Affiliation:
1. Department of Mathematics, Northwest Normal University , Lanzhou , P. R. China
Abstract
Abstract
In this article, we focus on the existence of positive solutions and establish a corresponding iterative scheme for a nonlinear fourth-order equation with indefinite weight and Neumann boundary conditions
y
(
4
)
(
x
)
+
(
k
1
+
k
2
)
y
″
(
x
)
+
k
1
k
2
y
(
x
)
=
λ
h
(
x
)
f
(
y
(
x
)
)
,
x
∈
[
0
,
1
]
,
y
′
(
0
)
=
y
′
(
1
)
=
y
‴
(
0
)
=
y
‴
(
1
)
=
0
,
\left\{\begin{array}{l}{y}^{\left(4)}\left(x)+\left({k}_{1}+{k}_{2}){y}^{^{\prime\prime} }\left(x)+{k}_{1}{k}_{2}y\left(x)=\lambda h\left(x)f(y\left(x)),\hspace{1em}x\in \left[0,1],\\ y^{\prime} \left(0)=y^{\prime} \left(1)={y}^{\prime\prime\prime }\left(0)={y}^{\prime\prime\prime }\left(1)=0,\\ \end{array}\right.
where
k
1
{k}_{1}
and
k
2
{k}_{2}
are constants,
λ
>
0
\lambda \gt 0
is a parameter,
h
(
x
)
∈
L
1
(
0
,
1
)
h\left(x)\in {L}^{1}\left(0,1)
may change sign, and
f
∈
C
(
[
0
,
1
]
×
R
+
,
R
)
f\in C\left(\left[0,1]\times {{\mathbb{R}}}^{+},{\mathbb{R}})
,
R
+
≔
[
0
,
∞
)
{{\mathbb{R}}}^{+}:= \left[0,\infty )
. We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we establish some new results of the existence of positive solutions to this problem if the nonlinearity
f
f
is monotone on
R
+
{{\mathbb{R}}}^{+}
. The technique for dealing with this article relies on a monotone iteration technique and Schauder’s fixed point theorem. Finally, an example is presented to illustrate the application of our main results.
Reference28 articles.
1. A. Cabada and R. R. Enguiça, Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Anal. 74 (2011), 3112–3122, https://doi.org/10.1016/j.na.2011.01.027.
2. X. L. Han and T. Gao, A priori bounds and existence of non-real eigenvalues of fourth-order boundary value problem with indefinite weight function, Electron J. Differential Equations 2016 (2016), no. 82, 1–9, https://ejde.math.txstate.edu/Volumes/2016/82/han.pdf.
3. Q. Z. Ma, On the existence of positive solutions of fourth-order semipositone boundary value problem, Gongcheng Shuxue Xuebao 19 (2002), 133–136.
4. J. J. Wang, C. H. Gao, and Y. Q. Lu, Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions, Quaest. Math. 45 (2022), no. 9, 1–29, https://doi.org/10.2989/16073606.2022.2036260.
5. Q. L. Yao, Monotone iterative technique and positive solutions of Lidstone boundary value problems, Appl. Math. Comput. 138 (2003), no. 1, 1–9, https://doi.org/10.1016/s0096-3003(01)00316-2.