Approximation results for local solution of the	initial value problems of nonlinear first order ordinary hybrid integrodifferential equations

Bapurao Dhage; Janhavi Dhage; Shyam Dhage

doi:10.26637/mjm1301/009

Abstract

In this paper, we establish a couple of approximation results for local existence and uniqueness of the solution of a initial value problem of nonlinear first order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2022) and Dhage {\em et al.} (2022). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with the help of a couple of numerical examples.

Keywords:

Integrodifferential equation, Hybrid fixed point principle, Dhage iteration method, Approximation theorem

Mathematics Subject Classification:

34A12, 34A34, 34A45

Authors Imprint References Funding Related Articles Downloads

Bapurao Dhage ``Kasubai, Gurukul Colony, Thodga Road,, Ahmedpur-413 515, Dist. Latur, Maharashtra, India https://orcid.org/0000-0002-2353-8618
Janhavi B. Dhage Kasubai, Gurukul Colony, Thodga Road, Ahmedpur - 413515, Dist. Latur, Maharashtra, India. https://orcid.org/0009-0001-7394-1229
Shyam Dhage Kasubai, Gurukul Colony, Thodga Road, Ahmedpur - 413515, Dist. Latur, Maharashtra, India.

Pages: 75-87
Date Published: 01-01-2025
Vol. 13 No. 01 (2025): Malaya Journal of Matematik (MJM)

G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll. Publ. New York 1967.

E.A. Coddington, An Introduction to Ordinary Differential Equations, Dover Publications Inc. New York, 1989.

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), 397-427. DOI: https://doi.org/10.5556/j.tkjm.45.2014.1512

B.C. Dhage, Two general fixed point principles and applications, J. Nonlinear Anal. Appl., 2016, (1) (2016), 23-27. DOI: https://doi.org/10.5899/2016/jnaa-00313

B.C. Dhage, A coupled hybrid fixed point theorem for sum of two mixed monotone coupled operators in a partially ordered Banach space with applications, Tamkang J. Math., 50(1) (2019), 1-36. DOI: https://doi.org/10.5556/j.tkjm.50.2019.2502

B.C. Dhage, Coupled and mixed coupled hybrid fixed point principles in a partially ordered Banach algebra and PBVPs of nonlinear coupled quadratic differential equations, Differ. Equ. Appl., 11(1) (2019), 1-85. DOI: https://doi.org/10.7153/dea-2019-11-01

B.C. Dhage, A Schauder type hybrid fixed point theorem in a partially ordered metric space with applications to nonlinear functional integral equations, J˜n¯an¯abha 52 (2) (2022), 168-181. DOI: https://doi.org/10.58250/jnanabha.2022.52220

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, 2 (2) (2013), 25-35.

B.C. Dhage, J.B. Dhage, S.B. Dhage, Approximating existence and uniqueness of solution to a nonlinear IVP of first order ordinary iterative differential equations, Nonlinear Studies, 29 (1) (2022), 303-314. DOI: https://doi.org/10.21203/rs.3.rs-2381198/v1

J.B. Dhage, B.C. Dhage, Approximating local solution of an IVP of nonlinear first order ordinary hybrid differential equations, Nonlinear Studies 30 (3) (2023), 721-732. DOI: https://doi.org/10.21203/rs.3.rs-2381198/v2

J.B. Dhage, B.C. Dhage, Approximating local solution of IVPs of nonlinear first order ordinary hybrid intgrodifferential equations, Malaya J. Mat., 11 (04) (2023), 344-355. DOI: https://doi.org/10.26637/mjm1104/002

S.B. Dhage, B.C. Dhage, J.B. Graef, Dhage iteration method for initial value problem for nonlinear first order integrodifferential equations, J. Fixed Point Theory Appl. 18 (2016), 309-325. DOI: https://doi.org/10.1007/s11784-015-0279-3

A. Granas, J.Dugundji, Fixed Point Theory, Springer 2003. DOI: https://doi.org/10.1007/978-0-387-21593-8

D. Gua, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, London 1988.

J. Huang, S. Jung, Y. Li, On Hyers-Ulam stability of nonlinear differential equations, Bull. Korean Math. Soc. 52 (2) (2015), 685-697. DOI: https://doi.org/10.4134/BKMS.2015.52.2.685

V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Academic Press, New York, 1969.

A.K. Tripathy, Hyers-Ulam stability of ordinary differential equations, Chapman and Hall / CRC, London, NY, 2021. 12 DOI: https://doi.org/10.1201/9781003120179