Existence of solutions to singular integral equations

Pages: 818 - 827, Issue Special, September 2009

 Abstract        Full Text (151.5K)              

Patricia J.Y. Wong - School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore (email)

Abstract: We consider the system of integral equations

$u_i(t)=int_0^Tg_i(t,s)[a_i(s,u_1(s),u_2(s),...,u_n(s))+b_i(s,u_1(s),u_2(s),...,u_n(s))]ds,$   $t \in [0,T],$   $1<=i<=n,$

where $T>0$ is fixed and the nonlinearities $a_i(t,u_1,u_2,\cdots,u_n)$ can be singular at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$ Criteria are established for the existence of fixed-sign solutions $(u_1^*,u_2^*,\cdots,u_n^*)$ to the above system, i.e., $\theta_iu_i^*(t)\geq 0$ for $t\in [0,T]$ and $1\leq i\leq n,$ where $\theta_i\in \{1,-1\}$ is fixed. We also include an example to illustrate the usefulness of the results obtained.

Keywords:  Fixed-sign solutions, system of singular integral equations
Mathematics Subject Classification:  Primary: 45B05, 45G15; Secondary: 45M20

Received: July 2008;      Revised: March 2009;      Published: September 2009.