Abstract
The existence results of multiple monotone and convex positive solutions for some fourthorder multipoint boundary value problems are established. The nonlinearities in the problems studied depend on all order derivatives. The analysis relies on a fixed point theorem in a cone. The explicit expressions and properties of associated Green's functions are also given.
MSC: 34B10; 34B15.
Keywords:
multipoint boundary value problem; positive solution; cone; fixed point1 Introduction
Boundary value problems for second and higher order nonlinear differential equations play a very important role in both theory and applications. For example, the deformations of an elastic beam in the equilibrium state can be described as a boundary value problem of some fourthorder differential equations. Owing to its importance in application, the existence of positive solutions for nonlinear second and higher order boundary value problems has been studied by many authors. We refer to recent contributions of Ma [13], He and Ge [4], Guo and Ge [5], Avery et al. [6,7], Henderson [8], Eloe and Henderson [9], Yang et al. [10], Webb and Infante [11,12], and Agarwal and O'Regan [13]. For survey of known results and additional references, we refer the reader to the monographs by Agarwal [14] and Agarwal et al. [15].
When it comes to positive solutions for nonlinear fourthorder ordinary differential equations, two point boundary value problems are studied extensively, see [1624]. Few papers deal with the multipoint cases. Furthermore, for nonlinear fourthorder equations, only the situation that the nonlinear term does not depend on the first, second and third order derivatives are considered, see [1623]. Few paper deals with the situation that lower order derivatives are involved in the nonlinear term explicitly. In fact, the derivatives are of great importance in the problem in some cases. For example, in the linear elastic beam equation (EulerBernoulli equation)
where u(t) is the deformation function, L is the length of the beam, f(t) is the load density, E is the Young's modulus of elasticity and I is the moment of inertia of the crosssection of the beam. In this problem, the physical meaning of the derivatives of the function u(t) is as follows: u^{(4)}(t) is the load density stiffness, u'''(t) is the shear force stiffness, u''(t) is the bending moment stiffness and the u'(t) is the slope. If the payload depends on the shear force stiffness, bending moment stiffness or the slope, the derivatives of the unknown function are involved in the nonlinear term explicitly.
In this paper, we are interested in the positive solution for fourthorder nonlinear differential equation
subject to multipoint boundary condition
or
where 0 < ξ_{1 }< ξ_{2 }< ⋯ < ξ_{m2 }< 1, β_{i }> 0, 1 = 1, 2, ..., m  2, , and f ∈ C([0, 1] × R^{4}, [0, +∞)).
One can see that all lower order derivatives are involved in the nonlinear term explicitly and the BCs are the mpoint cases. In this sense, the problems studied in this paper are more general than before. In the paper, multiple monotone and convex positive solutions for problems (1.1), (1.2) and (1.1), (1.3) are established. The results presented extend the study for fourthorder boundary value problems of nonlinear ordinary differential equations.
This paper is organized as follows. In Section 2, we present some preliminaries and lemmas. Section 3 is devoted to the existence of at least three convex and increasing positive solutions for problem (1.1), (1.2). In Section 4, we prove that there exist at least three convex and decreasing positive solutions for problem (1.1), (1.3).
2 Preliminaries and lemmas
In this section, some preliminaries and lemmas used later are presented.
Definition 2.1 The map α is said to be a nonnegative continuous convex functional on cone P of a real Banach space E provided that α : P → [0, +∞) is continuous and
Definition 2.2 The map β is said to be a nonnegative continuous concave functional on cone P of a real Banach space E provided that β : P → [0, +∞) is continuous and
Let γ, θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P and ψ be a nonnegative continuous functional on P. Then for positive numbers a, b, c and d, we define the following convex sets:
and a closed set
Lemma 2.1 [25] Let P be a cone in Banach space E. Let γ, θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional and ψ be a nonnegative continuous functional on P satisfying
such that for some positive numbers l and d,
for all . Suppose is completely continuous and there exist positive numbers a, b, c with a < b such that
(S_{1}) {x ∈ P (γ, θ, α, b, c, d)α(x) > b} ≠ ∅ and α(Tx) > b for x ∈ P (γ, θ, α, b, c, d);
(S_{2}) α(Tx) > b for x ∈ P (γ, α, b, d) with θ(Tx) > c;
(S_{3}) 0 ∉ R(γ, ψ, a, d) and ψ(Tx) < a for x ∈ R(γ, ψ, a, d) with ψ(x) = a.
Then T has at least three fixed points x_{1}, x_{2}, such that:
3 Positive solutions for problem (1.1), (1.2)
We begin with the fourthorder mpoint boundary value problem
where 0 < ξ_{1 }< ξ_{2 }< ⋯ < ξ_{m2 }< 1, β_{i }> 0, i = 1, 2, ..., m  2.
The following assumption will stand throughout this section:
Lemma 3.1 Denote ξ_{0 }= 0, ξ_{m 1 }= 1, β_{0 }= β_{m 1 }= 0, and y(t) ∈ C[0, 1]. Problem (3.1), (3.2) has the unique solution
where
for ξ_{i1 }≤ s ≤ ξ_{i}, i = 1, 2, ..., m 1.
Proof Let G(t, s) be the Green's function of problem x^{(4)}(t) = 0 with boundary condition (3.2). We can suppose
Considering the definition and properties of Green's function together with the boundary condition (3.2), we have
A straightforward calculation shows that
These give the explicit expression of the Green's function and the proof of Lemma 3.1 is completed.
Lemma 3.2 One can see that G(t, s) ≥ 0, t, s ∈ [0, 1].
Proof For ξ_{i1 }≤ s ≤ ξ_{i}, i = 1, 2, ..., m  1,
Then , 0 ≤ t, s ≤ 1. Thus G(t, s) is increasing on t. By a simple computation, we see
These ensures that G(t, s) ≥ 0, t, s ∈ [0, 1].
Lemma 3.3 Suppose x(t) ∈ C^{3}[0, 1] and
Furthermore x^{(4)}(t) ≥ 0 and there exist t_{0 }such that x^{(4)}(t_{0}) > 0. Then x(t) has the following properties:
Proof Since x^{(4)}(t) ≥ 0, t ∈ [0, 1], then x'''(t) is increasing on [0, 1]. Considering x'''(1) = 0, we have x'''(t) ≤ 0, t ∈ [0, 1]. Thus x''(t) is decreasing on [0, 1]. Considering this together with the boundary condition x''(1) = 0, we conclude that x''(t) ≥ 0. Then x(t) is convex on [0, 1]. Taking into account that x'(0) = 0, we get that
(1) From the concavity of x(t), we have
Multiplying both sides with β_{i }and considering the boundary condition, we have
Thus
(2) Considering the meanvalue theorem together with the concavity of x(t), we have
Multiplying both sides with β_{i }and considering the boundary condition, we have
Consequently
These give the proof of Lemma 3.3.
Remark Lemma 3.3 ensures that
Let Banach space E = C^{3}[0, 1] be endowed with the norm
Define the cone P ⊂ E by
Let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals γ, θ and the nonnegative continuous functional ψ be defined on the cone by
By Lemma 3.3, the functionals defined above satisfy
Denote
Assume that there exist constants 0 < a, b, d with a < b < λd such that
Theorem 3.1 Under assumptions (A_{1})(A_{3}), problem (1.1), (1.2) has at least three positive solutions x_{1}, x_{2}, x_{3 }satisfying
Proof Problem (1.1, 1.2) has a solution x = x(t) if and only if x solves the operator equation
Then
For , considering Lemma 3.3 and assumption (A_{1}), we have f(t, x(t), x'(t), x''(t), x'''(t)) ≤ d. Thus
Hence . An application of the ArzelaAscoli theorem yields that T is a completely continuous operator. The fact that the constant function x(t) = b/δ ∈ P(γ, θ, α, b, c, d) and α(b/δ) > b implies that
For x ∈ P(γ, θ, α, b, c, d), we have b ≤ x(t) ≤ b/δ and x'''(t) < d. From assumption (A_{2}), we see
Hence, by definition of α and the cone P, we can get
which means α(Tx) > b, ∀x ∈ P(γ, θ, α, b, b/δ, d). This ensures that condition (S1) of Lemma 2.1 is fulfilled.
Second, with (3.4) and b < λd, we have
for all x ∈ P(γ, α, b, d) with .
Thus, condition (S_{2}) of Lemma 2.1 holds. Finally we show that (S_{3}) also holds. We see ψ(0) = 0 < a and 0 ∉ R(γ, ψ, a, d). Suppose that x ∈ R(γ, ψ, a, d) with ψ(x) = a, then by the assumption of (A_{3}),
which ensures that condition (S_{3}) of Lemma 2.1 is fulfilled. Thus, an application of Lemma 2.1 implies that the fourthorder mpoint boundary value problem (1.1, 1.2) has at least three positive convex increasing solutions x_{1}, x_{2}, x_{3 }with the properties that
4 Positive solutions for problem (1.1), (1.3)
The following assumption will stand throughout this section:
Lemma 4.1 Denote ξ_{0 }= 0, ξ_{m1 }= 1, β_{0 }= β_{m1 }= 0, the Green's function of problem
is
for i = 1, 2, ..., m  1.
Proof Suppose that
Considering the definition and properties of Green's function together with the boundary condition (4.2), we have
Consequently
The proof of Lemma 4.1 is completed.
Lemma 4.2 One can see that H(t, s) ≥ 0, t, s ∈ [0, 1].
Proof For ξ_{i1 }≤ s ≤ ξ_{i}, i = 1, 2, ..., m  1,
Then , 0 ≤ t, s ≤ 1, which implies that H(t, s) is decreasing on t. The fact that
ensures that H(t, s) ≥ 0, t, s ∈ [0, 1].
Lemma 4.3 If x(t) ∈ C^{3}[0, 1],
and x^{(4)}(t) ≥ 0, there exists t_{0 }such that x^{(4)}(t_{0}) > 0, then
Proof It follows from the same methods as Lemma 3.3 that x(t) is convex on [0, 1]. Taking into account that x'(1) = 0, one can see that x(t) is decreasing on [0, 1] and
(1) From the concavity of x(t), we have
Multiplying both sides with β_{i }and considering the boundary condition, we have
Thus
(2) Considering the meanvalue theorem, we get
From the concavity of x similarly with above we know
Considering (4.3) together with (4.4) we have .
(3) For and x'(1) = 0, x''(1) = 0, we get
Thus
Remark We see that
Let Banach space E = C^{3}[0, 1] be endowed with the norm
Define the cone P ⊂ E by
Denote
Assume that there exist constants 0 < a, b, d with a < b < λ_{1}d such that
Theorem 4.1 Under assumptions (A_{4})(A_{6}), problem (1.1), (1.3) has at least three positive solutions x_{1}, x_{2}, x_{3 }with the properties that
Proof Problem (1.1), (1.3) has a solution x = x(t) if and only if x solves the operator equation
Then
For , considering Lemma 4.3 and assumption (A_{4}), we have
Thus
Hence and T_{1 }is a completely continuous operator obviously. The fact
that the constant function x(t) = b/δ_{1 }∈ P_{1}(γ, θ, α, b, c, d) and α(b/δ_{1}) > b implies that
This ensures that condition (S1) of Lemma 2.1 holds.
For x ∈ P_{1}(γ, θ, α, b, c, d), we have b ≤ x(t) ≤ b/δ_{1 }and x'''(t) < d. From assumption (A_{4}),
Hence, by definition of α and the cone P_{1}, we can get
which means α(T_{1}x) > b, ∀x ∈ P_{1}(γ, θ, α, b, b/δ, d).
Second, with (4.4) and b < λ_{1}d, we have
for all x ∈ P_{1}(γ, α, b, d) with .
Thus, condition (S_{2}) of Lemma 2.1 holds. Finally we show that (S_{3}) also holds. We see ψ(0) = 0 < a and 0 ∉ R(γ, ψ, a, d). Suppose that x ∈ R(γ, ψ, a, d) with ψ(x) = a, then by the assumption of (A_{6}),
which ensures that condition (S_{3}) of Lemma 2.1 is satisfied. Thus, an application of Lemma 2.1 implies that the fourthorder mpoint boundary value problem (1.1), (1.3) has at least three positive convex decreasing solutions x_{1}, x_{2}, x_{3 }satisfying the conditions that
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
We are indebted to the anonymous referee for a detailed reading and useful comments and suggestions, which allowed us to improve this work. This work was supported by the Anhui Provincial Natural Science Foundation (10040606Q50), National Natural Science Foundation of China (No.11071164), Shanghai Natural Science Foundation (No.10ZR1420800).
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