a fixed-point theoretic approach to the unique solvability of a high-order nonlinear fractional boundary value problem with integral conditions

This work is devoted to a rigorous analysis of the existence and uniqueness of solutions for a class of high-order nonlinear differential equations of fractional order. the considered problem is defined by a caputo fractional derivative and is augmented by a set of nonlocal boundary constraints. a key feature of these constraints is an integral condition that couples the behavior of the solution across its entire spatial domain, reflecting a global dependency. our primary analytical strategy is to recast the differential problem as a fixed-point equation for an equivalent integral operator. this is accomplished by first methodically constructing the green's function associated with the corresponding linear problem. with the integral operator established, the existence of a unique solution for the full nonlinear problem is then proven by leveraging the power of the banach contraction mapping principle. to demonstrate the practical relevance and applicability of our theoretical framework, a detailed illustrative example is presented and analyzed.