Positive solutions for a fourth-order three-point BVP with sign-changing Green ’ s function

This paper is concerned with the following fourth-order three-point boundary value problem { u(4)(t) = f ( t, u(t) ) , t ∈ [0, 1], u′(0) = u′′(0) = u′′′(η) = u(1) = 0, where η ∈ [ 1 3 , 1 ) . In spite of sign-changing Green’s function, for arbitrary positive integer n (≥ 2), we still obtain the existence of at least n− 1 decreasing positive solutions to the above problem by imposing some suitable conditions on f . The main tool used is the fixed point index theory.


Introduction
Boundary value problems (BVPs for short) of fourth-order ordinary differential equations have received much attention due to their striking applications in engineering, physics, material mechanics, fluid mechanics and so on.Many authors have studied the existence of single or multiple positive solutions to some fourth-order BVPs by using Banach contraction theorem, Guo-Krasnosel'skii fixed point theorem, Leray-Schauder nonlinear alternative, fixed point index theory in cones, monotone iterative technique, the method of upper and lower solutions, degree theory, critical point theorems in conical shells and so forth.However, it is necessary to point out that, in most of the existing literature, the Green's functions involved are nonnegative, which is an important condition in the study of positive solutions of BVPs.
Recently, there have been some works on positive solutions for second-order or third-order BVPs when the corresponding Green's functions are sign-changing.For example, Zhong and u + ρ 2 u = f (u), 0 < t < T, u(0) = u(T), u (0) = u (T), where 0 < ρ ≤ 3π 2T .The main tool used was the fixed point index theory of cone mapping.In 2008, for the singular third-order three-point BVP with an indefinitely signed Green's function where η ∈ 17  24 , 1 , Palamides and Smyrlis [14] discussed the existence of at least one positive solution.Their technique was a combination of the Guo-Krasnosel'skii fixed point theorem and properties of the corresponding vector field.In 2012, by applying the Guo-Krasnosel'skii and Leggett-Williams fixed point theorems, Sun and Zhao [17,18] obtained the existence of single or multiple positive solutions for the following third-order three-point BVP with signchanging Green's function For relevant results, one can refer to [3,4,9,10,13,15,16,19].It is worth mentioning that there are other type of achievements on either sign-changing or vanishing Green's functions which prove the existence of sign-changing solutions, positive in some cases, see [1,2,5,7,8,12].
Motivated and inspired by the above-mentioned works, in this paper, we are concerned with the following fourth-order three-point BVP with sign-changing Green's function By imposing some suitable conditions on f and η, we obtain the existence of at least n − 1 decreasing positive solutions to the BVP (1.1) for arbitrary positive integer n (≥ 2).
To end this section, we state some knowledge of the classical fixed point index for compact maps [6].
Let K be a cone in a Banach space X.If Ω is a bounded open subset of K (in the relative topology) we denote by Ω and ∂Ω the closure and the boundary relative to K. When D is an open bounded subset of X we write D K = D ∩ K, an open subset of K. Theorem 1.1.Let D be an open bounded set with D K = ∅ and D K = K.Assume that T : D K → K is a compact map such that x = Tx for x ∈ ∂D K .Then the fixed point index i K (T, D K ) has the following properties.
In the remainder of this paper, we always assume that η ∈ 1 3 , 1 .Now, if we let h then it is easy to know that h(x) is strictly decreasing on [0, 1], which together with h(0) > 0 and h(η) < 0 implies that there exists a unique x 0 ∈ (0, η) such that h(x 0 ) = 0. Obviously, x 0 is dependent on η.In fact, a direct calculation shows that , where For example, if we choose η = 1 3 , then x 0 ≈ 0.2572437.From now on, we suppose that θ ∈ (0, x 0 ] is a constant.
Theorem 3.1.Suppose that f : [0, 1] × [0, +∞) → [0, +∞) is continuous and satisfies the following conditions: (A 1 ) for any x ∈ [0, +∞), the mapping t → f (t, x) is decreasing; (A 2 ) for any t ∈ [0, 1], the mapping x → f (t, x) is increasing; (A 3 ) there exist three positive constants r i , i = 1, 2, 3 with r 1 < r 2 < r 3 such that either Then the BVP (1.1) has at least two decreasing positive solutions u 1 and u 2 satisfying Proof.Let K = u ∈ X : u(t) is decreasing and nonnegative on [0,1], and min Then it is easy to verify that K is a cone in X.Now, we define an operator T on K by Obviously, if u is a fixed point of the operator T, then u is a decreasing and nonnegative solution of the BVP (1.1).First, we assert that T : K → K. To see this, suppose u ∈ K. Then by (A 1 ), (A 2 ), Lemma 2.2 and (1) of Lemma 2.3, we get and for t ∈ (η, 1], it follows from u ∈ K, (A 1 ), (A 2 ) and η ∈ 1 3 , 1 that Thus, (Tu) (t) ≤ 0 for all t ∈ [0, 1], which shows that (Tu we know that (Tu)(t) is concave on [0, η], which together with 0 < θ ≤ x 0 < η and the fact that (Tu)(t) is decreasing and nonnegative on [0, 1] indicates that This proves that T : K → K.
Next, it follows from known textbook results, for example see Proposition 3.1 [11, p. 164], that T : K → K is compact.
Since the proof of the case when (b) of (A 3 ) is satisfied is similar, we only consider the case when (a) of (A 3 ) is fulfilled.Let On the one hand, for any u ∈ ∂Ω r i , i = 1, 3, we have which together with (1) of Lemma 2.3, (A 1 ), (A 2 ), (a) of (A 3 ) and the fact T : K → K implies that This indicates that Tu < u for any u ∈ ∂Ω r i , i = 1, 3. Hence, by (2) of Theorem 1.1, we get On the other hand, for any u ∈ ∂Ω r 2 , we have Let e(t) ≡ 1 for t ∈ [0, 1].Then it is obvious that e ∈ K \ {0}.Now, we prove that u = Tu + λe for all u ∈ ∂Ω r 2 and all λ ≥ 0. Suppose on the contrary that there exist u * ∈ ∂Ω r 2 and λ * ≥ 0 such that u * = Tu * + λ * e.Then it follows from u * ∈ K, Lemma 2.3, (3.2), (A 1 ), (A 2 ) and (a) of (A 3 ) that which is a contradiction.This shows that u = Tu + λe for all u ∈ ∂Ω r 2 and all λ ≥ 0. Hence, an application of (1) of Theorem 1.1 yields that Therefore, it follows from (3.1), (3.3) and (3) of Theorem 1.1 that T has fixed points u 1 and u 2 in K with r 1 < u 1 < r 2 < u 2 < r 3 , which are two desired decreasing positive solutions of the BVP (1.1).
Similarly, we can obtain the following more general result.
Then the BVP (1.1) has at least n − 1 decreasing positive solutions u i satisfying Example 3.3.Consider the following BVP In what follows, we verify that all the conditions of Theorem 3.1 are satisfied.
Next, we show that (b) of (A 3 ) holds, that is, there exist three positive constants r i , i = 1, 2, 3 with r 1 < r 2 < r 3 such that f (0, r 2 ) < For some examples, one can see the following table :