Bulletin of the Australian Mathematical Society

Research Article

EXISTENCE OF RESOLVENT FOR VOLTERRA INTEGRAL EQUATIONS ON TIME SCALES

MURAT ADıVARa1 c1 and YOUSSEF N. RAFFOULa2

a1 Izmir University of Economics, Department of Mathematics, 35330 Izmir, Turkey (email: murat.adivar@ieu.edu.tr)

a2 University of Dayton, Department of Mathematics, Dayton, OH 45469-2316, USA (email: youssef.raffoul@notes.udayton.edu)

Abstract

We introduce the concept of ‘shift operators’ in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equation

\[ x(t)=f(t)+\int _{t_{0}}^{t}a(t,s)x(s)\Delta s,\quad t_{0}\in \mathbb {T}^{\kappa }, \]

on time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.

(Received November 05 2009)

2000 Mathematics subject classification

  • primary 45D05; secondary 39A12

Keywords and phrases

  • existence;
  • resolvent;
  • shift operator;
  • time scales;
  • Volterra integral equation

Correspondence:

c1 For correspondence; e-mail: murat.adivar@ieu.edu.tr

Footnotes

This work was supported by the Scientific and Technological Research Council of Turkey.