Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions (Q392994)

The following nonlinear fractional three-point boundary value problem is considered \[ D^{\alpha}_{0+}u(t)+f(t,u(t))=0, \quad 0<t<1, \quad 2< \alpha \leq 3, \tag{1} \] \[ u(0)=u'(0)=0, \quad u'(1)=\int\limits_0^{\eta}u(s)ds, \tag{2} \] where \(D^{\alpha}_{0+}\) is the Riemann-Liouville fractional derivative and \(\eta\) satisfies the inequality \(0<\eta^{\alpha}<\alpha(\alpha-1)\). The basics results of the article are contained in the following two theorems: Theorem 3. Let the following two conditions are satisfied: (i)~\(f:[0,1]\times [0,\infty)\to[0,\infty)\) is continuous and nondecreasing with respect to the second variable, and \(f(t,u(t))\neq 0\) for \(t\in Z\subset [0,1]\) with \(\mu(Z)>0\) (\(\mu\) denotes the Lebesgue measure); (ii)~There exists \(0<\lambda<L^{-1}\) such that for \(u,v\in [0,\infty)\) with \(u \geq v\) and \(t \in [0,1]\), \(f(t,u)-f(t,v)\leq \lambda \ln(u-v+1)\). Then the BVP (1),(2) has a unique positive and strictly increasing solution \(u(t)\). Theorem 4. Let the following conditions are satisfied: \(f(t,u) \in C([0,1]\times [0,\infty),\mathbb{R}^+)\) is nondecreasing relative to \(u\), \(f(t,\rho(t))\neq 0\) for \(t\in (0,1)\), and there exists a positive constant \(\mu <1\) such that \[ k^{\mu}f(t,u)\leq f(t,ku) \quad \forall \quad 0\leq k\leq 1. \] Then the BVP (1),(2) has a positive solution \(u(t)\). For the proof, the method of lower and upper solutions, some properties of the Green function and a fixed point theorem in partially ordered sets are used.