Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition

For arbitrary two-point boundary condition, which makes the corresponding linear uniform problem well-posed, we obtain an existence and uniqueness result for the boundary value problem of finite beam deflection resting on arbitrary nonlinear non-uniform elastic foundation. The difference between the desired solution and the corresponding linear uniform one in L∞ sense is bounded explicitly in terms of given inputs of the problem. Our results seamlessly unify linear uniform and nonlinear non-uniform problems and lead to an iteration algorithm for uniformly approximating the desired deflection.


Introduction
Suppose that a beam with finite length 2l is horizontally put on an elastic foundation. Let E and I be the Young's modulus and the mass moment of inertia of the beam respectively, so that EI is the flexural rigidity of the beam. Throughout this paper, we assume that E, I, and l are fixed positive constants. From the classical Euler-Bernoulli beam theory [16], we have the following governing equation, which we denote by NDE(f , w), for the beam's vertical upward deflection u(x): Here, w(x) is a vertical downward load density on the beam, and -f (u(x), x) is the nonlinear and non-uniform elastic force density by the elastic foundation, which can depend on both the location x on the beam and the deflection u(x) at x. Beam deflection is one of the basic and important problems in structural mechanics and mechanical engineering, and it has a lot of applications [1,3,4,[7][8][9][10][11][12][13][15][16][17][18].
For given f in NDE(f , w), f u (u(x), x) corresponds to the nonlinear and non-uniform spring constant density of the elastic foundation. Let k be the maximal spring constant density of f at zero deflection, which is defined by (1.1) and is assumed to be positive. Then NDE(f , w) is a generalization of the following linear equation, which we denote by LDE(w): The elastic foundation represented by LDE(w) has the elastic force density -k · u(x), which is linear in the sense that it strictly follows Hooke's law and is uniform in the sense that its spring constant density k does not depend on the location x on the beam. Let gl(m, n, R) be the set of m×n matrices with real entries. For three times differentiable functions on [-l, l], we define the following linear operator B : C 3 [-l, l] → gl (8, The boundary value problem consisting of LDE(w) and BC(M, b) is well-posed if it has a unique solution.
In this paper, we analyze the nonlinear non-uniform boundary value problem consisting of NDE(f , w) and BC(M, b) for arbitrary boundary condition BC(M, b) which makes the corresponding linear uniform problem LDE(w) and BC(M, b) well-posed. We will obtain an existence and uniqueness result for this problem under physically realistic and presumably minimal assumptions. Specifically, we have three Assumptions (F), (A), and (B) on the inputs f , w, M, b of our problem. Assumption (F) is on the two-variable function f , which represents the elastic foundation in NDE(f , w). It essentially ensures that f is modeling a physically realistic elastic foundation.
The condition f (u, x) · u ≥ 0 means that the elastic force by the elastic foundation is restoring. The condition f u (u, x) ≥ 0 means that the nonlinear non-uniform spring constant density f u (u, x) of the elastic foundation is nonnegative, so that the magnitude of the elastic force increases as the magnitude of the deflection u increases.
There have been various attempts [1,3,4,[7][8][9][10][11][12][13]17] to generalize the classical linear uniform problem LDE(w) to nonlinear or non-uniform settings. For infinitely long beam, Choi and Jang [7] obtained an existence and uniqueness result for the following infinite version: of NDE(f , w) with assumptions similar to ours. Since the length of the beam they dealt with was infinite, it was sufficient for them to consider the boundary condition They showed the existence and the uniqueness of the solution to the boundary value problem consisting of (1.4) and (1.5) in some regions of C 0 (R) around the zero function.
Following the framework of [7], we will construct a nonlinear operator Ψ : , whose fixed points are solutions of our boundary value problem NDE(f , w) and BC(M, b). We will find out appropriate regions in L ∞ [-l, l] where Ψ is contractive, so that the desired nonlinear non-uniform deflection is guaranteed to exist in those regions by a generalization of Banach fixed point theorem [2]. Our results on finite beam are of more practical importance than those on infinite beam in [7] which are meaningful only in ideal situations.
A challenge with the finite beam problem is that there are a lot of possible well-posed boundary conditions BC(M, b). This is in sharp contrast to the infinite beam problem in [7], where it was sufficient to consider only one boundary condition (1.5). Note that Assumption (F), which is also assumed in [7] for x ∈ (-∞, ∞), implies that f (0, x) = 0 for every x. So the zero function is still a solution of the homogeneous boundary value problem consisting of (1.4) with w = 0 and (1.5). But the solution of the homogeneous boundary value problem NDE(f , 0) and BC(M, b) is not the zero function in general. So the effect of the boundary condition BC(M, b) is already nontrivial even without a nontrivial loading w. In fact, the solution of the linear uniform homogeneous boundary value problem LDE(0) and BC(M, b) is nonzero unless b = 0.
The boundary value problem LDE(w) and BC(M, b) is well-posed for any fixed w ∈ L ∞ [-l, l] and b ∈ gl(4, 1, R) if and only if the boundary value problem LDE(0) and BC(M, 0) is well-posed, in which case we just call the boundary matrix M ∈ gl(4, 8, R) well-posed. Up to a natural equivalence relation, the set of well-posed boundary matrices in gl (4, 8, R) is in one-to-one correspondence with a 16-dimensional algebra [6]. Hence, together with the 4-dimensional space gl(4, 1, R) of boundary values b, the set of different well-posed boundary conditions BC(M, b) forms a 20-dimensional space! a Starting from the inputs f , w, M, b of our problem, we will successively derive various quantities in effective ways. All of these quantities are explicitly computable from given inputs a priori. See Fig. 4 in Sect. 7 for a map of these derivations. The following are a few important end results.
• The upper bound D on the magnitude of linear uniform deflection. • The nonlinear operator Ψ : , which is continuous, strictly increasing, and ρ(0) = 0. It can be chosen with some freedom.  Theorem 1 will be proved in Sect. 6. It is important to note that the statements in Theorem 1 are not local ones such as "there exists something in some sufficiently small neighborhood". The constants r, R, ρ -1 (σ k) are explicitly computable from the inputs, as will be demonstrated by examples. See Fig. 1 for an illustration of Theorem 1. By Theorem 1(c), we always have . We justifiably call the region B(0, ρ -1 (σ k)) the deflection horizon since all the deflections appearing in our analysis cannot escape from it. The deflection horizon conforms to, and is an explicit quantification of, the physical observation that the equations LDE(w) and NDE(f , w) are designed for small deflections originally.
The dual radii r < R is an improved feature compared to [7].  and (B) will be shown to be satisfied for every w, M, b in this case. Note that Assumption (F) is automatically satisfied by f (u, x) = k · u. Thus Theorem 1 reproduces the well-known existence and uniqueness result for the linear uniform boundary value problem LDE(w) and BC(M, b) with no restriction on the inputs at all. This shows that Theorem 1 seamlessly covers the whole range of problems from linear uniform ones to nonlinear non-uniform ones.
Theorem 1(b) naturally leads to a numerical algorithm to approximate the nonlinear non-uniform deflection through iterations with the operator Ψ , which is also explicitly constructible from the inputs. The fact that our results are given in terms of the L ∞ -norm guarantees the approximation to be uniform.
The rest of the paper is organized as follows. In Sect. 2, quantities such as nonuniformity ratio η, nonlinearity function ρ, functional operator N , which measure nonlinearity and non-uniformity of given elastic foundation, are derived from the two-variable function f in NDE(f , w). In Sect. 3, the effects of arbitrary well-posed boundary condition BC(M, b) are encoded in the linear integral operator K M and the linear homogeneous deflection H M [b], which arise naturally from the linear uniform problem LDE(w) and BC(M, b). Here, Assumption (A) is explained in detail, and the elastic capacity σ is defined. The nonlinear operator Ψ is defined and analyzed in Sect. 4. In Sect. 5, Assump-tion (B) is explained in detail, and the radii r and R are derived explicitly from the inputs. The explicitness of these derivations is illustrated with concrete examples. In particular, it is shown that Theorem 1 can reproduce the classical existence and uniqueness result for the linear uniform problem. Theorem 1 is proved in Sect. 6, and some discussions on our results are given in Sect. 7.

Nonlinear non-uniform elastic foundation
For the rest of the paper, the function f in NDE(f , w) is supposed to satisfy Assumption (F) in Sect. 1.

Definition 2.1
Given f , the non-uniformity ratio η at zero deflection is defined by η is dimensionless and has the range 0 ≤ η ≤ 1. An elastic foundation which is uniform at zero deflection corresponds to the extreme case of η = 0. Non-uniformity of given elastic foundation increases as η increases. b The quantity f u (u, x)k amounts to the deviation of the spring constant density f u (u, x) from the corresponding linear uniform density k. Soρ, defined bŷ measures the nonlinearity in the spring constant density of given elastic foundation in terms of the magnitude t of deflection. Note thatρ becomes the zero function in the linear uniform case f (u, x) = k · u. It is clear from its definition (2.1) thatρ is nondecreasing. By Definition 2.1, (1.1), and (2.1), we haveρ(0) = max |x|≤l |f u (0,

Definition 2.2 Given f , a strictly increasing continuous function
For a given f , there are infinitely many possibilities for choosing a nonlinearity function ρ. For a nonlinearity function ρ, denote Since ρ is strictly increasing, we have s ρ > 0, and ρ always has the well-defined strictly increasing continuous inverse ρ -1 : A nonlinearity function ρ and its inverse ρ -1 are used to convert between the deflection variable t and the spring constant density variable s in Sect. 5.
, the quantity f (u, x)k · u corresponds to the nonlinear non-uniform part of the elastic force density f (u, x). Thus the following functional operator N embodies all the nonlinear non-uniform features of given elastic foundation.

As a consequence of Assumption
By the mean value theorem, we have for some τ between u(x) and v(x), and hence for some τ such that hence by (2.1) and (2.5), Thus the result follows from Definition 2.2.
Example 2.1 Suppose l ≥ π , and let the maximal spring constant density k at zero deflection in (1.1) is Since l ≥ π , we have hence the non-uniformity ratio η at zero deflection in Definition 2.1 is By (2.6) and (2.7), hence by (2.1) and (2.8), So by Definition 2.2, we can take ρ(t) = At 2 for t ≥ 0, where we put A = 3a(1 + ). With this ρ, we have s ρ = ∞, and its inverse ρ -1 : so that the elastic foundation is linear and uniform. Assumption (F) is clearly satisfied by f , and the non-uniformity ratio η at zero deflection is 0.
we can take any strictly increasing continuous ρ : The following choice of ρ will turn out to be useful.
Here, the constant σ > 0 is the elastic capacity defined in Definition 3.2, and c > 0 is the constant for converting deflection t to dimensionless ct. With this ρ, we have (2.10)

Boundary conditions and Assumption (A)
A It is well known [14] that, for each well-posed boundary matrix M, K M [w] has the integral form . It is shown in [6] that the space of equivalence classes of well-posed boundary matrices with respect to this relation, and hence the space of different integral operators K M , is in canonical one-to-one correspondence with the algebra gl(4, R) = gl(4, 4, R) whose dimension is 16.
is the L ∞ -norm of K M . The dimensionless quantity μ M is called the intrinsic L ∞ -norm of K M . μ M turns out to be an important quantity through which the boundary matrix M affects our problem.
For the rest of the paper, Assumption (A) μ M · η < 1 in Sect. 1 is supposed to be satisfied. Assumption (A) sets a mutual limit on the non-uniformity of the elastic foundation and the intrinsic L ∞ -norm of K M . Given an elastic foundation f , the possible boundary matrices M are restricted by the condition μ M < 1/η. The restriction gets looser as η becomes smaller.
In the extreme case when η = 0, there is no restriction on M. On the other hand, given a well-posed boundary matrix M, the non-uniformity ratio η is restricted by η < 1/μ M . This restriction gets looser as μ M becomes smaller. A critical phenomenon happens when μ M is less than 1, in which case there is no restriction on η since 0 ≤ η ≤ 1.
We call K M contractive if μ M < 1. The above observation shows that contractiveness of the operator K M for given boundary matrix M is critical in our problem. There are cases that are contractive [5] and cases that are not [6]. c Definition 3.2 Given f and M, the constant σ , called the elastic capacity, is defined by By Assumption (A) and the fact that μ M = 0 for every well-posed M, we always have 0 < σ < ∞. σ is dimensionless, hence the quantity σ k corresponds to spring constant density.   Proof Let α = 4 √ k/EI. Then LDE(w) is equivalent to u (4) = -α 4 u + α 4 k · w. So, by Definition 3.1, we have Suppose Ψ [u] = u. Then, by (4.6), hence u satisfies (4.5), which is equivalent to NDE(f , w). By In general, the nonlinear operator Ψ need not be contractive on the whole L ∞ [-l, l]. So it is crucial to find regions in L ∞ [-l, l] where Ψ becomes contractive. The following result will be useful for that purpose.
Thus we have f , w, M, b, and ρ, denote

Definition 5.2 Given
Note that the quantities represented by r and R are deflections. By (2.3), R = ∞, when s max = s ρ .
hence u * * ∈ B(u c , δ i 1 ) for some i 1 . Then, by Proposition 1, u * * is the unique fixed point of Φ in B(u c , δ i 1 ). It follows that u * * = u * since B(u c , r ) ⊂ B(u c , δ i 1 ). Thus u * is the unique fixed point of Φ in the open ball B(u c , R ). Suppose u 0 ∈ B(u c , R ), so that u 0 ∈ B(u c , δ i 2 ) for some i 2 by (6.5). Then, by Proposition 1, the sequence {u n } ∞ n=1 defined by u n = Φ[u n-1 ], n = 1, 2, 3, . . . , converges to u * . Thus it is sufficient to assume that Suppose R < ∞. By (6.5) and condition (a) for each i, So, by the continuity of Φ, we have Since 0 ≤ L i < 1 for i = 0, 1, 2, . . . , we have by (6.5), condition (b) for each i, and the continuity of Φ. Suppose that there exists another fixed point u * * of Φ in the closed ball B(u c , R ). Since u * , u * * are fixed points of Φ, we have Note that (u * + u * * )/2 is always contained in the open ball B(u c , R ). So, by (6.5), (u * + u * * )/2 ∈ B(u c , δ i 3 ) for some i 3 , hence we have since u * ∈ B(u c , r ) ⊂ B(u c , δ i 3 ) and 0 ≤ L i 3 < 1. By (6.7), since (u * + u * * )/2, u * * ∈ B(u c , R ). It follows from (6.8), (6.9), and (6.10) that u *u * * ∞ < u *u * * ∞ , which is a contradiction. Thus we conclude that u * is the unique fixed point of Φ in the closed ball B(u c , R ).
Let u 0 ∈ B(u c , R ), and let the sequence {u n } ∞ n=0 be defined by u n = Φ[u n-1 ], n = 1, 2, 3, . . . By (6.6), u n ∈ B(u c , R ) for n = 0, 1, 2, . . . Suppose that u n is in the sphere S(u c , R ) = {u ∈ L ∞ [-l, l] | uu c ∞ = R } for every n = 0, 1, 2, . . . Then there exist u * * in S(u c , R ) and a subsequence {u n k } ∞ k=0 of {u n } ∞ n=0 converging to u * * since S(u c , R ) is compact. It follows that u * * is a fixed point of Φ, which is a contradiction. So there exists n 0 such that u n 0 is in the open ball B(u c , R ), and u n 0 ∈ B(u c , δ i 4 ) for some i 4 by (6.5). Thus {u n } ∞ n=n 0 , hence {u n } ∞ n=0 converges uniformly to u * by Proposition 1.
The integral operator K M , the set of which amounts to 16-dimensional space, is a major mean by which the boundary matrix M affects our problem. Together with the other inputs, K M is used to construct the nonlinear operator Ψ . Its intrinsic L ∞ -norm μ M should satisfy Assumption (A) and determines the elastic capacity σ together with the non-uniformity ratio η of the elastic foundation. In particular, the contractiveness of K M is critical in Assumption (A).
The boundary matrix M also determines the linear operator H M , which in turn determines the linear uniform deflection L M [b, w] together with the boundary value b and the loading density w. L M [b, w] should satisfy Assumption (B) and determines the dual radii r and R.

Assumptions (F), (A), (B)
Assumption (F) can be considered as a minimal restriction on f in order to model physically realistic elastic foundations.
It is intuitively natural to imagine that too much of (i) or (ii) below would break the neat behavior, such as Theorem 1, of the resulting deflection.
(i) Nonlinearity and non-uniformity of given elastic foundation.
(ii) Loading density w and boundary value b. After all, the linear uniform equation LDE(w) and its nonlinear non-uniform generalization NDE(f , w) themselves would become physically unrealistic for too much of (i) or (ii). The introduction of Assumptions (A) and (B) is natural in this regard since these assumptions keep (i) and (ii) small enough to guarantee Theorem 1. What is important to note is that Assumptions (A), (B) provide explicit bounds which tell how small is enough. They also tell exactly which should be small among the various quantities that can be derived from the inputs f , w, M, b.
In fact, there are situations where Assumptions (A), (B) are not needed at all. Assumption (A) would not be needed for the following cases: • The non-uniformity ratio η is 0. Note from Definition 2.1 that η = 0 does not necessarily imply that the given elastic foundation is uniform. • The integral operator K M is contractive, i.e., the intrinsic L ∞ -norm μ M of K M is less than 1. Assumption (B) would not be needed for the following cases: • The constant D = ρ -1 (ŝ max ) in Definition 5.1 becomes ∞ or, equivalently,ŝ max = s ρ .
• The linear uniform deflection L M [b, w] is 0 or, equivalently, b = 0 and w = 0.

Nonlinearity function ρ
The nonlinearity function ρ is the only object which can be chosen with some freedom. As the nonlinearity of given elastic foundation is small, we can take smaller ρ, which would result in better bounds in general. Suppose that the nonlinearity of given elastic foundation is small enough so that we can choose ρ such that