In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second κ-variation into itself then its generator function satisfies a Matkowski condition. We also provide conditions for the existence and uniqueness of solutions of the Hammerstein and Volterra equations in this space.
Citation: Jurancy Ereú, Luz E. Marchan, Liliana Pérez, Henry Rojas. Some results on the space of bounded second κ-variation functions[J]. AIMS Mathematics, 2023, 8(9): 21872-21892. doi: 10.3934/math.20231115
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In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second κ-variation into itself then its generator function satisfies a Matkowski condition. We also provide conditions for the existence and uniqueness of solutions of the Hammerstein and Volterra equations in this space.
A function of bounded variation is a real-valued function in which total variation is finite. This class of functions was first introduced by Camile Jordan in [8] when he was dealing with the convergence of the Fourier series through a critical revision of Dirichlet's famous flawed proof. An interesting feature of bounded variation functions is that they form an algebra of functions whose first derivative exists almost everywhere. So, they have approximation properties and provide a natural framework for studying the theory of integration for a wider class of functions. The concept of bounded variation has become diversified and generalized in many ways in order to study more complex structures and properties (see e.g., [6,9,13]).
Ereú, López and Merentes [6] introduced the class of all functions of bounded second κ-variation on a closed interval [a,b] which is denoted by κBV2([a,b]) and is a generalization of the notion of bounded second variation in the sense of De la Vallée Poussin. It was obtained by distorting the classical notion through a function κ that measures lengths in the domain of functions and not in the range. Among other things, they equipped the set of functions of bounded second κ-variation with a norm and proved that κBV2([a,b]) is a Banach space under that norm.
The functions with certain notions of bounded variation are commonly employed to define generalized solutions for nonlinear problems involving functionals, integral equations and variational problems. Specifically, the fact that solutions of integral equations belong to a space of functions with some notion of bounded variation provides valuable insights into the properties of regularity, stability, good behavior and approximation associated with these solutions. These properties assume a vital role in theoretical analysis and numerical methods.
The equations of Hammerstein and Volterra type
x(t)=g(t)+λ∫IK(t,s)f(x(s))ds,λ∈I,t∈I=[0,b], | (1.1) |
x(t)=g(t)+∫taK(t,s)f(x(s))ds,t∈I=[a,b], | (1.2) |
where the integration is taken in the Lebesgue sense serve as models for a wide range of phenomena in diverse disciplines such as physics, engineering, economics and biology (see [5]). Consequently, there has been an escalating interest in exploring solutions for these integral equations in spaces of functions that incorporate specific notions of bounded variation or in spaces of absolutely continuous functions which is important since there the fundamental theorem of calculus is satisfied (see [2,3,11,14], among others).
In [14] conditions are studied in such a manner that Eqs (1.1) and (1.2) have solutions in the space of functions of bounded κ-variation. In this paper, we find conditions on f, g and K so that Eqs (1.1) and (1.2) have solutions in the space of functions bounded second κ-variation.
On the other hand, in problems related to existence of solutions of differential, integral or functional equations in spaces of functions it is required to consider the so-called Nemytskii operator or superposition operator in order to make use of the basic principles of nonlinear analysis. The natural problem related to the superposition operator H generated by a function h, consists in finding both necessary and sufficient conditions for the function h under which the superposition operator H maps a class of functions f:[a,b]→R into itself.
In 1982, J. Matkowski [12] showed that a composition operator maps the function space Lip([0,1],R) into itself and is globally Lipschitz if and only if its generator h has the form
h(x,y)=h1(x)y+h0(x),x∈[0,1], y∈R, | (1.3) |
for some h0,h1∈Lip([0,1],R). This result was extended to a lot of spaces such as spaces of functions with some notion of bounded variation (see e.g., [1,4,13]). In this paper, we establish that if the (non-autonomous) superposition operator maps the function space κBV2([a,b]) into itself and is globally Lipschitz then the generating function h satisfies Eq (1.3), with both h0 and h1 belonging to κBV2([a,b]).
This document is structured as follows. Section 2 on preliminaries provides the necessary results for the proof of the main theorems. Here it is shown that κBV2([a,b]) is contained in the space of absolutely continuous functions AC([a,b]). Section 3 studies the action of the superposition operator on κBV2([a,b]). Section 4 presents the existence and uniqueness results for the Hammerstein and Volterra integral equations on κBV2([a,b]), the proofs of which are based on the Banach Contraction Principle. Some applications are presented in Section 5. Finally, the conclusions in the last section.
In 1975, Korenblum [9] introduced the notion of bounded κ-variation of a function by studying the problem of representation for harmonic functions defined on the unit disk of the complex plane. This notion differs from other known variations in the fact that it maximizes ratios between Jordan's sums and the so-called κ-entropies generated by a distortion function κ.
Definition 2.1 ([9]). A function κ:[0,1]→[0,1] is said to be a distortion function if it satisfies the following properties:
i) κ is continuous with κ(0)=0 and κ(1)=1.
ii) κ is nondecreasing and concave.
iii) limx→0+κ(x)x=+∞.
All distortion function is subadditive; that is, if x,y∈[0,1] are such that x+y∈[0,1] then
κ(x+y)≤κ(x)+κ(y). |
Some examples of distortion functions are the following:
(1) κ(s)=sα, 0<α<1 is the simplest example.
(2) κ(s)=s(1−logs), 0<t≤1 is another typical example.
(3) κ(s)=(1−12logs)−1 is another one that arises in entropy theory.
Let [a,b]⊆R be an interval. From now on, Π([a,b]) denote the set of all partitions ξ={ti}ni=0 of the interval [a,b] such that a=t0<t1<…<tn−1<tn=b. For m>2, Πm([a,b]) denote the set of all partitions ξ={ti}ni=0 of the interval [a,b] such that a=t0<t1<…<tn−1<tn=b and n≥m−1.
Definition 2.2 ([9]). Let κ be a distortion function. The quantity
κ(ξ;[a,b]):=n∑i=1κ(ti−ti−1b−a) |
is called the κ-entropy of ξ relative to [a,b], where ξ={ti}ni=0∈Π([a,b]).
Notice that, by the subadditivity of κ, for any ξ={ti}ni=0∈Π([a,b]), we have
1=κ(1)≤n∑i=1κ(ti−ti−1b−a)=κ(ξ;[a,b]). | (2.1) |
Additionally, it can be easily derived that
κ(ξ;[a,b])≤nκ(1n), | (2.2) |
where the estimate is sharp and attained for equidistant partitions ξ (see [9]).
Definition 2.3 ([9]). Let κ be a distortion function. A function f:[a,b]→R is said to be of bounded κ-variation if
κV(f;[a,b]):=supξ∈Π([a,b])n∑i=1|f(ti)−f(ti−1)|κ(ξ;[a,b])<∞, |
where the supremum is taken over the set of all partitions ξ={ti}ni=0∈Π([a,b]).
In [6], we are provided with the concept of second κ-variation of a function which generalizes the notion of function of bounded second variation given by De la Vallée Poussin in 1908 while considering a distortion function in the same way as Korenblum did.
Definition 2.4 ([6]). A function f:[a,b]→R is said to be of bounded second κ-variation on [a,b] if
κV2(f;[a,b]):=supξ∈Π3([a,b])n−2∑i=0|f[ti+1,ti+2]−f[ti,ti+1]|κ(ξ;[a,b])<∞, |
where the supremum is taken over the set of all partitions ξ={ti}ni=0∈Π3([a,b]) and
f[ti+1,ti+2]=f(ti+2)−f(ti+1)ti+2−ti+1,i=0,…,n−2. |
The class of all functions defined on [a,b] of bounded second κ-variation is denoted by κBV2([a,b]). The following theorem characterizes those functions which second κ-variation is null.
Theorem 2.5 ([6]). Let f:[a,b]→R be a function. Then, κV2(f;[a,b])=0 if and only if there exist constants α,β∈R such that f(t)=αt+β.
The following remark guarantees that κBV2([a,b]) is known to be a vector space.
Remark 2.6 ([6]). Let f and g in κBV2([a,b]) and λ∈R. Then
κV2(f+λg;[a,b])≤κV2(f;[a,b])+|λ|κV2(g;[a,b]). |
Furthermore, κBV2([a,b]) is a Banach space.
Theorem 2.7 ([6]). κBV2([a,b]) equipped with the norm
‖f‖κBV2=|f(a)|+|f(b)|+κV2(f;[a,b]), |
is a Banach space.
In [6], it is also shown that the functions in the unit ball of the space (κBV2([a,b]),‖⋅‖) are uniformly majorized by a certain fixed continuous function. That is,
Theorem 2.8 ([6]). The continuous function pκ:[a,b]→R given by
pκ(t)=1+1b−a(b−t)(t−a)(κ(b−tb−a)+κ(t−ab−a)), | (2.3) |
is such that for all f∈κBV2([a,b]),
|f(t)|≤pk(t)‖f‖κBV2, for all t∈[a,b]. |
In particular, κBV2([a,b]) is a subspace of B([a,b]) the Banach space of all bounded functions on [a,b] with the sup norm.
Below, we show that κBV2([a,b])⊆AC([a,b]). Also we show that κBV2([a,b]) is closed under the pointwise product operation and in consequence the set of polynomial functions is contained in κBV2([a,b]).
A similar technique to the one used to prove lemma 4 in [15] and property (2.2) enable us to show the following property of functions of bounded second κ-variation.
Lemma 2.1. If f∈κBV2([a,b]), then f[y0,y1] is bounded for all y0, y1 in the interval [a,b].
Proof. First, consider a partition ξ={ti}ni=0∈Π4([a,b]). Then,
|f[ti+2,ti+3]−f[ti,ti+1]|=|f[ti+2,ti+3]−f[ti+1,ti+2]+f[ti+1,ti+2]−f[ti,ti+1]|≤|f[ti+2,ti+3]−f[ti+1,ti+2]|+|f[ti+1,ti+2]−f[ti,ti+1]|≤κ(ξ,[a,b])κV2(f;[a,b])+κ(ξ,[a,b])κV2(f;[a,b])=2κ(ξ,[a,b])κV2(f;[a,b]). | (2.4) |
Let c be chosen arbitrarily in (a,b) and let A=|f[a,c]|. The proof of this lemma depends on the position of y0 and y1 with respect to the points a, b, and c. So we consider the following six cases:
Case 1. a<y0<c<y1<b, Case 4. a<c≤y0<y1<b,
Case 2. a<y0<c<y1=b, Case 5. a=y0<c<y1<b,
Case 3. a<y0<y1≤c<b, Case 6. a=y0<c<y1=b.
Case 1. Suppose a<y0<c<y1<b. If y2 is any point such that y1<y2<b then a<y0<c<y1<y2<b. By (2.4) and (2.2) we have
|f[y0,y1]|=|f[y0,y1]−f[y1,y2]+f[y1,y2]−f[a,c]+f[a,c]|≤|f[y0,y1]−f[y1,y2]|+|f[y1,y2]−f[a,c]|+|f[a,c]|≤κ(ξ1,[a,b])κV2(f;[a,b])+2κ(ξ2,[a,b])κV2(f;[a,b])+A≤4κ(14)κV2(f;[a,b])+8κ(14)κV2(f;[a,b])+A=12κ(14)κV2(f;[a,b])+A:=K1 |
where ξ1 y ξ2 are the partitions a<y0<y1<y2<b and a<c<y1<y2<b, respectively.
Case 2. Suppose a<y0<c<y1=b. If y2 is any point such that c<y2<y1 then a<y0<c<y2<y1=b. It follows from (2.4) and (2.2) that
|f[y0,y1]|=|f[y0,y1]−f[a,y0]+f[a,y0]−f[y2,y1]+f[y2,y1]−f[a,c]+f[a,c]|≤|f[y0,y1]−f[a,y0]|+|f[a,y0]−f[y2,y1]|+|f[y2,y1]−f[a,c]|+|f[a,c]|≤κ(ξ3,[a,b])κV2(f;[a,b])+2κ(ξ4,[a,b])κV2(f;[a,b])+2κ(ξ5,[a,b])κV2(f;[a,b])+A≤[2κ(12)+12κ(13)]κV2(f;[a,b])+A:=K2 |
where the partitions ξ3,ξ4, and ξ5 are a<y0<y1=b, a<y0<y2<y1=b, and a<c<y2<y1=b, respectively. Consequently, |f[y0,y1]|≤K2.
Case 3. Suppose a<y0<y1≤c<b. If y2 is any point such that c<y2<b then a<y0<y1≤c<y2<b. By (2.4) and (2.2) we have
|f[y0,y1]|=|f[y0,y1]−f[y1,y2]+f[y1,y2]−f[y2,b]+f[y2,b]−f[a,c]+f[a,c]|≤|f[y0,y1]−f[y1,y2]|+|f[y1,y2]−f[y2,b]|+|f[y2,b]−f[a,c]|+|f[a,c]|≤κ(ξ6,[a,b])κV2(f;[a,b])+κ(ξ6,[a,b])κV2(f;[a,b])+2κ(ξ7,[a,b])κV2(f;[a,b])+A≤=[8κ(14)+6κ(13)]κV2(f;[a,b])+A:=K3, |
where ξ6 and ξ7 are the respective partitions a<y0<y1<y2<b and a<c<y2<b. So, |f[y0,y1]|≤K3.
The proof of the other cases is handled in a similar fashion and we omit the details. Since there is a finite number of cases we can ensure that f[y0,y1] is bounded.
Theorem 2.9. If f∈κBV2([a,b]) then f is Lipschitz on [a,b] and in consequence absolutely continuous on [a,b].
Proof. By lemma 2.1, there exists K>0 such that
|f(y1)−f(y0)y1−y0|=|f[y0,y1]|≤K |
for all y0,y1∈[a,b] with y0≠y1 which implies that f is Lipschitz continuous on [a,b] and thus f is absolutely continuous on the interval [a,b].
From now on, denote by Lba(f) the Lipschitz constant of a function f:[a,b]→R, i.e.,
Lba(f)=sup{|f(x2)−f(x1)x2−x1|:x1,x2∈[a,b],x1≠x2}. |
Proposition 2.1. If f,g∈κBV2([a,b]) then fg∈κBV2([a,b]).
Proof. Let ξ={ti}ni=0∈Π3([a,b]). We begin by considering the expression
(fg)[ti+1,ti+2]−(fg)[ti,ti+1]=f(ti+2)g(ti+2)−f(ti+1)g(ti+1)ti+2−ti+1−f(ti+1)g(ti+1)−f(ti)g(ti)ti+1−ti. |
By adding the terms ±f(ti+1)g(ti+2), ±f(ti)g(ti+1) we get
(fg)[ti+1,ti+2]−(fg)[ti,ti+1]=[f(ti+2)−f(ti+1)]g(ti+2)+f(ti+1)[g(ti+2)−g(ti+1)]ti+2−ti+1−[f(ti+1)−f(ti)]g(ti+1)+f(ti)[g(ti+1)−g(ti)]ti+1−ti=f(ti+2)−f(ti+1)ti+2−ti+1g(ti+2)−f(ti+1)−f(ti)ti+1−tig(ti+1)+f(ti+1)g(ti+2)−g(ti+1)ti+2−ti+1−f(ti)g(ti+1)−g(ti)ti+1−ti=f[ti+1,ti+2]g(ti+2)−f[ti,ti+1]g(ti+1)+f(ti+1)g[ti+1,ti+2]−f(ti)g[ti,ti+1]. |
Adding the following terms ±g(ti+2)f[ti,ti+1],±f(ti+1)g[ti,ti+1] and combining them yields
(fg)[ti+1,ti+2]−(fg)[ti,ti+1]=g(ti+2){f[ti+1,ti+2]−f[ti,ti+1]}+f(ti+1){g[ti+1,ti+2]−g[ti,ti+1]}+f[ti,ti+1]{g(ti+2)−g(ti+1)}+g[ti,ti+1]{f(ti+1)−f(ti)}=g(ti+2){f[ti+1,ti+2]−f[ti,ti+1]}+f(ti+1){g[ti+1,ti+2]−g[ti,ti+1]}+f[ti,ti+1]{g(ti+2)−g(ti)}. |
By theorem 2.9, f[y0,y1]≤Lba(f) and g[y0,y1]≤Lba(g) for all y0,y1∈[a,b] (lemma 2.1). Then,
|(fg)[ti+1,ti+2]−(fg)[ti,ti+1]|≤‖g‖∞|f[ti+1,ti+2]−f[ti,ti+1]|+‖f‖∞|g[ti+1,ti+2]−g[ti,ti+1]|+|f[ti,ti+1]||g[ti,ti+2]||ti+2−ti|≤‖g‖∞|f[ti+1,ti+2]−f[ti,ti+1]|+‖f‖∞|g[ti+1,ti+2]−g[ti,ti+1]|+Lba(f)Lba(g)|ti+2−ti|. |
Thus,
n−2∑i=0|(fg)[ti+1,ti+2]−(fg)[ti,ti+1]|κ(ξ;[a,b])<‖g‖∞n−2∑i=0|f[ti+1,ti+2]−f[ti,ti+1]|κ(ξ;[a,b])+‖f‖∞n−2∑i=0|g[ti+1,ti+2]−g[ti,ti+1]|κ(ξ;[a,b])+2Lba(f)Lba(g)(b−a)≤‖g‖∞κV2(f;[a,b])+‖f‖∞κV2(g;[a,b])+2Lba(f)Lba(g)(b−a). |
By theorem 2.8, κV2(fg;[a,b])<∞ and so fg∈κBV2([a,b]).
Corollary 2.1. If f is a polynomial function then f∈κBV2([a,b]).
Proof. Immediate consequence of theorem 2.5, proposition 2.1 and remark 2.6.
Let I,J⊆R be intervals and let JI denote the set of all functions φ:I→J. For a given function h:I×J→R the mapping H:JI→RI defined by
H(φ)(x)=h(x,φ(x)),φ∈JI |
is called a non-autonomous superposition operator (or Nemytskii operator) of a generator h.
In this section, we present a characterization of the action of a non-autonomous superposition operator on the space κBV2([a,b]) showing that if a superposition operator applies such space to itself and is globally Lipschitz then its generating function satisfies a so-called Matkowski condition.
Lemma 3.1. Let α,β∈[a,b] be such that α<β. Consider the function
ηα,β(t):={0,if a≤t≤αt−αβ−α,if α≤t≤β1,if β≤t≤b. |
For y1,y2∈R, with y1≠y2, define the functions φj:[a,b]→R,j=1,2, by
φj(t)=12[ηα,β(t)(y1−y2)+yj+y2],j=1,2. |
Then the functions φ1,φ2 are both of bounded second κ-variation.
Proof. Let ξ={ti}ni=0∈Π3([a,b]) such that a=t0<⋯<tr≤α<tr+1<⋯<tr+s≤β<tr+s+1<⋯<tn−1<tn=b.
Notice that by (2.1)
n−2∑i=0|φ1[ti+1,ti+2]−φ1[ti,ti+1]|κ(ξ;[a,b])≤n−2∑i=0|φ1[ti+1,ti+2]−φ1[ti,ti+1]|. |
We proceed to bound the right side of this inequality, by theorem 2.5, in the following way:
n−2∑i=0|φ1[ti+1,ti+2]−φ1[ti,ti+1]|≤|φ1(tr+1)−φ1(tr)tr+1−tr|+|φ1(tr+2)−φ1(tr+1)tr+2−tr+1−φ1(tr+1)−φ1(tr)tr+1−tr|+|φ1(tr+s+1)−φ1(tr+s)tr+s+1−tr+s−φ1(tr+s)−φ1(tr+s−1)tr+s−tr+s−1|+|φ1(tr+s)−φ1(tr+s−1)tr+s1−tr+s−1|≤2|φ1(tr+1)−φ1(tr)tr+1−tr|+|φ1(tr+2)−φ1(tr+1)tr+2−tr+1|+2|φ1(tr+s+1)−φ1(tr+s)tr+s+1−tr+s|+|φ1(tr+s)−φ1(tr+s−1)tr+s−tr+s−1|≤|y1−y2β−α|(tr+1−α)tr+1−tr+12|y1−y2β−α|+|y1−y22||1−tr+s−αβ−α|tr+s+1−tr+s+12|y1−y2β−α|≤2|y1−y2β−α|+|y1−y22||1−tr+s−αβ−α|tr+s+1−tr+s≤2|y1−y2β−α|+12|y1−y2β−α||β−tr+s|tr+s+1−tr+s≤52|y1−y2β−α|<∞. |
Thus, φ1∈κBV2([a,b]). It can be proved analogously that φ2∈κBV2([a,b]).
Theorem 3.1. Let h:[a,b]×R→R be a function such that for each x∈[a,b], h(x,⋅) is a continuous function. Suposse that the superposition operator H generated by h is a globally Lipschitzian map i.e., there is L≥0 such that
‖H(φ1)−H(φ2)‖κBV2≤L‖φ1−φ2‖κBV2,φ1,φ2∈κBV2([a,b]). | (3.1) |
Then, H transforms the space κBV2([a,b]) into itself if and only if h satisfies the {Matkowski's condition}
h(x,y)=h1(x)y+h0(x),x∈[a,b], y∈R, |
with h0,h1∈κBV2([a,b]).
Proof. Let us suppose that H transforms the space κBV2([a,b]) into itself. Let α,β∈[a,b] be such that a<α<β. For y1,y2∈R, with y1≠y2, consider the functions in lemma 3.1 given by
φj(t)=12[ηα,β(t)(y1−y2)+yj+y2],j=1,2. |
Notice that
φ1(α)=φ2(β)=12(y1+y2),φ2(α)=y2,andφ1(β)=y1. |
Also,
(φ1−φ2)(t)=y1−y22,t∈[a,b], |
is a constant function. So, by theorem 2.5
‖φ1−φ2‖κBV2=|(φ1−φ2)(a)|+|(φ1−φ2)(b)|+κV2(φ1−φ2;[a,b])=|y1−y2|. |
By lemma 3.1 φ1,φ2∈κBV2([a,b]). Then, by (3.1) we have
‖H(φ1)−H(φ2)‖κVB2=|h(a,φ1(a))−h(a,φ2(a))|+|h(b,φ1(b))−h(b,φ2(b))|+κV2(h(⋅,φ1(⋅))−h(⋅,φ2(⋅));[a,b])≤L(‖φ1−φ2‖κBV2)=L|y1−y2|. | (3.2) |
On the other hand,
κV2(h(⋅,φ1(⋅))−h(⋅,φ2(⋅));[a,b])≥|(h(β,φ1(β))−h(β,φ2(β)))−(h(α,φ1(α))−h(α,φ2(α)))β−α−(h(α,φ1(α))−h(α,φ2(α)))−(h(a,φ1(a))−h(a,φ2(a)))α−a|≥|h(β,φ1(β))−h(β,φ2(β))−h(α,φ1(α)+h(α,φ2(α))β−α|−|h(α,φ1(α))−h(α,φ2(α))−h(a,φ1(a))+h(a,φ2(a))α−a|. |
By (3.2) and the inequality above
|h(β,φ1(β))−h(β,φ2(β))−h(α,φ1(α)+h(α,φ2(α))β−α|−|h(α,φ1(α))−h(α,φ2(α))−h(a,φ1(a))+h(a,φ2(a))α−a|+|h(a,φ1(a))−h(a,φ2(a))|+|h(b,φ1(b))−h(b,φ2(b))|≤L|y1−y2| |
or equivalently
|h(β,y1)−h(β,y1+y22)−h(α,y1+y22)+h(α,y2)|−|h(α,y1+y22)−h(α,y2)−h(a,y1+y22)+h(a,y2)α−a||β−α|+|h(a,y1+y22)−h(a,y2)||β−α|+|h(b,y1)−h(b,y1+y22)||β−α|≤L|y1−y2||β−α|. | (3.3) |
Since H(φ)∈κBV2([a,b]) for every function φ∈κBV2([a,b]), putting φ(t)=y for t∈[a,b] we see that for every y∈R the function h(⋅,y)∈κBV2([a,b]). Then, h(⋅,y) is a continuous function. Taking limit as β→α in (3.3) we obtain
|h(α,y1)−h(α,y1+y22)−h(α,y1+y22)+h(α,y2)|≤0. |
Since α is arbitrary (and h(⋅,y) is continuos),
2h(x,y1+y22)=h(x,y1)+h(x,y2),∀x∈[a,b]. |
This shows that h(x,⋅) satisfies the Jensen's equation (see [10], p. 351). But h(x,⋅):R→R is continuous, the theorem 13.2.2 of [10] (p. 354) asserts that there exist h0,h1:[a,b]→R such that
h(x,y)=h1(x)y+h0(x),x∈[a,b],y∈R. | (3.4) |
Note that for each y∈R, we obtain that h(⋅,y)∈κBV2([a,b]). So, by equation (3.4) we have for y=0 and y=1
h0(x)=h(x,0)andh1(x)=h(x,1)−h0(x),∀x∈[a,b]. |
Therefore, h0,h1∈κBV2([a,b]).
Now, suppose that h0,h1∈κBV2([a,b]) and that the operator H is generated by the function h given by h(x,y)=h1(x)y+h0(x),x∈[a,b], y∈R. It follows from proposition 2.1 and remark 2.6 that H transforms the space κBV2([a,b]) into itself which completes the proof.
In this section, we study the existence and uniqueness of solutions in the class of functions of bounded second κ-variation of the nonlinear integral equations (1.1) and (1.2).
Note that the case a=0 in the Eq (1.2) is the Volterra-Hammerstein equation with λ=1.
The Banach Contraction Principle is used in the proofs of the main results to guarantee the existence and uniqueness of solutions of the integral equations.
Theorem 4.1 (Banach Contraction Principle). Let X be a complete metric space and let f:X→X be a contraction mapping. Then, f has a unique fixed point in X. Moreover, if B⊆X is a closed subset such that f(B)⊆B then f:B→B has a unique fixed point in B.
As conditions of existence and uniqueness for the solutions of the Eqs (1.1) and (1.2) on κBV2([a,b]) we set the following hypotheses: Let I=[a,b].
(H1) f:R→R is a locally Lipschitz function.
(H2) g:I→R is a function of bounded second κ-variation on I, that is, g∈κBV2(I).
(H3) K:I×I→R is a function such that κV2(K(⋅,s);I)≤M(s),for eachs∈I, where M:I→R is a Lebesgue integrable function. In addition, K(t,⋅) is Lebesgue integrable for each t∈I, and both K(a,⋅) and K(b,⋅) are L1 integrable functions.
Next, we estimate the second κ-variation of certain auxiliary functions.
Lemma 4.1. Let f and K satisfy hypotheses (H1) and (H3), respectively (with a=0). For every x∈κBV2([0,b]) let F(x) be the integral function given by F(x)(t)=∫b0K(t,s)f(x(s))ds, for all t∈[0,b]. Then,
κV2(F(x);[0,b])≤‖f‖∞∫b0M(s)ds<+∞. |
Proof. The function f∘x is continuous on [0,b] by (H1) and theorem 2.9, so that it is Lebesgue integrable. Since K(t,⋅) is Lebesgue integrable for all t∈[0,b] we have that K(t,⋅)f(x(⋅)) is Lebesgue integrable for all t∈[0,b]. Thus, the function F(x) is well defined.
Let ξ={ti}ni=0∈π3([0,b]). Then,
|F(x)[ti+1,ti+2]−F(x)[ti,ti+1]|=|F(x)(ti+2)−F(x)(ti+1)ti+2−ti+1−F(x)(ti+1)−F(x)(ti)ti+1−ti|=|∫b0[K(ti+2,s)−K(ti+1,s)]f(x(s))dsti+2−ti+1−∫b0[K(ti+1,s)−K(ti,s)]f(x(s))dsti+1−ti|≤sups∈[0,b]|f(x(s))||∫b0[K(⋅,s)[ti+2,ti+1]−K(⋅,s)[ti,ti+1]]ds|. |
Consequently, by (H3) we have
n−2∑i=0|F(x)[ti+1,ti+2]−F(x)[ti,ti+1]|κ(ξ,[0,b])≤sups∈[0,b]|f(x(s))|∫b0n−2∑i=0|(K(⋅,s)[ti+1,ti+2]−K(⋅,s)[ti,ti+1]|κ(ξ,[0,b])ds≤sups∈[0,b]|f(x(s))|∫b0M(s)ds. |
Taking supremum over the set of all partitions ξ∈π3([0,b]) yields
κV2(F(x);[0,b])≤‖f‖∞∫b0M(s)ds<+∞. |
The following result can be obtained by a similar technique to that used in the proofs of lemma above and theorem 2.9 to estimate κV2((F(x)−F(y));[0,b]) and by remark 2.6 and theorem 2.8.
Lemma 4.2. Let f and K satisfy hypotheses (H1) and (H3), respectively (with a=0). For every x∈κBV2([0,b]) let F(x) be defined as in the preceding lemma. Then, for all x,y∈κBV2([0,b]), {
κV2(λ(F(x)−F(y));[0,b])≤CpκLb0(f)|λ|‖x−y‖κBV2∫b0M(s)ds,λ∈[0,b] |
where Cpκ=supt∈[0,b]|pκ(t)| (see (2.3)).
The following remark is an immediate consequence of the lemma 2.1.
Remark 4.2. Let T={(t,s):a≤t≤b,a≤s≤t}, and let K:T→R be a function. If K(⋅,s)∈κBV2([s,b]) for each s∈[a,b] then there exists a function C which depends on s such that
K(⋅,s)[y0,y1]≤C(s), for all y0,y1∈[s,b]. |
Lemma 4.3. Let T={(t,s):a≤t≤b,a≤s≤t} and let K:T→R be a function such that K(⋅,s)∈κBV2([s,b]). If K(s,s)=0 for all s∈[a,b] or if there exists a function L:[a,b]→[0,+∞) such that |K(t,s)t−s|≤L(s) for all t∈[a,b] with t≠s. Then for
ˆK(t,s)={K(t,s),a≤s≤t0t<s≤b |
we have
κV2(ˆK(⋅,s);[a,b])≤2L(s)+3C(s)+κV2(K(⋅,s),[s,b]) | (4.1) |
where C(s) is guaranteed by remark 4.2. If K(s,s)=0 we put L(s)=0 in (4.1).
Proof. Let s∈[a,b], let ξ={ti}ni=0∈Π3([a,b]) and let 0≤r≤n−1 such that s∈[tr,tr+1] and s∉[tr−1,tr]. Then, by applying remark 4.2 we have
n−2∑i=0|ˆK(⋅,s)[ti+1,ti+2]−ˆK(⋅,s)[ti,ti+1]|=|K(tr+1,s)tr+1−tr|+|K(tr+2,s)−K(tr+1,s)tr+2−tr+1−K(tr+1,s)tr+1−tr|+n−2∑i=r+1|K(⋅,s)[ti+1,ti+2]−K(⋅,s)[ti,ti+1]|≤2|K(tr+1,s)tr+1−s|+C(s)+n−2∑i=r+1|K(⋅,s)[ti+1,ti+2]−K(⋅,s)[ti,ti+1]|. |
If K(s,s)=0 for all s∈[a,b] then
n−2∑i=0|ˆK(⋅,s)[ti,ti+1]−ˆK(⋅,s)[ti,ti+1]|≤2|K(tr+1,s)−K(s,s)tr+1−s|+n−1∑i=r+2|K(⋅,s)[ti,ti+1]−K(⋅,s)[ti,ti+1]|+C(s)≤3C(s)+n−2∑i=r+1|K(⋅,s)[ti+1,ti+2]−K(⋅,s)[ti,ti+1]|. |
So,
n−2∑i=0|ˆK(⋅,s)[ti,ti+1]−ˆK(⋅,s)[ti,ti+1]|κ(ξ,[a,b])≤3C(s)+κV2(K(⋅,s),[s,b]). |
Taking supremum over the set of all partitions ξ∈π3([a,b]) we have
κV2(ˆK(⋅,s),[a,b])≤3C(s)+κV2(K(⋅,s),[s,b]). |
Suppose there exists a function L such that |K(t,s)t−s|≤L(s) for each t∈[a,b] with t≠s. Then,
n−2∑i=0|ˆK(⋅,s)[ti,ti+1]−ˆK(⋅,s)[ti,ti+1]|≤C(s)+2L(s)+n−2∑i=r+2|K(⋅,s)[ti,ti+1]−K(⋅,s)[ti,ti+1]|, |
which implies
n−2∑i=0|ˆK(⋅,s)[ti,ti+1]−ˆK(⋅,s)[ti,ti+1]|κ(ξ,[a,b])≤C(s)+2L(s)+κV2(K(⋅,s),[s,b]). |
Taking supremum over the set of all partitions ξ∈π3([a,b]) yields
κV2(ˆK(⋅,s);[a,b])≤C(s)+2L(s)+κV2(K(⋅,s),[s,b]). |
Now we prove the main theorems of this section.
Theorem 4.3. Suppose that f, g and K satisfy hypotheses (H1), (H2), and (H3), respectively (with a=0). Then, there is a number τ>0 such that, for every λ with |λ|<τ, the Eq (1.1) has a unique solution in κBV2([0,b]).
Proof. Let ˆC=∫b0[|K(0,s)|+|K(b,s)|]ds<∞ (see (H3)) and let Cpκ be as in lemma 4.2. Let r>0 be such that ‖g‖κBV2<r and choose a number τ>0 such that
‖g‖κV2+τ‖f(x)‖∞[ˆC+∫b0M(s)ds]<rand | (4.2) |
τCpκLb0(f)[ˆC+∫b0M(s)ds]<1. | (4.3) |
For λ∈R with |λ|<τ define the function G:κBV2([0,b])→κBV2([0,b]) by
G(x)=g(x)+λF(x), |
where F(x)(t)=∫b0K(t,s)f(x(s))ds. By lemma 4.1 and remark 2.6 G is well defined.
Let ¯Br be the closed ball with center 0 and radius r in the space κBV2([0,b]). We begin by showing that G(¯Br)⊂¯Br. Let x∈¯Br. By remark 2.6 and the triangular inequality we have
‖G(x)‖κBV2≤‖g‖κBV2+|λ|‖F(x)‖κBV2. | (4.4) |
Now,
‖F(x)‖κBV2=|F(x)(0)|+|F(x)(b)|+κV2(F(x))=|∫b0K(0,s)f(x(s))ds|+|∫b0K(b,s)f(x(s))ds|+κV2(F(x);[0,b])≤‖f(x)‖∞ˆC+κV2(F(x);[0,b]). | (4.5) |
By lemma 4.1 the inequality (4.5) turns into the following
‖F(x)‖κBV2≤‖f(x)‖∞[ˆC+∫b0M(s)ds]. |
It follows from (4.4) and (4.2) that
‖G(x)‖κBV2≤‖g‖κBV2+|λ|‖f(x)‖∞[ˆC+∫b0M(s)ds]≤‖g‖κBV2+τ‖f(x)‖∞[ˆC+∫b0M(s)ds]<r. |
Thus, G(¯Br)⊂¯Br. Next we show that G is a contraction mapping. Let x,y∈¯Br. Then,
‖G(x)−G(y)‖κBV2=|G(x(0))−G(y(0))|+|G(x(b))−G(y(b))|+κV2(G(x)−G(y);[0,b])≤|λ|∫b0(|K(0,s)|+|K(b,s)|)|f(x(s))−f(y(s))|ds+κV2(λ(F(x)−F(y)),[0,b]). |
Because f∈κBV2([a,b]) is Lipschitz (H1) we infer from theorem 2.8 and lemma 4.2 that
‖G(x)−G(y)‖κBV2≤Lb0(f)|λ|‖x−y‖∞ˆC+CpκLb0(f)|λ|‖x−y‖κBV2∫b0M(s)ds≤CpκLb0(f)|λ|[ˆC+∫b0M(s)ds]‖x−y‖κBV2≤τCpκLb0(f)[ˆC+∫b0M(s)ds]‖x−y‖κBV2. |
By inequality (4.3) G is a contraction mapping and by theorem 4.1 G has a unique fixed point in ¯Br which is to say that there exists a unique x∈¯Br such that
g(t)+λ∫b0K(t,s)f(x(s))ds=x(t). |
Since r can be chosen as large as necessary x is the only solution of Eq (1.1) in κBV2([0,b]).
In addition to hypotheses H1 and H2, we consider
(H4) Let T={(t,s):a≤t≤b,a≤s≤t} and let K:T→R be a function such that K(b,⋅) is L1 integrable, K(t,⋅) is Lebesgue integrable for each t∈[a,b], K(⋅,s) satisfies the hypotheses of lemma 4.3 and
2L(s)+3C(s)+κV2(K(⋅,s);[s,b])≤m(s), |
where m:[a,b]→R is Lebesgue integrable.
Theorem 4.4. Let f, g and K be functions satisfying hypotheses (H1), (H2) and (H4), respectively. If
CpκLba(f)[∫ba|K(b,s)|ds+∫bam(s)ds]<1 | (4.6) |
with Cpκ=supt∈[a,b]|pκ(t)| (see (2.3)) then Eq (1.2) has only one solution in κBV2([a,b]).
Proof. Let r>0 such that
‖g‖κBV2+‖f(x)‖∞[∫ba|K(b,s)|ds+∫bam(s)ds]<r. | (4.7) |
Define the function ˜G:κBV2([a,b])→κBV2([a,b]) by ˜G(x)=g(x)+˜F(x) where ˜F(x)(t)=∫taK(t,s)f(x(s))ds with t∈[a,b], and consider the function
ˆK(t,s)={K(t,s),a≤s≤t0t<s≤b. |
By an argument similar to that used for F(x) in lemma 4.1 it follows that ˜F(x) is well defined. To see that ˜G is well defined we choose x∈κBV2([a,b]) and we estimate κV2(˜F(x);[a,b]). We will proceed in a similar way to that in the proof of lemma 4.1. Let ξ={ti}ni=0∈π3([a,b]). Then,
|˜F(x)[ti+1,ti+2]−˜F(x)[ti,ti+1]|=|˜F(x)(ti+2)−˜F(x)(ti+1)ti+2−ti+1−˜F(x)(ti+1)−˜F(x)(ti)ti+1−ti|=|∫ti+2aK(ti+2,s)f(x(s))ds−∫ti+1aK(ti+1,s)f(x(s))dsti+2−ti+1−∫ti+1aK(ti+1,s)−∫tiaK(ti,s)f(x(s))dsti+1−ti|=|∫baˆK(ti+2,s)f(x(s))ds−∫baˆK(ti+1,s)f(x(s))dsti+2−ti+1−∫baˆK(ti+1,s)f(x(s))ds−∫baˆK(ti,s)dsti+1−ti|≤sups∈[a,b]|f(x(s))||∫ba[ˆK(⋅,s)[ti+2,ti+1]−ˆK(⋅,s)[ti,ti+1]]ds|. |
Consequently, by lemma 4.3 and hypothesis (H4) we have
n−2∑i=0|˜F(x)[ti+1,ti+2]−˜F(x)[ti,ti+1]|κ(ξ,[a,b])≤sups∈[a,b]|f(x(s))|∫ban−2∑i=0|(ˆK(⋅,s)[ti+1,ti+2]−ˆK(⋅,s)[ti,ti+1]|κ(ξ,[a,b])ds≤sups∈[a,b]|f(x(s))|∫bam(s)ds. |
Taking supremum over the set of all partitions ξ∈π3([a,b]) yields
κV2(˜F(x))≤‖f‖∞(∫bam(s)ds)<+∞. | (4.8) |
Therefore, ˜G is well defined.
Note that if x,y∈κBV2([a,b]) then we can use the ideas above, (H1) and theorem 2.8 to deduce that
κV2(˜F(x)−˜F(y);[a,b])≤CpκLba(f)‖x−y‖κBV2∫bam(s)ds. | (4.9) |
We first show that ˜G(¯Br)⊂¯Br. By remark 2.6 and the triangular inequality we have
‖˜G(x)‖κBV2≤‖g‖κBV2+‖˜F(x)‖κBV2 | (4.10) |
but
‖˜F(x)‖κBV2=|˜F(x(a))|+|˜F(x(b))|+κV2(˜F(x);[a,b])≤‖f‖∞∫ba|K(b,s)|ds+κV2(˜F(x);[a,b]). | (4.11) |
Substituting (4.11) and (4.8) into the inequality (4.10) and by (4.7) we obtain
‖˜G(x)‖κBV2≤‖g‖κBV2+‖f‖∞[∫ba|K(b,s)|ds+∫bam(s)ds]<r. |
Now, let x,y∈¯Br. By inequality (4.9), theorem 2.8, hypothesis (H1) and (4.6) we have
‖˜G(x)−˜G(y)‖κBV2=|˜F(x)(b)−˜F(y)(b)|+κV2(˜F(x)−˜F(y);[a,b])≤CpκLba(f)[∫ba|K(b,s)|ds+∫bam(s)ds]‖x−y‖κBV2<‖x−y‖κBV2. |
So, ˜G is a contraction and by theorem 4.1 ˜G has a single fixed point at ¯Br that is there is a unique ˜x∈¯Br such that
g(t)+∫taK(t,s)f(˜x(s))ds=˜x(t). |
Since r can be chosen arbitrarily large ˜x is the only solution of Eq (1.2) in κBV2([a,b]).
Note the fact that the solutions of the Eqs (1.1) and (1.2) are in κBV2(I), implying that (1.1) and (1.2) are continuous on I by virtue of theorem 2.9.
In this section we give examples of application of theorem 4.3 and theorem 4.4.
Example 5.1. Several methods have been used to solve the Hammerstein integral equation. In [7], the nonlinear Hammerstein integral equation
x(t)=−512t+1+∫10stx2(s)ds,witht∈[0,1], | (5.1) |
is considered with an exact solution x(t)=1+13t. In that paper it can be observed that the Sinc-collocation method yields an indistinguishable approximation from the exact solution so the method is fairly accurate. In order to complement this approximate method we show that such solution is unique in the space of functions of bounded second κ-variation. Indeed, let f:R→R be defined by f(x)=x2 and g:[0,1]→R be defined by g(t)=−512t+1.
(1) It is clear that f is locally Lipschitz.
(2) By theorem 2.5 we have that g∈κBV2([0,1]).
(3) Let K(t,s)=ts for t,s∈[0,1]. Clearly, K(t,⋅) is Lebesgue integrable for all t∈[0,1]. By theorem 2.5
κV2(K(⋅,s),[0,1])=0≤s=M(s), |
with M Lebesgue integrable. Also,
∫10|K(t,s)|ds=∫10|ts|ds≤∫10|s|ds=12 |
then K(0,⋅) and K(1,⋅) are L1 integrable.
Therefore, all conditions of theorem 4.3 hold and so Eq (5.1) has a unique solution in κBV2([0,1]).
Example 5.2. Consider the Volterra-Hammerstein integral equation
x(t)=−14t5−23t4−56t3−t2+1+∫t0(ts−s2)x2(s)ds,with0≤t≤12. | (5.2) |
We will show that there is a unique solution in κBV2([0,12]) (which will be continuous). Let f:R→R be defined by f(x)=x2 and let g:[0,12]→R be defined by g(t)=−14t5−23t4−56t3−t2+1.
(1) It is clear that f is locally Lipschitz with L120(f)=1.
(2) Corollary 2.1 guarantees that g∈κBV2([0,12]).
(3) Let T={(t,s):0≤t≤12,0≤s≤t}, let K:T→R be given by K(t,s)=ts−s2 and let
ˆK(t,s)={K(t,s),0≤s≤t0,t<s≤12. |
Clearly, K(t,⋅) is Lebesgue integrable for all t∈[0,12] and K(12,⋅) is L1-integrable since
∫120|K(12,s)|ds=∫120|12s−s2|ds=148<+∞. |
Note that by theorem 2.5,
K(⋅,s)∈κV2([0,12]) |
and even more so κV2(K(⋅,s),[s,12])=0. Furthermore,
|K(⋅,s)[x,y]|=|K(x,s)−K(y,s)x−y|=|s|=sand|K(t,s)t−s|=|s|=s. |
Then, for C(s)=s and L(s)=s the hypothesis of lemma 4.3 are satisfied. In this way we have
κV2(ˆK(⋅,s);[0,12])≤2L(s)+3C(s)+κV2(K(⋅,s),[s,12])≤5s=m(s). |
It is evident that m is Lebesgue integrable.
(4) Finally, consider the distortion function κα:[0,1]→[0,1] given by κα(t)=tα (the entropy function corresponding to this distortion function is called Lipschitz entropy function). In the particular case α=13,
pκ(t)=1+2t(12−t)(κ(1−2t)+κ(2t))=1+(t−2t2)[3√1−2t+3√2t]. |
So, pκ attains its maximum value on t=14, i.e.
pκ(14)=1+3√48=1.19842<1.2=Cpκ. |
Since
∫120|K(12,s)|ds+∫120m(s)ds=3148=0.64583 |
we have
CpκL120(f)[∫120|K(12,s)|ds+∫120m(s)ds]<(1.2).(1)(0.64583)=0.77499<1. |
Therefore, all conditions of theorem 4.4 are satisfied and the equation (5.2) has a unique solution in κBV2([0,12]).
In this paper we proved that the space of polynomial functions is contained in the space of functions of bounded second κ-variation κBV2([a,b]) and that the latter is contained in the space of absolutely continuous functions which was a very useful result for the development of this work. In addition, we proved that the pointwise product is a closed operations in κBV2([a,b]). By using the technique of Kuczma for the Jensen's equation we also provided necessary conditions for the superposition operator to act on κBV2([a,b]). We then stated the theorem for the existence and uniqueness of the solutions of the Hammerstein and Volterra integral equations by means of Banach Contraction Principle. Finally, we presented some applications. 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 some new spaces of generalized bounded variation and that these results may be also a contribution to different areas which applications are modeled by this type of integral equations.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
We express our gratitude to the referees for their diligent review and valuable input.
The authors declare that there are no conflicts of interest regarding the publication of the paper.
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