Existence of solutions to a noncontractive caputo fractional delay integral boundary value problem

This paper establishes the existence of solutions for a class of Caputo fractional delay integral boundary value problems (CFDIBVPs) in the Banach space AC¹([0, 1], ℝⁿ). The presence of nonlocal boundary conditions and delay arguments produces an associated integral operator that is generally noncontractive. Using a decomposition into condensing components and Sadovskii’s fixed point theorem, we obtain existence results without invoking the Banach contraction principle. Precise estimates in the AC¹([0, 1])-norm verified continuity, boundedness, and the self-map property of the operator. An explicit example illustrates how delay and boundary terms determined the operator constant and prevented classical contractivity. The analysis extends fixed-point methods for fractional systems with memory and delay, showing that the boundary and delay structures, rather than the fractional order alone, govern the solvability of the CFDIBVP.