Positive solutions of second-order problem with dependence on derivative in nonlinearity under Stieltjes integral boundary condition

In this paper, we investigate the second-order problem with dependence on derivative in nonlinearity and Stieltjes integral boundary condition { −u′′(t) = f (t, u(t), u′(t)), t ∈ [0, 1], u(0) = α[u], u′(1) = 0, where f : [0, 1]×R+ ×R+ → R+ is continuous and α[u] is a linear functional. Some inequality conditions on nonlinearity f and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C1[0, 1]. The conditions allow that f (t, x1, x2) has superlinear or sublinear growth in x1, x2. Some examples are given to illustrate the theorems respectively under multi-point and integral boundary conditions with sign-changing coefficients.


Introduction
In this paper, we discuss the existence of positive solutions for second-order boundary value problem (BVP) with dependence on derivative in nonlinearity and Stieltjes integral boundary conditions −u (t) = f (t, u(t), u (t)), t ∈ [0, 1], where α denotes linear functional given by α[u] = 1 0 u(t)dA(t) involving Stieltjes integral with suitable function A of bounded variation.
Corresponding author.Email: gwzhang@mail.neu.edu.cn,gwzhangneum@sina.comRecently, Li [7] considered the existence of solutions for second order boundary value problem −u (t) = f (t, u(t), u (t)), t ∈ [0, 1], u(0) = 0, u(1) = 0, ( where f : [0, 1] × R + × R → R + is continuous.Under the conditions that the nonlinearity f (t, x 1 , x 2 ) may be superlinear or sublinear growth on x 1 and x 2 , the existence of positive solutions are obtained.It should be remarked that the constant π 2 in the discussion plays an important role and is the first eigenvalue of the linear eigenvalue problem corresponding to BVP (1.2) which is related to the spectral radius of linear operator.The results of [7] extend those of [14] in which only sublinear problem was treated.The fourth-order problem with fully nonlinear terms was also investigated in [8].
Webb and Infante [11] gave a unified method of establishing the existence of positive solutions for a large number of local and nonlocal boundary problems by applying the theory of fixed point index on cones when f does not depend on u .They dealt with the boundary problems involving Stieltjes integrals with signed measures and their results included the multipoint and integral boundary conditions as special cases.We also refer some other relevant articles, for example, [3,[9][10][11][12][13] and references therein.In these works the nonlinearity is independent of derivative term.
Inspired and motived by those previous works, we investigate BVP (1.1) in which not only the nonlinearity depends on derivative term but also the boundary conditions involves Stieltjes integral.Some inequality conditions on nonlinearity f and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to BVP (1.1) by the theory of fixed point index on a special cone in C 1 [0, 1].The conditions allow that f (t, x 1 , x 2 ) has superlinear or sublinear growth in x 1 , x 2 .The readers can also refer to [4,5,15] for some pertinent questions.
The organization of this paper is as follows.In Section 2, we give some lemmas useful for our main results and present a reproducing cone P and a cone K which play important roles in calculating fixed point indexes of nonlinear operator.In Section 3, we discuss the existence of positive solutions to problem mentioned above and give its complete proof.At last in Section 4, some examples are given to illustrate the theorems respectively under multi-point and integral boundary conditions with sign-changing coefficients.

Preliminaries
Let C 1 [0, 1] denote the Banach space of all continuously differentiable functions on [0, 1] with the norm u C 1 = max{ u C , u C }.We first make the assumption: As shown by Webb and Infante [11] BVP (1.1) has a solution if and only if there exists a solution in C 1 [0, 1] for the following integral equation where and α[u] = 1 0 u(t)dA(t).We set We also impose the following hypotheses: (C 2 ) A is of bounded variation and Adopting the notations and ideas in [11], define the operator S as and thus S can be written in the form as follows, where where Define two cones in C 1 [0, 1] and several linear operators as follow.
where a i , b i (i = 1, 2) are nonnegative constants and τ ∈ (0, 1).We write u v equivalently v u if and only if v − u ∈ P, to denote the cone ordering induced by P.
Proof.From (2.3), (2.4) and (C 1 )-(C 3 ) we have for u ∈ P that (Su)(t) ≥ 0 and It is easy to see from (C 1 ) that S : Let F is a bounded set in P and there exists M > 0 such that u C 1 ≤ M for all u ∈ F. By (C 1 ) and Lemma 2.1 we have that ∀u ∈ F and t ∈ [0, 1], thus S(F) and S (F) =: {v : v (t) = (Su) (t), u ∈ F} are equicontinuous.Therefore S : P → C 1 [0, 1] is completely continuous by the Arzelà-Ascoli theorem and so are For u ∈ P it follows from Lemma 2.1 that and hence for t ∈ [0, 1], (Su From (C 1 )-(C 3 ) it can easily be checked that α[Su] ≥ 0 and (Su) (1) = 0. Thus S : P → K.
By the same way, we have L i : P → K (i = 1, 2, 3).

Lemma 2.3 ([11]
).If (C 1 )-(C 3 ) hold, then S and T have the same fixed points in K.As a result, BVP (1.1) has a solution if and only if S has a fixed point.

Main results
In order to prove the main theorems, we need the following properties of fixed point index, see for example [1,2].

Lemma 3.1.
Let Ω be a bounded open subset of X with 0 ∈ Ω and K be a cone in X.If A : K ∩ Ω → K is a completely continuous operator and µAu = u for u ∈ K ∩ ∂Ω and µ ∈ [0, 1], then the fixed point index i(A, K ∩ Ω, K) = 1.

Lemma 3.2.
Let Ω be a bounded open subset of X and K be a cone in X.If A : K ∩ Ω → K is a completely continuous operator and there exists v 0 ∈ K \ {0} such that u − Au = νv 0 for u ∈ K ∩ ∂Ω and ν ≥ 0, then the fixed point index i(A, K ∩ Ω, K) = 0.
Recall that a cone P in Banach space X is said to be reproducing if X = P − P.

Lemma 3.3 (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, where 0 denotes the zero element in X.
Then BVP (1.1) has at least one nondecreasing positive solution.
Proof.Let W = {u ∈ K : u = µSu, µ ∈ [0, 1]} where S and K are respectively defined in (2.3) and (2.6).We first assert that W is a bounded set.In fact, if u ∈ W, then u = µSu for some µ ∈ [0, 1].From (2.7) and (3.1) we have that Obviously v ∈ P and it is easy to see from (2.4) that Because of the spectral radius r(L 1 ) < 1, we know that I − L 1 has a bounded inverse operator (I − L 1 ) −1 which can be written as Since L 1 (P) ⊂ K ⊂ P by Lemma 2.2, we have (I − L 1 ) −1 (P) ⊂ P which implies the inequality u (I − L 1 ) −1 v. Therefore, It is easy to verify that P is a solid cone, i.e. the interior point set P = ∅, then P is reproducing (cf.[1,2,6]).Since L 2 : P → K ⊂ P and r(L 2 ) ≥ 1, it follows from Lemma 3.3 that there exists 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.
Otherwise, there exist u 0 ∈ K ∩ ∂Ω r and ν 0 ≥ 0 such that u 0 − Su 0 = ν 0 ϕ 0 , and it is clear Set But r(L 2 ) ≥ 1, so u 0 (ν 0 + ν * )ϕ 0 , which is a contradiction to the definition of ν * .Therefore u − Au = νϕ 0 for u ∈ K ∩ ∂Ω r and ν ≥ 0. From Lemma 3.2 it follows that i(S, K ∩ Ω r , K) = 0. Making use of the properties of fixed point index, we have that and hence S has at least one fixed point in K. Therefore, BVP (1.1) has at least one nondecreasing positive solution by Lemma 2.3.
If the following Nagumo condition is fulfilled, i.e.
(F 5 ) for any M > 0 there is a positive continuous function H M (ρ) on R + satisfying then BVP (1.1) has at least one nondecreasing positive solution.
Proof.(i) First we prove that µSu = u for u ∈ K ∩ ∂Ω r and µ ∈ [0, 1].In fact, if there exist and (3.6) that and from (2.4) that thus (I − L 2 )u 1 0. Because of the spectral radius r(L 2 ) < 1, we know that I − L 2 has a bounded inverse operator (I − L 2 ) −1 : P → P and u 1 (3.9)By (3.7) it is easy to see that Similar to the proof in Lemma 2.2, we know that S 1 : P → K is completely continuous.Let R > max{r, M 1 } and we will show that .12) Our last example is the following problem with integral boundary condition in which one should notice that cos(πt) is sign-changing over [0, 1]: