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We show that the sublinearity hypothesis of some well-known existence results on multipoint Boundary Value Problems (in short BVPs) may allow the existence of infinitely many solutions by using Tietze extension theorem. This is a qualitative result which is of concern in Applied Analysis and can motivate more research on the conditions that ascertain the existence of multiple solutions to sublinear BVPs. The idea of the proof is of independent interest since it shows a constructive way to have ordinary differential equations with multiple solutions.

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 [

Let us base our terminology on those of Degla [

It follows that

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

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

We have the following.

Let

Moreover let

Then there exist a positive integer

For a proof of this theorem, we will use the following.

If

For the sake of simplicity we take

Moreover, choose

For

Therefore by (

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank very much the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) for its hospitality during his second visit as a Regular Associate.