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A Note on a Beam Equation with Nonlinear Boundary Conditions
Boundary Value Problems volume 2011, Article number: 376782 (2011)
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.
The fourth-order differential equation
arises naturally in the study of the displacement of 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 conditions (BCs) that describe the controls at the ends of the beam. In particular, Gupta  studied, along other sets of local homogeneous BCs, the problem
that models a bar with the left end being simply supported (hinged) and the right end being sliding clamped. This problem, and its generalizations, has been studied previously by Davies and coauthors , Graef and Henderson  and Yao .
Multipoint versions of this problem do have a physical interpretation. For example, the four-point boundary conditions
model a bar where the displacement and the bending moment at are zero, and there are relations, not necessarily linear, between the shearing force and the angular attitude at and the displacement in two other points of the beam.
In this paper we establish new results on the existence of positive solutions for the fourth-order differential equation (1.1) subject to the following nonlocal nonlinear boundary conditions:
where are nonnegative continuous functions and are linear functionals given by
involving Riemann-Stieltjes integrals.
The conditions (1.5)-(1.6) cover a variety of cases and include, as special cases when , 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, [15–20] and references therein.
The study of positive solutions of BVPs that involve Stieltjes integrals has been done, in the case of positive measures, in [21–24]. Signed measures were used in [12, 25]; here, as in [21, 22], due to some inequalities involved in our theory, the functionals are assumed to be given by positive measures.
A standard methodology to solve (1.1) subject to local BCs is to find the corresponding Green's function and to rewrite the BVP as a Hammerstein integral equation of the form
However, for nonlocal and nonlinear BCs the form of Green's functions can become very complicated. In the case of linear, nonlocal BCs, an elegant approach is due to Webb and Infante , 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
with suitable functions .
Infante studied in [26, 27] the case of one nonlinear BC and in  a thermostat model with two nonlinear controllers. The approach used in  relied on an extension of the results of , valid for equations of the type
and gives a simple general method to avoid long technical calculations.
In our paper the approach of  is applied to BVP (1.1)–(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 . We observe that our results are new even for the local BCs, when . 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
of which we seek an equivalent integral formulation of the form
Due to the nature of these particular BCs, the Green's function can be constructed (as in ) by means of an auxiliary second-order BVP, namely,
The solutions of the BVP (2.3) can be written in the form
Therefore the function in (2.2) is given by
we obtain the following formulation for the Green's function:
We now look for a suitable interval , a function , and a constant such that
Since is continuous on and for , a natural choice could be
here we look for a better , since this enables us to weaken the growth requirements on the nonlinearity .
An upper bound for is obtained by finding for each fixed . Since for , is a nondecreasing function of that attains its maximum, for each fixed , when .
Therefore, for , we have
Now, one can see that the derivative of the function with respect to is non-positive for all , that is, the function is a non-increasing function of . Therefore, for an arbitrary , we have
We now turn our attention to the BVP (1.1)–(1.6)
and we show that we can study this problem by means of a perturbation of the Hammerstein integral equation (2.2).
In order to do this, we look for the (unique) solutions of the linear problems
We observe that, for an arbitrary , we have
where , and therefore
By a solution of the BVP (1.1)–(1.6) we mean a solution of the perturbed integral equation
and we work in a suitable cone in the Banach space of continuous functions defined on the interval endowed with the usual supremum norm.
Our assumptions are the following:
satisfies Carathéodory conditions, that is, for each , is measurable, for almost every , is continuous, and for every , there exists an -function such that
, almost everywhere, and ;
are positive continuous functions such that there exist with
for every ;
are positive bounded linear functionals on given by
involving Stieltjes integrals with positive measures ;
, and .
It follows from this last hypothesis that
The above hypotheses enable us to utilize the cone
for an arbitrary and
We observe, as in , that leaves invariant and is compact. We give the proof in the Carathéodory case for completeness.
If the hypotheses hold, then maps into . Moreover, is a compact map.
Take such that . Then we have, for ,
Then we obtain
Hence we have . Moreover, the map is compact since it is a sum of two compact maps: the compactness of is well known, and since , and are continuous, the perturbation maps bounded sets into bounded subsets of a 1-dimensional space.
Note that .
We employ the following numbers:
and we note
The proofs of the following results can be immediately deduced from the analogous results in , 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 to be Carathéodory 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 on the set .
Suppose that hold. Assume that there exist such that
Then the fixed point index, , is .
Now, we give conditions which imply that the fixed point index is on the set .
Suppose hold. Assume that there exists such that
The two lemmas above lead to the following result on existence of one or two positive solutions for the integral equation (2.20). Note that, if the nonlinearity has a suitable oscillatory behavior, it is possible to state, with the same arguments as in , a theorem on the existence of more than two positive solutions.
Suppose hold. Let and be as in (2.26). Then (2.20) has one positive solution in if either of the following conditions holds:
there exist with such that (2.34) is satisfied for and (2.33) is satisfied for ;
there exist with such that (2.33) is satisfied for and (2.34) is satisfied for .
Equation (2.20) has at least two positive solutions in if one of the following conditions hold.
there exist , with , such that (2.34) is satisfied for , (2.33) is satisfied for , and (2.34) is satisfied for ;
there exist , with and , such that (2.33) is satisfied for , (2.34) is satisfied for , and (2.33) is satisfied for .
3. Optimal Constants and Examples
Consider the differential equation
with the BCs (1.4)–(1.6).
The value is given by direct calculation as follows:
We seek the "optimal" for which is a minimum. This type of problems has been tackled in the past, for example, in the second-order case for heat-flow problems in  and for beam equations (under different BCs) in [9, 12, 13].
The kernel is a positive, nondecreasing function of , thus
Since is a nondecreasing function of , we have
Such maximum is attained at . Thus the "optimal" interval , for which is a minimum, is the interval ; this gives and .
From Theorem 2.4, it is possible to state results for the existence of several nonnegative solutions for the homogeneous BVP
For example, with and , the BVP (3.5) has at least two positive solutions in if there exist , such that , and .
These results are new and improve and complement the previous ones. Gupta  and Yao  studied the problem with more general nonlinearity but established existence results only. Davies and co-authors  and Graef and Henderson  obtain the existence of multiple positive solutions for a -order differential equation subject to our boundary conditions in the case . In  the choice gives the values and which replace our constants and in the growth conditions of . The growth conditions of the nonlinearity in Theorem in  cannot be compared with ours, but we do not require the restriction .
The next examples illustrate the applicability of our results. Firstly we consider, as an illustrative example, the case of a nonlinear -point problem.
Consider the differential equation
with the BCs
where and, as in , for
In this case we have , , , ,
We now fix , and show that all the constants that appear in (2.33) and in (2.34) can be computed. This choice leads to
Moreover we have
and conditions (2.34) and (2.33) read and . Since holds, from Theorem 2.4 it follows that this BVP has a nontrivial solution in . A nonlinearity that easily verifies , for example, is the function for every and every .
We now give an example with continuously distributed positive measures.
Consider the differential equation
with the BCs
with and , , as in the previous example. In this case, we obtain
The Condition becomes
We now fix . This choice leads to
and conditions (2.34) and (2.33) read and .
4. Other Nonlinear BCs
So far we have discussed in detail the case of the BCs (1.4)–(1.6), but the same approach may be applied to (1.1) subject to the nonlinear BCs
As in , where a different set of BCs were investigated, we point out that these nonlocal boundary conditions can be interpreted as feedback controls: for example, the BCs (4.1) can be seen as a control on the displacement at the left end and a device handling the shear force at .
Table 1 illustrates how the choice of the BCs affects the functions and the constants .
Since one can see that
the cone , given by the constant , varies according to the nonhomogeneous BCs considered. This affects also, in a natural manner, conditions (2.33) and (2.34).
Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis 1988, 26(4):289-304. 10.1080/00036818808839715
Davis JM, Erbe LH, Henderson J: Multiplicity of positive solutions for higher order Sturm-Liouville problems. The Rocky Mountain Journal of Mathematics 2001, 31(1):169-184. 10.1216/rmjm/1008959675
Graef JR, Henderson J:Double solutions of boundary value problems for th-order differential equations and difference equations. Computers & Mathematics with Applications 2003, 45(6–9):873-885.
Yao Q: An existence theorem for a nonlinear elastic beam equations with all order derivatives. Journal of Mathematical Study 2005, 38(1):24-28.
Eggesperger M, Kosmatov N: Positive solutions of a fourth-order multi-point boundary value problem. Communications in Mathematical Analysis 2009, 6(1):22-30.
Graef JR, Qian C, Yang B: A three point boundary value problem for nonlinear fourth order differential equations. Journal of Mathematical Analysis and Applications 2003, 287(1):217-233. 10.1016/S0022-247X(03)00545-6
Graef JR, Yang B: Positive solutions to a multi-point higher order boundary value problem. Journal of Mathematical Analysis and Applications 2006, 316(2):409-421. 10.1016/j.jmaa.2005.04.049
Henderson J, Ma D: Uniqueness of solutions for fourth-order nonlocal boundary value problems. Boundary Value Problems 2006, 2006:-12.
Infante G, Pietramala P: A cantilever equation with nonlinear boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations 2009, (15):1-14.
Karna BK, Kaufmann ER, Nobles J: Comparison of eigenvalues for a fourth-order four-point boundary value problem. Electronic Journal of Qualitative Theory of Differential Equations 2005, (15):-9.
Kelevedjiev PS, Palamides PK, Popivanov NI: Another understanding of fourth-order four-point boundary-value problems. Electronic Journal of Differential Equations 2008, (47):-15.
Webb JRL, Infante G: Non-local boundary value problems of arbitrary order. Journal of the London Mathematical Society 2009, 79(1):238-258.
Webb JRL, Infante G, Franco D: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. Proceedings of the Royal Society of Edinburgh. Section A 2008, 138(2):427-446.
Yude J, Guo Y: The existence of countably many positive solutions for nonlinear nth-order three-point boundary value problems. Boundary Value Problems 2005, 2005:-18.
Alves E, Ma TF, Pelicer ML: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3834-3841. 10.1016/j.na.2009.02.051
Amster P, Cárdenas Alzate PP: A shooting method for a nonlinear beam equation. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(7):2072-2078. 10.1016/j.na.2007.01.032
Cabada A, Minhós FM: Fully nonlinear fourth-order equations with functional boundary conditions. Journal of Mathematical Analysis and Applications 2008, 340(1):239-251. 10.1016/j.jmaa.2007.08.026
Cabada A, Pouso RL, Minhós FM: Extremal solutions to fourth-order functional boundary value problems including multipoint conditions. Nonlinear Analysis: Real World Applications 2009, 10(4):2157-2170. 10.1016/j.nonrwa.2008.03.026
Franco D, O'Regan D, Perán J: Fourth-order problems with nonlinear boundary conditions. Journal of Computational and Applied Mathematics 2005, 174(2):315-327. 10.1016/j.cam.2004.04.013
Minhós F: Location results: an under used tool in higher order boundary value problems. In Proceedings of the International Conference on Boundary Value Problems, 2009, Mathematical Models in Engineering, Biology, and Medicine Edited by: Cabada A, Liz E, Nieto JJ. 1124: 244-253.
Infante G: Nonlocal boundary value problems with two nonlinear boundary conditions. Communications in Applied Analysis 2008, 12(3):279-288.
Infante G, Pietramala P: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(3-4):1301-1310. 10.1016/j.na.2008.11.095
Infante G, Webb JRL: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proceedings of the Edinburgh Mathematical Society. Series II 2006, 49(3):637-656. 10.1017/S0013091505000532
Karakostas GL, Tsamatos PCh: Existence of multiple positive solutions for a nonlocal boundary value problem. Topological Methods in Nonlinear Analysis 2002, 19(1):109-121.
Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society. Second Series 2006, 74(3):673-693. 10.1112/S0024610706023179
Infante G: Positive solutions of differential equations with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems. Series A 2003, (supplement):432-438.
Infante G: Nonzero solutions of second order problems subject to nonlinear BCs. In Dynamic Systems and Applications, Vol. 5. Dynamic, Atlanta, Ga, USA; 2008:222-226.
Lan K, Webb JRL: Positive solutions of semilinear differential equations with singularities. Journal of Differential Equations 1998, 148(2):407-421. 10.1006/jdeq.1998.3475
Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 1976, 18(4):620-709. 10.1137/1018114
Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. Journal of the London Mathematical Society. Second Series 2001, 63(3):690-704. 10.1112/S002461070100206X
Webb JRL: Optimal constants in a nonlocal boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2005, 63(5–7):672-685.
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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Pietramala, P. A Note on a Beam Equation with Nonlinear Boundary Conditions. Bound Value Probl 2011, 376782 (2011). https://doi.org/10.1155/2011/376782
- Nonlinear Boundary Condition
- Nonlinear Controller
- Integral Boundary Condition
- Multiple Positive Solution
- Hammerstein Integral Equation