- Research Article
- Open Access
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).
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