On attractivity and positivity of solutions for functional integral equations of fractional order (Q474263)

Summary: We investigate a class of functional integral equations of fractional order given by \[ x(t)= q(t)+f_{1}(t,x(\alpha_{1}(t)), x(\alpha_{2}(t))) +\frac{f_{2}(t,x(\beta_{1}(t)), x(\beta_{2}(t)))}{\Gamma(\alpha)}\times\int_{0}^{t}(t-s)^{\alpha-1}f_{3}(t,s,x(\gamma_{1}(s)), x(\gamma_{2}(s)))ds. \] Sufficient conditions for the existence, global attractivity, and ultimate positivity of solutions of the equations are derived. The main tools include the techniques of measures of noncompactness and a recent measure theoretic fixed point theorem of Dhage. Our investigations are placed in the Banach space of continuous and bounded real-valued functions defined on unbounded intervals. Moreover, two examples are given to illustrate our results.