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Abstract and Applied Analysis
Volume 2010, Article ID 836347, 17 pages
http://dx.doi.org/10.1155/2010/836347
Research Article

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

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

Received 11 February 2010; Accepted 17 June 2010

Academic Editor: H. Bevan Thompson

Copyright © 2010 Aneta Sikorska-Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove existence theorems for integro-differential equations , , , , where denotes a time scale (nonempty closed subset of real numbers ), and is a time scale interval. The functions are weakly-weakly sequentially continuous with values in a Banach space , 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 and satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.

1. Introduction

A time scale is a nonempty closed subset of real numbers , with the subspace topology inherited from the standard topology of .

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 , , , where .

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 [311]. 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 [1221], 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], -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 where , denotes a time scale, denotes a time scale interval, is a Banach space, and denotes the pseudo-derivative of .

We use a new (Henstock-Kurzweil-Pettis--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 implies the existence of such integrals over all subintervals of but not for all measurable subsets of this interval, so the theory of such integrals on does not follows from general theory on .

In [29] Cichoń introduce a definition of the Henstock-Kurzweil delta integral (HK--integral) and HL delta integral (HL--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 [3840]. 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 for -difference equations.)

Our result extends the existence of pseudosolutions not only to the discrete intervals with uniform step size (Z) but also to the discrete intervals with nonuniform step size (.

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 . For example: (1)for , we have , , and , and (1.1) becomes the Cauchy integrodifferential equation: (2)for , we have , , and , and (1.1) becomes the Cauchy problem:

We assume that the functions 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 be a Banach space and let be the dual space. Moreover, let denote the space of all continuous functions from to endowed with the topology and denotes the space of all -continuous functions from the time scale interval to .

By we denote the Lebesgue measure on . 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 family of functions is said to be uniformly absolutely continuous in the restricted sense on or in short uniformly if for every there exists such that for every in and for every finite or infinite sequence of nonoverlapping intervals with and satisfying , one has where denotes the oscillation of over (i.e., ).

A family of functions is said to be uniformly generalized absolutely continuous in the restricted sense on or uniformly if is the union of a sequence of closed sets such that on each the function is uniformly .

Definition 2.2. A function is said to be weakly continuous if it is continuous from to endowed with its weak topology. A function , where and are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent sequence , the sequence is weakly convergent in . If a sequence tends weakly to in we will denote it by .

Theorem 2.3 (see [42]). Let be a metrizable locally convex topological vector space. Let be a closed convex subset of , and let be a weakly-weakly sequentially continuous map of into itself. If for some the implication holds for every subset of , then 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 [35, 711] and references therein).

A time scale is a nonempty closed subset of real numbers , with the subspace topology inherited from the standard topology of . If are points in , then a time scale interval we denote by and . Other types of intervals are approached similarly. By a subinterval of we mean the time scale subinterval.

Definition 2.4. The forward jump operator and the backward jump operator are defined by and , respectively. We put (i.e., if has a maximum and (i.e., if has a minimum . The jump operators and allow the classification of points in time scale in the following way: is called right dense, right scattered, left dense, left scattered, dense and isolated if and , respectively.

Definition 2.5. One says that is right-dense continuous (rd-continuous) if is continuous at every right-dense point and exists and is finite at every left-dense point .

Definition 2.6. Fix . Let . Then one defines the -derivative by

Remark 2.7. The -derivative satisfies (1) is the usual derivative if (2) is the usual forward difference operator if ,(3) is the -derivative if . 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 space .

We will use the notation , where is called the graininess function and , where is called the left-graininess function.

We say that is a -gauge for time scale interval provided on , on , , and for all .

We say that a partition for a time scale interval given by with for and is -fine if

Definition 2.8 (see [29]). A function is Henstock-Kurzweil--integrable on -HK integrable in short) if there exists a function , defined on the subintervals of , satisfying the following property: given there exists a positive function on such that is -fine division of a , one has

Definition 2.9 (see [29]). A function is Henstock-Lebesgue--integrable on -HL integrable in short) if there exists a function , defined on the subintervals of , satisfying the following property: given there exists a positive function on such that is -fine division of a , one has

Remark 2.10. We note that, by the triangle inequality, if is -HL integrable it is also -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 function is Pettis--integrable ( integrable for short) if (1) is Lebesgue--integrable on (2)

Now, we present a new definition of the integral on time scales which is a generalization for both the Pettis--integral and the Henstock-Kurzweil--integral.

Definition 2.12 (see [29]). The function is Henstock-Kurzweil-Pettis--integrable (--integrable for short) if (1) is Henstock-Kurzweil--integrable on (2)The function will be called a primitive of and by we will denote the Henstock-Kurzweil-Pettis--integral of on the interval .

In [29] the authors give examples of Henstock-Kurzweil-Pettis--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 each -subinterval , if the integral exists, then one has where denotes the close convex hull of the set .

Theorem 2.14 (see [45]). Let and assume that are -HKP integrable on . Let be a primitive of . If one assumes that: (1) a.e. on (2)for each the family is uniformly on (i.e., weakly uniformly on ), (3)for each the set is equicontinuous on then is -HKP integrable on and tends weakly in to for each

Theorem 2.15 (see [5] Gronwall's inequality). Suppose that and . Then implies that

(III)The De Blasi measure of weak noncompactness is one of our fundamental tool in this paper (see [46]).

Let be a bounded nonempty subset of .

The measure of weak noncompactness is defined by where is a set of weakly compact subsets of and is a norm unit ball in .

Some properties of the measure of weak noncompactness are known [46]: (1)if , then ; (2), where denotes the weak closure of ; (3) if and only if is relatively weakly compact; (4); (5), (; (6); (7), where denotes the convex hull of .

We will need the following lemmas.

Lemma 2.16 (see [47]). Let be bounded subsets of the Banach space If , then where and

The lemma below is an adaptation of the corresponding result of Ambrosetti (see [48]).

Lemma 2.17. Let be a family of strongly equicontinuous functions. Let , for and . Then where denotes the measure of weak noncompactness in and the function is continuous.

Proof. (I) First we prove the equality: .
Since by the first property of the measure of noncompactness and consequently By strong equicontinuity of we deduce that for , there exists such that for and for all .
We divide the interval in the following way: , Since is closed, we have . If some then .
As where , we have
By the properties of the measure of weak noncompactness and Lemma 2.16, we obtain Since the above inequality holds, for any , we have Hence, from (2.13) and (2.18), we conclude that
(II) The proof of the equality is similar to the proof of Lemma of Ambrosetti (see [48]), where we choose points as in part (I) of our proof.
(III) Now we prove that the function is continuous.
Let because , where and the sum is taken in the Minkowski sense.
By the property (vi) of the measure of weak noncompactness, we have This implies that By equicontinuity of we obtain the continuity of .

3. Main Results

Now, we will consider equivalently integral problem where , denotes a time scale, , denotes a time scale interval, is a Banach space, and integrals are taken in the sense of -HKP integrals.

Fix and consider the problem Let us introduce a definition.

Definition 3.1. Let and let . The function is a pseudo-derivative of on if for each the real-valued function is -differentiable almost everywhere on and almost everywhere on .

Regarding the above definition it is clear that the left-hand side of (3.2) can be rewritten to the form , where denotes the pseudo-derivative.

To obtain the existence result for our problem it is necessary to define a notion of a solution.

Definition 3.2. A function is said to be a pseudosolution of the problem (1.1) if it satisfies the following conditions: (1) is function, (2), (3)for each there exists a set with measure zero, such that for each , A continuous function is said to be a solution of the problem (3.1) if it satisfies

Because we consider a new type of integral and a new type of solutions is necessary to prove that each solution of problem (1.1) is the solution of problem (3.1).

Let is a continuous solution of (3.1). Fix . By definition, is function and . Since, for each and a.e , and the last is -HK integrable, so is differentiable a.e.

Moreover, . Thus satisfies (3.1).

Now assume that is function and . By the definition of -HKP integrals there exists an function such that and Hence We obtain -a.e. and then .

Let Moreover, let , , , .

Theorem 3.3. Assume, that for each uniformly function the functions: are -HKP integrable, and are weakly-weakly sequentially continuous functions. Suppose that there exist constants such that for each subset of , for each subset of , where
Moreover, let and be equicontinuous, equibounded and uniformly on . Then there exists a pseudosolution of the problem (1.1) on , for some and

Proof. Fix an arbitrary . Recall, that a set of continuous functions defined on a time scale interval is equicontinuous on if for each there exists such that for all whenever , for each . Thus, for each there exists such that for all whenever and . As a result, there exists a number , such that We will show that the operator is well defined and maps into .
To see this note for any , such that , for any and we have so and as a result Thus .
We will show, that the operator is weakly-weakly sequentially continuous. By Lemma of [34] a sequence is weakly convergent in to if and only if tends weakly to for each , so if in then in for and by Theorem 2.14 (see our assumptions on we have weakly in , for each . Moreover, because is weakly-weakly sequentially continuous, we have in , for each . Thus Theorem 2.14 (see our assumptions on ) implies weakly in for each so Lemma of [34] guarantees that in with its weak topology.
Suppose that satisfies the condition . We will prove that is relatively weakly compact and so (2.1) is satisfied. Since . Then is equicontinuous. By Lemma 2.17, is continuous on .
For fixed we divide the interval into parts in the following way , Since is closed, we have . If some then .
For fixed we divide the interval into parts: ,, .
Let . By Lemma 2.17 and the continuity of there exists , such that By Theorem 2.13 and the properties of the -HKP integral we have for that where , .
Using (3.8), (3.9) and properties of the measure of noncompactness we obtain Since , so , for .
Using Lemma 2.17 we obtain Since we obtain for .

Using Ascoli`s theorem we have that is relatively weakly compact.

By Theorem 2.3 the operator has a fixed point. This means that there exists a pseudosolution of the problem (1.1).

Theorem 3.4. Assume, that for each uniformly function , the functions are -HKP integrable and are weakly-weakly sequentially continuous. Suppose that there exists a constant and a continuous function such that where Moreover, let and be equicontinuous and uniformly on . Then there exists a pseudo solution of the problem (1.1) on , for some

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 of , where is defined in Theorem 3.3. In this case notice for and as in Theorem 3.3 we have Let , . Then Fix . From the equicontinuity of we may choose large enough so that and so Since is arbitrary small and is bounded, we have that is arbitrary small. Therefore Thus, by Gronwall's inequality for -integrals [5] we have that Using Ascoli`s theorem we deduce that is relatively weakly compact.

By Theorem 2.14 the operator has a fixed point. This means that there exists a pseudo solution of the problem (1.1).

Remark 3.5. The conditions in Theorems 3.3 and 3.4 can be also generalized to the Sadovskii condition [28], the Szufla condition [49], and the others and can be replaced by some axiomatic measure of weak noncompactness.

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