SOLVABILITY FOR A NONLINEAR THREE-POINT BOUNDARY VALUE PROBLEM WITH p-LAPLACIAN-LIKE OPERATOR AT RESONANCE

Let φ be an odd increasing homeomorphism from R onto R which satisfies φ(0) = 0 and let f : [a,b]×R×R → R be a function satisfying Carathéodory conditions. Separated two-point and periodic boundary value problems containing the nonlinear operator (φ(u′))′, or its more particular form, the so-called p-Laplace operator, have received a lot of attention lately (cf. [6, 7, 8, 14, 15] and the references therein). On the other hand, three-point (or m-point) boundary value problems for the case when (φ(u′))′ = u′′, that is, the linear operator, have been considered by many authors (cf. [3, 9, 10, 12, 13]). The purpose of this paper is to study the following three-point boundary value problem which contains the nonlinear operator (φ(u′))′, ( φ(u′) )′ = f (t,u,u′), u′(a) = 0, u(η) = u(b), (1.1)

for a.e. t ∈ [a, b] and all (u, v) ∈ R 2 . Suppose further that there exists an M > 0 such that (1.12) Then the boundary value problem (1.4) has at least one solution in C 1 [a, b] provided that The proofs of Theorems 1.1 and 1.2 are direct applications of Theorems 3.1 and 3.2, respectively.
In Section 4, we prove some existence results with the help of time-mapping techniques as in [6,7,8]. Our main purpose here is to obtain existence results with one-sided growth restrictions for the three-point boundary value problem. Conditions of this type have been considered by Schmitt [20], Mawhin and Ward [18], and Fernandes and Zanolin [4] for the periodic case and the second-order linear differential operator, by de Figueiredo and Ruf in [1] for the second-order linear differential operator and Neumann boundary conditions, and by Manásevich and Zanolin in [16] for the one-dimensional p-Laplacian and Dirichlet boundary value conditions.
We introduce here a technical condition for the homeomorphism φ which will be used in Section 4 in order to guarantee some properties of the time-mapping for non-homogeneous operators (see [6]).
We say that φ satisfies the lower σ -condition if for any σ > 1, lim inf s→+∞ φ(σ s) φ(s) > 1. (1.14) We end this section by stating a theorem which is a consequence of Theorem 4.3 in Section 4 and which illustrates the type of results that we will obtain in that section. We first give the following definitions. For q ∈ L 1 (a, b), we set (1. 15) 194 Solvability for nonlinear three point Also from [2], we recall that the number π p , which will be used below, is defined by π p := 2(p − 1) 1/p 1 0 ds (1 − s p ) 1/p = 2(p − 1) 1/p (π/p) sin(π/p) .
(1.16) Theorem 1.3. Consider the problem where η ∈ (a, b) and q ∈ L 1 (a, b), with q m , q m defined in (1.15) such that The function g : R → R is continuous and satisfies (1.20) then problem (1.17) has at least one solution.
The proof of this theorem will be given in Section 4.

Abstract formulation and a deformation lemma
We begin this section by developing a general continuation theorem for the solvability of problem (1.1). Assume that f * : Then, problem (1.1) has a solution in¯ .
Proof. If (1.1) has a solution in ∂ , then there is nothing to prove, hence we suppose that (1.1) has no solutions belonging to ∂ . This assumption combined with (i) implies that there are no solutions to (2.1) in ∂ for 0 < λ ≤ 1. We show next that (2.1), for λ ∈ (0, 1], is equivalent to an abstract equation. Indeed, define the operator * : We note that for u ∈ C 1 [a, b] and λ ∈ [0, 1], it holds that f * (·, u(·), u (·), λ) ∈ L 1 . Thus the mapping s → s a f * (τ, u(τ ), u (·), λ) dτ is absolutely continuous and hence the operator * is well defined since φ −1 ( s a f * (τ, u(τ ), u (τ ), λ) dτ ) is continuous. Now, by integrating the equation in (2.1) and using the boundary conditions, we find that if u is a solution of (2.1), then it satisfies Next, for λ ∈ (0, 1], assume that u is a solution to (2.5), that is, u satisfies This shows that, for λ ∈ (0, 1], any solution of (2.5) (equivalently (2.7)) is actually a solution to the boundary value problem (2.1). Setting (u) := * (u, 1), we observe that u is a solution of (1.1) if and only if it is a fixed point of .
Standard arguments show that * is a completely continuous operator. Moreover, assumption (i) of Lemma 2.1 can be restated as (2.11) We show next that this is also true for λ = 0. We note from (2.4) that * (u, 0)(t), t ∈ [a, b], is a real constant for each u ∈ C 1 [a, b]. Thus, if for some u ∈ ∂ , u = * (u, 0), (2.12) then, for all t ∈ [a, b], we have that u(t) = s ∈ R, and so u(a) = s. Hence, from (2.7), with λ = 0, which implies that F (s) = 0, for s ∈ R ∩ ∂ , contradicting assumption (ii) of Lemma 2.1. In this manner we have verified that Then, from the homotopy invariance property of the Leray-Schauder degree, it follows that where 0 = ∩R. In this form we obtain that the mapping = * (·, 1) has at least one fixed point in and hence that problem (1.1) has at least one solution in .

First existence results
Consider the boundary value problem (1.1) given in Section 1. We have the following result.
is continuous and satisfies the following conditions.
In our second application we consider the boundary value problem where η ∈ (a, b).
Theorem 3.2. Assume that q ∈ L 1 (a, b), and that q m and q m defined in (1.15) satisfy (1.18). Suppose also that f : [a, b] × R × R → R is Carathéodory and satisfies the following conditions. (i) There are nonnegative functions d 1 , d 2 , and r in L 1 (a, b) such that it follows that problem (3.19) has at least one solution u ∈ C 1 [a, b].
where η ∈ (a, b), and λ ∈ (0, 1). By (3.20), we obtain that and hence as in Theorem 3.1, from (3.26), we find that Next, let u be a solution of (3.26) for some λ ∈ (0, 1). We claim that there is at ∈ [a, b], such that Indeed, integrating the equation of (3.26), we find that Hence if u(t) ≥ d for all t ∈ [a, b], then from the first condition in hypothesis (ii), we find that

t, u(t), u (t) dt < q m (t − a), (3.32)
and hence which again cannot be. Hence in the case that the solution is a constant, say u(t) = c, then necessarily |c| ≤ d.
In this form we find that Combining this inequality with (3.28), it follows that and thus from (3.25) there must be a z 0 > 0 such that |u| ∞ ≤ z 0 . Hence combining with (3.28), we find that there is R 0 > R (R as in hypothesis (iii)), so that for allR , then for all λ ∈ (0, 1), problem

Existence results via time-mapping
In this section we will consider the problem where η ∈ (a, b), g is Carathéodory, and q ∈ L 1 (a, b). In this respect the following obvious modification of Lemma 2.1 will be used. Let g * : [a, b]×R×[0, 1] → R be a function which satisfies the Carathéodory conditions and is such that and for λ ∈ (0, 1], consider the problem (4.5)

Then, problem (4.3) has a solution in¯ .
In our following step we show that under certain conditions on g * solutions to (4.3) which are bounded from above or from below are in fact bounded. See [18,19], for analogous results in the periodic case for the linear operator, that is, φ(s) = s, and [6] for the Neumann case.
Proof. Let u be a solution to (4.3) for some λ ∈ (0, 1). Since the proof of (i) is entirely similar to that of (3.29) of Theorem 3.2, it will be omitted. Also, and as in that theorem, we continue the proof assuming that u is a nonconstant solution.
To prove (ii) we only consider the case max u ≤ R, since the argument in the case min u ≥ −R is completely similar. Thus suppose t 1 and t 2 are, respectively, two points in [a, b] where u reaches its absolute maximum and minimum. We note that t 1 and t 2 belong to [a, b) and thus u (t 1 ) = 0, and u (t 2 ) = 0. We assume t 1 < t 2 , with a similar argument for the other case. Integrating the equation of (4.3) on [t 1 , t 2 ], we find that Then, by (4.9) and hypothesis (4.6), which in turn implies that (4.11) Since g * satisfies the Carathéodory conditions, we find that and where µ = µ R ∈ L 1 (a, b). Then, the last integral in (4.11) can be bounded from above by B (µ(t) − q m ) dt, and thus Now, by integrating the equation of (4.3) on [t 1 , t], we find first that (4.14) and then, using (4.13), we find a constant C 2 (R) such that We observe that |u| ∞ is reached at t 1 or t 2 . Also we note that there must be a point t 3 ∈ (t 1 , t 2 ) such that |u(t 3 )| ≤ d. Thus by integrating (4.15) from t 1 to t 3 , and assuming first that |u| ∞ is reached at t 1 , we obtain that where, without loss of generality, we have taken ρ(R) ≥ R. Since a similar argument applies if |u| ∞ is reached at t 2 , the proof of (ii) is completed. Finally we prove (iii) by using Lemma 4.1. Let R ≥ d be such that there is no solution u to (4.3), with λ ∈ (0, 1) and max u = R (the other case being analogous). Let ρ(R) be the bound given in (ii), and define For λ ∈ (0, 1), suppose u is a solution to (4.3). We claim that u ∈ ∂ . Indeed, if u ∈¯ , then −R 1 ≤ u(t) ≤ R for all t ∈ [a, b] and thus by our hypotheses, u(t) < R. Now from the choice of R 1 > ρ(R) ≥ R we have that u(t) > −R 1 , concluding that −R 1 < u(t) < R for all t ∈ [a, b]. Thus u ∈ and (i) of Lemma 4.1 is satisfied.
Next we note that ∩ R = (−R 1 , R) and ∂ ∩ R = {−R 1 , R}. Also by hypothesis (4.6), it follows that  We continue by reviewing some basic facts concerning time-mappings. Thus consider the equation where h : R → R satisfies lim s→+∞ h(s) sgn(s) = +∞. This equation can be equivalently written as the autonomous system for all t ∈ R. Hence y(t) > 0 and from (4.22), u (t) > 0 for all t ∈ (0, T ). Thus 0 < u(t) < R for all t ∈ (0, T ), with y(T ) = 0. Therefore we obtain where r denotes the right inverse of * , that is, the inverse of the restriction of * to [0, +∞). Then, from the first equation in system (4.22), . (4.32) We call this function T h so far defined for large positive values of R, the timemapping of h with respect to φ, or simply the time-mapping of h. In a similar form we can define T h for large negative values. Indeed by assuming that u min = u(−T ) = −R,R > 0, we havẽ where now l denotes the left inverse of * . We note that in our case l (s) = − r (s), since we are assuming φ is odd.
Next define a one-parameter family of functions by    it follows from (4.34) and the definition of e that hypothesis (4.6) of Lemma 4.2 holds, thus the conclusion of that lemma applies to problem (4.43).