Existence and non-existence of positive solutions for a class of fractional boundary value problems
Abstract
In this paper, we consider a boundary value problem consisting of the nonlinear fractional differential equation
\begin{equation*}
-D_{0^+}^{\alpha}u + a D_{0^+}^{\gamma}u = f(t,u), \quad 0 < t < 1,
\end{equation*}
with nonlocal boundary conditions
\begin{equation*}
\qquad D_{0^+}^{\beta}u(0)= 0,\quad D_{0^+}^{\alpha-\gamma}u(1) = au(1),\quad u'(1) = 0,
\end{equation*}
where, $2<\gamma<\alpha\leq 3$, $0\leq \beta<\alpha-\gamma$, $0\leq a< \Gamma(\alpha-\gamma+1)$ and $f(t,u)\in C([0,1]\times [0,\infty),[0,\infty))$ and $D_{0^+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $\alpha$. The associated Green's function is derived in terms of the generalized Mittag-Leffler functions and shown that it satisfies certain properties. An attempt has been made to establish the existence and non-existence of positive solutions by using Leggett-Williams fixed point theorems on a cone in a Banach space. The results obtained in this paper extended and generalizes the result of \cite{6}. Finally, we provide a couple of examples to illustrate the validation of established results.
\begin{equation*}
-D_{0^+}^{\alpha}u + a D_{0^+}^{\gamma}u = f(t,u), \quad 0 < t < 1,
\end{equation*}
with nonlocal boundary conditions
\begin{equation*}
\qquad D_{0^+}^{\beta}u(0)= 0,\quad D_{0^+}^{\alpha-\gamma}u(1) = au(1),\quad u'(1) = 0,
\end{equation*}
where, $2<\gamma<\alpha\leq 3$, $0\leq \beta<\alpha-\gamma$, $0\leq a< \Gamma(\alpha-\gamma+1)$ and $f(t,u)\in C([0,1]\times [0,\infty),[0,\infty))$ and $D_{0^+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $\alpha$. The associated Green's function is derived in terms of the generalized Mittag-Leffler functions and shown that it satisfies certain properties. An attempt has been made to establish the existence and non-existence of positive solutions by using Leggett-Williams fixed point theorems on a cone in a Banach space. The results obtained in this paper extended and generalizes the result of \cite{6}. Finally, we provide a couple of examples to illustrate the validation of established results.
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