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An integral boundary value problem of conformable fractional integro-differential equations with a parameter
Chengbo Zhai,Yuqing Liu
Keywords:positive solution; conformable fractional derivative; integro-differential equations; fixed point theorem of generalized concave operators
Abstract:
      In this article, we consider some properties of positive solutions for a new conformable fractional integro-differential equation with integral boundary conditions and a parameter $$\left\{ \begin{array}{l} T_{\alpha}u(t)+\lambda f(t,u(t),I_{\alpha}u(t))=0,t\in[0,1],\\ u(0)=0,u(1)=\beta\int_{0}^{1}u(t)dt ,\beta\in[\frac 32,2),\end{array}\right.\eqno(1.1)$$ where $\alpha\in(1,2]$, $\lambda$ is a positive parameter, $T_{\alpha}$ is the usual conformable fractional derivative and $I_{\alpha}$ is the conformable fractional integral, $f:[0,1]\times\mathbf{R^{+}}\times\mathbf{R^{+}}\rightarrow \mathbf{R^{+}} $ is a continuous function, where $\mathbf{R^{+}}=[0,+\infty)$. 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 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.