Fixed Point Methods for Nonlinear Integral Equations Associated with Fractional Differential Problems

Hybrid fractional differential equations are useful mathematical models to describe the problems, in which the fractional-order behavior interacts with nonlinear functions and integral operators. The existence of solutions for these equations can be difficult to prove due to the combined effects of the fractional derivatives, the nonlinear terms and the structure of the underlying function space. To overcome this problem, an operator-based framework is developed in the study based on Dhage’s fixed-point theorem and monotone iterative techniques. The corresponding differential problem is reduced to the nonlinear integral equation and the existence of a mild solution is studied by imposing suitable conditions on the operators. The properties like monotonicity, partial continuity, boundedness and compactness in the partially ordered Banach space are incorporated in the analysis. Under the given assumptions the existence of a mild solution is proved as well as the convergence of the sequence of successive approximations to that solution monotonically.