DHAGE ITERATION METHOD FOR APPROXIMATING
THE POSITIVE SOLUTIONS OF IVPS FOR NONLINEAR
FIRST ORDER QUADRATIC NEUTRAL FUNCTIONAL
DIFFERENTIAL EQUATIONS WITH DELAY AND MAXIMA
Bapurao C. Dhage1, Shyam B. Dhage2, John R. Graef3 1,2 Kasubai, Gurukul Colony
Ahmedpur - 413 515, Dist: Latur, Maharashtra, INDIA 3Department of Mathematics
University of Tennessee at Chattanooga
Chattanooga, TN 37403, USA
In this paper, the authors prove an existence and approximation theorem for positive solutions to a nonlinear first order quadratic hybrid neutral functional differential equations with delay and maxima under mixed geometric, algebraic, and topological conditions. They employ the Dhage iteration method embodied in a hybrid fixed point principle of Dhage (2014) involving the product of two operators in a partially ordered Banach algebra.
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
S. Chandrasekher, Radiative Transfer, Dover Publications, New York (1960).
 K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985).
 B.C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl., 2 (2010), 465-486.
 B.C. Dhage, Fixed point theorems in ordered Banach algebras and applications, PanAmer. Math. J., 9, No 4 (1999), 93-102.
 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.
 B.C. Dhage, Partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math., 45, No 4 (2014), 397-427.
 B.C. Dhage, Nonlinear D-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malays. J. Mat. Sci., 3, No 1 (2015), 62-85.
 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.
 B.C. Dhage, Approximating solutions of nonlinear periodic boundary value problems with maxima, Cogent Mathematics, 3 (2016), # 1206699.
 B.C. Dhage, Dhage iteration method for nonlinear first order ordinary hybrid differential equations with mixed perturbation of second type with maxima, J. Nonlinear Funct. Anal., 2016 (2016), Article ID 31.
 B.C. Dhage, Dhage iteration method in the theory of initial value problems of nonlinear first order hybrid differential equations, Malays. J. Mat. Sci., 5, No 4 (2017), to appear.
 B.C. Dhage and S.B. Dhage, Approximating solutions of nonlinear pbvps of hybrid differential equations via hybrid fixed point theory, Indian J. Math., 57, No 1 (2015), 103-119.
 B.C. Dhage and S.B. Dhage, Approximating solutions of nonlinear first order ordinary differential equations, GJMS Special Issue for Recent Ad- vances in Mathematical Sciences and Applications - 13 (GJMS), 2, No 2 (2013), 25-35.
 B.C. Dhage and S.B. Dhage, Approximating positive solutions of nonlinear first order ordinary hybrid differential equations, Cogent Mathematics, 2 (2015), # 1023671.
 B.C. Dhage and S.B. Dhage, Approximating positive solutions of PBVPs of nonlinear first order ordinary hybrid differential equations, Appl. Math. Lett., 46 (2015), 133-142.
 S.B. Dhage and B.C. Dhage, Dhage iteration method for approximating positive solutions of PBVPs of nonlinear hybrid differential equations with maxima, Intern. Jour. Anal. Appl., 10, No 2 (2016), 101-111.
 S.B. Dhage and B.C. Dhage, Dhage iteration method for approximating positive solutions of nonlinear first order ordinary quadratic differential equations with maxima, Nonlinear Anal. Forum, 16, No 1 (2016), 87-100.
 B.C. Dhage, S.B. Dhage, and J.R. Graef, Dhage iteration method for initial value problems for nonlinear first order hybrid integro-differential equations, J. Fixed Point Theory Appl., 18 (2016), 309-326.
 B.C. Dhage, S.B. Dhage, and J.R. Graef, Dhage iteration method for nonlinear first order hybrid differential equations with a mixed perturbation of the second type, Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal., 24 (2017), 159-180.
 B.C. Dhage, N.S. Jadhav, and A.Y. Shete, Hybrid fixed point theorem with PPF dependence in Banach algebras and applications to quadratic differential equations, J. Math. Comput. Sci., 5 (2015), 601-614.
 J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Berlin (1977).
 S. Heikkil¨a and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Dekker, New York (1994).
 A.R. Magomedov, On some questions about differential equations with maxima, Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tehn. Mat. Nauk, 1 (1977), 104-108 (in Russian).
 D.V. Mule and B.R. Ahirrao, Approximating solution of an initial and periodic boundary value problems for first order quadratic functional differential equations, Int. J. Pure Appl. Math., 113, No 2 (2017), 251-271.
 A.D. Myshkis, On some problems of the theory of differential equations with deviating argument, Russian Math. Surveys, 32 (1977), 181-210.