Archive
Special Issues

Volume 9, Issue 3, June 2020, Page: 64-69
Approximating Solutions of Non Linear First Order Abstract Measure Differential Equations by Using Dhage Iteration Method
Dnyanoba Maroti Suryawanshi, Dayanand Science College Latur, Swami Ramanand Teerth Marathwada University, Nanded, India
Sidheshwar Sangram Bellale, Dayanand Science College Latur, Swami Ramanand Teerth Marathwada University, Nanded, India
Pratiksha Prakash Lenekar, Dayanand Science College Latur, Swami Ramanand Teerth Marathwada University, Nanded, India
Received: Apr. 15, 2020;       Accepted: Apr. 30, 2020;       Published: Jun. 4, 2020
Abstract
In this paper we have proved the approximating solutions of the nonlinear first order abstract measure differential equation by using Dhage’s iteration method. The main result is based on the iteration method included in the hybrid fixed point theorem in a partially ordered normed linear space. Also we have solved an example for the applicability of given results in the paper. Sharma [2] initiated the study of nonlinear abstract differential equations and some basic results concerning the existence of solutions for such equations. Later, such equations were studied by various authors for different aspects of the solutions under continuous and discontinuous nonlinearities. The study of fixed point theorem for contraction mappings in partial ordered metric space is initiated by different authors. The study of hybrid fixed point theorem in partially ordered metric space is initiated by Dhage with applications to nonlinear differential and integral equations. The iteration method is also embodied in hybrid fixed point theorem in partially ordered spaces by Dhage [12]. The Dhage iteration method is a powerful tool for proving the existence and approximating results for nonlinear measure differential equations. The approximation of the solutions are obtained under weaker mixed partial continuity and partial Lipschitz conditions. In this paper we adopted this iteration method technique for abstract measure differential equations.
Keywords
Abstract Measure Differential Equation, Dhage Iteration Method, Existence Theorem, Extremal Solutions, Approximation of Solution, Hybrid Fixed Point Theorem
Dnyanoba Maroti Suryawanshi, Sidheshwar Sangram Bellale, Pratiksha Prakash Lenekar, Approximating Solutions of Non Linear First Order Abstract Measure Differential Equations by Using Dhage Iteration Method, Applied and Computational Mathematics. Vol. 9, No. 3, 2020, pp. 64-69. doi: 10.11648/j.acm.20200903.13
Reference
[1]
M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964.
[2]
R. R. Sharma, An Abstract Measure Differential Equations, proc. Amer. Math. Soc. 32 (1972) 503-510.
[3]
S. R. Joshi, A system of abstract measure delay differential equations J. Math. Phys. Sci. 13 (1979) 497-506.
[4]
G. R. Shendge, S. R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hun 41 (1983) 53-54.
[5]
B. C. Dhage, On abstract measure integro-differential equations. J. Math. Phys. Sci. 20 (1986) 367–380.
[6]
B. C. Dhage On system of abstract measure integro-differential inequalities and applications, Bull Inst. Math. Acad. Sin. 18 (1989), 53-54.
[7]
S. Heikkiläand, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker Inc., New York 1994.
[8]
G. A. Granas. J. Dugundji, Fixed Point Theory, Springer Verlag. 2003.
[9]
B. C. Dhage, S. S. Bellale, Abstract measure integro-differential equations, Global J. Math. Anal. 1 (1-2) (2007) 91-108
[10]
B. C. Dhage, and S. S. Bellale, Existence theorem for perturbed Abstract measure differential equations. Nonlinear Analysis 71 (2009) 319-328.
[11]
D. D. Bainov, S. Hristova, Differential Equations with Maxima, Chapman & Hall/CRC Pure andApplied Mathematics, 2011.
[12]
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.
[13]
S. S. Bellale, Hybrid Fixed point theorem for abstract measure differential equation, World Academy Of Science, Engineering and Technology, 73 (2013) 782-785.
[14]
B. C. Dhage, Approximating Solutions of nonlinear first order ordinary differential equations, Global Journal of mathematical sciences 3 (2014), 1-9.
[15]
B. C. Dhage, A new monotone iteration principle in the theory of nonlinear first order integro differential equations, Nonlinear Studies, 22 (3) (2015), 397-417.
[16]
S. B. Dhage, A. D. Kadam, Dhage Iteration method for initial value problems of nonlinear second order hybrid functional differential equations. Electronic Journal of Mathematical Analysis and Applications Vol (1), Jan. 2018 pp. 79-93.
[17]
D. M. Suryawanshi, S. S. Bellale, Dhage Iteration Method for Non-Linear first order abstract measure integro differential equations with linear perturbation. International Journal of Mathematics Trends and Technology (IJMTT)-Volume 64 issue 2 Dec 2018, 115-129.
[18]
D. M. Suryawanshi, S. S. Bellale, Dhage Iteration Method for Non-Linear first order abstract measure differential equations with linear perturbation. International Journal of Mathematics Trends and Technology (IJMTT)-Volume 65 issue 2 Feb 2019, 139-149.
[19]
D. M. Suryawanshi, S. S. Bellale, Iteration method for approximating solutions of perturbed abstract measure differential equations. Journal of Emerging Technologies and Innovative. April 2019, 746-752.