Positive solutions for first-order nonlinear Caputo-Hadamard fractional relaxation differential equations. (English) Zbl 1499.34170

Summary: This article concerns the existence and uniqueness of positive solutions of the first-order nonlinear Caputo-Hadamard fractional relaxation differential equation \[ \begin{cases} \mathfrak{D}_1^{\alpha}(x(t) -g(t,x(t))) +wx (t) = f(t,x(t)), \quad 1<t \leq e,\\ x(1) = x_0 > g(1,x_0) >0, \end{cases} \] where \(0<\alpha \leq 1\). In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ the Krasnoselskii fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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