Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form \vskip -4mm \begin{align*} (P_{a,b})\quad \left\{\!\!\! \begin{array}{l} D^{\alpha }u(x) +f(x,u(x))=0,\quad x\in (0,1) ,\smallskip \u(0)=u(1)=0,\quad D^{\alpha -3}u(0)=a,\quad u'(1)=-b,% \end{array}% \right. \end{align*}% \vskip-2mm \nd where $3<\alpha \leq 4,$ $D^{\alpha }$ is the standard Riemann-Liouville fractional derivative and $a,b$ are nonnegative constants. First the authors suppose that $f(x,t)=-p(x)t^{\sigma },$ with $\sigma \in \left( -1,1\right) $ and $p$ being a nonnegative continuous function that may be singular at $x=0$ or $x=1$ and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch\"{a}uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem $(P_{0,0}).$ Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that $a,b$ are nonnegative constants such that $a+b>0$ and $f(x,t)=t\varphi (x,t),$ with $\varphi (x,t)$ being a nonnegative continuous function in $(0,1)\times 0,\infty )$ that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution $u$ to problem $% (P_{a,b})$, which behaves like the unique solution of the homogeneous problem corresponding to $(P_{a,b}).$ Some examples are given to illustrate the existence results.