Approach to a Fifth-Order Boundary Value Problem, via Sperner's Lemma

We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.


Introduction
In this paper we study the boundary value problem under the following assumptions: (A1) F is continuous and positive; i.e.
       0,1 0, , 0, 1 In recent years, boundary-value problems for second and higher order differential equations have been extensively studied.Erbe and Wang [1] used a Green's function and the Krasnoselskii's fixed point theorem in a cone to prove the existence of a positive solution of the boundary value problem Their technique assumed that the nonlinearity grew either superlinearly or sublinearly.The growth assumptions and calculations involving the Green's function followed by an application of Krasnoselskii's Theorem yielded the result.
Recently an increasing interest in studying the existence of solution and positive solutions to boundary-value problems for higher order differential equation is observed, see for example [2][3][4][5][6][7].Especially, Graef and Yang [4] Hao et al. [8] Ge and Bai [9] and Kelevedjiev, Palamides and Popivanov [7] proved the existence of results on nonlinear boundary-value problem for fourth order equations.We are aware of limited number of works that study the boundary value problem for fifth order differential equations.We mention the work of Doronin and Larkin [10], which deals with the one-dimensional Kawahara equation that is a nonlinear fifth-order ODE with a convective nonlinearity, while Odda [11] obtains solution of 5th order differential equations under some conditions using a fixed-point theorem.Also, we refer to the works El-Shahed, Al-Mezel [12] and Noor, Mohyud-Din [13,14].
Our analysis of problem (1) will combine the wellknown Kneser's theorem with the Sperner's lemma principle.The aim of this paper is to use Sperner's lemma as an alternative to the classical methodologies based on fixed point theory or degree theory under simple assumptions.
Let us recall some basic notions and results from the theory of simplex, which we will subsequently need.Let The points are called vertices of it and the simplex We make use of the following Sperner's (see [15]).
For completeness, we recall the well-known Kneser's Theorem.
Theorem 1 ([16]): Consider a system with continuous.Let be a continuum (compact and connected) in and let be the family of solutions of (2) emanating from .If any solution  , then the set (cross-section)

Main Results
The change of variable    u x x t    reduces the boundary value problem (1) to: and 0 1 0 where We may extend the nonlinearity as From the sing property of F , we have We will initially study the following boundary value problem Remark 1: The boundary value problem ( 4)-( 5) defines a vector field, the properties of which will be crucial for our study.More specifically, let us look at the By the sign condition on and f   c t , we obtain that Thus any trajectory , 0, emanating from any point in the fourth quarter: "evolves'' naturally, initially (when ) toward the negative u -semi-axis and then (when ) toward the negative u -semi-axis.Setting a certain growth rate on f (say superlinearity), we can control the vector field, so that some trajectories will satisfy the given boundary conditions.These properties will be referred to as the nature of the vector field throughout the rest of the paper.
The hypotheses on the nonlinearity , , , lim min uniformly for every   , y z in any compact subset of .


The following result will be useful in our study of the problem (4)- (5).
Lemma 2: is a solution of the boundary value problem (4)-( 5) which satisfies that: then   0, 0 Remark 2: From the above Lemma we have that every solution of the boundary value problem (4)-( 5) is positive, provided that (6) holds.

Proof of Main Results
Proof of Lemma 2: Arguing by contradiction, suppose that there exists where is the symmetric point of t with respect to t  (i.e.
a contradiction to the fact that .
In view of the assumptions (H1) and (H2) there exist and such that: and for every , , , , Claim 1: There exists a region , which depends on 0 and V r  such that any solution of the problem (4), which emanates from every initial point of , satisfies   If we take the region V where every initial point then any solution for the boundary value problem ( 4) Moreover, because the derivative , is decreasing we obtain 7) and the Taylor's formula, we take the contradiction, hence In addition, again from (7) and the Taylor's formula, we obtain Let us fix a point   0 , A u u V    and let   0 , 0 .B u By the definition of B, every ( denotes the set of solutions of (4) emanating from the initial point B), has the property that   0.

  u
Claim 2: There exists a region U which depends on 0 0 and , R r  such that any solution   u u t  of the problem (4), which emanates from every initial point of U, satisfies If we take the region U where every initial point satisfies then any solution of problem (4) emanating from satisfies Arguing by contradiction, assume .Then, since the function , and, from the Taylor's formula, we have Using ( 8) and ( 9) we take the contradiction The Claim 2 is true.
From (10) we have From (11) and the Taylor's formula it follows that By the Kneser's Theorem 1 and the Claims 1 and 2 there exist points 1    2 1 0, for some solution By the Kneser's Theorem 1 and the Claim 1 and since for some solution . As in proof of Claim 1, by the sign property of f and c we have On the other hand, we have From ( 14) and ( 15) we take the contradiction.This proves Claim 4.
We consider now the sets From the Claim 4 we have and from the Claims 2 and 4 we have . 1 2   , otherwise we don't have anything to prove.
Recalling that S is the simplex with vertices where denotes a solution for the problem (4) emanating from the corresponding initial point in .
  u t S We have from the Claim 1, from the nature of the vector field and from the Claim 2.
Γ Take a point of the phase [ , ]  A B then 1) either and  then we have a contradiction from the Claim 4.
Consequently, we have On the other hand, let point of the phase [ , then then from the Claim 2 we have is a suitable closed covering of S that satisfies the hypotheses of Sperner's lemma.Thus, there exists an initial point such that .
The case that we have two solutions     2 1 0 u  has been addressed by Palamides, Infante and Pietramala [17].They approached the continuous nonlinearity by a sequence of locally Lipschitz functions and then each such a Lipschitz boundary value problem ensure the existence of a solution.Finally the well-known Kamke theorem may be applied, to get a solution of the boundary value problem (3), as a limit solution.
This means that the corresponding solution is a solution of the boundary value problem ( 4 Then it is known (see for example [9]) that (18) has the solution , a solution for the initial boundary value problem (1) is given by the last formula.

Claim 3 :
Let us fix another point   Every solution of the boundary value problem (4) emanating from any initial point

3 :
)-(5).QED Proof Theorem From the Remark 3 we have a positive solution for the boundary value problem (

s
Consequently in view of the transformation