Boundary value problems for functional difference equations
The authors apply the topological transversality in a form based on the nonlinear alternative of Leray and Schauder to obtain solutions for a second-order functional difference equation. Let \(a\) and \(b\) be integers with \(a< b\). We write \([a, b]= \{a, a+1,\dots, b\}\) and \([a, b)= \{a, a+ 1,\dots, b- 1\}\). For a function \(f: [a, b]\to (- \infty, \infty)\) we define its norm, \(|f|_{[a, b]}\), by \(|f|_{[a, b]}= \max_{a\leq x\leq b} |f(x)|\). Define the space of functions, \(F_N\), by \(F_N= \{x\mid x: [- N, 0]\to (- \infty, \infty)\}\). Then \((F_N, |\cdot|_{[- N, 0]})\) is a Banach space. The authors prove the existence of solutions of boundary value problems for the second-order functional difference equation (1) \(\Delta^2 x(t- 1)= f(t, x_{t- 1}, \Delta x(t- 1))\), \(t\in [1, T+ 1]\), satisfying (2) \(x(s)= \phi(s)\), \(s\in [- N, 0]\), (3) \(x(T+ 2)= A\), where \(f: [1, T+ 1]\times F_N\times (- \infty, \infty)\to (-\infty, \infty)\) is continuous, \(\phi\in F_N\), and \(A\in (- \infty, \infty)\), and where \(T\) and \(N\) are integers with \(T> N\). In particular, they prove the following result: Theorem 1. Let \(f: [1, T+ 1]\times F_N\times (- \infty, \infty)\to (- \infty, \infty)\) be continuous, and assume there exists a \(K> 0\) such that, for each \(0\leq \lambda\leq 1\), \(|x|_{[0, T+ 2]}\leq K\), for all solutions \(x\) of the boundary value problem (4) \(\Delta^2 x(t- 1)= \lambda f(t, x_{t- 1}, \Delta x(t- 1))\), \(t\in [1, T+ 1]\), (5) \(x(s)= \phi(s)\), \(s\in [- N, 0]\), (6) \(s(T+ 2)= \lambda A\). Then, the boundary value problem (1)--(3) has at least one solution. The authors obtain two generalizations of Theorem 1. The first generalization involves Sturm-Liouville boundary conditions, and the second generalization is for \(n\)th-order equations.
nonlinear alternative of Leray and Schauder
second-order functional difference equation
scientific article; zbMATH DE number 915096
scientific article