Infinite many positive solutions for nonlinear first-order BVPs with integral boundary conditions on time scales. (English) Zbl 1292.34088

Li, Yongkun; Sun, Lijuan

Topol. Methods Nonlinear Anal. 41, No. 2, 305-321 (2013).

Summary: We investigate the existence of infinite many positive solutions for the nonlinear first-order BVP with integral boundary conditions \[ \begin{aligned} x^\Delta(t)+ p(t) x^\sigma(t)& = f(t, x^\sigma(t)),\quad t\in (0,T)_{\mathbb{T}},\\ x(0)-\beta x^\sigma(T) &= \alpha \int^{\sigma(T)}_0 x^\sigma(s)\,\Delta g(s),\end{aligned} \] where \(x^\sigma= x\circ\sigma\), \(f: [0,T]_{\mathbb{T}}\times \mathbb{R}^+\to \mathbb{R}^+\) is continuous, \(p\) is regressive and rd-continuous, \(\alpha,\beta\geq 0\), \(g:[0,T]_{\mathbb{T}}\to \mathbb{R}\) is a nondecreasing function. By using the fixed-point index theory and a new fixed-point theorem in a cone, we provide sufficient conditions for the existence of infinite many positive solutions to the above boundary value problem on time scale \(\mathbb{T}\).

MSC:

34N05	Dynamic equations on time scales or measure chains
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

time scale; boundary value problem; positive solution; integral boundary condition; fixed-point theorem