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Existence of positive solutions to a nonpositive elastic beam equation with both ends fixed
Boundary Value Problems volume 2012, Article number: 56 (2012)
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.
1 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 [1–10] 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 [12–14].
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 \stackrel{\u0304}{\phi}\in P\backslash \left\{\theta \right\}, and g* ∈ P*\{θ}, such that
Fix \stackrel{\u0304}{\phi}\in P\backslash \left\{\theta \right\}, g* ∈ P*\{θ} such that (2.3) holds. For δ > 0, let
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 \stackrel{\u0304}{\phi}\in P\backslash \left\{\theta \right\}, g* ∈ P*\{θ} and δ > 0 such that (2.3) holds, and B maps P into P(g*, δ).
Remark 2.3 Let B\phi ={\int}_{a}^{b}k\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)\phi \left(y\right)dy, where k(x, y) ∈ C([a, b] × [a, b]), k(x, y) ≥ 0, φ ∈ C[a, b]. Suppose that (2.3) holds and there exists v(x) ∈ P\{θ} such that
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
Fx\ge {a}_{1}x{y}_{1},\phantom{\rule{1em}{0ex}}\forall x\in P;(2.4) 
(ii)
there exist 0 ≤ a _{2} ≤ r^{} ^{1}(B), y _{2} ∈ P such that
Fx\ge {a}_{2}x{y}_{2},\phantom{\rule{1em}{0ex}}\forall x\in \left(P\right).(2.5)
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 A:\stackrel{\u0304}{\mathrm{\Omega}}\to P a completely continuous operator. Suppose that A has no fixed point on ∂Ω. If

(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 p\left(t\right)=\frac{2}{3}\text{min}\left\{{t}^{2},{\left(1t\right)}^{2}\right\}.
Lemma 3.1 Let H\left(t\right)=\frac{1}{2}{t}^{2}{\left(1t\right)}^{2} and q\left(s\right)=\frac{2}{3}{s}^{2}{\left(1s\right)}^{2}. Then
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 {u}_{0}\left(t\right)=H\left(t\right)=\frac{1}{2}{t}^{2}{\left(1t\right)}^{2}, t ∈ [0,1]. For any u ∈ P\{θ}, by Lemma 3.1
we have
Take arbitrarily 0<{\epsilon}_{0}<\frac{1}{2}, then
Let \zeta ={\int}_{{\epsilon}_{0}}^{1{\epsilon}_{0}}q\left(s\right)u\left(s\right)ds>0, η = u > 0. Then
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}) \underset{u\to +\infty}{\text{lim}}\frac{f\left(t,u\right)}{u}=+\infty.
Theorem 3.1 Suppose that (H_{1}) and (H_{2}) hold. Then for any \mathrm{\lambda}\in \left(\frac{{\mathrm{\lambda}}_{1}}{\alpha},\frac{{\mathrm{\lambda}}_{1}}{\iota}\right), BVP (P) has at least one nontrivial solution, where λ_{1} = r^{}^{1}(B) is the only eigenvalue of B, B is denoted by (3.3), ι = max{β, γ}.
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 p\left(t\right)=\frac{2}{3}\text{min}\left\{{t}^{2},{\left(1t\right)}^{2}\right\}\in P\backslash \left\{\theta \right\} such that
Notice that Bu\left(t\right)={\int}_{0}^{1}G\left(t,s\right)u\left(s\right)ds, where G(t, s) ≥ 0, G(t, s) ∈ C([0,1] × [0,1]). From Lemma 3.2 B is a u_{0} bounded linear operator. By Lemma 2.1 we have r(B) ≠ 0 and λ_{1} = r^{}^{1}(B) is the only eigenvalue of B. Then there exist \stackrel{\u0304}{\phi}\in P\backslash \left\{\theta \right\} and g* ∈ P*\{θ} such that (2.3) holds. Notice that λ > 0, from Remark 2.3, λB satisfies H condition.
By (3.4) and (3.5), there exist r > 0, M > 0 and 0<\epsilon <\text{min}\left\{\frac{\alpha \mathrm{\lambda}{\mathrm{\lambda}}_{1}}{\mathrm{\lambda}},\frac{{\mathrm{\lambda}}_{1}\beta \mathrm{\lambda}}{\mathrm{\lambda}},\frac{{\mathrm{\lambda}}_{1}\gamma \mathrm{\lambda}}{\mathrm{\lambda}}\right\} such that
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 {r}^{1}\left({B}_{1}\right)=\frac{{\mathrm{\lambda}}_{1}}{\mathrm{\lambda}}. Obviously, for any \mathrm{\lambda}\in \left(\frac{{\mathrm{\lambda}}_{1}}{\alpha},\frac{{\mathrm{\lambda}}_{1}}{\iota}\right), a_{1}> r^{}^{1}(B_{1}), a_{2}< r^{}^{1}(B_{1}). From Lemma 2.2 there exists R_{0}> 0 such that for any R > max{R_{0}, r},
Let B_{2} = λ(γ + ε)B. From (3.8) we have λAu ≤ B_{2}u, also r\left({B}_{2}\right)=\frac{\mathrm{\lambda}\left(\gamma +\epsilon \right)}{{\mathrm{\lambda}}_{1}}\le 1. Without loss of generality we assume that λA has no fixed point on ∂T_{ r } , where T_{ r } = {u ∈ C[0,1]  u < r}. By Lemma 2.3 we have
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 \mathrm{\lambda}\in \left(\frac{{\mathrm{\lambda}}_{1}}{\alpha},\frac{{\mathrm{\lambda}}_{1}}{\beta}\right) and \mathrm{\lambda}\ne \frac{{\mathrm{\lambda}}_{1}}{\rho}, BVP (P) has at least one nontrivial solution.
Proof. Since f(t, 0) ≡ 0, ∀t ∈ [0,1], then Aθ = θ. By (3.11) we have that the Frechet derivative {A}_{\theta}^{\prime} of A at θ exists and
Notice that \mathrm{\lambda}\ne \frac{{\mathrm{\lambda}}_{1}}{\rho}, then 1 is not an eigenvalue of \mathrm{\lambda}{A}_{\theta}^{\prime}. By the famous LeraySchauder theorem there exists r > 0 such that
where κ is the sum of algebraic multiplicities for all eigenvalues of \mathrm{\lambda}{A}_{\theta}^{\prime} lying in the interval (0, 1). From the proof of Theorem 3.1 we have that (3.9) holds for any \mathrm{\lambda}\in \left(\frac{{\mathrm{\lambda}}_{1}}{\alpha},\frac{{\mathrm{\lambda}}_{1}}{\beta}\right). By (3.9) and (3.12), λA has at least one nontrivial fixed point. Thus problem (P) has at least one nontrivial solution. □
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
G\left(t,s\right)\ge \delta G\left(\tau ,s\right),\phantom{\rule{1em}{0ex}}\forall t\in {D}_{0},\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}\tau \in D; 
(ii)
f(t, u) is bounded from below and \underset{u\to +\infty}{\text{lim}}f\left(t,u\right)=+\infty uniformly holds for t ∈ D _{0}. Then for any λ* > 0, there exists R = R(λ*) > 0 such that if 0 < λ_{0} ≤ λ*, φ _{0} ≥ R and φ _{0} = λ_{0} Aφ _{0}, then φ _{0}(x) ≥ 0, where A is denoted by (3.2).
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), \stackrel{\u0304}{\mathrm{\lambda}}=r{\left(mG\right)}^{1}. For any 0<\mathrm{\lambda}<\stackrel{\u0304}{\mathrm{\lambda}}, u ∈ ∂T_{ r } , we have
Thus
From (4.2) and (4.3), we have that for any 0<\mathrm{\lambda}<\stackrel{\u0304}{\mathrm{\lambda}}, there exist u_{λ,} ∈ C[0,1], u_{λ} > r such that u_{λ} = λA_{1}u_{λ}. In order to apply Lemma 4.1 we claim that
In fact, if (4.4) doesn't hold, then there exist λ _{ n } > 0, {u}_{{\mathrm{\lambda}}_{n}}\in C\left[0,1\right] such that λ _{ n } → 0, r<\left\right{u}_{{\mathrm{\lambda}}_{n}}\left\right<c (c > 0 is a constant) and
Since A_{1} is completely continuous, then \left\{{u}_{{\mathrm{\lambda}}_{n}}\right\} has a subsequence converging to u* ∈ C[0,1]. Assume, without loss of generality, that it is \left\{{u}_{{\mathrm{\lambda}}_{n}}\right\}. Taking n → +∞ in (4.5), we have u* = θ, which is a contradiction to \left\right{u}_{{\mathrm{\lambda}}_{n}}\left\right>r>0. Hence (4.4) holds.
Let D = [0,1], D_{0} = [t_{1}, t_{2}] ⊂ (0, 1) ⊂ D,\delta =\underset{{t}_{1}\le t\le {t}_{2}}{\text{min}}p\left(t\right)>0. By (G_{3})
From (H_{3}) and the definition of f_{1}, we have
By Lemma 4.1 there exists R=R\left(\stackrel{\u0304}{\mathrm{\lambda}}\right)>0 such that if 0<\mathrm{\lambda}\le \stackrel{\u0304}{\mathrm{\lambda}}, u_{λ} ≥ R and u_{λ} = λA_{1}u_{λ}, then u_{λ}(t) ≥ 0. By (4.4), there exists {\mathrm{\lambda}}^{*}<\stackrel{\u0304}{\mathrm{\lambda}} such that if 0 < λ ≤ λ*, u_{λ} ≥ r and u_{λ} = λA_{1}u_{λ}, then u_{λ} ≥ R. Thus u_{λ}(t) ≥ 0. By the definitions of A_{1} and f_{1} we have
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 + u arctanu + 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 \mathrm{\lambda}\in \left(\frac{{\mathrm{\lambda}}_{1}}{\alpha},\frac{{\mathrm{\lambda}}_{1}}{\beta}\right) and \mathrm{\lambda}\ne \frac{{\mathrm{\lambda}}_{1}}{\rho}, BVP (P_{1}) has at least one nontrivial solution.
Example 5.2 Consider the fourthorder BVP
In this example, f\left(t,u\right)=\left(\sqrt{t}+1\right){u}^{2}\sqrt[3]{u}, then
(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.
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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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Lu, H., Sun, L. & Sun, J. Existence of positive solutions to a nonpositive elastic beam equation with both ends fixed. Bound Value Probl 2012, 56 (2012). https://doi.org/10.1186/16872770201256
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DOI: https://doi.org/10.1186/16872770201256