On <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-7.2259pt" id="M1" height="18.6253pt" version="1.1" viewBox="-0.0657574 -11.3994 44.8644 18.6253" width="44.8644pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-140" d="M671 0V28C608 36 593 47 562 130C495 309 432 486 368 665L336 656C270 481 202 311 134 139C98 47 84 39 20 28V0H241V28C167 38 156 47 178 115C217 236 276 383 327 530H329C375 398 437 223 484 92C499 47 489 37 424 28V0H671Z"/></g><g transform="matrix(.012,0,0,-0.012,14.26,4.134)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g><g transform="matrix(.017,0,0,-0.017,22.115,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.017,0,0,-0.017,32.428,0)"><path id="g113-87" d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"/></g></svg>-</span>Solutions of Some Nonlinear Integral Equations on the Plane

Ereú, Jurancy; Pérez, Liliana; Rodríguez, Luz

doi:https://doi.org/10.1155/2022/5482688

International Journal of Differential Equations

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5482688 | https://doi.org/10.1155/2022/5482688

On -Solutions of Some Nonlinear Integral Equations on the Plane

Jurancy Ereú,¹Liliana Pérez,²and Luz Rodríguez³

Academic Editor: Jaume Giné

Received12 Aug 2021

Revised21 Feb 2022

Accepted31 Mar 2022

Published28 May 2022

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:Raising to the power , by part (c) of Lemma 2, and by the hypothesis , we have the following:Taking supremum over ,It is shown analogously as follows:Let us then computewhereNote thatwithBy Jensen’s inequality, and by the fact that f is locally Lipschitz, we have the following:By raising both sides of the inequality above to the power , and by part (c) of Lemma 2,Taking supremum over , and by the hypothesis , we obtain as follows:So, we conclude from the inequalities (58) and (59), and (66) thatand the proof is complete.
The following lemma is a direct consequence of the triangle inequality.

Lemma 6. Suppose the hypotheses , , hold. Let be the function defined by , where is as in Lemma 4 and where is a bounded interval. Then,

Lemma 7. Let , and let be a function with , and . Then, forwe have thatwhere

Proof. Let , where and sequences of nonoverlapping intervals of and , respectively, where and . Let us consider . We know thatFirst, we determine . Since is a sequence of nonoverlapping intervals , let and for some , with . Let us consider , then,By raising to the power , taking supremum over , and applying the property if , we obtain as follows:It is shown analogously thatTo compute , let for some with , and let be an arbitrary fixed point. SoNow, we can display the terms of summation and substitute the values of according to (70) to obtain as follows:Raising to the power of , taking supremum over , and applying the property , we have as follows:It follows from (75) and (76), and (78) thattakingyieldswhich completes the proof.

3. Study of Solutions of Integral Equations

In this section, we prove the main theorems of this work, which guarantee the existence and uniqueness of the solutions of (1) and (2) in the space of functions of bounded variation in the Shiba sense in the plane. In addition to the hypotheses considered in Section 2, let us consider the following hypotheses:(i) Let , and is a function such that

almost everywhere, , where is a integrable function and is Lebesgue integrable over for every .

Theorem 5. Suppose the hypotheses , , and hold. Then, there exists a number such that for every , equation (1) has a unique solution in defined on .

Proof. Let , where and be sequences of nonoverlapping intervals of and , respectively, where and . Let such that , and denotes the Lipschitz constant of corresponding to the cube . denotes the closed ball with center 0 and radius in the space , that is,Using part (c) of Lemma 2, choose a number such thatandwhere is guaranteed by part (c) of Lemma 2, and is the area of the rectangle . In order to prove this theorem, the Banach contraction principle is needed (Theorem 1). Fix and define the operator 4 byIt can be verified that it is well defined, as in the proof of Lemma 4. We show first that . Indeed, for every , it follows from Lemma 6 thatFor the second term in the right hand side of the inequality above, we have as followsBy Lemma 4, the inequality (88) can be rewritten as follows:Substituting the inequality (89) into (87) yields the following:Therefore, . To show now that is a contraction mapping, let , that is, . So,It is straightforward to compute thatBy the fact that is locally Lipschitz, and by part (c) of Lemma 2, we have the following:Lemma 5 implies as follows:Moreover, it follows from the inequality (94) thatThus, is a contraction mapping, by the inequality (86). Theorem 1 implies that has a unique fixed point in , which is to say that there exists a unique such thatTherefore, is a unique solution of (1).

Theorem 6. Suppose the hypotheses , , and hold. Then, there exists a rectangle such that the equation (2) has a unique solution in .

Proof. Let and be denoted as in the proof of the preceding theorem. Let us denote with and , the closed ball with center 0 and radius in the space , that is,Choose with such thatwhere the existence of are assured by part (c) of Lemmas 2 and 7, respectively, and is the area of the rectangle . Let , where and are sequences of nonoverlapping intervals of and , respectively, with and . We define the functions and the operator by, whereWith , it can be verified that it is well defined, as in the proof of Lemma 4.
As in the proof of Theorem 5, we will use the Banach contraction principle (Theorem 1) to prove this theorem. We begin by showing that . So, let ,It can easily be shown thatThus,However,In addition,To compute , we proceed as in the proof of Lemma 4. Let , for some , with and let us consider , . Then,Raising both sides of the inequality above to the power yields the following:Now,By taking supremum over in the inequality (107), by Lemma 7 and the Hypothesis , we conclude as follows:It is shown analogously as follows:andBy the inequalities (109)–(111), we obtain the following:Substituting the inequality (113) into (104) leads to the following:Next, we show that is a contraction mapping. For , we have as follows:Now we compute the variation . Let and for some , then, for By part (c) of Lemma 2, the inequality (108), the Hypothesis , and raising both sides of the inequality above to the power , we obtain the following:An argument analogous to the one we used above shows the following:Therefore,The inequality (98) implies that is a contraction mapping; and by Theorem 1, has a unique fixed point in , that is, there exists a unique such thatHence, is a unique solution of the (2).

4. An Application

The field of integral and integrodifferential equations is an important subject in many subfields of applied mathematics since the mathematical formulation of physical phenomena, as well as dynamical models for chemical reactors or fluid dynamics contain integrodifferential equations, which are transformed specifically into Hammerstein–Volterra equations. As just a few of these equations can be solved explicitly, it is often necessary to use numerical methods to find those solutions, as such it has been studied in [18–22] and [23]. However, it should be pointed out that in most of these papers the solutions of the studied equations are found, but are not guaranteed to be unique. The next example shows the existence and uniqueness of a Hammerstein–Volterra equation defined on the plane, in which are verified each of the assumptions of Theorem 6.

Example 1. Let , and let the functions , and be defined by , and , respectively. We want to show that the nonlinear Hammerstein–Volterra integral equationhas a unique solution .
In order to solve this problem, we verify the hypotheses of Theorem 6.(1) is measurable in , to show that belongs to , let , where and be sequences of nonoverlapping intervals of and , respectively, where and . To determine , and , let us first compute From Lemma 5 in [24] we have the following: Hence, Thus, . An argument entirely analogous to that above leads to . On the other hand, Now, using the inequality , the mean value theorem, and the inequality (122) to obtain Then, Thus, and in consequence, .(2)We need to show now that is locally a Lipschitz on . Indeed, let , , and . Then So there exists such that Moreover, therefore is locally a Lipschitz constant on .Hence, is Lebesgue integrable. On the other hand,We proceed as in Part 1 of this example to show that , and so it is evident that is integrable. Thus, all the conditions of Theorem 6 are satisfied, therefore, there exists a rectangle such that the (120) has a unique solution in .

5. Conclusion

In this paper, we proved existence-uniqueness theorems for nonlinear Hammerstein, Hammerstein–Volterra, and Volterra integral equations in the space of functions of bounded variation in the sense of Shiba on the plane. In addition, we showed that this space is a Banach space with the norm . For the proofs of the main theorems, the Banach contraction principle was used. We also presented an application problem of some of the previously proven theorems. We hope the ideas and techniques used in this paper may be an inspiration to readers that are interested in studying these nonlinear integral equations in new spaces of generalized bounded variation, and that these results may be also a contribution to different areas whose applications are modeled by this type of integral equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of the paper.

Acknowledgments

The authors are very grateful to the referees for their helpful comments and constructive suggestions which enabled the authors to improve the first version of this manuscript.

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Copyright © 2022 Jurancy Ereú et al. 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.

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