Some results on the study of -Hilfer type fuzzy fractional differential equations with time delay

Year 2022, Volume: 4 Issue: 2, 65 - 76, 31.12.2022

R. Vivek D. Vivek Vivek , Kangarajan K. , Elsayed Elsayed

https://doi.org/10.47086/pims.1168552

Abstract

This paper is concerned with the finite-time stability of -Hilfer type fuzzy fractional differential equations (FFDEs) with time delay. By applying standard theorems and a hypothetical condition, we explore the existence of solution and stabilty results.

Keywords

Fuzzy differential equations, Finite-time stability, Existence, Fixed point, Hilfer fractional derivative

Supporting Institution

-

Project Number

-

Thanks

-

References

• O.P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15(2012), 700-711.
• B. Ahmad, J.J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory, 13(2013), 329-336.
• K. Balachandran, S. Kiruthika, J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Mathematica Scientia., 3(33)(2013), 712-720.
• X.K. Cao, J.R. Wang, Finite-time stability of a class of oscillating systems with two delays, Math. Methods Appl. Sci., 41(13)(2018), 4943-4954.
• F.F. Du, J.G. Lu, Finite-time stability of fractional-order fuzzy cellular neural networks with time-delays, Fuzzy Set. Syst., 438(2022), 107-120.
• O.S. Fard, M. Salehi, A survey on fuzzy fractional variational problems, J. Comput. Appl. Math., 271(2014), 71-82.
• [7] K.M. Furati, N.D. Kassim, N.E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(2012), 1616-1626.
• A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• R. Hilfer, Applications of Fractional Calculus in Physics, Singapore: World Scientific, 2000.
• N.V. Hoa, H. Vu, T.M. Duc, Fuzzy fractional differential equations under Caputo- Katugampola fractional derivative approach, Fuzzy Set. Syst., 375(2019), 70-99.
• Y. Jiang, J. Qiu, F. Meng, Existence and finite-time stability results of fuzzy Hilfer-Katugampola fractional delay differential equations 1, J. Intell. Fuzzy Syst., 2022, 1-10.
• K. Kanagarajan, R. Vivek, D. Vivek, E.M. Elsayed, Existence Results for Fuzzy Differential Equations with Hilfer Fractional Derivative, Ann. Commun. Math., 5(1)(2022), 38-54.
• U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(4)(2014), 1-15.
• A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam, 207, 2006.
• K. Kotsamran, W. Sudsutad, C. Thaiprayoon, J. Kongson, Analysis of a nonlinear Hilfer fractional integrodifferential equation describing cantilever beam model with nonlinear boundary conditions, Fractal. Fract., 5(177), 2021.
• M.M. Li, J.R. Wang, Finite-time stability and relative controllability of Riemann-Liouville fractional delay differential equations, Math. Methods Appl. Sci., 42(18)(2019), 6607-6623.
• Q. Li, D.F. Luo, Z.G. Luo, Q.X. Zhu, On the novel finite-time stability results for uncertain fractional delay differential equations involving noninstantaneous impulses, Math. Probl. Eng., 9097135(2019), 2019.
• H.V. Long, N.T.K. Son, N.V. Hoa, Fuzzy fractional partial differential equations in partially ordered metric spaces, Iran. J. Fuzzy Syst., 14(2017), 107-126.
• H.V. Long, N.P. Dong, An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncer tainty, Fixed Point Theory Appl., 20(2018), 37.
• D.F. Luo, Z.G. Luo, Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses, Math. Slovaca., 70(5)(2020), 1231-48.
• S.K. Ntouyas, A survey on existence results for boundary value problems of Hilfer fractional differential equations and inclusions, Foundations., 1(2021), 63-98.
• E.C. Oliveira, J.A.T. Machado, A Riew of Definitions for Fractional Derivatives and Integral, Math. Probl. Eng., 2014(238)(459), 2014.
• D.S. Oliveira, E.C. Oliveira, Hilfer-Katugampola fractional derivative, Comput. Appl. Math., 37(2018), 3672-3690.
• I. Podlubny, Fractional differential equations, Acadamic Press, San Diego, 1999.
• C. Promsakon, S.K. Ntouyas, J. Tariboon, Hilfer-Hadamard nonlocal integro multipoint fractional boundary value problems, J. Funct. Spaces., 2021(9)(8031524), 2021.
• S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications, New York: Gordon and Breach, 1993.
• K. Shah, A. Ali, S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Math. Methods Appl. ci., 41(17)(2018), 8329-8343.
• L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. Theor., 71(2009), 1311-1328.
• J.V.C. Sousa, E.C. Oliveira, On the -Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60(2018), 72-91.
• J.V.C. Sousa, E.C. Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the -Hilfer operator, J. Fixed Point Theory Appl., 20(2018), 1-21.
• D. Vivek, K. Kanagarajan, E.M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15(2018), 1-15.
• D. Vivek, K. Kanagarajan, S. Sivasundaram, Dynamics and stability of pantograph equations via hilfer fractional derivative, Nonlinear Studies., 23(4)(2016), 685-698.
• R. Vivek, E.M. Elsayed, K. Kanagarajan, D. Vivek, Qualitative Analysis of Quaternion Fuzzy Fractional Differential Equations with -Hilfer Fractional Derivative, Pure. Appl. Anal., 6(2022), 2022.
• H. Vu, N.V. Hoa, Hyers-ulam stability of fuzzy fractional volterra integral equations with the kernal functions via successive approximation method, Fuzzy set syst., 419(2021), 67-98.
• J.Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 2015(266), 850-859.