R E S E A R C H
Open Access
Positive solutions of singular beam
equations with the bending term
Xuemei Zhang
1*and Meiqiang Feng
2*Correspondence: [email protected] 1Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, People’s Republic of China
Full list of author information is available at the end of the article
Abstract
Using a new technique for dealing with the bending term of beam equations, we consider the existence and multiplicity of positive solutions for a beam equation. Besides achieving new results, upper and lower bounds for these positive solutions will also be provided. The results are shown by using a novel technique and fixed point theories.
Keywords: positive solution; singularity; beam equation with the bending term; fixed point theorem
1 Introduction
In this paper, we shall investigate the existence and multiplicity of positive solutions for the fourth order differential equation with integral boundary conditions
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
y()(t) =ω(t)F(t,y(t),y(t)), <t< ,
y() =y() =h(s)y(s)ds,
ay() –by() =g(s)y(s)ds,
ay() +by() =g(s)y(s)ds,
(.)
wherea,b> ,F: [, ]×R×R→Ris continuous, and theyinFis the bending moment term which represents bending effect. Problem (.) is often referred to as the deformation of an elastic beam under a variety of boundary conditions; for details, see [–]. Most research papers on beam equations consider nonlinear terms thatF in (.) involvesy
only, and derivative-dependent nonlinearities are seldom tackled; see [–, –] to name a few.
Some classical tools such as fixed point theorems in cones [, –, , ], the method of lower and upper solutions [, ], and the monotone iterative method [, ] and the theory of critical point theory and variational methods [, , ] have been widely used to study beam equations.
Some new techniques via appropriate transformation are proved to be very effective in studying the solvability of differential equations. Such techniques have attracted the atten-tion of Zhanget al.[] and Wong [],etc.In [], Zhanget al.considered the existence
of positive solutions of the following problems:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
x(n)(t) +f(t,x(t),x(t), . . . ,x(n–)(t)) =θ, t∈J,t=t
k,k= , , . . . ,m, x(n–)|
t=tk= –Ik(x
(n–)(t
k)), k= , , . . . ,m, x(i)() =θ, i= , , . . . ,n– ,
x(n–)() =x(n–)() =
g(t)x(n–)(t)dt,
(.)
whereJ= [, ].
Firstly, by means of the transformation
x(n–)(t) =y(t),
the authors converted (.) into
x(n–)(t) =y(t), t∈J,
x(i)() =θ, i= , , . . . ,n– ,
and ⎧ ⎪ ⎨ ⎪ ⎩
–y(t) =f(t,x(t),x(t), . . . ,x(n–)(t)), t∈J,t=t
k, y|t=tk= –Ik(y(tk)), k= , , . . . ,m,
y() =y() =g(t)y(t)dt.
(.)
Then it follows from Lemma . in [] that they converted the results obtained for prob-lem (.) to the counterpart for probprob-lem (.).
In [], Wong transformed the following problems:
(–)my(m+)(t) =F(t,y(t),y(t)), <t< ,
y() = , y(k–)() =y(k–)() = , ≤k≤m, (.) into
y(t) =x(t), t∈J,
y() = ,
and
(–)mx(m)(t) =F(t,y(t),y(t)), <t< ,
x(k–)() =x(k–)() = , ≤k≤m (.)
by usingy(t) =x(t). So the existence of a solution of the complementary Lidstone bound-ary value problem (.) follows from the existence of a solution of the Lidstone boundbound-ary value problem (.). We notice that the above paper requires thatFsatisfies some assump-tions of monotonicity which are essential for the technique used.
estimates on the norms of these solutions will also be provided. Secondly, we transform problem (.) into a differential system without bending term,i.e., the technique to deal with bending term is completely different from that of [, –]. Finally, it is pointed out that we do not need any monotone assumption onF, which is weaker than the corre-sponding assumptions onFin [].
On the other hand, boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [], semiconductor problems [], hydrody-namic problems [] and so on. It is interesting to point out that such problems include two, three, multi-point and nonlocal boundary value problems as special cases and have been extensively studied in the last ten years (see, for example, [–]). Therefore, it is important to study fourth order elasticity problems with integral boundary conditions.
The rest of the paper is organized as follows. In Section , we provide some preliminaries and lemmas. In particular, we transform problem (.) into a differential system without the bending moment term. In Section , the main results will be stated and proved.
2 Preliminaries
To establish the existence of positive solutions for problem (.), let us list the following assumptions, which will hold throughout this paper:
(H) ω∈C((, ), [, +∞))with <
ω(s)ds<∞andωdoes not vanish on any
subinter-val of(, );
(H) F∈C([, ]×[, +∞)×(–∞, ], [, +∞));
(H) g,h∈L[, ]are nonnegative andμ∈[,a),ν∈[, ), where
μ=
g(s)ds, ν=
h(s)ds. (.)
Lemma .[] Assume that(H)holds.Then,for any x∈C[, ],the boundary value
problem
–y(t) =x(t), <t< ,
y() =y() =h(s)y(s)ds (.) has a unique solution y given by
y(t) =
H(t,s)x(s)ds, (.)
where
H(t,s) =G(t,s) +
–ν
G(s,τ)h(τ)dτ, (.)
G(t,s) =
t( –s), ≤t≤s≤,
s( –t), ≤s≤t≤. (.)
Taking into account (.) and (.), problem (.) reduces to the following problem:
x(t) = –ω(t)F(t,H(t,s)x(s)ds, –x(t)), <t< ,
Lemma .[] Assume that(H)-(H)hold.Then boundary value problem(.)has a
unique solution x given by
x(t) =
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds, (.)
where
H(t,s) =G(t,s) +
a–μ
G(s,τ)g(τ)dτ, (.)
G(t,s) =
d
(b+as)(b+a( –t)) if≤s≤t≤,
(b+at)(b+a( –s)) if≤t≤s≤, (.)
d=a(b+a).
If problem (.) has a solutionx∗, then by (.) problem (.) has a solution given by
y∗(t) =
H(t,s)x∗(s)ds.
So the existence of a solution of problem (.) follows from the existence of a solution of problem (.).
Lemma .[] For t,s∈J,we have the following results:
G(t,t)G(s,s)≤G(t,s)≤G(s,s)≤
, (.)
db
≤G
(t,s)≤G(s,s)≤
d(b+a)
. (.)
Lemma .[] Let(H)hold.Then we obtain the following results:
ρG(s,s)≤H(t,s)≤γG(s,s)≤
γ, ∀t,s∈J, (.)
ρ≤H(t,s)≤H(s,s)≤ρ, ∀t,s∈J, (.)
where
γ =
–ν, ρ=
G(τ,τ)h(τ)dτ
–ν , ρ= bγ
a+ b,
ρ=
γ(b+a)
a+ b , γ=
a–μ.
It is clear from (.) thaty∗ ≤γx∗; moreover, ifx∗is positive, so isy∗.
LetE=C[, ]. It is well known thatEis a real Banach space with the norm · defined byx=maxt∈J|x(t)|.
Define an operatorT:E→Eas follows:
(Tx)(t) =
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
LetCbe a cone inEwhich is defined as
C=x∈E:x≥,x(t)≥δx,t∈J,
where
δ=ρ
ρ
. (.)
It is easy to see thatEis a closed convex cone ofE.
Lemma . Let(H)-(H)hold.Then we have T(C)⊂C,and T:C→C is completely
continuous.
Proof In view of condition (.), we see that
Tx=max
t∈J
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≤ρ
ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds.
Moreover, it follows from (.) and (.) that
(Tx)(t) =
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≥ρ
ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≥ ρ
ρ
ρ
ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≥δTx, t∈J. (.)
This proves thatT(C)⊂C.
Next, by standard methods and the Ascoli-Arzelà theorem, one can prove thatT:C→
Cis completely continuous.
To obtain positive solutions of problem (.), the following fixed point theorem in cones, which can be found in [], p., is fundamental.
Lemma . Let P be a cone in a real Banach space E.Assume,are bounded open
sets in E with∈,¯⊂.If
A:P∩(¯\)→P
is completely continuous such that either
(a) Ax ≤ x,∀x∈P∩∂andAx ≥ x,∀x∈P∩∂,or
(b) Ax ≥ x,∀x∈P∩∂andAx ≤ x,∀x∈P∩∂,
3 Main results
In this section, we apply Lemma . to establish the existence of positive solutions for problem (.). We begin by introducing the following conditions onF(t,u,v):
(H) There exist two positive constantsr,Rwith(γ + )r< (σ+δ)Rsuch that:
F(t,u,v)≤
ρη
r, ∀t∈J,|u|+|v| ≤ γ
+
r, (.)
F(t,u,v)≥
ρηδ
R, ∀t∈J,|u|+|v| ≥(σ+δ)R, (.)
where
η=
ω(s)ds, σ=ρδβ, β=
s( –s)ds= .
Theorem . Assume that(H)-(H)hold.Then we have the following conclusions:
(i) Problem(.)has(at least)one positive solutionx∈Csuch that
δr≤x(t)≤
δR, t∈J. (.)
(ii) Problem(.)has(at least)one positive solutionysuch that
⎧ ⎪ ⎨ ⎪ ⎩
y(t) =H(t,s)x(s)ds, t∈J;
y ≤γx;
y(t)≥σx, t∈J.
(.)
We further have
σ δr≤y(t)≤ γ
δR, t∈J. (.)
Proof LetTbe the cone preserving, completely continuous operator that was defined by (.).
Letx∈Cwithx=r. Then ≤x(t)≤r,t∈J, and ≤H(s,τ)x(τ)dτ≤ γr. And
hence, forx∈Cwithx=r, we have
H(s,τ)x(τ)dτ
+–x(t)≤ γ +
r.
Then it follows from (.) that
Tx=max
t∈J
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≤ρη
ρη
r=r. (.)
Now if we let={x∈C:x<r}, then (.) shows that
Further, let
R=
δR, (.)
and
=
x∈C:x<R
.
Thenx∈Candx=Rimply
x(t)≥δx(s), t,s∈J,
that is,
x(t)≥δR=R, t∈J.
Hence,x(t)≥Rfor allt∈J, and
H(s,τ)x(τ)dτ≥ρ
G(τ,τ)x(τ)dτ≥ρβδx=σx.
So
H(s,τ)x(τ)dτ
+–x(t)≥(σ+δ)R.
Using condition (.), it follows fromx∈Candx=Rthat
Tx=max
t∈J
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds≥ρη
ρηδ
R=R
δ =R,
that is,x∈∂implies
Tx ≥ x. (.)
It now follows from Lemma . that problem (.) has (at least) one positive solution
x∈ ¯\satisfying (.).
It is observed from (.) that problem (.) has (at least) one positive solutionysuch that
y=
H(t,s)x(s)ds, t∈Jandy ≤
γ
x.
Moreover, sincex∈C, we get fort∈J
y(t) =
H(t,s)x(s)ds
≥ρ
≥ρβδx
=σx.
Then we get (.).
Further, it follows from (.) and (.) that (.) holds.
In Theorem . we assume the following condition onf(t,u,v):
(H) There exist two positive constantsr,Rwith(γ+ )r<Rsuch that:
F(t,u,v)≥
ρη(σ+δ)
|u|+|v|, ∀t∈J,|u|+|v| ≤ γ
+
r, (.)
F(t,u,v)≤ ρη(γ+ )
|u|+|v|, ∀t∈J,|u|+|v| ≥R, (.)
and write
M= max
∀t∈J,|u|+|v|≤RF(t,u,v). (.)
Theorem . Assume that(H)-(H)and(H)hold.Then we have the following
conclu-sions:
(i) Problem(.)has(at least)one positive solutionx∈Csuch that
δr≤x(t)≤max{R, ρηM}, t∈J. (.)
(ii) Problem(.)has(at least)one positive solutionysuch that(.)holds.We further have
σ δr≤y(t)≤max
γR
(σ+δ),
ρηMγ
, t∈J. (.)
Proof Letx∈Cwithx=r. Then ≤x(t)≤r,t∈J, and ≤H(s,τ)x(τ)dτ ≤ γr.
And hence forx∈Cwithx=r, we have
(σ+δ)r≤
H(s,τ)x(τ)dτ
+–x(t)≤ γ +
r,
and it follows from condition (H) that
Tx=max
t∈J
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≥ρη
ρη(σ+δ)
(σ+δ)r=r, (.)
that is,x∈∂implies that
Next, we turn to (.) and (.). From (.) and (.), we have
F(t,u,v)≤M+ ρη(γ+ )
|u|+|v|, (t,u,v)∈J×[,∞)×(–∞, ]. (.)
Further, let
R>max
ρηM,
R σ+δ
, (.)
and
=
x∈C:x<R
.
Notice that forx∈∂we have
R< (σ+δ)R≤
H(s,τ)x(τ)dτ
+–x(t)≤ γ +
R.
Thus, forx∈∂, it follows from (.) that
Tx=max
t∈J
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
≤ρ
ω(s) M+ ρη
R
ds
≤ρη M+
ρη
R
<R,
that is,x∈∂implies
Tx<x. (.)
It now follows from Lemma . that problem (.) has (at least) one positive solution
x∈ ¯\satisfying (.).
It follows from (.) that problem (.) has (at least) one positive solutiony. Similar to the proof of (.), one can show thatysatisfies (.).
Theorem . Assume that(H)-(H), (.)of(H)and(.)of(H)hold.In addition,
letting f satisfies the following condition:
(H) Letl,ζ andLsatisfy
<l< γ +
l<ζ< γ +
ζ<δL<L.
If
max
∀t∈J,ζ≤|u|+|v|≤(γ+)ζ
F(t,u,v) <
ρη
ζ,
(i) Problem(.)has(at least)two positive solutionsx,x∈Csuch that
δl≤x(t) <ζ<
γ
+
ζ<x(t)≤L, t∈J. (.)
(ii) Problem(.)has(at least)two positive solutionsy,ysuch that fori= , ,
⎧ ⎪ ⎨ ⎪ ⎩
yi(t) =
H(t,s)xi(s)ds, t∈J;
yi ≤ γxi; yi(t)≥σxi, t∈J.
(.)
We further have
y(t) >σ δl, t∈J;
y ≤γL.
(.)
Proof If (.) of (H) holds, similar to the proof of (.), we can prove that
Tx ≥ x, x∈C,x=l. (.)
If (.) of (H) holds, similar to the proof of (.), we have
Tx ≥ x, x∈C,x=L. (.)
Finally, we show that
Tx<x, x∈C,x=ζ. (.)
In fact, forx∈Cwithx=ζ, we have
x(t)≤ x=ζ,
and
σ ζ≤
H(s,τ)x(τ)dτ≤
γ
ζ.
Therefore,
ζ < (σ+ )ζ ≤
H(s,τ)x(τ)dτ
+–x(s)≤ γ +
ζ,
and hence it follows from (H) that
Tx=max
t∈J
H(t,s)ω(s)F s,
H(s,τ)x(τ)dτ, –x(s)
ds
<ρη
ρη
ζ=ζ,
Applying Lemma . to (.), (.) and (.) yields that problem (.) has (at least) two positive solutions x, x with x ∈C¯l,ζ ={x∈C,l≤ x<ζ}, x ∈Cζ,L¯ ={x∈C,
(γ+ )ζ <x ≤L}. Hence, since forx∈Cwe havex(t)≥δx,t∈J, it follows that
(.) holds.
Similar to the proof of (.) and (.), one can show that (.) and (.) hold.
Remark . In Theorems .-., we generalize the results of [, –] in three main directions as follows:
() Upper and lower bounds for these positive solutions are given. () Estimates on the norms of positive solutions are considered.
() The method to deal with the bending term of beam equations in this paper is completely different from that of [, –], which opens a new technique to study the beam equations with the bending term.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Author details
1Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, People’s Republic of China.2School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, People’s Republic of China.
Acknowledgements
This work is sponsored by the project NSFC (11301178), the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201511232018). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.
Received: 25 November 2014 Accepted: 12 May 2015
References
1. Gupta, CP: Existence and uniqueness theorems for a bending of an elastic beam equation. Appl. Anal.26, 289-304 (1988)
2. Agarwal, RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986) 3. Liu, LS, Zhang, XG, Wu, YH: Positive solutions of fourth-order nonlinear singular Sturm-Liouville eigenvalue problems.
J. Math. Anal. Appl.326, 1212-1224 (2007)
4. Graef, JR, Qian, C, Yang, B: A three point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl.287, 217-233 (2003)
5. Anderson, DR, Avery, RI: A fourth-order four-point right focal boundary value problem. Rocky Mt. J. Math.36(2), 367-380 (2006)
6. Ma, TF: Positive solutions for a beam equation on a nonlinear elastic foundation. Math. Comput. Model.39, 1195-1201 (2004)
7. Yao, QL: Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right. Nonlinear Anal.69, 1570-1580 (2008)
8. Bai, ZB: The method of lower and upper solutions for a bending of an elastic beam equation. J. Math. Anal. Appl.248, 95-202 (2000)
9. Zhang, XG, Liu, LS: Positive solutions of fourth-order four-point boundary value problems withp-Laplacian operator. J. Math. Anal. Appl.336, 1414-1423 (2007)
10. Zhang, XM, Feng, MQ, Ge, WG: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal.69, 3310-3321 (2008)
11. Zhang, XM, Ge, WG: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput.219, 3553-3564 (2012)
12. Sun, J, Wang, X: Monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. Math. Probl. Eng. (2011). doi:10.1155/2011/609189
13. Yang, L, Chen, H, Yang, X: The multiplicity of solutions for fourth-order equations generated from a boundary condition. Appl. Math. Lett.24, 1599-1603 (2011)
14. Cabada, A, Tersian, S: Multiplicity of solutions of a two point boundary value problem for a fourth-order equation. Appl. Math. Comput.219, 5261-5267 (2013)
16. Ma, R, Zhang, J, Fu, S: The method of lower and upper solutions for fourth-order two-point boundary value problems. J. Math. Anal. Appl.215, 415-422 (1997)
17. Li, Y: On the existence of positive solutions for the bending elastic beam equations. Appl. Math. Comput.189, 821-827 (2007)
18. Zhang, XM, Ge, WG: Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput. Math. Appl.58, 203-215 (2009)
19. Zhang, XM, Yang, XZ, Ge, WG: Positive solutions ofnth-order impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal.71, 5930-5945 (2009)
20. Wong, PJY: Triple solutions of complementary Lidstone boundary value problems via fixed point theorems. Bound. Value Probl.2014, 125 (2014)
21. Cannon, JR: The solution of the heat equation subject to the specification of energy. Q. Appl. Math.21, 155-160 (1963)
22. Ionkin, NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ.13, 294-304 (1977)
23. Chegis, RY: Numerical solution of a heat conduction problem with an integral boundary condition. Liet. Mat. Rink.24, 209-215 (1984)
24. Boucherif, A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal.70, 364-371 (2009)
25. Wang, YQ, Liu, LS, Wu, YH: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal.74, 3599-3605 (2011)
26. Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal., Real World Appl.9, 1727-1740 (2008)
27. Webb, JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal.47, 4319-4332 (2001)
28. Jiang, JQ, Liu, LS, Wu, YH: Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. Appl. Math. Comput.215, 1573-1582 (2009)
29. Liu, LS, Hao, XA, Wu, YH: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model.57, 836-847 (2013)
30. Liu, LS, Liu, BM, Wu, YH: Nontrivial solutions for higher-orderm-point boundary value problem with a sign-changing nonlinear term. Appl. Math. Comput.217, 3792-3800 (2010)