MULTIPLICITY RESULTS FOR FOURTH ORDER PROBLEMS RELATED TO THE THEORY OF DEFORMATIONS BEAMS

. The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams ﬁxed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two diﬀerent particular functions.

1. Introduction. The study of fourth-order boundary value problems are useful for material mechanics because this kind of problems usually characterize the deformations of an elastic beam. They have been studied by many authors via various methods, such as Leray-Schauder continuation method, topological degree theory, shooting method, fixed point theorems on cones, critical point theory, the lower and upper solutions method or spectral theory, see for example [4,6,3,13,11,16,2] and references therein.
Motivated by the above works, in this paper we study the existence and multiplicity of positive solutions for the fourth order equation: subject to the perturbed functional boundary conditions: Here f is such that M ∈ [−m 4 0 , m 4 1 ) and λ is a positive parameter bounded from above by a constant that will be introduced later.
The boundary conditions (4) model the deflection of beam fixed in 0 and has some mechanism in 1 which controls the displacement according to the feedback from devices measuring the displacements along parts of the beam.
A standard approach to study positive solutions of a boundary value problem such as (3) -(4) consists on finding the corresponding Green's function G M and seek solutions as fixed points of the following integral operator: in the cone P = {u ∈ C(I), u ≥ 0 on I} of non-negative functions in the space C(I) endowed with the usual supremum norm.
We will obtain the expression of the Green's function G M related to the linear equation (1) coupled to the functional boundary conditions (4) as a combination of the expression of g M , the Green's function related to Problem (1) - (2). In this case, we will give the exact values on the positive parameter λ for which G M remains negative on (0, 1) × (0, 1), whenever M ∈ [−m 4 0 , m 4 1 ). Moreover there is no λ > 0 for which G M < 0 when M > m 4 1 . In the case of M < −m 4 0 we have that it must exists a set of values of λ where such property is fulfilled, but this case has not been considered in this paper, and remains as an open problem.
Following the arguments of [20,21], to ensure the positiveness of the solutions at some subinterval [a, b] of [0, 1], it is convenient to work in a smaller cone than P , namely, To construct such cone, it is usually convenient to establish the following type of inequality for some function φ ∈ C(I), such that φ(s) > 0 for all s ∈ (0, 1).
. It is possible to use this approach in our situation but, as we will see, the explicit form of Green's function G M is very complicated for M = 0, and so, condition (C) becomes hard to check.
So, we look for a condition on the line of the following one introduced in [7, Page 86]: (N g ) There is a continuous function φ(t) > 0 for all t ∈ (0, 1) and k 1 , k 2 ∈ L 1 (I), such that k 1 (s) < k 2 (s) < 0 for a. e. s ∈ I, satisfying To this end, to avoid long computations concerning the expression of the Green's function, we will study the limits at s = 0 and s = 1 of the quotient G M (t, s)/ G M (1, s).
The paper is organized as follows: in Section 2, we provide some necessary background material such as the compression/expansion and Leggett-Williams fixed point theorems in cones together with some properties of the Green's function associated to the two-point homogeneous boundary value problem (1)-(2). In Section 3, the explicit expression of the Green's function related equation (1) coupled to the integral boundary condition (4) and deduce some additional properties concerning its constant sign. Section 4 is devoted to prove the existence of countably many positive solutions for the nonlinear problem 2. Preliminaries. In this section we introduce some preliminary results which will be used along the paper. First, we provide some background definitions cited from cone theory in Banach spaces. After that, we introduce some definitions and properties of the Green's function g M related to problem (1) -(2). Definition 2.1. Let E be a real Banach space. A nonempty convex closed set K ⊂ E is said to be a cone provided that (i) αu ∈ K for all u ∈ K and all α ≥ 0; (ii) u, −u ∈ K implies u = 0.
In the sequel, we enunciate the celebrated compression/expansion Krasnoselskii's fixed point theorem: Theorem 2.2. [12]. Let K be a cone and T : K → K a completely continuous operator and 0 < r < R. Moreover, if one of the following conditions are fulfilled: (i) T u ≤ u for any u ∈ K with u = r and T u ≥ u for any u ∈ K with u = R, or (ii) T u ≥ u for any u ∈ K with u = r and T u ≤ u for any u ∈ K with u = R, then operator T has a fixed point in K such that r ≤ x ≤ R.

ALBERTO CABADA AND ROCHDI JEBARI
In order to enunciate the Leggett-Williams fixed point theorem, we introduce the following concepts. Definition 2.3. A map β is said to be a nonnegative, continuous, concave functional on a cone K of a real Banach space E, if β : K → [0, ∞) is continuous, and for all x, y ∈ K and t ∈ I. Definition 2.4. Let P be a cone in a real Banach space E, 0 < a < b and let β be a nonnegative continuous concave functional on K. Define the convex sets P r and P (β, a, b) by Theorem 2.5. (Leggett-Williams fixed point theorem) (see [15]) Let A : P c → P c be completely continuous operator and β be a nonnegative continuous concave Then A has at least three fixed points x 1 , x 2 , x 3 in P c such that By the other hand, we point out that problem has no solution if and only the following equality holds M > 0 and tan In any other case, it has a unique solution, denoted by w M , which is given by the following expression: It is not difficult to verify that w M (t) > 0 for all t ∈ I if and only if M < m 4 1 . Moreover, by denoting

PROBLEMS RELATED TO THE THEORY OF DEFORMATIONS BEAMS 493
we have that it is given by the following expression: if m > 0 and M = −m 4 . (9) Moreover, we enunciate the following result concerning the expression of the Green's function g M , related to the linear Problem (1) - (2). The proof can be found in [4] Lemma 2.6. Let σ ∈ C(I) and M ∈ R be such that (7) does not hold. Then problem has a unique solution given by Here, for M = −m 4 < 0, we have If M = 0, it is given by Moreover, when M = m 4 > 0 it follows the expression Moreover As a direct consequence of previous result, we deduce the following properties for function g M : Corollary 1. Function g M defined in Lemma 2.6 satisfies the following properties: , for all t ∈ (0, 1). On the other hand, if we consider the following boundary value problem: in [7, Section 1.4] or [8,9] one can see that Problem (11) is just the adjoint of Problem (10). So, the eigenvalues of both problems coincide and Green's function g * M related to this problem satisfies that g * M (t, s) = g M (s, t) for all t, s ∈ I. As a direct consequence, we have that is the unique solution of the following boundary value problem: We point out that, by direct computations, it is possible to obtain the explicit expression of function z M . However, it is, specially when M > 0, too long, which makes it very difficult to deal with.
3. Expression of the Green's function. In this section we will obtain the explicit expression of the Green's function related to the equation (1) coupled to boundary conditions (4). The result is the following.
has a unique solution if and only if In such a case, it is given by the following expression and w M and C M are defined in (8) and (9) respectively.
Proof. Let v M and w M be the unique solutions of Problems (10) and (6) respectively.
Then, it is clear that u M = v M +λ w M 1 0 u M (s) ds is the unique solution of problem (14). As a consequence, for all t ∈ I, the following equalities are fulfilled: Let A M = 1 0 u M (τ ) dτ , then, from the previous equality, we deduce that or, which is the same, Replacing this value in (17), we arrive at the following expression for function u M : u M (t) = − Proof. Since M ∈ [−m 4 0 , m 4 1 ) we have that g M < 0 and, as a direct consequence of λ ∈ (0, 1/C M ) and the fact that w M > 0 on I for all M < m 4 1 , we conclude, from (16), that G M (t, s) > 0 for all (t, s) ∈ (0, 1) × (0, 1). Now, we denote by It is clear that function ϕ(t, s) is continuous on [0, 1] × (0, 1). Using the properties of g M showed in Lemma 2.6 and those of z M explained in previous section, by means of L'Hôpital rule, we deduce, for all t ∈ (0, 1): Thus, Analogously, if t ∈ (0, 1), we have The limits l 1 (t) and l 2 (t) exist and are finite, so ϕ has removable discontinuities at s = 0, 1, and we can extend it to a function ϕ ∈ C(I × I).

Corollary 2. Let G M (t, s) be Green's function related to problem (3)-(4)
given by expression (16). Then if M ∈ [−m 4 0 , m 4 1 ) and λ ∈ (0, 1/C M ) we have that for all positive constant δ ∈ (0, 1) there exists γ(δ) ∈ (0, 1) for which the following inequality is fulfilled: Proof. The result follows from the fact that function h is continuous on I and strictly positive on (0, 1]. Remark 1. If, instead of problem (1), (4), we consider the linear equation (1) coupled to the adjoint integral boundary conditions: It is immediate to verify that u is a solution of problem (1) So, previous results can be directly adapted to this problem under a simple change of variables.
The same comment is valid for the results concerning nonlinear problems proved in next section.
4. Nonlinear problems. This section is devoted to prove existence and multiplicity of solutions of the nonlinear problem (3) -(4). To this, we will work on the Banach space C(I) endowed with the supremum norm u = max t∈I |u(t)|.
The following result is a direct consequence of the results showed in previous sections. Proof. From the non-negativeness of functions f and G M we deduce that T (u)(t) ≥ 0 for all t ∈ I and u ∈ K. The regularity of both functions allow us to deduce the completely continuous character of operator T as a direct application of Arzela-Ascoli Theorem [12]. Let u ∈ K, by (18), we have, for all t ∈ I, that the following inequalities are fulfilled for all t ∈ I T (u)(t) = So, T (u) ∈ K for all u ∈ K and the proof is complete.
In the sequel, for any pair δ, γ satisfying (20) we introduce the following cone as follows: As in the proof of Lemma 4.2, one can verify the following result. .
In order to use Theorem 2.5, let β : K → [0, +∞) be a functional defined by: Then, it is easy to see that β is a nonnegative continuous and concave functional on K, moreover, for each u ∈ K, one has β(u) ≤ u .
For convenience, we introduce the following notations:

ALBERTO CABADA AND ROCHDI JEBARI
Our first existence result is the following: Theorem 4.5. Choose 0 < γ < 1/R and let a, b, c in R be such that 0 < a < b < b γ R ≤ c. Assume that B ∈ (Λ 3 , +∞) and A ∈ (0, Λ 4 ), the following properties hold: Then, if condition (H 0 ) holds, the boundary value problem (3)-(4) has at least two positive solutions u 1 and u 2 in Proof. First, let us prove that the operator T maps P c into itself. Indeed, if u ∈ P c , then u ≤ c.
Thus, from hypothesis (H 2 ), we have Hence, T (u) ≤ c, that is, T : P c → P c . In the same way, condition (H 2 ) implies that condition (A 2 ) of Theorem 2.5 holds.
Let us see now that condition (A 1 ) of Theorem 2.5 is also fulfilled. Clearly, if u(t) = b γ R then, β(u) > b and u ≤ b γ R , that is Moreover, from (H 3 ), Therefore, condition (A 3 ) in Theorem 2.5 is also satisfied. By Theorem 2.5, there exist three nonnegative solutions u 1 , u 2 and u 3 such that u 1 < a, β(u 2 ) > b and u 3 > a with β(u 3 ) < b.
Since, we cannot ensure that u 1 ≡ 0 and u 2 and u 3 are positive on (0, 1], the result is concluded. then, if we replace the hypothesis (H 1 )-(H 3 ) of Theorem 4.5 by the following ones: (H n,1 ) For all 1 ≤ l ≤ n and u ∈ R such that u ∈ [0, c l ], we have (H n,2 ) For all 1 ≤ j ≤ n − 1 and u ∈ R such that Bb j < Ac j+1 and u ∈ b j , bj γ R , we have we obtain the following result: Proof. In order to prove Theorem 4.6, observe that for n = 1, we know from (H 2 ) that T : P c1 → P c1 . Then it follows from Schauder fixed point theorem that (3)-(4) has at least one positive solution in P c1 . Moreover, for n = 2, it is clear that Theorem 4.5 holds (with a = c 1 , b = b 1 and c = c 2 ). Then, we can obtain three positive solutions x 2 , x 3 , and x 4 . Along this way, we can finish the proof by the induction method. To this aim, we suppose that there exist numbers b j (1 ≤ j ≤ n) and c l (1 ≤ l ≤ n + 1) such that and (H n+1,1 ) and (H n+1,2 ) hold true. We know by the inductive hypothesis that (3)-(4) has at least 2n − 1 positive solutions u i (i = 1, 2, ..., 2n − 1) in P cn . At the same time, it follows from Theorem 4.5, (H n+1,1 ) and (H n+1,2 ) that (3)-(4) has at least three positive solutions u, v and w in P cn+1 such that u < c n , β(v) > b n and w > c n with β(w) < b n . Obviously, v and w are not in P cn . Therefore, (3)-(4) has at least 2n + 1 nonnegative solutions in P cn+1 . By using condition (H 4 ), Problem (3)-(4) has at least 2n + 1 positive solutions.

5.
Examples. In the sequel, we will obtain the different bounds and results for the particular case when M = 0. That is, we want to prove the existence of multiple positive solutions of the problem: subject to the boundary conditions: with 0 < λ < 4. Now, let us obtain the correspondent δ, γ and R. We have to calculate the related Green's function. By means of the Mathematica program developed in Cabada et al [5] we obtain Thus, clearly, for this case By denoting we have and for all t ∈ 1 2 , 1 , ∂ ϕ(t, α(t)) ∂s = 0.