Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator

Abstract Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.


Introduction
There is an increasing interest in the existence of solutions to boundary value problems for finite difference equations with the p.x/ Laplacian operator in the last decades. These kinds of problems like (1) play a fundamental role in different fields of research, because of their applications in many fields, they can model various phenomena arising from the study of elastic mechanics [33], electrorheological fluids [15] and image restoration [14] and other fields such as biological neural networks, cybernetics, ecology, control systems, economics, computer science, physics, finance, artificial and many others. Important tools in the study of nonlinear difference equations are fixed point methods in cone; see [2,19], and upper and lower solution techniques; see, for instance, [24,25] and references therein. It is well known that critical point theory is an important tool to deal with the problems for differential equations. For background and recent results for nonlinear discrete boundary value problems, we refer the reader to [5-7, 10, 12, 13, 16, 17, 20, 27] and the seminal papers [3,4] and references therein.
In this paper, based on a local minimum theorem (Theorem 2.5) due to Ricceri [30], we ensure an exact interval of the parameter , in which the problem (1) admits at least three solutions.
As an example, here, we point out the following special case of our main results.
Theorem 1.1. Assume T 2 is a fixed integer number and there exists a positive constant d < 1 such that, ; lim sup jtj!0 Then, there exists r > 0 with the following property: for every continuous function g W OE1; T R ! R, there exists ı > 0 such that, for each 2 OE0; ı, the problem (2) has at least three solutions whose norms are less than r.
The rest of this paper is arranged as follows. In Section 2, we recall some basic definitions and the main tool (Theorem 2.5.), and in Section 3, we provide our main results that contain several theorems, and finally, we illustrate the results by giving examples.

Preliminaries
Let us introduce some notations that will be used later. Let T 2 be a fixed positive integer, OE1; T is the discrete interval f1; : : : ; T g. In order to give the variational formulation of the problem (1), we introduce, equipped with the norm kuk WD then by Weighted H R older's inequality, one can conclude that where, In the space W we can also consider the Luxemburg norm [8], Since W has finite dimension, the two last norms are equivalent. Therefore there exist constants L 1 > 0 and L 2 > 1 such that Now, let ' W W ! R be given by the formula It is easy to check that for any u 2 W the following properties hold: Let, F .k; t/ WD R t 0 f .k; /d ; G.k; t / WD R t 0 g.k; /d ; for every k 2 OE1; T and t 2 R. In the sequel, we will use the following inequality Proof. Let u be in W . Then by the discrete HR older inequality, for every k 2 OE1; T , we get for all k 2 OE1; T . Lemma 2.2. 1. There exists a positive constant C 1 such that for all u 2 W with kuk > 1, Proof. Let u 2 W be fixed. By a similar argument as in [28], we set We define for each k 2 OE0; T C 1, . It follows that ju.k 1/j; ju.k/j < 1 for each k 2 OE1; T C 1.
To study the problem (1), we consider the functional I ; W W ! R defined by An easy computation ensures that I ; is of class C 1 on W with for all u; v 2 W . .u/.v/ D 0. Thus, for every v 2 W , and taking v.0/ D v.T C 1/ D 0 and summation by parts into account, one has Since v 2 W is arbitrary, one has for every k 2 OE1; T . Therefore, u is a solution of (1). So by bearing in mind that u is arbitrary, we conclude that every critical point of the functional I ; in W is exactly a solution of the problem (1). Also, if u 2 W is a solution of problem (1), one can show that u is a critical point of I ; . Indeed, by multiplying the difference equation in problem (1) by v.k/ as an arbitrary element of W and summing and using the fact that we have I 0 ; .u/.v/ D 0, hence u is a critical point for I ; . Thus the vice versa holds and the proof is completed.
Our main tool is a local minimum theorem (Theorem 2.5) due to Ricceri (see [30,Theorem 2]), which is recalled below. We refer to the papers [22,30,32] in which Theorem 2.5 has been successfully employed for the existence of at least three solutions for two-point boundary value problems. First, we give the following definition.
Definition 2.4. If X is a real Banach space, we denote by W X the class of all functionalsˆW X ! R possessing the following property: if fu n g is a sequence in X converging weakly to u 2 X and lim inf n!1ˆ. u n / Äˆ.u/, then fu n g has a subsequence converging strongly to u. For instance, if X is uniformly convex and g W OE0; C1OE! R is a continuous, strictly increasing function, then, by a classical result, the functional u ! g.kuk/ belongs to the class W X .
Our main tool reads as follows: . Let X be a separable and reflexive real Banach space andˆW X ! R be a coercive, sequentially weakly lower semicontinuous C 1 functional, belonging to W X , bounded on each bounded subset of X and whose derivative admits a continuous inverse on X ; J W X ! R a C 1 functional with compact derivative. Assume thatˆhas a strict local minimum assume that˛<ˇ. Then, for each compact interval OEa; b 1 ; 1 OE (with the conventions 1 0 D C1; 1 C1 D 0) there exists r > 0 with the following property: for every 2 OEa; b and every C 1 functional ‰ W X ! R with compact derivative, there exists ı > 0 such that, for each 2 OE0; ı, the equationˆ0.x/ D J 0 .x/ C ‰ 0 .x/ has at least three solutions whose norms are less than r.
then either u is positive or u Á 0.
By a similar argument used in [5, Theorem 2.3], we obtain the next result which guarantees the same conclusion of the preceding Strong maximum principle, independently of the sign of the operator. Lemma 2.7. Fix u 2 W such that, if u.k/ Ä 0, it follows that .w.k 1/ p.k 1/ .u.k 1/// C q.k/ p.k/ .u.k// D 0: Then either u is positive or u Á 0.

Main results
Theorem 3.1. Assume that there exist constants M > 0, m > 0, d 2 .0; 1/, and N a; N b > 0 and s > p C such that, jt j s C jtj p C /; for every .k; jt j/ 2 OE1; T OE0; m. (B2) F .k; t/ < N b.1 C jt j p /; for every .k; jt j/ 2 OE1; T OEM; 1/. Then, for each compact interval OEa; b ƒ WD 1 ; 2 OE, where there exists r > 0 with the following property: for every 2 OEa; b and for every continuous function g W OE1; T R ! R, there exists ı > 0 such that, for each 2 OE0; ı, the problem (1) has at least three solutions whose norms are less than r.
Proof. Our aim it to apply Theorem 2.5 to our problem. To this end, we observe that due to (B0) the interval 1 ; 2 OE is non-empty, so fix N in OEa; b 1 ; 2 OE, take X D W , and putˆ; ‰ as follows: for every u 2 W . So I N ; N Dˆ N J N ‰. An easy computation ensures thatˆ, J and ‰ turn out to be of class  (1). Hence, to prove our result, it is enough to apply Theorem 2.5. We knowˆis a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X , and ‰ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Also the functionalˆis coercive. Indeed, by (3) for kuk > L 2 , we have kuk p. / > kuk L 2 > 1. Also by (5), '.u/ > kuk p p. / > kuk p L p 2 . Therefore by (4), one haŝ In addition,ˆhas a strict local minimum 0 withˆ.0/ D J.0/ D 0. We show that for N 2 OEa; b ƒ WD 1 ; 2 OE to be fixed,˛< 1 N ,ˇ> 1 N . Let d 2 .0; 1/ be fixed and put and F .k; d /: On the other hand, from the conditions (B1) and (B2), and bearing (6) in mind, we get and by Lemma 2.2 (2) lim sup By (12) and (13) By (11) and (14), we deduce that˛<ˇ. All the assumptions of Theorem 2.5 are satisfied, so by applying that theorem, the conclusion follows. Hence, the proof is complete. Now, we present an example to illustrate the results of Theorem 3.1. for any k 2 OE1; 10, where t C D maxf0; tg. Therefore,  Figure 1), one can conclude (B1) holds. It is clear that (B2) holds. there exists r > 0 with the following property: for every 2 OEa; b and every positive continuous function g W OE1; 10 R ! R, there exists ı > 0 such that, for each 2 OE0; ı, the problem 8 < : .e k.10 k/ k 2 C1 p.k 1/ .u.k 1// C p.k/ .u.k// D f .k; u C .k// C g.k; u.k//; u.0/ D u.11/ D 0; k 2 OE1; 10, has at least three solutions whose norms are less than r, hence by Lemma 2.6 whose signs are positive.
The next result reads as follows.
Theorem 3.3. Assume that there exists a constant d > 1 such that, Then, for each compact interval OEa; b ƒ WD 1 ; 2 OE, where there exists r > 0 with the following property: for every 2 OEa; b and every continuous function g W OE1; T R ! R, there exists ı > 0 such that, for each 2 OE0; ı, the problem (1) has at least three solutions whose norms are less than r.
Proof. Our aim it to apply Theorem 2.5 to our problem. To this end, we observe that due to (C0) the interval 1 ; 2 OE is non-empty, so fix N in 1 ; 2 OE, take X D W , and putˆ; ‰ and J , as given in (8) (1). We know thatˆ, ‰ and J satisfy the regularity assumptions of Theorem 2.5. Hence, to prove our result, it is enough to apply Theorem 2.5.
We show that for N 2 OEa; b ƒ WD 1 ; 2 OE to be fixed,˛< 1 N ,ˇ> 1 N . Let d > 1 be fixed and put N v.k/ D d for every k 2 OE1; T and N v. q.k//: In view of (C1), there exist 0 < r 1 < 1 < r 2 such that F .k; t / < jtj p C ; for any .k; t / 2 OE1; T OE r 1 ; r 1 ; F .k; t / < jt j p < jtj p C ; for any .k; t / 2 OE1; T R n .OE r 2 ; r 2 /: Since F is continuous, then F .k; t / is bounded for any .k; t / 2 OE1; T .OEr 1 ; r 2 [ OE r 2 ; r 1 /, so we can choose C 2 > 0 and s > p C such that F .k; t / < jtj p C C C 2 jtj s ; for all .k; t / 2 OE1; T R; and bearing (6) in mind, we get and by Lemma 2.2 (2) lim sup Again since F is continuous, then F .k; t / is bounded for any .k; t / 2 OE1; T OE r 2 ; r 2 , so we can choose C 3 > 0 such that F .k; t / < jtj p C C 3 ; for all .k; t / 2 OE1; T R; and bearing (6) By (16) and (17) By (15) and (18), we deduce that˛<ˇ. All the assumptions of Theorem 2.5 are satisfied, so by applying that theorem, the conclusion follows. Hence, the proof is complete. Now, we present an example to illustrate the results of Theorem 3.3.  for every k 2 OE1; 10, has at least three solutions whose norms are less than r. Finally we present the problem (1), in which the function f .k; u/ has separable variables and D 1.
Theorem 3.6. Let f 0 W OE1; T ! R be a non-negative, non-zero and essentially bounded function such that and f 1 W R ! R be a non-negative and continuous function and F 1 . / D R 0 f 1 .x/dx for every 2 R. Assume that there exists a positive constant d < 1 such that, (D0) Then, there exists r > 0 with the following property: for every continuous function g W OE1; T R ! R, there exists ı > 0 such that, for each 2 OE0; ı, the problem for every k 2 OE1; T , has at least three solutions whose norms are less than r.
Proof. We can choose D 1 2 OEa; b such that OEa; b . 1 ; 1 /. Take X D W , and putˆ, ‰ and J as given in (8) and (9), so I 1; N Dˆ J N ‰ and the solutions of the equation I 0 1; N Dˆ0 J 0 N ‰ 0 D 0 are exactly the solutions for problem (19). Hence, to prove our result, it is enough to apply Theorem 2.5. We show that,˛< 1,ˇ> 1. Put N v.k/ D d 2 .0; 1/ for every k 2 OE1; T and N v.0/ D N v.T C 1/ D 0. In view of (D0) and (10),ˇ> Applying similar argument as in the proof of Theorem 3.3, and taking into account (D1) and Lemma 2.
By (20) and (21), we deduce that˛<ˇ. All the assumptions of Theorem 2.5 are satisfied, so by applying that theorem, the conclusion follows. Hence, the proof is complete. Finally, we present an example of Theorem 1.1.  Thus, .A2/ holds. Therefore, by using Theorem 1.1, there exists r > 0 such that for every continuous function g W OE1; T R ! R, there exists ı > 0 such that, for each 2 OE0; ı, the problem (2) has at least three solutions whose norms are less than r.