Seshadev Padhi, B. S. R. V. Prasad, Satyam Narayan Srivastava, Shasanka Dev Bhuyan
abstract:
In this paper, we apply the monotone iteration method to establish the existence
of a positive solution for the fractional differential equation
\begin{equation*}
D_{0+}^{\alpha}u(t)+q(t)f(t,u(t))=0, \;\; 0<t<1,
\end{equation*}
together with the boundary conditions (BCs)
\begin{equation*}
u(0)=u'(0)=\cdots=u^{n-2}(0)=0, \;\;
D_{0+}^{\beta}u(1)=\int\limits_{0}^{1}h(s,u(s))\,dA(s),
\end{equation*}
where $n>2$, $n-1<\alpha\leq n$, $\beta \in [1,\alpha-1]$, $D_{0+}^{\alpha}$ and
$D_{0+}^{\beta}$ are the standard Riemann-Liouville fractional derivatives of
order $\alpha$ and $\beta$, respectively, and $f,h:[0,1]\times[0,\infty)\to
[0,\infty)$ are continuous functions. The sufficient condition provided in this
paper is new, interesting and easy to verify. Our conditions do not require the
sublinearity or superlinearity on the nonlinear functions $f$ and $h$ at $0$ or
$\infty$. The paper is supplemented with examples illustrating the applicability
of our result.
Mathematics Subject Classification: 34B08, 34B10, 34B15, 34B18
Key words and phrases: Fractional differential equations, Riemann-Liouville derivative, boundary value problems, positive solutions, monotone iteration method