• 대한수학회논문집
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• 제24권1호
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• pp.29-40
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• 2009

Motivated by [Linear Algebra and its Appl. 420(2007), 218-227] and [Linear Algebra and its Appl. 425(2007), 171-183], we, in this paper, study the solvability of periodic boundary value problems of higher order nonlinear functional difference equations with p-Laplacian. Sufficient conditions for the existence of at least one solution of this problem are established.

• 대한수학회보
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• 제52권5호
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• pp.1517-1533
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• 2015

In this paper, we prove the existence of at least three nontrivial solutions to nonlinear discrete boundary value problems $$\{^{-{\Delta}_{p,{\omega}}u(x)+V(x){\mid}u(x){\mid}^{q-2}u(x)=f(x,u(x)),x{\in}S,}_{u(x)=0,\;x{\in}{\partial}S}$$, involving the discrete p-Laplacian on simple, nite and connected graphs $\bar{S}(S{\cup}{\partial}S,E)$ with weight ${\omega}$, where 1 < q < p < ${\infty}$. The approach is based on a suitable combine of variational and truncations methods.

• 충청수학회지
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• 제24권4호
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• pp.639-663
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• 2011

Motivated by Agarwal and O'Regan ( Boundary value problems for general discrete systems on infinite intervals, Comput. Math. Appl. 33(1997)85-99), this article deals with the discrete type BVP of the infinite difference equations. The sufficient conditions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main results. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multi-fixed-point theorems can be extended to treat BVPs for infinite difference equations. The strong Caratheodory (S-Caratheodory) function is defined in this paper.

• 대한수학회지
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• 제47권1호
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• pp.135-163
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• 2010

Motivated by [Science in China (Ser. A Mathematics) 36 (2006), no. 7, 721?732], this article deals with the following discrete type BVP $\LARGE\left\{{{\;{\Delta}[{\phi}({\Delta}x(n))]\;+\;f(n,\;x(n\;+\;1),{\Delta}x(n),{\Delta}x(n + 1))\;=\;0,\;n\;{\in}\;[0,N],}}\\{\;{x(0)-{\sum}^m_{i=1}{\alpha}_ix(n_i) = A,}}\\{\;{x(N+2)-\;{\sum}^m_{i=1}{\beta}_ix(n_i)\;=\;B.}}\right.$ The sufficient conditions to guarantee the existence of at least three positive solutions of the above multi-point boundary value problem are established by using a new fixed point theorem obtained in [5]. An example is presented to illustrate the main result. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multifixed-point theorems can be extended to treat nonhomogeneous BVPs. The emphasis is put on the nonlinear term f involved with the first order delta operator ${\Delta}$x(n).