Abstract
This article is concerned with the existence of nontrivial solutions for a nonpositive fourthorder twopoint boundary value problem (BVP) and the existence of positive solutions for a semipositive fourthorder twopoint BVP. In mechanics, the problem describes the deflection of an elastic beam rigidly fixed at both ends. The method to show our main results is the topological degree and fixed point theory of nonlinear operator on lattice.
Mathematics Subject Classification 2010: 34B18; 34B16; 34B15.
Keywords:
lattice; topological degree; fixed point; nontrivial solutions and positive solutions; elastic beam equations1 Introduction
The purpose of this article is to investigate the existence of nontrivial solutions and positive solutions of the following nonlinear fourthorder twopoint boundary value problem (for short, BVP)
where λ is a positive parameter, f : [0,1] × R^{1 }→ R^{1 }is continuous.
Fourthorder twopoint BVPs are useful for material mechanics because the problems usually characterize the deflection of an elastic beam. The problem (P) describes the deflection of an elastic beam with both ends rigidly fixed. The existence and multiplicity of positive solutions for the elastic beam equations has been studied extensively when the nonlinear term f : [0,1] × [0, +∞) → [0, +∞) is continuous, see for example [110] and references therein. Agarwal and Chow [1] investigated problem (P) by using of contraction mapping and iterative methods. Bai [3] applied upper and lower solution method and Yao [9] used GuoKrasnosel'skii fixed point theorem of cone expansioncompression type. However, there are only a few articles concerned with the nonpositive or semipositive elastic beam equations. Yao [11] considered the existence of positive solutions of semipositive elastic beam equations by constructing control functions and a special cone and using fixed point theorem of cone expansioncompression type. In this article, we assume that f : [0,1] × R^{1 }→ R^{1}, which implies the problem (P) is nonpositive (or semipositive particularly). By the topological degree and fixed point theory of superlinear operator on lattice (the definition of lattice will be given in Section 2), we obtain the existence of nontrivial solutions for the nonpositive BVP (P) and the existence of positive solutions for the semipositive BVP (P).
2 Preliminaries
Let E be an ordered Banach space in which the partial ordering ≤ is induced by a cone P ⊂ E, θ denote the zero element of E. P is called solid if int P ≠ ∅, i.e., P has nonempty interior. P is called a generating cone if E = P  P . For the concepts and properties about the cones we refer to [1214].
We call E a lattice in the partial ordering ≤, if for arbitrary x, y ∈ E, sup{x, y} and inf{x, y} exist. For x ∈ E, let
which are called the positive and the negative part of x, respectively. Take x = x^{+ }+ x^{}, then x ∈ P , and x is called the module of x. One can see [15] for the definition and the properties about the lattice.
For convenience, we use the following notations:
then
Remark 2.1 If E is a lattice, then P is a generating cone.
Definition 2.1 [[16], Definition 3.2, p. 929]. Let D ⊂ E and F : D → E be a nonlinear operator. F is said to be quasiadditive on lattice, if there exists y_{0 }∈ E such that
where x_{+ }and x_{ }are defined by (2.1).
Remark 2.2 We point out that the condition (2.2) appears naturally in the applications of nonlinear differential equations and integral equations.
Let
and f : [a, b] × R^{1 }→ R^{1}. Consider the Nemytskii operator
Set P = {x ∈ C[a, b]  x(t) ≥ 0}, then E = C[a, b] is a lattice in the partial ordering which is induced by P . For any x ∈ C[a, b], it is evident that
and hence x(t) = x(t). By Remark 3.1 in [16], we know that there exists y_{0 }∈ C[a, b] such that Fx = Fx_{+ }+ Fx_{ }+ y_{0}, ∀x ∈ C[a, b].
Suppose that B is a linear operator and A = BF . It follows that
which means that A is quasiadditive on lattice.
Definition 2.2 [[17], Definition, p. 261]. Let B : E → E be a linear operator. B is said to be a u_{0}bounded linear operator, if there exists u_{0 }∈ P\{θ}, such that for any x ∈ P\{θ}, there exist a natural number n and real numbers ζ, η > 0, such that
Lemma 2.1 [[18], Theorem 4.2.2, p. 122]. Let P be a generating cone and B a u_{0}bounded completely continuous linear operator. Then the spectral radius r(B) ≠ 0 and r^{}^{1}(B) is the only positive eigenvalue corresponding to positive eigenvectors and B has no other eigenvectors except those corresponding to r^{}^{1}(B).
Let B : E → E be a positive completely continuous linear operator, r(B) a spectral radius of B, B* the conjugated operator of B, and P* the conjugated cone of P. Since P ⊂ E is a generating cone, according to the famous KreinRutman theorem (see [14]), if r(B) ≠ 0, then there exist
Fix
then P (g*, δ) is also a cone in E.
Definition 2.3 [[19], Definition, p. 528]. Let B be a positive linear operator. B is said to satisfy H condition, if there exist
Remark 2.3 Let
and v(x)g*(x) ≢ 0, then B satisfies H condition (see [19]).
Lemma 2.2 [[16], Theorem 3.1, p. 929]. Let P be a solid cone, A : E → E be a completely continuous operator satisfying A = BF, where F is quasiadditive on lattice, B is a positive bounded operator satisfying H condition. Suppose that
(i) there exist a_{1 }> r^{}^{1}(B), y_{1 }∈ P such that
(ii) there exist 0 ≤ a_{2 }≤ r^{}^{1}(B), y_{2 }∈ P such that
Then there exists R_{0 }> 0 such that deg(I  A, T_{R} , θ) = 0 for any R > R_{0 }, where T_{R }= {x ∈ C[0, 1] : x < R}.
Lemma 2.3 [[16], Theorem 3.3, p. 932]. Let Ω be a bounded open subset of E, θ ∈ Ω, and
(i) there exists a positive bounded linear operator B such that Ax ≤ Bx, for all x ∈ ∂Ω;
(ii) r(B) ≤ 1.
Then deg(I  A, Ω, θ) = 1.
3 Existence of nontrivial solutions for the nonpositive BVP (P)
In the sequel we always take E = C[0,1] with the norm u = max_{0≤t≤1 }u(t) and P = {u ∈ C[0, 1]  u(t) ≥ 0, 0 ≤ t ≤ 1}. Then P is a solid cone in E and E is a lattice under the partial ordering ≤ induced by P.
A solution of BVP (P) is a fourth differentiable function u : [0,1] → R such that u satisfies (P). u is said to be a positive solution of BVP (P) if u(t) > 0, 0 < t < 1. Let G(t, s) be Green's function of homogeneous linear problem u^{(4)}(t) = 0, u(0) = u(1) = u'(0) = u'(1) = 0. From Yao [11] we have
and
(G_{1}) G(t, s) ≥ 0, 0 ≤ t, s ≤ 1;
(G_{2}) G(t, s) = G(s, t);
(G_{3}) G(t, s) ≥ p(t)G(τ; s), 0 ≤ t, s, τ ≤ 1, where
Lemma 3.1 Let
Proof. Since G(0, s) = G(1, s) = 0, 0 ≤ s ≤ 1, H(0) = H(1) = 0, then q(s)H(t) = G(t, s) = H(t) holds for t = 0 and t = 1. If 0 < t ≤ s ≤ 1 and t < 1, then
and
Similarly, (3.1) holds for 0 ≤ s ≤ t < 1 and t > 0. The proof is complete. □
It is well known that the problem (P) is equivalent to the integral equation
Let
Lemma 3.2 Let B be defined by (3.3). Then B is a u_{0}bounded linear operator.
Proof. Let
we have
Take arbitrarily
Let
This indicates that B : E → E is a u_{0}bounded linear operator. □
From Lemma 2.1 we have r(B) ≠ 0 and r^{}^{1}(B) is the only eigenvalue of B. Denote λ_{1 }= r^{}^{1}(B).
Now let us list the following conditions which will be used in this article:
(H_{1}) there exist constants α and β with α > β ≥ 0 satisfying
(H_{2}) there exists a constant γ ≥ 0 satisfying
(H_{3})
Theorem 3.1 Suppose that (H_{1}) and (H_{2}) hold. Then for any
Proof. Let (Fu)(t) = f(t, u(t)). Then A = BF, where A is denoted by (3.2). By Remark 2.2, F is quasiadditive on lattice. Applying the ArzelaAscoli theorem and a standard argument, we can prove that A : E → E is a completely continuous operator.
Now we show that λA = λBF has at least one nontrivial fixed point, which is the nontrivial solution of BVP (P).
On account of (G_{3}) we have that
Notice that
By (3.4) and (3.5), there exist r > 0, M > 0 and
By (3.6) and (3.7), we have (2.4) and (2.5) hold, where a_{1 }= α  ε, a_{2 }= β + ε.
Let B_{1 }= λB. Then
Let B_{2 }= λ(γ + ε)B. From (3.8) we have λAu ≤ B_{2}u, also
It is easy to see from (3.9) and (3.10) that λA has at least one nontrivial fixed point. Thus problem (P) has at least one nontrivial solution. □
Remark 3.1 If α = +∞, β = γ = 0, then for any λ > 0 problem (P) has at least one nontrivial solution.
Theorem 3.2 Suppose that (H_{1}) holds. Assume f(t, 0) ≡ 0, ∀t ∈ [0,1] and
Then for any
Proof. Since f(t, 0) ≡ 0, ∀t ∈ [0,1], then Aθ = θ. By (3.11) we have that the Frechet derivative
Notice that
where κ is the sum of algebraic multiplicities for all eigenvalues of
4 Existence of positive solutions for the semipositive BVP (P)
In many real problems, the positive solution is more significant. In this section we will study this kind of question.
Lemma 4.1 [[20], Theorem 1, p. 90]. Let D = [a, b]. Suppose
(i) G(t, s) is a symmetric kernel. And there exist D_{0 }⊂ D, mesD_{0 }≠ 0 and δ > 0 such that
(ii) f(t, u) is bounded from below and
Theorem 4.1 Suppose that (H_{3}) holds. Then there exists λ* > 0 such that for any 0 < λ < λ* BVP (P) has at least one positive solution.
Proof. By (H_{3}) there exists a constant b > 0 such that
Let
From (4.1) f_{1 }is bounded from below. Let
Then A_{1 }: E → E is a completely continuous operator.
From the proof of Theorems 2.7.3 and 2.7.4 in Sun [18], there exists R_{0 }> 0 such that for any R > R_{0},
Take 0 < r < R_{0}. Let m = max_{0≤t≤1,u<r}f_{1}(t, u), G = max_{0≤s,t≤1 }G(t, s),
Thus
From (4.2) and (4.3), we have that for any
In fact, if (4.4) doesn't hold, then there exist λ_{n }> 0,
Since A_{1 }is completely continuous, then
Let D = [0,1], D_{0 }= [t_{1}, t_{2}] ⊂ (0, 1) ⊂ D,
From (H_{3}) and the definition of f_{1}, we have
By Lemma 4.1 there exists
And so u_{λ}(t) is a positive solution of problem (P). □
Remark 4.1 In Theorem 4.1, without assuming that f(t, u) ≥ 0 when u ≥ 0, we obtain the existence of positive solutions for the semipositive BVP (P).
Remark 4.2 Noticing that, in this article, we only study the existence of positive solutions for BVP (P), which is irrelevant to the value of f(t, u) when u ≤ 0, we only need to suppose that f(t, u) is bounded from below when u ≥ 0. In fact, f(t, u) may be unbounded from below when u ≤ 0.
5 Two examples
In order to illustrate possible applications of Theorems 3.2 and 4.1, we give two examples.
Example 5.1 Consider the fourthorder BVP
In this example, f(t, u) = sinu + uarctanu + πu, then
Take α = 3π/2, β = π/2, ρ = π +1. Then (5.1), (5.2) indicate (H_{1}), (3.11) hold, repectively. Notice that α > ρ > β > 0 and f(t, 0) ≡ 0, ∀t ∈ [0,1], by Theorem 3.2 for any
Example 5.2 Consider the fourthorder BVP
In this example,
(5.3) means (H_{3}) holds. By Theorem 4.1 there exists λ* > 0 such that for any 0 < λ < λ* BVP (P_{2}) has at least one positive solution.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179, 11026203), NSF (BK2011202) of the Jiangsu Province, NSF (09XLR04) of the Xuzhou Normal University and NSF (2010KY07) of the Suqian College.
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