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The goal of this work is to look at generalised contractions and show that they may be used to derive fixed point theorems in generalised metric spaces. Furthermore, we'll also use fixed point theorems to show that the outcome of the ordinary differential equation exists and is exclusive (ODE). Fixed point theorem applies to modified Lipschitz condition. First, we have proved that generalised condition of Boyd Wong theorem and using this concept we have modified Lipschitz condition by adding constant which is suitable for solving differential equation. [ABSTRACT FROM AUTHOR]

In this paper, by using the Guo-Krasnoselskii theorem, we investigate the existence and nonexistence of positive solutions of a system of integral equation with parameters which can be seen as an effective generalization of various types of systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations. We give a general approach of positive solutions to cover various systems of boundary value problems in a unified way, which avoids treating these problems on a case-by-case basis. Under some growth conditions imposed on the nonlinear term, we obtain explicit ranges of values of parameters with which the problem has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to consider existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations. [ABSTRACT FROM AUTHOR]

In this article we derive existence and comparison results for discontinuous non-absolute functional integral equations of Volterra type in an ordered Banach space which has a regular order cone. The obtained results are then applied to first-order impulsive differential equations. [ABSTRACT FROM AUTHOR]

In this paper, a new class of order cones in the space of continuous functions is introduced. The result unifies some previous work in studying the existence of solutions for differential equations using the compression cone techniques and fixed point theorems. It is shown that the method is more adaptable, particularly in dealing with changing sign Green's functions. Applications are illustrated by examples. Limitations of such a new method are also discussed. [ABSTRACT FROM AUTHOR]