Countably many solutions of a fourth order boundary value problem, Electron

We apply xed point theorems to obtain sucien t conditions for existence of innitely many solutions of a nonlinear fourth order boundary value problem

EJQTDE, 2004 No. 12, p. 1 The paper is organized in the following fashion.In the introduction we briefly discuss the background of the problem and give an overview of related results.In section 2 we introduce the assumptions on the inhomogeneous term of (1), discuss the properties of the Green's function of the homogeneous (1), (2), and state the theorems that we use to obtain our main results presented in sections 3 and 4.
Fixed point theorems have been applied to various boundary value problems to show the existence of multiple positive solutions.An overview of numerous such results can be found in Guo and Lakshmikantham [4] and Agarwal, O'Reagan and Wong [1].
The study of sufficient conditions for the existence of infinitely many positive solutions was originated by Eloe, Henderson and Kosmatov in [3].The authors of [3] Their approach was based on applications of cone-theoretic theorems due to Krasnosel'skiȋ and Leggett-Williams [16].For applications of the latter see Davis and Henderson [2] and Henderson and Thompson [6] and the references therein.Later, in [13,14], the author obtained infinitely many solutions for the second order BVP where α, β, γ, δ ≥ 0, αγ + αδ + βγ > 0. In addition, we point out that [13,14] only dealt with a very special choice of a singular (L 1 ) integrable EJQTDE, 2004 No. 12, p. 2 function a(t).The study of infinitely many solutions was further developed by Kaufmann and Kosmatov [9,10] to extend the results [13,14] to the general case of a(t) ∈ L p [0, 1] for p ≥ 1.In [9], a(t) was taken to possess countably many singularities (or to be an infinite series of singular functions) and, in [10], a(t) was selected in the form of a finite product of singular functions.It is also relevant to our discussion to mention [8,11,12] devoted to BVP's on time scales and three-point BVP's.
In this note, we extend the results of Yao [18] (with λ = 1).We also generalize and refine the results of [3] (with n = 4, k = 2) by obtaining sharper sufficient conditions for existence of infinitely many solutions of (1), (2).

Auxiliaries and fixed point theorems
The Green's function of Definition 2.1 Let B be a Banach space and let K ⊂ B be closed and nonempty.Then K is said to be a cone if 1. αu + βv ∈ K for all u, v ∈ K and for all α, β ≥ 0, and 2. u, −u ∈ K implies u ≡ 0.
We let B = C[0, 1] with the norm u = max t∈[0,1] |u(t)|.In the sequel of our note we take τ ∈ [0, 1  2 ) and define our cone where c τ = 2 3 τ 4 .We define an operator T : B → B by The required properties of T are stated in the next lemma.
Proof: By Arzela-Ascoli theorem, T is completely continuous.Now we show that it is cone-preserving.To this end, if s With the estimate above we have Fixed points of T are solutions of (1), (2).The existence of a fixed point of T follows from theorems due to Krasnosel'skiȋ and Leggett-Williams.Now we state the former.

and let
be a completely continuous operator such that either EJQTDE, 2004 No. 12, p. 4 Then T has a fixed point in K ∩ (Ω 2 \ Ω 1 ).
To introduce Leggett-Williams fixed point theorem we need more definitions.

Definition 2.4
The map α is said to be a nonnegative continuous concave functional on a cone K of a (real) Banach space B provided that α : K → [0, ∞) is continuous and for all u, v ∈ K and 0 ≤ t ≤ 1. Definition 2.5 Let 0 < a < b be given and α be a nonnegative continuous concave functional on a cone K. Define convex sets The following fixed point theorem due to Leggett and Williams enables one to obtain triple fixed points of an operator on a cone.Then T has at least three fixed points u 1 , u 2 , and u 3 such that u 1 < a, b < α(u 2 ), and To obtain some of the norm inequalities in Theorems 2.3 and 2.6 we employ Hölder's inequality.EJQTDE, 2004 No. 12, p. 5 and 1 p The following assumptions on the inhomogeneous term of (2) will stand throughout this paper: (A1) f is nonnegative and continuous; (A2) lim t→t 0 a(t) = ∞, where 0 < t 0 < 1; (A3) a(t) is nonnegative and there exists m > 0 such that a(t) ≥ m a.e. on [0, 1]; Any fixed points of T are now positive.
We will need to employ some estimates on (3) that are given below.One can readily see that max The function attains its maximum on the interval [0, 1] at t = 1 2 and max t∈[0,1] EJQTDE, 2004 No. 12, p. 6 and its minimum on the interval [τ, 1 − τ ] at t = τ, 1 − τ so that Using the fact that (7) attains its minimum on the interval [τ , τ ] ⊂ (0, 1 2 ) at one of the end-points and defining we get that for all τ ∈ [τ , τ ].It follows from ( 6) that max One can also easily see from ( 5) that max G q (t, s) ds Remark: Other estimates on (3) (used in construction of cones) can be found in [3,18].
Assume that f satisfies (H2) and Then the boundary value problem ( 1), ( 2) has infinitely many solutions Proof: We now use (10) and repeat the argument above.
Our last result corresponds to the case of p = 1.
4 Positive solutions and Legett-Williams fixed point theorem In this section we only consider the case of p > 1.The existence theorems corresponding to the cases of p = 1 and p = ∞ are similar to the next theorem and are omitted.
For our cone we now choose and our nonnegative continuous concave functionals on K are defined by EJQTDE, 2004 No. 12, p. 11 where M 1 is as in Theorem 3.1.Suppose that f satisfies ) and l(τ * , τ 1 ) is given by (8).
Then the boundary value problem (1), (2) has three infinite families of solutions {u 1k } ∞ k=1 , {u 2k } ∞ k=1 , and Proof: As in Definition 2.5, set for each k ∈ N, We use (H5) and (H7) and repeat the argument leading to (12) to see that Thus, the condition (C2) of Theorem 2.6 is satisfied.As in Definition 2.5, set

Theorem 2 . 6
Let T : B c → B c be a completely continuous operator and let α be a nonnegative continuous concave functional on a cone K such that α(u) ≤ u for all u ∈ B c .Suppose there exist 0 < a < b < d ≤ c such that (C1) {u ∈ P (α, b, d)| α(u) > b} = ∅ and α(T u) > b for u ∈ P (α, b, d), (C2) T u < a for u ≤ a, and (C3) α(T u) > b for u ∈ P (α, a, b) with T u > d.