Inequalities of Green’s functions and positive solutions to nonlocal boundary value problems

In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in C2[0, 1], and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.


Introduction and preliminaries
In this paper, we discuss the existence of positive solutions to fourth-order boundary value problems (BVPs): (4) (t) = f (t, u(t), u (t), u (t)), t ∈ [0, 1], involving Stieltjes integrals with B i , A i of bounded variation. They model the deflection of beam equations with the nonlinearities including the slope u and bending moment u . The boundary conditions of Stieltjes integrals imply that the mechanism at the end points depends on the feedback along parts of the beam to control the displacement. By monotone iteration method, the cantilever beam equation containing the slope term u (4) (t) = f t, u(t), u (t) was considered by Alves et al. [1] and Yao [18] separately under the boundary conditions where g is a continuous function, and Respectively based on fixed point index method and global bifurcation technique, Li [10] and Ma [12] were devoted to the beam equations involving the bending moment with the hinged ends ⎧ ⎨ ⎩ u (4) (t) = f (t, u(t), u (t)), t ∈ (0, 1), Li [11] was concerned with the cantilever beam equation where f : [0, 1] × R 3 + × R -→ R + is continuous. The existence of positive solutions is obtained if the superlinear or sublinear growth conditions are satisfied for the nonlinearity. However, the boundary conditions in [1,[10][11][12]18] are all local. Webb et al. [17] considered the existence of positive solutions for the beam equation In these conditions α[u] = 1 0 u(s) dA(s) is given by Stieltjes integral. Infante and Pietramala [6] studied the existence of positive solutions for cantilever beam equation (4) (t) = g(t)f (t, u(t)), t ∈ (0, 1), where k 0 is a nonnegative constant, α [u] is Stieltjes integral, and B is a nonnegative continuous function. The nonlinearity f in [6,17] is not affected by the slope and bending moment, and the authors used the method of fixed point index on cone. We also refer to some other articles, for instance, [2,5,7,9,14,19].
Recently, the authors in [13] investigated the existence of positive solutions to the following problems: are linear functionals involving Stieltjes integrals of signed measures, and the nonlinearities f , g satisfy the Nagumo condition, which restricts f and g on u to quadric growth for the superlinear case, as in Li [11]. If the nonlinearities in (1.3) and (1.4) are independent of u , the restriction of quadric growth can certainly be rid of. However, the boundary conditions in (1.1) and (1.2) are different from those in (1.3) and (1.4). Especially the third derivatives with respect to t of their Green's functions corresponding to (1.1) and (1.2) may be sign-changing while they are not corresponding to (1.3) and (1.4), which plays an important part when estimating the norms in [13]. When BVPs are converted to integral equations, a general method due to Webb and Infante [16] is applied to use the theory of fixed point index on cones in C 2 [0,1]. Some examples are given for beam equations subject to mixed integral and multipoint boundary conditions with sign-changing coefficients. We also cite [15] in which a different approach is applied to the existence of positive solutions for the problem If the nonempty subset P in Banach space X satisfies the following conditions: (i) it is a closed convex set, (ii) λx ∈ P for any λ > 0, x ∈ P, and (iii) ±x ∈ P ⇔ x = 0 (0 stands for the zero element in X), then P is said to be a cone in X. A cone P is called reproducing if X = P -P. It is well known that if P is a solid cone, i.e., the interior point setP = ∅, P is reproducing. Now we state some properties of fixed point index (see [3,4]).

Lemma 1.1
Let Ω be a bounded open subset of Banach space X with 0 ∈ Ω and P be a cone in X. If S : P ∩ Ω → P is a completely continuous operator and μSu = u for u ∈ P ∩ ∂Ω and μ ∈ [0, 1], then the fixed point index i(S, P ∩ Ω, P) = 1.

Lemma 1.2
Let Ω be a bounded open subset of Banach space X and P be a cone in X. If S : P ∩ Ω → P is a completely continuous operator and there exists v 0 ∈ P \ {0} such that u -Su = νv 0 for u ∈ P ∩ ∂Ω and ν ≥ 0, then the fixed point index i(S, P ∩ Ω, P) = 0. Lemma 1.3 (Krein-Rutman) Let P be a reproducing cone in Banach space X and L : X → X be a completely continuous linear operator with L(P) ⊂ P. If the spectral radius r(L) > 0, then there exists ϕ ∈ P \ {0} such that Lϕ = r(L)ϕ.

Lemma 1.4 ([8])
Let P be a cone in Banach space X and L : X → X be a completely continuous linear operator with L(P) ⊂ P. If there exist v 0 ∈ P \ {0} and λ 0 > 0 such that Lv 0 ≥ λ 0 v 0 in the sense of partial ordering induced by P, then there exist u 0 ∈ P \ {0} and λ 1 ≥ λ 0 such that Lu 0 = λ 1 u 0 .
Throughout this paper, denote the Banach space that consists of all second-order continuously differentiable functions on [0, 1] by X = C 2 [0, 1] and the norm by u C 2 = max{ u C , u C , u C }.

Inequalities of Green's function and positive solutions for (1.1)
For BVP (1.1) we make the assumption: (  where We put forward the following hypotheses:  According to the following two inequalities: we have, for t, s ∈ [0, 1], Moreover, the next two inequalities Finally from the two inequalities it follows that Define the subsets in C 2 [0, 1] as follows: Clearly both P and K are cones, and it is easy to check that P is a solid cone. Denote the cone ordering induced by P, u v for u, v ∈ X if and only if vu ∈ P and equivalently v u. Now we define linear operators in C 2 [0, 1]: where a i , b i , c i (i = 1, 2) are nonnegative constants. Similar to [16], we have the following Lemma 2.2 by Lemma 2.1.
(ii) In this step we construct a homotopy and find a subset Ω R in order to compute the fixed point index later.
(vi) From (2.17) and (2.18) it follows that i(S, K ∩ Ω R , K) = 0 and Hence S has one fixed solution, i.e., BVP (1.1) has one positive solution in K .
We may suppose that S has no fixed points in K ∩ ∂Ω r and will show that u -Su = νϕ 0 for u ∈ K ∩ ∂Ω r and ν ≥ 0.
As an example, we consider the fourth-order boundary problem under mixed multipoint and integral boundary conditions with sign-changing coefficients: . We estimate some coefficients, and Matlab is used in some places.

2)
where and Proof According to the following two inequalities: we have, for t, s ∈ [0, 1], Moreover, the next two inequalities κ i (s), imply, for t, s ∈ [0, 1], Finally, from the two inequalities it follows that for t, s ∈ [0, 1].
Define the subsets in C 2 [0, 1] as follows: 1, 2, 3, 4) . (3.5) Clearly both P and K are cones, and it is easy to check that P is a solid cone. Now we define linear operators in C 2 [0, 1]: where a i , b i , c i (i = 1, 2) are nonnegative constants. Similar to [16], we have the following Lemma 3.2 by Lemma 3.1.

Proof Let
here L i (i = 1, 2) are defined in (3.6). Since all the terms are nonnegative in the first and second derivatives of k S (t, s) with respect to t, we also have that, for u ∈ C 2 [0, 1] and t ∈  Proof Take a 2 = 3/2, b 2 = c 2 = 1, r < 1, it is easy to check that (3.10) and (3.13) for i = 2 are satisfied. Now take a 1 = 5040, b 1 = 160, c 1 = 990, it is clear that We also have