# OnΛpBV-Solutions of Some Nonlinear Integral Equations on the Plane

Authors: Jurancy Ereú; Liliana Pérez; Luz Rodríguez
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

In this paper, we define the space of functionsΛp-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, ΛpBVIab,ℝ.


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## Body

## 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, BVIab, on the plane, which are defined by the following:(1)ux=vx+λ∫IabKx,yfy,uydy,λ∈ℝ,x∈Iab=a1,b1×a2,b2.and(2)ux=vx+∫I0xKx,yfy,uydy,x∈I0b=0,b1×0,b2.respectively, wheref is Lebesgue integrable and a=a1,a2, b=b1,b2∈ℝ2. Under the following hypotheses.(i)
H1v:Iab⟶ℝ is a function of two variables in the space of functions of bounded variation, BVIab,ℝ.(ii)
H2f:Iab×ℝ⟶ℝ is a locally Lipschitz function.(iii)
H3K:Iab×Iab⟶ℝ is a function such that TVK·,α:I≤Mα almost everywhere, α=α1,α2∈Iab, M:Iab⟶ℝ is a Lebesgue integrable function and Kβ,·,· is L-integrable for every β=β1,β2∈Iab,moreover, together with some additional hypotheses, they showed that there exists a number ρ>0 such that for every λ, with |λ|<ρ, the (1) has a unique solution in BVIab,ℝ, defined on Iab.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 ΛpBVIab,ℝ, with p≥1. 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)
H^1v:Ia¯b¯⟶ℝ is a function in ΛpBVIab,ℝ.(ii)
H^2f:Ia¯b¯×ℝ⟶ℝ is a locally Lipschitz function.(iii)
H^3K:Ia¯b¯×Ia¯b¯⟶ℝ is a function such that TVΛpK·,·,α≤Mα almost everywhere,  α=α1,α2∈Ia¯b¯,  M:Ia¯b¯⟶ℝ+ is a Lp integrable function and Kβ,·,· is Lebesgue integrable for all β=β1,β2∈Ia¯b¯.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}.LetX be a complete metric space and f:X⟶X be a contraction mapping, then f has a unique fixed point. Even more, let X be a complete metric space and B⊆X be a closed subset such that fB⊆B. If f:B⟶B be a contraction mapping, then f has a unique fixed point in B.Definition 1.
LetL be the class of all nondecreasing sequences Λ′=λn′, n=1,2,… of positive numbers such that ∑n=1+∞1/λn′ diverges. Let Λ=Λ1,Λ2 where Λ1,Λ2∈L and Ia¯b¯=a,b×c,d with a¯=a,c,b¯=b,d∈ℝ2, such that a<b, c<d and I=a,b, J=c,d. For 1≤p<+∞ and f:Ia¯b¯⟶ℝ,(1)
Fixx2∈J=c,d, and let x1,y1⊂a,b. The partial Λp-variation of f in the sense of Shiba on x1,y1 is defined as follows:(3)VΛpx1y1f·,x2=supIi∑i=1mfIi,x2pλi11/p,whereIii=1m is a sequence of non-overlapping intervals Ii=ai,bi⊂x1,y1 and fIi,x2:=fbi,x2−fai,x2.(2)
Fixx1∈I=a,b, and let x2,y2⊂c,d. The partial Λp-variation of f in the sense of Shiba on x2,y2 is defined by the following:(4)VΛpx2y2fx1,·=supJj∑j=1nfx1,Jjpλj21/p,whereJjj=1n is a sequence of non-overlapping intervals Jj=cj,dj⊂x2,y2 and fx1,Jj:=fx1,dj−fx1,cj.(3)
The Hardy–VitaliΛp variation of the function f:Ia¯b¯⟶ℝ in the sense of Shiba is defined by the following:(5)VΛp2f,Ia¯b¯,ℝ=supIi×Jj∑i=1m∑j=1nΔfIi,Jjpλi1λj21/p,whereIii=1m, Jjj=1n are sequences of nonoverlapping intervals Ii=ai,bi⊂a,b and Jj=cj,dj⊂c,d, with(6)ΔfIi,Jj:=fai,cj+fbi,dj−fai,dj−fbi,cj.(4)
The totalΛp-variation of the function f:Ia¯b¯⟶ℝ in the sense of Shiba is defined as follows(7)TVΛpf,Ia¯b¯,ℝ=TVΛpf=VΛpabf·,c+VΛpcdfa,·+VΛp2f,Ia¯b¯,ℝ.Observation 1.
Clearly,VΛp2f,Ia¯b¯,ℝ, VΛpf·,x2, VΛpx2y2fx1,· are non-negative and hence, TVΛpf too.Observation 2.
We denote byΛpBVIa¯b¯,ℝ=ΛpBVIa¯b¯ the set of functions with finite total Λp-variation in the sense of Shiba, that is,(8)ΛpBVIa¯b¯,ℝ=f:Ia¯b¯→ℝsuch thatfis measurable andTVΛpf,Ia¯b¯,ℝ<+∞.
The next theorem asserts that the set of functionsΛpBVIa¯b¯,ℝ is a vector space, the proof is consequence of the proven properties in [17].Theorem 2.
Letf and g in ΛpBVIa¯b¯, with 1≤p<+∞, and let α be any real constant. Then,(1)
TVΛpαf=αTVΛpf.(2)
TVΛpf+g≤TVΛpf+TVΛpg.Lemma 1.
Letf:Ia¯b¯⟶ℝ be a function in the space ΛpBVIa¯b¯,ℝ. f is a constant function if and only if TVΛpf,Ia¯b¯,ℝ=0.Proof.
Supposef is constant. By Definition 1, it is fulfilled that TVΛpf,Ia¯b¯,ℝ=0.
Suppose now thatTVΛpf,Ia¯b¯,ℝ=0. From Observation 1, it follows that(9)VΛpabf·,c=VΛpcdfa,·=VΛ2f,Ia¯b¯,ℝ=0.
LetIii=1m and Jjj=1n be sequences of nonoverlapping intervals Ii=ai,bi⊂a,b and Jj=cj,dj⊂c,d, with i=1,…,m and j=1,…,n, then(10)∑i=1mfbi,c−fai,cpλi11/p=0,∑j=1nfa,dj−fa,cjpλj21/p=0.
In particular, for the sequences of nonoverlapping intervals,Iii=12=a,x,x,b and Jjj=12=c,y,y,d of a,b and c,d, respectively, we have as follows, by (9) that(11)fa,c=fx,c=fb,cfor allx∈a,b,(12)fa,c=fa,y=fa,dfor ally∈c,d.
Analogously, for any partition of non-overlapping intervalsIi=ai,bi⊂a,b and Jj=cj,dj⊂c,d, we have the following:(13)∑i=1m∑j=1nfai,cj+fbi,dj−fai,dj−fbi,cjpλi1λj21/p=0.
In particular, it holds for the partition considered above, i.e., we obtain the following:(14)fa,c+fx,y−fa,y−fx,c=0.
Finally, substituting (11) and (12) into (14), we conclude that fx,y=fa,c for all x,y∈Ia¯b¯. Thus, f is constant on Ia¯b¯. □Definition 2.
Letf∈ΛpBVIa¯b¯, where 1≤p<+∞. The function ⋅ΛpBVIa¯b¯ is defined in the space ΛpBVIa¯b¯ as ⋅ΛpBVIa¯b¯:ΛpBVIa¯b¯⟶ℝ given by the following:(15)fΛpBVIa¯b¯=fa¯+TVΛpf,Ia¯b¯,ℝ.Theorem 3.
Letf∈ΛpBVIa¯b¯, where 1≤p<+∞. The function ⋅ΛpBVIa¯b¯ defined above is a norm on the space ΛpBVIa¯b¯.Proof.
It is easy to verify that⋅ΛpBVIa¯b¯ satisfies the properties of a norm on the space ΛpBVIa¯b¯.Lemma 2.
Leta¯=a,c and b¯=b,d, let Ia¯b¯=a,b×c,d be a closed rectangle in ℝ2, and let f:Ia¯b¯⟶ℝ be a function. If f∈ΛpBVIa¯b¯, then(a)
there exists aC^>0 such that fa¯−fb¯≤C^⋅TVΛpf,Ia¯b¯, for all f.(b)
TVΛpf,Ia¯x≤TVΛpf,Ia¯b¯ for all x∈Ia¯b¯,(c)
there existsC>0 such that supx∈Ia¯b¯fx≤CfΛpBVIa¯b¯, for all f.Proof.
(a)
Supposef∈ΛpBVIa¯b¯. For I=a,b and J=c,d, we have the following:(16)fb,c−fa,cλ111/p≤VΛpabf·,c,(17)fa,d−fa,cλ121/p≤VΛpcdfa,·,(18)fb,d+fa,c−fa,d−fb,cλ11λ121/p≤VΛp2f,Ia¯b¯.Let us compute the value offb¯−fa¯,(19)fb¯−fa¯=fb,d−fa,c=fb,d−fa,c+fa,c−fa,c+fb,c−fb,c+fa,d−fa,d≤fb,d+fa,c−fa,d−fb,c+fa,d−fa,c+fb,c−fa,c.By the inequalities (16), (17), and (18), we have the following:(20)fb¯−fa¯≤λ11λ121/pVΛp2f,Ia¯b¯+λ121/pVΛpcdfa,·+λ111/pVΛpabf·,c≤maxλ11λ121/p,λ121/p,λ111/pVΛp2f,Ia¯b¯+VΛpcdfa,·+VΛpabf·,c=C^⋅TVΛpf,Ia¯b¯,whereC^=maxλ11λ121/p,λ121/p,λ111/p.(b)
Letx=x1,x2∈Ia¯b¯ and Ia¯x=a,x1×c,x2⊂Ia¯b¯. Let us consider Iii=1m a sequence of nonoverlapping intervals Ii=ai,bi⊂a,x1, and Jjj=1n be a sequence of nonoverlapping intervals Jj=cj,dj⊂c,x2. By adding x1,b to the sequence of nonoverlapping intervals Iii=1m and x2,d to the sequence of nonoverlapping intervals Jjj=1n, we obtain that the following inequality holds(21)∑i=1mfbi,c−fai,cpλi1≤∑i=1mfbi,c−fai,cpλi1+fbi,c−fx1,cpλm+11.Raising to the power1/p yields  VΛpax1f·,c≤VΛpabf·,c    and analogously,(22)VΛpcx2fa,·≤VΛpcdfa,·.Let us consider the sequences of nonoverlapping intervalsIii=1m and Jjj=1n of a,x1 and c,x2, respectively, and by an argument analogous to that above, we can add x1,b and x2,d to nonoverlapping interval sequences Iii=1m and Jjj=1n, respectively, to get the sequences of nonoverlapping intervals of a,b and c,d. Then,(23)∑i=1m∑j=1nΔfIi,Jjpλi1λj2≤∑i=1m∑j=1nΔfIi,Jjpλi1λj2+fb,x2+fx1,c−fb,c−fx1,x2pλm+11λn+12+fb,d+fx1,x2−fb,x2−fx1,dpλm+21λn+22+fa,x2+fx1,d−fa,d−fx1,x2pλm+31λn+32.Raising to the power1/p yieldsVΛp2f,Ia¯x≤VΛp2f,Ia¯b¯.Hence,(24)VΛpax1f·,c+VΛpcx2fa,·+VΛp2f,Ia¯x≤VΛpabf·,c+VΛpcdfa,·+VΛp2f,Ia¯b¯.Therefore,TVΛpf,Ia¯x≤TVΛpf,Ia¯b¯ for all x∈Ia¯b¯.(c)
It follows from Part (a) and (b) that there exists aC^>0 such that(25)fx−fa¯≤C^⋅TVΛpf,Ia¯b¯,
withx∈Ia¯b¯. Thus,(26)fx≤fa¯+C^⋅TVΛpf,Ia¯b¯≤Cfa¯+TVΛpf,Ia¯b¯whereC=max1,C^.
Taking supremum over allx∈Ia¯b¯ leads to the following:(27)supx∈Ia¯b¯fx≤CfΛpBVIa¯b¯.Lemma 3.
Leta¯=a,c, b¯=b,d and Ia¯b¯=a,b×c,d be a closed rectangle in ℝ2, and let f:Ia¯b¯⟶ℝ be a function. If f∈ΛpBVIa¯b¯, then f is bounded.Proof.
The proof is an immediate consequence of part (c) of Lemma2.Theorem 4.
LetIa¯b¯ be a closed rectangle in ℝ2. Then, ΛpBVIa¯b¯,ℝ,⋅ΛpBVIa¯b¯ is a Banach space.Proof.
Theorem 3 shows that⋅ΛpBVIa¯b¯ is a norm on the space ΛpBVIa¯b¯. We need to prove that ΛpBVIa¯b¯ is a complete space. Suppose fss≥1 is a Cauchy sequence in ΛpBVIa¯b¯, which is to say that there exists N>0 such that(28)fs−frΛpBVIa¯b¯<ε6,fors,r≥N.
That is,(29)fs−fra+TVΛpfs−fr,Ia¯b¯<ε6,fors,r≥N.
Thus,(30)TVΛpfs−fr,Ia¯b¯<ε6,for everys,r≥N.
Sincefs∈ΛpBVIa¯b¯, then TVΛpfs,Ia¯b¯<+∞ for every s≥1. It follows from Part (c) of Lemma 2 that there exists C>0 such that fs−fr∞≤Cfs−frΛpBVIa¯b¯ for all s,r≥1, so that the sequence fss≥1 is Cauchy uniformly on Ia¯b¯. Because this space is complete, there exists a function f:Ia¯b¯⟶ℝ with f∞<+∞ such that fs⟶f uniformly on Ia¯b¯. To see that f∈ΛpBVIa¯b¯, let us consider the sequences of non-overlapping intervals Iii=1m and Jjj=1n of a,b and c,d, respectively, where Ii=ai,bi⊂a,b and Jj=cj,dj⊂c,d, then(31)∑i=0mfs−fbi,c−fs−fai,cpλi11/p=∑i=0mfs−limr⟶+∞frbi,c−fs−limr⟶+∞frai,cpλi11/p=limr⟶+∞∑i=0mfs−frbi,c−fs−frai,cpλi11/p.
Now, we know that(32)∑i=0mfs−frbi,c−fs−frai,cpλi11/p≤VΛpabfs−fr≤TVΛpfs−fr,Ia¯b¯<ε6.
Therefore,(33)∑i=0mfs−frbi,c−fs−frai,cpλi11/p=limr⟶+∞∑i=0mfs−frbi,c−fs−frai,cpλi11/p=liminfr⟶+∞∑i=0mfs−frbi,c−fs−frai,cpλi11/p≤ε6.
Taking supremum overIii=1m, we have the following:(34)VΛpabfs−f≤ε6.
For everys≥N. Similarly(35)VΛpcdfs−f≤ε6VΛp2fs−f≤ε6.
By the inequalities (34) and (35), for every s≥N we have the following:(36)TVΛpfs−f≤ε2.
Thus,fs−f∈ΛpBVIa¯b¯ for every s≥N. Since fs∈ΛpBVIa¯b¯ and ΛpBVIa¯b¯ is a vector space, f∈ΛpBVIa¯b¯. Now, we can conclude that fss≥1 converges to f in the norm .ΛpBVIa¯b¯. Indeed, for every s≥N,(37)fs−fΛpBVIa¯b¯=fsa−fa+TVΛpfs−f.
On the other hand, by the inequality (29),(38)fsa−fra<ε6,fors,r≥N.
So, fixings and taking limit respect to r, we have as follows:(39)fsa−fa=fsa−limr⟶+∞fra≤ε6.
Using (36) and (39) into (37)(40)fs−fΛpBVIa¯b¯≤ε6+ε2<ε.
Therefore,ΛpBVIa¯b¯,ℝ,⋅ΛpBVIa¯b¯ is a Banach space.Lemma 4.
Suppose the hypothesesH^2 and H^3 hold. Let the integral function F be defined by Fux=∫Ia¯b¯Kx,yfy,uydy for all u∈ΛpBVIa¯b¯,ℝ and all x∈Ia¯b¯, and let A be the area of the rectangle Ia¯b¯. Then,(41)TVΛpFu≤3Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯Mypdy1/p<+∞.Proof.
Letf:Ia¯b¯×ℝ⟶ℝ be as in hypothesis H^2, u∈ΛpBVIa¯b¯,ℝ and x∈Ia¯b¯, then Hx=fx,ux is measurable and bounded in Ia¯b¯, hence Hx is Lebesgue integrable. By the hypothesis H^3, Kx,· is Lebesgue integrable for all x∈Ia¯b¯. Thus, Fu is well defined. Let Iii=1m and Jjj=1n be sequences of nonoverlapping intervals of a,b and c,d, respectively, where Ii=ai,bi⊂a,b and Jj=cj,dj⊂c,d. We know that the following:(42)TVΛpFu=VΛpabFu·,c+VΛpcdFua,·+VΛp2Fu,Ia¯b¯,ℝ.
Let us study each of these variations separately.(43)∑i=1mFubi,c−Fuai,cpλi1=∑i=1m∫Ia¯b¯Kbi,c,yfy,uydy−∫Ia¯b¯Kai,c,yfy,uydypλi1=∑i=1m∫Ia¯b¯Kbi,c,y−Kai,c,yfy,uydypλi1.
Becausep≥1, the function xp is convex on ℝ. Moreover, by the hypothesis, K is Lebesgue integrable and also f·,u· is bounded, and so(44)Kbi,c,·−Kai,c,·f·,u·
is a Lebesgue integrable. Applying Jensen’s inequality yields as follows:(45)∑i=1mFubi,c−Fuai,cpλi1≤Ap−1∑i=1m∫Ia¯b¯Kbi,c,y−Kai,c,yfy,uypdyλi1≤Ap−1supy¯∈Ia¯b¯fy¯,uy¯p∫Ia¯b¯∑i=1mKbi,c,y−Kai,c,ypλi1dy.
Raising both sides of the inequality above to the power1/p yields as follows:(46)∑i=1mFubi,c−Fuai,cpλi11/p≤Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯∑i=1mKbi,c,y−Kai,c,ypλi1dy1/p.
Taking supremum overIii=1m in the inequality above, and using the hypothesis H^3, i.e., that TVΛpK·,y≤My where M is Lp integrable, we obtain the following:(47)VΛpabFu·,c≤Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯Mypds1/p<+∞.
Analogously, it is shown that(48)VΛpcdFua,·≤Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯Mypds1/p<+∞.
Now, we computeVΛp2Fu,Ia¯b¯,ℝ:(49)∑i=1m∑j=1nFuai,cj+Fubi,dj−Fuai,dj−Fubi,cjpλi1λj2=∑i=1m∑j=1n∫Ia¯b¯ΔKIi,Jj,yfy,uydypλi1λj2,where ΔKIi,Jj,y=Kai,cj,y+Kbi,dj,y−Kai,dj,y−Kbi,cj,y. Applying Jensen’s inequality, we obtain the following:(50)∑i=1m∑j=1nΔFuIi,Jjpλi1λj2≤Ap−1∑i=1m∑j=1n∫Ia¯b¯ΔKIi,Jj,yfy,uypdyλi1λj2≤Ap−1supy¯∈Ia¯b¯fy¯,uy¯p∫Ia¯b¯∑i=1m∑j=1nΔKIi,Jj,ypλi1λj2dy.
Raising both sides of the inequality to the power1/p yields the following:(51)∑i=1m∑j=1nΔFuIi,Jjpλi1λj21/p≤Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯∑i=1m∑j=1nΔKIi,Jj,ypλi1λj2dy1/p.
Taking supremum overIi×Jj, and by the hypothesis H^3, we have the following:(52)VΛp2Fu,Ia¯b¯,ℝ≤Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯Mypds1/p<+∞.
It follows then from the inequalities (47), (48), and (52) that(53)TVΛpFu≤3Ap−1/psupy¯∈Ia¯b¯fy¯,uy¯∫Ia¯b¯Mypdy1/p<+∞.
which completes the proof.Lemma 5.
Suppose the hypothesesH^2 and H^3 hold. Let Fux be the integral function for all u∈ΛpBVIa¯b¯,ℝ and all x∈Ia¯b¯, defined as in the preceding lemma. Then, for all u1,u2∈ΛpBVIa¯b¯,(54)TVΛpλFu1−Fu2≤3Ap−1/pCLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mypdy1/p,λ∈ℝ,whereC is guaranteed by part (c) of Lemma 2, A is the area of the rectangle Ia¯b¯, and Lr is the Lipschitz constant associated with f restricted to the interval Ia¯b¯×−r,r.Proof.
ConsiderIii=1m,  Jjj=1n sequences of nonoverlapping intervals of a,b and  c,d, respectively, where Ii=ai,bi⊂a,b and Jj=cj,dj⊂c,d. Let u1,u2∈ΛpBVIa¯b¯ and let r>0 such that u1∞,u2∞<r. Lr denotes the Lipschitz constant of f corresponding to the cube Ia¯b¯×−r,r. We know that(55)TVΛpλFu1−Fu2=VΛpabλFu1−Fu2·,c+VΛpcdλFu1−Fu2a,·+VΛp2λFu1−Fu2,Ia¯b¯,ℝ.
Let us study again each of these variations separately. To calculate the variationVΛpabλFu1−Fu2·,c we can proceed similarly as in the proof of Lemma 4. So, we obtain the following:(56)∑i=1mλFu1−Fu2bi,c−λFu1−Fu2ai,cpλi1≤Ap−1∑i=1m∫Ia¯b¯λpKbi,c,y−Kai,c,ypfy,u1y−fy,u2ypdyλi1≤Ap−1Lrλsupy¯∈Ia¯b¯u1y¯−u2y¯p∫Ia¯b¯∑i=1mKbi,c,y−Kai,c,ypλi1dy.
Raising to the power1/p, by part (c) of Lemma 2, and by the hypothesis H^3, we have the following:(57)∑i=1mλFu1−Fu2bi,c−λFu1−Fu2ai,cpλi11/p≤Ap−1/pLrλCu1−u2ΛpBVIa¯b¯∫Ia¯b¯∑i=1mKbi,c,y−Kai,c,ypλi1ds1/p≤CAp−1/pLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mspds1/p.
Taking supremum overIii=1m,(58)VΛpabλFu1−Fu2≤CAp−1/pLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mspds1/p.
It is shown analogously as follows:(59)VΛpcdλFu1−Fu2≤CAp−1/pLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mspds1/p.
Let us then compute(60)VΛp2λFu1−Fu2,Ia¯b¯,ℝ=supIi×Jj∑i=1m∑j=1nΔλFu1−Fu2Ii,Jjpλi1λj21/p,where(61)ΔλFu1−Fu2Ii,Jj=λFu1−Fu2ai,cj+λFu1−Fu2bi,dj−λFu1−Fu2ai,dj−λFu1−Fu2bi,cj.
Note that(62)∑i=1m∑j=1nΔλFu1−Fu2Ii,Jjpλi1λj2=∑i=1m∑j=1nλ∫Ia¯b¯ΔKIi,Jj,yfy,u1y−fy,u2ydypλi1λj2,
with(63)ΔKIi,Jj,y=Kai,cj,y+Kbi,dj,y−Kai,dj,y−Kbi,cj,y.
By Jensen’s inequality, and by the fact thatf is locally Lipschitz, we have the following:(64)∑i=1m∑j=1nΔλFu1−Fu2Ii,Jjpλi1λj2≤Ap−1∑i=1m∑j=1n∫Ia¯b¯λpΔKIi,Jj,yfy,u1y−fy,u2ypdyλi1λj2≤Ap−1∑i=1m∑j=1n∫Ia¯b¯λpΔKii,jj,ypLrpu1y−u2ypdyλi1λj2≤Ap−1∑i=1m∑j=1nλLrsupy¯∈Ia¯b¯|u1y−u2y|p∫Ia¯b¯ΔKIi,Jj,ypλi1λj2dy=Ap−1λLrsupy¯∈Ia¯b¯u1y¯−u2y¯p∫Ia¯b¯∑i=1m∑j=1nΔKIi,Jj,ypλi1λj2dy.
By raising both sides of the inequality above to the power1/p, and by part (c) of Lemma 2,(65)∑i=1m∑j=1nΔλFu1−Fu2Ii,Jjpλi1λj21/p≤λLrCAp−1/pu1−u2ΛpBVIa¯b¯∫Ia¯b¯∑i=1m∑j=1nΔKIi,Jj,ypλi1λj2dy1/p.
Taking supremum overIi×Jj, and by the hypothesis H^3, we obtain as follows:(66)VΛp2λFu1−Fu2,Ia¯b¯,ℝ≤CAp−1/pLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mypds1/p.
So, we conclude from the inequalities (58) and (59), and (66) that(67)TVΛpλFu1−Fu2≤3CAp−1/pLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mypdy1/p<+∞,
and the proof is complete.
The following lemma is a direct consequence of the triangle inequality.Lemma 6.
Suppose the hypothesesH^1, H^2, H^3 hold. Let G:ΛpBVIa¯b¯⟶ΛpBVIa¯b¯ be the function defined by Gux=vx+λFux, where Fu is as in Lemma 4 and λ∈I where I is a bounded interval. Then,(68)GuΛpBVIa¯b¯≤vΛpBVIa¯b¯+λFuΛpBVIa¯b¯.Lemma 7.
LetT=t′,s′:t′∈I0b¯,s′∈I0t′, and let K:T⟶ℝ be a function with K·,·,s∈ΛpBVIsb¯, s∈I0b¯ and p≥1. Then, for(69)K^t′,s=Kt′,s,t′∈Isb¯,0,t′∈I0b¯\Isb¯,
we have that(70)TVΛpK^·,·,s,I0b¯≤2C˜Ks,s+TVΛpK·,·,s,Isb¯,where(71)C˜=max1λ111/p+1λ121/p+1λ11λ121/p,3+1λ111/p+1λ121/p.Proof.
LetIi×Jj, where Iii=1m and Jjj=1n sequences of nonoverlapping intervals of 0,b and 0,d, respectively, where Ii=ai,bi⊂0,b and Jj=cj,dj⊂0,d. Let us consider s=s1,s2∈I0b¯. We know that(72)TVΛpK^·,·,s,I0b¯=VΛpK^·,0,s+VΛpK^0,·,s+VΛp2K^·,·,s.
First, we determineVΛp0bK^·,0,s. Since Iii=1m is a sequence of nonoverlapping intervals Ii=ai,bi⊂0,b, let t′=br,0 and s1∈ar,br for some r, with 1≤r≤m. Let us consider s=s1,0∈0,b×0, then,(73)∑i=1mK^bi,0,s−K^ai,0,spλi1=K^b1,0,s−K^a1,0,spλ11+K^b2,0,s−K^a2,0,spλ21+⋯+K^br,0,s−K^ar,0,spλr1+⋯+K^bm,0,s−K^am,0,spλm1=Kbr,0,spλr1+Kbr+1,0,s−Kar+1,0,spλr+11+⋯+Kbm,0,s−Kam,spλm1≤Kbr,0,s−Ks1,0,s+Ks1,0,spλr1+Kbr+1,0,s−Kar+1,0,spλr+11+⋯+Kbm,0,s−Kam,0,spλm1≤2pKs1,0,spλr1+2pKbr,0,s−Ks1,0,spλr1+Kbr+1,0,s−Kar+1,0,spλr+11+⋯+Kbm,0,s,s−Kam,0,spλm1≤2pKs1,0,spλ11+2pVΛp0bpK·,0,s,s,b=2pKs,spλ11+VΛp0bpK·,0,s,s,b.
By raising to the power1/p, taking supremum over Iii=1m, and applying the property a+b1/p≤a1/p+b1/p if 1/p<1, we obtain as follows:(74)VΛp0bK^·,0,s,0,b≤2Ks,sλ111/p+VΛp0bK·,0,s,s,b≤2Ks,sλ111/p+TVΛpK·,0,s,s,b.
It is shown analogously that(75)VΛp0dK^0,·,s,0,d≤2Ks,sλ121/p+TVΛpK0,·,s,s,s,d.
To computeVΛp2K^·,·,s,I0b¯, let t′=br,dl for some r,l with 0≤r≤m,0≤l≤n, and let s∈ar,br×cl,dl be an arbitrary fixed point. So(76)∑i=1m∑j=1nΔK^Ii,Jj,spλi1λj2=∑i=1m∑j=1nK^ai,cj,s+K^bi,dj,s−K^ai,dj,s−K^bi,cj,spλi1λj2.
Now, we can display the terms of summation and substitute the values ofK^ according to (70) to obtain as follows:(77)∑i=1m∑j=1nΔK^Ii,Jj,spλi1λj2=Kbr,dl,spλr1λl2+Kbr,dl+1,s−Kbr,cl+1,spλr1λl+12+⋯+Kbr,dn,s−Kbr,cn,spλr1λn2+Kbr+1,dl,s−Kar+1,dl,spλr+11λl2++Kbr+1,dl+1,s+Kar+1,cl+1,s−Kbr+1,cl+1,s−Kar+1,dl+1,spλr+11λl+12+⋯+Kbr+1,dn,s+Kar+1,cn,s−Kbr+1,cn,s−Kbr+1,cl+1,spλr+11λn2+⋯+Kbm,dl,s−Kam,dl,spλm1λl2+Kbm,dl+1,s+Kbm,dl+1,s−Kbm,dl+1,s−Kbm,dl+2,spλm1λl+12+⋯+Kbm,dn,s+Kam,cn,s−Kam,dn,s−Kbm,cn,spλm1λn2=Kbr,dl,spλr1λl2+Kbr,dl+1,s−Kbr,cl+1,spλr1λl+12+Kbr,dl+2,s−Kbr,cl+2,spλr1λl+22+⋯+Kbr,dn,s−Kbr,cn,spλr1λn2+Kbr,dl+1,s−Kar+1,dl,spλr+11λl2+Kbr+2,dl,s−Kar+2,dl,spλr+21λl2+⋯+Kbm,dl,s−Kam,dl,spλm1λl2+Kbr+1,dl+1,s+Kar+1,cl+1,s−Kbr+1,cl+1,s−Kar+1,dl+1,spλr+11λl+12+⋯+Kbr+1,dn,s+Kar+1,cn,s−Kbr+1,cn,s−Kar+1,dn,spλr+11λn2+⋯+Kbm,dl+1,s+Kam,cl+1,s−Kbm,cl+1,s−Kam,dl+1,spλm1λl+12+⋯+Kbm,dn,s+Kam,cn,s−Kbm,cn,s−Kam,dn,spλm1λn2=Kbr,dl,spλr1λl2+∑j=l+1nKbr,dj,s−Kbr,cj,spλr1λj2+∑i=r+1mKbi,dl,s−Kai,dl,spλi1λl2+∑i=r+1m∑j=l+1nKbi,dj,s+Kai,cj,s−Kai,dj,s−Kbi,cj,spλi1λj2≤2pKs,spλ11λ12+2pKbr,dl,s−Ks,spλr1λl2+2p∑j=l+1nKbr,dj,s−Kbr,cj,spλr1λj2+2p∑i=r+1mKbi,dl,s−Kai,dl,spλi1λl2+2p∑i=r+1m∑j=l+1nKbi,dj,s+Kai,cj,s−Kai,dj,s−Kbi,cj,spλi1λj2≤2pKs,spλ11λ12+1λ12VΛp0bK·,dl,s,Isb¯p+1λ11VΛp0dKbr,·,s,Isb¯p+VΛp2K·,·,s,Isb¯p.
Raising to the power of1/p, taking supremum over Ii×Jj, and applying the property a+b1/p≤a1/p+b1/p, we have as follows:(78)VΛp2K^·,·,s≤2Ks,sλ11λ121/p+1λ111/p+1λ121/p+1TVΛpK·,·,s,Isb.
It follows from (75) and (76), and (78) that(79)TVΛpK^·,·,s≤21λ111/p+1λ121/p+1λ11λ121/pKs,s+3+1λ111/p+1λ121/pTVΛpK·,·,s,Isb,
taking(80)C˜=max1λ111/p+1λ121/p+1λ11λ121/p,3+1λ111/p+1λ121/p,
yields(81)TVΛpK^·,·,s≤2C˜Ks,s+TVΛpK·,·,s,Isb,
which 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)
H^4 Let T=t′,s′:t′∈I0b¯,s′∈I0t′, and K:T⟶ℝ is a function such that(82)Ks′,s′+TVΛpK·,·,s′,Is′b¯≤ms′,almost everywhere,s′∈I0b¯, where m:I0b¯⟶ℝ+ is a Lp integrable function and Kt′,·,· is Lebesgue integrable over I0t′ for every t′∈I0b¯.Theorem 5.
Suppose the hypothesesH^1, H^2, andH^3 hold. Then, there exists a number ρ>0 such that for every λ<ρ, equation (1) has a unique solution in ΛpBVIa¯b¯,ℝ defined on Ia¯b¯.Proof.
LetIi×Jj, where Iii=1m and Jjj=1n be sequences of nonoverlapping intervals of a,b and c,d, respectively, where Ii=ai,bi⊂a,b and Jj=cj,dj⊂c,d. Let r>0 such that vΛpBVIa¯b¯<r, and Lr denotes the Lipschitz constant of f corresponding to the cube Ia¯b¯×−r,r. B¯r denotes the closed ball with center 0 and radius r in the space ΛpBVIa¯b¯, that is,(83)B¯r=x∈ΛpBVIa¯b¯/xΛpBVIa¯b¯<r¯.
Using part (c) of Lemma2, choose a number ρ>0 such that(84)vΛpBVIa¯b¯+3ρsupy¯∈Ia¯b¯,u¯≤Crfy¯,u¯∫Ia¯b¯Ka,ydy+Ap−1/p∫Ia¯b¯Mypdy1/p<r.and(85)CρLr∫Ia¯b¯Ka,ydy+3Ap−1/p∫Ia¯b¯Mypdy1/p<1,where C is guaranteed by part (c) of Lemma 2, and A is the area of the rectangle Ia¯b¯. In order to prove this theorem, the Banach contraction principle is needed (Theorem 1). Fix λ<ρ and define the operator 4G:ΛpBVIa¯b¯,ℝ⟶ΛpBVIa¯b¯,ℝ by(86)Gux=vx+λ∫Ia¯b¯Kx,yfy,uydy.
It can be verified that it is well defined, as in the proof of Lemma4. We show first that GB¯r⊂B¯r. Indeed, for every u∈B¯r, it follows from Lemma 6 that(87)GuΛpBVIa¯b¯≤vΛpBVIa¯b¯+|λ|FuΛpBVIa¯b¯.
For the second term in the right hand side of the inequality above, we have as follows(88)FuΛpBVIa¯b¯=Fua+TVΛpFu=∫Ia¯b¯Ka,yfy,uydy+TVΛpFu≤supy¯∈Ia¯b¯,u¯≤Crfy¯,u¯∫Ia¯b¯Ka,ydy+TVΛpFu.
By Lemma4, the inequality (88) can be rewritten as follows:(89)FuΛpBVIa¯b¯≤supy¯∈Ia¯b¯,u¯≤Crfy¯,u¯∫Ia¯b¯Ka,ydy+3Ap−1/p∫Ia¯b¯Mypdy1/p.
Substituting the inequality (89) into (87) yields the following:(90)GuΛpBVIa¯b¯≤vΛpBVIa¯b¯+λsupy¯∈Ia¯b¯,u¯≤Crfy¯,u¯∫Ia¯b¯Ka,ydy+3Ap−1/p∫Ia¯b¯Mypdy1/p≤vΛpBVIa¯b¯+3ρsupy¯∈Ia¯b¯,u¯≤Crfy¯,u¯∫Ia¯b¯Ka,ydy+Ap−1/p∫Ia¯b¯Mypdy1/p<r.
Therefore,GB¯r⊂B¯r. To show now that G is a contraction mapping, let u1,u2∈B¯r, that is, u1,u2∈ΛpBVIa¯b¯. So,(91)Gu1−Gu2ΛpBVIa¯b¯=Gu1a−Gu2a+TVΛpGu1−Gu2=λFu1a−Fu2a+TVΛpλFu1−Fu2.
It is straightforward to compute that(92)Fu1a−Fu2a=∫Ia¯b¯Ka,yfy,u1y−fy,u2ydy≤∫Ia¯b¯Ka,yfy,u1y−fy,u2ydy.
By the fact thatf is locally Lipschitz, and by part (c) of Lemma 2, we have the following:(93)Fu1a−Fu2a≤LrCu1−u1ΛpBVIa¯b¯∫Ia¯b¯Ka,ydy.
Lemma5 implies as follows:(94)TVΛpλFu1−Fu2≤3CAp−1/pLrλu1−u2ΛpBVIa¯b¯∫Ia¯b¯Mypdy1/p,
Moreover, it follows from the inequality (94) that(95)Gu1−Gu2ΛpBVIa¯b¯≤CρLr∫Ia¯b¯Ka,ydy+3Ap−1/p∫Ia¯b¯Mypdy1/pu1−u2ΛpBVIa¯b¯.
Thus,G is a contraction mapping, by the inequality (86). Theorem 1 implies that G has a unique fixed point in B¯r, which is to say that there exists a unique u∈B¯r⊂ΛpBVIa¯b¯,ℝ such that(96)vx+λ∫Ia¯b¯Kx,yfy,uydy=ux.
Therefore,u is a unique solution of (1).Theorem 6.
Suppose the hypothesesH^1, H^2, and H^4 hold. Then, there exists a rectangle ℛ⊂I0b such that the equation (2) has a unique solution in ΛpBVℛ.Proof.
Letr and Lr be denoted as in the proof of the preceding theorem. Let us denote ℛ=0,d¯1×0,d¯2=I0d¯ with d¯=d¯1,d¯2 and d¯1,d¯2>0, B¯r the closed ball with center 0 and radius r in the space ΛpBVI0d¯, that is,(97)B¯r=x∈ΛpBVℛ/xΛpBVℛ<r¯.
Choosed¯=d¯1,d¯2 with d¯1,d¯2>0 such that(98)vΛpBVI0d¯+6C˜Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0d¯ms′pds′1/p<r6CC˜Ap−1/pLr∫I0d¯ms′pds′1/p<1,where the existence of C,C˜ are assured by part (c) of Lemmas 2 and 7, respectively, and A is the area of the rectangle I0d¯. Let Ii×Jj, where Iii=1m and Jjj=1n are sequences of nonoverlapping intervals of 0,d¯1 and 0,d¯2, respectively, with Ii=ai,bi⊂0,d¯1 and Jj=cj,dj⊂0,d¯2. We define the functions K^ and the operator G˜:ΛpBVI0d¯,ℝ⟶ΛpBVI0d¯,ℝ    by(99)K^t′,s=Kt′,s′,t′∈Is′b¯,0t′∈I0b¯∖Is′b¯,G˜ut′=vt′+F˜ut′, where(100)F˜ut′=∫I0t′Kt′,s′fs′,us′ds′.
Witht′∈I0d¯, it can be verified that it is well defined, as in the proof of Lemma 4.
As in the proof of Theorem5, we will use the Banach contraction principle (Theorem 1) to prove this theorem. We begin by showing that G˜B¯r⊂B¯r. So, let u∈B¯r,(101)G˜uΛpBVI0d¯=v0+F˜u0+TVΛpv+F˜u=v0+∫I00K0,s′fs′,us′ds′+TVΛpv+F˜u=v0+TVΛpv+F˜u.
It can easily be shown that(102)TVΛpv+F˜u≤TVΛpv+TVΛpF˜u.
Thus,(103)G˜uΛpBVI0d¯≤v0+TVΛpv+TVΛpF˜u=vΛpBVI0d¯+TVΛpF˜u,
However,(104)TVΛpF˜u=VΛp0d1F˜u+VΛp0d2F˜u+VΛp2F˜u,I0d¯,ℝ.
In addition,(105)F˜uΛpBVI0d¯=F˜u0+VΛpF˜u=TVΛpF˜u.
To computeTVΛpF˜u, we proceed as in the proof of Lemma 4. Let t′=br,0,   s1′∈ar,br for some r, with 1≤r≤m and let us consider s′=s1′,0, d¯=d¯1,0∈0,b×0. Then,(106)∑i=1mF˜ubi,0−F˜uai,0pλi1≤Ap−1∑i=1m∫I0t′Kbi,0,s′−Kai,0,s′fs′,us′pds′λi1≤Ap−1∑i=1m∫I0d¯K^bi,0,s′−K^ai,0,s′fs′,us′pds′λi1≤Ap−1∑i=1msups¯∈I0d¯,u¯≤Crfs¯,u¯p∫I0d¯K^bi,0,s′−K^ai,0,s′pλi1ds′=Ap−1sups¯∈I0d¯,u¯≤Crfs¯,u¯p∫I0d¯∑i=1mK^bi,0,s′−K^ai,0,s′pλi1ds′.
Raising both sides of the inequality above to the power1/p yields the following:(107)∑i=1mF^ubi,0−F^uai,0pλi11/p≤Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0d¯∑i=1mK^bi,0,s′−K^ai,0,s′pλi1ds′1/p.
Now,(108)∑i=1mK^bi,0,s′−K^ai,0,s′pλi1≤VΛpK^·,0,s′p≤TVΛpK^·,·,s′p.
By taking supremum overIi in the inequality (107), by Lemma 7 and the Hypothesis H^4, we conclude as follows:(109)VΛp0d1F^u≤2C˜Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0d¯ms′pds′1/p.
It is shown analogously as follows:(110)VΛp0d2Fu≤2C˜Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0dms′pds′1/p
and(111)VΛp2Fu,I0d,ℝ≤2C˜Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0dms′pds′1/p.
By the inequalities (109)–(111), we obtain the following:(112)TVΛpF˜u≤6C˜Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0dms′pds′1/p.
Substituting the inequality (113) into (104) leads to the following:(113)G˜uΛpBVI0d¯≤vΛpBVI0d¯+6C˜Ap−1/psups¯∈I0d¯,u¯≤Crfs¯,u¯∫I0d¯ms′pds′1/p<r.
Next, we show thatG˜u is a contraction mapping. For u1,u2∈B¯r, we have as follows:(114)G˜u1−G˜u2ΛpBVI0d¯=TVΛpF˜u1−F˜u2.
Now we compute the variationVΛpF˜u1−F˜u2. Let t′=bi,0 and s1′∈ai,bi for some i, then, for s′=s1′,0(115)∑i=1mF˜u1−F˜u2bi,0−F˜u1−F˜u2ai,0pλi1=Ap−1∑i=1m∫I0t′Kbi,0,s′−Kai,0,s′pfs′,u1s′−fs′,u2s′pds′λi1≤Ap−1Lrp∑i=1m∫I0d¯u1s′−u2s′pK^bi,0,s′−K^ai,0,s′pds′λi1≤Ap−1Lru1−u2ΛpBVI0d¯p∫I0d¯∑i=1mK^bi,0,s′−K^ai,0,s′pλi1ds′.
By part (c) of Lemma2, the inequality (108), the Hypothesis H^4, and raising both sides of the inequality above to the power 1/p, we obtain the following:(116)VΛp0d1F˜u1−F˜u2≤2LrCC˜Ap−1/pu1−u2ΛpBVI0d¯∫I0d¯ms′pds′1/p.
An argument analogous to the one we used above shows the following:(117)VΛpF˜u1−F˜u2≤2LrCC˜Ap−1/pu1−u2ΛpBVI0d¯∫I0d¯ms′pds′1/pVΛp2F˜u1−F˜u2≤2LrCC˜Ap−1/pu1−u2ΛpBVI0d¯∫I0d¯ms′pds′1/p.
Therefore,(118)G˜u1−G˜u2ΛpBVI0d¯≤6LrCC˜Ap−1/p∫I0d¯ms′pds′1/pu1−u2ΛpBVI0d¯.
The inequality (98) implies that G˜ is a contraction mapping; and by Theorem 1, G˜ has a unique fixed point in B¯r, that is, there exists a unique u˜∈B¯r⊂ΛpBVI0d¯ such that(119)vx+∫I0xKx,sfs,u˜sds=u˜x,withx∈I0d.
Hence,u˜ 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.
LetI01=0,1×0,1, and let the functions v:I01⟶ℝ, f:I01×ℝ⟶ℝ and K:I01×I01⟶ℝ be defined by vx1,x2=cosx1+x2, fy1,y2,y3=y12+y22+y32 and Kx,y=e2y1+y2sinx1+x2, respectively. We want to show that the nonlinear Hammerstein–Volterra integral equation(120)ux=vx+∫I01Kx,yfy,uydy,withx∈I01
has a unique solutionu∈ΛpBVI01,ℝ.
In order to solve this problem, we verify the hypotheses of Theorem6.(1)
vx1,x2=cosx1+x2 is measurable in I01, to show that vx1,x2 belongs to ΛpBVI01,ℝ, let Ii×Jj, where Iii=1m and Jjj=1n be sequences of nonoverlapping intervals of 0,1 and 0,1, respectively, where Ii=ai,bi⊂0,1 and Jj=cj,dj⊂0,1. To determine VΛp01v·,0, VΛpv0,· and VΛp2v,I01, let us first compute(121)∑i=1mvbi,0−vai,0pλi1=∑i=1mcosbi−cosaipλi1≤∑i=1mcosbi−cosaipλ1.From Lemma 5 in [24] we have the following:(122)xp≤pbp−1xforx∈0,bwith1≤p<+∞,Hence,(123)∑i=1mvbi,0−vai,0pλi1≤∑i=1mpbi−aiλ1=pλ1<+∞.Thus,VΛp01v.,0<+∞. An argument entirely analogous to that above leads to VΛp01v0,.<+∞. On the other hand,(124)Δvbi,djp=vai,cj+vbi,dj−vai,dj−vbi,cjp=cosai+cj+cosbi+dj−cosai+dj−cosbi+cjp=cosbi−cosaicosdj−coscj−sinbi−sinaisindj−sincjp.Now, using the inequalitya+bp≤2pap+bp, the mean value theorem, and the inequality (122) to obtain(125)Δvbi,djp≤2pcosbi−cosaipcosdj−coscjp+2psinbi−sinaipsindj−sincjp≤2pbi−aipdj−cjp+2pbi−aipdj−cjp≤p22p+1bi−aidj−cj.Then,(126)∑i=1m∑j=1nΔvbi,djpλi1λj2≤∑i=1m∑j=1np22p+1bi−aidj−cjλ1λ2=p22p+1λ1λ2∑i=1m∑j=1nbi−aidj−cj=p22p+1λ1λ2<+∞,Thus,VΛp2v,I01<+∞ and in consequence, v∈ΛpBVI01,ℝ.(2)
We need to show now thatfy1,y2,y3=y12+y22+y32 is locally a Lipschitz on I01×ℝ. Indeed, let r>0,  y1,y2,y^1,y^2∈I01, and y3,y^3∈−r,r. Then(127)fy1,y2,y3−fy^1,y^2,y^3=y12−y^12+y22−y^22+y32−y^32≤y1+y^1y1−y^1+y2+y^2y2−y^2+y3+y^3y3−y^3≤max2,2ry1−y^1+y2−y^2+y3−y^3=Lry1,y2,y3−y1,y2,y3.So there existsLr=max2,2r such that(128)fy1,y2,y3−fy1,y2,y^3≤Lry1,y2,y3−y^1,y^2,y^3,Moreover, thereforef is locally a Lipschitz constant on I01×ℝ.(129)∫I01Kx,ydy=∫01∫01sinx1+x2e2y1+y2dy1dy2=e4−2e2+14sinx1+x2.
Hence,Kx,·,· is Lebesgue integrable. On the other hand,(130)TVΛpK·,·,α,I01=TVΛpsinx1+x2e2y1+y2,I01=e2y1+y2TVΛpsinx1+x2,I01=My.
We proceed as in Part 1 of this example to show thatTVΛpsinx1+x2,I01<+∞, and so it is evident that My is Lp integrable. Thus, all the conditions of Theorem 6 are satisfied, therefore, there exists a rectangle ℛ⊂I01 such that the (120) has a unique solution in ΛpBVI01,ℝ.


## 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 normfΛpBVIa¯b¯=fa¯+TVΛpf,Ia¯b¯,ℝ. 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.



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*Source: 5482688-2022-05-28.xml*