Existence and approximate solutions for hybrid fractional integro-differential equations

Bapurao C. Dhage; Shyam B. Dhage; Sotiris K. Ntouyas

doi:10.26637/mjm402/002

Abstract

In this paper we prove existence and approximations of the solutions for initial value problems of non-linear hybrid fractional differential equations, using the operator theoretic technique in a partially ordered metric space proved by Dhage.

Keywords:

Fractional differential equation, Fixed point theorem, Dhage iteration method, Existence and uniqueness theorems

Mathematics Subject Classification:

534A08, 47H07, 47H10

Authors Imprint References Funding Related Articles Downloads

Bapurao C. Dhage Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur, Maharashtra, India.
Shyam B. Dhage Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur, Maharashtra, India.
Sotiris K. Ntouyas Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece.

Pages: 195-204
Date Published: 01-04-2016
Vol. 4 No. 02 (2016): Malaya Journal of Matematik (MJM)

B. Ahmad, A. Alsaedi, B. Alghamdi, Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real World Appl. 9 (2008), 1727-1740. DOI: https://doi.org/10.1016/j.nonrwa.2007.05.005

B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. Appl. Anal., 2009 (2009), Article ID 494720, 9 pages. DOI: https://doi.org/10.1155/2009/494720

B. Ahmad, S. K. Ntouyas, Existence results for nonlinear fractional differential equations with four-point nonlocal type integral boundary conditions, Afr. Diaspora J. Math. 11 (2011), 29-39. DOI: https://doi.org/10.1155/2011/107384

B. Ahmad, S.K. Ntouyas, Some existence results for boundary value problems for fractional differential inclusions with non-separated boundary conditions, Electron. J. Qual. Theory Differ. Equ. 2010, No. 71, $1-17$. DOI: https://doi.org/10.14232/ejqtde.2010.1.71

B. Ahmad, S. K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ., Volume 2011, Article ID 107384, 11 pages. DOI: https://doi.org/10.1155/2011/107384

B. Ahmad, S. Sivasundaram, On four-point nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order, Appl. Math. Comput. 217 (2010), 480-487. DOI: https://doi.org/10.1016/j.amc.2010.05.080

M. Ammi, E. El Kinani, D. Torres, Existence and uniqueness of solutions to functional integro-differential fractional equations, Electron. J. Diff. Equ., Vol. 2012 (2012), No. 103, pp. 1-9.

Z.B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72 (2010), 916-924. DOI: https://doi.org/10.1016/j.na.2009.07.033

K. Balachandran, J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal. 72 (2010) 4587-4593. DOI: https://doi.org/10.1016/j.na.2010.02.035

M. El Borai, M. Abbas, On some integro-differential equations of fractional orders involving Carathéodory nonlinearities, Int. J. Mod. Math. 2 (2007), 41-52.

B.C. Dhage, Basic results in the theory of hybrid differential equations with mixed perturbations of second type, Funct. Diff. Equ. 19 (2012), 1-20.

B.C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl. 2 (2010), 465-486. DOI: https://doi.org/10.7153/dea-02-28

B.C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differ. Equ. Appl. 5 (2013), 155-184. DOI: https://doi.org/10.7153/dea-05-11

B.C. Dhage, Partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45(4) (2014), 397-426. DOI: https://doi.org/10.5556/j.tkjm.45.2014.1512

B.C. Dhage, Nonlinear $mathcal{D}$-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malaya J. Mat. 3(1)(2015), 62-85. DOI: https://doi.org/10.26637/mjm301/007

B.C. Dhage, Operator theoretic techniques in the theory of nonlinear hybrid differential equations, Nonlinear Anal. Forum 20 (2015), 15-31.

B.C. Dhage, A new monotone iteration principle in the theory of nonlinear fractional differential equations, Intern. J. Anal. Appl. 8(2) (2015), 130-143.

B.C. Dhage, S.B. Dhage, Approximating solutions of nonlinear first order ordinary differential equations, GJMS Special Issue for Recent Advances in Mathematical Sciences and Applications-13, GJMS Vol. 2, No. 2, (2014), 25-35. DOI: https://doi.org/10.1080/23311835.2015.1023671

B.C. Dhage, S.B. Dhage, S.K. Ntouyas, Approximating solutions of nonlinear hybrid differential equations, Appl. Math. Lett. 34 (2014), 76-80. DOI: https://doi.org/10.1016/j.aml.2014.04.002

B.C. Dhage, S.K. Ntouyas, Existence results for boundary value problems for fractional hybrid differential inclucions, Topol. Methods Nonlinear Anal. 44 (2014), 229-238. DOI: https://doi.org/10.12775/TMNA.2014.044

S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York 1994.

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

S. Sun, Y. Zhao, Z. Han, Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4961-4967. DOI: https://doi.org/10.1016/j.cnsns.2012.06.001

Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011), 1312-1324 DOI: https://doi.org/10.1016/j.camwa.2011.03.041