On the functional integral equations of mixed type and integro-differential equations of fractional orders

摘要

The existence of monotonic solution for the functional integral equation of the mixed typex(t) = g(t) + integral(0)(1) k(1)(t,s)f(1) (s, integral(0)(s) k(2)(s,tau)f(2)(tau,x(tau)) dtau) dswill be proved in L-1 [0, 1]. Such kind of equations plays a significant role in many branches of mathematical physics, engineering and economics. On the other hand, our considered equation are the general form of Hammerstein and Volterra integral equations (see [4] and [5]).As an application we prove the existence of solution for the initial value problem of the fractional order integro-differential equation.dx(t)/dt = g(t) + integral(0)(1) k(1)(t,s)f(1)(s,D(beta)x(s))ds, beta is an element of (0,1]. (C) 2003 Elsevier Inc. All rights reserved.