### 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

$$

\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),

\endcases

$$

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 $\mathbb{T}$.

first-order BVP with integral boundary conditions

$$

\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),

\endcases

$$

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 $\mathbb{T}$.

#### Keywords

Time scale; boundary value problem; positive solution; integral boundary condition

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.