Nonlocal type asymptotic behavior for solutions of second order difference equations
- Cristóbal González^{1} and
- Antonio Jiménez-Melado^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-016-0962-8
© González and Jiménez-Melado 2016
Received: 4 May 2016
Accepted: 4 September 2016
Published: 13 September 2016
Abstract
Keywords
MSC
1 Introduction
Throughout this paper \(\mathbb{N}\), \(\mathbb{R}\), and \(\mathbb{R}_{+}\) will denote the usual sets of numbers, i.e., nonnegative integers, reals, and nonnegative reals, respectively. Also, \((X,\Vert{\cdot}\Vert_{X})\) will denote an arbitrary (real) Banach space, whose (open) balls (with center \(x\in X\) and radius r) are \(B(x,r)\), or \(B_{X}(x,r)\) depending on whether the base space needs emphasis or not. The closure, the convex hull, and the closed convex hull of a given set \(A\subset X\) are written A̅, \(\operatorname{co}(A)\), and \(\overline{\operatorname{co}}(A)\), respectively. Sequences (or double sequences) will be understood as mappings defined on \(\mathbb{N}\) (or \(\mathbb{N} \times\mathbb{N}\)), thus avoiding as much as possible the use of subscripts or superscripts. For instance, a sequence in X will be a mapping \(\mathbf {x}: \mathbb{N}\to X\), whose nth component is \(\mathbf {x}(n)\). The operator forward difference, Δ, applies sequences into sequences. Thus, if x is a sequence, then Δx is a sequence that component-wise is given by \(\Delta \mathbf {x}(n) = \mathbf {x}(n+1) -\mathbf {x}(n)\). Rather than considering the whole space of sequences on X, we shall be working with \(\ell _{\infty}(X)\), the usual space of bounded sequences on X, which becomes a Banach space under the sup norm: \(\Vert{ \mathbf {x}}\Vert_{\infty}= \sup_{n}\Vert{ \mathbf {x}(n)}\Vert_{X}\).
The study of difference equations in the context of Banach spaces has always attracted the attention of several authors (see for instance [1] and the references therein). In this paper, we follow this line of work.
2 A fixed point theorem
- (MNC1)
for all bounded \(A\subset X\), \(m(A)=0\) if and only if A̅ is compact;
- (MNC2)
for all bounded \(A\subset X\), \(m(A)=m(\overline{A})\);
- (MNC3)
for all bounded \(A,B\subset X\), \(m(A\cup B)=\max\{ m(A),m(B)\}\).
- (K1)
for all bounded \(A\subset X\) and all scalars λ, \(\alpha(\lambda A) = |\lambda|\alpha(A)\);
- (K2)
for all bounded \(A,B\subset X\), \(\alpha(A+B)\leq \alpha(A)+\alpha(B)\);
- (K3)
for all bounded \(A\subset X\), \(\alpha (\operatorname{co} (A) ) =\alpha(A)\).
Compact mappings are particular cases of m-condensing mappings. These are continuous mappings T that send bounded sets A with \(m(A)>0\) into bounded sets \(T(A)\) with \(m({T(A)}) < m(A)\). In this respect, in 1967, Sadovskiĭ [10] obtained an extension of Schauder’s fixed point theorem for condensing mappings, namely, if the measure of noncompactness is such that \(m(A) = m (\operatorname{co}(A) )\) for any bounded set \(A\subset X\), then any condensing mapping has a fixed point, whenever it leaves invariant a nonempty, bounded, closed, and convex set. Again, the invariance of the set can be relaxed via a Leray-Schauder type fixed point theorem: if the measure of noncompactness is unaltered after applying convex hulls, we see that any condensing mapping has a fixed point if it satisfies the Leray-Schauder boundary condition (LS) on \(\overline{B}_{X}(0,R)\), for some \(R>0\). For proofs of these results, at least for the Kuratowski measure of noncompactness, see, for instance, [11] and [12].
From now on, we restrict ourselves to the Kuratowski measures of noncompactness \(\alpha_{X}\) in X, and \(\alpha_{\infty}\) in \(\ell _{\infty}(X)\), respectively. In 2003, the authors [2] used Sadovskiĭ fixed point theorem in order to obtain a result as regards fixed points for mappings on sequence spaces.
Theorem A
Let \(\mathbf {K}\subset\ell_{\infty}(X)\) be a nonempty, bounded, closed, and convex set. Assume that a mapping \(\mathbf {T}:\mathbf {K}\to \mathbf {K}\) is given by \(\mathbf {T} = (T_{0},T_{1},\ldots)\), where each \(T_{n}\) is \((\alpha _{\infty},\alpha_{X})\)-condensing, and \(\lim_{n} \operatorname{diam}(T_{n}\mathbf {K}) =0\). Then T has a fixed point in K.
The proof just consisted in showing that T is \(\alpha _{\infty}\)-condensing. Now, we shall need a result which is a Leray-Schauder counterpart of Theorem A, in order to relax the existence of a T-invariant set. This observation is just stated as our first result, for which there is no need to repeat the proof.
Theorem 1
Let \(\mathbf {T}:\ell_{\infty}(X)\to\ell_{\infty}(X)\), \(\mathbf {T} = (T_{0},T_{1},\ldots)\), be such that maps bounded sets into bounded sets. Assume that each \(T_{n}\) is \((\alpha_{\infty},\alpha_{X})\)-condensing, and that \(\lim_{n} \operatorname{diam}T_{n}(\mathbf {\mathcal{U}}) =0\) for each bounded set \(\mathbf {\mathcal{U}}\in\ell_{\infty}(X)\). Then T is \(\alpha_{\infty}\)-condensing.
If, besides, T satisfies the Leray-Schauder boundary condition (LS) on \(\overline{B}_{\ell_{\infty}(X)}(\mathbf {0},R)\), for some \(R>0\), then T has a fixed point.
3 Solutions for a summation equation
Observe that (2) is just the case \(P(n,k) = \sum_{j=n}^{k-1} \frac{1}{q(j)}\), for \(k\geq n+1\).
- (H1)
\(g: \ell_{\infty}(X) \to X\) is compact and bounded.
- (H2)There exists a sequence \(p:\mathbb{N}\to\mathbb {R}_{+}\) such that$$ P(n,k)\leq p(k),\quad \mbox{for all }n\in\mathbb{N}\mbox{ and all }k\geq n+1. $$
- (H3)There exist \(\varphi: \mathbb{R}_{+}\to\mathbb{R}_{+}\), nondecreasing, and \(a:\mathbb{N}\to\mathbb{R}_{+}\) such that(This implies that each \(f(n,\cdot)\) maps bounded sets into bounded sets.)$$ \bigl\Vert {f(n,x)}\bigr\Vert _{X} \leq a(n) \varphi\bigl({ \Vert {x}\Vert _{X}}\bigr), \quad \mbox{for all }n\in\mathbb{N} \mbox{ and all }x\in X. $$
- (H4)
\(\varphi(s) >0\) for all \(s>0\), and \(\int _{1}^{\infty}\frac{1}{\varphi(s)} \,ds = +\infty\).
- (H5)
\(\sum_{n=0}^{\infty}p(n) a(n) <\infty\).
- (H6)For \(k\in\mathbb{N}\), \(f(k,\cdot)\) is continuous on X, there exists \(L_{k}\geq0\) such that$$ \alpha_{X}\bigl({f(k,A)}\bigr)\leq L_{k} \alpha_{X}(A),\quad \mbox{for all bounded } A\subset X, \mbox{and } \sum_{k=0}^{\infty}p(k) L_{k} < 1. $$
Observe that if each \(f(k,\cdot)\) were a compact function then we could take \(L_{k}=0\) for all k in (H6). Also, notice that (H3) and (H5) tell us that \(\sum_{k} p(k)f(k,\mathbf {x}(k))\) converges absolutely for each \(\mathbf {x}\in\ell_{\infty}(X)\), and it does it uniformly on each bounded set of \(\ell_{\infty}(X)\). Precisely, this convergence condition will help us to show that any solution of (1) is also a solution of (2). Recall that we mentioned before that the other direction was straightforward.
Proposition 2
Let \(\mathbf {x}\in\ell_{\infty}(X)\) be a solution of (1). Put \(P(n,k) = \sum_{j=n}^{k-1} \frac{1}{q(j)}\) (for \(k\geq n+1\)), and \(p(k) = \sum_{j=0}^{k-1} \frac{1}{q(j)}\) (for \(k\geq1\)). Assume that conditions (H3) and (H5) hold, as well as \(\lim_{k} p(k) =\infty\). Then x is also a solution of (2).
Remark 1
Proof
Let us now state and prove the main result of this paper.
Theorem 3
Under conditions (H1)-(H6), equation (E) has a solution in \(\ell_{\infty}(X)\).
Proof
From now on, we assume that the sum on the right hand side of equation (E) is not always 0. Define \(\mathbf {T}: \ell_{\infty}(X)\to\ell_{\infty}(X)\) as \(\mathbf {T} = \mathbf {G}+ \mathbf {S}\) where \(\mathbf {G}(\mathbf {x}) = \mathbf {c}^{g(\mathbf {x})}\), and each component of \(\mathbf {S} = (S_{0}, S_{1},\ldots)\) is \(S_{n}(\mathbf {x}) = -\sum_{k=n+1}^{\infty}P(n,k) f({k,\mathbf {x}(k)})\). Observe that G is well defined, and so is each \(S_{n}\) because conditions (H2), (H3), and (H5) tell us that the series defining \(S_{n}\) converges absolutely for each \(\mathbf {x}\in\ell_{\infty}(X)\), and does it uniformly on each bounded set of \(\ell_{\infty}(X)\). Clearly, by (HH1), G is compact and bounded.
The plan for the rest of the proof consists in showing that S is \(\alpha_{\infty}\)-condensing. This will be achieved using the first part of Theorem 1. From here, it will be clear that \(\mathbf {T} = \mathbf {G} + \mathbf {S}\) is \(\alpha _{\infty}\)-condensing. Finally, we shall show that T satisfies a Leray-Schauder boundary condition (LS) on some closed ball, concluding that T has a fixed point, which necessarily is a solution to equation (E).
The diameter condition on a bounded set, say contained in a ball of radius R, also follows from the above argument, for the right hand side in (4) goes to zero as \(n\to\infty\).
According to our route map, to finish the proof, we just need to show that T satisfies the Leray-Schauder boundary condition on \(\overline{B}_{\ell_{\infty}(X)}(0,R)\) for some \(R>0\). To find what conditions must satisfy such \(R>0\), assume that there exist \(\lambda>1\) and \(\mathbf {y}\in \ell_{\infty}(X)\), with \(\Vert{ \mathbf {y}}\Vert_{\infty}=R\) and \(\mathbf {T}\mathbf {y} = \lambda \mathbf {y}\).
The technique we are about to use now consists in applying an infinite discrete analog of the well-known Gronwall-Bellman-Bihari inequality, which has extensively been studied by Pachpatte (see, for instance, [13]), especially for finite difference equations. We just show the method for the sake of completeness.
As an easy consequence of this theorem, we present now a result about existence of solutions for the second order difference problem (1).
Corollary 1
With \(P(n,k) = \sum_{j=n}^{k-1} \frac{1}{q(j)}\), for \(k\geq n+1\), where q is a sequence of positive numbers, under conditions (H1)-(H6), the second order difference problem (1) has a solution \(\mathbf {x}\in\ell_{\infty}(X)\).
For the proof, first find a solution of the corresponding difference equation (E) or, for the case, of equation (2). Then this solution is also a solution of (1).
Remark 2
As we mentioned before, Corollary 1 includes the well-known problem on the existence of asymptotically constant solutions, just taking \(g(\mathbf {x})=C\), with C a constant. Obviously, the situation covered by our result is more general, and this can be seen considering a bounded, continuous, and nonconstant \(g:\ell_{\infty}(X)\to X\). To obtain an explicit example of this type, take a continuous and bounded map \(g_{0}:\mathbb{R}\to\mathbb{R}\) and \(u\in X\), and define \(g:\ell _{\infty}(X)\to X\) as \(g(\mathbf {x})=g_{0}(\Vert \mathbf {x}\Vert_{\infty}) u\), or \(g(\mathbf {x})=g_{0}(\Vert \mathbf {x}(1)\Vert_{X}) u\). Then it is easily checked that g is bounded and compact.
Declarations
Acknowledgements
Research was partially supported by the Spanish (Grant MTM2014-52865-P) and regional Andalusian (Grant FQM210) Governments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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