Volterra and Functional Differential Equations

COSINE FAMILIES, PRODUCT SPACES AND INITIAL VALUE PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

Samuel M. Rankin, III

Department of Mathematics

West Virginia University

Morgantown, West Virginia

DOI: 10.1201/9781003420026-22

1. INTRDOUCTION

Consider the initial boundary value problem

$$\begin{array}{c}{\text{y}}_{\text{tt}}{\text{(x,t)=y}}_{\text{xx}}\text{(x,t),}\text{0<x<π,}t\in ℝ\\ {\text{y}}_{\text{tt}}\text{(0,t)}={\int }_0^{\pi }\text{f(x)y(x,t)dx+u(t)},t\in R,\text{u}\in {\text{L}}_{\text{10c}}^{\text{2}}(0,\infty ),\text{f}\in {\text{L}}^{\text{2}}(0,\pi )\\ \text{y(π,t)}=0,\text{t}=ℝ\\ \text{y(x,0)}={\text{y}}_{\text{0}}\text{(x)},{\text{y}}_{\text{t}}{\text{(x,0)=y}}_{\text{1}}\text{(x),}{\text{y}}_{\text{0}}{\text{,y}}_{\text{1}}{\text{=L}}^{\text{2}}\text{(0,π)}\\ \text{y(0,0)}={\text{z}}_{\text{0}},{\text{y}}_{\text{t}}{\text{(0,0)=z}}_{\text{1}}\text{,}{\text{z}}_{\text{0}}{\text{,z}}_{\text{1}}\in ℝ\end{array}$$(1)

Define the operators:

$$\begin{array}{ll}\hfill & \text{A}:\text{D}(\text{A}\hfill \end{array}$$