Existence and multiple solutions for a second-order difference boundary value problem via critical point theory

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1 J. Math. Anal. Appl. 36 (7) Existence and multiple solutions for a second-order difference boundary value problem via critical point theory Haihua Liang a,b,, Peixuan Weng b a Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou, Guangdong 51665, PR China b School of Mathematical Sciences, South China Normal University, Guangzhou 51631, PR China Received 5 July 5 Available online 17 April 6 Submitted by D. O Regan Abstract In this paper, the critical point theory is employed to establish existence and multiple solutions for a second-order difference boundary value problem. 6 Elsevier Inc. All rights reserved. Keywords: Second-order; Difference equation; Boundary value problem; Existence and multiple solutions; Critical points 1. Introduction Let R and Z be the sets of real numbers and integers, respectively. For any a,b Z, a<b, denote [a,b] ={a,a + 1,...,b}. Assume that N is a given positive integer with N>. We consider the second-order difference boundary value problem (briefly BVP) (p 1 x 1 ) + q x + f(,x ) =, [1,N], (1.1) x = x N, p x = p N x N. (1.) Project supported by the NNSF of China (157164) and Natural Science Foundation of Guangdong Province (11471). * Corresponding author. address: haiihuaa@tom.com (H. Liang). -47X/$ see front matter 6 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa

2 51 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) As usual, denotes the forward difference operator defined by x = x +1 x, x = ( x ), p and q R for all [1,N]. We will assume throughout this paper that (A) For any [1,N], f(, ) : [1,N] R R is continuous. (B) p N. Early in 1999, by employing a fixed point theorem in cone, the existence of positive periodic solutions for the BVP was investigated by Atici and Guseinov [1]. In 3, by using the upper and lower solution method, Atici and Cabada [] considered Eq. (1.1) with p = 1 subject to the boundary value condition x = x N, x = x N and obtained the existence and uniqueness results. See [1,] for more details. There are many other literature dealing with the similar second-order difference equation subject to various boundary value conditions. We refer to [1 8] and references therein. However, we note that these results were usually obtained by analytic techniques and various fixed point theorems. For example, the upper and lower solution method [5 7], the conical shell fixed point theorems [1,3], the Brouwer and Schauder fixed point theorems [,4,7], topological degree theory [8]. As we now, the critical point theory has played an important role in dealing with the existence and multiple results for differential equations, which include the boundary value problems, please see [9 11] and references given therein. However, few research has been done to use such a powerful tool to handle the difference BVP. Very recently, in [1], Agarwal, Perera and O Regan have employed the mountain pass lemma to study the following discrete equation y( 1) + f (,y() ) =, [1,N], under Dirichlet boundary value conditions, and have obtained the existence of multiple solutions. To the best of our nowledge, this wor is the one among a few which deal with difference problems via variational method. The aim of this paper is to apply some basic theorems in critical point theory to establish existence and multiple results. For a fuller treatment on critical point theory used here we refer the reader to [13,14].. Auxiliary results and variational framewor Let E be a real Banach space and J C 1 (E, R). A critical point of J is a point x E where J (x ) = and a critical value is a number c such that J(x ) = c. The following is the definition of Palais Smale condition. Definition.1. We say that J satisfies the Palais Smale condition if every sequence {x n } E, such that J(x n ) is bounded and J (x n ) asn +, has a converging subsequence. The following lemmas play an important role in proving our main results. Lemma.1. [13, p. 43] Let E be a real reflexive Banach space, and let J be wealy lower (upper) semicontinuous such that ( ) lim J(x)=+ lim J(x)=. x x

3 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) Then, there exists x E such that ( ) J(x ) = inf J(x) J(x ) = sup J(x). x E x E Furthermore, if J has bounded linear Gâteaux derivative, then J (x ) =. Denote by θ the zero of E and by S n 1 the (n 1)-dimensional unit sphere; we have Lemma.. [14, p. 53] Let E be a real Banach space, J C 1 (E, R) with J even, bounded from below and satisfying the P S condition. Suppose J(θ)=, and there is a set K E such that K is homeomorphic to S n 1 by an odd map, and sup K J<. Then J possesses at least n distinct pairs of critical points. We will now establish the variational functional of BVP. Let R N be the N-dimensional Hilbert space with the usual inner product and the usual norm ( N ) 1 (x, y) = x i y i, x = xi, x,y R N. i=1 i=1 Define a functional J on R N as J(x)= 1 [ p 1 x 1 + q x + F(,x ) ], (.1) where F(,u)= u f(,t)dt and x = x N. Obviously, J(θ)=. Let E = { } χ χ = (x,x 1,...,x N,x N+1 ), where x = x N,p N x N = p x. Then by the assumption (B) it is easy to see that E is isomorphic to R N. In the following, when we say x R N, we always imply that x can be extended to χ E if it is necessary. Now we claim that if x T = (x 1,x,...,x N ) R N is a critical point of J, then χ = (x,x 1,...,x N,x N+1 ) E is precisely a solution of BVP. Indeed, for every u, v R N it results F(,u+ sv) = F(,u)+ u+sv Moreover, for any x,y R N,wehave J(x+ sy) = J(x)+ s u f(,t)dt = F(,u)+ svf(,u + θ sv), θ (, 1). [ p 1 x 1 y 1 + q x y + y f(,x + θ sy ) ] + s [ p 1 y 1 + q y ]. Hence ( J (x), y ) [ = p 1 x 1 y 1 + q x y + y f(,x ) ]. (.)

4 514 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) On the other hand, we have from the boundary value condition p N x N = p x, y N = y that [ (p 1 x 1 ) ] y = (p x y p 1 x 1 y ) N 1 = = p x y + p N x N y N p x y 1 p 1 x 1 y 1. So, if J (x) =, then from (.) we have [ (p 1 x 1 ) + q x + f(,x ) ] y =. Note that y R N is arbitrary, hence we obtain (p 1 x 1 ) + q x + f(,x ) =, [1,N]. Therefore, if x is a critical point of J, then χ is exactly the solution of BVP. 3. Main results Now we establish the existence of at least one solution to problem BVP. Theorem 3.1. Assume that there exists a constant δ with δ>4p q such that inf δ, [1,N] u u (3.1) where { p = max p }, [1,N] q = min }. [1,N] (3.) Then BVP has at least one solution. Proof. By (3.1), there exists a constant M>such that u M implying δ ε, for [1,N], (3.3) u where ε = 1 (δ 4p + q) >. We first consider the case u. If u>m, we have from (3.3) that (δ ε)u, and then F(,u)= u f(,s)ds= M M f(,s)ds+ f(,s)ds+ u M u M f(,s)ds (δ ε)s ds

5 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) where C 1 = inf [1,N] { M If u M, then = δ ε ( u M ) + δ ε u + C 1, } f(,s)ds δ ε M. F(,u)= δ ε u + F(,u) δ ε δ ε u + inf u M [1,N] so there is a constant C such that { u u M f(,s)ds f(,s)ds δ ε u F(,u) δ ε u + C, u. If u, by a similar argument we also obtain that there is a constant C 3 such that F(,u) δ ε u + C 3. Thus we get C R such that F(,u) δ ε u + C, u R, [1,N], (3.4) where C is a constant. Moreover, for every x R N,wehave }, J(x)= 1 p 1 x q x + F(,x ) p p p x 1 + q ( x + x 1 ) q + x + q ( δ ε = p + q = ε 4 x + NC. x + F(,x ) x + δ ε ) x + NC x + δ ε x + NC x + NC

6 516 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) Note that ε>, thus J(x) + as x +, i.e., J(x) is a coercive map. In view of Lemma.1, we now that there exists at least one x R N such that J ( x) =, hence BVP has at least one solution. We complete the proof. Corollary 3.1. Suppose that (1) uf (, u) holds for u large sufficiently; () there exist δ 1 > and α>1 such that inf lim [1,N] u + Then BVP has at least one solution. u α δ 1. (3.5) Proof. By conditions (1) and (), it is easy to see that inf lim =+, [1,N] u u so the assumption of Theorem 3.1 is satisfied, and the conclusion comes from Theorem 3.1 immediately. Remar. Suppose that the conditions in Theorem 3.1 are satisfied and that there exists [1,N] such that f(,), then BVP has at least one nonzero solution. In order to establish our next theorem, we need to give some notations. We rewrite J(x) as follows: J(x)= 1 xt Ax + 1 q x + F(,x ), (3.6) where A = p + p 1 p 1 p p 1 p 1 + p p p p + p p N + p N 1 p N 1 p p N 1 p N 1 + p and x T denotes the transpose of x. It is clear that is an eigenvalue of A. Letη = (1, 1,...,1) T and Y = span{η}. Evidently the eigenspace of A associated to is Y. Set A N 1 = p + p 1 p 1 p 1 p 1 + p p p p + p p N 3 + p N p N p N p N + p N 1 N N. (N 1) (N 1).

7 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) Assume that {p } N = satisfies p > for [,N 1], then it is easy to chec that A N 1 is positive-definite. Hence ran A = ran A N 1 = N 1, which implies that A are positive semidefinite and all eigenvalues of A is positive except for. Denote these eigenvalues of A by λ 1,λ,...,λ N, where λ i >, i = 1,,...,N 1; λ N =. Corresponding to each eigenvalue λ i, i = 1,,...,N, there exist eigenvectors η i (i = 1,,...,N) such that Aη j = λ j η j and (η i,η j ) = {, i j, 1, i = j, i, j = 1,,...,N. Let X = span{η 1,η,...,η N 1 }, then R N = X Y. Thus for any x R N, there exists unique {b j } N j=1, such that x = N j=1 b j η j. Moreover, it is clear that x = N j=1 bj.let λ = min i [1,N 1] {λ i }, λ = max i [1,N] {λ i }, then λ λ>, and x T Ax = λ j bj λ x. (3.7) j=1 Lemma 3.1. Assume that p > for [,N 1], and (i) there exist constants β 1,β,...,β N such that lim u u β, [1,N]; (ii) r := min [1,N] {q + β } > λ. Then J satisfies the P S condition. Proof. Suppose that {x (n) } n=1 RN with {J(x (n) )} is bounded and J (x (n) ) asn +. By (.) we find hence ( J (x), x ) = p 1 x 1 [ + q x + x f(,x ) ] [ q x (n) = x T Ax + + x (n) [ q x + x f(,x ) ], f (,x (n) )] ( = x (n) ) T Ax (n) + ( J ( x (n)),x (n)) We have from (i) that there exists a constant C> such that u (β ε )u C, u R, where ε = r λ, thus λ x (n) + ( J ( x (n)),x (n)). (3.8)

8 518 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) [ q x (n) + x (n) f (,x (n) )] N (q + β ε ) (n) x NC (r ε ) x (n) NC = r + λ x (n) NC. In view of (3.8) we obtain r + λ x (n) NC λ x (n) + ( J ( x (n)),x (n)) or r λ x (n) ( J ( x (n)),x (n)) + NC J ( x (n)) x (n) + NC. Note that J (x (n) ) asn and r λ >, so {x (n) } is bounded. Since R N is N-dimensional Hilbert space, the above argument implies that there exists a converging subsequence of {x (n) }, thus the proof of Lemma 3.1 is completed. Theorem 3.. Assume that is odd on its second variable u, and the conditions of Lemma 3.1 hold. In addition, suppose that (iii) there exist constants μ 1,μ,...,μ N such that lim u u μ, [1,N]; (iv) r := max [1,N] {q + μ } <λ. Then BVP has at least (N 1) distinct solutions. Proof. Since is odd in u, then J is an even functional. Moreover, according to Lemma 3.1, J satisfies the P S condition. Let ε 1 = λ r, it follows from condition (iii) that there exists ρ>such that μ + ε 1, for u ρ, u and therefore F(,u) μ + ε 1 u, u ρ. (3.9) For any x = (x 1,x,...,x N ) X,wehavex = N 1 i=1 b iη i and x =( N 1 i=1 b i ) 1,so N 1 N 1 x T Ax = λ i bi λ bi = λ x. (3.1) i=1 i=1 Consequently, if x ρ, then

9 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) J(x)= 1 xt Ax + 1 q x + F(,x ) λ x + q + μ + ε 1 λ x + r + ε 1 x x = λ r ε 1 x = ε 1 x. (3.11) Let σ = ε 1 ρ, then J(x) σ <forx K, where K = { x X x =ρ }. It is clear that K is homeomorphic to S N by odd map. On the other hand, by condition (i), it is easy to see that there exists a constant C such that F(,u) β ε u C, u R, (3.1) where ε = r λ. For any x R N,wehavefrom(3.1) that J(x)= 1 xt Ax + 1 q x + F(,x ) = 1 λ b + 1 q x + F(,x ) λ b + 1 (q + β ε )x NC λ x + r ε x NC = ε x NC. (3.13) Thus inf x R N J(x)>. By Lemma. and the above result, J possesses at least (N 1) distinct pairs of critical points, i.e., BVP has at least (N 1) distinct solutions. Corollary 3.. Assume that f(, u) = and p >, =, 1,...,N 1. In addition, suppose that (a) conditions (1), () in Corollary 3.1 hold; (b) lim u u ; (c) max [1,N] {q } <λ. Then the BVP possesses at least (N 1) distinct solutions.

10 5 H. Liang, P. Weng / J. Math. Anal. Appl. 36 (7) Proof. Let q = min [1,N] {q }, β = λ q, by assumptions (1), () in Corollary 3.1, it is obvious that inf lim β. [1,N] u + u Let β = β ( = 1,,...,N), then r = min [1,N] {q + β }=β + q = λ> λ. So both (i) and (ii) in Lemma 3.1 are satisfied. Moreover, by conditions (b) and (c) it is easy to verify that (iii), (iv) are true. Thus the conclusion comes from Theorem 3. immediately. We can get from Corollary 3. directly Corollary 3.3. Assume that p >, q and f(t,u)= m i=1 g(t) u α i sgn u, where g(t) is continuous with g(t) > and α i > 1 for each i [1,N]. Then the BVP possesses at least (N 1) distinct solutions. References [1] F.M. Atici, G.Sh. Guseinov, Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl. 3 (1999) [] F.M. Atici, A. Cabada, Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. Math. Appl. 45 (3) [3] R.P. Agarwal, D. O Regan, Nonpositone discrete boundary value problems, Nonlinear Anal. 39 () [4] R.P. Agarwal, D. O Regan, A fixed-point approach for nonlinear discrete boundary value problems, Comput. Math. Appl. 36 (1998) [5] R.P. Agarwal, D. O Regan, Boundary value problems for discrete equations, Appl. Math. Lett. 1 (1997) [6] Z. Wan, Y.B. Chen, S.S. Cheng, Monotone methods for discrete boundary value problem, Comput. Math. Appl. 3 (1996) [7] J. Henderson, H.B. Thompson, Existence of multiple solutions for second-order discrete boundary value problems, Comput. Math. Appl. 43 () [8] H.B. Thompson, C. Tisdell, Systems of difference equations associated with boundary value problems for secondorder systems of ordinary differential equations, J. Math. Anal. Appl. 48 () [9] G. Bonanno, A Minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl. 7 () 1 9. [1] Y.Sh. Tao, G.Zh. Gao, Existence, uniqueness and approximation of classical solutions to nonlinear two-point boundary value problems, Nonlinear Anal. 5 () [11] X.B. Shu, Y.T. Xu, Existence of multiple solutions for a class of second-order ordinary differential equations, Electron. J. Differential Equations 137 (4) [1] R.P. Agarwal, K. Perera, D. O Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (4) [13] D.J. Guo, Nonlinear Functional Analysis, Shandong Publishing House of Science and Technology, Jinan, 1 (in Chinese). [14] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.