Bulletin of the Australian Mathematical Society

Research Article

THE SET OF SOLUTIONS OF INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACES

RAVI P. AGARWALa1, DONAL O’REGANa2 and ANETA SIKORSKA-NOWAKa3

a1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA (email: agarwal@fit.edu)

a2 Department of Mathematics, National University of Ireland, Galway, Ireland (email: donal.oregan@nuigalway.ie)

a3 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland (email: anetas@amu.edu.pl)

Abstract

In this paper, we first prove an existence theorem for the integrodifferential equation

\begin{equation}\label {eqast} \xcases x'(t)=f\bigg (t,x(t), \int _{0}^{t}k(t,s,x(s))\,ds\bigg ) \\ x(0)=x_{0} \endxcases \hspace *{-.9pc} ,\quad t\in I_{a} =[0,a],\ a\in R_{+} , \end{equation}

(*)

where f,k,x are functions with values in a Banach space E and the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the set S of all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where $I_{d} \subset I_{a} $.

(Received March 17 2008)

2000 Mathematics subject classification

  • 34D09;
  • 34D99

Keywords and phrases

  • integral equations;
  • existence theorem;
  • pseudo-solution;
  • set of solutions;
  • measure of noncompactness;
  • Henstock–Kurzweil–Pettis integral