A THIRD-ORDER 3-POINT BVP. APPLYING KRASNOSEL’SKIĬ’S THEOREM ON THE PLANE WITHOUT A GREEN’S FUNCTION.

Consider the three-point boundary value problem for the 3rd order differential equation: � x 000 (t) = �(t)f(t,x(t), x 0 (t),x 00 (t)), 0 < t < 1, x(0) = x 0 (�) = x 00 (1) = 0, under positivity of the nonlinearity. Existence results for a positive and con- cave solution x(t), 0 � t � 1 are given, for any 1/2 < � < 1. In addition, without any monotonicity assumption on the nonlinearity, we prove the exis- tence of a sequence of such solutions with lim n!1 ||xn|| = 0. Our principal tool is a very simple applications on a new cone of the plane of the well-known Krasnosel'skiuo's fixed point theorem. The main feature of this aproach is that, we do not use at all the associated Green's function, the necessary positivity of which yields the restriction � 2 (1/2,1). Our method still guarantees that the solution we obtain is positive.

In the above papers there are no assumptions for singularity of the nonlinearity f at the point u = 0. Zhang and Wang [29] and recently Liu [18] obtained some existence results for a singular nonlinear second order 3-point boundary-value problem, for the case where only singularity of q(t) at t = 0 or t = 1 is permitted. Other applications of Krasnosel'skiȋ's fixed point theorem to semipositone problems can, for example, be found in [1]. Further recently interesting results have been proved in [4], [11], or [26]. Anderson and Avery [2] and Anderson [3], proved that there exist at least three positive solutions to the BVP (1.1) (below) and the analogous discrete one respectively, by using the Leggett-Williams fixed point theorem. Yao in [28] and Haiyan and Liu in [10], using the Krasnosel'skiȋ's fixed point theorem showed the existence of multiple solutions to the BVP (1.1). More similar results can be found in Du et al [6] and also in Feng and Webb [7].
Recently, Du et al [5] via the coincidence degree of Mawhin, proved existence for the BVP at the resonance case. In an also recent paper Sun [25], obtained existence of infinitely many positive solutions to the BVP (1. 1) u (t) = λα (t) f (t, u(t)), 0 < t < 1, u (0) = u (η) = u (1) = 0, η ∈ (1/2, 1) mainly under superlinearity on the nonlinearity f of the type There exist two positive constants θ, R = r such that where M and N are also constants. Sun, in order to obtain his existence results applied the classical Krasnosel'skii fixed-point theorem on cone expansion-compression type and furthermore to prove his multiplicity results he assumed monotonicity of the nonlinearity with respect the second variable.
Very recently there have been several papers on third-order boundary value problems. Hopkins and Kosmatov [12], Li [17], Liu et al [19,20], Guo et al [9] and Kang et al [22] have all considered third-order problems. Graef and Yang [8] and Wong [27] consider three-point focal problems, while Palamides and Smyrlis [23] consider the three-point boundary conditions In this work, motivated by the above mentioned papers and especially the ones of Sun [25] and Palamides and Smyrlis [23], we suppose a superlinearity-type growth rate of f (t, u, u , u ) at both the origin u = 0 and u = +∞. The emphasis in this paper is mainly to apply the well-known Krasnosel'skiȋ's fixed point theorem just on the plane, using in this way an alternative to the classical methodologies, in which as it is common, a Banach space of functions is used. We combine the above Krasnosel'skii's theorem with properties of the associating vector field, defined on the phase plane and this results in the use of similar quite natural hypothesis.
Furthermore we prove existence of infinitely many positive solutions for the more general boundary value problem and at the same time, we eliminate at all the related monotonicity assumption on the nonlinearity in [25]. EJQTDE, 2008 No. 14, p. 2

Preliminaries
Consider the third-order nonlinear boundary value problem (E), where we assume (within this paper) that η ∈ (1/2, 1), the continuous functions α (t) , t ∈ (0, 1) and F ∈ C (Ω, [0, +∞)) are nonnegative and Ω = Then, a vector field is defined with crucial properties for our study. More precisely, considering the (x , x ) phase semi-plane (x > 0), we easily check that x, x , x ) ≥ 0. Thus, any trajectory (x (t), x (t)), t ≥ 0, emanating from any point in the fourth quadrant: "evolutes" in a natural way, when x (t) > 0, toward the negative x −semi-axis. Then, when x (t) ≤ 0, the trajectory "evolutes" toward the negative x −semi-axis and finally it stays asymptotically in the second quadrant. As a result, assuming a certain growth rate on f (e.g. a superlinearity), we can control the vector field in a way that assures the existence of a trajectory satisfying the given boundary conditions. These properties, which will be referred as "the nature of the vector field", combined with the Krasnosel'skii's principle, are the main tools that we will employ in our study. In this paper, we employ a simple cone on the phase plane. First we recall the next definition: EJQTDE, 2008 No. 14, p. 3 For example, the above fourth quadrant on the plane R 2 is a cone. We need a preliminary result from the fixed point theory, which will be our base for all results in this paper.
Precisely will apply the well known Krasnosel'skiȋ's fixed point theorem in cones.
Lemma 1. Let E be a Banach space and K * ⊂ E a cone in E. Assume that Ω 1 and Ω 2 are open subsets of E with 0 ∈ Ω 1 andΩ 1 ⊂ Ω 2 . Let be a completely continuous operator. We assume furthermore either Then T has a fixed point in K * ∩ (Ω 2 \Ω 1 ) .
Consider the third-order nonlinear three-point boundary value problem: where f is a continuous extension of F, i.e.
Remark 1. By the sign property of F, it follows that Then for any initial value (u 0 , u 0 ) with u 0 ≥ −2u 0 . EJQTDE, 2008 No. 14, p. 4 Proof. By the Taylor's Formula and (3.3), we get u (t) > 0 for all t in a (right) neighborhood of t = 0. Assume that there exists a t * ∈ (0, 1) such that Given that u 0 ≥ −2u 0 , we get, noticing the sign of the nonlinearity Assume throughout of this paper, that 0 < θ < 1/2 and there exist positive constants r 0 and R 0 with such that for every 0 < r ≤ r 0 and any R ≥ R 0 , Proof. We choose (without loss of generality) Furthermore, EJQTDE, 2008 No. 14, p. 5 Thus by the mean value theorem, . Consequently in view of the Remark 1 and the assumption (A 1 ), we obtain the contradiction On the other hand, again by Taylor's formula and condition (A 1 ), We recall choices (3.4) and r 0 1 + η 2 ≤ ηR 0 and fix the obtained initial point Proposition 2. The derivative of every solution u = u (t) of (3.1) emanating from any initial point P 1 = (u 1 , u 1 ) ∈ [A, B] (we denote in the sequel such a choice by u ∈ X (P 1 )) satisfies Proof. We assume on the contrary that u (η) ≤ 0 and notice that it follows that Consider now the two possible cases: EJQTDE, 2008 No. 14, p. 6 This clearly implies that u (t) ≥ 0, 0 ≤ t ≤ t * and furthermore we have In view of (3.6) and Taylor's formula, we get the contradiction • Let us assume now that u (1) ≥ 0. Then there exists at ∈ (0, 1] with u t = 0 and u (t) ≤ 0, 0 ≤ t ≤t.
As above we conclude immediately that the function u (t) , 0 ≤ t ≤t is decreasing. If u t > 0, then, in view of the nature of vector field, we obtain u (t) > 0, 0 ≤ t ≤ 1, a contradiction to u (η) ≤ 0. Hence u t ≤ 0 and thus we get a point t * ≤t such that Then as above, Taylor's formula also leads to another contradiction u (t * ) > 0. where ||y|| = max 0≤t≤1 y (t) .
Proof. Since y (t) ≥ 0, the function y (t) is nondecreasing. So noticing y (1) ≤ 0, this implies that y (t) ≤ 0, 0 < t < 1. Now due to the concavity of y (t), for any µ, t 1 and t 2 in [0, 1] , we have Moreover using the assumption y (η) ≤ 0, we conclude that there is a t * ∈ (0, η) such that y (t * ) = 0 and ||y|| = y (t * ) . Therefore The next result is crucial for the sequence of our theory. Then No. 14, p. 7 Proof. Suppose that there is a T ∈ (η, 1) such that u (t) > 0, t ∈ (0, T ) , u (T ) = 0 and u (t) < 0, t ∈ (T, 1]. Since η ∈ (1/2, 1) , we get 2η − T ≥ 0. Consider then, two symmetric with respect to η, partitions The map u = u (t) , t ∈ [0, 1] is nondecreasing and thus we get In addition, since the map u = u (t) , 0 ≤ r ≤ T is continuous (and bounded), we can choose the max{r i − r i−1 : i = 1, 2, ..., k} small enough and given that 2η − T ≥ 0, we obtain  Proof. We will show (extending partially the conclusion of previous Proposition 2) first that If not, then proceeding as in the proof of Proposition 2, we have u (t * ) = η −1 R 0 for some t * ∈ (0, 1], u (t) ≥ η −1 R 0 , t ∈ (0, t * ) . Then we get the contradiction (see EJQTDE, 2008 No. 14, p. 8 Hence, given that u (t) ≤ u A , 0 ≤ t ≤ 1, we obtain and this yields Remark 3. We need some concepts, in the sequel, concerning the case where initial value problems have not a unique solution. Consider a set-valued mapping F , which maps the points of a topological space X into compact subsets of another one Y. F is upper semi-continuous (usc) at x 0 ∈ X iff for any open subset V in Y with F (x 0 ) ⊆ V, there exists a neighborhood U of x 0 such that F (x) ⊆ V , for every x ∈ U. Let P be any initial point such that every solution u ∈ X (P ) is defined on the interval [0, η] . Then, by the well-known Knesser's property (see [13,24]), the cross-section X (η; P ) = {u (η) , u (η) , u (η)) : u ∈ X (P )} is a continuum (compact and connected set) in R 3 , the same being its projections {u (η) : u ∈ X (P )} and {u (1) : u ∈ X (P )} . Furthermore the image of a continuum under an upper semi-continuous map K is again a continuum. Also considering the set-valued mapping we notice (see [24]) that it is an upper semi-continuous mapping. Obviously, if an IVP has a unique solution, then this map is simply continuous.
Lemma 5. Let P 1 = (u 0 , u 1 ) be a point in the face [K, B] such that u (η) = 0, for some solution u = u (t) emanating from the initial point P 1 i.e. u ∈ X (P 1 ). Then Proof. We notice firstly that such a point P 1 always exists, because of Propositions 1 and since by the sign of the nonlinearity, u (η) > 0 for every u ∈ X (B). Indeed in view of the Remark 3, the image of the segment [K, B] under the map X , that is is a continuum. Hence its projection {u (η) : u ∈ X (P ) : P ∈ [K, B]} crosses the negative u −semi axis of the phase-plane.

Recalling the Remark 3, we state the next
The upper semicontinuity of the map K yields the existence of an open ball U (P 0 ) centering at P 0 , such that for all P ∈ U (P 0 ) But this clearly means that (3.9) u (η) < 0, ∀u ∈ X (U (P )) .
Hence P 0 is an interior point of Ω 1 , that is Ω 1 is an open set, a contradiction. Similarly someone can prove that Ω 2 is also open.
Then in view of assumptions (A 1 ) and (A 2 ), we may apply once again the Krasnosel'skiȋ's theorem on the triangle [K 1 , A 1 , B 1 ] , to obtain another positive solution u = u 2 (t) of the BVP (3.1), (3.2). By the construction of [K 1 , A 1 , B 1 ] and (4.1), it is obvious that u = u 2 (t) is different than the solution u = u 1 (t) , 0 ≤ t ≤ 1, obtained in the previous Theorem 1.

Discussion
If we assume that both functions α (t) and f (t, x, y, z) are negative, we may easily demonstrate similar existence and multiplicity results. Indeed, considering the (x , x ) face semi-plane (x ≤ 0), we easily check that x = α (t) f (t, x, x , x ) < 0. Thus, any trajectory (x (t), x (t)), t ≥ 0, emanating from any point in the second quadrant {(x , x ) : x < 0, x > 0} "evolutes" in a natural way, when x (t) < 0, toward the positive x −semi-axis and then, when x (t) ≥ 0 toward the positive x −semi-axis. As a result, under a certain growth rate on f , we can control the vector field in a way that assures the existence of a trajectory satisfying the given boundary conditions. Let's notice that in present situation, the obtaining solution (x (t) , x (t)) is convex, in contrast to the previous case, where it is concave (see Fig. 1).
Furthermore we could easily get analogous results, for the case when the nonlinearity is sublinear.