 Applied Mathematics, 2011, 2, 993-998 doi:10.4236/am.2011.28137 Published Online August 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Approach to a Fifth-Order Boundary Value Problem, via Sperner’s Lemma Panos K. Palamides, Evgenia H. Papageorgiou Helleni c Naval Academy, Piraeus, Greece E-mail: ppalam@otenet.gr, epap@snd.edu.gr Received November 29, 2010; revised May 31, 2011; accepted June 7, 2011 Abstract We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlin-earity 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 ap-proach is strongly based on the associated vector field. Keywords: Fifth-Order Differential Equation, Vector Field, Kneser’s Theorem, Sperner’s Lemma 1. Introduction In this paper we study the boundary value problem     5411224,, ,01ξ00and01 0xtctFtxtxtx ttax xxxxx x        (1) under the following assumptions: (A1) F is continuous and positive; i.e. 0,10,, 0,FC; (A2) is continuous and positive; i.e. c0,1 ,0,;cC (A3) 121,1 ,,,,,,0,2a with 1201, and 21:pa a  0.1In recent years, boundary-value problems for second and higher order differential equations have been extensively studied. Erbe and Wang  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  ,,0xtftxt t 0   000,11axbxcx dx  Their technique assumed that the nonlinearity grew ei-ther 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 prob-lems for higher order differential equation is observed, see for example [2-7]. Especially, Graef and Yang  Hao et al.  Ge and Bai  and Kelevedjiev, Palamides and Po-pivanov  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 , which deals with the one-dimensional Kawahara equation that is a non-linear fifth-order ODE with a convective nonlinearity, while Odda  obtains solution of 5th order differential equations under some conditions using a fixed-point theo-rem. Also, we refer to the works El-Shahed, Al-Mezel  and Noor, Mohyud-Din [13,14]. Our analysis of problem (1) will combine the well- known Kneser’s theorem with the Sperner’s lemma princi-ple. 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 01,,,mpp p be 1m affinely independent points of the m-dimensional Euclidean space . Then the simple m01[,,Spp,mp] is defined by P. K. PALAMIDES ET AL. 994 ii,111:0 with1 andmmmiiiiSpp p  The points are called vertices of it and the simplex 01,,,mpp p01[,,,kii ipp p]0kmC[, is a phase of If 2 then 2-dimensional sim-plex is the triangle .S01,,pApB[,, ]Sppp p012 , ].ABC We make use of the following Sperner’s (see ). Lemma 1: If be a closed m-simplex with vertices and be a closed covering of such that each closed phase of is containing in the corresponding union then the intersection is nonempty. mT kE01,, ,mee emTmT01iiEE01,,,mEE Ei01,,,kiiiee e0miiEFor completeness, we recall the well-known Kneser’s Theorem. Theorem 1 (): Consider a system ,, ,Ω:= ,xf txtxab (2) with continuous. Let be a continuum (compact and connected) in f0E0Ω:, Ω:txt a and let be the family of solutions of (2) emanating from . If any solution E0E00xE is defined on the interval [,a], then the set (cross-section) 00,: :ExxE is a continuum in . n 2. Main Results The change of variable uxx t reduces the boundary value problem (1) to:   ,, ,,0,1and 010ut ctFtutututtuu u  (3) where  12112d1dtxtt sussats usp s We may extend the nonlinearity as ,, ,,0, ,,0ftuuuFtuuu From the sing property of F, we have 3,, ,0,,, ,0,1ftuu utuu u We will initially study the following boundary value problem ,, ,,0,1ut ctFtutututt  (4) 0uu u10 (5) Remark 1: The boundary value problem (4)-(5) de-fines a vector field, the properties of which will be cru-cial for our study. More specifically, let us look at the ,uu face semi-plane 0u By the sign condition on and fct, we obtain that Thus any tra-jectory 0u t t,utu,0, emanating from any point in the fourth quarter: ,: 0,0uu uu  “evolves’’ naturally, initially (when 0ut) toward the negative u-semi-axis and then (when 0ut) 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 re-ferred to as the nature of the vector field throughout the rest of the paper. The hypotheses on the nonlinearity 0,1, 0,fC  are the following: (H1) It is superlinear at origin; that is 010,,,lim max0txftxyzx uniformly for every ,yz in any compact subset of . 2(H2) It is superlinear at infinitive; that is 01,,,lim minxtftxyzx uniformly for every ,yz in any compact subset of . 2The following result will be useful in our study of the problem (4)-(5). Lemma 2: If uut is a solution of the boundary value problem (4)-(5) which satisfies that: 0, 01ut t (6) then 0, 01ut t. Remark 2: From the above Lemma we have that every solution of the boundary value problem (4)-(5) is positive, provided that (6) holds. Theorem 2: If the hypotheses (H1)-(H2) hold then the boundary value problem (4) has a positive solution. Remark 3: The above positive function 00uut solves the boundary value problem (3). Our existence theorem reads as follows. Theorem 3: If the hypotheses (A1)-(A2) and (H1)- Copyright © 2011 SciRes. AM P. K. PALAMIDES ET AL. 995(H2) hold, then the boundary value problem (1) has a positive solution. 3. Proof of Main Results Proof of Lemma 2: Arguing by contradiction, suppose that there exists ,1T such that:  0, 0,0and 0, ,1utt TuT utt T0 We have 2T. We consider 2,tT and [,t]T where is the symmetric point of t with respect to t (i.e. 2tt). Because of the concavity of and the map is in-creasing and negative we obtain, uut,0uut1t ut ut d'0 and we have   020ddddTTTut tut tuttuTuttu tt   a contradiction to the fact that . 0uT QED Proof Theorem 2: In view of the assumptions (H1) and (H2) there exist and such that: 00r00R1) For 10M where 0d1Mct t and for every 0,,,0,10,,0txyz R000,rRr we have  0,,,1,,,ftxyz rxftxyzxMM M (7) 2) For 10,N where 10,21dNcttand for every  0220000,,,0,1,22,,txyz RRRRR  we have 00,,,1,,,ftxyzxNRRxftxyz NNN (8) Claim 1: There exists a region , which depends on 0 and Vr such that any solution of the prob-lem (4), which emanates from every initial point of , satisfies uutV0 and0,0,1uutt If we take the region V where every initial point 00,uuV satisfies 2000001andurRu r then any solution for the boundary value problem (4) which emanates from 00,uu , satisfies 0u and 0,0,1 .ut t  We proceed by contradiction, suppose that u 10. By the sign property of f and we have, c0,ut  0t,ut1which implies that the function 0t1 is increasing, so there exists a such that (0t,1)000,utand R uttt 0, Which implies that 00,[0,utRtR tt ) Moreover, because the derivative , is decreasing we obtain ut0, .tt000 0,0,uturt tuttuur 0  From (7) and the Taylor’s formula, we take the con-tradiction, hence  10010000000,, ,d0ututcstfstustustust srut cstsutrurM      0d In addition, again from (7) and the Taylor’s formula, we obtain 00120200 01,,,0uuudscsfsus us ussuur  This proves Claim 1. Let us fix a point 0,Auu V and let 0,0.Bu By the definition of B, every (uBB de-notes the set of solutions of (4) emanating from the ini-tial point B), has the property that 0.uClaim 2: There exists a region U which depends on 00 and ,Rr such that any solution uut of the problem (4), which emanates from every initial point of U, satisfies 0,0,0,1and1uRut tu 0 Copyright © 2011 SciRes. AM 996 P. K. PALAMIDES ET AL. If we take the region U where every initial point satisfies *0,uu U200*02RRuu  then any solution of problem (4) emanating from satisfies *0,uu0,0,0,1and1uRuttu0 Arguing by contradiction, assume . Then, since the function , 10u 1ut 0t1 is increasing we obtain , . That means the function , is decreasing. It follows that 0ut 01t0tut 20*2,0 1,0 1Rut ututt   and, from the Taylor’s formula, we have *0120'' 0*0 *01,,,utu tutscst f stustu stustsRuutuu d So, we have 00for every 0,1Rut t So, we obtain 2002,0 1RRut t  hence we get  100dtutusstR Moreover, because of the fact that , uut[0,1]t is increasing, we obtain  101minxut uRR0  (9) Using (8) and (9) we take the contradiction  1001000001 ,,,,, ,d0,a contradictionuucsfsusususucsfsususus suR  ds The Claim 2 is true. Let us fix another point ΓΓΓ,uu U. We consider the Simplex ,,Γ.SAB Claim 3: Every solution of the boundary value prob-lem (4) emanating from any initial point 11Δ,[Γ,Β]uu satisfies 0.uFor 11Δ,[Γ,Β]uu we have 0100Γ01110 Γ0uu uuuu uuuu uu   (10) From (10) we have 010111Γ000010ΓΓ011100uu uuuu uuuuuuuuu uuuu    (11) From (11) and the Taylor’s formula it follows that 1112011111,,,0uuudscsfsus us ussuuuu This proves Claim 3. By the Kneser’s Theorem 1 and the Claims 1 and 2 there exist points 12Δ,Δ[Α,Γ] such that 10,for some solutionΔuu (12) 210,for some solutionΔuu  (13) By the Kneser’s Theorem 1 and the Claim 1 and since 0u for uBthere exists point 3Δ[,]AB such that 10, for some solutionΔuu Claim 4: If 0ΔΔ,[,uu AB] such that 0u, for some solution Δu then 10u.Arguing by contradiction, assume 10u . As in proof of Claim 1, by the sign property of f and c we have 0,ut  0t1 which implies that the func-tion ,ut0 1t is increasing, so there exists a (0,1)t such that 00and0 0,utR uttt  Which implies that 000,0,rutRtRtt so, we have  Δ1010ΔΔ0Δ0001,,,dut uts cstfstustustustsrut cstsutrurA    d Copyright © 2011 SciRes. AM P. K. PALAMIDES ET AL. 997we obtain Δ00ur (14) On the other hand, we have 0Δ00Δ0Δ0(),,,duuuscsfsus us ussuuuu  we obtain 0Δ0uu (15) From (14) and (15) we take the contradiction. This proves Claim 4. We consider now the sets 111Δ,: 0,1CuuSu u0 and 211Δ,: 0,1CuuSu u0 From the Claim 4 we have and from the Claims 2 and 4 we have . 1C2CCCWe suppose that 12, otherwise we don’t have anything to prove. Recalling that S is the simplex with vertices 00,,Auu0,0 ,Buand Γ0Γ,.uu We define the closed sets 0000Γ00:, :0,10,:10:, :0ABuEuSuuEuSuEuSuuu    where denotes a solution for the problem (4) emanating from the corresponding initial point in . ut SWe have from the Claim 1, from the nature of the vector field and from the Claim 2. ΓAEΔBBEΓEΓTake a point of the phase [,]AB then 1) either and 10u 0thenu ΔEE .AABE2) or then 10u ΔBABEEE 3) or and 0u10u then we have a con-tradiction from the Claim 4. Consequently, we have ,ABABEE (16) On the other hand, let point of the phase [, then Δ Γ]A1) either and 10u 0thenu ΓΔAAEE .E 2) or then 0uΓΓΔAEEE3) or 0u and 10u  then 12CC that is a contradiction. Consequently, we have Γ,ΓAAEE (17) Finally, if Δ[Γ,]B then from the Claim 2 we have 0uΔEE which implies that ΓΓ Therefore ΓΓB is a suitable closed covering of S that satisfies the hypotheses of Sperner’s lemma. Thus, there exists an initial point such that .ΔBEEE00,ΔuuEE.EEΓΓBThe case that we have two solutions ,Δuu12110u of the problem (4) with , 10u  and 20u, 210u has been addressed by Palamides, Infante and Pietramala . They approached the con-tinuous 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 00 Δuu  is a solution of the boundary value problem (4)-(5). QED Proof Theorem 3: From the Remark 3 we have a pos-itive solution for the boundary value problem (3). 0We consider the boundary value problem u 01122,0 1ξ00xt uttax xxx  (18) Then it is known (see for example ) that (18) has the solution  1210120d1d,01txtt su ssats uspt s Consequently in view of the transformation utxt, a solution for the initial boundary value problem (1) is given by the last formula. 4. Acknowledgements The authors wish to thank the referee for his/her helpful remarks. 5. References  L. H. Erbe and H. Wang, “On Existence of Positive Solu-Copyright © 2011 SciRes. AM P. K. PALAMIDES ET AL. Copyright © 2011 SciRes. AM 998 tions of Ordinary Differential Equations,” Proceedings of the American Mathematical Society, Vol. 120, 1994, pp. 743-748. doi:10.1090/S0002-9939-1994-1204373-9  D. R. Anderson and J. M. 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