NONDECREASING SOLUTIONS OF A FRACTIONAL QUADRATIC INTEGRAL EQUATION OF URYSOHN-VOLTERRA TYPE

In this paper we study a very general quadratic integral equation of fractional order. We show that the quadratic integral equations of fractional orders has at least one monotonic solution in the Banach space of all real functions defined and continuous on a bounded and closed interval. The concept of a measure of noncompactness related to monotonicity, introduced by J. Bana´s and L. Olszowy, and a fixed point theorem due to Darbo are the main tools in carrying out our proof. In fact we generalize, improve the results of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005), 112–119]. Also, we extend and generalize the results of the paper [J. Bana´s and B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl. 332 (2007), 1370–1378].