Approximation methods in the theory of hybrid differential equations with linear perturbations of second type
Keywords:Hybrid differential equation, Existence theorem, Upper and lower solutions, Monotone iterative technique.
AbstractIn this paper, some existence theorems for the extremal solutions are proved for an initial value problem of nonlinear hybrid differential equations via constructive methods. The monotone iterative techniques for initial value problems of first order hybrid differential equations are developed and it is shown that the sequences of successive iterations defined in a certain way converge to the minimal and maximal solutions of the hybrid differential equations.
B. C. Dhage, A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear Studies, 13 (2006), 343--354.
B. C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J., 44(2004), 145--155.
B. C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Diff. Equ.& Appl., 2(2010), 465--486.
B. C. Dhage, Theoretical approximation methods for hybrid differential equations, Dynamic Systems and Applications, 20 (2011), 455--478.
B. C. Dhage and V. Lakshmikatham, Basic results on hybrid differential equations, Nonlinear Analysis: Hybrid Systems, 4(2010), 414--424.
B. C. Dhage and Namdev S. Jadhav, Basic results on hybrid differential equations with linear perturbation of second type, Tamkang J. Math., 44 (2) (2013), 171--186.
G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, New York, 1985.
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Academic Press, New York, 1969.