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

Volume 8, Issue 3, pp 174--183 http://dx.doi.org/10.22436/jnsa.008.03.01

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Authors

Faxing Wang - TongDa College of Nanjing University of Posts and Telecommunications, 225127 Yangzhou, China. Ying Zheng - College of Science, Nanjing University of Posts and Telecommunications, 210046 Nanjing, China.

Abstract

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.

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Keywords

• Coincidence point
• first-order discrete nonlocal problem
• contraction
• lower and upper solutions
• connected sets.

MSC

•  47H10
•  54H25

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