On the existence and uniqueness of solutions for a ψ-Caputo fractional boundary value problem with fractional boundary conditions
Resumo
The main purpose of the present work is the investigation of the existence and uniqueness of solutions to an implicit nonlinear boundary value problem
of fractional differential equations involving ψ-Caputo fractional derivative and subject to fractional boundary conditions. The results of existence and uniqueness are proved thanks to Banach and Scheafer's fixed point theorems. Finally, we present appurtenant examples to illustrate the obtained results.
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Referências
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Proc. Indian Acad. Sci. (Math. Sci.) $\ 129\ (2019),\ 65.$
\bibitem{alm} \ R. Almeida, A Caputo fractional derivative of a function
with respect to another function. Commun. Nonlinear Sci. Numer. Simul. $44,\
460-481\ (2017).$
\bibitem{alm2} R. Almeida, Fractional differential equations with mixed
boundary conditions. Bull. Malays. Math. Sci. Soc. $2019,\ 42,\ 1687-1697.$
\bibitem{alm3} R. Almeida, A. B. Malinowska, T. Odzijewicz, Optimal
Leader-Follower control for the fractional opinion Formation Model. J.
Optim. Theory Appl. $2019,\ 182,\ 1171-1185.$
\bibitem{cap} M. Caputo, M. Fabrizio, A new definition of fractional
derivative without singular kernel. Prog. Fract. Differ. Appl. $1(2),\
73-85\ (2015)$.
\bibitem{D} k. Deimling, Nonlinear functional analysis, Springer, Berlin $%
1985$.
\bibitem{ben} C. Derbazi , Z. Baitiche , M. Benchohra , A. Cabada, Initial
value problem for nonlinear fractional differential equations with $\mathbf{%
\psi }$-Caputo derivative via monotone iterative technique. Axioms. $2020,\
9\ (2)\ 1-13.$
\bibitem{13} R. Hilfer, Applications of fractional calculus in Physics.
World Scientific Publishing Co, Singapore, $2003.$
\bibitem{gam} Y. Y. Gambo, F. Jarad, D. Baleanu, T. Abdeljawad, On Caputo
modification of the Hadamard fractional derivatives. Adv. Differ. Equ. $%
2014,\ 10\ (2014).$
\bibitem{jar} F. Jarad, T. Abdeljawad, Generalized fractional derivatives
and Laplace transforms. Discrete Contin. Dyn. Syst. Ser. S $13\ (3),709-722\
(2020).$
\bibitem{kat} U. Katugampola, A new approach to a generalized fractional
integral. Appl. Math. Comput. $218,\ 860-865$ $(2011).$
\bibitem{kil} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and
applications of fractional differential equations. Elsevier, Amsterdam $%
(2006).$
\bibitem{19} R. L. Magin, Fractional calculus in Bioengineering. Begell
House Inc. Publisher, $2006.$
\bibitem{set} B. Samet, H. Aydi, Lyapunov-type inequalities for an
anti-periodic fractional boundary value problem involving $\mathbf{\psi }$%
-Caputo fractional derivative. J. Inequal. Appl. 2018, 2018, 286.
\bibitem{sam} S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional
integrals and derivatives, translated from the $1987$ Russian original.
Gordon and Breach, Yverdon, $1993.$
\bibitem {li}S. Solhi, E. Anaama, A. Kajouni, K. Hilal, Existence and
uniqueness of solutions for nonlinear fractional boundary value problems with
$\mathbf{\psi}-$Caputo derivatives, Bol. Soc. Paran. Mat. $2025\ (43)\ 1:1-12.$
\bibitem{sou} J. V. C. Sousa, E. C. Oliveira, On the $\mathbf{\psi }$-Hilfer
fractional derivative. Commun. Nonlinear Sci. Numer. Simul. $60$, $72-91\
(2018).$
Publicado
2026-03-21
Seção
Artigos
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