Fractional differential equations with nonlocal boundary conditions involving initial and final segments of the given domain
DOI:
https://doi.org/10.31489/2026m2/4-16Keywords:
fractional differential equations, boundary value problems, Caputo fractional derivative operator, periodic/anti-periodic segmental boundary data, existence, uniqueness, nonlocal boundary conditions, fixed pointAbstract
We explore the existence and uniqueness criteria for solutions of a Liouville–Caputo fractional differential equation with the nonlinearity containing the unknown function as well as its lower order fractional derivative, and supplemented with a set of nonlocal fractional boundary data with respect to initial and final segments of the given domain. Integral boundary conditions offer an effective approach to model the flow and drag phenomena in arbitrary shaped vessels, heat conduction, biomedical computational fluid dynamics, engineering problems, etc. The notion of segmental type nonlocal fractional integral boundary conditions introduced in this paper is novel and specializes to periodic/anti-periodic boundary data under a suitable choice of the parameters involved in these conditions (see the second last paragraph of Introduction). We apply Krasnosel’ski˘i’s fixed point theorem and Leray-Schauder’s nonlinear alternative to prove two existence results for the problem at hand, while the uniqueness of its solutions is established via Banach’s contraction mapping principle. Examples are constructed for illustrating the obtained results. Our work is useful in the given configuration as it leads to a new direction for research on fractional boundary value problems. The paper concludes with some interesting observations.
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