boundary value problems

In this paper, by employing fixed point index theory and Leray-Schauder degree theory, we obtain the existence and multiplicity of sign-changing solutions for nonlinear second-order differential equations with integral boundary value conditions.


Introduction
In this paper, we are concerned with the existence of sign-changing solutions for the following nonlinear second-order integral boundary-value problem (BVP for short) x(0) = 0, x(1) = 1 0 a(s)x(s)ds, where f ∈ C(R, R), a ∈ L[0, 1] is nonnegative with 1 0 a 2 (s)ds < 1. Nonlocal boundary value problems have been well studied especially on a compact interval.For example, Gupta et al. have made an extensive study of multi-point boundary value problems in [4,5,6].Many researchers have studied positive solutions for multi-point boundary value problems, and obtained sufficient conditions for existence, see [4,5,6,15,28] and the references therein.Nodal solutions for multi-point boundary value problems have also been paid much attention by some authors, see [13,14,17,19] for reference.
Boundary-value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics.Moreover, boundary-value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems.They include two, three, multi-point and integral boundary-value problems as special cases, see [7,8,21,22].For boundary value problems with other integral boundary conditions and comments on their importance, we refer the reader to the papers by Karakostas and Tsamatos [7,8], Yuhua Li and Fuyi Li [11], Lomtatidze and Malaguti [12], Webb and Infante [21,22] and Yang [26,27] and the references therein.
In [21,22], Webb and Infante used fixed point index theory and formulated a general method for solving problems with integral boundary conditions of Riemann-Stieltjes type.In [22] they studied the existence of multiple positive solutions of nonlinear differential equations of the form with boundary conditions including the following where α, β are linear functionals on C[0, 1] given by with A, B functions of bounded variation.By giving a general approach to cover all of these boundary conditions (and others) in a unified way, their work included much previous work as special cases and improved the corresponding results.We should note that the work of Webb and Infante does not require the functionals α[u], β[u] to be positive for all positive u.
Recently, Xu [24] considered the following second-order multi-point boundary value problem where . By using fixed point index theory, under some suitable conditions, they obtained some existence results for multiple solutions including sign-changing solutions.In [17], Rynne carefully investigated the spectral properties of the linearisation of BVP (2) which are used to prove a Rabinowitz-type global bifurcation theorem for BVP (2).Then nodal solutions of the above problem are obtained by using this global bifurcation theorem.Thereafter, employing similar method to [24], Pang, Dong and Wei [16], Wei and Pang [23], Li and Liu [10] proved the existence of multiple solutions to some fourth two-point and multi-point boundary value problems, respectively.
Recently, utilizing the fixed point index theory and computing eigenvalues and the algebraic multiplicity of the corresponding linear operator, Li and Li [11] obtained some existence results for sign-changing solutions for the following integral boundary-value problem EJQTDE, 2010 No. 44, p. 2 where f, g ∈ C(R, R).
To the authors' knowledge, there are few papers that have considered the existence of multiple sign-changing solutions for integral boundary value problems.Motivated by [11], [17], [19], [21], [22], the purpose of this paper is to investigate sign-changing solutions for BVP (1) following the method formulated by Xu in [24].Obviously, BVP (1) can not be included in BVP (3).We will show that BVP (1) has at least six different nontrivial solutions when f satisfies certain conditions: two positive solutions, two negative solutions and two sign-changing solutions.Moreover, if f is also odd, then the BVP (1) has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.
We shall organize this paper as follows.In Section 2, some preliminaries and lemmas are given including the study of the eigenvalues of the linear operator A ′ (θ) and A ′ (∞).The main results are proved by using the fixed point index and Leray-Schauder degree method in section 3. A concrete example is given to illustrate the application of the main results in Section 4.

Preliminaries and several lemmas
Then, (E, • ) is a Banach space and P is a cone of E. Let Define operators K, F and A as follows where Denote the set of eigenvalues of the operator K by σ(K), since K is a completely continuous operator, σ(K) is countable.Throughout this paper, we adopt the following hypotheses.
(H 2 ) For each of the sets α 0 σ(K) and β 0 σ(K), the number of elements greater than 1 is even, where α 0 σ(K) = {y : The main results of this paper are the following.
Theorem 1 Suppose that (H 1 ) − (H 3 ) hold.Then integral boundary-value problem (1) has at least two sign-changing solutions.Moreover, the integral boundary-value problem (1) also has at least two positive solutions and two negative solutions.
Theorem 2 Suppose that (H 1 ) − (H 3 ) hold, and, f is odd, i.e., f (−x) = −f (x) for all x ∈ R. Then integral boundary-value problem (1) has at least eight different nontrivial solutions, which are four sign-changing solutions, two positive solutions and two negative solutions.
Before giving the proofs of Theorems 1 and 2, we list some preliminary lemmas.
if and only if x ∈ C[0, 1] is a solution of the integral equation Remark 1. Lemma 1 was shown in [1] by a direct calculation, a more general results for a Riemann-Stieltjes integral boundary value condition was shown in [21,22] by a simple method.Obviously, A, K : E → E are all completely continuous operators.By Lemma 1, we know that x is a solution of BVP (1) if and only if x is a fixed point of operator A.
Proof.For any ε > 0, by (4), there exists δ > 0 such that for any 0 It is easy to see from (H 1 ) that f (0) = 0.Then, for any x ∈ E with x < δ, by Lemma 2, we have This implies On the other hand, for any x ∈ E, x < δ, Thus, By ( 10) and ( 11),we have which means that A is Fréchet differentiable at θ and A ′ (θ) = α 0 K.For any ε > 0, by (4), there exists R > 0 such that As a consequence, This implies that EJQTDE, 2010 No. 44, p. 5 Similarly, we can show that By ( 12) and ( 13), we have Consequently, Lemma 4 Let β be a positive number.Then the sequence of positive eigenvalues of the linear operator βK is countable.Moreover, the algebraic multiplicity of each positive eigenvalue of the operator βK is equal to 1.
Proof.The set of positive eigenvalues of the linear operator βK is countable because To prove all eigenvalues are simple we can, without loss of generality, take β = 1.An eigenvector x of K corresponding to λ > 0 is a nonzero solution of the problem Next, we show that the algebraic multiplicity of each positive eigenvalue of the operator K is equal to 1.To do this we will show that for each eigenvalue λ = 1/k 2 , where k is a solution of ( 14), the following inclusion is valid, The solution of this problem is of the form c 1 sin(kt) + c 2 t cos(kt), where C = 0 implies c 2 = 0. Since the term sin(kt) satisfies the boundary condition at t = 1 so must the term t cos(kt), that is, Noting that , from ( 14) and ( 15) we have This contradiction proves that we must have C = 0 so the eigenvalue is simple.
We recall three well known lemmas, they can be found, for example, in [2].
Lemma 6 Let θ ∈ Ω and A : P ∩ Ω → P be completely continuous.Suppose that Then i(A, P ∩ Ω, P ) = 1.Lemma 8 Let A : P → P be completely continuous.Suppose that A is differentiable at θ and ∞ along P and 1 is not an eigenvalue of A ′ + (θ) and A ′ + (∞) corresponding to a positive eigenfunction.(i) If A ′ + (θ) has a positive eigenfunction corresponding to an eigenvalue greater than 1, and Aθ = θ.Then there exists τ > 0 such that i(A, P ∩ B(θ, r), P ) = 0 for any 0 < r < τ .
(ii) If A ′ + (∞) has a positive eigenfunction which corresponds to an eigenvalue greater than 1.Then there exists ζ > 0 such that i(A, P ∩ B(θ, R), P ) = 0 for any R > ζ.
We now prove an important result for our work.
Proof.We prove only conclusion (i), conclusion (ii) can be proved in a similar way.By Lemma 3, and the corresponding eigenfunction is sin(kt).By assumption, there are an even number of eigenvalues of α 0 K greater than 1, in particular, α 0 /k 2 1 > 1 where k 1 is the smallest positive root of (16).We show that k 1 ∈ (0, π) so that the corresponding eigenfunction is positive on (0, 1).
Define a real function F by a(s) sin(ks)ds.
Lemma 10 Let A be a completely continuous operator, let x 0 ∈ E be a fixed point of A and assume that A is defined in a neighborhood of x 0 and Fréchet differentiable at x 0 .If 1 is not an EJQTDE, 2010 No. 44, p. 8 eigenvalue of the linear operator A ′ (x 0 ), then x 0 is an isolated singular point of the completely continuous vector field I − A and for small enough r > 0 where k is the sum of the algebraic multiplicities of the real eigenvalues of A ′ (x 0 ) in (1, +∞).
Lemma 11 Let A be a completely continuous operator which is defined on a Banach space E. Assume that 1 is not an eigenvalue of the asymptotic derivative.The completely continuous vector field I − A is then nonsingular on spheres S ρ = {x : x = ρ} of sufficiently large radius ρ and where k is the sum of the algebraic multiplicities of the real eigenvalues of A ′ (∞) in (1, +∞).
To simplify the proof of the main results, we will use the following lemma.
Proof of Theorem 2 According to Theorem 1, the BVP (1) has at least six different nontrivial solutions It follows from f (−x) = −f (x) for all x ∈ R that −x 5 and −x 6 are also solution for the BVP (1).Denote x 7 = −x 5 , x 8 = −x 6 .It is clear x 7 and x 8 are two sign-changing solutions, too.So, BVP (1) has eight different nontrivial solutions x i , i = 1, 2, • • • , 8.

An example
Consider the following nonlinear second-order integral boundary-value problem (BVP) where where Proof of the conclusion.Let β be a positive number.First we show that the sequence of positive eigenvalues of the operator βK 1 is of the form Let λ be a positive eigenvalue of the linear operator βK 1 , and x ∈ E\{θ} be an eigenfunction corresponding to the eigenvalue λ.Then we have By Lemma 1, we obtain λ β x ′′ + x = 0.
That is The differential equation (41) has roots ± β λ i.Thus, the general solution of (41) can be expressed by Applying the boundary value condition x(0) = 0, we obtain that C 1 = 0, and so the general solution can be reduced to Since the positive solutions of the equation (1 and the eigenfunction corresponding to the eigenvalue β λn is x n (t) = C sin t λ n , t ∈ [0, 1], where C is a nonzero constant.Next, we check that all the conditions of Theorem 2 hold.Take a(s) = s.It is clear that Thus, we have proved that (H 3 ) holds for r = 2.As a consequence, our conclusion follows from Theorem 2. Remark 3. It is clear that κ 1 (t, s) ≤ 3 2 s(1 − s).Thus, the constant γ = 7 4 in this example is not optimal and the optimal constant is 3  2 .

= 1 0 1 0
a(s)x(s)ds.Therefore the eigenfunctions are scalar multiples of sin(kt) and λ = 1/k 2 where k is one of the positive solutions of the equation sin(k) = a(s) sin(ks)ds.

Lemma 12 (
[16]) Let P be a solid cone of a real Banach space E, Ω be a relatively bounded open subset of P , A : P → P be a completely continuous operator.If all fixed point of A in Ω is the interior point of P , there exists an open subset O of E such that O ⊂ Ω and deg(I − A, O, 0) = i(A, Ω, P ).
The conclusion of Theorem 2 holds for integral boundary value problem (37).