Existence and Nonexistence of Positive Solutions for Fractional Boundary Value Problems with Lidstone-Inspired Fractional Conditions

Jeffrey W. Lyons (), Jeffrey T. Neugebauer and Aaron G. Wingo
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Jeffrey W. Lyons: Department of Mathematical Sciences, The Citadel, 171 Moultrie Street, Charleston, SC 29409, USA
Jeffrey T. Neugebauer: Department of Mathematics and Statistics, Eastern Kentucky University, 521 Lancaster Avenue, Richmond, KY 40475, USA
Aaron G. Wingo: Independent Researcher, Marieta, GA 30060, USA

Mathematics, 2025, vol. 13, issue 8, 1-15

Abstract: This paper investigates the existence and nonexistence of positive solutions for a class of nonlinear Riemann–Liouville fractional boundary value problems of order α + 2 n , where α ∈ ( m − 1 , m ] with m ≥ 3 and m , n ∈ N . The conjugate fractional boundary conditions are inspired by Lidstone conditions. The nonlinearity depends on a positive parameter on which we identify constraints that determine the existence or nonexistence of positive solutions. Our method involves constructing Green’s function by convolving the Green functions of a lower-order fractional boundary value problem and a conjugate boundary value problem and using properties of this Green function to apply the Guo–Krasnosel’skii fixed-point theorem. Illustrative examples are provided to demonstrate existence and nonexistence intervals.

Keywords: fractional derivative; Lidstone; fixed-point theorem; existence; nonexistence; convolution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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