New approach on the solutions of nonlinear q-hybrid integro-differential equations

The momentous objective in this paper is to prove existence as well as approximations of solution in hybrid initial value problem with fractional q-derivative in Caputo sense. The main approach is based on the operator theoretic technique in a partially ordered metric space. The results are obtained under weaker partially conditions and partially Lipschitz conditions via Dhage iteration method. Examples illustrating the results are also presented.