A Unifying Framework for Nonlocal Boundary Value Problems

This paper develops a unified existence and uniqueness theory for one‐dimensional nonlocal boundary value problems with coefficients depending on a global functional of the solution. Using tailored cones and annular regions, we derive explicit inequalities that determine solvability ranges and provide a natural eigenvalue interpretation for the model parameter. The framework covers integral norms, convolution‐type operators, and variable growth functionals, and yields structural conditions guaranteeing nonnegative solutions and, under monotonicity assumptions, uniqueness. A numerical study of a logistic‐type nonlocal problem with ‐dependent diffusion coefficient illustrates the theory. Self‐consistent solutions are computed with a collocation BVP solver, verified to satisfy all cone constraints and physical bounds with robust margins. The results connect the abstract cone construction to concrete parameter ranges and provide a foundation for future well‐posedness and numerical analyses of more singular nonlocal problems.