Volume 2011, Article ID 376782,14pages doi:10.1155/2011/376782
Research Article
A Note on a Beam Equation with Nonlinear
Boundary Conditions
Paolamaria Pietramala
Dipartimento di Matematica, Universit`a della Calabria, Arcavacata di Rende 87036, Cosenza, Italy
Correspondence should be addressed to Paolamaria Pietramala,[email protected]
Received 14 May 2010; Revised 12 July 2010; Accepted 31 July 2010
Academic Editor: Feliz Manuel Minh ´os
Copyrightq2011 Paolamaria Pietramala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present new results on the existence of multiple positive solutions of a fourth-order differential equation subject to nonlocal and nonlinear boundary conditions that models a particular stationary state of an elastic beam with nonlinear controllers. Our results are based on classical fixed point index theory. We improve and complement previous results in the literature. This is illustrated in some examples.
1. Introduction
The fourth-order differential equation
u4t gtft, ut, t∈0,1, 1.1
arises naturally in the study of the displacementuutof an elastic beam when we suppose that, along its length, a load is added to cause deformations. This classical problem has been widely studied under a variety of boundary conditionsBCsthat describe the controls at the ends of the beam. In particular, Gupta1studied, along other sets of local homogeneous BCs, the problem
u0 0, u0 0, u1 0, u1 0 1.2
Multipoint versions of this problem do have a physical interpretation. For example, the four-point boundary conditions
u0 0, u0 0, u1 H1uτ, u1 H2uξ 0 1.3
model a bar where the displacementuand the bending momentuatt0 are zero, and there are relations, not necessarily linear, between the shearing forceuand the angular attitudeu
att1 and the displacementuin two other points of the beam.
In this paper we establish new results on the existence of positive solutions for the fourth-order differential equation1.1subject to the following nonlocal nonlinear boundary conditions:
u0 0, u0 0, 1.4
u1 H1α1u, 1.5
u1 H2α2u 0, 1.6
where H1, H2 are nonnegative continuous functions and α1·, α2· are linear functionals
given by
α1u
1
0
usdA1s, α2u
1
0
usdA2s, 1.7
involving Riemann-Stieltjes integrals.
The conditions1.5-1.6cover a variety of cases and include, as special cases when
H1w H2w w, multipoint and integral boundary conditions. These are widely studied
objects in the case of fourth-order BVPs; see, for example,5–14. BCs of nonlinear type also have been studied before in the case of fourth-order equations; see, for example,15–20and references therein.
The study of positive solutions of BVPs that involve Stieltjes integrals has been done, in the case ofpositivemeasures, in21–24.Signedmeasures were used in12,25; here, as in
21,22, due to some inequalities involved in our theory, the functionalsαi·are assumed to
be given by positive measures.
A standard methodology to solve1.1subject to local BCs is to find the corresponding Green’s functionkand to rewrite the BVP as a Hammerstein integral equation of the form
ut
1
0
kt, sgsfs, usds. 1.8
Infante12, where a unified method is given to study a large class of problems. This is done via an auxiliary perturbed Hammerstein integral equation of the form
ut γ1tα1u γ2tα2u
1
0
kt, sgtfs, usds, 1.9
with suitable functionsγ1, γ2.
Infante studied in26,27the case of one nonlinear BC and in21a thermostat model with two nonlinear controllers. The approach used in21relied on an extension of the results of25, valid for equations of the type
ut γ1tH1α1u γ2tH2α2u
1
0
kt, sgsfs, usds, 1.10
and gives a simple general method to avoid long technical calculations.
In our paper the approach of21is applied to BVP1.1–1.6: we rewrite this BVP as a perturbed Hammerstein integral equation, and we prove the existence of multiple positive solutions under a suitable oscillatory behavior of the nonlinearity f. We observe that our results are new even for the local BCs, whenH1α1u H2α2u 0. We illustrate our
theory with some examples. We also point out that this approach may be utilized for other sets of nonlinear BCs that have a physical interpretation, this is done in the last section.
2. The Boundary Value Problem
We begin by considering the homogeneous BVP
u4t gtft, ut, u0 u0 u1 u1 0, t∈0,1, 2.1
of which we seek an equivalent integral formulation of the form
ut
1
0
kt, sgsfs, usds. 2.2
Due to the nature of these particular BCs, the Green’s functionk can be constructed
as in4by means of an auxiliary second-order BVP, namely,
ut gtft, ut 0, u0 0, u1 0, t∈0,1. 2.3
The solutions of the BVP2.3can be written in the form
ut
1
0
where
Gt, s
⎧ ⎨ ⎩
s, s≤t,
t, s > t. 2.5
Therefore the functionkin2.2is given by
kt, s
1
0
Gt, vGv, sdv. 2.6
In order to use the approach of 21, 25, 28, we need to use some monotonicity properties ofk. Now, since
Gt, vGv, s v2χ0,tvχ0,sv vsχ0,tvχs,1v
tvχt,1vχ0,sv tsχt,1vχs,1v,
2.7
we obtain the following formulation for the Green’s function:
kt, s
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
s
6
−s2−3t26t, s≤t,
t
6
−t2−3s26s, s > t.
2.8
We now look for a suitable intervala, b⊂0,1, a functionΦs, and a constantcΦ>0 such
that
kt, s≤Φs, for everyt, s∈0,1×0,1,
kt, s≥cΦΦs, for everyt, s∈a, b×0,1.
2.9
Sincekis continuous on0,1×0,1andkt, s>0 fort, s∈0,1×0,1, a natural choice could be
Φs max
t,s∈0,1×0,1kt, s, cΦt,s∈mina,b×0,1
kt, s
Φs ; 2.10
here we look for a betterΦ, since this enables us to weaken the growth requirements on the nonlinearityf.
An upper bound fork is obtained by finding maxt∈0,1kt, sfor each fixed s. Since
∂/∂tkt, s ≥ 0 fort, s ∈ 0,1×0,1,k is a nondecreasing function oftthat attains its maximum, for each fixeds, whent1.
Therefore, fort, s∈0,1×0,1, we have
kt, s≤k1, s s
6 −s
Now, one can see that the derivative of the functionkt, s/Φswith respect tosis non-positive for alls∈0,1, that is, the functionkt, s/Φsis a non-increasing function of
s. Therefore, for an arbitrarya, b⊂0,1, we have
kt, s≥cΦΦs, for everyt, s∈a, b×0,1, 2.12
where
cΦ:min
a≤t≤b
kt,1
Φ1
ka,1
Φ1
1
2a 3−a
2. 2.13
We now turn our attention to the BVP1.1–1.6
u4t gtft, ut, u0 u0 u1−H
1α1u u1 H2α2u 0, 2.14
and we show that we can study this problem by means of a perturbation of the Hammerstein integral equation2.2.
In order to do this, we look for theuniquesolutions of the linear problems
γ14t 0, γ10 γ10 0, γ11 1, γ11 0,
γ24t 0, γ20 γ20 γ21 0, γ21 10
2.15
that are
γ1t t, γ2t −1
6t
3 1
2t. 2.16
We observe that, for an arbitrarya, b⊂0,1, we have
γ11, min
t∈a,bγ1t γ1a a,
γ2
1
3, tmin∈a,bγ2t γ2a −
1 6a
3 1
2a,
2.17
whereu:sup{|ut|:t∈0,1}, and therefore
γ1t≥cγ1γ1, γ2t≥cγ2γ2, for everyt∈a, b, 2.18
with
cγ1:
mint∈a,bγ1t
γ1 a, cγ2:
mint∈a,bγ2t
γ2 −a3
2 3a
By a solution of the BVP 1.1–1.6 we mean a solution of the perturbed integral equation
ut γ1tH1α1u γ2tH2α2u
1
0
kt, sgtfs, usds:Tut, 2.20
and we work in a suitable cone in the Banach spaceC0,1of continuous functions defined on the interval0,1endowed with the usual supremum norm.
Our assumptions are the following:
C1f : 0,1×0,∞ → 0,∞satisfies Carath´eodory conditions, that is, for eachu,
t → ft, uis measurable, for almost every t,u → ft, u is continuous, and for everyr >0, there exists anL∞-functionφr :0,1 → 0,∞such that
ft, u≤φrt for almost allt∈0,1and allu∈0, r; 2.21
C2g Φ∈L10,1,g≥0 almost everywhere, and
1
0Φsgsds >0;
C3H1, H2 are positive continuous functions such that there exist h11, h12, h21, h22 ∈ 0,∞with
h11v≤H1v≤h12v, h21v≤H2v≤h22v, 2.22
for everyv≥0;
C4α1·, α2·are positive bounded linear functionals onC0,1given by
αiu 1
0
usdAis, i1,2, 2.23
involving Stieltjes integrals withpositivemeasuresdAi;
C5D2 : 1−h12α1γ11 −h22α2γ2−h12h22α1γ2α2γ1 > 0, h12α1γ1 < 1 and
h22α2γ2<1.
It follows from this last hypothesis that
D1:
1−h11α1
γ1
1−h21α2
γ2
−h11h21α1
γ2
α2
γ1
The above hypotheses enable us to utilize the cone
K
u∈C0,1:u≥0, min
t∈a,but≥cu
, 2.25
for an arbitrarya, b⊂0,1and
c:mincΦ, cγ1, cγ2
a, 2.26
and to use the classical fixed point index for compact mapssee e.g.,29or30.
We observe, as in21, thatT leavesKinvariant and is compact. We give the proof in the Carath´eodory case for completeness.
Lemma 2.1. If the hypothesesC1–C4hold, thenTmapsKintoK. Moreover,Tis a compact map.
Proof. Takeu∈Ksuch thatu ≤r. Then we have, fort∈0,1,
Tut γ1tH1α1u γ2tH2α2u
1
0
kt, sgsfs, usds
≤γ1tH1α1u γ2tH2α2u
1
0Φ
sgsφrsds,
2.27
therefore
Tu ≤γ1H1α1u γ2H2α2u
1
0
Φsgsφrsds. 2.28
Then we obtain
min
t∈a,bTut≥cγ1γ1H1α1u cγ2γ2H2α2u cΦ
1
0Φ
sgsφrs≥cTu. 2.29
Hence we haveTu ∈ K. Moreover, the mapT is compact since it is a sum of two compact maps: the compactness of01kt, sgsfs, usdsis well known, and sinceγ1, γ2, H1, and
H2are continuous, the perturbationγ1tH1α1u γ2tH2α2umaps bounded sets into
bounded subsets of a 1-dimensional space.
Forρ >0, we use, as in23,31, the following bounded open subsets ofK:
Kρ
u∈K:u< ρ, Vρ
u∈K: min
t∈a,but< ρ
. 2.30
We employ the following numbers:
f0,ρ : sup
0≤u≤ρ, 0≤t≤1
ft, u
ρ , fρ,ρ/c :ρ≤u≤ρ/c, ainf ≤t≤b
ft, u ρ ,
1
m :tsup∈0,1
1
0
kt, sgsds, 1
Ma, b :t∈infa,b
b
a
kt, sgsds,
2.31
and we note
Kis: 1
0
kt, sdAit, i1,2. 2.32
The proofs of the following results can be immediately deduced from the analogous results in21, where the proofs involve a careful analysis of fixed point index and utilize order-preserving matrices. The only difference here is that we allow nonlinearity f to be Carath´eodory instead of continuous. The lines of proof are not effected and therefore the proofs are omitted.
Firstly we give conditions which imply that the fixed point index is 0 on the setVρ.
Lemma 2.2. Suppose thatC1–C5hold. Assume that there existρ >0such that
fρ,ρ/c
cγ1γ1
D1
1−h21α2
γ2
cγ2γ2
D1
h11α2
γ1
1
0
K1sgsds
cγ1γ1
D1
h21α1
γ2
cγ2γ2
D1
1−h11α1
γ1
1
0
K2sgsds 1 Ma, b
>1.
2.33
Then the fixed point index,iKT, Vρ, is0.
Now, we give conditions which imply that the fixed point index is 1 on the setKρ.
Lemma 2.3. SupposeC1–C5hold. Assume that there existsρ >0such that
f0,ργ1
D2
1−h22α2
γ2
γ2
D2
h12α2
γ1
1
0
K1sgsds
γ1
D2
h22α1
γ2 γ2 D2
1−h12α1
γ1
1
0
K2sgsds 1 m
<1.
2.34
TheniKT, Kρ 1.
Theorem 2.4. SupposeC1–C5hold. Leta, b⊂0,1andcbe as in2.26. Then2.20has one
positive solution inKif either of the following conditions holds:
S1there existρ1, ρ2 ∈ 0,∞withρ1 < ρ2such that 2.34is satisfied forρ1 and2.33is
satisfied forρ2;
S2there existρ1, ρ2 ∈0,∞withρ1 < cρ2such that2.33is satisfied forρ1and2.34is
satisfied forρ2.
Equation2.20has at least two positive solutions inKif one of the following conditions hold.
D1there exist ρ1, ρ2, ρ3 ∈ 0,∞, withρ1 < ρ2 < cρ3, such that 2.34is satisfied forρ1, 2.33is satisfied forρ2, and2.34is satisfied forρ3;
D2there existρ1, ρ2, ρ3 ∈0,∞, withρ1 < cρ2andρ2 < ρ3, such that2.33is satisfied for
ρ1,2.34is satisfied forρ2, and2.33is satisfied forρ3.
3. Optimal Constants and Examples
Consider the differential equation
u4t ft, ut, t∈0,1, 3.1
with the BCs1.4–1.6.
The value 1/mis given by direct calculation as follows:
1
m tsup∈0,1
1
0
kt, sds max
t∈0,1
t
24 t
3−4t28 5
24. 3.2
We seek the “optimal”a, bfor whichMa, bis a minimum. This type of problems has been tackled in the past, for example, in the second-order case for heat-flow problems in32and for beam equationsunder different BCsin9,12,13.
The kernelkis a positive, nondecreasing function oft, thus
1
Ma, b tmin∈a,b b
a
kt, sds
b
a
ka, sds
b
a
a
6 −a
2−3s26sds. 3.3
Sincekis a nondecreasing function oft, we have
max
0<a<b≤1
1
Ma, b
max
0<a≤1
1
a
a
6 −a
2−3s26sdsmax 0<a≤1
a
6 2a
3−4a22. 3.4
Such maximum is attained ata 1/2. Thus the “optimal” intervala, b, for whichMa, b
is a minimum, is the interval1/2,1; this givesM1/2,1 48/5 andc1/2.
Remark 3.1. From Theorem 2.4, it is possible to state results for the existence of several nonnegative solutions for the homogeneous BVP
For example, witha, b 1/2,1 andc 11/16, the BVP 3.5has at least two positive solutions inKif there existρ1< ρ2< cρ3, such thatf0,ρ1 <4.8,fρ2,ρ2/c>9.6 andf
0,ρ3 <4.8.
These results are new and improve and complement the previous ones. Gupta1and Yao4studied the problem with more general nonlinearity but established existence results only. Davies and co-authors2and Graef and Henderson3obtain the existence of multiple positive solutions for a 2nth-order differential equation subject to our boundary conditions in the casen2. In2the choicea, b 1/4,3/4gives the valuesη4 andμ164which
replace our constantsmandM1/2,1in the growth conditions off. The growth conditions of the nonlinearityf fuin Theorem 3.1 in3cannot be compared with ours, but we do not require the restrictionf0/0.
The next examples illustrate the applicability of our results. Firstly we consider, as an illustrative example, the case of a nonlinear 4-point problem.
Example 3.2. Consider the differential equation
u4t ft, ut, t∈0,1, 3.6
with the BCs
u0 0, u0 0, u1 H1uτ, u1 H2uξ 0, 3.7
whereτ, ξ∈0,1and, as in22, fori1,2
Hiw ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
1
4iw, 0≤w≤1,
1 8iw
1
8i, w≥1.
3.8
In this case we haveh11 1/8,h121/4,h211/16,h221/8,
α1
γ1
τ, α1
γ2
−1
6 τ
3−3τ, α 2
γ1
ξ, α2
γ2
−1
6 ξ
3−3ξ,
1
0
K1sds
1
0
kτ, sds τ
24 τ
3−4τ28,
1
0
K2sds
1
0
kξ, sds ξ
24 ξ
3−4ξ28.
We now fixτ 1/4,ξ 1/2 and show that all the constants that appear in2.33and in2.34can be computed. This choice leads to
α1
γ1
1
4, α1
γ2
47
384, α2
γ1
1
2, α2
γ2
11
48, D2 45373
49152. 3.10
Moreover we have
1
0
K1sds 497
6144,
1
0
K2sds 19
128, 3.11
and conditions2.34and2.33read f0,ρ1 < 2.8587 andf
ρ2,ρ2/c > 6.9470. SinceS1holds,
fromTheorem 2.4it follows that this BVP has a nontrivial solution inK. A nonlinearity that easily verifiesS1, for example, is the functionft, u u3 for every 0 < ρ1 <
√
2.8587 and everyρ2>
√
6.9470.
We now give an example with continuously distributed positive measures.
Example 3.3. Consider the differential equation
u4t ft, ut, t∈0,1, 3.12
with the BCs
u0 0, u0 0, u1 H1
1
0
α1usds
, u1 H2
1
0
α2usds
0,
3.13
withα1, α2>0 andHi,i1,2, as in the previous example. In this case, we obtain
α1
γ1
α1
2 , α1
γ2
5α1
24, α2
γ1
α2
2 , α2
γ2
5α2
24 , 3.14
and fori1,2
Kis 1
0
kt, sdAit 1
0
αikt, sdtαi
1 24s
4−1
6s
3 1
3s
,
1
0
Kisds
2αi
15.
The ConditionC5becomes
α1<8, α2<
192
5 , 1− 5 192α2−
1
8α1>0. 3.16
We now fixα14, α28. This choice leads to
α1
γ1
2, α1
γ2
5
6, α2
γ1
4, α2
γ2
5
3, D2 7 24,
1
0
K1sds 8
15,
1
0
K2sds 16
15
3.17
and conditions2.34and2.33readf0,ρ1<0.30713 andf
ρ2,ρ2/c >1.1166.
4. Other Nonlinear BCs
So far we have discussed in detail the case of the BCs1.4–1.6, but the same approach may be applied to1.1subject to the nonlinear BCs
u0 H1α1u, u0 H2α2u 0, u1 0, u1 0, 4.1
or
u0 H1α1u, u0 0, u1 H2α2u, u1 0, 4.2
or
u0 H1α1u, u0 0, u1 0, u1 H2α2u 0, 4.3
or
u0 0, u0 H1α1u 0, u1 H2α2u, u1 0, 4.4
or
u0 0, u0 H1α1u 0, u1 0, u1 H2α2u 0. 4.5
Table1
BCs1.4–1.6 BCs4.1 BCs4.2 BCs4.3 BCs4.4 BCs4.5
γ1t t 1 1 1 −t22t −t22t
γ2t −t 3
6 t
2 −
t2
2 t t −
t3 6
t
2 t −
t3 6
t 2
cγ1 a 1 1 1 2a−a
2 2a−a2
cγ2 −
a3 2
3a
2 2a−a
2 a −a3
2 3a
2 a −
a3 2
3a 2
Table 1 illustrates how the choice of the BCs affects the functions γ1, γ2 and the
constantscγ1, cγ2.
Since one can see that
1>2a−a2>−a3
2 3a
2 > a, a∈0,1, 4.6
the coneK, given by the constantc, varies according to the nonhomogeneous BCs considered. This affects also, in a natural manner, conditions2.33and2.34.
Acknowledgments
The author would like to thank professor Salvatore Lopez of the Faculty of Engineering, University of Calabria, for shedding some light on the physical interpretation of this problem. The author wishes to thank the anonymous referees for their constructive comments.
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