International Journal of Nonlinear Analysis and Applications
Dhage iteration principle for IVPs of nonlinear first order impulsive differential equations
Document Type : Research Paper
Authors
Kasubai, Gurukul Colony, Thodga Road, Ahmedpur-413 515, Dist: Latur, Maharashtra, India
Abstract
In this paper we prove the existence and approximation theorems for the initial value problems of first order nonlinear impulsive differential equations under certain mixed partial Lipschitz and partial compactness type conditions. Our results are based on the Dhage monotone iteration principle embodied in a hybrid fixed point theorem of Dhage involving the sum of two monotone order preserving operators in a partially ordered Banach space. The novelty of the present approach lies the fact that we obtain an algorithm for the solution. Our abstract main result is also illustrated by indicating a numerical example.
Keywords
[1] D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, 1993.
[2] B.C. Dhage, Local fixed point theory for sum of two operators in Banach spaces, Fixed Point Theory, 4(1) (2003) 49–60.
[3] B.C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Diff. Eq. Appl. 5 (2013) 155–184.
[4] B.C. Dhage, Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45 (2014) 397–426.
[5] B.C. Dhage, Nonlinear D-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malaya J. Mat. 3 (2015) 62–86.
[6] B.C. Dhage, Some generalizations of a hybrid fixed point theorem in a partially ordered metric space and nonlinear functional integral equations, Differ. Equ. Appl. 8 (2016) 77–97.
[7] B.C. Dhage, Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations, Comm. Opt. Theory 2017 (2017), Article ID 32, pp. 22.
[8] B.C. Dhage, Dhage iteration method for approximating solutions of a IVP of nonlinear first order functional differential equations, Malaya J. Mat. 5(4) (1917), 680-689.
[9] B.C. Dhage, The Dhage iteration method for initial value problems of first order hybrid functional integrodifferential equations, Recent Advances in Fixed point Theory and Applications, Nova Science Publishers Inc., 2017
[10] B.C. Dhage, A coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space with applications, Differ. Equ. Appl. 9(4) (2017), 453-477.
[11] B.C. Dhage, Some variants of two basic hybrid fixed point theorem of Krasnoselskii and Dhage with applications, Nonlinear Studies 25 (3) (2018), 559-573.
[12] 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.
[13] 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, Diff. Eq. Appl. 11(1) (2019) 1–85.
[14] B.C. Dhage, Dhage monotone iteration principle for IVPs of nonlinear first order impulsive differential equations, J˜n¯an¯abha, 50(1) (2020) 265–278. +.
[15] B.C. Dhage, Dhage monotone iteration method for for PBVPs of nonlinear first order impulsive differential equations, Electronic J. Math. Anal. Appl. 9(1) (2021) 37–51.
[16] 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) (2014) 25–35.
[17] S.B. Dhage, B.C. Dhage, Dhage iteration method for Approximating positive solutions of PBVPs of nonlinear hybrid differential equations with maxima, Int. Jour. Anal. Appl. 10(2) (2016) 101–111.
[18] S. B. Dhage, B. C. Dhage and 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.
[19] S.B. Dhage, B.C. Dhage and J.B. Graef, Dhage iteration method for approximating the positive solutions of IVPs for nonlinear first order quadratic neutral functional differential equations with delay and maxima, Intern. J. Appl. Math. 31(6) (2018) 1–21.
[20] B. C. Dhage and D. Otrocol, Dhage iteration method for approximating solutions of nonlinear differential equations with maxima, Fixed Point Theo. 19(2) (2018) 545–556.
[21] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Berlin, 1977.
[2] B.C. Dhage, Local fixed point theory for sum of two operators in Banach spaces, Fixed Point Theory, 4(1) (2003) 49–60.
[3] B.C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Diff. Eq. Appl. 5 (2013) 155–184.
[4] B.C. Dhage, Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45 (2014) 397–426.
[5] B.C. Dhage, Nonlinear D-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malaya J. Mat. 3 (2015) 62–86.
[6] B.C. Dhage, Some generalizations of a hybrid fixed point theorem in a partially ordered metric space and nonlinear functional integral equations, Differ. Equ. Appl. 8 (2016) 77–97.
[7] B.C. Dhage, Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations, Comm. Opt. Theory 2017 (2017), Article ID 32, pp. 22.
[8] B.C. Dhage, Dhage iteration method for approximating solutions of a IVP of nonlinear first order functional differential equations, Malaya J. Mat. 5(4) (1917), 680-689.
[9] B.C. Dhage, The Dhage iteration method for initial value problems of first order hybrid functional integrodifferential equations, Recent Advances in Fixed point Theory and Applications, Nova Science Publishers Inc., 2017
[10] B.C. Dhage, A coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space with applications, Differ. Equ. Appl. 9(4) (2017), 453-477.
[11] B.C. Dhage, Some variants of two basic hybrid fixed point theorem of Krasnoselskii and Dhage with applications, Nonlinear Studies 25 (3) (2018), 559-573.
[12] 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.
[13] 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, Diff. Eq. Appl. 11(1) (2019) 1–85.
[14] B.C. Dhage, Dhage monotone iteration principle for IVPs of nonlinear first order impulsive differential equations, J˜n¯an¯abha, 50(1) (2020) 265–278. +.
[15] B.C. Dhage, Dhage monotone iteration method for for PBVPs of nonlinear first order impulsive differential equations, Electronic J. Math. Anal. Appl. 9(1) (2021) 37–51.
[16] 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) (2014) 25–35.
[17] S.B. Dhage, B.C. Dhage, Dhage iteration method for Approximating positive solutions of PBVPs of nonlinear hybrid differential equations with maxima, Int. Jour. Anal. Appl. 10(2) (2016) 101–111.
[18] S. B. Dhage, B. C. Dhage and 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.
[19] S.B. Dhage, B.C. Dhage and J.B. Graef, Dhage iteration method for approximating the positive solutions of IVPs for nonlinear first order quadratic neutral functional differential equations with delay and maxima, Intern. J. Appl. Math. 31(6) (2018) 1–21.
[20] B. C. Dhage and D. Otrocol, Dhage iteration method for approximating solutions of nonlinear differential equations with maxima, Fixed Point Theo. 19(2) (2018) 545–556.
[21] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Berlin, 1977.
[22] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, London 1988.
[23] S. Heikkil¨a, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York 1994.
[24] V. Lakshmikantham, D.D. Bainov, and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Pub. Co., Singapore, 1989.
[25] A.N. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
[23] S. Heikkil¨a, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York 1994.
[24] V. Lakshmikantham, D.D. Bainov, and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Pub. Co., Singapore, 1989.
[25] A.N. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
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Volume 12, Issue 2
November 2021Pages 219-235
- Receive Date: 10 February 2019
- Revise Date: 08 November 2019
- Accept Date: 08 November 2019