# Integrodifferential Equations on Time Scales with  Henstock-Kurzweil-Pettis Delta Integrals

Authors: Aneta Sikorska-Nowak
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We prove existence theorems for integro-differential equationsxΔ(t)=f(t,x(t),∫0tk(t,s,x(s))Δs), x(0)=x0, t∈Ia=[0,a]∩T, a∈R+, where T denotes a time scale (nonempty closed subset of real numbers R), and Ia is a time scale interval. The functions f,k are weakly-weakly sequentially continuous with values in a Banach space E, and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.


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## Body

## 1. Introduction

A time scaleT is a nonempty closed subset of real numbers R, with the subspace topology inherited from the standard topology of R.The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [1]), that is, when T=R, T=N, T=qN0={qt:t∈N0}, where q>1.Time scale (or a measure chain) was introduced by Hilger in his Ph.D. thesis in 1988, [2]. Since the time Hilger formed the definitions of a derivative and integral on a time scale, several authors have extended on various aspects of the theory [3–11]. Time scales have been shown to be applicable to any field that can be described by means of discrete or continuous models.In this paper we consider an integrodifferential equation. As is known, ordinary integrodifferential equations, an extreme case of integrodifferential equations on time scales [12–21], find many applications in various mathematical problems; see Corduneanu's book [22] and references therein for details.In [23] the authors extend such results to the integrodifferential equations on time scales and therefore obtained corresponding criteria which can be employed to study the difference equation of Volterra type [21, 24], q-difference equations of Volterra type, and so forth.In [25] the authors proved a new comparison result and develop the monotone iterative technique to show the existence of extremal solutions of the periodic boundary value problems of nonlinear integrodifferential equation on time scales.We extend this result by proving the existence of a pseudosolution of the problem(1.1)xΔ(t)=f(t,x(t),∫0tk(t,s,x(s))Δs),t∈Ia=[0,a]∩T,a∈R+,x(0)=x0,
where f:Ia×E×E→E,k:Ia×Ia×E→E,x:Ia→E, T denotes a time scale, 0∈T,Ia denotes a time scale interval, (E,∥·∥) is a Banach space, and xΔ denotes the pseudoΔ-derivative of x.We use a new (Henstock-Kurzweil-Pettis-Δ-integral), more general concept of integral on time scales. This new integral generalizes the Henstock-Kurzweil delta integral, which has been studied by Peterson and Thompson in [26], Avsec et al. in [27] and the Pettis integral [28]. These integrals are important when we consider the weak topology on a Banach space.The Henstock-Kurzweil delta integral contains the Riemann delta, Lebesgue delta and Bochner delta integrals as special cases. These integrals will enable time scale researchers to study more general dynamic equations. In [26] it is shown that there are highly oscillatory functions that are not delta integrable on a time scale, but are Henstock-Kurzweil delta integral.Let us remark that the existence of the Henstock-Kurzweil integral over[a,b] implies the existence of such integrals over all subintervals of [a,b] but not for all measurable subsets of this interval, so the theory of such integrals on T does not follows from general theory on R.In [29] Cichoń introduce a definition of the Henstock-Kurzweil delta integral (HK-Δ-integral) and HL delta integral (HL-Δ-integral) on Banach spaces for checking the existence of solutions of differential (or dynamic) equations in Banach spaces. He presented also a new definition of the Henstock-Kurzweil-Pettis delta integral on time scales.The study for weak solutions of Cauchy differential equations in Banach spaces was initiated by Szép [30] and theorems on the existence of weak solutions of this problem were proved by Cramer et al. [31], Kubiaczyk [32], Kubiaczyk and Szufla [33], Mitchell and Smith [34], Szufla [35], and Cichoń and Kubiaczyk [36]. There are also some existence theorems for the Volterra and Urysohn integral equations [37] on time scales.Similar methods for solving existence problems for difference equations in Banach spaces equipped with its weak topology were studied, for instance, in [38–40]. In particular, the importance of conditions expressed in terms of the weak topology was considered in [38].We will unify both cases and using the weak topology, we will obtain the first result for pseudosolutions of an integrodifferential dynamic problem. (This is new also forq-difference equations.)Our result extends the existence of pseudosolutions not only to the discrete intervals with uniform step size (hZ) but also to the discrete intervals with nonuniform step size (Kq).We note that (1.1) in its general form involves some different types of differential and difference equations depending on the choice of the time scale T. For example: (1)
forT=R, we have σ(t)=t, μ(t)=0, and xΔ(t)=x′(t), and (1.1) becomes the Cauchy integrodifferential equation:
(1.2)x′(t)=f(t,x(t),∫0tk(t,s,x(s))ds),t∈R,x(0)=x0;(2)
forT=N, we have σ(n)=n+1, μ(n)=1, and xΔ(n)=Δx(n)=x(n+1)-x(n), and (1.1) becomes the Cauchy problem:
(1.3)Δx(t)=f(t,x(t),∫0tk(t,s,x(s))ds),t∈N,x(0)=x0.We assume that the functionsf,k are weakly-weakly sequentially continuous with values in a Banach space and satisfy some regularity conditions expressed in terms of the De Blasi measure of weak noncompactness. We introduce a weakly sequentially continuous operator associated to an integral equation which is equivalent to (1.1). There exist many important examples of mappings which are weakly sequentially continuous but not weakly continuous. The relations between weakly sequentially continuous and weakly continuous mappings are studied by Ball [41].Adopting the fixed point theorem for weakly sequentially continuous mappings given by Kubiaczyk [42] and the properties of measures of weak noncompactness, we are able to study the existence results for problem (1.1)


## 2. Preliminaries

Let(E,∥·∥) be a Banach space and let E* be the dual space. Moreover, let (C(Ia,E),ω) denote the space of all continuous functions from Ia to E endowed with the topology σ(C(Ia,E),C(Ia,E)*) and Crd(Ia,E) denotes the space of all rd-continuous functions from the time scale interval Ia to E.ByμΔ we denote the Lebesgue measure on T. For a precise definition and basic properties of this measure we refer the reader to [43].We now gather some well-known definitions and results from the literature, which we will use throughtout this paper.Definition 2.1 (see [44]).
A family𝔽 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 ɛ>0 there exists η>0 such that for every F in 𝔽 and for every finite or infinite sequence of nonoverlapping intervals {[ai,bi]} with ai,bi∈A and satisfying ∑iμΔ([ai,bi])<η, one has ∑iω(F,[ai,bi])<ɛ where ω denotes the oscillation of F over [ai,bi] (i.e., ω(F,[ai,bi])=sup{|F(r)-F(s)|:r,s∈[ai,bi]}).A family𝔽 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 Ai such that on each Ai the function F is uniformly AC*(Ai).Definition 2.2.
A functionf:Ia→E is said to be weakly continuous if it is continuous from Ia to E endowed with its weak topology. A function g:E→E1, where E and E1 are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent sequence (xn)⊂E, the sequence (g(xn)) is weakly convergent in E1. If a sequence xn tends weakly to x0 in E we will denote it by xn→ωx0.Theorem 2.3 (see [42]).
LetE be a metrizable locally convex topological vector space. Let D be a closed convex subset of E, and let F be a weakly-weakly sequentially continuous map of D into itself. If for some x∈D the implication
(2.1)V¯=conv¯({x}∪F(V))⇒Visrelativelyweaklycompact,
holds for every subset V of D, then F has a fixed point.(I)
To understand the so-called dynamic equations and follow this paper easily, we present some preliminary definitions and notations of time scales which are very common in the literature (see [3–5, 7–11] and references therein).A time scaleT is a nonempty closed subset of real numbers R, with the subspace topology inherited from the standard topology of R. If a,b are points in T, then a time scale interval I we denote by I=[a,b]={t∈T:a≤t≤b} and Ia={t∈T:0≤t≤a}. Other types of intervals are approached similarly. By a subinterval Ib of Ia we mean the time scale subinterval.Definition 2.4.
The forward jump operatorσ:T→T and the backward jump operator ρ:T→T are defined by σ(t)=inf{s∈T:s>t} and ρ(t)=sup{s∈T:s<t}, respectively. We put inf∅=supT (i.e., σ(M)=M if T has a maximum M) and sup∅=infT (i.e., ρ(m)=m if T has a minimum m). The jump operators σ and ρ allow the classification of points in time scale in the following way: t is called right dense, right scattered, left dense, left scattered, dense and isolated if σ(t)=t,σ(t)>t,ρ(t)=t,ρ(t)<t,ρ(t)=t=σ(t) and ρ(t)<t<σ(t), respectively.Definition 2.5.
One  says thatk:T→E is right-dense continuous (rd-continuous) if k is continuous at every right-dense point t∈T and lims→t-k(s) exists and is finite at every left-dense point t∈T.Definition 2.6.
Fixt∈T. Let f:I→E. Then one defines the Δ-derivative fΔ(t) by
(2.2)fΔ(t)=lims→tf(σ(t))-f(s)σ(t)-s.Remark 2.7.
TheΔ-derivative satisfies (1)
fΔ=f′ is the usual derivative if T=R,(2)
fΔ=Δf is the usual forward difference operator if T=Z,(3)
fΔ=Dqf is the q-derivative if T=qN0={qt:t∈N0},q>1.Hence, time scales allows us unify the treatment of differential and difference equations (and many more time scales).(II)
We need to define some integrals which are important, when we consider the weak topology on a Banach spaceE.We will use the notationη(t):=σ(t)-t, where η is called the graininess function and ϑ(t):=t-ρ(t), where ϑ is called the left-graininess function.We say thatδ=(δL,δR) is a Δ-gauge for time scale interval [a,b] provided δL(t)>0 on (a,b], δR(t)>0 on [a,b), δL≥0, δR≥0 and δR≥η(t) for all t∈[a,b).We say that a partitionD for a time scale interval [a,b] given by(2.3)D={a=t0≤ξ1≤t1≤⋯≤tn-1≤ξn≤tn=b}
with ti>ti-1 for 1≤i≤n and ti,ξi∈T is δ-fine if(2.4)ξi-δL(ξi)≤ti-1<ti≤ξi+δR(ξi)for1≤i≤n.Definition 2.8 (see [29]).
A functionf:[a,b]→E is Henstock-Kurzweil-Δ-integrable on [a,b](Δ-HK integrable in short) if there exists a function F:[a,b]→E, defined on the subintervals of [a,b], satisfying the following property: given ɛ>0 there exists a positive function δ on [a,b] such that D={[u,v],ξ} is δ-fine division of a [a,b], one has
(2.5)∥∑Df(ξ)(v-u)-(F(v)-F(u))∥<ɛ.Definition 2.9 (see [29]).
A functionf:[a,b]→E is Henstock-Lebesgue-Δ-integrable on [a,b](Δ-HL integrable in short) if there exists a function F:[a,b]→E, defined on the subintervals of [a,b], satisfying the following property: given ɛ>0 there exists a positive function δ on [a,b] such that D={[u,v],ξ} is δ-fine division of a [a,b], one has
(2.6)∑D∥f(ξ)(v-u)-(F(v)-F(u))∥<ɛ.Remark 2.10.
We note that, by the triangle inequality, iff is Δ-HL integrable it is also Δ-HK integrable. In general, the converse is not true. For real-valued functions, the two integrals are equivalent.Definition 2.11 (see [28, 29]).
The functionf:Ia→E is Pettis-Δ-integrable ((P) integrable for short) if (1)
∀x*∈E*,x*f is Lebesgue-Δ-integrable on Ia,(2)
∀A⊂I,∃g∈E,∀x*∈E*,x*g=(Δ-L)∫Ax*f(s)Δs.Now, we present a new definition of the integral on time scales which is a generalization for both the Pettis-Δ-integral and the Henstock-Kurzweil-Δ-integral.Definition 2.12 (see [29]).
The functionf:Ia→E is Henstock-Kurzweil-Pettis-Δ-integrable (HKP-Δ-integrable for short) if (1)
∀x*∈E*,x*f is Henstock-Kurzweil-Δ-integrable on Ia,(2)
∀t∈Ia∀x*∈E*,x*g(t)=(Δ-HK)∫0tx*f(s)Δs.The function g will be called a primitive of f and by g(t)=(Δ-HKP)∫0tf(s)Δs we will denote the Henstock-Kurzweil-Pettis-Δ-integral of f on the interval Ia.In [29] the authors give examples of Henstock-Kurzweil-Pettis-Δ-integrable functions which are not integrable in the sense of Pettis and Henstock-Kurzweil on time scales.Theorem 2.13 (see [29] mean value theorem).
For eachΔ-subinterval [c,d]⊂[a,b], if the integral (Δ-HKP)∫cdy(s)Δs exists, then one has
(2.7)(Δ-HKP)∫cdy(s)Δs∈μΔ([c,d])·conv¯y([c,d]),
where conv¯y([c,d]) denotes the close convex hull of the set y([c,d]).Theorem 2.14 (see [45]).
Letfn,f:Ia→E and assume that fn:Ia→E are Δ-HKP integrable on Ia. Let Fn be a primitive of fn. If one assumes that: (1)
∀x*∈E*,x*fn(t)→x*f(t)μΔ a.e. on Ia,(2)
for eachx*∈E* the family G={x*Fn:n=1,2,…} is uniformly ACG* on Ia (i.e., weakly uniformly ACG* on Ia),(3)
for eachx*∈E* the set G is equicontinuous on Ia,then f is Δ-HKP integrable on Ia and ∫0tfn(s)Δs tends weakly in E to ∫0tf(s)Δs for each t∈Ia.Theorem 2.15 (see [5] Gronwall's inequality).
Suppose thatu,g,h∈Crd(Ia,E) and h≥0. Then
(2.8)u(t)≤g(t)+∫0th(τ)u(τ)Δτ,foreacht∈Ia
implies that
(2.9)u(t)≤(g(t)+∫0tg(τ)h(τ)Δτ)exp(∫0th(τ)Δτ),foreacht∈Ia.(III)
The De Blasi measure of weak noncompactness is one of our fundamental tool in this paper (see [46]).LetA be a bounded nonempty subset of E.The measure of weak noncompactnessβ(A) is defined by(2.10)β(A)=inf{t>0:thereexistsC∈KωsuchthatA⊂C+tB0},
where Kω is a set of weakly compact subsets of E and B0 is a norm unit ball in E.Some properties of the measure of weak noncompactnessβ(A) are known [46]: (1)
ifA⊂B, then β(A)≤β(B);(2)
β(A)=β(Aw¯), where Aw¯ denotes the weak closure of A;(3)
β(A)=0 if and only if A is relatively weakly compact;(4)
β(A∪B)=max{β(A),β(B)};(5)
β(λA)=|λ|β(A), (λ∈R);(6)
β(A+B)≤β(A)+β(B);(7)
β(conv¯(A))=β(conv(A))=β(A), where conv(A) denotes the convex hull of A.We will need the following lemmas.Lemma 2.16 (see [47]).
LetE1,E2 be bounded subsets of the Banach space E. If ∥E1∥=sup{∥x∥:x∈E1}<1, then
(2.11)β(E1+E2)≤β(E2)+∥E1∥β(K(E2,1)),
where K(E2,1)={x:D(E2,x)<1} and D(E2,x)=inf{∥x-y∥:y∈E2}.The lemma below is an adaptation of the corresponding result of Ambrosetti (see [48]).Lemma 2.17.
LetH⊂C(Ia,E) be a family of strongly equicontinuous functions. Let H(t)={h(t)∈E,h∈H}, for t∈Ia and H(Ia)=⋃t∈IaH(t). Then
(2.12)βC(H)=supt∈Iaβ(H(t))=β(H(Ia)),
where βC(H) denotes the measure of weak noncompactness in C(Ia,E) and the function t↦β(H(t)) is continuous.Proof.
(I) First we prove the equality:supt∈Iaβ(H(t))=β(H(Ia)).
SinceH(t)⊂H(Ia) by the first property of the measure of noncompactness β(H(t))≤β(H(Ia)) and consequently(2.13)supt∈Iaβ(H(t))≤β(H(Ia)).
By strong equicontinuity of H we deduce that for ɛ>0, there exists δ>0 such that |t-s|<δ⇒∥u(t)-u(s)∥<ɛ for t,s∈Ia and for all u∈H.
We divide the intervalIa={t∈T:0≤t≤a} in the following way: t0=0,(2.14)t1=sups∈Ia{s:s≥t0,s-t0<δ},t2=sups∈Ia{s:s≥t1,s-t1<δ},⋮tn=sups∈Ia{s:s≥tn-1,s-tn-1<δ}.
Since T is closed, we have ti∈Ia. If some ti+1=ti then ti+2=inf{t∈T:t≥ti+1}.
As(2.15)u(t)=u(ti)+u(t)-u(ti)∈u(ti)+ɛK(0,1),
where K(0,1)={x:∥x∥<1}, we have
(2.16)u(t)∈⋃i=1nH(ti)+ɛK(0,1),H(Ia)⊂⋃i=1nH(ti)+ɛK(0,1).
By the properties of the measure of weak noncompactness and Lemma2.16, we obtain(2.17)β(H(Ia))≤β(⋃i=1nH(ti))+∥ɛK(0,1)∥·β(K(⋃i=1nH(ti),1))<supti∈Iaβ(H(ti))+ɛβ(K(H(Ia),1))≤supt∈Iaβ(H(t))+ɛβ(K(H(Ia),1)).
Since the above inequality holds, for any ɛ>0, we have
(2.18)β(H(Ia))≤supt∈Iaβ(H(t)).
Hence, from (2.13) and (2.18), we conclude that β(H(Ia))=supt∈Iaβ(H(t)).
(II) The proof of the equalityβC(H)=supt∈Iaβ(H(t)) is similar to the proof of Lemma 2.1 of Ambrosetti (see [48]), where we choose points ti as in part (I) of our proof.
(III) Now we prove that the functiont↦β(H(t)) is continuous.
Letv(t)=β(H(t)) because H(t)⊂H(t)-̇H(s)+̇H(s)⊂H(t)-̇H(s)+H(s), where H(t)-̇H(s)+̇H(s)={y(t):y(t)=y(t)-y(s)+y(s):y∈H} and the sum is taken in the Minkowski sense.
By the property (vi) of the measure of weak noncompactness, we have(2.19)β(H(t))≤β(H(t)-̇H(s))+β(H(s)).
This implies that
(2.20)|β(H(t))-β(H(s))|≤β(H(t)-̇H(s))≤δ(H(t)-̇H(s))=supx,y∈H{∥(x(t)-x(s))-(y(t)-y(s))∥}≤supx,y∈H{∥x(t)-x(s)∥+∥y(t)-y(s)∥}.
By equicontinuity of H we obtain the continuity of v(t).


## 3. Main Results

Now, we will consider equivalently integral problem(3.1)x(t)=x0+∫0tf(z,x(z),∫0zk(z,s,x(s))Δs)Δz,t∈Ia=[0,a]∩T
where f:Ia×E×E→E,k:Ia×Ia×E→E,x:Ia→E, T denotes a time scale, 0∈T, Ia denotes a time scale interval, (E,∥·∥) is a Banach space, and integrals are taken in the sense of Δ-HKP integrals.Fixx*∈E*and consider the problem(3.2)(x*x)Δ=x*(f(t,x(t),∫0tk(t,s,x(s)))Δs).
Let us introduce a definition.Definition 3.1.
LetF:I→E and let A⊂I. The function f:A→E is a pseudoΔ-derivative of F on A if for each x*∈E* the real-valued function x*F is Δ-differentiable μΔ almost everywhere on A and (x*F)Δ=x*fμΔ almost everywhere on A.Regarding the above definition it is clear that the left-hand side of (3.2) can be rewritten to the form x*(xΔ), where xΔ denotes the pseudoΔ-derivative.To obtain the existence result for our problem it is necessary to define a notion of a solution.Definition 3.2.
A functionx:Ia→E is said to be a pseudosolution of the problem (1.1) if it satisfies the following conditions: (1)
x(·) is ACG* function,(2)
x(0)=x0,(3)
for eachx*∈E* there exists a set A(x*) with μΔ measure zero, such that for each t∉A(x*),
(3.3)(x*x)Δ(t)=x*(f(t,x(t),∫0tk(t,s,x(s))Δs)). A continuous function x:Ia→E is said to be a solution of the problem (3.1) if it satisfies
(3.4)x(t)=x0+∫0tf(z,x(z),∫0zk(z,s,x(s))Δs)Δz,foreveryt∈Ia.Because we consider a new type of integral and a new type of solutions is necessary to prove that each solutionx of problem (1.1) is the solution of problem (3.1).Letx is a continuous solution of (3.1). Fix x*∈E*. By definition, x is ACG* function and x(0)=x0. Since, for each x*∈E* and μΔ a.e t∈Ia, (x*x(t))Δ=x*(xΔ(t))=x*(f(t,x(t),∫0tk(t,s,x(s))Δs)) and the last is Δ-HK integrable, so is differentiable μΔ a.e.Moreover,∫0tx*(f(z,x(z),∫0zk(z,s,x(s)Δs))Δz=∫0t(x*x(s))ΔΔs=x*(x(t)-x0). Thus x(t) satisfies (3.1).Now assume thaty is ACG* function and y(0)=x0. By the definition of Δ-HKP integrals there exists an ACG* function G such that G(0)=x0 and(3.5)x*(GΔ(t))=x*(f(t,y(t),∫0tk(t,s,y(s))Δs))μΔa.e.
Hence(3.6)x*(y(t))=x0+∫0tx*(f(z,y(z),∫0zk(z,s,y(s))Δs)Δz)=x0+∫0tx*(GΔ)(s)Δs=x0+x*(G(t))-x*(G(0))=x*(G(t)).
We obtain y=G,μΔ-a.e. and then yΔ(t)=f(t,y(t),∫0tk(t,s,y(s))Δs).Let(3.7)B={x∈E:∥x∥≤∥x0∥+p,p>0},B̃={x∈(C(Ia,E),ω):x(0)=x0,∥x∥≤∥x0∥+p,p>0}.
Moreover, let F(x)(t)=x0+∫0tf(z,x(z),∫0zk(z,s,x(s))Δs)Δz, t∈Ia, K={F(x):x∈B̃}, K1={∫0zk(z,s,x(s))Δs:z∈It=[0,t]∩T,t∈Ia,x∈B̃}.Theorem 3.3.
Assume, that for each uniformlyACG* function x:Ia→E the functions: k(·,s,x(s)),f(·,x(·),∫0(·)k(·,s,x(s))Δs) are Δ-HKP integrable, f and k are weakly-weakly sequentially continuous functions. Suppose that there exist constants d1,d2,d3>0 such that
(3.8)β(f(I×A×C))≤d1·β(A)+d2·β(C),
for each subset A,C of B,I⊂Ia,
(3.9)β(k(I×I×X))≤d3·β(X)
for each subset X of B,I⊂Ia, where f(I×A×C)={f(t,x1,x2):(t,x1,x2)∈I×A×C},k(I×I×X)={k(t,s,x):(t,s,x)∈I×I×X}.
Moreover, letK and K1 be equicontinuous, equibounded and uniformly ACG* on Ia. Then there exists a pseudosolution of the problem (1.1) on Ic, for some c∈T,0<c≤a and 0<c·d1+c2·d2·d3<1.Proof.
Fix an arbitraryp≥0. Recall, that a set K of continuous functions F(x)∈K defined on a time scale interval Ia is equicontinuous on Ia if for each ɛ>0 there exists δ>0 such that ∥F(x)(t)-F(x)(τ)∥<ɛ for all x∈B̃ whenever |t-τ|<δ,t,τ∈Ia, for each F(x)∈K. Thus, for each ɛ>0 there exists δ>0 such that ∥∫τtf(z,x(z),∫0zk(z,s,x(s))Δs)Δz∥<ɛ for all x∈B̃ whenever |t-τ|<δ and t,τ∈Ia. As a result, there exists a number c, 0<c≤a such that
(3.10)∥∫0tf(z,x(z),∫0zk(z,s,x(s))Δs)Δz∥≤p,t∈Ic,x∈B̃.
We will show that the operator F is well defined and maps B̃ into B̃.
To see this note for anyx*∈E*, such that ∥x*∥≤1, for any x∈B̃ and t∈Ic we have(3.11)|x*F(x)(t)|≤|x*x0|+|x*∫0tf(z,x(z),∫0zk(z,s,x(s))ds)Δz|≤∥x*∥∥x0∥+∥x*∥∥∫0tf(z,x(z),∫0zk(z,s,x(s))ds)Δz∥≤∥x0∥+p,
so
(3.12)sup{|x*F(x)(t)|:x*∈E*,∥x*∥≤1}≤∥x0∥+p
and as a result
(3.13)∥F(x)(t)∥≤∥x0∥+p.
Thus F(x)(t)∈B̃.
We will show, that the operatorF is weakly-weakly sequentially continuous. By Lemma 9 of [34] a sequence xn(·) is weakly convergent in C(Ic,E) to x(·) if and only if xn(t) tends weakly to x(t) for each t∈Ic, so if xn→ωx in C(Ic,E) then k(t,s,xn(s))→ωk(t,s,x(s)) in E for t∈Ic and by Theorem 2.14 (see our assumptions on K1) we have limn→∞∫0tk(z,s,xn(s))ds=∫0tk(z,s,x(s))ds weakly in E, for each t∈Ic. Moreover, because f is weakly-weakly sequentially continuous, we have(3.14)f(t,xn(t),∫0tk(t,s,xn(s))Δs)→ωf(t,x(t),∫0tk(t,s,x(s))Δs)
in E, for each t∈Ic. Thus Theorem 2.14 (see our assumptions on K) implies F(xn)(t)→F(x)(t) weakly in E for each t∈Ic so Lemma 9 of [34] guarantees that F(xn)→F(x) in C(Ic,E) with its weak topology.
Suppose thatV⊂B̃ satisfies the condition V¯=conv¯({x}∪F(V)). We will prove that V is relatively weakly compact and so (2.1) is satisfied. Since V⊂B̃,F(V)⊂K. Then V⊂V¯=conv¯({x}∪F(V)) is equicontinuous. By Lemma 2.17, t↦v(t)=β(V(t)) is continuous on Ic.
For fixedt∈Ic we divide the interval [0,t] into m parts in the following way t0=0,(3.15)t1=sups∈Ia{s:s≥t0,s-t0<δ},t2=sups∈Ia{s:s≥t1,s-t1<δ},⋮tn=sups∈Ia{s:s≥tn-1,s-tn-1<δ}.
Since T is closed, we have ti∈Ia. If some ti+1=ti then ti+2=inf{t∈T:t≥ti+1}.
For fixedz∈[0,t] we divide the interval [0,z] into m parts: z0=0,(3.16)z1=sups∈[0,t]{s:s≥z0,s-z0<δ},z2=sups∈[0,t]{s:s≥z1,s-z1<δ},⋮zn=sups∈[0,t]{s:s≥zn-1,s-zn-1<δ}suchthatμΔ(Ij)=jzm,j=0,1,…,m, Ij=[zj,zj+1].
LetV([zj,zj+1])={u(s):u∈V,zj≤s≤zj+1,j=0,1,…,m-1}. By Lemma 2.17 and the continuity of v there exists sj∈Ij=[zj,zj+1], such that(3.17)β(V([zj,zj+1]))=sup{β(V(s)):zj≤s≤zj+1}:=v(sj).
By Theorem 2.13 and the properties of the Δ-HKP integral we have for x∈V that
(3.18)F(x)(t)=x0+∑i=0m-1∫Jif(z,x(z),∑j=0m-1∫Ijk(z,s,x(s))Δs)Δz∈x0+∑i=0m-1μΔ(Ji)conv¯f(Ji,V(Ji),∑j=0m-1(zj+1-zj)conv¯k(Ij,Ij,V(Ij))),
where Ji=[ti,ti+1], i=0,1,…,m-1.
Using (3.8), (3.9) and properties of the measure of noncompactness we obtain(3.19)β(F(V)(t))≤∑i=0m-1μΔ(Ji)β(f(Ji,V(Ji),∑j=0m-1μΔ(Ij)conv¯k(Ij,Ij,V(Ij))))≤∑i=0m-1μΔ(Ji)β(f(Ji,V(Ji),∑j=0m-1μΔ(Ij)conv¯k(Ij,Ij,V(Ij))))≤∑i=1m-1μΔ(Ji)·d1·β(V(Ji))+∑i=1m-1μΔ(Ji)·d2·β(∑j=0m-1μΔ(Ij)·conv¯k(Ij,Ij,V(Ij)))≤∑i=1m-1μΔ(Ji)·d1·β(V(Ic))+∑i=1m-1μΔ(Ji)·d2·∑j=0m-1μΔ(Ij)·β(k(Ij,Ij,V(Ij)))≤β(V(Ic))·d1·c+∑i=1m-1μΔ(Ji)·d2·∑j=0m-1μΔ(Ij)·d3·β(V(Ij))≤β(V(Ic))·d1·c+β(V(Ic))·d2·d3·c2=β(V(Ic))(d1·c+d2·d3·c2).
Since V¯=conv¯({x}∪F(V)), β(V(t))=β(conv¯({x}∪F(V(t)))) so β(V(t))≤β(V(Ic))(d1·c+d2·d3·c2), for t∈Ic.
Using Lemma2.17 we obtain(3.20)β(V(Ic))≤β(V(Ic))(d1·c+d2·d3·c2).
Since 0<c·d1+c2·d2·d3<1 we obtain v(t)=β(V(t))=0 for t∈Ic.Using Ascoli`s theorem we have thatV is relatively weakly compact.By Theorem2.3 the operator F has a fixed point. This means that there exists a pseudosolution of the problem (1.1).Theorem 3.4.
Assume, that for each uniformlyACG* function x:Ia→E, the functions k(·,s,x(s)),f(·,x(·),∫0(·)k(·,s,x(s))ds) are Δ-HKP integrable and k,f are weakly-weakly sequentially continuous. Suppose that there exists a constant d>0 and a continuous function c1:Ia→R+ such that
(3.21)β(f(I,A,C))≤d·β(C),foreachA,C⊂B,I⊂Ia,β(k(I,I,X))≤sups∈Ic1(s)β(X),foreachX⊂B,I⊂Ia,
where
(3.22)f(I,A,C)={f(t,x1,x2):(t,x1,x2)∈I×A×C},k(I,I,X)={k(t,s,x):(t,s,x)∈I×I×X}.
Moreover, let K and K1 be equicontinuous and uniformly ACG* on Ia. Then there exists a pseudo solution of the problem (1.1) on Ic, for some c∈T,0<c≤a.Proof.
The first part of the proof is the same as that of the proof of the previous theorem. It remains to show the relative compactness ofV, where V is defined in Theorem 3.3. In this case notice for t∈Ic and zj as in Theorem 3.3 we have
(3.23)β(V(t))≤∑i=0m-1μΔ(Ji)·d·β(∑j=0m-1μΔ(Ij)·conv¯k(Ij,Ij,V(Ij)))≤∑i=0m-1μΔ(Ji)·d·∑j=0m-1μΔ(Ij)·β(k(Ij,Ij,V(Ij)))≤∑i=0m-1μΔ(Ji)·d·∑j=0m-1μΔ(Ij)·sups∈Ijc1(s)β(V(Ij)).
Let sups∈Ijc1(s)=c1(pj), pj∈Ij. Then
(3.24)β(V(t))≤c·d·∑j=0m-1μΔ(Ij)·c1(pj)v(sj)=c·d(∑j=0m-1μΔ(Ij)·c1(pj)v(pj)+∑j=0m-1μΔ(Ij)(c1(pj)(v(sj)-v(pj)))).
Fix ɛ>0. From the equicontinuity of V we may choose m large enough so that v(sj)-v(pj)<ɛ and so
(3.25)β(V(t))≤c·d(∑j=0m-1μΔ(Ij)·c1(pj)v(pj)+∑j=0m-1zmc1(pj)·ɛ)≤c·d(∑j-0m-1μΔ(Ij)c1(pj)v(pj)+z·ɛ·max0≤k≤m-1c1(pk)).
Since ɛ>0 is arbitrary small and z·max0≤k≤m-1c1(pk) is bounded, we have that z·ɛ·max0≤k≤m-1c1(pk) is arbitrary small. Therefore
(3.26)v(t)=β(V(t))≤c·d·∫0tc1(s)v(s)Δs,t∈Ic.
Thus, by Gronwall's inequality for Δ-integrals [5] we have that
(3.27)v(t)=β(V(t))=0,fort∈Ic.
Using Ascoli`s theorem we deduce that V is relatively weakly compact.By Theorem2.14 the operator F has a fixed point. This means that there exists a pseudo solution of the problem (1.1).Remark 3.5.
The conditions in Theorems3.3 and 3.4 can be also generalized to the Sadovskii condition [28], the Szufla condition [49], and the others and β can be replaced by some axiomatic measure of weak noncompactness.



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*Source: 836347-2010-07-27.xml*