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Existence of multiple positive solutions for \(m\)-point fractional boundary value problems on an infinite interval. (English) Zbl 1235.34023

Summary: We consider the following \(m\)-point fractional boundary value problem on an infinite interval \[ \begin{gathered} D^\alpha_{0+}u(t)+a(t)f(t,u(t))=0,\qquad 0<t<+\infty,\\ u(0)=u'(0)=0,\;D^{\alpha-1}u(+\infty)=\sum^{m-2}_{i=1}\beta_iu(\xi_i),\end{gathered} \] where \(2<\alpha<3\), \(D^\alpha_{0+}\) is the standard Riemann-Liouville fractional derivative, \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<+ \infty\), \(\beta_i\geq 0\), \(i=1,2,\dots,m-2\) satisfies \(0<\sum^{m-2}_{i=1}\beta_i\xi_i^{\alpha-1}<\Gamma (\alpha)\). Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. As applications, examples are presented to illustrate the main results.

MSC:

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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