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Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments. (English) Zbl 1235.34209

This paper deals with the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments \[ D^\alpha_0 u(t)+ a(t)(u(\theta(t)))= 0, \;0<t<1,\;n-1<\alpha\leq n \] satisfying \[ u^{(i)} (0)=0,\;i=0,1,2,\ldots, n-2,\quad [ D_0^\beta u(t)]_{t=1} =0,\;1\leq \beta\leq n-2, \] where \(n>3\) (\(n\in \mathbb{N}\)), \(D^\alpha_0\) is the standard Riemann-Liouville fractional derivative of order \(\alpha\), \(f: [0, \infty) \to [0, \infty) \) and \(\theta : (0, 1) \to (0, 1]\) are continuous functions.
The proof uses the Banach contraction principle and the well-known Guo-Krasnosel’skii fixed point theorem.

MSC:

34K37 Functional-differential equations with fractional derivatives
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K10 Boundary value problems for functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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