Two point fractional boundary value problems with a fractional boundary condition
Suppose \(n\in \mathbb{N}\) such that \(n\geq 5\), \(\alpha\in (n-1, n]\) and \(\lambda>0\) is a parameter. Consider \(f:[0, \infty)\rightarrow [0, \infty)\) and \(g:[0, 1]\rightarrow [0, \infty)\) two continuous functions such that \(\displaystyle\int_0^1g(t)dt>0\). Then the existence and non-existence of the positive solution \(u\) of the following fractional differential equation is considered: \[ D_{0+}^{\alpha}u = \lambda g(t) f(u),\quad t\in (0, 1)\tag{1} \] with respect to the boundary conditions: \[ u^{(i)}(0) = 0, \quad i= 0, 1,\dots, n-4,\quad D_{0+}^{\alpha-2}u(0)=0, \quad D_{0+}^{\alpha-2}u(1)=0,\quad D_{0+}^{\beta}u(1)=0\tag{2} \] or \[ u^{(i)}(0) = 0,\quad i= 0, 1,\dots, n-4,\quad D_{0+}^{\alpha-2}u(0)=0,\quad D_{0+}^{\alpha-1}u(1)=0,\quad D_{0+}^{\beta}u(1)=0,\tag{3} \] where \(\beta\in[1, n-3]\) and \(D_{0+}^{\alpha}, D_{0+}^{\beta}\) are the Riemann-Liouville fractional derivatives of order \(\alpha\), \(\beta\) respectively. Further, in another section the problem of existence and non-existence of positive solution \(u\) of the following fractional differential equation is discussed: \[ D_{0+}^{\mu}u + \lambda g(t) f(u)=0,\quad t\in (0, 1)\tag{4} \] with the boundary conditions \[ u^{(i)}(0)= 0,\quad i= 0, 1,\dots, m-2, \quad D_{0+}^{\beta}u(1)=0,\tag{5} \] where \(\beta\in [1, m-1]\), \(\mu\in (m-1, m]\), \(m\in \mathbb{N}\), \(m\geq 3\). Throughout this work the authors establish the existence and non-existence of the positive solution \(u\) for the pairs \((1, 2)\), \((1,3)\) and \((4, 5)\) of fractional differential equations with boundary conditions. The authors apply Krasnosel'skii's fixed point theorem to obtain these results.
scientific article; zbMATH DE number 6958685
scientific article