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Existence Of Nonnegative Solutions For Discrete Robin Boundary Value Problems With Sign-Changing Weight, Yan Zhu

Turkish Journal of Mathematics

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.