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).
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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