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

and Applied Analysis 3 which implies g (j) (a i ) = 0 for i = 1, . . . , m; j = 0, . . . , k i − 1, [Lg] (a i ) = g (n) (a i ) + n−1


Introduction
BVPs occur in most of the branches of sciences, engineering, and technology, for example, boundary layer theory in fluid mechanics, heat power transmission theory, space technology, and also control and optimization theory.Concretely, BVPs manifest themselves through the modelling of the motion of a particle 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 lamina of different materials, the deflection of a beam, and the transverse displacement of an elastically imbedded rail to a distributed transverse load, and so forth [1].In particular higher order linear differential equations subjected to multipoint boundary conditions, of which we are concerned, arise in the modelling of many phenomena of physical or technological nature such as the deflection of a curved beam, the three-layer beam, and the steam supply control slide [2].
Let us base our terminology on those of Degla [3], Elias [4], and Coppel [5].Let , , and  1 , . . .,   be positive integers such that 2 ≤  ≤ ∑  =1   = , and let  =  1 < ⋅ ⋅ ⋅ <   =  be  real numbers.We will denote by  the Levin polynomial defined by () = ∏  =1 ( −   )   and we will deal with disconjugate th-order differential operators on [, ] of the form where the coefficients  1 , . . .,   are given real-valued continuous functions on [, ]; for instance,  =  () .The disconjugacy of the higher order differential linear operator  means that every nontrivial solution of the ordinary differential equation  = 0 has less than  zeros counting their multiplicities.This means also that  has a Polya factorization; that is, there exist  smooth positive functions where  = / (cf.[5]).It follows that  admits Green's function associated with the Boundary Value Problems: Abstract and Applied Analysis and so for every  ∈ C([, ]), there exists a unique solution of the BPVs: Besides, we will also adopt the notations for any  ∈ C([, ]).
Many authors have proved the existence of at least one nontrivial solution for sublinear Boundary Value Problems that can be transformed to the model problem under various boundary conditions and where is sublinear with respect to  uniformly on ; that is, lim see [3,[6][7][8] and the references therein.Some authors have shown the existence of multiple solutions (sometimes by introducing a parameter); see [9][10][11][12][13][14][15] and the references therein.But results about the cases of sublinear Boundary Value Problems of any order with infinitely many solutions are scarce; see [15,16].In this paper, using a new basic topological idea in the Theory of Differential Equations, we would like to underline that, under the hypothesis that  is sublinear with respect to  uniformly on , infinitely many solutions may occur.This will be achieved by starting adequately with infinitely many functions satisfying the boundary condition and by constructing a sublinear (in fact bounded) function  for which a sequence of these functions satisfies the Boundary Value Problem:  () =  (,  ()) ,  1 ≤  ≤   ,  () (  ) = 0,  = 1, . . ., ;  = 0, . . .,   − 1. (9)

The Result
We have the following.Then there exist a positive integer ℓ 0 and a bounded positive continuous function such that all the functions  + /ℓ with ℓ ≥ ℓ 0 satisfy the single nonlinear problem with the boundary condition of ( 4); that is, For a proof of this theorem, we will use the following.
Lemma 2 (see [3]). and the homogeneous Hermite -point conditions Proof of Theorem 1.For the sake of simplicity we take  to be the solution of and so  is -times differentiable with and furthermore, according to the above lemma, we have inf(/) > 0. Hence by setting  =  2+1 , it is clear that  ⋅  is positive on ⋃ −1 =1 (  ,  +1 ) and moreover  () (  ) = 0, 1 ≤  ≤ , and all  = 0, . . ., , which implies  () (  ) = 0 for  = 1, . . ., ;  = 0, . . .,   − 1, Now set and  :=  +  0 = 3/2.Moreover, choose ℓ 0 ≥ 1 such that (1/ℓ 0 )‖‖ ∞ ≤  = 1/2.Thus for all ℓ ≥ ℓ 0 we have, on the one hand, and on the other hand, Note at once that  0 is the graph of  and that, for every  ∈ Γ,   is compact and so Hence Besides it is not hard to see that  is closed (in fact compact) because every sequence of elements of  has a subsequence that is either contained in some fixed   0 (which is compact as the graph of a continuous function on a compact set) or distributed into infinitely many   in which case it has an adherent point in  0 ⊂ .
Therefore by ( 29), ( 27), (24), and the continuity of the functions   , we have a well-defined and continuous map completing the proof.