Existence results for nonlocal Cauchy problem of nonlinear $\psi-$Caputo type fractional differential equations via topological degree methods

Year 2022, , 270 - 279, 30.06.2022

Alı El Mfadel , Said Melliani , Elomari M'hamed

https://doi.org/10.31197/atnaa.1059793

Cited By: 19

Abstract

This manuscript is devoted to the investigation of the existence results of fractional Cauchy problem for
some nonlinear ψ−Caputo fractional differential equations with non local conditions. By applying fixed
point theorems, some results of topological degree theory for condensing maps and some fractional analysis
techniques, we establish some new existence theorems. As application, a nontrivial example is given to
illustrate our theoretical results.

Keywords

$\psi-$fractional integral, $\psi-$Caputo fractional derivative, fixed point, Kuratowski measure, topological degree theory

References

• [1] R.P. Agarwal, Y. Zhou, J. Wang and X. Luo. Fractional functional differential equations with causal operators in Banach spaces, Mathematical and Compututer Modelling 54 (2011), 1440-1452.
• [2] 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 (2016), 1-15.
• [3] R.P. Agarwal, S. Hristova and D. O'Regan. A survey of Lyapunov functions, stability and impulsive Caputo fractional di?erential equations, Fract. Calc. Appl. Anal 19 (2016), 290-318.
• [4] R. Almeida. Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), 460-481.
• [5] R. Almeida, A.B. Malinowska, and 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 41(1) (2018), 336-352.
• [6] D. Baleanu, H. Jafari, H. Khan and S.J Johnston. Results for mild solution of fractional coupled hybrid boundary value problems, Open Math. 13 (2015), 601-608.
• [7] D. Baleanu, H. Khan, H. Jafari and R.A. Alipour. On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Advances in Di?erence Equations 1(2015), 1-14.
• [8] A. Belarbi, M. Benchohra, A. Ouahab. Uniqueness results for fractional functional di?erential equations with infinite delay in Frechet spaces, Appl. Anal. 58 (2006), 1459-1470.
• [9] M. Caputo. Linear models of dissipation whose Q is almost frequency independent, International Journal of Geographical Information Science 13(5)(1967), 529-539.
• [10] B.C. Dhage. On α−condensing mappings in Banach algebras. Math. Student 63(1994)146-152.
• [11] B.C. Dhage. On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 18(2005) 273-280.
• [12] Z. Smith, Fixed point methods in nonlinear analysis (2014).
• [13] B.C. Dhage and V. Lakshmikantham. Basic results on hybrid differential equations. Nonlinear Anal. Hybrid 4(2010) 414-424.
• [14] K. Diethelm. The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin, 2010.
• [15] K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, New York, (1985).
• [16] J. W. Green, and F. A. Valentine, On the arzela-ascoli theorem, Mathematics Magazine, 34(1961)199-202.
• [17] A. El Mfadel, S. Melliani and 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).
• [18] A. El Mfadel, S. Melliani and 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).
• [19] A. El Mfadel, S. Melliani and M. Elomari. Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives. Journal of Mathematics, (2021).
• [20] F. Isaia. On a nonlinear integral equation without compactness, Acta. Math. Univ.Comenianae, 75 (2006), 233-240.
• [21] T.L. Guo and M. Jiang. Impulsive fractional functional differential equations, Comput. Math. Appl., 64 (2012), 3414-3424.
• [22] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical studies 204. Ed van Mill. Amsterdam. Elsevier Science B.V. Amsterdam, (2006).
• [23] R. Hilfer, Applications of Fractional Calculus in Physics Singapore, (2000).
• [24] Y. Luchko and J.J. Trujillo. Caputo-type modification of the Erdelyi-Kober fractional derivative, Fract. Calc. Appl. Anal. 10( 2007), 249-267.
• [25] F. Mainardi, Fractals and Fractional Calculus Continuum Mechanics, Springer Verlag.(1997).
• [26] S K. Ntouyas. boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions, Opuscula Math. 331 (2013), 117-138.
• [27] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press. New York. 1999.
• [28] J. Schauder, Der Fixpunktsatz in Functionalraiumen, Studia Math. 2(1930), 171-180.
• [29] G.M. N'Guérékata, A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods and Applications 70 (5) (2009), 1873-1876.
• [30] J. Zhao, P. Wang and W. Ge. Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces, Nonlinear Sci. Numer. Simul. 16(2011), 402-413.
• [31] S. Zhang. Existence of solutions for a boundary value problem of fractional order, Acta Math. Scientia 6 (2006), 220-228.
• [32] W. Zhon and W. Lin: Nonlocal and multiple-point boundary value problem for fractional differential equations, Mathematics with Applications 593 (2010), 1345-1351.