Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

We prove two existence theorems for the integrodifferential equation of mixed type: x′(t) = f (t,x(t), t0k1(t,s)g(s,x(s))ds, ∫ a 0k2(t,s)h(s,x(s))ds), x(0) = x0, where in the first part of this paper f , g, h, x are functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil (HK). In the second part f , g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functions f , g, h, x satisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.


Introduction
The Henstock-Kurzweil integral encompasses the Newton, Riemann, and Lebesgue integrals [1][2][3].A particular feature of this integral is that integrals of highly oscillating functions such as F (t), where F(t) = t 2 sint −2 on (0,1] and F(0) = 0, can be defined.This integral was introduced by Henstock and Kurzweil independently in 1957-1958 and has since proved useful in the study of ordinary differential equations [4][5][6][7].In the paper, [8] Cao defined the Henstock integral in a Banach space, which is a generalization of the Bochner integral.The Pettis integral is also a generalization of the Bochner integral [9].This notion is strictly relative to weak topologies in Banach spaces.

International Journal of Mathematics and Mathematical Sciences
The paper is divided into two main sections.In Section 1, we prove some existence theorem for the problem x (t) = f t,x(t), t 0 k 1 (t,s)g s,x(s) ds, a 0 k 2 (t,s)h s,x(s) ds , x(0) = x 0 , x 0 ∈ E, t ∈ I a = [0,a], a ∈ R + , (1.1) where E is a Banach space with the norm • , f , g, h, x are functions with values in a Banach space E, k j , j = 1,2 are real-valued functions, and integrals are taken in the sense of HL.
In Section 2, we prove some existence theorem for the problem (1.1), where f , g, h, x are functions with values in a Banach space E, weakly-weakly sequentially continuous, and k j , j = 1,2 are real-valued functions.The integrals are taken in the sense of Henstock-Kurzweil-Pettis.
Our fundamental tools are the Kuratowski measure of noncompactness [19] and the measure of weak noncompactness developed by De Blasi [20].
For any bounded subset A of E, we denote by α(A) the Kuratowski measure of noncompactness of A, that is, the infimum of all ε > 0, such that there exists a finite covering of A by sets of diameter smaller than ε.
The De Blasi measure of weak noncompactness β(A) is defined by where K ω is the set of weakly compact subsets of E and B 0 is the norm unit ball in E.
The properties of the measure of noncompactness α(A) are as follows: , where conv(A) denotes the convex extension of A.
The properties of the weak measure of noncompactness β are analogous to the properties of the measure of noncompactness α(A) (see [21]).
We now gather some well-known definitions and results from the literature, which we will use throughout this paper.
, where E 1 and E 2 are Banach spaces, is L 1 -Carathéodory, if the following conditions hold: A. Sikorska-Nowak and G. Nowak 3 Definition 1.2.A function f : I a → E is said to be weakly continuous if it is continuous from I a to E endowed with its weak topology.A function g : E → E 1 , where E and E 1 are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent sequence (x n ) in E, the sequence (g(x n )) is weakly convergent in E 1 .
When the sequence x n tends weakly to x 0 in E, we will write x n ω − → x 0 .
Definition 1.3 [1,3].A family Ᏺ of functions F is said to be uniformly absolutely continuous in the restricted sense on A ⊆ [a,b] or in short uniformly AC * (A) if, for every ε > 0, there exists η > 0 such that for every F in Ᏺ and for every finite or infinite sequence of nonoverlapping intervals A family Ᏺ of functions F is said to be uniformly generalized absolutely continuous in the restricted sense on [a,b] or uniformly ACG * if [a,b] is the union of a sequence of closed sets A i such that on each A i the family Ᏺ is uniformly AC * (A i ).
In the proof of the main theorem in Section 1, we will apply the following fixed point theorem.
Theorem 1.4 [22].Let D be a closed convex subset of E, and let F be a continuous map from D into itself.If for some x ∈ D the implication that holds for every countable subset V of D, then F has a fixed point.
In Section 2 we will apply the following theorem.
Theorem 1.5 [23].Let X be a metrizable locally convex topological vector space.Let D be a closed convex subset of X, and let F be a weakly-weakly sequentially continuous map from D into itself.If for some x ∈ D the implication that holds for every subset V of D, then F has a fixed point.

Henstock-Kurzweil and Henstock-Kurzweil-Pettis integrals in Banach spaces
In this part, we present the definitions of Henstock-Kurzweil and Henstock-Kurzweil-Pettis integrals and properties of these integrals which we will use in the proof of the main theorems.
The letter P will be used to denote finite collections of nonoverlapping tagged intervals.Let be such a collection in [a,b].Then, (i) the points {s i : 1 ≤ i ≤ n} are called the tags of P; ) is subordinate to δ for each i, then we write P is sub δ; (2.2) if there exists a real number L with the following property: for each ε > 0, there exists a positive function a,b],E)) if there exists a vector z ∈ E with the following property: for every ε > 0, there exists a positive function We remark that this definition includes the generalized Riemann integral defined by Gordon [25].In a special case, when δ is a constant function, we get the Riemann integral.
A. Sikorska-Nowak and G. Nowak 5 Remark 2.5.We note by triangle inequality that In general, the converse is not true.For real-valued functions, the two integrals are equivalent.
Definition 2.6 [9].The function f : where I is an arbitrary subinterval of I a and |I| is the length of I.
If the integral is taken in the sense of HL, then the proof is similar to that of [27,Lemma 2.1.3].The proof for HKP integral is presented in [24].Theorem 2.9 [8].
.., is a sequence of HL integrable functions satisfying the following conditions: We remark that this theorem for Denjoy-Bochner integrals is mentioned in [28] without proof.It is also true for HL integrals.The proof is similar to that of [3,Theorem 7.6] (see also [29,Theorem 1.8]).
Theorem 2.11 [26].Let f : I a → E and assume that f n : I a → E, n ∈ N, are HKP integrable on I a .For each n ∈ N, let F n be a primitive of f n .If it is assumed that a.e. on I a , (ii) for each x * ∈ E * , the family G = (x * F n : n = 1,2,...) is uniformly ACG * on I a (i.e., weakly uniformly ACG * on I a ), (iii) for each x * ∈ E * , the set G is equicontinuous on I a , then f is HKP integrable on I a and t 0 f n (s)ds tends weakly in E to t 0 f (s)ds for each t ∈ I a .

An existence result for integrodifferential equations
It is well known that Henstock's lemma plays an important role in the theory of the Henstock-Kurzweil integral in the real-valued case.On the other hand, in connection with the Henstock-Kurzweil integral for Banach-space-valued functions, Cao pointed out in [8] that Henstock's lemma holds for the case of finite dimensions, but it does not always hold in infinite dimensions.
In this section, we will use the HL integral which satisfies Henstock's lemma and which is more general than the Bochner integral.
Our fundamental tool is a Kuratowski measure of noncompactness α.It is necessary to remark that the following lemma is true.Lemma 3.1 [30].Let H ⊂ C(I a ,E) be a family of strongly equicontinuous functions.Let, for t ∈ I a , H

(t) = {h(t) ∈ E, h ∈ H}. Then, α(H(I a )) = sup t∈Ia α(H(t)) and the function t → α(H(t)) is continuous.
Observe that the problem (1.1) is equivalent to the integral equation [31]: To obtain the existence result, it is necessary to define a notion of a solution.Definition 3.2.An ACG * function x : I a → E is said to be a solution of the problem (1.1) if it satisfies the following conditions: (i) For x ∈ C(I a ,E), we define the norm of x by: Note that these sets are closed and convex.
Define the operator F : C(I a ,E) → C(I a ,E) by A. Sikorska-Nowak and G. Nowak 7 where integrals are in the sense of HL.Let Let r(K) be the spectral radius of the integral operator K defined by where the kernel k ∈ C(I a × I a ;R), u ∈ C(I a ;E) and c denotes any fixed value in I a , a > 0.
Theorem 3.4.Assume that for each ACG * function x : Assume that there exist p 0 > 0 and positive constants L, L 1 , and d 1 , such that α g(I,X) ≤ Lα(X) for I ⊂ I a , for every where g(I,X) = {g(t, x(t)) : t ∈ I, x ∈ X}, h(I,X) = {h(t, x(t)) : t ∈ I, x ∈ X}, f (t,A,C,D) = { f (t,x 1 ,x 2 ,x 3 ) : (x 1 ,x 2 ,x 3 ) ∈ A × C × D} and α denotes the Kuratowski measure of noncompactness.Moreover, let Γ(p 0 ) be equicontinuous, equibounded, and uniformly ACG * on I a .Then, there exists at least one solution of the problem (1.1) on I c , for some 0 < c ≤ a, such that Proof.By equicontinuity and equiboundedness of Γ(p 0 ) there exists a number c, 0 < c ≤ a such that By our assumptions the operator F is well defined and maps B(p 0 ) into B(p 0 ): Using Theorem 2.10, we deduce that F is continuous.
Suppose that V ⊂ B(p 0 ) satisfies the condition V = conv({x} ∪ F(V )), for some x ∈ B(p 0 ).We will prove that V is relatively compact, thus (1.3) is satisfied.Theorem 1.4 will ensure that F has a fixed point.
Let, for For fixed t ∈ I c we divide the interval [0, t] into m parts: ..,m − 1.By Lemma 3.1 and the continuity of v there exists For fixed z ∈ I c we divide the interval [0, z] into m parts: ..,m − 1.By Lemma 3.1 and the continuity of v there exists Furthermore, we divide the interval [0, c] into m parts: ..,m − 1.By Lemma 3.1 and the continuity of v there exists By Definition 2.7 and the properties of the HL integral, we have where k(I,J) = {k(t, s) : A. Sikorska-Nowak and G. Nowak 9 Using (3.5) and the properties of measure of noncompactness α, we have (3.12) Let us observe that if then Because (3.16) For j = 0,1,...,m − 1 there exists q j = 0,1,...,m − 1 such that k 1 (I i ,I j ) ≤ k 1 (I qj ,I j ).So z j+1 − z j k 1 I qj ,I j v s j , where s j ∈ I j . (3.17) Hence, By the continuity of v we have v(s j ) − v(p j ) < ε 1 and ε 1 → 0 as m → ∞.Therefore, , by the property of the measure of noncompactness, we have Because this inequality holds for every t∈ I c and L • d 1 • c • r(K) < 1, so by applying Gronwall's inequality, we conclude that α(V (t)) = 0 for t ∈ I c .Hence Arzela-Ascoli's theorem implies that the set V is relatively compact.Consequently, by Theorem 1.4, F has a fixed point which is a solution of the problem (1.1).
Similarly, if A. Sikorska-Nowak and G. Nowak 11 then we prove that α(F(V ds and we conclude that the set V is relatively compact.By Theorem 1.4, F has a fixed point which is a solution of the problem (1.1).

An existence result for integrodifferential equations in weak sense
In this part, we prove a theorem for the existence of pseudosolutions to the Cauchy problem in Banach spaces.Functions f , g, h, x will be assumed Henstock-Kurzweil-Pettis integrable but this assumption is not sufficient for the existence of solutions.We impose a weak compactness-type conditions expressed in terms of measures of weak noncompactness.Throughout this part, (E, • ) will denote a real Banach space, E * the dual space.
Unless otherwise stated, we assume that " " denotes the Henstock-Kurzweil-Pettis integral.Fix x * ∈ E * and consider the equation Now, we can introduce the following definition.

Definition 2 . 1 .
Let δ be a positive function defined on the interval [a,b].A tagged interval (x,[c,d]) consists of an interval [c,d] ⊂ [a,b] and a point x ∈ [c,d].The tagged interval defined on the subintervals of [a,b], satisfying the following property: given ε > 0, there exists a positive function δ on [a,b] such that if

z 0 k 1
(z,s)g s,x(s) ds, a 0 k 2 (z,s)h s,x(s) ds dz for t ∈ I a .(3.1) Assume that there exist p 0 > 0 and positive constants L, L 1 , and d such thatβ g(I,X) ≤ Lβ(X) for I ⊂ I a , for every X ⊂ B p 0 , β h(I,X) ≤ L 1 β(X) for I ⊂ I a , for every X ⊂ B(p 0 ), β f (t,A,C,D) ≤ d • max β(A),β(C),β(D) for every A,C,D ⊂ B p 0 , t ∈ I a ,(4.4)where the sets g(I,X)h(I,X) and f (t,A,C,D) are defined as in Theorem 3.4 and β denotes the De Blasi measure of weak noncompactness.Moreover, let Γ(p 0 ) be equicontinuous and uniformly ACG * on I a .Then, there exists at least one pseudosolution of the problem(1.1)on I c , for some 0 < c ≤ a, such that d • c • L • r(K) < 1 and d • c < 1.Proof.By equicontinuity of Γ(p 0 ), there exists a number c, 0 < c ≤ a, such that t 0 f z,x(z), z 0 k 1 (z,s)g s,x(s) ds, c 0 k 2 (z,s)h s,x(s) ds dz ≤ p 0 , where x ∈ B p 0 , t ∈ I c .

( 4 . 5 )
Indeed, for any x * ∈ E * , such that x * ≤ 1 and for any x ∈ B(p 0 ), we havex * F(x)(t) = x * x 0 + x * z,s)g s,x(s) ds, c 0 k 2 (z,s)h s,x(s) ds dz ≤ x * x 0 + x * z,s)g s,x(s) ds, c 0 k 2 (z,s)h s,x(s) ds dz ≤ x 0 + p 0 .(4.6)From heresup{ x * F(x)(t) : x * ∈ E * , x * ≤ 1 ≤ x 0 + p 0 , F(x)(t) ≤ x 0 + p 0 ,(4.7)so F(x)(t) ∈ B(p 0 ).We will show, that the operator F is weakly-weakly sequentially continuous.By [32, Lemma 9], a sequence x n (•) is weakly convergent in C(I c ,E) to x(•) if and only if x n (t) tends weakly to x(t) for each t ∈ I c .Because g(s,•) and h(s,•) are weaklyweakly sequentially continuous, so if x n ω − → x in (C(I c ,E),ω), then g(s,x n (s)) ω − → g(s,x(s)) and h(s,x n (s)) ω − → h(s,x(s))in E for t ∈ I c and by Theorem 2.11 we have lim n→∞ z 0 k 1 (t,s)g s,x n (s) ds = z 0 k 1 (t,s)g s,x(s) ds (4.8) weakly in E for each t ∈ I c and lim n→∞ c 0 k 2 (t,s)h s,x n (s) ds = c 0 k 2 (t,s)h s,x(s) ds (4.9) ds, where (L) A denotes the Lebesgue integral over A. Now, we present a definition of an integral which is a generalization for both Pettis and Henstock-Kurzweil integrals.Definition 2.7 [26].The function f : I a → E is Henstock-Kurzweil-Pettis integrable (HKP integrable for short) if there exists a function g : I a → E with the following properties: (i) ∀x * ∈E * x * f is Henstock-Kurzweil integrable on I a ; (ii) ∀t∈I a ∀x * ∈E * x * g(t) = (HK) t 0 x * f (s)ds.This function g will be called a primitive of f and by g(a) = a 0 f (t)dt we will denote the Henstock-Kurzweil-Pettis integral of f on the interval I a .Theorem 2.8 (Mean value theorem).If the function f : I a → E is HK (or HKP) integrable, then Pettis integrable, f , g, and h are weakly-weakly sequentially continuous functions.Let k 1 , k 2 : I a × I a → R + be measurable functions such that k 1 (t,•), k 2 (t,•) are continuous.