Construction of upper and lower solutions for singular discrete initial and boundary value problems via inequality theor
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e 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,
(1.1)
and the singular discrete initial value problem
∆u(k − 1) = q(k) f k,u(k) , u(0) = 0,
k ∈ N = {1,... ,T },
(1.2)
where ϕ p (s) = |s| p−2 s, p > 1, ∆u(k − 1) = u(k) − u(k − 1), T ∈ {1,2,... }, N + = {0,1,...,T }, and u : N + → R. Throughout this paper, we will assume f : N × (0, ∞) → 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 × (0, ∞) → R is continuous if it is continuous as a map of the topological space N × (0, ∞) 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 u = maxk∈N + u(k). By a solution to (1.1) (resp., (1.2)) we mean a u ∈ C(N + , R) such that u satisfies (1.1) (resp., (1.2)) for i ∈ 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
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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 ∈ {1,2,...} be fixed and suppose the following conditions are satisfied: f : N × (0, ∞) −→ R is continuous,
(1.3)
q ∈ C N,(0, ∞) ,
(1.4)
there exists a function α ∈ C N + , R with α(0) = α(T + 1) = 0,
α > 0 on N such that
q(k) f k,α(k) ≥ −∆ ϕ p α(k − 1)
(1.5)
for k ∈ N,
there exists a function β ∈ C(N + , R) with 1 β(k) ≥ α(k), β(k) ≥ for k ∈ N + with n0 q(k) f k,β(k) ≤ −∆ ϕ p β(k − 1) for k ∈ N.
(1.6)
Then (1.1) has a solution u ∈ C(N + , R) with u(k) ≥ α(k) for k ∈ N + . In [1], the following result has been proved. Theorem 1.3. Let n0 ∈ {1,2,... } be fixed and suppose the following conditions are satisfied: f : N × (0, ∞) −→ R is continuous,
(1.7)
q ∈ C N,(0, ∞) ,
(1.8)
there exists a function α ∈ C N + , R with α(0) = 0,
α > 0 on N such that
q(k) f k,α(k) ≥ ∆α(k − 1)
(1.9)
for k ∈ N,
there exists a function β ∈ C N + , R with 1 β(k) ≥ α(k), β(k) > for k ∈ N + with n0 q(k) f k,β(k) ≤ ∆β(k − 1) for k ∈ N.
(1.10)
Then (1.2) has a solution u ∈ C(N + , R) with u(k) ≥ α(k) for k ∈ N + . Also some results from the literature, which will be needed in
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