AN INTEGRAL BOUNDARY VALUE PROBLEM OF CONFORMABLE INTEGRO-DIFFERENTIAL EQUATIONS WITH A PARAMETER

In this article, we consider some properties of positive solutions for a new conformable integro-differential equation with integral boundary conditions and a parameter $ \left\{ \begin{array}{l} T_{\alpha}u(t)+\lambda f(t,u(t),I_{\alpha}u(t)) = 0,t\in[0,1],\\ u(0) = 0,u(1) = \beta\int_{0}^{1}u(t)dt ,\beta\in[\frac 32,2), \ \end{array}\right.\nonumber $ where $ \alpha\in(1,2] $, $ \lambda $ is a positive parameter, $ T_{\alpha} $ is the usual conformable derivative and $ I_{\alpha} $ is the conformable integral, $ f:[0,1]\times\bf{R^{+}}\times\bf{R^{+}}\rightarrow \bf{R^{+}} $ is a continuous function, where $ \bf{R^{+}} = [0,+\infty) $. We use a recent fixed point theorem for monotone operators in ordered Banach spaces, and then establish the existence and uniqueness of positive solutions for the boundary value problem. Further, we give an iterative sequence to approximate the unique positive solution and some good properties of positive solution about the parameter $ \lambda $. A concrete example is given to better demonstrate our main result.