NONLOCAL INITIAL VALUE PROBLEMS FOR HILFER-TYPE FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS WITH IMPULSE

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Published: 2021-10-05

Page: 1031-1039


D. VIVEK

Department of Mathematics, PSG College of Arts & Science, Coimbatore-641014, India.

K. KANAGARAJAN

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts & Science, Coimbatore-641020, India.

E. M. ELSAYED *

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

*Author to whom correspondence should be addressed.


Abstract

In this note, we verify the existence of solutions to nonlocal initial value problems for Hilfer-type fractional hybrid differential equations with impulsive condition. Then, we use prerequisites of Hilfer fractional calculus and the standard fixed point theorem due to Dhage for deriving the existence results in the weighted space of continuous functions. An example is presented to illustrate the theory results.

Keywords: Hilfer fractional derivative, Existence, Fixed point, Fractional hybrid differential equations.


How to Cite

VIVEK, D., KANAGARAJAN, K., & ELSAYED, E. M. (2021). NONLOCAL INITIAL VALUE PROBLEMS FOR HILFER-TYPE FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS WITH IMPULSE. Asian Journal of Advances in Research, 4(1), 1031–1039. Retrieved from https://mbimph.com/index.php/AJOAIR/article/view/2507

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References

Hilfer R. Application of fractional calculus in physics. World Scientific, Singapore; 1999.

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, Elsevier. 2006;204.

Podlubny I. Fractional differential equations. New York, Academic Press; 1999.

Ahmad B, Sotiris K. Ntouyas, An Existence Theorem for Fractional Hybrid Differential Inclusions of Hadamard Type with Dirichlet Boundary Conditions. Abstr. Appl. Anal; 2014. Article ID 705809, 1-7.
DOI: 10.1155/2014/705809.

Ahmad B, Ntouyas SK. Initial value problems for hybrid Hadamard fractional differential equations. Electron. J. Differential Equations. 2014;161:1-8.

Sun S, Zhao Y, Han Z, Li Y. The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012;17:4961-4967.

Zhao Y, Sun S, Han Z, Li Q. Theory of fractional hybrid differential equations. Comput. Math. Appl. 2011;62(3):1312-1324.

Hilfer R, Luchko Y, Tomovski Z. Operational method for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative. Fract. Calc. Appl. Anal. 2009;12:229-318.

Kamocki, Rafal and Obczynski, Cezary. On fractional Cauchy-type problems containing Hilfer’s derivative. Electron. J. Qual. Theory Differ. Equ. 2016;50:1-12.

Chanakarn Kiataramkul, Sotiris K. Ntouyas, Jessada Tariboon, Existence Results for ψ-Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and
Inclusions. Advances in Mathematical Physics; 2021. Article ID 9044313, 12pages.

Furati KM, Kassim MD, Tatar NE. Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 2012;64(6):1616-1626.

Furati KM, Kassim MD, Tatar NE. Non-existence of global solutions for a differential equation involving Hilfer fractional derivative. Electron. J. Differential Equations. 2013;235:1-10.

Gu H, Trujillo JJ. Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 2015;257:344-354.

Sousa JV, Kucche KD, Capelas de Oliveira E. On the Ulam-Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator. Mathematical Methods in the Applied Sciences. 2019;42(9):3021-3032.

Wang JR, Zhang Y. Nonlocal initial value problems for differential equations with Hilfer fractional derivative. Appl. Math. Comput. 2015;266:850-859.

Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, Engl. Trans. from the Russian; 1987.

Vivek D, Kanagarajan K, Sivasundaram S. Dynamics and stability of pantograph equations via Hilfer fractional derivative. Nonlinear Stud. 2016;23(4):685-698.

Bainov DD, Simeonov PS. Systems with impulsive effect. Horwood, Chichester; 1989.

Benchohra M, Henderson J, Ntouyas SK. Impulsive differential equations and inclusions. Hindawi Publishing Corporation, New York. 2006;2.

Lakshmikantham V, Bainov DD, Simeonov PS. Theory of impulsive differential equations. World Scientific, Singapore; 1989.

Bajo I, Liz E. Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 1996;204:65-73.

Abbas S, Benchohra M, Sivasundaram S. Dynamics and Ulam stability for Hilfer type fractional differential equations. Nonlinear Stud. 2016;4:627-637.

Vivek D, Baghani O, Kanagarajan K. Existence results for hybrid fractional differential equations with Hilfer fractional derivative, (to appear-2017).

Wang JR, Zhang Y. On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 2015;39:85-90.

Dhage BC. On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 2005;18(3):273-280.

Dhage BC, Lakshmikantham V. Basic results on hybrid differential equations. Nonlinear Anal. Hybrid Syst. 2010;4(3):414-424.