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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 $u''=u'\sum^m_{k=1}a_k(t)+f(t,u,u',u'),\quad u'(0)=0,\quad u(1)=\Phi(u)-\Lambda(u').$

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