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| Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems |
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Citation： |
Imed BACHAR,Habib MÂAGLI,Faten TOUMI,Zagharide ZINE EL ABIDINE.Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems[J].Chinese Annals of Mathematics B,2016,37(1):1~28 |
| Page view： 3392
Net amount： 1773 |
Authors： |
Imed BACHAR; Habib M?AGLI;Faten TOUMI;Zagharide ZINE EL ABIDINE |
Foundation： |
This project was funded by the National Plan for Science, Technology and Innovation(MAARIFAH), King Abdulaziz City for Science and
Technology, Kingdom of Saudi Arabia, Award Number (No.13-MAT2137-02) |
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| Abstract： |
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. |
Keywords： |
Fractional differential equation, Positive solution, Fractional
Green's function, Karamata function, Perturbation arguments,
Sch\"{a}uder fixed point theorem |
Classification： |
34A08, 34B15, 34B18, 34B27 |
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