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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Jeffrey W. Lyons & Jeffrey T. Neugebauer & Aaron G. Wingo, 2025. "Existence and Nonexistence of Positive Solutions for Fractional Boundary Value Problems with Lidstone-Inspired Fractional Conditions," Mathematics, MDPI, vol. 13(8), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:8:p:1336-:d:1637981
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