I. INTRODUCTION

In this article, we aim to study the existence of a monotonic positive solution for fourth-order three-point Nonlinear BVP with changing sign Green's function

where α ∈ [0,2), f ∈ C(...) where $\alpha \in [0,2),f\in C([0,1]\times [0,+\mathrm{\infty }),[0,+\mathrm{\infty }))$ and $\eta \in [\frac{1}{2},1)$. By using iterative methods, We can still obtain the existence of a monotonic positive solution under certain suitable conditions of $f$.

In recent decades, The differential equations come from various fields of a mathematical applied and physics, for example, in the deflection of curved beams with constant or varying cross-sections, triple-layer beams, electromagnetic waves or gravity-driven currents, etc. [1].

Recently, when the corresponding Green's function is undergoing sign conversion, there has been some work on the positive solution of the second and third-order BVP. For example, in [8] the existence of at least one positive solution of the following second-order periodic BVP with positive and negative transformation Green's function studied by Zhong and An

where $\eta \in (\frac{17}{24},1)$, Palamide and Smirlis [9] discussed the existence of at least one positive solution. Their technique is a combination of GuoKrasnosel'ski fixed point theory and the corresponding vector field characteristics. In 2012, Sun and Zhao [10], [11] obtained single or multiple positive solutions with three-point positive and negative BVP by applying the fixed point theory of Guo-Krasnosel'skii

Motivated. Through the above work, this article will study BVP (1.1)Through an iterative method. Throughout this article, we always assume $\alpha \in [0,2)$ and $\eta \in [\frac{1}{2},1)$. Although the corresponding Green function is changing its sign, under certain suitable conditions, we can still obtain the existence of the monotonic positive solution of BVP (1.1) on $f$. Moreover, our iterative scheme starts with a zero function, This means that iterative scheme is feasible.

In this article, by applying an iterative approach, we always assume that $\alpha \in [0,2)$ and $\eta \in [\frac{1}{2},1)$.

We first recall the following fixed point of Krasnoselskii's type.

Theorem 1.1. Let $E$ be a Banach space and $K$ be a cone in $E$. Assume that ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_1$ are bounded open subsets of $E$ such that $0\in {\mathrm{\Omega }}_1$, ${\bar{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2$, and let $A:K\cap ({\mathrm{\Omega }}_2\setminus {\mathrm{\Omega }}_1)\to K$ be a completely continuous operator such that either

(1) $\Vert Au\Vert \le \Vert Au\Vert$ for $u\in K\cap \partial {\mathrm{\Omega }}_1$ and $\Vert Au\Vert \ge \Vert u\Vert$ for $u\in K\cap \partial {\mathrm{\Omega }}_2$ or (2) $\Vert Au\Vert \ge \Vert Au\Vert foru\in K\cap \partial {\mathrm{\Omega }}_1$ and $\Vert Au\Vert \le \Vert u\Vert$ for $u\in K\cap \partial {\mathrm{\Omega }}_2$

• A. has a fixed point in $K\cap ({\bar{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)$.

II. PRELIMINARIES

For the BVP

we have the following lemma

Lemma 2.1. The BVP (2.1) has only trivial solution.

Proof. It is simple to check. for any $y\in C[0,1]$, we consider the boundary value problems

After a direct computation, one may obtain the expression of Greens function $G(t,s)$ of the BVP (2.2) as following:

Proof. Integrating four times the linear problem gives us that

The conditions $u^{\prime }(0)=u^{‴}(0)=0$ implies that

and the conditions $u(1)=0$ this means

Next, $\alpha u(0)+u^{″}(\eta )=0$ is rewritten Such as

whence

form The conditions $u(1)=0$ we have

If we substitute(1.3) with the expression from above and simplify, we get that

Finally, we obtain that

As a result, we have that For $s\ge \eta$

and

Remark 2.1. It is not difficult to verify that $G(t,s)$ has the following characteristics:

Moreover, if $s\ge \eta$, then

if , then

therefore, if we let $\delta =max|G(t,s)|:t,s\in [0,1]$ then

Now, let Banach space $E=C[0,1]$ is equipped with the $\Vert u\Vert =\underset{t\in [0,1]}{max}|u(t)|$.

$K=\{y\in C[0,1]:y(t)\}$ is nonnegative and decreasing on [0, 1]. Then $K$ is a cone in $C[0,1]$.

Note that this order relationship is inducesan in $E$ by defining $uv$ if and only if $u-v\in K$

In the remainder of this paper, we always assume that $f:C[0,1]\times [0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is continuous and satisfies the following conditions:

(F1) For each $u\in [0,+\mathrm{\infty })$, the mapping $t\mapsto f(t,u)$ is decreasing;

(F2) For each $t\in [0,1]$, the mapping $u\mapsto f(t,u)$ is increasing.

Obviously, if $u$ is a fixed point of $A$ in $K$, then $u$ is a nonnegative and decreasing solution of the BVP (1.1).

Lemma 2.2. Let $A:K\to K$ is completely continuous.

Proof. let $u\in K$. Then, for $0\le t\le \eta$, we have

which together with $(F1)$ and $(F2)$ implies that