Basic results in the theory of hybrid differential equations with linear perturbations os second type

Abstract

Abstract. 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.

Study of Nonlocal Boundary Value Problems of Non-Integer Order Hybrid Differential Equations

“…The results were further extended to the class of first order initial value problem for hybrid differential equation with linear perturbation of second type (Dhage and Jadhav, 2013).…”

Approximation methods in the theory of hybrid differential equations with linear perturbations of second type

“…The following existence result is proved in Dhage and Jadhav [6] via a fixed point technique formulated in Dhage [2]. …”

“…However, to the best of our knowledge, there is no such fixed point theorem or method developed so far for the hybrid differential equations without further assumptions on the nonlinearities involved in the equations. Recently, Dhage and Jadhav [6] and Dhage and Lakshmikantham [5] have proved some basic results for the hybrid differentials equations of first order with the linear and quadratic perturbations of second type. In this paper, using the ideas from Lakshmikantham and Leela [8], Dhage [4] and Ladde et al [7], we establish some theoretical approximation results for the extremal solutions of hybrid differential equations between the given lower and upper solutions.…”

“…The HDE (1.1) is a linear perturbation of second type of an initial value problem of first order nonlinear differential equations and has been discussed in Dhage and Jadhav [6] for existence theory for different aspects of the solutions. The details of different types of nonlinear perturbations of a differential equation appears in Dhage [3].…”

On applications of coupled fixed point theorem in hybrid differential equations of arbitrary order

“…which are Lebegue integrable bounded by a Lebesgue integrable function on J [5]. Some of study studied the existence and uniqueness theorems of the solution of the ordinary first-order hybrid differential equation with perturbation of second type [36].…”