EXISTENCE, UNIQUENESS AND BOUNDEDNESS OF SOLUTIONS FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

It is well known that the study of many processes of the natural sciences can be reduced to solve integro-differential equations with variable boundaries. Recently, studies on certain problems of the environment, such as the corona virus, the emergence of new diseases, and diseases associated with mutations of viruses, have become relevant. A solution to such problems is associated with finding solutions of integro-differential equations. For the last few years, researchers have been paying attention to the newly discovered fractional operators involving nonsingular kernels. The Caputo fractional derivative is the one of these operators which has captured the interest of scientists the most because of the many interesting results reported when this derivative is used in modelling some real-world phenomena. However, the theory of these operators is still to be addressed. In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinear Caputo fractional Volterra-Fredholm integro-differential equations with nonlocal conditions. The desired results are proved by using theory of fractional calculus aid of fixed point theorems due to Banach and Krasnoselskii in Banach spaces.