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Limit properties of positive solutions of fractional boundary value problems. (English) Zbl 1308.34104
Summary: We investigate the sequence of fractional boundary value problems \[ ^cD^{\alpha_n}u=\sum^m_{k=1}a_k(t)^cD^{\mu_{k,n}}u+f(t,u,u',^cD^{\beta_n}u),\quad u'(0)=0,\quad u(1)=\Phi(u)-\Lambda(u'), \] where \(\lim_{n\to\infty}\alpha_n=2\), \(\lim_{n\to\infty}\beta_n=1\), \(\lim_{n\to\infty}\mu_{k,n}=1\), \(a_k\in C[0,1]\) (\(k=1,2,\dots,m\)), \(f\in C([0,1]\times \mathcal{D})\), \(\mathcal{D}\subset \mathbb{R}^3\), and \(\Phi,\Lambda:C[0,1]\to \mathbb{R}\) are linear functionals. \(^cD\) is the Caputo fractional derivative. It is proved, by the Leray-Schauder degree theory, that for each \(n\in \mathbb{N}\) the problem has a positive solution \(u_n\), and that there exists a subsequence \(\{u_{n'}\}\) of \(\{u_n\}\) converging to a positive solution of the differential boundary value problem \[ u''=u'\sum^m_{k=1}a_k(t)+f(t,u,u',u'),\quad u'(0)=0,\quad u(1)=\Phi(u)-\Lambda(u'). \]

34K37 Functional-differential equations with fractional derivatives
34K10 Boundary value problems for functional-differential equations
26A33 Fractional derivatives and integrals
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