Research Article Building Infinitely Many Solutions for Some Model of Sublinear Multipoint Boundary Value Problems

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1 Abstract and Appied Anaysis Voume 2015, Artice ID , 4 pages Research Artice Buiding Infinitey Many Soutions for Some Mode of Subinear Mutipoint Boundary Vaue Probems Guy Aymard Dega 1,2 1 The Abdus Saam Internationa Centre for Theoretica Physics (ICTP), Trieste, Itay 2 Institut de Mathematiques et de Sciences Physiques (IMSP), 01 BP 613 Porto-Novo, Benin Correspondence shoud be addressed to Guy Aymard Dega; gdega@ictp.it Received 17 Juy 2015; Accepted 15 September 2015 Academic Editor: Maoan Han Copyright 2015 Guy Aymard Dega. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. We show that the subinearity hypothesis of some we-known existence resuts on mutipoint Boundary Vaue Probems (in short BVPs) may aow the existence of infinitey many soutions by using Tietze extension theorem. This is a quaitative resut which is of concern in Appied Anaysis and can motivate more research on the conditions that ascertain the existence of mutipe soutions to subinear BVPs. The idea of the proof is of independent interest since it shows a constructive way to have ordinary differentia equations with mutipe soutions. 1. Introduction BVPsoccurinmostofthebranchesofsciences,engineering, and technoogy, for exampe, boundary ayer theory in fuid mechanics, heat power transmission theory, space technoogy, and aso contro and optimization theory. Concretey, BVPs manifest themseves through the modeing of the motion of a partice under the action of a force, the diffusion of heat generated by positive temperature-dependent sources, the distribution of shear deformation in a beam formed by a few amina of different materias, the defection of a beam, and the transverse dispacement of an easticay imbedded rai to a distributed transverse oad, and so forth [1]. In particuar higher order inear differentia equations subjected to mutipoint boundary conditions, of which we are concerned, arise in the modeing of many phenomena of physica or technoogica nature such as the defection of a curved beam, the three-ayer beam, and the steam suppy contro side [2]. LetusbaseourterminoogyonthoseofDega[3],Eias [4], and Coppe [5]. Let n, m, andk 1,...,k m be positive integers such that 2 m m i=1 k i =n,andeta=a 1 < < a m =bbe m rea numbers. We wi denote by P the Levin poynomia defined by P(t) = m i=1 (t a i) k i and we wi dea with disconjugate nth-order differentia operators on [a, b] of the form Lx := x (n) +p 1 (t) x (n 1) + +p n (t) x, (1) where the coefficients p 1,...,p n are given rea-vaued continuous functions on [a, b]; forinstance,lx = x (n).the disconjugacy of the higher order differentia inear operator L means that every nontrivia soution of the ordinary differentia equation Lx=0has ess than n zeros counting their mutipicities. This means aso that L has a Poya factorization; that is, there exist n smooth positive functions V i C n i+1 ([a, b]), 1 i n,suchthat Lx = V 1 V n D 1 V n D D 1 V 1 x for every x C n ([a, b]), where D = d/dt (cf. [5]). It foows that L admits Green s function associated with the Boundary Vaue Probems: Lx=0, (3) x (j) (a i ) = 0, 1 i m, 0 j k i 1, (2)

2 2 Abstract and Appied Anaysis and so for every f C([a, b]), there exists a unique soution of the BPVs: Lx = f, (4) x (j) (a i )=0, 1 i m, 0 j k i 1. Besides, we wi aso adopt the notations f = max a t b f(t) and Moreover et g C n ([a 1,a m ]) be such that P (t) g (t) >0 fort m 1 (a i,a i+1 ), i=1 g (j) (a i )=0 for i=1,...,m; j=0,...,k i 1, [Lg] (a i )=0 for i=1,...,m. (10) S f (t) = { f (t) { f (t) if f (t) =0 { 0 if f (t) =0 for any f C([a, b]). Many authors have proved the existence of at east one nontrivia soution for subinear Boundary Vaue Probems thatcanbetransformedtothemodeprobem (5) Ly = F(t, y) (6) under various boundary conditions and where F:[a, b] R R + (7) is subinear with respect to y uniformy on t;thatis, im (max y + a t b F(t, y) y )=0, im (min F(t, y) y 0 a t b y )=+ ; see [3, 6 8] and the references therein. Some authors have shown the existence of mutipe soutions (sometimes by introducing a parameter); see [9 15] and the references therein. But resuts about the cases of subinear Boundary Vaue Probems of any order with infinitey many soutions are scarce; see [15, 16]. In this paper, using a new basic topoogica idea in the Theory of Differentia Equations, we woud ike to underine that, under the hypothesis that f is subinear with respect to y uniformy on t, infinitey many soutions may occur. Thiswibeachievedbystartingadequateywithinfinitey many functions satisfying the boundary condition and by constructing a subinear (in fact bounded) function F for which a sequence of these functions satisfies the Boundary Vaue Probem: Ly (t) =F(t,y(t)), a 1 t a m, y (j) (a i )=0, i=1,...,m; j=0,...,k i The Resut We have the foowing. Theorem 1. Let ψ C n ([a 1,a m ]) be such that Lψ > 0 everywhere in [a 1,a m ] and satisfy the boundary condition of (4); for exampe, the unique soution ψ of Ly = 1 and y (j) (a i )= 0, i=1,...,m; j=0,...,k i 1. (8) (9) (e.g., the function g=ψ 2n+1 ). Then there exist a positive integer 0 and a bounded positive continuous function F:[a 1,a m ] R (0, ) (11) such that a the functions ψ+g/with 0 satisfy the singe noninear probem with the boundary condition of (4); that is, Ly = F(t, y) (12) y (j) (a i )=0, i=1,...,m; j=0,...,k i 1. (13) For a proof of this theorem, we wi use the foowing. Lemma 2 (see [3]). If L is disconjugate, then there exists a continuous function φ C([a 1,a m ]) positive on m 1 i=1 (a i,a i+1 ) with φ/ P having a positive infimum such that S P (t) y (t) φ(t) y, a 1 t a m, (14) for every y C n ([a 1,a m ]) satisfying the differentia inequaity Ly 0 (15) and the homogeneous Hermite m-point conditions y (j) (a i )=0, 1 i m; 0 j k i 1. (16) Proof of Theorem 1. For the sake of simpicity we take ψ to be the soution of Lx=1, x (j) (a i ) = 0, 1 i m, 0 j k i 1, and so ψ is n-times differentiabe with Lψ=1>0, ψ (j) (a i )=0, 1 i m, 0 j k i 1, (17) (18) and furthermore, according to the above emma, we have inf(ψ/p) > 0. Hencebysettingg = ψ 2n+1,itiscearthat P gis positive on m 1 i=1 (a i,a i+1 ) and moreover g (k) (a i )=0, 1 i m, and a k=0,...,n, (19)

3 Abstract and Appied Anaysis 3 which impies g (j) (a i )=0 for i=1,...,m; j=0,...,k i 1, [Lg] (a i )=g (n) n 1 (a i )+ Now set j=0 p n j (a i )g (j) (a i )=0 for i=1,...,m. α:= 1 2 min {[Lψ] (t),a 1 t a m }= 1 2, β 0 := max {[Lψ] (t),a 1 t a m }=1, (20) (21) and β:=α+β 0 =3/2. Moreover, choose 0 1such that (1/ 0 ) Lg α = 1/2.Thusfora 0 we have, on the one hand, Ly =Lψ+ Lg Lψ andontheotherhand, yieding Ly =Lψ+ Lg Moreover et us set Aso set β 0 + Lg Lg = 1 2, (22) = 3 2, (23) 1 2 Ly 3 2. (24) Γ:={0} { 1, N, 0}, y μ := ψ + μg for every μ Γ. (25) A μ := {(t, y μ (t)), a 1 t a m } for each μ Γ, (26) A:= A μ. μ Γ (27) Note at once that A 0 is the graph of ψ and that, for every μ Γ, A μ is compact and so A μ \ m i=1 {(a i,0)}is open in A. For μ 1 =μ 2 in Γ,wehave (s, z) A μ1 A μ2 (s) =y μ2 (s) (s), (μ 1 μ 2 )g(s) =0, (s), g(s) =0, s {a 1,a 2,...,a m }, z=0. with μ 1 =μ 2 (28) Hence A μ1 A μ2 ={(a i,0); i=1,...,m} for μ 1,μ 2 Γwith μ 1 =μ 2. (29) Besides it is not hard to see that A is cosed (in fact compact) because every sequence of eements of A has a subsequence that is either contained in some fixed A μ0 (which is compact as the graph of a continuous function on a compact set) or distributed into infinitey many A μ in which case it has an adherent point in A 0 A. Therefore by (29), (27), (24), and the continuity of the functions y μ, we have a we-defined and continuous map f:a [a 1,a m ] R [α, β] (0, ), characterized by the reation where α= 1 2, β = 3 2, (30) f (t, x) := Ly μ (t) if (t, x) A μ. (31) But, by the we-known Tietze extension theorem (aso known as Tietze-Uryshon-Brouwer theorem) [17, 18], this map f has a continuous extension F:[a 1,a m ] R [α, β] (0, ). (32) It foows that for every μ Γwe have Ly μ (t) =F(t,y μ (t)), a 1 t a m, y (j) μ (a i)=0, i=1,...,m; j=0,...,k i 1, competing the proof. Confict of Interests (33) The author decares that there is no confict of interests regarding the pubication of this paper. Acknowedgment The author woud ike to thank very much the Abdus Saam Internationa Centre for Theoretica Physics (ICTP, Trieste, Itay) for its hospitaity during his second visit as a Reguar Associate. References [1] R. P. Agarwa, Boundary Vaue Probems from Higher Order Differentia Equations, Word Scientific, Singapore, [2] M. Gregus, ThirdOrderLinearDifferentiaEquations,Mathematics and Its Appications, D. Reide, [3] G. A. Dega, A unifying maximum principe for conjugate boundary vaue probems, Advanced Noninear Studies, vo.1, no. 1, pp , [4] U. Eias, Osciation Theory of Two-Term Differentia Equations, vo. 396 of Mathematics and Its Appications,Kuwer Academic Pubishers, Dordrecht, The Netherands, 1997.

4 4 Abstract and Appied Anaysis [5] W. A. Coppe, Disconjugacy, vo. 220 of Lecture Notes in Mathematics, Springer, New York, NY, USA, [6] L. H. Erbe and H. Wang, On the existence of positive soutions of ordinary differentia equations, Proceedings of the American Mathematica Society,vo.120,no.3,pp ,1994. [7]P.W.EoeandJ.Henderson, Positivesoutionsfor(n 1, 1) conjugate boundary vaue probems, Noninear Anaysis, Theory, Methods & Appications, vo.28,no.10,pp , [8] H. Brezis and L. Oswad, Remarks on subinear eiptic equations, Noninear Anaysis. Theory, Methods & Appications,vo. 10,no.1,pp.55 64,1986. [9] L.H.Erbe,S.C.Hu,andH.Wang, Mutipepositivesoutions of some boundary vaue probems, Mathematica Anaysis and Appications,vo.184,no.3,pp ,1994. [10] R. Ma, Mutipe positive soutions for noninear m-point boundary vaue probems, Appied Mathematics and Computation,vo.148,no.1,pp ,2004. [11] J. M. Davis, J. Henderson, and P. J. Y. Wong, Genera idstone probems: mutipicity and symmetry of soutions, Mathematica Anaysis and Appications,vo.251,no.2,pp , [12] R. I. Avery and J. Henderson, Three symmetric positive soutions for a second-order boundary vaue probem, Appied Mathematics Letters,vo.13,no.3,pp.1 7,2000. [13] Z. Liu and F. Li, Mutipe positive soutions of noninear two-point boundary vaue probems, Mathematica Anaysis and Appications,vo.203,no.3,pp ,1996. [14] G. Dega, Positive noninear eigenvaue probems for conjugate BVPs, Noninear Anaysis: Theory, Methods & Appications, vo. 55,no.5,pp ,2003. [15] J. Coye, P. W. Eoe, and J. Henderson, Bifurcation from infinity and higher order ordinary differentia equations, Mathematica Anaysis and Appications, vo.195,no.1,pp.32 43, [16] Q. Yao, Existence, mutipicity and infinite sovabiity of positive soutions to a noninear fourth-order periodic boundary vaue probem, Noninear Anaysis. Theory, Methods & Appications,vo.63,no.2,pp ,2005. [17] J. Dugundji, Topoogy, Wm. C. Brown Company Pubishers, Dubuque, Iowa, USA, [18] H. Eschrig, Topoogy and Geometry for Physics,vo.822of Lecture Notes in Physics, Springer, Berin, Germany, 2011.

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