Existence results of a singular fractional differential equation with perturbed term. (English) Zbl 1395.34002
Summary: The boundary value problem
\[
D^{{\alpha}}u(t)+{\mu} a(t) f(t,u(t))-q(t)=0,
\]
\[ u(0)=u^{\prime}(0)=u^{(n-2)}(0)=0,\quad u(1)=\lambda \int_{0}^{1}u(s) ds \] is studied, where \(\mu\) is a positive parameter, \(f:[0,1]\times[0;+\infty)\rightarrow[0;+\infty)\) and \(a:(0,1)\rightarrow [0,+\infty)\) are continuous functions, while \(q:(0,1)\rightarrow [0,+\infty)\) is a measurable function. The case, where the function \(a\) has singularities at the points \(t=0\) and \(t=1\), is admissible. Conditions are found guaranteeing, respectively, the existence of at least one and at least two positive solutions. Examples are gives.
\[ u(0)=u^{\prime}(0)=u^{(n-2)}(0)=0,\quad u(1)=\lambda \int_{0}^{1}u(s) ds \] is studied, where \(\mu\) is a positive parameter, \(f:[0,1]\times[0;+\infty)\rightarrow[0;+\infty)\) and \(a:(0,1)\rightarrow [0,+\infty)\) are continuous functions, while \(q:(0,1)\rightarrow [0,+\infty)\) is a measurable function. The case, where the function \(a\) has singularities at the points \(t=0\) and \(t=1\), is admissible. Conditions are found guaranteeing, respectively, the existence of at least one and at least two positive solutions. Examples are gives.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
