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Positive solutions of BVPs for third-order discrete nonlinear difference systems

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Abstract

This paper is concerned with the following system

$$\Delta ^3u_i(k)+f_i(k,u_1(k),u_2(k),\ldots,u_n(k))=0,\quad{}k\in [0,T],\ i=1,2,\ldots,n,$$

with the Dirichlet boundary condition

$$u_i(0)=u_i(1)=u_i(T+3)=0,\quad{}i=1,2,\ldots,n.$$

Some results are obtained for the existence, multiplicity and nonexistence of positive solutions to the above system by using nonlinear alternative of Leray-Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Leggett-Williams fixed point theorem. In particular, it proves that the above system has N positive solutions under suitable conditions, where N is an arbitrary integer.

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Authors and Affiliations

  1. School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, People’s Republic of China

    Rui Zhang

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  1. Rui Zhang
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Correspondence to Rui Zhang.

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Zhang, R. Positive solutions of BVPs for third-order discrete nonlinear difference systems. J. Appl. Math. Comput. 35, 551–575 (2011). https://doi.org/10.1007/s12190-010-0378-7

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  • DOI: https://doi.org/10.1007/s12190-010-0378-7

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