compact subset of C(K , X ). We claim that F is weakly sequentially continuous.
for each t ∈ K , and with the same argument we obtain F x n F x in C(K , X ). So, F is weakly sequentially continuous. It follows using Theorem 2.3 that F : H −→ H has a fixed point and so S = ∅. Because S ⊂ F(Q), F(S) = S and F is -condensing, we have (S) = 0 and so S is a relatively weakly compact subset of C(K , X ). Also, by the sequentially weak continuity of F, the set S is weakly sequentially closed. Let x ∈ Q, be weakly adherent to S. Since S w , the weak closure of S in C(K , X ), is weakly compact, by the Eberlein-Šmulian theorem [14,Theorem 8.12.4,p. 549], there exists a sequence {x n } ⊂ S such that x n x, so x ∈ S. Hence S is a weakly closed subset of Q. Therefore, S is weakly compact in C(K , X ).
Remark 2.7 Theorem 2.6 is a special case of Theorem 12 in [16], namely where the Banach space is C(K , X ).

Main result
Let I = [0, 1] and X be a real Banach space. In this section, we investigate topological structure of the set of solutions in weak sense of following nonlinear Volterra type integral equation (1.1), x ∈ C(I, X ) and involving the Henstock-Kurzweil-Pettis integral [7,8].
First, we introduce the concept of Henstock-Kurzweil-Pettis integrability and give some related facts which are useful in the sequel.
Remark 3.2 This definition includes the generalized Riemann integral defined by Gordon [17]. In a special case, when δ is a constant function, we get the Riemann integral.
We will also use the following equi-integrability notion, specific to the HK integrability that allows to obtain a Vitali-type convergence result.
The generalization of the Pettis integral obtained by replacing the Lebesgue integrability of the functions by the Henstock-Kurzweil integrability produces the Henstock-Kurzweil-Pettis integral (for the definition of Pettis integral, see [12]).
This function g will be called a primitive of f and by g(T ) = T 0 f (t)dt we will denote the Henstock-Kurzweil-Pettis integral of f on the interval [0, T ].
Remark 3.5 (i) Any HK-integrable function is HKP-integrable. The converse is not true (see an example in [15]). Thus, the family of all Kurzweil-Henstock-Pettis integrable functions is larger than the family of all Kurzweil-Henstock integrable ones. (ii) Since each Lebesgue integrable function is HK-integrable, we find that any Pettis integrable function is HKP-integrable. The converse is not true (see also [15]).
For b > 0 we denote by B b = {y ∈ X such that y ≤ b}, D b = {z ∈ C(I, X ) such that z ≤ b}, and integrals are taken in the sense of (HKP) integrals. The closed unit ball of the dual X * is denoted by B(X * ). (1) h is weakly continuous on I .
Then, there exists an interval J = [0, a] ⊂ I such that the set of solutions of (1.1) defined on J is non-empty and weakly compact in the space C(J, X ).
First notice that for x ∈ C(I, X ), the family is HK-equi-integrable [see (3.2)]. Since for t ∈ I the function s −→ G(t, s) is of bounded variation then by [24,Lemma 25] and assumption (4), the function is HKP-integrable on [0, t] and thus the operator F makes sense.
We assert that F : B −→ B is weakly-weakly sequentially continuous.

Let us firstly prove that the values of F are in B.
For any x * ∈ B(X * ), for any x ∈ B

G(t, s) f (s, x(s), T x(s))ds
From here Next, we will prove that F( B) is a strongly equicontinuous subset. Let 0 ≤ t 1 < t 2 ≤ a and x ∈ B.
We suppose without loss of generality that F x (t 1 ) = F x (t 2 ). By the Hahn-Banach theorem, there exists x * ∈ X * , such that x * = 1 and So, the result follows from hypotheses (1), (2) and inequality (3.6). 3. Now we will show that F is weakly-weakly sequentially continuous. Let (x n (.)) n a weakly convergent sequence to x in B. Then by Theorem 2.4, x n (t) x(t) for each t ∈ [0, a]. Let s ∈ [0, a] and x * ∈ X * . We have T x(s). Therefore, the operator T is weakly-weakly sequentially continuous on B. Moreover, because f is weakly-weakly sequentially continuous, so for each s ∈ [0, a]. Now, for each t ∈ [0, a], applying Theorem 5 in [11] and Lemma 25 in [24] to the sequence (G(t, .) f (., x n (.), T x n (.)) n , we find that the function and by Theorem 2.4 the operator F is weakly-weakly sequentially continuous. Next, we consider Q = convF( B). Because F( B) is bounded and strongly equicontinuous, so Q is a weakly closed bounded and strongly equicontinuous subset of B. Clearly F(Q) ⊂ Q. We claim that Because V is bounded and strongly equicontinuous, we have by Lemma 2.5(a) that sup t∈J β(V (t)) = β(V ) = β(V (J )). For fixed t ∈ J , we divide the interval [0, t] into n parts: 0 = t 0 < t 1 < · · · < t n = t and put T i = [t i−1 , t i ]. By Henstock-Kurzweil-Pettis integral mean value theorem [8], we obtain Using the properties of the measure of weak noncompactness, we have if |τ − s| < η, |q − s| < η, q, s, τ ∈ [0, t]. So, taking t i in the manner that |t i − t i−1 | < η and by (3.7) we infer that As the last inequality is satisfied for every ε > 0, we get β(F(V )(t)) ≤ sup (HK) t 0 |G(t, s)| L(s, β(V ))ds, t ∈ J .
Applying again Lemma 2.5(a) for the bounded strongly equicontinuous subset F(V ), we obtain that β(F(V )) = sup t∈J {F(V )(t)}. Accordingly, by (3.5) β(F(V )) ≤ sup (HK) t 0 |G(t, s)| L(s, β(V ))ds, t ∈ J < β(V ), so, F is β-condensing. Now, applying Theorem 2.6, we infer that F has a fixed pint in Q ⊂ B. Therefore, S the fixed point set of F in B is non-empty. Finally, we have F(S) = S ⊂ Q, so β(F(S)) = 0 and hence S is relatively weakly compact subset of B. Because the operator F is weakly-weakly sequentially continuous, so S is weakly sequentially closed. The use of Eberlein-Šmulian theorem [14,Theorem 8.12.4,p. 549], proves that S is weakly compact. This achieves the proof.
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