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Fixed point techniques in cone theory with applications to difference equations
Abstract
First, a second order vector boundary value problem for difference equation, $-\Delta\sp2y(t-1)=f(t,y(t)), 1\le t\le b+1,\alpha y(0)-\beta y(0)=0,\gamma y(b+1)+\delta\Delta y(b+1)=0,$ is considered, where $b\ge2$ is a fixed integer, for each t in the discrete interval $\lbrack1, b+1\rbrack\equiv\{1,2,\...,b+1\}, f$ is an n dimensional vector function which is continuous with respect to the vector variable y, and $\alpha,\beta,\gamma\ge0,\delta>0$ are constants. The assumption $\rho\equiv\alpha\gamma(b+1)+\alpha\delta+\beta\gamma>0?$ gives the existence and positivity of the Green's function G(t, s) for the corresponding homogenous boundary value problem. Various properties of G(t,s) are given. We use cone theory results in a Banach space to prove the existence of at least two positive solutions for this boundary value problem. Next, a formula for the Green's matrix G(t,s) for the vector boundary value problem $-\Delta\lbrack P(t-1)\Delta y(t-1)\rbrack=0,\alpha y(0)-\beta y(0)=0,\gamma y(b+1)+\delta\Delta y(b+1)=0$, is derived under the assumption that $D\equiv\alpha\gamma\sum\sbsp{\tau=0}{b}P\sp{-1}(\tau)+ \alpha\delta P\sp{-1}(b+1)+\beta\gamma P\sp{-1}(0)$ is nonsingular. Using various properties of G(t, s) and abstract fixed point theorems in cone theory, the existence of positive solutions of a certain nonlinear vector boundary value problem is proved. Finally, existence of positive solutions for (n, n)-focal boundary value problems is given for a $2n\sp{th}$ order difference equation. The techniques involve the theory of operators on a Banach space. Sign properties of the Green's function are determined in order that the fixed point theorem of cone expansion and compression can be applied.
Subject Area
Mathematics
Recommended Citation
Atici, Ferhan, "Fixed point techniques in cone theory with applications to difference equations" (1995). ETD collection for University of Nebraska-Lincoln. AAI9600725.
https://digitalcommons.unl.edu/dissertations/AAI9600725