Existence of nonnegative solutions for discrete Robin boundary value problems with sign-changing weight

Authors

YAN ZHU

DOI

10.3906/mat-2012-34

Abstract

In this paper,~we are concerned with the following discrete problem first $$\left\{ \begin{array}{ll} -\Delta^{2}u(t-1)=\lambda p(t)f(u(t)), &t\in[1,N-1]_{\mathbb{Z}},\\ \Delta u(0)=u(N)=0,\\ \end{array} \right. $$ where $N>2$~is an integer,~$\lambda>0$~is a parameter,~$p:[1,N-1]_{\mathbb{Z}}\rightarrow\mathbb{R}$~is a sign-changing function,~$f:[0,+\infty)\rightarrow[0,+\infty)$~is a continuous and nondecreasing function.~$\Delta u(t)=u(t+1)-u(t)$,~$\Delta^{2}u(t)=\Delta(\Delta u(t))$.~By using the iterative method and Schauder's fixed point theorem,~we will show the existence of nonnegative solutions to the above problem. Furthermore, we obtain the existence of nonnegative solutions for discrete Robin systems with indefinite weights.

Keywords

Robin boundary value problems, Green's function, iterative method, nonnegative solutions, system

First Page

864

Last Page

877

Recommended Citation

ZHU, YAN (2021) "Existence of nonnegative solutions for discrete Robin boundary value problems with sign-changing weight," Turkish Journal of Mathematics: Vol. 45: No. 2, Article 15. https://doi.org/10.3906/mat-2012-34
Available at: https://journals.tubitak.gov.tr/math/vol45/iss2/15