ON MONOTONIC SOLUTIONS OF SOME NONLINEAR FRACTIONAL INTEGRAL EQUATIONS

Abstract

t is well known that nonlinear integral and differential equations create an important branch of nonlinear analysis. A lot of nonlinear problems arising from areas of the real world are generally represented with integral and differential equations. Especially, integral and differential equations of fractional order play a very important role in modelling of some problems in physics, mechanics and other fields in natural sciences. For instance, these equations are used in describing of some problems in theory of neutron transport, the theory of radioactive transfer, the kinetic theory of gases [18], the traffic theory and so on.

In this study, we examine the solvability of the following nonlinear integral equation of fractional order in C[0,a] which is the space of real valued and continuous functions defined on the interval [0, a]

We present some sufficient conditions for existence of nondecreasing solutions of the above equation. Then using a Darbo type fixed point theorem associated with the measure of noncompactness we prove that this equation has at least one nondecreasing solution in C[0,a] . Finally we give some examples to show that our result is applicable.