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Periodic forcing for some difference equations in Hilbert spaces.

1 Introduction

Let H be a real Hilbert space with inner product (x,x) and the induced Hilbertian norm [parallel] x [parallel]. Let A : D(A) [subset] H [right arrow] H be a (possibly multivalued) maximal monotone operator. Consider the difference equation

[DELTA][u.sub.n] + [c.sub.n]A[u.sub.n+1] [contains as member] [f.sub.n], n = 0,1, ..., (E)

where ([c.sub.n]) [subset] (0, [infinity]), ([f.sub.n]) [contains as member] H are p-periodic sequences for a positive integer p and [DELTA] is the difference operator defined as usual, i.e., [DELTA][u.sub.n] = [u.sub.n+1] - [u.sub.n].

We shall investigate some conditions that guarantee the existence of periodic solutions to equation (E) as well as the weak or strong convergence of any solution to a periodic one, as n [right arrow] [infinity].

The problem we investigate in the present paper is the discrete analogue of the periodic forcing problem for the "continuous" equation

u'(t)+ Au(t) [contains as member] f (t), t > 0,

studied by J. B. Baillon and A. Haraux [2].

Recently Djafari Rouhani and Khatibzadeh [5] formulated essentially Theorem 1 below that shows that the weak convergence result stated by Baillon and Haraux [2] for the case of a subdifferential operator A has a discrete counterpart for a general maximal monotone operator A. Here we provide a simpler proof of Theorem 1 by using old existing results due to Browder and Petryshyn [4] and Opial (see Lemma 3 below). In addition, we formulate a strong convergence result (Theorem 2) and discuss some examples to illustrate the two theorems.

2 Preparatory Lemmas

In what follows we need the following lemmas.

Lemma 1 ([4]). Let X be a uniformly convex Banach space and let Q be a nonexpansive mapping of X into X (i.e., Q is Lipschitzian with constant 1). Then Q has a fixed point if and only if for any specific [x.sub.0] [member of] X the sequence [x.sub.n] = [Q.sup.n][x.sub.0]) is bounded in X.

Lemma 2 ([4]). Let H be a Hilbert space and let Q : H [right arrow] H be a nonexpansive mapping such that the set F of its fixed points is nonempty and Q is asymptotically regular (i.e., [Q.sup.n+1] x - [Q.sup.n]x right arrow 0 strongly in H as n [right arrow] [infinity] for each x [member of] H). Then, [for all][x.sub.0] [member of] H, every weak cluster point of the sequence [x.sub.n] = [Q.sup.n][x.sub.0] belongs to F.

Lemma 3 (Opial's Lemma, see, e.g., [6], p. 5). Let H be a real Hilbert space and let F be a nonvoid subset of H. Assume that ([x.sub.n]) is a sequence in H satisfying:

(i) the lim [parallel][x.sub.n] - q[parallel] = [rho](q) exists, [for all]q [member of] F;

(ii) any weak cluster point of([x.sub.n]) belongs to F.

Then, there exists a p [member of] F such that [x.sub.n] [right arrow] p weakly in H.

Lemma 4 (see, e.g., [6], p. 42). If A : D(A) [subset] R [right arrow] R is maximal monotone, then there exists a lower semicontinuous (LSC) convex function [phi] : R [right arrow] (-[infinity], +[infinity]] such that A is the subdifferential of [phi]: A = [partial derivative][phi].

3 Main Results

The following theorem is the discrete analogue of the 1977 Baillon and Haraux result [2] and was essentially stated by Djafari Rouhani and Khatibzadeh in a recent paper [5].

Theorem 1. Assume that A : D(A) [subset] H [right arrow] H is a maximal monotone operator. Let [c.sub.n] > 0 and [f.sub.n] [member of] H be p-periodic sequences, i.e., [c.sub.n+p] = [c.sub.n], [f.sub.n+p] = [f.sub.n] (n = 0,1, ...), for a given positive integer p. Then equation (E) has a bounded solution ifand only ifit has at least one p-periodic solution. In this case all solutions of (E) are bounded and for every solution ([u.sub.n]) there exists a p-periodic solution ([[omega].sub.n]) of (E) such that

[u.sub.n] - [[omega].sub.n] [right arrow] 0, weakly in H, as n [right arrow] [infinity].

Moreover, every two periodic solutions differ by an additive constant vector.

Proof. Consider the initial condition

[u.sub.0] = x, (IC)

for a given x [member of] H. We can rewrite equation (E) in the form:

[u.sub.n+1] - [u.sub.n] + [c.sub.n]A[u.sub.n+1] [contains as member] [f.sub.n].

The solution of the problem (E)-(IC) is calculated successively from

[u.sub.n+1] = [(I + [c.sub.n]A).sup.-1] ([u.sub.n] + [f.sub.n]), n = 0,1, ...,

in a unique manner, which will give a unique solution [([u.sub.n]).sub.n[greater than or equal to]0].

If a solution ([u.sub.n]) of (E) is bounded (in particular periodic), then any other solution ([[??].sub.n]) of (E) is bounded, because

[parallel][u.sub.n] - [[??].sub.n][parallel] [less than or equal to] [parallel][u.sub.0] - [[??].sub.0][parallel] [for all]n = 0,1, ... (1)

Set Q: H [right arrow] H,

Qx = [u.sub.p;x],

where ([u.sub.n;x]) is the solution of (E) starting from x: [u.sub.0] = x. From (1) it follows that Q is nonexpansive. We also have [Q.sup.n]x = [u.sub.np;x], n = 0,1,.... Thus [([Q.sup.n]x).sub.n [greater than or equal to] 0] is a bounded sequence for all x [member of] H. Obviously H is uniformly convex so, by Lemma 1, there is an x* [member of] H such that Qx* = x*, i.e., [u.sup.*.sub.p;x*] = [u.sup.*.sub.0] = x*, where ([u.sup.*.sub.n]) is the solution of (E) starting from x*. In fact, x* [member of] D(A). Since both ([c.sub.n]) and ([f.sub.n]) are p-periodic sequences, this implies

[u.sup.*.sub.n+p] = [u.sup.*.sub.n] [for all]n = 0,1, ...

So the first part of the theorem is proved. For the second part we shall use Lemmas 2 and 3. Let F be the set of all fixed points of Q. According to the first part of the theorem, F is nonempty if and only if all the solutions of (E) are bounded. Assume that F is nonempty. Let [u.sub.0] [member of] F, i.e., the corresponding solution [([u.sub.n]).sub.n [greater than or equal to] 0] of equation (E) is p-periodic. Let ([z.sub.n]) be an arbitrary solution of (E) (which is bounded). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each m [member of] {0,1, ... , p - 1} and [for all]k = 1,2,....

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m [member of] {0,1, ... , p - 1},

where C is a constant, independent of m.

In particular (for m = 0) the sequence [([z.sub.kp]).sub.k[greater than or equal to] 0] satisfies the first condition of Opial's Lemma (Lemma 3). For the other condition of Lemma 3, we can use Lemma 2. Obviously, [w.sub.n] : = [z.sub.n] - [u.sub.n] satisfies

[w.sub.n] - [w.sub.n+1] [member of] [c.sub.n](A[z.sub.n+1] - A[u.sub.n+1]), n [greater than or equal to] 0.

Since A is monotone, we have

0 [less than or equal to] ([w.sub.n] - [w.sub.n+1], [w.sub.n+1]), n [greater than or equal to] 0.

In particular, ([parallel][w.sub.n][parallel]) is nonincreasing. From

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we derive

[[SIGMA].sup.[infinity].sub.n=0][parallel][w.sub.n+1] - [w.sub.n][[parallel].sup.2] [less than or equal to] [parallel][w.sub.0][[parallel].sup.2].

Therefore, [w.sub.n+1] - [w.sub.n] [right arrow] 0 as n [right arrow] [infinity], which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so Q is asymptotically regular, since [z.sub.0] is an arbitrary vector. It follows by Lemma 2 that any weak cluster point of [z.sub.kp] = [Q.sup.k][z.sub.0] belongs to F. Thus Lemma 3 implies that [z.sub.kp] converges weakly to some [[omega].sub.0] [member of] F, as k [infinity]. Let ([[omega].sub.n]) be the periodic solution corresponding to [[omega].sub.0]. By the reasoning above ([z.sub.n+1] - [[omega].sub.n+1]) - ([z.sub.n] - [[omega].sub.n]) converges strongly to 0, as n [right arrow] [infinity]. Therefore, [z.sub.kp+m] - [[omega].sub.m] converges weakly to 0 as k [right arrow] [infinity], for all m [member of] {0,1,p - 1}.

In fact, for all n = 0,1,we have n = kp + m, with m [member of] {0,1, ... , p - 1} and k [right arrow ] [infinity] as n [right arrow] [infinity]. Thus, [z.sub.n] - [[omega].sub.n] = [z.sub.kp+m] - [w.sub.m] converges weakly to 0 as n [right arrow] [infinity].

Now, let ([[omega]'.sub.n]) be another periodic solution of (E). By the above reasoning, ([z.sub.kp+m+1] - [[omega]'.sub.m+1]) - ([z.sub.kp+m] - [[omega]'.sub.m]) [right arrow] 0 as k [right arrow] [infinity], strongly in H, for all m [member of] {0, 1, p - 1}. Therefore, [[omega].sub.m+1] - [[omega]'.sub.m+1] = [[omega].sub.m] - [[omega]'.sub.m] and thus [[omega].sub.m] - [[omega]'.sub.m] = [[omega].sub.0] - [[omega]'.sub.0] = Const., for all m [member of] {1, p - 1}, showing that any two periodic solutions differ by an additive constant. The proof is complete.

Open Problem 1. In general the periodic solution is not unique, i.e., F is not a singleton (see Example 1 below). Can one characterize the periodic solution ([[omega].sub.n]) associated with ([u.sub.n]) in Theorem 1?

Open Problem 2. If in Theorem 1 A is the subdifferential of a proper, convex and lower semicontinuous function [phi] : H [right arrow] (-[infinity], +[infinity]] and F is nonempty (i.e., [phi] has at least a minimum point), then it is easy to see that

[phi]([u.sub.kp+m]) [right arrow] [phi]([[omega].sub.m]), (2)

as k [right arrow] [infinity], for m = 0, 1, p - 1. Indeed, assuming for simplicity that A is single-valued, we have for all m [member of] {1, 2, ... , p}

[phi]([u.sub.kp+m]) - [phi]([[omega].sub.m]) [less than or equal to] (A[u.sub.kp+m] - A[[omega].sub.m] + A[[omega].sub.m], [u.sub.kp+m] - [[omega].sub.m] =

[1/[c.sub.m-1]](([u.sub.kp+m-l] - [[omega].sub.m-l]) - ([u.sub.kp+m] - [[omega].sub.m]), [u.sub.kp+m] - [[omega].sub.m]) + (A[[omega].sub.m], [u.sub.kp+m] - [[omega].sub.m]),

which implies

lim[sup.sub.k[right arrow][infinity]] [phi]([u.sub.kp+m]) [less than or equal to] [phi]([[omega].sub.m]).

Therefore (2) holds since [phi] is lower semicontinuous.

Question: What can one say about the rate of convergence in (2)?

Remark 1. Strong convergence in Theorem 1 is not true in general. Indeed, if [f.sub.n] = 0 and [c.sub.n] = c > 0, [[for all].sub.n] [greater than or equal to] 0, then (E) has a bounded solution if and only if all its solutions are bounded. In this case (E) has a 1-periodic solution, i.e. a constant solution, [u.sub.n] = [u.sub.0]: 0 [member of] A[u.sub.0]. It is known that if A is the subdifferential of a proper, convex, lower semicontinuous function, [A.sup.-1]0 [not equal to] [empty set], then every solution [([z.sub.n]).sub.n [greater than or equal to] 0] of

[z.sub.n+1] - [z.sub.n] + cA[z.sub.n+1] [contains as member] 0, n = 0, 1, ...

converges weakly to a point of [A.sup.-1]0, but not strongly in general (see Baillon's counterexample [1]). However, strong convergence is possible in some cases, for instance, if either [(I + A).sup.-1] is a compact operator or if A is strongly monotone, i.e., there is a constant a > 0, such that

([x.sub.1] - [x.sub.2], [y.sub.1] - [y.sub.2]) [greater than or equal to] a[[parallel][x.sub.1] - [x.sub.2][parallel].sup.2], [for all] [x.sub.i] [member of] D(A), [y.sub.i] [member of] A[x.sub.i], i = 1,2. In the latter case, we can state the following result:

Theorem 2. Assume that A : D(A) [subset] H [right arrow] H is a maximal monotone operator, that is also strongly monotone (with a constant a > 0). Let [c.sub.n] > 0 and [f.sub.n] [member of] H be p-periodic sequences for a given positive integer p. Then Equation (E) has a unique p-periodic solution ([[omega].sub.n]) and for every solution ([u.sub.n]) of (E) we have

[u.sub.n] - [[omega].sub.n] [right arrow] 0, strongly in H, as n [right arrow] [infinity].

Proof. Note that [c.sub.n] [greater than or equal to] min{[c.sub.k] : 0 [less than or equal to] k [less than or equal to] p - 1} =: c > 0. Since A is strongly monotone (hence coercive), it follows by Theorem 2 in [3] that all solutions of equation (E) are bounded. Therefore, by the argument used for the first part of

Theorem 1, there exists a p-periodic solution ([[omega].sub.n]) of equation (E). If ([u.sub.n]) is an arbitrary solution of (E), we have

[u.sub.n] - [[omega].sub.n] [member of] [u.sub.n+1] - [[omega].sub.n+1] + [c.sub.n] (A[u.sub.n+1] - A[[omega].sub.n+1]), n = 0,1, ...

Multiplying this equation by [u.sub.n+1] - [[omega].sub.n+1] we easily get

(1 + ac) [parallel][u.sub.n+1] - [[omega].sub.n+1][parallel] [less than or equal to] [parallel][u.sub.n] - [[omega].sub.n[parallel], n = 0, 1, ... ,

which implies

[parallel][u.sub.n] - [[omega].sub.n][parallel] [less than or equal to] [(1 + ac).sup.-n] [parallel][u.sub.0] - [[omega].sub.0][parallel], n = 0,1, ...

Therefore, ([[omega].sub.n]) is the unique p-periodic solution of (E) and [u.sub.n] - [[omega].sub.n] [right arrow] 0, strongly in H, as claimed.

4 Examples

If A is maximal monotone and coercive (i.e., there exists a v [member of] H such that (w,v - [v.sup.*])/[parallel] v [parallel] [right arrow] [infinity], for v [member of] D(A), w([member of] Av, [parallel] v [parallel] [right arrow] [infinity]), then equation (E) has a periodic solution (equivalently, all its solutions are bounded) for all p-periodic sequences ([c.sub.n]) [subset] (0, +[infinity]) and ([f.sub.n]) [infinity] H. Indeed, in this case ([f.sub.n]) is bounded, and [c.sub.n] [greater than or equal to] min{[c.sub.k] : 0 [less than or equal to] k [less than or equal to] p - 1} =: c > 0, so the assertion follows from Theorem 2 in [3]. This is not the case in general, as the following simple example shows:

Example 1. Let H = R, A = [partial derivative][phi] = [phi]', [phi] : R [right arrow] R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[c.sub.n] = 1 for all n = 0, 1, If ([f.sub.n]) is the 3-periodic sequence defined by [f.sub.3k] = -3, [f.sub.3k+1] = 3, [f.sub.3k+2] = -1 for k = 0, 1, ... , then for [u.sub.0] = 1 equation (E) has a 3-periodic solution ([[omega].sub.n]), [[omega].sub.3k] = 1, [[omega].sub.3k+1] = -1, [[omega].sub.3k+2] = 2, k = 0,1,.... It turns out that ([[omega].sub.n) is the unique 3-periodic solution of equations (E). This follows easily by using the fact that any 3-periodic solution has the form ([[omega].sub.n] + c), c [member of] R. By Theorem 1, every solution ([u.suub.n]) of equation (E) tends asymptotically to ([[omega].sub.n]): [u.sub.3k] [right arrow] 1, [u.sub.3k+1] [right arrow] - 1, [u.sub.3k+2] [right arrow] 2 ask [right arrow] [infinity] (see Figure 1).

On the other hand, if ([f.sub.n]) is the 3-periodic sequence defined by [f.sub.3k] = 2, [f.sub.3k+1] = 3, [f.sub.3k+2] = -2 for k = 0,1, ... , then all solutions of (E) are unbounded. Indeed, it is easy to show that there exists an unbounded solution of equation (E) and thus, according to Theorem 1, all solutions of(E) are unbounded.

Now, let us show that for some periodic sequences ([f.sub.n]) the set of periodic solutions of (E) is not a singleton. For example, if ([f.sub.n]) is the 3-periodic sequence defined by [f.sub.3k] = 0.5, [f.sub.3k+1] = 1.5, [f.sub.3k+2] = -2 for k = 0,1, ... , then the sequence ([[omega].sub.n]) defined by [[omega].sub.3k] = c, [[omega].sub.3k+1] = 0.5 + c, [[omega].sub.3k+2] = 2 + c, k = 0,1, ... , is a 3-periodic solution of (E) for all c [greater than or equal to] 0.

In what follows we investigate some applications of the results presented in the previous section.

Example 2. Consider in R the following parabolic type difference equation:

[[DELTA].sub.m][u.sub.m,n] + [f.sub.m+1] ([u.sub.m+1,n]) [contains as member] [L.sub.n][u.sub.m+1,n] + [q.sub.m,n,] m = 0, 1, ... , n = 1,2, ... ([E.sub.p])

with the condition

[u.sub.m,0] = 0, m = 0, 1, ... , (D)

where [L.sub.n] denotes the discrete Laplace operator,

[L.sub.n][u.sub.m+1,n] = [[DELTA].sup.2.sub.n][u.sub.m+1,n+1] = [u.sub.m+1,n+1] - 2[u.sub.m+1,n] + [u.sub.m+1,n-1], (3)

[q.sub.m,n] is a double real sequence, which is p-periodic with respect to m, and [f.sub.m] : D([f.sub.m]) [subset] R [right] R (m = 1,2, ...) are (possibly multivalued) maximal monotone mappings.

Consider the real Hilbert space

H = [l.sup.2](R) = {u = ([u.sub.1], [u.sub.2], ...) : [[infinity].summation of (n=1)] [[absolute value of [u.sub.n].sup.2] < [infinity]}

with the usual inner product

(u,v) := [[infinity]summation of (n=1)] [u.sub.n][v.sub.n] [for all]u, v [member of] H.

Define on H the operator [A.sub.1]:= - L, i.e., [A.sub.1] ([([v.sub.n]).sub.n[greater than or equal to]1]) = [(-[v.sub.n+1] + 2[v.sub.n] - [v.sub.n-1]).sub.n[greater than or equal to]1], where [v.sub.0] = 0. We also define [A.sub.2] : D([A.sub.2]) = [[PI].sup.[infinity].sub.n=1] D([f.sub.n]) [subset] H [right arrow] H,

[A.sub.2]v := ([f.sub.1]([v.sub.1]),[f.sub.2]([v.sub.2]), v = ([v.sub.1],[v.sub.2], ...) [member of] D([A.sub.2]).

Thus Equation ([E.sub.p]) with condition (D) can be written in the form

[DELTA][u.sub.m] + A[u.sub.m+1] = [q.sub.m], m = 0, 1, ... , ([E.sup.*.sub.p])

with

[u.sub.m] := [([u.sub.m,n]).sub.n[greater than or equal to]1], [q.sub.m] := [([q.sub.m,n]).sub.n[greater than or equal to]1],

where A = [A.sub.1] + [A.sub.2].

Operator [A.sub.1] is everywhere defined, linear and strictly monotone (positive):

([A.sub.1]v, v) = [[absolute value of [v.sub.1]].sup.2] + [[absolute value of [v.sub.1] - [v.sub.2]].sup.2] + [[absolute value of [v.sub.2] - [v.sub.3]].sup.2] + ... > 0,

for all v different from zero. Moreover, [A.sub.1] is symmetric:

([A.sub.1]u, v) = (u, [A.sub.1]v) = [[infinity].summation over (n=0)] ([u.sub.n+1] - [u.sub.n])([v.sub.n+1] - [v.sub.n]),

where u, v [member of] H, [u.sub.0] = [v.sub.0] = 0. Then [A.sub.1] is the subdifferential of [[phi].sub.1], [[phi].sub.1] (v) = (1/2)([A.sub.1]v,v), v [member of] H.

Assume that 0 [member of] D([f.sub.m]) for all m = 1, 2, .... Then, [A.sub.2] is maximal monotone in H. Moreover, [A.ub.2] is cyclically monotone in H, since all [f.sub.m] are so in R (cf. Lemma 4), i.e., [A.sub.2] is a subdifferential. Since D([A.sub.1]) = H, it follows that A = [A.sub.1] + [A.sub.2] is a maximal (cyclically) monotone operator, and furthermore A is strictly monotone. Therefore, if [([q.sub.m]).sub.m[greater than or equal to]0] is a p-periodic sequence in H then, the conditions specified in Theorem 1 are satisfied for equation ([E.sup.*.sub.p]). If a p-periodic solution of equation ([E.sup.*.sub.p]) exists, it is unique. Denote it [([[omega].sub.m]).sub.m[greater than or equal to]0]. For any other solution ([u.sub.m]) we have [u.sub.m] - [[omega].sub.m] [right arrow] 0, weakly in H, as m [right arrow] [infinity]. In particular, if [u.sub.m] = [([u.sub.m,n]).sub.n[greater than or equal to]1], we have [u.sub.m,n] - [[omega].sub.m,n] [right arrow] 0, as m [right arrow] [infinity], for each n = 1,2,....

Remark 2. Note that [A.sub.1] is just strictly monotone, not strongly monotone. Indeed, strong monotonicity would imply that [A.sub.1] is surjective, which is not the case

(e.g., the sequence (-1/n) [member of] H does not belong to the range of [A.sub.1]). Example 3. Consider in R the following difference equation:

[[DELTA].sub.m][u.sub.m,n] + [f.sub.m+1]([u.sub.m+1,n]) [subset] [L.sub.n][u.sub.m+1,n] + [q.sub.m,n],

m = 0, 1, ... , n = 1, 2, ... , N, (4)

with the Dirichlet type conditions

[u.sub.m,0] = 0 = [u.sub.m,N+1], m = 0, 1, ... , (5)

where N is a positive integer, [q.sub.m+pn] = [q.sub.m,n] for all m = 0,1, ... , n = 1, ... N for a given positive integer p, and [f.sub.m] satisfy the same assumptions as in Example 2. We can choose H to be the Euclidean space [R.sup.N]. [A.sub.1] and [A.sub.2] are defined similarly. In this case, [A.sub.1] is even strongly monotone, so (according to Theorem 2) the above equation has a unique p-periodic solution, [[omega].sub.m+pn] = [[omega].sub.mn], m = 0,1, ... , n = 1,2, ... , N and any other solution [u.sub.m,n] converges to it: [u.sub.m,n] - [[omega].sub.mn] [right arrow] 0,as m [right arrow] [infinity], for each n = 1,2, ... N.

Remark 3. Choosing convenient mappings [f.sub.m] we can obtain solutions with specific desired properties. E.g., if for all m [f.sub.m] is the subdifferential of the indicator function of [0,[infinity]), then the corresponding solutions have nonnegative components: [u.sub.m,n] [greater than or equal to] 0.

Remark 4. If fact, in the above examples, [A.sub.2] could be a general maximal monotone operator from the corresponding space H into itself. Even more, if R is replaced by a general Hilbert space, then all the above reasonings work with slight modifications.

Acknowledgements. The authors thank Professor Hadi Khatibzadeh for providing them with paper [5] and for interesting conversations.

Received by the editors in August 2012.

Communicated by J. Mawhin.

References

[1] J.B. Baillon, Un exemple concernant le comportement asymptotique de la solution du probleme du/dt + [partial derivative][phi](u) [contains as member] 0, J. Funct. Anal. 28 (1978), pp. 369-376.

[2] J.B. Baillon and A. Haraux, Comportement a l'infini pour les equations devolution avec forcing periodique, Archive Rat. Mech. Anal. 67 (1977), pp. 101-109.

[3] O.A. Boikanyo and G. Morosanu, Modified Rockafellar's algorithms, Math. Sci. Res. J. 13 (2009), pp. 102-122.

[4] F. Browder and W.V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), pp. 571-576.

[5] B. Djafari Rouhani and H. Khatibzadeh, Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing, J. Difference Eqns Appl., DOI:10.1080/10236198.2012.658049.

[6] G. Morosanu, Nonlinear Evolution Equations and Applications, D. Reidel, Dordrecht-Boston-Lancaster-Tokyo, 1988.

Department of Mathematics and its Applications, Central European University, Budapest, Hungary

Email: morosanug@ceu.hu

Bolvadin Vocational School, Afyon Kocatepe University, Afyonkarahisar, Turkey

Email: fozpinar@aku.edu.tr
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