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)+∫10k1(t,s)f1s,∫0sk2(s,τ)f2(τ,x(τ))dτdswill be proved in L1[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)+∫10k1(t,s)f1(s,Dβx(s))ds,β∈(0,1].