On a generalized fractional differential Cauchy problem

Mesfin  Etefa; Gaston M.N. Guerekata; Pierre  Ngnepieba; Olaniyi  S. Iyiola

doi:10.26637/mjm1101/006

Abstract

Qualitative results for abstract problems are very important in understanding mathematical analysis on which any application is possible. The focus of this paper is twofold: first, we investigate the existence and uniqueness of mild solutions to a generalized Cauchy problem for the nonlinear differential equation with non-local conditions in a Banach space $X$. This is achievable using some fixed point theorems in infinite dimensional spaces. Secondly, we study the stability results of the system in the sense of Ulam-Hyers-Rassias. Our results improve and generalize most recent related results in the literature.

Keywords:

$\kappa$-Hilfer operator, Cauchy problem, mild solution, existence theory, Krasnoselski theorem, Ulam-Hyers-Rassias stability

Mathematics Subject Classification:

26A33, 34A12, 34G20

Authors Imprint References Funding Related Articles Downloads

Mesfin Etefa Department of Mathematics, NEERLab Laboratory, Bowie State University, 14000 Jericho Park Rd., Bowie, MD 20715, USA.
Gaston M.N. Guerekata University Distinguished Professor, Department of Mathematics, NEERLab Laboratory, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA.
Pierre Ngnepieba College of Science and Technology, Florida A\& M University, Tallahassee, FL USA.
Olaniyi S. Iyiola Department of Mathematics, Clarkson University, Potsdam, NY, USA.

Pages: 80-93
Date Published: 01-01-2023
Vol. 11 No. 01 (2023): Malaya Journal of Matematik (MJM)

S. Abbas, M. Benchohra, G.M. N'GuÉrékata, Topics in Fractional Differential Equations, Developments in Mathematics, 27. Springer, New York, 2012.

S. Abbas, M. Benchohra, G.M. N'GuÉrÉKATA, Advanced Fractional Differential and Integral Equations, Mathematics Research Developments, Nova Science Publishers, Inc, New Yorj, 2015.

S. Abbas, M. Benchohra, G.M. N'Guéré́ata, Instantaneous and noninstantaneous impulsive integrodifferential equations in Banach spaces, Abstr. Appl. Anal. 2020, 1-8.

L. BysZEWSKI, Theorems about the existence and uniqueness of solutions of a semilinear nonlocal Cauchy problem, J. Math. Anal. Appl., 162(1991), 494-505.

L. D. Baleanu, G.C. Wu, S.D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solitons Fractals, 102(2017), 99-105.

L. D. Baleanu, Z.B. Guvenc, J.T Machado, New Trends in Nanotechnology and Fractional Calculus Applications Springer: New York, NY, USA, (2010).

K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179(1993), 630-637.

O.S. IYiola, F.D. Zaman, A fractional diffusion equation model for cancer tumor, Am. Inst. Phys. Adv., 2014(4), 107121.

O.S. IyIOLA, B. Oduro, T. Zabilowicz, B. IyIOLA, D. KEnEs, System of time fractional models for COVID19: modeling, analysis and solutions, Symmetry, 2021(13), 787.

D. Kumar, A.R. Seadawy, A.K. Joardar, System of Time Fractional Models for COVID-19: Modeling, Analysis and Solutions, Chin. J. Phys., 2018(56), 75-85

J. Vanterler da C. Sousa, E. Capelas Oliveira, On the $kappa$-Hilfer fractional derivatives, Common Nonlinear Sci Numer Simul., 60(2018), 72-91.

G. M. N'GuÉrÉKATA, A Cauchy problem for some fractional abstract differential equation with non local condition, Nonlinear Analysis, 70(2019), 1873-1879.

Vanterler da C. Sousa, E. Capelas Oliveira, On the Ulam-Hyers-Rassias stability for non-linear fractional differential equations using the $kappa$-Hilfer operator, J. Fixed Point Theory Appl., 20(3)(2018), 96113.

J. VAnterler da C. Sousa, E. Capelas OliveIra, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81(2018), 50-56.

J. Vanterler da C. Sousa, E. Capelas Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $kappa-$ Hilfer operator, Diff. Equ. & Appl., 11(1)(2019), 87-106.

M.S. ABdo, S.K. PANCHAL, Fractional integrodifferential equations involving $kappa$-Hilfer fractional derivative, Adv. Appl. Math. Mech., 11(2019), 338-359.

R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech., 51(2)(1984), 299-307.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, New Jersey, 2000.

R. Hilfer, L. Anton, Fractional master equations and fractal time random walks, Phys. Rev., (1995), 51, R848-R851.

K.B. Lima, J. Vanterler da C. Souza, E. Capelas de Oliveira, Ulam-Hyers type stability for $kappa$-Hilfer fractional differential equations with impulses and delay, Computational and Applied Maths, (2021) 40:293.

F. MaInARdi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press: London, UK, 2010.

H. Nasrolahpour, A note on fractional electrodynamics, Commun. Nonlinear. Sci. Numer. Simul., 18(2013), 2589-2593.

F. Norouzi, G.M. N'GuÉrékata, A study of $kappa$-Hilfer differential system with applications in financial crisis, Chaos, Solitons and Fractals:X, 6(2021), 100056.

F. Norouzi, G.M. N'GuÉrÉKATA, Existence results to a $kappa$-Hilfer neutral fractional evolution equations with infinite delay, Nonauton. Dyn. Syst., 8(1)(2021), 101-124.

C.J. Sousa, Capelas, E. De Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integrodifferential equation, Appl. Math. Lett., 81(2018), 50-56.