A new approach to the theory of functional integral equations of fractional order (Q615901)

The aim of the paper is to present a new approach to the theory of functional integral equations of fractional order of the form \[ x(t)=h(t)+\frac{f_1(t,x(t))}{\Gamma(\alpha)}\int_{0}^{t}\frac{u(t,s,x(s))}{(t-s)^{1-\alpha}} ds, \quad t\in [0,1],\quad \alpha\in (0,1). \] That approach depends on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type \[ x(t)=h(t)+\int_{a}^{t} u(t,s,x(s))d_sg(t,s), \quad t\in [a,b], \] where \(g\) is defined on the square \([a,b]\times [a,b]\) and satisfies some conditions on this square. The authors develop the theory concerning the existence of solutions of this equation using the technique associated with measures of noncompactness and the theory of functions with bounded variation. It turns out that the study of functional integral equations of Volterra-Stieltjes type is more convenient and effective than the study of functional integral equations of fractional order. An example illustrating the approach is also discussed.