Positive solution of a system of integral equations with applications to boundary value problems of differential equations

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.