Infinite many positive solutions for nonlinear first-order BVPs with integral boundary conditions on time scales

Abstract

In this paper, we investigate the existence of infinite many positive solutions for the nonlinear first-order BVP with integral boundary conditions $$ \begin{cases} x^{\Delta}(t)+p(t)x^{\sigma}(t)=f(t,x^{\sigma}(t)), & t\in (0,T)_{\mathbb{T}}, \\ x(0)-\beta x^{\sigma}(T)=\alpha\int_{0}^{\sigma(T)}x^{\sigma}(s)\Delta g(s), \end{cases} $$ where $x^{\sigma}=x\circ\sigma$, $f\colon [0,T]_{\mathbb{T}}\times\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ is continuous, $p$ is regressive and rd-continuous, $\alpha,\beta\geq0$, $g\colon [0,T]_{\mathbb{T}}\rightarrow \mathbb{R}$ 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 $\mathbb{T}$.