R E S E A R C H
Open Access
The method of lower and upper solutions
for fourth order equations with the Navier
condition
Ruyun Ma
*, Jinxiang Wang and Dongliang Yan
*Correspondence: [email protected] Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China
Abstract
The aim of this paper is to explore the method of lower and upper solutions in order to give some existence results for equations of the form
y(4)(x) + (k1+k2)y(x) +k1k2y(x) =f(x,y(x)
)
, x∈(0, 1), with the Navier conditiony(0) =y(1) =y(0) =y(1) = 0
under the conditionk1< 0 <k2<
π
2. The main tool is the Schauder fixed pointtheorem.
MSC: 34B10; 34B18
Keywords: elastic beam; fourth order equations; lower and upper solutions; Green function
1 Introduction
The aim of this paper is to explore the method of lower and upper solutions in order to give some existence of solutions for equations of the form
y()(x) + (k+k)y(x) +kky(x) =f
x,y(x), x∈(, ), (.)
with the Navier condition
y() =y() =y() =y() = . (.)
Such boundary value problems appear, as it is well known [–], in the theory of hinged beams.
Recently, Vrabel [] studied problem (.), (.) under the assumption
(H) kandkare two constants with
k<k< . (.)
He constructed the Green function for the linear problem
L(y)(x)≡y()(x) + (k+k)y(x) +kky(x) = , x∈(, ), y() =y() =y() =y() = ,
(.)
and proved its non-negativity and established the method of lower and upper solutions for (.), (.).
Definition .([]) The functionα∈C[, ] is said to be alower solutionfor (.), (.) if
Lα(x)≤fx,α(x) forx∈(, ), (.)
and
α()≤, α()≤, α()≥, α()≥. (.)
Anupper solutionβ∈C[, ] is defined analogously by reversing the inequalities in (.),
(.).
Theorem A([, Theorem ]) Let(H)hold.Suppose that for problem(.), (.)there exist a lower solutionαand an upper solutionβsuch that
α(x)≤β(x) for x∈[, ]. (.)
If f : [, ]×R→Ris continuous and satisfies
f(x,u)≤f(x,u) forα(x)≤u≤u≤β(x)and x∈[, ], (.) then there exists a solution y(x)for(.), (.)satisfyingα(x)≤y(x)≤β(x)for≤x≤.
Of course the natural question is what would happen if (H) is replaced with the condi-tion
(H) k< <k.
Roughly speaking, for some kind of second order boundary value problems, it is well known that the existence of a lower solutionαand an upper solutionβ, which are well or-dered, that is,α≤β, implies the existence of a solution between them (see []). However, the use of lower and upper solutions in boundary value problems of the fourth order, even for the simple boundary conditions (.), is heavily dependent on the positiveness prop-erties for the corresponding linear operators, see the counterexample in [, Remark .].
It is the purpose of this paper to establish the method of lower and upper solutions for fourth order problem (.), (.) under condition (H). To do that, we study the positive-ness properties of the solutions of the nonhomogeneous linear problems
Ly(x) = , x∈(, ),
y() = , y() =y() =y() = ,
and
Ly(x) = , x∈(, ),
y() = , y() =y() =y() = .
(.)
Since the general solution ofLy= under (H) is different from that under (H), we de-termine the sign of solution of (.) via its equivalent second order systems.
In [], Cabada et al. have extensively studied the positiveness properties of the operator
Ly=y()–My
with the homogeneous boundary value conditions (.) as well as the more general nonho-mogeneous boundary value conditions, and then applied the positiveness properties in a systematic way to obtain existence theorems in the presence of lower and upper solutions allowing the case where they are not ordered. Obviously, Cabada et al. [] only dealt with the case that
k+k= (.)
in (.) and (.).
For the related results on the existence and multiplicity of positive solutions or sign-changing solutions for fourth order problems, see Bai and Wang [], Chu and O’Regan [], Cid et al. [], Drábek and Holubová [, ], Hernandez and Manasevich [], Korman [], Liu and Li [], Ma et al. [–], Rynne [, ], Schröder [], Webb et al. [], Yang [] and Yao [] and the references therein.
The rest of the paper is arranged as follows. In Section , we show that the Green func-tion of (.) possesses the positiveness properties under the condifunc-tionk< <k<π.
Finally, in Section , we develop the method of lower and upper solutions for (.), (.) under some monotonic condition on the nonlinearityf, and give some applications of our main results.
2 Green function in the casek1< 0 <k2
LetE=C[, ] be the Banach space of continuous functions defined on [, ] with its usual normal · . Denote
k= –r, k=m
with somer> andm> . Let us consider
y(x) +m–ry(x) –rmy(x) = , x∈(, ),
y() =y() =y() =y() = .
(.)
Define a linear operatorL:D(L)→E
with the domain
D(L) :=y∈C[, ] :y() =y() =y() =y() = . Firstly, we construct the Green functionG(x,s) forLy= . Define a linear operator
Ly:=y–ry, D(L) :=
y∈C[, ] :y() =y() = . The Green function ofLy= is
G(t,s) = ⎧ ⎨ ⎩
sinh(rt)sinh(r(–s))
rsinhr , ≤t≤s≤, sinh(rs)sinh(r(–t))
rsinhr , ≤s≤t≤.
Define a linear operator
Ly:=y+my, D(L) :=
y∈C[, ] :y() =y() = . The Green function ofLy= is
G(t,s) = ⎧ ⎨ ⎩
sin(mt)sin(m(–s))
msinm , ≤t≤s≤, sin(ms)sin(m(–t))
msinm , ≤s≤t≤.
Obviously,
Ly=L◦Ly,
and the Green function ofLy= is
G(x,s) :=
G(x,t)G(t,s)dt, (x,s)∈[, ]×[, ],
which can be explicitly given by
G(x,s) =
⎧ ⎨ ⎩
m+r[
sin(mx)sin(m(s–)) msinm +
sinh(rx)sinh(r(–s))
rsinhr ], ≤x≤s≤,
m+r[sin(msm)sin(sinmm(x–))+sinh(rsr)sinh(sinhrr(–x))], ≤s≤x≤.
(.)
Theorem . Let m∈(,π)and r∈(,∞).Then G(x,s)≥, (x,s)∈[, ]×[, ].
Proof It is an immediate consequence of the facts that form∈(,π),
G(t,s)≥, (t,s)∈[, ]×[, ],
and forr∈(,∞),
3 Method of lower and upper solutions
In this section, we will establish the method of lower and upper solutions for (.), (.) in the casek< <k.
Denote
gα(x) =Lα(x)–fx,α(x), gβ(x) =Lβ(x)–fx,β(x), x∈[, ]. (.)
Then
gα(x)≤, gβ(x)≥, x∈[, ]. (.)
Now letvα(x) be the solution of
Lvα(x) = , x∈(, ),
vα() =α(), vα() =α(), vα() =α(), vα() =α().
(.)
Thenvα(x) is uniquely determined as
vα(x) =α()w(x) +α()w( –x) +α()χ(x) +α()χ( –x), (.)
wherew(x) is the unique solution of the nonhomogeneous problem
L(y) = , y() = , y() =y() =y() = , (.)
and it can be explicitly given by
w(x) = m
r+m
sinh[r( –x)]
sinhr +
r r+m
sin[m( –x)]
sinm , (.)
χ(x) is the unique solution of the nonhomogeneous problem
L(y) = , y() = , y() = , y() =y() = , (.)
and it can be explicitly given by
χ(x) = (r+m)
sinh[r( –x)]
sinhr –
(r+m)
sin[m( –x)]
sinm .
Letvβ(x) be the solution of
Lvβ(x) = , x∈(, ),
vβ() =β(), vβ() =β(), vβ() =β(), vβ() =β().
Thenvβ(x) is uniquely determined as
Lemma .
() Let <r<∞and <m<π.Thenw(x) > forx∈(, ). () Let <r<∞and <m<π.Thenχ(x) < forx∈(, ).
Proof () Sincer( –x) > and –∞<m( –x) <πforx∈(, ), it follows from (.) that
w(x) > forx∈(, ).
() Obviously, (.) is equivalent to the system
Lχ=Z, χ() = ,χ() = , (.)
LZ= , Z() = ,Z() = . (.)
It is easy to see from (.) and the factG(t,s) > for (t,s)∈(, )×(, ) that Z(x) > forx∈[, ).
Combining this with (.) and using the factG(t,s) > for (t,s)∈(, )×(, ), we deduce
thatχ(x) < in (, ).
From Lemma . and the definitions ofvαandvβ, it follows that
vα(x)≤, vβ(x)≥, x∈[, ]. (.)
Now, for a lower solutionαof (.), (.), we have the following implications:
L(α(x)) =fx,α(x)+gα(x)
⇒ α(x) =vα(x) +
G(x,s)fs,α(s)ds+
G(x,s)gα(s)ds
⇒ α(x)≤Tα(x) on [, ],
and, by a similar way, we obtainβ(x)≥Tβ(x) on [, ], whereT:C[, ]→C[, ] is the
operator defined by
Tφ(x) =
G(x,s)fs,φ(s)ds, ≤x≤, (.)
where the Green functionGis as in (.). It is easy to check that (.), (.) is equivalent to the operator equation
y=Ty. (.)
As a direct consequence of the Schauder fixed point theorem [, Theorem ], we have the following lemma.
Lemma . Let there exist a constant M such that
f(x,y)≤M
Theorem . Let k< <k<π.Suppose that for problem(.), (.)there exist a lower solutionαand an upper solutionβsuch that
α(x)≤β(x) for x∈[, ].
If f : [, ]×R→Ris continuous and satisfies
f(x,u)≤f(x,u) forα(x)≤u≤u≤β(x),and x∈[, ], (.) then there exists a solution y(x)for(.), (.)satisfying
α(x)≤y(x)≤β(x) for≤x≤. (.)
Proof Define the functionFon [, ]×Rby setting
F(x,y) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
f(x,β(x)), y>β(x),
f(x,y), α(x)≤y≤β(x),
f(x,α(x)), y<α(x).
SinceFis continuous and bounded on [, ]×R, by Lemma ., there exists a solutiony
of the problem
L(y) =F(x,y),
y() =y() =y() =y() = .
We now show that inequality (.) is true. We have
Ly(x) –β(x)=Ly(x)–Lβ(x)≤Fx,y(x)–fx,β(x)≤.
ThusL(y(x) –β(x)) =h(x)≤ forx∈[, ], that is, from Theorem . and (.)
y(x) –β(x) = –vβ(x) +
G(t,s)h(s)ds≤ forx∈[, ].
By a similar way,
Ly(x) –α(x)=Ly(x)–Lα(x)≥Fx,y(x)–fx,α(x)≥.
ThusL(y(x) –α(x)) =h(x)≥ forx∈[, ], that is, from Theorem . and (.)
y(x) –α(x) = –vα(x) +
G(t,s)h(s)ds≥ forx∈[, ].
Therefore,α(x)≤y(x)≤β(x) forx∈[, ], and accordingly,yis a solution of (.), (.).
Remark . In the case|k|>k, the assertions of Theorem . can be deduced from
Habets and Sanchez [, Theorem .].
Remark . Let us consider the problem
u()(x) – u(x) + u(x) =u+sinx, x∈(, ),
u() =u() =u() =u() = .
(.)
It is easy to verify thatf(x,u) =u+sinx,k
= – andk= , and
α(x)≡–, β(x)≡
satisfy all of the conditions in Theorem .. Therefore, (.) has a solutionusatisfying
–≤u(x)≤, x∈[, ].
Acknowledgements
This work was supported by NSFC (No. 11671322) and NSFC (No. 11361054). The authors are very grateful to the anonymous referees for their valuable suggestions.
Funding
Not applicable.
Abbreviations
Not applicable.
Availability of data and materials
Not applicable.
Competing interests
All of the authors of this article claim that together they have no competing interests.
Authors’ contributions
RM and JW completed the main study together. RM wrote the manuscript, DY checked the proofs process and verified the calculation. Moreover, all the authors read and approved the last version of the manuscript.
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Received: 30 May 2017 Accepted: 2 October 2017 References
1. Gupta, CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal.26(4), 289-304 (1988)
2. Lazer, AC, McKenna, PJ: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev.32, 537-578 (1990)
3. Schröder, J: Operator Inequalities. Mathematics in Science and Engineering, vol. 147. Academic Press, New York (1980)
4. Vrabel, R: On the lower and upper solutions method for the problem of elastic beam with hinged ends. J. Math. Anal. Appl.421(2), 1455-1468 (2015)
5. Cabada, A, Cid, J, Sanchez, L: Positivity and lower and upper solutions for fourth order boundary value problems. Nonlinear Anal.67(5), 1599-1612 (2007)
6. Bai, Z, Wang, H: On positive solutions of some nonlinear fourth order beam equations. J. Math. Anal. Appl.270, 357-368 (2002)
7. Chu, J, O’Regan, D: Positive solutions for regular and singular fourth-order boundary value problems. Commun. Appl. Anal.10, 185-199 (2006)
8. Cid, JA, Franco, D, Minhós, F: Positive fixed points and fourth-order equations. Bull. Lond. Math. Soc.41, 72-78 (2009) 9. Drábek, P, Holubová, G: Positive and negative solutions of one-dimensional beam equation. Appl. Math. Lett.51, 1-7
(2016)
11. Hernandez, GE, Manasevich, R: Existence and multiplicity of solutions of a fourth order equation. Appl. Anal.54, 237-250 (1994)
12. Korman, P: Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proc. R. Soc. Edinb., Sect. A134(1), 179-190 (2004)
13. Liu, X, Li, W: Existence and multiplicity of solutions for fourth-order boundary value problems with parameters. J. Math. Anal. Appl.327, 362-375 (2007)
14. Ma, R: Existence of positive solutions of a fourth-order boundary value problem. Appl. Math. Comput.168, 1219-1231 (2005)
15. Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. Anal.59, 225-231 (1995)
16. Ma, R: Nodal solutions for a fourth-order two-point boundary value problem. J. Math. Anal. Appl.314(1), 254-265 (2006)
17. Ma, R: Nodal solutions of boundary value problems of fourth-order ordinary differential equations. J. Math. Anal. Appl.
319(2), 424-434 (2006)
18. Ma, R, Xu, J: Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. Nonlinear Anal. TMA72(1), 113-122 (2010)
19. Rynne, BP: Infinitely many solutions of superlinear fourth order boundary value problems. Topol. Methods Nonlinear Anal.19(2), 303-312 (2002)
20. Rynne, BP: Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems. J. Differ. Equ.188, 461-472 (2003)
21. Webb, JRL, Infante, G, Franco, D: Positive solutions of nonlinear fourth-order boundary value problems with local and non-local boundary conditions. Proc. R. Soc. Edinb., Sect. A138(2), 427-446 (2008)
22. Yang, Z: Existence and uniqueness of positive solutions for higher order boundary value problem. Comput. Math. Appl.54(2), 220-228 (2007)
23. Yao, Q: Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Appl. Math. Lett.17(2), 237-243 (2004)
