Approximating Solutions of Nonlinear Abstract Measure First Order Differential Equations via Hybrid Fixed Point Theory

Approximating Solutions of Nonlinear Abstract Measure First Order Differential Equations via Hybrid Fixed Point Theory

  IJETT-book-cover  International Journal of Engineering Trends and Technology (IJETT)          
  
© 2017 by IJETT Journal
Volume-49 Number-6
Year of Publication : 2017
Authors : S.S.Bellale, G.B.Dapke
DOI :  10.14445/22315381/IJETT-V49P253

Citation 

S.S.Bellale, G.B.Dapke "Approximating Solutions of Nonlinear Abstract Measure First Order Differential Equations via Hybrid Fixed Point Theory", International Journal of Engineering Trends and Technology (IJETT), V49(6),348-357 July 2017. ISSN:2231-5381. www.ijettjournal.org. published by seventh sense research group

Abstract
In this article we prove the existence and approximations of solutions of AMDE of first order ordinary nonlinear hybrid differential equations. We relay our results on Dhage iterations principle or method embodied in a recent hybrid fixed point theorem of Dhage [B.C. Dhage, on some nonlinear alternatives of Leray-Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11-23] under the mixed generalized Lipschitz and Caratheodory Conditions. The existence and approximations of solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage given in the above mentioned reference on ordered Banach spaces.

Reference
[1] S.S.Bellale; Hybrid Fixed point theorem for Abstract Measure Differential Equation,Word Academy of Science, Engineering and Technology, 73(2013) 782-785.
[2] S.S.Bellale , S.B.Birajdar; Existence Theorem for Nonlinear Functional Two Point Boundary Value of problem in Banach Algebra, International Journal Of Universal Mathematics and Mathematical Sciences, 1 (1) (2015), 47-58.
[3] S.S.Bellale , G.B.Dapke; Existence Theorem And Extremal Solutions For Perturbed Measure Differential Equations with Maxima, International Journal Of Mathematical Archive – 7 (10), 2016, 1-11.
[4] S.S.Bellale, G.B.Dapke; Approximate Solutions For Perturbed Measure Differential Equations with Maxima, International Journal of Engineering Sciences And Research Technology, 5(9), sep, 2016.
[5] S.S.Bellale , S.B. Birajdar; on Quadratic Abstract Measure Integro-Differential Equations, Journal of Computations And Modeling, 5 (3) (2015), 99-122.
[6] B.C.Dhage , S.S.Bellale; Existence Theorem For Perturbed Abstract Measure Differential Equations, Nonlinear Analysis, 71(2009), e319-e328.
[7] B.C.Dhage, S.S.Bellale; Local Asymptotic Stability For Nonlinear Quadratic Functional Integral Equations, Electronics Journal Of Qualitive Theory Of Differential Equations, 10 (2008),1-23.
[8] B.C.Dhage, S.S. Bellale; Abstract Measure Integro-Differential Equations, Global.J.Ana.1 (1-2) (2007) 91-108.
[9] B.C.Dhage ,S.B.Dhage; Approximating Solutions Of Nonlinear PBVPS Of Second-Order Differential Equations Via Hybrid Fixed Point Theory, Electronic Journal Of Differential Equations, Vol. (2015), No. 20 , pp. 1-10.
[10] B.C. Dhage; On Some Nonlinear Alternatives Of Leray-Schauder Type And Functional Integral Equations, Arch.Math.(Brno) 42 (2006) 11-23.
[11] B.C.Dhage; Hybrid Fixed Point Theory In Partially Ordered Normed Linear Spaces And Applications To Fractional Integral Equations, Differ.Equ.Appl. 6 (2014), 165-186.
[12] B.C.Dhage; Global Attractivity Results For Comparable Solutions Of Nonlinear Hybrid Fractional Integral Equations, Differ, Equ. Appl. 6 (2014) , 165-186.
[13] B.C.Dhage; Partially Condensing Mappings In Ordered Normed Linear Spaces And Applications To Functional Integral Equations, Tamkang J.Math. 45 (4) (2014), 397-426. Doi:10.5556/j.tkjm.45.2014.1512.
[14] B.C.Dhage; Nonlinear D- Set Contraction Mappings In Partially Ordered Normed Linear Spaces And Applications To Fuctional Hybrid Integral Equations, Malaya J.Mat. 3(1) (2015)
[15] A.Granas, J. Dugundji; Fixed Point Theory, Springer Verlag, 2003.
[16] S.Heikkila, V.Lakshikantham; Monotone Iterative Technique for Discontinuous Nonlinear Differential equations, Pacific J.Math. 52 (2) (1974) 489-498.
[17] P.J.Torres; Existence Of One-Signed Periodic Solutions Of Some Second-Order Differential Equations Via a Krasnoselskii Fixed Point Theorem, J. Differential Eqautions 190 (2003), 643-662.

Keywords
Abstract measure differential equation, hybrid fixed point theory, approximate Solution.