Existence and Multiple Solutions for Nonlinear Second-Order Discrete Problems with Minimum and Maximum
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Research Article Existence and Multiple Solutions for Nonlinear Second-Order Discrete Problems with Minimum and Maximum Ruyun Ma and Chenghua Gao College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China Correspondence should be addressed to Ruyun Ma, [email protected] Received 15 March 2008; Revised 6 June 2008; Accepted 19 July 2008 Recommended by Svatoslav Stanek ˇ Consider the multiplicity of solutions to the nonlinear second-order discrete problems with } A, max{uk : minimum and maximum: Δ2 uk−1 fk, uk, Δuk, k ∈ T, min{uk : k ∈ T } B, where f : T × R2 →R, a, b ∈ N are fixed numbers satisfying b ≥ a 2, and A, B ∈ R are k∈T {a, a 1, . . . , b − 1, b}. satisfying B > A, T {a 1, . . . , b − 1}, T Copyright q 2008 R. Ma and C. Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction {a, a 1, . . . , b − 1, b}. Let Let a, b ∈ N, a 2 ≤ b, T {a 1, . . . , b − 1}, T : u | u : T −→ R , E
1.1
let and for u ∈ E, uE maxuk.
1.2
E : u | u : T −→ R ,
1.3
uE maxuk.
1.4
k∈T
Let
and for u ∈ E, let k∈T
and E, respectively, and that the finite dimensionality It is clear that the above are norms on E of these spaces makes them Banach spaces.
2
Advances in Difference Equations
In this paper, we discuss the nonlinear second-order discrete problems with minimum and maximum: Δ2 uk − 1 f k, uk, Δuk , k ∈ T, 1.5 A, B, 1.6 min uk : k ∈ T max uk : k ∈ T where f : T × R2 → R is a continuous function, a, b ∈ N are fixed numbers satisfying b ≥ a 2 and A, B ∈ R satisfying B > A. Functional boundary value problem has been studied by several authors 1–7 . But most of the papers studied the differential equations functional boundary value problem 1–6 . As we know, the study of difference equations represents a very important field in mathematical research 8–12 , so it is necessary to investigate the corresponding difference equations with nonlinear boundary conditions. Our ideas arise from 1, 3 . In 1993, Brykalov 1 discussed the existence of two different solutions to the nonlinear differential equation with nonlinear boundary conditions x h t, x, x , t ∈ a, b , 1.7 min ut : t ∈ a, b A, max ut : t ∈ a, b B, where h is a bounded function, that is, there exists a constant M > 0, such that |ht, x, x | ≤ M. The proofs in 1 are based on the technique of monotone boundary conditions developed in 2 . From 1, 2 , it is clear that the results of 1 are valid for functional differential equations in general form and for some cases of unbounded right-hand side of the equation see 1, Remark 3 and 5 , 2, Remark 2 and 8 . In 1998, Stanˇek 3 worked on the existence of two different solutions to the nonlinear differential equation with nonlinear boundary con
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