Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary value problem. (English) Zbl 1253.35201

Summary: We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system \(-\mathcal D^\alpha_{\mathbf{t}} x(t) = f(t, x(t), x'(t), x''(t), \dots, x^{(n-2)}(t))\), \(0 < t < 1\), \(x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0\), \(x^{(n-2)}(1) = \int^1_0 x^{(n-2)}(s)dA(s)\) where \(n - 1 < \alpha \leq n\), \(n \in \mathbb N\), and \(n \geq 2, \mathcal D^\alpha_{\mathbf{t}}\) is the standard Riemann-Liouville derivative, \(\int^1_0 x(s)dA(s)\) is the linear functional given by the Riemann-Stieltjes integrals, \(A\) is a function of bounded variation, and \(dA\) can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder’s fixed-point theorem and upper and lower solution method.


35R11 Fractional partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35B09 Positive solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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