Abstract
The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equation , , where is a parameter and is Lipschitz continuous and has three real zeros . In particular we prove that for each sufficiently small there exists a solution such that is increasing, and . The problem is motivated by some models arising in hydrodynamics.
Highlights
In particular we prove that for each sufficiently small h > 0 there exists a solution {x n }∞n 0 such that {x n }∞n 1 is increasing, x 0 x 1 ∈ L0, 0, and limn → ∞x n > L
In 1 we have shown that 1.1 is a discretization of differential equations which generalize some models arising in hydrodynamics or in the nonlinear field theory; see 2– 6
In 1, we have described the set of all solutions of problem 1.1, 1.6, where x 0 B, x 1 B, B ∈ L0, 0
Summary
We will investigate the following second-order non-autonomous difference equation xn 1 xn n n1 x n −x n−1. Let us note that f ∈ Liploc L0, ∞ means that for each L0, A ⊂ L0, ∞ there exists KA > 0 such that |f x − f y | ≤ KA|x − y| for all x, y ∈ L0, A. For each values B, B1 ∈ L0, ∞ there exists a unique solution {x n }∞n 0 of 1.1 satisfying the initial conditions x 0 B, x 1 B1. In this paper, using 1 , we will prove that for each sufficiently small h > 0 there exists at least one B ∈ L0, 0 such that the corresponding solution of problem 1.1 , 1.6 fulfils x0. We would like to point out that recently there has been a huge interest in studying the existence of monotonous and nontrivial solutions of nonlinear difference equations. A lot of other interesting references can be found therein
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