Existence Results for $\aleph$-Caputo Fractional Boundary Value Problems with $p$-Laplacian Operator

Year 2024, , 61 - 71, 30.06.2024

Özlem Batit Özen

https://doi.org/10.53570/jnt.1472049

Abstract

This study delves into the investigation of positive solutions for a specific class of $\aleph$-Caputo fractional boundary value problems with the inclusion of the p-Laplacian operator. In this research, we use the theory of the fixed point theory within a cone to establish the existence results for solutions of nonlinear $\aleph$-Caputo fractional differential equations involving the p-Laplacian operator. These findings not only advance the theoretical understanding of fractional differential equations but also hold promise for applications in diverse scientific and engineering disciplines. Furthermore, we provide a clear and illustrative example that serves to reinforce the fundamental insights garnered from this investigation.

Keywords

Fractional differential equation, boundary value problem, $p$-Laplacian operator, fixed point theorem

References

• A. E. Mfadel, S. Melliani, M. Elomari, Existence and uniqueness results for $\psi$-caputo fractional boundary value problems involving the $p$ Laplace operator, UPB Scientific Bulletin Series A: Applied Mathematics and Physics 94 (2022) 37-46.
• M. S. Abdo, A. G. Ibrahim, S. K. Panchal, Nonlinear implicit fractional differential equation involving $\psi$-Caputo fractional derivative, Proceeding of the Jangjeon Mathematical Society 22 (3) (2019) 387-400.
• R. Almeida, A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation 44 (2017) 460-481.
• M. S. Abdo, S. K. Panchal, A. M. Saeed, Fractional boundary value problem with $\psi$-Caputo fractional derivative, Proceedings Mathematical Sciences 129 (65) (2019) 64-78.
• M. S. Abdo, S. K. Panchal, Fractional integro-differential equations involving $\psi$-Hilfer fractional derivative, Advances in Applied Mathematics and Mechanics 11 (2) (2019) 338-359.
• H. A. Wahash, S. K. Panchal, M. S. Abdo, Existence and stability of a nonlinear fractional differential equation involving a $\psi$-Caputo operator, Advances Theory of Nonlinear Analysis and Its Application 4 (4) (2020) 266-278.
• S. Wang, Z. Bai, Existence and uniqueness of solutions for a mixed $p$ Laplace boundary value problem involving fractional derivatives, Advances in Difference Equations 694 (2020) 1-9.
• C. Bai, Existence and uniqueness of solutions for fractional boundary value problems with $p$ Laplacian operator, Advances in Difference Equations 4 (2018) 1-12.
• A. Alsaedi, M. Alghanmi, B. Ahmad, B. Alharbi, Uniqueness results for a mixed $p$ Laplacian boundary value problem involving fractional derivatives and integrals with respect to a power function, Electronic Research Archive 31 (1) (2022) 367-385.
• A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
• R. Almeida, A. B. Malinowska, M. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Mathematical Methods in the Applied Sciences 41 (1) (2018) 336-352.
• Y. Sun, Z. Zeng, J. Song, Existence and uniqueness for the boundary value problems of nonlinear fractional differential equation, Applied Mathematics 8 (3) (2017) 312-323.
• R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports 339 (1) (2000) 1-77.
• R. L. Bagley, P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, American Institute of Aeronautics and Astronautics Journal 23 (6) (1985) 918-925.
• R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Computers and Mathematics with Applications 59 (5) (2010) 1586-1593.