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Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with \(p\)-Laplacian. (English) Zbl 1181.26019

Summary: We consider the existence of positive solutions for three-point fractional differential equation boundary value problems with \(p\)-Laplacian
\[ D_{0+}^\gamma (\varphi_p(D_{0+}^\alpha u(t)))+ f(t,u(t))=0, \quad 0<t<1, \]
\[ u(0)=0, \quad u(1)= au(\xi), \quad D_{0+}^\alpha u(0)=0, \]
where \(0<\gamma\leq 1\), \(1<\alpha\leq 2\), \(0\leq a\leq 1\), \(0<\xi<1\), \(D_{0+}^\alpha\) is the standard Riemann-Liouville differentiation, and \(f:[0,1]\times [0,+\infty)\to [0,+\infty)\) is continuous. \(\varphi_p(s)= |s|^{p-2}s\), \(p>1\), \((\varphi_p)^{-1}= \varphi_q\), \(\frac 1p+\frac 1q=1\). By using Krasnoselskii’s fixed point theorem and Leggett-Williams’ theorem, some sufficient conditions for the existence of positive solutions to the above boundary value problems are obtained. We remove the condition that the solutions are concave functions. Two examples are given to illustrate the effectiveness of our results.

MSC:

26A33 Fractional derivatives and integrals
35K05 Heat equation
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