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Dhage B.C.,Kasubai | Lakshmikantham V.,Florida Institute of Technology
Nonlinear Analysis: Hybrid Systems | Year: 2010

In this paper, an existence theorem for hybrid nonlinear differential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental differential inequalities are also established which are utilized to prove the existence of extremal solutions and a comparison result. © 2009 Elsevier Ltd.

Dhage B.C.,Kasubai | Jadhav N.S.,Kasubai
Tamkang Journal of Mathematics | Year: 2013

In this paper, some basic results concerning the strict and nonstrict differential inequalities and existence of the maximal and minimal solutions are proved for a hybrid differential equation with linear perturbations of second type.

In this paper, the author introduces a notion of partially condensing mappings in a partially ordered normed linear space and proves some hybrid fixed point theorems under certain mixed conditions of algebra, analysis and topology. The applications of abstract results presented here are given to some nonlinear functional integral equations for proving the existence as well as global attractivity of the comparable solutions under certain monotonicity conditions. The abstract theory presented here is very much useful to develop the algorithms for the solutions of some nonlinear problems of analysis and allied areas of mathematics. A realization of of our hypotheses is also indicated by a numerical example.

In 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 theminimal andmaximal solutions of the hybrid differential equations.

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