International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 375634, 18 pages

doi:10.1155/2012/375634

Research Article

Fourth-Order Four-Point Boundary Value Problem: A Solutions Funnel Approach

Panos K. Palamides¹ and Alex P. Palamides²

¹Naval Academy of Greece, 45110 Piraeus, Greece

²Technological Educational Institute of Piraeus, Department of Electronic Computer Systems, P.Ralli and Thivon 250, 12240 Athens, Greece

Received 9 April 2012; Revised 26 May 2012; Accepted 10 June 2012

Academic Editor: Michael Tom

Copyright © 2012 Panos K. Palamides and Alex P. Palamides. 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.

Abstract.

We investigate the existence of positive or a negative solution of several classes of four-point boundary-value problems for fourth-order ordinary differential equations. Although these problems do not always admit a (positive) Green's function, the obtained solution is still of definite sign. Furthermore, we prove the existence of an entire continuum of solutions. Our technique relies on the continuum property (connectedness and compactness) of the solutions funnel (Kneser's Theorem), combined with the corresponding vector field.

1. Introduction

In recent years, boundary-value problems for second and higher order differential equations have been extensively studied. They are used to describe a large number of physical, biological, and chemical phenomena. The work of Timoshenko [1] on elasticity, the monograph by Soedel [2], the paper by Palamides [3] on deformation of elastic membrane, and the work of Dulàcska [4] on the effects of soil settlement are rich sources of such applications.

Pietramala [5] presented some 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.

In [6], Erbe and Wang by using a Green's function and Krasnoselskii's fixed point theorem in a cone proved the existence of a positive solution of the following boundary value problem:

under positivity and sub or superlinearity of the nonlinearity and moreover .

The literature for previous BVP is voluminous. Suggestively, we refer to [7–11] and the references therein. The monograph of Agarwal et al. [12] contains excellent surveys of known results.

Recently, an increasing interest in studying the existence of solutions and positive solutions to boundary-value problems for higher order differential equations is observed, see for example [13–17]. Especially, Graef and Yang [15] and Hao et al. [18] proved existence results on nonlinear boundary-value problem for fourth-order equations.

Also, Ge and Bai [19] by using a fixed point theorem due to Krasnoselskii and Zabreiko in [20] investigated the fourth-order nonlinear boundary value problem as

Precisely, they proved the next result.

Theorem 1.1. Assume that

are nonnegative constants satisfying , and ; the nonlinearity can be separated as , where are continuous;

 and . Moreover, there exists some such that , and there exists a continuous nonnegative function such that for each ;, where are constants depending on and . Then the BVP (1.2) admits at least one nontrivial solution .

Similarly Cui and Zou [21], on the base of a fixed point theorem, studied the BVP

to the case where is monotone and either superlinear or sublinear, mainly under the assumption

Finally, Infante and Pietramala [22] proved some results on the existence of positive solutions for some cantilever fourth-order differential equation subject to nonlocal and nonlinear boundary conditions.

Restricting our consideration on the linear case, notice as far as the author is awake, that only the conditions in (1.2) have been studied, where the constants are nonnegative.

In a recent paper, Kelevedjev et al. [23] proved the existence of a positive and/or a negative solution for the boundary value problem (1.2), mainly under superlinearity conditions on the nonlinearity. Moreover, they exhibited existence results to the following differential equation:

subject to the boundary conditions as

Moreover, in an interesting paper [24], Anderson and Avery, by applying a generalization of the Leggett-Williams fixed point theorem, proved the existence of at least three positive solutions to the BVP as

In this paper, we relax the assumptions in [23] and extend the above results to the case where the constants and are not necessarily positive and the nonlinearity is asymptotically linear and not necessarily separated, that is we study the BVP

with . Instead of the assumptions and/or in [20], we need only the condition

Also the constants and (see the assumptions – below) satisfy different (see (2.7)) inequalities than the following one:

in [23]. Moreover, the obtained solutions are of definite sign.

Similarly, we give existence results for several boundary value problems of type

where the constants are chosen suitable positive. At these cases, we do not use the related Green's function. Thus, our approach is applicable in cases where we cannot construct a Green's function or at least it is not of definite sign. For example, instead of the above linear boundary conditions, we could choose the following nonlinear one:

At such cases, the Krasnoselskii's fixed point theorem in a cone seems not to be applicable. Furthermore, we prove the existence of an entire continuum of solutions  (Remark 2.3) and, to the best of our knowledge, this is the first result of this type of multiplicity.

Remark 1.2. Assume the nonlinearity is nonnegative. The differential Equation (1.9) defines a vector field, the properties of which will be crucial for our study. More specifically, assuming that , we set . Let us now look (see the Figure 1, case “Th 2.2”) at the phase quadrant . By the sign condition on , we obtain that . Thus any trajectory , crossing the semiline as

at time , “evolutes” naturally, toward the negative semiline

Setting a certain growth rate on (say superlinearity), we can control the vector field, so that some trajectory reaches on at the time . Whenever the nonlinearity is nonpositive or/and the sign of constants different, trajectories have an analogous behavior.

Figure 1: Vector fields and Trajectories of BVP’s solutions.

The technique presented here is different to those in the above mentioned papers. Actually, we rely on the above “properties of the vector field” and the Kneser's property (continuum) of the cross-sections of the solutions funnel. For completeness, we restate the well-known Kneser's theorem.

Theorem 1.3 (see [25]). Consider a system , with continuous. Let be a continuum (compact and connected) in and let be the family of all solutions of emanating from . If any solution is defined on the interval , then the set (cross-section)

is a continuum in .

2. Main Results

Consider the boundary value problem following:

Setting , the boundary value problem - reduces to

where

Remark 2.1. We note that for all , we have

Moreover, whenever we are interested in nonnegative solutions, without loss of generality, we may extend the nonlinearity as

and if we are asking for nonpositive solutions, we may set

We will use the following assumptions.

The nonlinearity is a continuous and positive function, that is,

it is asymptotically linear at the origin and at infinity, that is,

Similarly, is a continuous and positive function, that is,

it is asymptotically linear at the origin and at infinity, that is,

Consider now the boundary value problem -.

Theorem 2.2. Assume – hold and furthermore that

Then the boundary value problem - admits a positive and concave solution , provided that

Furthermore,

Proof. By the asymptotic linearity of at (see the assumption and (2.3)–(2.7), for any there is an such that

Consider any positive number , such that

and choose a positive , where

We assert that for and any solution , where , it follows that

Indeed in view of Remark 1.2, let us assume that there exists such that

Then by the Taylor's formula, (2.10) and (2.14), we get , such that

Consequently,

contrary to the choice of in (2.12). Hence, the assertion in (2.13) is proved. Moreover, if there is such that

then

a contradiction to (2.12). Thus

Consequently,

On the other hand, in order to prove the opposite inequality, we assume on the contrary that for every solution , we have

Then,

In view of the assumption , we chose and such that

Then, there exists , such that

Assume that there is a solution and such that

We assert first that

Indeed, by (2.22) and (2.25), we obtain

and then (2.26) follows.

By (2.24)-(2.25), we obtain

Consequently, (2.26) yields

a contradiction to the choice of . Thus,

Consequently, by the choice of and (2.24), we get

Hence, we conclude that

contrary to the assertion (2.21).

Finally, consider the segment (recall that and )

and furthermore the cross-section

of the solutions funnel emanating from the segment . By the definition of the function , (2.20), and (2.32), it is clear (recall that ) that

This means that there is a point such that and thus a solution satisfying the second boundary condition in .

Furthermore, by the above analysis, the obtained solution is negative. We extend on the entire interval as follows:

Then, the function , is negative and continuous. In view of the transformation , we consider the boundary value problem as

It is well known that the Green function of it is

Consequently (see (2.3)), the desired positive and concave solution of the boundary value problem - is given by the formula as

Remark 2.3. Since we can extend the solution with infinite ways, we can immediately obtain an entire continuum of solutions for the boundary value problem -.

Consider the boundary conditions

Corollary 2.4. Assume – hold, where the interval is replaced by , and furthermore

Then the boundary value problem – has a positive and concave solution , provided that

Furthermore,

Proof. Consider the BVP

where

and is any real map.

Since satisfies the conditions –, it is obvious that the same conditions are fulfilled by . Hence, in view of Theorem 2.2, the BVP (2.43) admits a positive and concave solution . Thus, setting

we get

Consequently, , is a solution of the differential equation . Furthermore the boundary conditions in (2.43) and the transformation guarantee that the boundary conditions also hold. Hence, is the requite solution of –.

Remark 2.5. Obviously, we could give an analytical proof (similar to the one given at Theorem 2.2) to the above Corollary 2.4, as well as to the following ones.

Consider now the differential equation as

and then the boundary value problem –,

Corollary 2.6. Assume – hold and furthermore that

Then the boundary value problem – admits a negative and convex solution , provided that (2.7)-(2.8) holds. Furthermore,

Proof. Consider the BVP

where, for any real function ,

Then, since the conditions – are performed by , satisfies the assumptions – (applied on the interval ). Hence, by Theorem 2.2, (2.49) admits a positive and concave solution . We set

Then

that is, the map , is a solution of . Furthermore, since

(and similarly for the other) the boundary conditions in are satisfied by the solution . Consequently, is the desired solution of the boundary value problem –

Remark 2.7. By the following formula:

it is clear that the map , is a negative solution of the BVP (2.49). Hence we get also the positive solution as

of the boundary value problem –. The same reasoning folds for all the following results.

Consider the BVP –.

Corollary 2.8. Assume – hold, and furthermore that

Then the boundary value problem – admits a negative and convex solution , provided that (2.7)-(2.8) holds. Furthermore,

Proof. Consider the BVP

where

Then, since the conditions – are performed by , satisfies the assumptions – of Theorem 2.2. Hence, that BVP admits a positive and concave solution . We set

Then

that is, the map , is a solution of . Furthermore, since

the first (and similar the other) of the boundary conditions in is satisfied. Consequently, , is the desired solution of the boundary value problem –.

Consider now the boundary value problem –.

Corollary 2.9. Assume – hold and furthermore that . Then the boundary value problem – admits a positive and concave solution , provided that (2.41) holds. Furthermore,

Proof. Consider any function We define a map by

Since satisfies the conditions –, we easily check that suits the same conditions. In view of Theorem 2.2, consider a solution , of the following BVP:

We notice that and set

Then, we obtain

that is, the function is a solution of the differential equation . Moreover, the boundary conditions are satisfied by the function , since the solution fulfils the boundary conditions (2.66). Consequently, is the requite solution of –.

Corollary 2.10. Assume – hold and furthermore that . Then the boundary value problem – admits a positive and concave solution , provided that (2.41) holds. Furthermore,

Proof. It follows by Corollary 2.9, via the following transformation:

Here , is a positive solution of the BVP as

where

Indeed, for ,

that is, , is a solution of differential equation . Finally, , is a desired solution of BVP -.

Similarly, we may prove the next results.

Corollary 2.11. Assume – hold and furthermore,

Then the boundary value problem – has a negative and convex solution , provided that (2.7)-(2.8) holds. Furthermore,

Corollary 2.12. Assume – hold and furthermore

Then the boundary value problem – has a negative and convex solution , provided that (2.41) holds. Furthermore,

Corollary 2.13. Assume – hold and furthermore

Then the boundary value problem – has a negative and convex solution , provided that (2.7)-(2.8) holds. Furthermore,

Corollary 2.14. Assume that holds and the nonlinearity is a sublinear map, that is, , . Then, the boundary value problem - admits a positive and concave solution.

Proof. Obviously, – are fulfilled. Thus, Theorem 2.2, yields the result.

Remark 2.15. It is obvious that the results of all Corollaries, given above, still hold, under a sublinearity assumption of the nonlinearity (see [23]).

Remark 2.16. We could replace the following lines (initial and terminal points):

with any suitable continuum and , respectively. At such a case, the existence of the related Green's function is uncertain. For example, we may consider the following boundary conditions:

Then, the proof can be carried out word perfect.

Disclosure

This paper is in final form and no version of it will be submitted for publication elsewhere.

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