Lower and upper solutions for a discrete first-order nonlocal problems at resonance :

Wang, Faxing ¹   ; Zheng, Ying ²

¹ TongDa College of Nanjing University of Posts and Telecommunications, 225127 Yangzhou, China
² College of Science, Nanjing University of Posts and Telecommunications, 210046 Nanjing, China

Journal of nonlinear sciences and its applications, Tome 8 (2015) no. 3, p. 174-183

Résumé

We discuss the existence of solutions for the discrete first-order nonlocal problem

$ \begin{cases} \Delta u(t - 1) = f(t, u(t)),\quad t \in \{1, 2, ... , T\},\\ u(0) +\Sigma_{i=1}^m \alpha_iu(\xi_i) = 0, \end{cases} $

where $f : \{1,..., T\} \times \mathbb{R}\rightarrow \mathbb{R}$ is continuous, $T > 1$ is a fixed natural number, $\alpha_i \in (-\infty; 0],\, \xi_i \in \{1,...,T\}(i = 1,..., m; 1 \leq m \leq T; m \in \mathbb{N})$ are given constants such that $\Sigma_{i=1}^m \alpha_i+ 1 = 0$. We develop the methods of lower and upper solutions by the connectivity properties of the solution set of parameterized families of compact vector fields.

Détail

DOI : 10.22436/jnsa.008.03.01

Classification : 47H10, 54H25
Keywords: Coincidence point, first-order discrete nonlocal problem, contraction, lower and upper solutions, connected sets.

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