On -Solutions of Some Nonlinear Integral Equations on the Plane

Abstract

In this paper, we define the space of functions -bounded variation on the plane and endow it with a norm under which it is a Banach space. In addition, we study some nonlinear integral equations and providing conditions for the functions and kernel involved in such equations under which we guarantee the existence and uniqueness in the space of functions of bounded variation in the sense of Shiba on the plane, .

1. Introduction

The integral equations that have often been the subject of intensive study are those of Volterra and Hammerstein. These equations are used as mathematical models of multiple physical phenomena such as the behavior of electromagnetic fluids; furthermore, the solution of some boundary value problems for partial differential equations is usually expressed as the solution of a Hammerstein integral equation. On the other hand, the functions of bounded variation are adapted to the study of parameter identification problems such as coefficients of an elliptical or parabolic operator. These functions are also useful for studying image recovery problems. One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere and are frequently used to define generalized solutions to nonlinear problems involving functionals, and partial differential equations in mathematics, physics, and engineering. In recent decades, the solutions of this type of integral equations have been studied by various authors in various spaces of bounded variation, for example, in the space of the functions of bounded variation in the Jordan sense and in the Waterman sense, see [1, 2], in addition to other generalized spaces of bounded variation, some of these have been studied in [3–16].

In this article, we study this type of nonlinear equations in the space of functions of bounded variation in the Shiba sense in the plane; first, we prove that this space endowed with a norm is a Banach space and we verify the existence and uniqueness of the solutions of these equations. For this, we use Banach’s fixed point theorem, which is a very useful tool to demonstrate the existence of solutions of differential equations in the analysis of dynamic systems, and even in the study of iterative methods in numerical calculation.

In [16] Aziz, Leiva, and Merentes studied the solutions for the nonlinear Hammerstein equation and for the Volterra-Hammerstein integral equation in the space of functions of bounded variation, , on the plane, which are defined by the following:

and

respectively, where is Lebesgue integrable and , . Under the following hypotheses.(i) is a function of two variables in the space of functions of bounded variation, .(ii)H₂ is a locally Lipschitz function.(iii)H₃ is a function such that almost everywhere, , is a Lebesgue integrable function and is L-integrable for every ,moreover, together with some additional hypotheses, they showed that there exists a number such that for every , with , the (1) has a unique solution in , defined on .

This research work is motivated by the paper [16]. Here, we consider the Hammerstein equation (1) and the Volterra–Hammerstein integral (2); under certain hypotheses, the solutions of these equations are studied in the classes of functions of bounded variation in the sense of Shiba on the plane , with . This paper is structured as follows: Section 2 focuses on preliminaries, which gives a round-up of the necessary results for proofs of the main theorems. Section 3 contains the existence and uniqueness results for the Hammerstein equation, Volterra–Hammerstein, and Volterra integral equation. In Section 4 an application of the main results is presented and finally, the conclusions.

2. Preliminaries

This section is based on the preliminary results that are fundamentals for the development of the main theorems.

From now on, we will introduce the following hypotheses to study the solutions of the nonlinear integral (1) and (2):(i) is a function in .(ii) is a locally Lipschitz function.(iii) is a function such that almost everywhere,  ,  is a integrable function and is Lebesgue integrable for all .

Below, we present a summary of definitions, theorems, lemmas, and remarks that are used in the proofs of the main theorems, and are stated here so the paper is self-contained.

Theorem 1 {Banach Contraction Principle}.

Let be a complete metric space and be a contraction mapping, then has a unique fixed point. Even more, let be a complete metric space and be a closed subset such that . If be a contraction mapping, then has a unique fixed point in .

Definition 1. Let be the class of all nondecreasing sequences , of positive numbers such that diverges. Let where and with , such that , and , . For and ,(1)Fix , and let . The partial -variation of in the sense of Shiba on is defined as follows: where is a sequence of non-overlapping intervals and .(2)Fix , and let . The partial -variation of in the sense of Shiba on is defined by the following: where is a sequence of non-overlapping intervals and .(3)The Hardy–Vitali variation of the function in the sense of Shiba is defined by the following: where , are sequences of nonoverlapping intervals and , with(4)The total -variation of the function in the sense of Shiba is defined as follows

Observation 1. Clearly, , , are non-negative and hence, too.

Observation 2. We denote by the set of functions with finite total -variation in the sense of Shiba, that is,The next theorem asserts that the set of functions is a vector space, the proof is consequence of the proven properties in [17].

Theorem 2. Let and in , with , and let be any real constant. Then,(1).(2).

Lemma 1. Let be a function in the space . is a constant function if and only if .

Proof. Suppose is constant. By Definition 1, it is fulfilled that .
Suppose now that . From Observation 1, it follows thatLet and be sequences of nonoverlapping intervals and , with and , thenIn particular, for the sequences of nonoverlapping intervals, and of and , respectively, we have as follows, by (9) thatAnalogously, for any partition of non-overlapping intervals and , we have the following:In particular, it holds for the partition considered above, i.e., we obtain the following:Finally, substituting (11) and (12) into (14), we conclude that for all . Thus, is constant on . □

Definition 2. Let , where . The function is defined in the space as given by the following:

Theorem 3. Let , where . The function defined above is a norm on the space .

Proof. It is easy to verify that satisfies the properties of a norm on the space .

Lemma 2. Let and , let be a closed rectangle in , and let be a function. If , then(a)there exists a such that , for all .(b) for all ,(c)there exists such that , for all .

Proof. (a)Suppose . For and , we have the following: Let us compute the value of , By the inequalities (16), (17), and (18), we have the following:(b)Let and . Let us consider a sequence of nonoverlapping intervals , and be a sequence of nonoverlapping intervals . By adding to the sequence of nonoverlapping intervals and to the sequence of nonoverlapping intervals , we obtain that the following inequality holds Raising to the power yields      and analogously, Let us consider the sequences of nonoverlapping intervals and of and , respectively, and by an argument analogous to that above, we can add and to nonoverlapping interval sequences and , respectively, to get the sequences of nonoverlapping intervals of and . Then, Raising to the power yields. Hence, Therefore, for all .(c)It follows from Part (a) and (b) that there exists a such thatwith . Thus,Taking supremum over all leads to the following:

Lemma 3. Let , and be a closed rectangle in , and let be a function. If , then is bounded.

Proof. The proof is an immediate consequence of part (c) of Lemma 2.

Theorem 4. Let be a closed rectangle in . Then, is a Banach space.

Proof. Theorem 3 shows that is a norm on the space . We need to prove that is a complete space. Suppose is a Cauchy sequence in , which is to say that there exists such thatThat is,Thus,Since , then for every . It follows from Part (c) of Lemma 2 that there exists such that for all , so that the sequence is Cauchy uniformly on . Because this space is complete, there exists a function with such that uniformly on . To see that , let us consider the sequences of non-overlapping intervals and of and , respectively, where and , thenNow, we know thatTherefore,Taking supremum over , we have the following:For every . SimilarlyBy the inequalities (34) and (35), for every we have the following:Thus, for every . Since and is a vector space, . Now, we can conclude that converges to in the norm . Indeed, for every ,On the other hand, by the inequality (29),So, fixing and taking limit respect to , we have as follows:Using (36) and (39) into (37)Therefore, is a Banach space.

Lemma 4. Suppose the hypotheses and hold. Let the integral function be defined by for all and all , and let be the area of the rectangle . Then,

Proof. Let be as in hypothesis , and , then is measurable and bounded in , hence is Lebesgue integrable. By the hypothesis , is Lebesgue integrable for all . Thus, is well defined. Let and be sequences of nonoverlapping intervals of and , respectively, where and . We know that the following:Let us study each of these variations separately.Because , the function is convex on . Moreover, by the hypothesis, is Lebesgue integrable and also is bounded, and sois a Lebesgue integrable. Applying Jensen’s inequality yields as follows:Raising both sides of the inequality above to the power yields as follows:Taking supremum over in the inequality above, and using the hypothesis , i.e., that where is integrable, we obtain the following:Analogously, it is shown thatNow, we compute :where . Applying Jensen’s inequality, we obtain the following:Raising both sides of the inequality to the power yields the following:Taking supremum over , and by the hypothesis , we have the following:It follows then from the inequalities (47), (48), and (52) thatwhich completes the proof.

Lemma 5. Suppose the hypotheses and hold. Let be the integral function for all and all , defined as in the preceding lemma. Then, for all ,

where is guaranteed by part (c) of Lemma 2, is the area of the rectangle , and is the Lipschitz constant associated with restricted to the interval .

Proof. Consider ,  sequences of nonoverlapping intervals of and  , respectively, where and . Let and let such that . denotes the Lipschitz constant of corresponding to the cube . We know thatLet us study again each of these variations separately. To calculate the variation we can proceed similarly as in the proof of Lemma 4. So, we obtain the following: