Existence Results for a Multi-Point $p-$Laplacian Boundary Value Problem of Fractional Order

Year 2025, Volume: 17 Issue: 2 , 396 - 407 , 30.12.2025

Tuğba Şenlik Çerdik

https://doi.org/10.47000/tjmcs.1578171

https://izlik.org/JA79YM33CW

Abstract

This study investigates sufficient conditions to guarantee the existence of positive solutions for a fractional boundary value problem with integral boundary conditions. While there has been limited research on Riemann–Liouville fractional boundary value problems involving $p-$Laplacian operators and nonlinear terms with fractional derivatives of unknown functions, this work contributes to filling that gap. By employing Bai–Ge’s fixedpoint theorem and the corresponding Green’s function, we establish the existence of multiple positive solutions. An illustrative example is also provided to support the theoretical findings.

Keywords

Positive solution , fixed-point theorem , fractional derivative

References

• Abdo, M.S., Panchal, S.K., Saeed, A.M., Fractional boundary value problem with ψ-Caputo fractional derivative, Proceedings-Mathematical Sciences, 129(2019).
• Basua, D., Positive solutions for nonlinear fractional boundary value problems, Int. J. Dynam. Control, 12(2024), 283–291.
• Bai, Z., Sun, W., Existence and multiplicity of positive solutions for singular fractional boundary value problems, Computers and Mathematics with Applications, 63(9)(2012), 1369–1381.
• Bai, Z., Ge, W., Existence of three positive solutions for some second-order boundary value problems, Comput. Math. Appl., 48(2004), 699–707.
• Bai, C., Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 24(2008), 1–10.
• Batik, S., Deren. F.Y., Semipositone Fractional Boundary Value Problems with n Point Fractional Integral Boundary Conditions, Miskolc Mathematical Notes,23(1)(2022),93–104.
• Cabada, A., Wang, G., Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389(1)(2012), 403–411.
• Cabada, A., Hamdi, Z., Nonlinear fractional differential equations with integral boundary value conditions, Applied Mathematics and Computation, 228(2014), 251–257.
• Chai, G., Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator, Bound. Value Probl., 2012(2012).
• Cerdik, T.S., Hamal, N.A., Deren, F.Y., Existence of solutions for nonlinear fractional differential equations with m-point integral boundary conditions, Dynam. Systems Appl., 24(2015), 283–294.
• Deren, F.Y., Cerdik, T.S., Agarwal, R.P., Existence criteria of positive solutions for fractional p-Laplacian boundary value problems, Filomat, 34(11)(2022), 3789–3799.
• Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• Jiafa, X., Jiang, J., O’Regan, D., Positive solutions for a class of p-Laplacian Hadamard fractional-order three-point boundary value problems, Mathematics, 8(3)(2020), 308.
• Khan, H., Alzabut, J., Gulzar, H., Existence of solutions for hybrid modified ABC-fractional differential equations with p-Laplacian operator and an application to a waterborne disease model, Alexandria Engineering Journal, 70(2023), 665–672.
• Khan, H., Alzabut, J., Alkhazzan, A., Qualitative dynamical study of hybrid system of pantograph equations with nonlinear p-laplacian operator in banach’s space, Results in Control and Optimization, 15(2024), 100416.
• Kilbas, A.A., Srivastava, H.M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V, Amsterdam, 2006.
• Lakshmikantham, V., Leela, S., Devi, J.V., Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
• Li, Y., Li, G., Positive solutions of p-Laplacian fractional differential equations with integral boundary value conditions, J. Nonlinear Sci. Appl., 9(2016), 717–726.
• Liu, X., Jia, M., The positive solutions for integral boundary value problem of fractional p-Laplacian equation with mixed derivatives, Mediterr. J. Math., 14(2017), 94.
• Lyons, J.W., Neugebauer, J.T., Existence of a positive solution for a singular fractional boundary value problem with fractional boundary conditions using convolution and lower order problems, Turk J Math., 45(2021), 125–138.
• Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
• Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
• Rao, S.N., Meetei, M.Z., Nonexistence of positive solutions for a systems of coupled fractional bvps with p-Laplacian, TWMS J. App. Eng. Math. V., 9(3)(2019), 597–608.
• Rezapour, S., Thabet, S.T.M., Matar, M.M., Alzabut, J., Etemad, S., Some existence and stability criteria to a generalized FBVP having fractional composite p-Laplacian operator, Journal of Function Spaces, 2021(1), 9554076.
• Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
• Sousa, J.V.D.C., De Oliveira, E.C., On the ψ-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60(2018), 72–91.
• Vong, S., Positive solutions of singular fractional differential equations with integral boundary conditions, Mathematical and Computer Modelling, 57(5-6)(2013), 1053–1059.
• Wang, G., Liu, S., Agarwal, R.P., Zhang, L., Positive solutions of integral boundary value problem involving Riemann-Liouville fractional derivative, J. Fract. Calc. Appl., 4(2)(2013), 312–321.
• Wu, J., Zhang, X., Liu, L., Wu, Y., Twin iterative solutions for a fractional differential turbulent flow model, Bound. Value Probl., 98(2016).
• Yaslan, I., Gunendi, M., Higher order multi-point fractional boundary value problems with integral boundary conditions, Int. J. Nonlinear Anal. Appl., 9(1)(2018), 247–260.
• Zhang, X., Wang, L., Sun, Q., Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Applied Mathematics and Computation, 226(2014), 708–718.
• Zhou, C., Existence and uniqueness of positive solutions to higher-order nonlinear fractional differential equation with integral boundary conditions, Electron. J. Differential Equations, 2012(234), 1–11.
• Zhang, S.Q., Positive solutions for boundary value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ., 2006(2006), 1–12.