Existence and Multiple Solutions for Nonlinear Second-Order Discrete Problems with Minimum and Maximum
© R. Ma and C. Gao. 2008
Received: 15 March 2008
Accepted: 19 July 2008
Published: 29 July 2008
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.
where is a bounded function, that is, there exists a constant , such that . The proofs in  are based on the technique of monotone boundary conditions developed in . From [1, 2], it is clear that the results of  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)]).
But as far as we know, there have been no discussions about the discrete problems with minimum and maximum in literature. So, we use the Borsuk theorem  to discuss the existence of two different solutions to the second-order difference equation boundary value problem (1.5), (1.6) when satisfies
So, in the rest part of this paper, we only deal with BVP (1.5), (2.4).
Similarly, we can obtain the following lemma.
If , then ; if , then . Equation (2.24) is obvious.
We only prove that (2.27) holds when Case 1 occurs, (if Case 2 occurs, it can be similarly proved).
If Case 1 holds, we divide the proof into two cases.
From (2.26), (2.55), and (2.56), the assertion is proved.
At last, we prove (c).
Then (c) is proved.
3. The Main Results
Next, we need to prove BVPs (1.5), (3.2), and (1.5) and (3.3) have solutions, respectively.
we can obtain a solution of BVP (1.5) and (3.3).
This work was supported by the NSFC (Grant no. 10671158), the NSF of Gansu Province (Grant no. 3ZS051-A25-016), NWNU-KJCXGC, the Spring-sun Program (no. Z2004-1-62033), SRFDP (Grant no. 20060736001), and the SRF for ROCS, SEM (2006).
- Brykalov SA: Solutions with a prescribed minimum and maximum. Differencial'nye Uravnenija 1993, 29(6):938-942. translation in Differential Equations, vol. 29, no. 6, pp. 802–805, 1993MathSciNetGoogle Scholar
- Brykalov SA: Solvability of problems with monotone boundary conditions. Differencial'nye Uravnenija 1993, 29(5):744-750. translation in Differential Equations, vol. 29, no. 5, pp. 633–639, 1993MathSciNetGoogle Scholar
- Stančk S: Multiplicity results for second order nonlinear problems with maximum and minimum. Mathematische Nachrichten 1998, 192(1):225-237. 10.1002/mana.19981920113MathSciNetView ArticleGoogle Scholar
- Staněk S: Multiplicity results for functional boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 1997, 30(5):2617-2628. 10.1016/S0362-546X(97)00215-0MATHMathSciNetView ArticleGoogle Scholar
- Whyburn WM: Differential equations with general boundary conditions. Bulletin of the American Mathematical Society 1942, 48: 692-704. 10.1090/S0002-9904-1942-07760-3MATHMathSciNetView ArticleGoogle Scholar
- Ma R, Castaneda N: Existence of solutions of boundary value problems for second order functional differential equations. Journal of Mathematical Analysis and Applications 2004, 292(1):49-59. 10.1016/j.jmaa.2003.11.044MATHMathSciNetView ArticleGoogle Scholar
- Cabada A:Extremal solutions for the difference -Laplacian problem with nonlinear functional boundary conditions. Computers & Mathematics with Applications 2001, 42(3–5):593-601.MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Nonlinear discrete Sturm-Liouville problems at resonance. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(11):3050-3057. 10.1016/j.na.2006.09.058MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Bifurcation from infinity and multiple solutions for some discrete Sturm-Liouville problems. Computers & Mathematics with Applications 2007, 54(4):535-543. 10.1016/j.camwa.2007.03.001MATHMathSciNetView ArticleGoogle Scholar
- Ma R, Luo H, Gao C: On nonresonance problems of second-order difference systems. Advances in Difference Equations 2008, 2008:-11.Google Scholar
- Sun J-P: Positive solution for first-order discrete periodic boundary value problem. Applied Mathematics Letters 2006, 19(11):1244-1248. 10.1016/j.aml.2006.01.007MATHMathSciNetView ArticleGoogle Scholar
- Gao C:Existence of solutions to -Laplacian difference equations under barrier strips conditions. Electronic Journal of Differential Equations 2007, 2007(59):1-6.Google Scholar
- Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.MATHView ArticleGoogle Scholar
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