Existence of positive solutions for Riemann-Liouville fractional order three-point boundary value problem. (English) Zbl 1332.34012
Summary: We study the following fractional order three-point boundary value problem
\[
\begin{gathered} D^{q_1}_{0^+} u(t)+ f(t,u(t))= 0,\qquad t\in[0,1],\\ u(0)-\alpha D^{q_2}_{0^+} u(o)= D^{q_2}_{0^+} u(\eta)=\beta u(1)+\gamma D^{q_3}_{0^+} u(1)= 0,\end{gathered}
\]
where \(D^{q_i}_{0^+}\), \(i= 1,2,3\), ar the standard Riemann-Liouville fractional order derivatives with \(2< q_i< 3\), \(0< q_2\leq 1\), \(1< q_3< 2\) and \(\alpha> 0\), \(\beta> 0\), \(\eta\in(0,1)\) and \(f: [0,1]\times[0, \infty]\to[0,\infty]\) is continuous. By using several well-known fixed-point theorems in a cone, the existence of at least one and two positive solutions is obtained. Some examples are presented to illustrate the main results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34A40 | Differential inequalities involving functions of a single real variable |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
fractional differential equations; positive solution; fixed-point theorems; Green’s function; coneReferences:
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