Existence and uniqueness results of nonlinear hybrid Caputo-Fabrizio fractional differential equations with periodic boundary conditions
Abstract
In this manuscript , we establish the existence and uniqueness of solutions for nonlinear hybrid fractional differential equations involving Caputo-Fabrizio fractional derivatives of order $\varrho\in(0,1)$. The proofs are based on Banach fixed point theorem and some basic concept of Caputo-Fabrizio fractional analysis. As an application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.
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R. P. Agarwal, Y. Zhou, J. Wang, X. Luo, Fractional functional differential equations with causal operators in Banach spaces. Mathematical and Compututer Modelling,1440-1452,(2011).
A. El Mfadel, S. Melliani, Others Existence results for anti-periodic fractional coupled systems with p- Laplacian operator via measure of noncompactness in Banach spaces, Journal Of Mathematical Sciences, 1-14, (2023).
A. El Mfadel, S. Melliani, M. Elomari, Existence of anti-periodic solutions for ψ-Caputo-type fractional p-Laplacian problems via Leray–Schauder degree theory,Analysis, (2023).
R. P. Agarwal, S. K. Ntouyas, B. Ahmad and A. K. Alzahrani,Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments,Advances in Difference Equations ,1-15,(2016) .
F. Mainardi, Fractals and Fractional Calculus Continuum Mechanics Springer Verlag, (1997).
R. P. Agarwal, S. Hristova, D. O’Regan,A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal,290-318 , (2016) .
K. Diethelm,The Analysis of Fractional Differential Equations ,Springer-Verlag, Berlin, (2010).
M. R. Ammi, E. El Kinani, and D. F. Torres, Existence and uniqueness of solutions to functional integro-differential fractional equations, Electronic Journal of Differential Equations, 1-9,(2012).
R. Almeida,Caputo fractional derivative of a function with respect to another function,Commun. Nonlinear Sci. Numer. Simul,460-481, (2017).
R. Almeida, A.B. Malinowska, M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Mathematical Methods in the Applied Sciences, 336-352,(2018).
D. Baleanu, H. Jafari, H. Khan, S. J Johnston. Results for mild solution of fractional coupled hybrid boundary value problems, Open Math,601-608, (2015) .
D. Baleanu, H. Khan, H. Jafari, R. A. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Advances in Difference Equations ,1-14,(2015).
A. Belarbi, M. Benchohra, A. Ouahab. Uniqueness results for fractional functional differential equations with infinite delay in Frechet spaces, Appl. Anal, 1459-1470 ,(2006).
T. A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett,85-88,(1998) .
D. Kumar, J. Singh, D. Baleanu,Numerical computation of a fractional model of differential-difference equation, Journal Of Computational And Nonlinear Dynamics, (2016).
M. Caputo,Linear models of dissipation whose Q is almost frequency independent, International Journal of Geographical InformationScience, 529-539 ,(1967).
J. Losada, J. Nieto,Properties of a new fractional derivative without singular kernel,Progr.Fract. Differ. Appl, 87-92, (2015)
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress In Fractional Differentiation and Applications, 73-85,(2015).
M. Ma, D.Baleanu, Y. Gasimov,X.Yang, New results for multidimensional diffusion equations in fractal dimensional space, Rom. J. Phys, 784-794 ,(2016).
B. C. Dhage,Basic results in the theory of hybrid differential equations with mixed perturbations of second type, Funct, Differ. Equ, 87-106, (2012).
B. C. Dhage, V. Lakshmikantham,Basic results on hybrid differential equations, Nonlinear Anal. Hybrid, 414-424,(2010).
A. El Mfadel, S. Melliani, and M. Elomari,Existence and uniqueness results for Caputo fractional boundary value problems involving the p-Laplacian operator U.P.B. Sci. Bull. Series A, 37-46,(2022).
A. El Mfadel, S. Melliani, M. Elomari,New existence results for nonlinear functional hybrid differential equations involving the ψ− Caputo fractional derivative, Results in Nonlinear Analysis, 78-86,(2022).
A. El Mfadel, S. Melliani, M. Elomari,Existence results for nonlocal Cauchy problem of nonlinear ψ−Caputo type fractional differential equations via topological degree methods, Advances in the Theory of Nonlinear Analysis and its Application, (2022).
A. El Mfadel, S. Melliani, M. Elomari, On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative, Journal of Function Spaces, (2021).
A. El Mfadel, S. Melliani, M. Elomari,A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative, International Journal of Mathematics and Mathematical Sciences,( 2021).
A. El Mfadel, S. Melliani, M. Elomari,Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives, Journal of Mathematics, (2021).
J. W. Green, F. A. Valentine,On the arzela-ascoli theorem, Mathematics Magazine,199-202, (1961) .
N. Derdar,Nonlinear implicit Caputo-Fabrizio fractional hybrid differential equations, International Journal Of Nonlinear Analysis And Applications, (2023).
T. L. Guo, M. Jiang,Impulsive fractional functional differential equations, Comput.Math.Appl,3414-3424, (2012).
A. A. Kilbas, H.M. Srivastava, J.J. Trujillo. Theory and Applications of Fractional Differential Equations, North-Holland Mathematical studies ,204,(2006) .
R. Hilfer, Applications of Fractional Calculus in Physics ,Singapore, (2000).
H. Lu, S. Sun, D. Yang, H. Teng,Theory of fractional hybrid differential equations with linear perturbations of second type, Boundary Value Problems, 1-16,(2013).
Y. Luchko, J.J. Trujillo,Caputo-type modification of the Erdelyi-Kober fractional derivative, Fract. Calc. Appl. Anal, 249-267,( 2007).
I. Podlubny,Fractional Differential Equations ,Mathematics in Science and Engineering, Academic Press, (1999).
H. A. Wahash, S.K. Panchal, M.S. Abdo, Existence and stability of a nonlinear fractional differential equation involving a ψ-Caputo operator, Advances in the Theory of Nonlinear Analysis and its Application,266-278,(2020).
Y. Zhao, S. Sun, Z. Han, Q. Li,Theory of fractional hybrid differential equations, Computers and Mathematics with Applications,1312-1324,(2011) .
G. Nchama,Properties of Caputo-Fabrizio fractional operators, New Trends In Mathematical Sciences, 1-25,(2020).
S. Dragomir,M. City,Some Gronwall type inequalities and applications,(2002).
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