Existence of the positive solutions for a tripled system of fractional differential equations via integral boundary conditions

Abstract

The purpose of this paper, is studying the existence and
nonexistence of positive solutions to a class of a following tripled
system of fractional differential equations.
\begin{eqnarray*} \left\{ \begin{array}{ll}
D^{\alpha}u(\zeta)+a(\zeta)f(\zeta,v(\zeta),\omega(\zeta))=0, \quad
\quad u(0)=0,\quad u(1)=\int_0^1\phi(\zeta)u(\zeta)d\zeta, \\ \\
D^{\beta}v(\zeta)+b(\zeta)g(\zeta,u(\zeta),\omega(\zeta))=0, \quad
\quad v(0)=0,\quad v(1)=\int_0^1\psi(\zeta)v(\zeta)d\zeta,\\ \\
D^{\gamma}\omega(\zeta)+c(\zeta)h(\zeta,u(\zeta),v(\zeta))=0,\quad
\quad \omega(0)=0,\quad
\omega(1)=\int_0^1\eta(\zeta)\omega(\zeta)d\zeta,\\ \end{array}
\right.\end{eqnarray*} \\ where $0\leq \zeta \leq 1$, $1<\alpha,
\beta, \gamma \leq 2$, $a,b,c\in C((0,1),[0,\infty))$, $ \phi, \psi,
\eta \in L^1[0,1]$ are nonnegative and $f,g,h\in
C([0,1]\times[0,\infty)\times[0,\infty),[0,\infty))$ and $D$ is the
standard Riemann-Liouville fractional derivative.\\
Also, we provide some examples to demonstrate the validity of our
results.

Keywords

• Tripled System
• fractional differential equation
• integral boundary conditions
• existence and nonexistence of positive solutions

Kaynakça

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