A Study on the Stability and Solvability of Pantograph-Type Equations with \((k,\psi)\)-Caputo proportional fractional derivative operator
Analysis of Existence, Uniqueness and Stability under Nonlocal Boundary Conditions
Abstract
This study investigates a class of pantograph-type equations involving the \((k,\psi)\)-Caputo proportional fractional derivative, subject to nonlocal fractional integral boundary conditions. The existence and uniqueness of solutions are established through the application of Banach’s and Krasnoselskii’s fixed point theorems. Furthermore, various forms of Ulam stability are analyzed. To illustrate the theoretical results and demonstrate their applicability, a numerical example is provided.Downloads
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References
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{\em Existence and Uniqueness of Solutions for Nonlinear Fractional Boundary Value Problems with $\psi$-Caputo Derivatives}, Bol. Soc. Paran. Mat. (43)1, 1-12, (2025).
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\end{thebibliography}
\bibitem{1} Houas, M.,
{\em Solvability and stability of neutral Caputo-Hadamard fractional pantograph-type differential equations}, Acta Univ. Apul. Math. Inform. 68, 83-98, (2021).
\bibitem{2} Khaminsou, B., Thaiprayoon, C., Sudsutad, W., $\&$ Jose, S. A.,
{\em Qualitative analysis of a proportional Caputo fractional pantograph differential equation with mixed nonlocal conditions}, Nonlinear Funct. Anal. Appl. 197-223 (2021).
\bibitem{3} Bouzid, H., et al.
{\em Existence and Ulam stability results of hybrid Langevin pantograph $\psi$-fractional coupled systems}, Filomat 39.10, 3401-3424, (2025).
\bibitem{4} Granas, A., Dugundji, J.,
{\em Fixed point theory}, Springer, (2003).
\bibitem{5} Sudsutad, W., Kongson, J., $\&$ Thaiprayoon, C.,
{\em On generalized (k, $\psi$)-Hilfer proportional fractional operator and its applications to the higher-order Cauchy problem}, Bound. Value Probl. 1, 83, (2024).
\bibitem{6} D{\i}az, R., Pariguan, E.,
{\em On hypergeometric functions and Pochhammer k-symbol}, J. Divulgaciones Matem{\'a}ticas. 15(2), 179-192, (2007).
\bibitem{7} Solhi, S., Anaama, E., Kajouni, A., and Hilal, K.,
{\em Existence and Uniqueness of Solutions for Nonlinear Fractional Boundary Value Problems with $\psi$-Caputo Derivatives}, Bol. Soc. Paran. Mat. (43)1, 1-12, (2025).
\bibitem{8} Kumar, D., $\&$ Baleanu, D.,
{\em Fractional calculus and its applications in physics}, Front. Phys. 7, 81, (2019).
\bibitem{9} Simpson, R., et al.
{\em Fractional calculus as a mathematical tool to improve the modeling of mass transfer phenomena in food processing}, Food Eng. Rev. 5, 45-55, (2013).
\bibitem{10} Luchko, Yury F., et al.
{\em Fractional models, non-locality, and complex systems}, Comput. Math. Appl. 59(3), 1048-1056, (2010).
\bibitem{11} Harikrishnan, S., Elsayed, E. M., Kanagarajan1, K.,
{\em Existence and uniqueness results for fractional pantograph equations involving $\psi$-Hilfer fractional derivative}, Dyn. Contin. Discret. Impuls. Syst. 25, 319-328, (2018).
\bibitem{12} Arhrrabi E., Elomari M., Melliani S. and Chadli L. S.,
{\em Existence and stability of solutions for a coupled problem of fuzzy fractional pantograph stochastic differential equations}, Asia Pac. J. Math. (9)20, (2022).
\end{thebibliography}
Published
2025-09-30
Section
Advanced Computational Methods for Fractional Calculus
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