Nonlocal type asymptotic behavior for solutions of second order difference equations

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R E S E A R C H

Open Access

Nonlocal type asymptotic behavior for

solutions of second order difference

equations

Cristóbal González and Antonio Jiménez-Melado

*

*_{Correspondence: [email protected]}

Dept. Análisis Matemático, Estadística e I.O., y Matemática Aplicada, Fac. Ciencias, Univ. Málaga, Málaga, 29071, Spain

Abstract

In this work we consider a discrete nonlocal problem of the following type:

(qx)(n) +f(n+ 1,x(n+ 1)) = 0,

x(∞) =g(x),

in the context of an arbitrary Banach space (X,·X), and we give suﬃcient conditions

that ensure the existence of solutions to it.

In order to present our result, we shall need to study conditions that ensure the existence of solutions with a nonlocal asymptotic behavior for the following equation:

x(n) =g(x) – ∞

k=n+1 k–1

j=n

1

q(j)f

(k

,x(k)

)

.

MSC: 39A22; 39B99; 47N20

Keywords: nonlocal problem; asymptotic behavior; second order diﬀerence equation; Leray-Schauder type ﬁxed point theorem

1 Introduction

Throughout this paperN,R, andR+will denote the usual sets of numbers,i.e., nonnegative

integers, reals, and nonnegative reals, respectively. Also, (X,·X) will denote an arbitrary (real) Banach space, whose (open) balls (with centerx∈Xand radius r) areB(x,r), or BX(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 setA⊂Xare writtenA,co(A), andco(A), respectively. Sequences (or double sequences) will be understood as mappings deﬁned on N(orN×N), thus avoiding as much as possible the use of subscripts or superscripts. For instance, a sequence inXwill be a mapping x :N→X, whosenth component is x(n). The operator forward diﬀerence,, applies sequences into sequences. Thus, if x is a sequence, thenxis a sequence that component-wise is given byx(n) = x(n+ ) – x(n). Rather than considering the whole space of sequences onX, we shall be working with_∞(X), the usual

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space of bounded sequences onX, which becomes a Banach space under the sup norm:

x∞=sup_nx(n)X.

The study of diﬀerence equations in the context of Banach spaces has always attracted the attention of several authors (see for instance [] and the references therein). In this paper, we follow this line of work.

Forqa given sequence of positive real numbers andf anX-valued function deﬁned on N×X, we look, in this paper, for conditions that ensure the existence of boundedX-valued sequences x, solutions of the following problem:

⎧ ⎨ ⎩

(qx)(n) +f(n+ , x(n+ )) = ,

x(∞) =g(x). ()

Here, x(∞) =limn→∞x(n) andgis anX-valued function defined on∞(X). Thisnonlocal asymptotic behavior, as opposed tolocalasymptotic behavior given by x(∞) = constant, is the interesting part of our study and makes it different from what, we believe, has been done before. For instance, in [], a quite general set of hypotheses was given in order to ensure the existence of solutions with a specified local asymptotic behavior for the above equation (see also [–]). Those hypotheses were enough in order to use the Sadovski˘ı fixed point theorem, but when we tried to adapt them to the nonlocal setting, we did not see how to do it without imposing conditions difficult to verify. In the end, we have realized that a Leray-Schauder type fixed point theorem is the appropriate tool to obtain a good result for this nonlocal setting. The basics of this theory are briefly explained in Section , together with a result on fixed points of mappings defined on sequence spaces which is similar to Theorem  in [].

In Section , we shall study the following summation equation:

x(n) =g(x) –

∞

k=n+

k–

j=n  q(j)f

k, x(k), ()

whose solutions, as one can easily check, are also solutions of the problem (), because the convergence of the series is implicitly assumed in this direction. In the other direction, one needs to provide conditions ensuring the convergence of the series in order to see that the solutions to problem () are also solutions of the above summation equation. Actually, an equation a bit more general than () will be the main object of our study. The paper will be complete once we present the corresponding result to obtain solutions of problem ().

2 A ﬁxed point theorem

Compactness is a usual ingredient when trying to obtain results about fixed points of map-pings. Without going any further, the well-known Schauder fixed point theorem asserts that a compact mapping,i.e., a continuous mapping that transforms bounded sets into rel-atively compact ones, has a fixed point, whenever it leaves invariant a nonempty, bounded, closed, and convex set. The invariance of the set can be relaxed if we refer to an also well-known Leray-Schauder type fixed point theorem, which states that a compact mappingT has a fixed point if it satisfies the Leray-Schauder boundary condition onB(,R), for some R> , that is, if there existsR>  such that

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A natural generalization of compactness in these results deal with the concept of mea-sures of noncompactness. A typical one is the measure of noncompactness of Kuratowski, defined for each bounded setAas the infimumα(A) of theε>  for whichAadmits a finiteε-cover. (By anε-cover ofAwe mean a collection of setsFi,i∈I, of diameter less thanε, whose union containsA.)

α(A) =inf

ε>  :A⊆ n

i=

Fi,diam(Fi) <ε

.

In general, a measure of noncompactness inXis a mappingmthat assigns to each bounded set inXa nonnegative real number, such that

(MNC) for all boundedA⊂X,m(A) = if and only ifAis compact; (MNC) for all boundedA⊂X,m(A) =m(A);

(MNC) for all boundedA,B⊂X,m(A∪B) =max{m(A),m(B)}.

A property deduced easily from (MNC) is that if A⊆B are bounded, then m(A)≤

max{m(A),m(B)}=m(A∪B) =m(B). Additionally, the measure of noncompactness of Ku-ratowski also satisﬁes the following useful properties:

(K) for all boundedA⊂Xand all scalarsλ,α(λA) =|λ|α(A); (K) for all boundedA,B⊂X,α(A+B)≤α(A) +α(B); (K) for all boundedA⊂X,α(co(A)) =α(A).

Compact mappings are particular cases of m-condensing mappings. These are con-tinuous mappingsT that send bounded setsAwith m(A) >  into bounded setsT(A) with m(T(A)) <m(A). In this respect, in , Sadovski˘ı [] obtained an extension of Schauder’s fixed point theorem for condensing mappings, namely, if the measure of non-compactness is such thatm(A) =m(co(A)) for any bounded setA⊂X, then any condens-ing mappcondens-ing 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 con-vex hulls, we see that any condensing mapping has a fixed point if it satisfies the Leray-Schauder boundary condition (LS) onBX(,R), for someR> . For proofs of these results, at least for the Kuratowski measure of noncompactness, see, for instance, [] and [].

From now on, we restrict ourselves to the Kuratowski measures of noncompactnessαX inX, andα∞in∞(X), respectively. In , the authors [] used Sadovski˘ı ﬁxed point

the-orem in order to obtain a result as regards ﬁxed points for mappings on sequence spaces.

Theorem A LetK⊂_∞(X)be a nonempty,bounded,closed,and convex set.Assume that a mappingT: K→Kis given byT= (T,T, . . .),where each Tnis(α∞,αX)-condensing, andlim_ndiam(TnK) = .ThenThas a ﬁxed point inK.

The proof just consisted in showing that T isα_∞-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  LetT:∞(X)→∞(X), T = (T,T, . . .),be such that maps bounded sets into

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If,besides, Tsatisﬁes the Leray-Schauder boundary condition(LS)on B_∞(X)(,R),for

some R> ,thenThas a ﬁxed point.

3 Solutions for a summation equation

As we said in the Introduction, we will study an equation a bit more general than (), in the context of the Banach spaceX,

x(n) =g(x) –

∞

k=n+

P(n,k)fk, x(k), (E)

whereg:∞(X)→X,P:N×N→R+, andf:N×X→X.

Observe that () is just the caseP(n,k) =_jk₌–_n _q₍_j₎, fork≥n+ .

In order to ensure the existence of solutions to this equation we have found that the following conditions ﬁt our requirements and are quite general:

(H) g:∞(X)→Xis compact and bounded.

(H) There exists a sequencep:N→R+such that

P(n,k)≤p(k), for alln∈Nand allk≥n+ .

(H) There existϕ:R+→R+, nondecreasing, anda:N→R+such that

f(n,x)_X≤a(n)ϕxX

, for alln∈Nand allx∈X.

(This implies that eachf(n,·)maps bounded sets into bounded sets.) (H) ϕ(s) > for alls> , and_∞ 

ϕ(s)ds= +∞.

(H) ∞_n_=p(n)a(n) <∞.

(H) Fork∈N,f(k,·)is continuous onX, there existsLk≥such that

αX

f(k,A)≤LkαX(A), for all boundedA⊂X,and

∞

k=

p(k)Lk< .

Observe that if eachf(k,·) were a compact function then we could takeLk=  for allk in (H). Also, notice that (H) and (H) tell us that_kp(k)f(k, x(k)) converges absolutely for each x∈∞(X), and it does it uniformly on each bounded set of∞(X). Precisely, this

convergence condition will help us to show that any solution of () is also a solution of (). Recall that we mentioned before that the other direction was straightforward.

Proposition  Letx∈∞(X)be a solution of ().Put P(n,k) =kj=–nq(j) (for k≥n+ ),

and p(k) =k_j_=– 

q(j) (for k≥).Assume that conditions(H) and(H)hold,as well as

limkp(k) =∞.Thenxis also a solution of ().

Remark  The hypothesis_j_q₍_j₎=∞(p(k)→ ∞) becomes natural if we think ofq(j) as equal to  for allj. Equation () would then be

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Proof For ﬁxednandN≥nwe have

k≥n+ , the series in the resulting right hand side converges absolutely, by (H) and (H). So, taking limits asN→ ∞, we conclude that there exists the limity∈Xofq(N)x(N) asN→ ∞.

We claim that this limit must be y= . For, assuming y= , by Hahn-Banach, a continuous (real) linear functional H exists on X such that H(y) = . This shows that q(N)H(x(N))→H(y) = , and hence, there existsN∈Nsuch thatH(x(N)) > __q₍_N₎

leaving us with a contradiction at the moment of taking limits asN→ ∞, for the left hand side converges toH(x(∞)) =H(g(x)) while the right hand side goes to∞because, by hypothesis,_j_q₍_j₎=∞.

Once the claim is proved, () yields

x(n) = 

Let us now state and prove the main result of this paper.

Theorem  Under conditions(H)-(H),equation(E)has a solution in∞(X).

Proof First we get rid of the case in which, for one reason or another, the sum in the right hand side of equation (E) is always missing (equal to  for alln),

x(n) =g(x).

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compact and bounded, showing thatco(g˜(X)) is ag˜-invariant nonempty, bounded, closed, and convex set. So by Schauder’s ﬁxed point theorem, g˜ has a ﬁxed pointx∈X and,

consequently, the corresponding constant sequence cx_{is a solution of our equation.}

From now on, we assume that the sum on the right hand side of equation (E) is not always . Deﬁne T :∞(X)→∞(X) as T = G + S where G(x) = cg(x), and each component

of S = (S,S, . . .) isSn(x) = –

∞

k=n+P(n,k)f(k, x(k)). Observe that G is well deﬁned, and

so is eachSnbecause conditions (H), (H), and (H) tell us that the series deﬁningSn converges absolutely for each x∈∞(X), and does it uniformly on each bounded set of ∞(X). Clearly, by (HH), G is compact and bounded.

The plan for the rest of the proof consists in showing that S isα_∞-condensing. This will be achieved using the first part of Theorem . From here, it will be clear that T = G+ S isα∞-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).

We initiate our route map showing that S = (S,S, . . .) satisﬁes the conditions in

The-orem . We start checking that S maps bounded sets into bounded sets. If R> , and x∈∞(X),x∞≤R, then conditions (H), (H), (H), and the fact thatϕis

nondecreas-The diameter condition on a bounded set, say contained in a ball of radiusR, also follows from the above argument, for the right hand side in () goes to zero asn→ ∞.

proving the continuity ofSnat x.

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of A ontoX, and notice thatαX(Ak)≤α∞(A), for ifε>  is such that there exists a ﬁnite ε-cover of A, then thekth projections of this cover provide a ﬁniteε-cover (inXthis time) ofAk. Also notice that

Sn(A) =

and that this last set is bounded; so, using (H) and the algebraic properties of the Kura-towski measure of noncompactness, we have for anyN∈N,N≥n+ ,

According to our route map, to finish the proof, we just need to show that T satisfies the Leray-Schauder boundary condition onB_∞(X)(,R) for someR> . To find what

condi-tions must satisfy suchR> , assume that there existλ>  and y∈_∞(X), withy_∞=R and Ty =λy.

By (H),gis bounded, so there isMg>  such thatg(x)X≤Mgfor all x∈∞(X). Now,

forn∈N, using the hypotheses at our disposal,

y(n)_X=

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Letz:N→Rbe deﬁned, for eachn, as the rightmost member of (). Observe ﬁrst that

Observe that is a strictly increasing function from [,∞) onto [,∞), because ∞

 

ϕ(s)ds=∞, so

– _{is a well-deﬁned strictly increasing function from [,}_∞_{) onto}

[,∞). Therefore, forn∈N, the contradiction of assuming alsoy_∞=R. This ﬁnishes the proof of the theorem.

As an easy consequence of this theorem, we present now a result about existence of solutions for the second order diﬀerence problem ().

Corollary  With P(n,k) =k_j₌–_n _q₍_j₎,for k≥n+ ,where q is a sequence of positive num-bers,under conditions(H)-(H),the second order diﬀerence problem()has a solution x∈∞(X).

For the proof, first find a solution of the corresponding difference equation (E) or, for the case, of equation (). Then this solution is also a solution of ().

Remark  As we mentioned before, Corollary  includes the well-known problem on the existence of asymptotically constant solutions, just takingg(x) =C, withC a constant. Obviously, the situation covered by our result is more general, and this can be seen con-sidering a bounded, continuous, and nonconstantg:∞(X)→X. To obtain an explicit

example of this type, take a continuous and bounded mapg:R→Randu∈X, and

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper, and all authors read and approved the ﬁnal manuscript.

Acknowledgements

Research was partially supported by the Spanish (Grant MTM2014-52865-P) and regional Andalusian (Grant FQM210) Governments.

Received: 4 May 2016 Accepted: 4 September 2016

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