On the Global Solvability of a Class of Fourth-Order Nonlinear Boundary Value Problems

In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains. Mathematics Subject Classification: 49G99, 73H05, 73K15


INTRODUCTION
The Euler buckling load of simply supported straight elastic beam subject to an end axial compressive load P can be modeled by the equation: (2) where L is the length of the beam, E the Young's modulus, and I the area moment of inertia. Integrating (1) two twice gives: 1 2 . EIv Pv c c x ′′ + = + Applying boundary conditions (2), we get: 1 2 The boundary value problem (1), (2), then reduces to: with the boundary conditions: The general solution of (3) is: ( ) cos sin , where A and B are arbitrary constants to be determined so that the conditions (4) are satisfied. This gives the following system of two equations in A and : Therefore, the boundary value problem (1), (2) can be thought of as the following eigenvalue problem: 0, 0 , with eigenpairs given by: 2 ( ) sin , , 1,2,3,...
) v x is called the first eigenfunction or the first buckling mode and 1 2 crit P EI L λ = is the well-known Euler critical buckling load, sometimes also called the onset buckling load, see [11] for details.
The above classical Euler buckling load is derived based on the classical Hooke's law: E σ ε = and the assumption that during the deformation, the crosssections of the column remains perpendicular to the center line. Modern advances in manufacturing of metal made available in the markets the following high strength materials satisfying a more general constitutive equation: which is known as the Hollomon's power-law. Here, σ stands for the true stress ε true strain, n the strain hardening index, and K the strength coefficient. The differential equation for the Hollomon's power-law beam subjected to an end axial compressive load P can be written as: where is the first moment of area inertia. We consider here the solvability of (6) with one of the pin-pin (PP), the pin-slide (PS) or the slide-slide (SS) boundary conditions: If n KI in (6) is constant, then equations (6), (7) can be written as the following eigenvalue problems:  [11] can be applied to imply that for each and K n, (8) has the following solutions corresponding to the three boundary conditions: (3) For the SS boundary condition: corresponding to the critical n n n n n n n n n In the above, sin ( ) n t is a generalized sine function and [3], [10], and [13] for definitions and properties of the generalized sine function. If , and for 1 n ≠ the corresponding onset buckling load is given by ( ) cr P n . This paper is concerned with the global existence and uniqueness of solutions to the following boundary value problems: These boundary value problems model the deflection of a power-law column subject to axial load and lateral force, with nonlinear foundation constrain. Our objectives are to extend the results established in [1] for the n=1 case. The main conclusion of our result here is to show that the linear lateral constrain in (9) cause a shift of the critical buckling load. In the following sections of this paper, we only consider the global existence and uniqueness of solution for the differential equations in (9) for the "PP Condition." Similar results can be established for the other boundary conditions however, for simplicity and brevity, we omit the proofs.

ASSUMPTIONS AND PRELIMINARY RESULTS
Throughout the rest of this paper we will use the following set notation: It can be easily seen that is a Banach space. H We will make the following assumptions: where ( ) and ( , , , ) are continous, and ( , , , ) ( , ) defines a map : (0,1) (0,1) which is continuous.
The proof of our main result of the next section consists of verifying the conditions of a corollary of Leray-Schauder Fixed Point Theorem which we state here as the following lemma.
Provided that: Therefore, the Arzelà-Ascoli theorem applies and the operator K is compact. With the notation: we note that:

GLOBAL EXISTENCE OF SOLUTIONS
In this section we consider the global solvability of the boundary value problem (9) (with the (PP) boundary condition) in the following two theorem.
where K is the operator defined in Proposition 2.1. We will prove the existence of a solution of (15) by verifying the conditions of Lemma 2.1.

5.
The results of this paper generalize the previous results of the papers [1] and [2] for elastic materials for the elasto-plastic materials based on the Hollomon s power-l ′ CONCLUSIONS aw. Similar results for material following other power laws, for instant Ludwick and Ramberg and Osgood laws, as well as for other elasto-plastic basic structures such as rings, plates and arches are under preparation.