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CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS

CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS VIA INEQUALITY THEORY ¨ HAISHEN LU AND DONAL O’REGAN Received 25 May 2004 We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign. 1. Introduction An upper- and lower-solution theory is presented for the singular discrete boundary value problem −∆ ϕ p ∆u(k − 1) = q(k) f k,u(k) , k ∈ N = {1,.,T }, u(0) = u(T + 1) = 0, and the singular discrete initial value problem ∆u(k. | CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS VIA INEQUALITY THEORY HAISHEN LU AND DONAL O REGAN Received 25 May 2004 1.1 1.2 We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign. 1. Introduction An upper- and lower-solution theory is presented for the singular discrete boundary value problem -A ỹp A k- 1 q k f k u k k e N 1 . T u 0 u T 1 0 and the singular discrete initial value problem Au k - 1 q k f k u k k e N 1 . T u 0 0 where Ọp s s p-2s p 1 Au k - 1 u k - u k -1 T e 1 2 . N 0 1 . T and u N R. Throughout this paper we will assume f N X 0 to R is continuous. As a result our nonlinearity f k u may be singular at u 0 and may change sign. Remark 1.1. Recall a map f N X 0 to R is continuous if it is continuous as a map of the topological space N X 0 to into the topological space R. Throughout this paper the topolopy on N will be the discrete topology. We will let C N R denote the class of map u continuous on N discrete topology with norm Huh maxkeN u k . By a solution to 1.1 resp. 1.2 we mean a u e C N R such that u satisfies 1.1 resp. 1.2 for i e N and u satisfies the boundary resp. initial condition. It is interesting to note here that the existence of solutions to singular initial and boundary value problems in the continuous case have been studied in great detail in Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 2 2005 205-214 DOI 10.1155 ADE.2005.205 206 Discrete initial and boundary value problems the literature see 2 4 5 6 7 9 10 11 and the references therein . However only a few papers have discussed the discrete singular case see 1 3 8 and the references therein . In 7 the following result has been proved. Theorem 1.2. Let n0 e 1 2 . be fixed and suppose the following conditions are satisfied f N X 0 to R is continuous 1.3 q e