Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function ()
1. Introduction
Boundary value problems (BVPs for short) of fourth-order ordinary differential equations have received much attention due to their striking applications in engineering, physics, material mechanics, fluid mechanics and so on. Many authors have studied the existence of single or multiple positive solutions to some fourth-order BVPs by using Banach contraction theorem, Guo-Krasnosel’skii fixed point theorem, Leray-Schauder nonlinear alterative, fixed point index theory in cones, monotone iterative technique, the method or upper and lower solutions, degree theory, critical point theorems in conical shells and so forth see [1] [2] [3] [4] [5] .
However, it is necessary to point out that, in most of the existing literature, the Green’s function involved is nonnegative, which is an important condition in the study of positive solutions of BVPs.
Recently, there have been some works on positive solutions for second-order or third-order BVPs when the corresponding Green’s functions are sign-changing. For example, Gao, Zhang and Ma [6] studied the following second-order periodic BVP with sign-changing Green’s function
where
,
is continuous,
is continuous and
is a parameter. The main tool used was the Leray-Schauder fixed point theorem. In 2013 [7] , by applying iterative technique, Sun and Zhao discussed the existence of monotone positive for the following third-order three-point BVP with sign-changing Green’s function
Motivated and inspired by the above-mentioned works, in this paper, we are concerned with the following fourth-order three-point BVP with sign-changing Green’s function
(1.1)
We will study as follows: calculating the corresponding Green function; studying the properties of Green function; constructing the proper cone; defining the proper operator; by applying iterative technique, we obtain the existence of the positive solution for the above problem.
Theorem 1.1. Let E be a Banach space and let K be a cone in E. Assume that
and
are bounded open subsets of E such that
,
, and let
be a completely continuous operator such that either;
1)
for
and
for
or
2)
for
and
for
.
Then T has a fixed point in
.
2. Preliminaries
In this paper, we always assume that
is continuous and satisfies the following conditions;
(H1) for each
, the mapping
is decreasing;
(H2) for each
, the mapping
is increasing.
Lemma 2.1. [8] Let
. Then for any given
, the BVP
has a unique solution
where
(2.1)
Lemma 2.2. Green’s function defined by (2.1)
has the following properties;
1)
for
and
for
.
2)
.
Proof. Since (1) is obvious, we only prove (2). If
, then we have
If
,
which together with the
implies that
Let
be equipped with the norm
and
Then it is easy to check that X is a Banach space and P is a cone in X.
Introduce an order relation
in X by defining
if and only if
, we define an operator T on P by
Of course, if u is a fixed point of T in P, then u is a decreasing nongetative solution of BVP (1.1). Besides, because of
and literature [8] ,
for
and
, so
. More, it follows from known textbook results, for example see proposition [4] , that
is completely continuous.
In the following sections and f satisfies the following conditions;
(H3) there exists positive constant r such that
;
(H4) there exists two positive constant
and
such that
Note;
,
,
if and only if
.
Lemma 2.3. Let
. Then
.
Proof. Let
, then
which together with the conditions (H1) - (H3) and (2) of Lemma 2.2, we get
this indicates
, in view of
. Hence
.
3. Main Results
Theorem 3.1. If we construct a iterative sequence
,
, where
, for
, then
converges to
in X and
is a decreasing positive solution of BVP (1.1).
Proof. In view of
and
imply
,
, therefore
is a bounded set. Because of T is completely continuous operator, set
is relatively compact.
By introduce prove
First, it is obvious that
, which shows that
. Next, we assume that
. Then it follows from (H2), we have
It follows from (H2) and (H4) that
hence,
that is
which indicates that
. Thus, we have shown that
.
Since
is relatively compact and monotone, there exist a
. Such that
which together with the continuity of T and the fact that
,
implies that
. This indicates that
is an increasing nonnegative solution of (1.1). Moreover in view of
,
, we know that zero function is not a solution of (1.1), which shows that
is a positive solution of (1.1).
Ethical Approval
We certify that this manuscript is original and has not been published and will not be submitted elsewhere for publication while being considered by boundary value problem. No data have been fabricated or manipulated (including images) to support our conclusions. And authors whose names appear on the submission have contributed sufficiently to the scientific work and therefore share collective responsibility and accountability for the results.
Authors Contributions
Yue Junrui and Zhang Yun wrote the main manuscript text and Bai Qingyue calculated all the conclusions of the article. All authors reviewed the manuscript.
Foundation Item
Education Science Program of Shanxi Province. The application of fractional differential equation in the immunization of an infectious disease model with SVIR (2023L491).
Availability of Data and Materials
Data sharing is not applicable to this article as no new data were created or analyzed is this study.
I declare the research results obtained in the research work of the authors of the papers submitted. To the best of my knowledge, this paper does not contain any research results that have been published or written by other individuals or groups, except those that have been noted and cited. Individuals and groups who have made significant contributions to the study of this paper have been clearly described in the paper.