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Three solutions of constant sign for a system of discrete equations

Authors:

Ravi P. Agarwal Donal O’Regan Patricia J. Y. Wong

Affiliation:

(1) Department of Mathematical Sciences, Florida Institute of Technology, 32901-6975 Melbourne, FL, U.S.A.;(2) Department of Mathematics, National University of Ireland, Galway, Ireland;(3) School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore, Singapore

Abstract:

We consider the following system of discrete equations

$u_i left( k right) = sumlimits_{ell = 0}^N {g_i left( {k,ell } right)P_i left( {ell ,u_1 left( ell right),u_2 left( ell right),...,u_n left( ell right)} right),k in left{ {0,1,...,T} right},1 leqslant i leqslant n}$

. Criteria for the existence of three constant-sign solutions of the system will be developed. To illustrate the generality of the results obtained, we include applications to several well-known boundary value problems. Parallel results are also established for a system on {0,1,...}

$u_i left( k right) = sumlimits_{ell = 0}^infty {g_i left( {k,ell } right)P_i left( {ell ,u_1 left( ell right),u_2 left( ell right),...,u_n left( ell right)} right),k in left{ {0,1,...} right},1 leqslant i leqslant n}$

Keywords:

constant-sign solutions system of discrete equations boundary value problems

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