Successive Iteration and Positive Solutions for Nonlinear 𝑚-Point Boundary Value Problems on Time Scales

Abstract

We study the existence of positive solutions for a class of 𝑚-point boundary value problems on time scales. Our approach is based on the monotone iterative technique and the cone expansion and compression fixed point theorem of norm type. Without the assumption of the existence of lower and upper solutions, we do not only obtain the existence of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.

1. Introduction

The purpose of this paper is to consider the existence of positive solutions and establish the corresponding iterative schemes for the following 𝑚-point boundary value problems (BVP) on time scales:

𝑢^Δ∇(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,1]_𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖),𝑚≥3. (1.1)

The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Some preliminary definitions and theorems on time scales can be found in [2–5] which are good references for the calculus of time scales.

In recent years, by applying fixed point theorems, the method of lower and upper solutions and critical point theory, many authors have studied the existence of positive solutions for two-point and multipoint boundary value problems on time scales, for details, see [2, 3, 6–18] and references therein. However, to the best of our knowledge, there are few papers which are concerned with the computational methods of the multipoint boundary value problems on time scales. We would like to mention some results of Sun and Li [16], Aykut Hamel and Yoruk [12], Anderson and Wong [10], Wang et al. [18], and Jankowski [13], which motivated us to consider the BVP (1.1).

In [16], Sun and Li considered the existence of positive solutions of the following dynamic equations on time scales:

𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇),𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝛼𝑢(𝜂)=𝑢(𝑇), (1.2) where 𝛽,𝛾≥0, 𝛽+𝛾>0,𝜂∈(0,𝜌(𝑇)),0<𝛼<𝑇/𝜂. They obtained the existence of single and multiple positive solutions of (1.2) by using a fixed point theorem and the Leggett-Williams fixed point theorem, respectively.

Very recently, in [12], Aykut Hamel and Yoruk discussed the following dynamic equation on time scales:

𝑢^Δ∇(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,1]⊂𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖),𝑚≥3. (1.3) They obtained some results for the existence of at least two and three positive solutions to the BVP (1.3) by using fixed point theorems in a cone and the associated Green's function.

In related paper, in [10], Anderson and Wong studied the second-order time scale semipositone boundary value problem with Sturm-Liouville boundary conditions or multipoint conditions as in

(𝑝𝑢^Δ)^∇(𝑡)+𝜆𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(𝑎,𝑏]_𝐓,𝛼𝑢(𝑎)−𝛽(𝑝𝑢^Δ)(𝑎)=0,𝛾𝑢^𝜎(𝑏)+𝛿(𝑝𝑢^Δ)(𝑏)=0,or𝛼𝑢(𝑎)−𝛽(𝑝𝑢^Δ)(𝑎)=^𝑛_𝑖=1𝜙_𝑖(𝑝𝑢^Δ)(𝑡_𝑖),𝛾𝑢^𝜎(𝑏)+𝛿(𝑝𝑢^Δ)(𝑏)=^𝑛_𝑖=1𝜓_𝑖(𝑝𝑢^Δ)(𝑡_𝑖). (1.4)

On the other hand, the method of lower and upper solutions has been effectively used for proving the existence results for dynamic equations on time scales. In [18], Wang et al. considered a method of generalized quasilinearization, with even-order 𝑘(𝑘≥2) convergence, for the BVP

−(𝑝(𝑡)𝑥^Δ)^∇+𝑞(𝑡)𝑥^𝜎=𝑓(𝑡,𝑥^𝜎)+𝑔(𝑡,𝑥^𝜎),𝑡∈[𝑎,𝑏]_𝐓,𝜏₁𝑥(𝜌(𝑎))−𝜏₂𝑥^Δ(𝜌(𝑎))=0,𝑥(𝜎(𝑏))−𝜏₃𝑥(𝜂)=0. (1.5) The main contribution in [18] relaxed the monotone conditions on 𝑓^(𝑖)(𝑡,𝑥),𝑔^(𝑖)(𝑡,𝑥)(1<𝑖<𝑘) including a more general concept of upper and lower solution in mathematical biology, so that the high-order convergence of the iterations was ensured for a larger class of nonlinear functions on time scales.

In [13], Jankowski investigated second-order differential equations with deviating arguments on time scales of the form

−𝑥^ΔΔ(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥(𝛼(𝑡)))≡(𝐹𝑥)(𝑡),𝑡∈𝐽,𝑥(0)=𝑘₁∈𝐑,𝑥(𝑇)=𝑘₂∈𝐑. (1.6) They formulated sufficient conditions, under which such problems had a minimal and a maximal solution in a corresponding region bounded by upper-lower solutions.

We would also like to mention the result of Yao [19]. In [19], Yao considered the positive solutions to the following two classes of nonlinear second-order three-point boundary value problems:

𝑢^′′(𝑡)+𝑓(𝑡,𝑢(𝑡),𝑢^′𝑢(𝑡))=0,0≤𝑡≤1,_′′(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,0≤𝑡≤1,𝑢(0)=0,𝛼𝑢(𝜂)=𝑢(1), (1.7) where both 𝜂 and 𝛼 are given constants satisfying 0<𝜂<1,0<𝛼<1/𝜂. By improving the classical monotone iterative technique of Amann [20], two successive iterative schemes were established for the BVP (1.7). It was worth stating that the first terms of the iterative schemes were constant functions or simple functions. We note that Ma et al. [21] and Sun et al. [22, 23] have also applied the similar methods to 𝑝-laplacian boundary value problems with 𝐓=𝐑.

In this paper, we will investigate the iterative and existence of positive solutions for the BVP (1.1), by considering the “heights” of the nonlinear term 𝑓 on some bounded sets and applying monotone iterative techniques on a Banach space, we do not only obtain the existence of positive solutions for the BVP (1.1), but also give the iterative schemes for approximating the solutions. We should point out that the monotone condition imposed on the nonlinear term 𝑓 will play crucial role in obtaining the iterative schemes for approximating the solutions. In essence, we combine the method of lower and upper solutions with the cone expansion and compression fixed point theorem of norm type. The idea of this paper comes from Yao [19, 24, 25].

Let 𝐓 be a time scale which has the subspace topology inherited from the standard topology on 𝐑. For each interval 𝐼 of 𝐑, we define 𝐼_𝐓=𝐼∩𝐓.

For the remainder of this article, we denote the set of continuous functions from [0,1]_𝐓 to 𝐑 by 𝐶([0,1]_𝐓,𝐑). Let 𝐶([0,1]_𝐓,𝐑) be endowed with the ordering 𝑥≤𝑦 if 𝑥(𝑡)≤𝑦(𝑡) for all 𝑡∈[0,1]_𝐓, and ‖𝑢‖=max_{𝑡∈[0,1]𝐓}|𝑢(𝑡)| is defined as usual by maximum norm. The 𝐶([0,1]_𝐓,𝐑) is a Banach space.

Throughout this paper, we will assume that the following assumptions are satisfied:

(𝐻₁)𝛽,𝛾≥0,0<𝛽+𝛾≤1,𝜉_𝑖∈(0,𝜌(1))_𝐓 for 𝑖=1,2,…,𝑚−2 with 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<𝜌(1);(𝐻₂)∑^𝑚−2_𝑖=1𝛼_𝑖∈(0,1) with 𝛼_𝑖∈(0,+∞) for 𝑖=1,2,…,𝑚−2 and 𝑑=𝛽(1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜉_𝑖)+𝛾(1−∑^𝑚−2_𝑖=1𝛼_𝑖)>0; (𝐻₃)𝑓∶[0,1]_𝐓×[0,+∞)→[0,+∞) is continuous.

2. Preliminaries and Several Lemmas

To prove the main results in this paper, we will employ several lemmas. These lemmas are based on the linear boundary value problem

𝑢^Δ∇(𝑡)+ℎ(𝑡)=0,𝑡∈[0,1]_𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖),𝑚≥3. (2.1)

Lemma 2.1 (see [12]). It holds 𝑑=𝛽(1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜉_𝑖)+𝛾(1−∑^𝑚−2_𝑖=1𝛼_𝑖)≠0; then the Green's function for the BVP −𝑢^Δ∇(𝑡)=0,𝑡∈[0,1]_𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖),𝑚≥3, (2.2) is given by 1𝐺(𝑡,𝑠)=𝑑(𝛽𝑠+𝛾)[(1−𝑡)−^𝑚−2_𝑗=1𝑎_𝑗(𝜉_𝑗−𝑡)],if0≤𝑡≤1,0≤𝑠≤𝜉₁,𝑠≤𝑡;(𝛽𝑠+𝛾)(1−𝑡)−^𝑚−2_𝑗=𝑖𝛼_𝑗(𝜉_𝑗−𝑡)(𝛽𝑠+𝛾)+^𝑖−1_𝑗=1𝛼_𝑗(𝛽𝜉_𝑗+𝛾)(𝑡−𝑠),if𝜉_𝑟−1≤𝑡≤𝜉_𝑟,2≤𝑟≤𝑚−1,𝜉_𝑖−1≤𝑠≤𝜉_𝑖,2≤𝑖≤𝑟,𝑠≤𝑡;(𝛽𝑡+𝛾)[(1−𝑠)−^𝑚−2_𝑗=𝑖𝛼_𝑗(𝜉_𝑗−𝑠)],if𝜉_𝑟−1≤𝑡≤𝜉_𝑟,2≤𝑟≤𝑚−2,𝜉_𝑖−1≤𝑠≤𝜉_𝑖,𝑟≤𝑖≤𝑚−2,𝑡≤𝑠;(𝛽𝑡+𝛾)(1−𝑠),if0≤𝑡≤1,𝜉_𝑚−2≤𝑠≤1,𝑡≤𝑠. (2.3) Here for the sake of convenience, one writes ∑^𝑚2_𝑖=𝑚1ℎ(𝑖)=0 for 𝑚₂<𝑚₁.

Lemma 2.2 (see [12]). Assume that conditions (𝐻₁)-(𝐻₃) are satisfied. Then(i)𝐺(𝑡,𝑠)≥0 for 𝑡,𝑠∈[0,1]_𝐓; (ii)there exist a number Ψ∈(0,1) and a continuous function 𝜃∶[0,1]_𝐓→𝐑⁺ such that 𝐺(𝑡,𝑠)≤𝜃(𝑠)for𝑡,𝑠∈[0,1]_𝐓,𝐺(𝑡,𝑠)≥Ψ𝜃(𝑠)for𝑡∈[𝜉₁,1]_𝐓,𝑠∈[0,1]_𝐓, (2.4) where ∑𝜃(𝑠)=max{1,^{𝑚−2𝑖=1}𝛼_𝑖𝜉₁}(𝛽𝑠+𝛾)(1−𝑠)𝑑,1Ψ=max{1,∑^𝑚−2_𝑖=1𝛼_𝑖/𝜉₁}×min_{2≤𝑠≤𝑚−2}{(𝛽𝜉₁+𝛾)(1−𝜉_𝑚−2),^𝑚−2_𝑗=1𝛼_𝑗(1−𝜉_𝑗),^𝑠−1_𝑗=1𝛼_𝑗(𝛽𝜉_𝑗+𝛾)+^𝑚−2_𝑗=𝑠𝛼_𝑗(1−𝜉_𝑗)}. (2.5)

Let 𝐁=𝐶([0,1]_𝐓,𝐑). It is easy to see that the BVP (1.1) has a solution 𝑢=𝑢(𝑡) if and only if 𝑢 is a fixed point of the operator equation:

𝑇𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠. (2.6) Denote

𝐾={𝑢∈𝐁∶𝑢isnonnegative,concave,andmin_{𝑡∈[𝜉1,1]𝐓}𝑢(𝑡)≥Ψ‖𝑢‖}, (2.7) where Ψ is the same as in Lemma 2.2. By [12, Lemma  3.1], we can obtain that 𝑇(𝐾)⊂𝐾 and 𝑇∶𝐾→𝐾 is completely continuous.

3. Successive Iteration and One Positive Solution for (1.1)

For notational convenience, we denote

𝐴=[max_{𝑡∈[0,1]𝐓}¹₀𝐺(𝑡,𝑠)∇𝑠]⁻¹,𝐵=[max_{𝑡∈[0,1]𝐓}¹_𝜉1𝐺(𝑡,𝑠)∇𝑠]⁻¹. (3.1) Constants 𝐴,𝐵 are not easy to compute explicitly. For convenience, we can replace 𝐴 by 𝐴^′, 𝐵 by 𝐵^′, where

𝐴^′=[¹₀𝜃(𝑠)∇𝑠]⁻¹,𝐵^′=[Ψ¹_𝜉1𝜃(𝑠)∇𝑠]⁻¹. (3.2) Obviously, 0<𝐴^′<𝐴<𝐵<𝐵^′.

Theorem 3.1. Assume (𝐻₁)-(𝐻₃) hold, and there exist two positive numbers 𝑎,𝑏 with 𝑏<𝑎 such that(𝐶₁)max{𝑓(𝑡,𝑎)∶𝑡∈[0,1]_𝐓}≤𝑎𝐴, min{𝑓(𝑡,Ψ𝑏)∶𝑡∈[𝜉₁,1]_𝐓}≥𝑏𝐵;(𝐶₂)𝑓(𝑡,𝑢₁)≤𝑓(𝑡,𝑢₂) for any 𝑡∈[0,1]_𝐓,0≤𝑢₁≤𝑢₂≤𝑎. Then the BVP (1.1) has at least one positive solution 𝑢^∗ such that 𝑏≤‖𝑢^∗‖≤𝑎 and lim_𝑛→∞𝑇^𝑛𝑢=𝑢^∗, that is, 𝑇^𝑛𝑢 converges uniformly to 𝑢^∗ in [0,1]_𝐓, where 𝑢(𝑡)≡𝑎,𝑡∈[0,1]_𝐓.

Remark 3.2. The iterative scheme in Theorem 3.1 is 𝑢₁=𝑇𝑢,𝑢_𝑛+1=𝑇𝑢_𝑛,𝑛=1,2,…. It starts off with constant function 𝑢(𝑡)≡𝑎,𝑡∈[0,1]_𝐓.

Proof of Theorem 3.1. Denote 𝐾[𝑏,𝑎]={𝑢∈𝐾∶𝑏≤‖𝑢‖≤𝑎}. If 𝑢∈𝐾[𝑏,𝑎], then max_{𝑡∈[0,1]𝐓}𝑢(𝑡)≤𝑎,min_{𝑡∈[𝜉1,1]𝐓}𝑢(𝑡)≥Ψ‖𝑢‖≥𝑏Ψ. (3.3) By Assumptions (𝐶₁) and (𝐶₂), we have 𝑓(𝑡,𝑢(𝑡))≤𝑓(𝑡,𝑎)≤𝑎𝐴,𝑡∈[0,1]_𝐓;𝑓(𝑡,𝑢(𝑡))≥𝑓(𝑡,𝑏Ψ)≥𝑏𝐵,𝑡∈[𝜉₁,1]_𝐓. (3.4) It follows that ‖𝑇𝑢‖=max_{𝑡∈[0,1]𝐓}|¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠|≤𝑎𝐴max_{𝑡∈[0,1]𝐓}¹₀𝐺(𝑡,𝑠)∇𝑠=𝑎;‖𝑇𝑢‖≥max_{𝑡∈[0,1]𝐓}¹_𝜉𝑖𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠≥𝑏𝐵max_{𝑡∈[0,1]𝐓}¹_𝜉𝑖𝐺(𝑡,𝑠)∇𝑠=𝑏. (3.5) Thus, we assert that 𝑇∶𝐾[𝑏,𝑎]→𝐾[𝑏,𝑎].
Let 𝑢(𝑡)≡𝑎,𝑡∈[0,1]_𝐓, then 𝑢∈𝐾[𝑏,𝑎]. Let 𝑢₁=𝑇𝑢, then 𝑢₁∈𝐾[𝑏,𝑎]. Denote 𝑢_𝑛+1=𝑇𝑢_𝑛,𝑛=1,2,…. Since 𝑇(𝐾[𝑏,𝑎])⊂𝐾[𝑏,𝑎], we have 𝑢_𝑛∈𝑇(𝐾[𝑏,𝑎])⊂𝐾[𝑏,𝑎],𝑛=1,2,…. Since 𝑇 is completely continuous, we assert that {𝑢_𝑛}^∞_𝑛=1 has a convergent subsequence {𝑢_𝑛𝑘}^∞_𝑘=1 and there exists 𝑢^∗∈𝐾[𝑏,𝑎], such that 𝑢_𝑛𝑘→𝑢^∗.
Now, since 𝑢₁∈𝐾[𝑏,𝑎], we have
𝑢₁(𝑡)≤‖𝑢₁‖≤𝑎=𝑢(𝑡),𝑡∈[0,1]_𝐓. (3.6) By Assumption (𝐶₂), 𝑢₂(𝑡)=𝑇𝑢₁=(𝑡)₁₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢₁≤(𝑠))∇𝑠₁₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠=𝑇𝑢(𝑡)=𝑢₁(𝑡). (3.7) By the induction, then 𝑢_𝑛+1(𝑡)≤𝑢_𝑛(𝑡),𝑡∈[0,1]_𝐓,𝑛=1,2,…. (3.8) Hence, 𝑇^𝑛𝑢=𝑢_𝑛→𝑢^∗. Applying the continuity of 𝑇 and 𝑢_𝑛+1=𝑇𝑢_𝑛, we get 𝑇𝑢^∗=𝑢^∗. Since ‖𝑢^∗‖≥𝑏>0 and 𝑢^∗ is a nonnegative concave function, we conclude that 𝑢^∗ is a positive solution of the BVP (1.1).

Corollary 3.3. Assume that (𝐻₁)-(𝐻₃) hold, and the following conditions are satisfied:(𝐶^′₁)lim_𝑙→0min_{𝑡∈[𝜉1,1]𝐓}𝑓(𝑡,𝑙)/𝑙>Ψ⁻¹𝐵,lim_𝑙→+∞max_{𝑡∈[0,1]𝐓}𝑓(𝑡,𝑙)/𝑙<𝐴 (particularly, lim_𝑙→0min_{𝑡∈[𝜉1,1]𝐓}𝑓(𝑡,𝑙)/𝑙=+∞,lim_𝑙→+∞max_{𝑡∈[0,1]𝐓}𝑓(𝑡,𝑙)/𝑙=0);(𝐶^′₂)𝑓(𝑡,𝑢₁)≤𝑓(𝑡,𝑢₂) for any 𝑡∈[0,1]_𝐓,𝑢₁≤𝑢₂,𝑢₁,𝑢₂∈[0,+∞). Then the BVP (1.1) has at least one positive solution 𝑢^∗∈𝐾 and there exists a positive number 𝑎 such that lim_𝑛→∞𝑇^𝑛𝑢=𝑢^∗, that is, lim_𝑛→∞sup_{𝑡∈[0,1]𝐓}|𝑇^𝑛𝑢(𝑡)−𝑢^∗(𝑡)|=0, (3.9) where 𝑢(𝑡)≡𝑎,𝑡∈[0,1]_𝐓.

Theorem 3.4. Assume (𝐻₁)-(𝐻₃) hold, and the following conditions are satisfied:(𝐷₁) there exists 𝑎>0 such that 𝑓(𝑡,⋅)∶[0,𝑎]→(0,+∞) is nondecreasing for any 𝑡∈[0,1]_𝐓 and max{𝑓(𝑡,𝑎)∶𝑡∈[0,1]_𝐓}≤𝑎𝐴; (𝐷₂)𝑓(𝑡,0)>0, for any 𝑡∈[0,1]_𝐓. Then the BVP (1.1) has one positive solution 𝑢^∗ such that 0<‖𝑢^∗‖≤𝑎 and lim_𝑛→∞𝑇^𝑛0=𝑢^∗, that is, 𝑇^𝑛0 converges uniformly to 𝑢^∗ in [0,1]_𝐓. Furthermore, if there exists 0<𝜔<1 such that |𝑓(𝑡,𝑙₂)−𝑓(𝑡,𝑙₁)|≤𝜔𝐴|𝑙₂−𝑙₁|,𝑡∈[0,1]_𝐓,0≤𝑙₁,𝑙₂≤𝑎. (3.10) Then ‖𝑇^𝑛+10−𝑢^∗‖≤𝜔^𝑛/(1−𝜔)‖𝑇0‖.

Proof. Denote 𝐾[0,𝑎]={𝑢∈𝐾∶‖𝑢‖≤𝑎}. Similarly to the proof of Theorem 3.1, we can know that 𝑇∶𝐾[0,𝑎]→𝐾[0,𝑎]. Let 𝑢₁=𝑇0, then 𝑢₁∈𝐾[0,𝑎]. Denote 𝑢_𝑛+1(𝑡)=𝑇𝑢_𝑛,𝑛=1,2,…. Copying the corresponding proof of Theorem 3.1, we can prove that 𝑢_𝑛+1(𝑡)≥𝑢_𝑛(𝑡),𝑡∈[0,1]_𝐓,𝑛=1,2,…. (3.11)
Since 𝑇 is completely continuous, we can get that there exists 𝑢^∗∈𝐾[0,𝑎] such that 𝑢_𝑛→𝑢^∗. Applying the continuity of 𝑇 and 𝑢_𝑛+1(𝑡)=𝑇𝑢_𝑛, we can obtain that 𝑇𝑢^∗=𝑢^∗. We note that 𝑓(𝑡,0)>0,forall𝑡∈[0,1]_𝐓, it implies that the zero function is not the solution of the problem (1.1). Therefore, 𝑢^∗ is a positive solution of (1.1).
Now, since
|𝑓(𝑡,𝑙₂)−𝑓(𝑡,𝑙₁)|≤𝜔𝐴|𝑙₂−𝑙₁|,𝑡∈[0,1]_𝐓,0≤𝑙₁,𝑙₂≤𝑎. (3.12) If 𝑢₁,𝑢₂∈𝐾[0,𝑎] and 𝑢₂(𝑡)≥𝑢₁(𝑡), 𝑡∈[0,1]_𝐓, then ‖𝑇𝑢₂−𝑇𝑢₁‖=max_{𝑡∈[0,1]𝐓}|¹₀𝐺(𝑡,𝑠)[𝑓(𝑠,𝑢₂(𝑠))−𝑓(𝑠,𝑢₁(𝑠))]∇𝑠|≤𝜔𝐴max_{𝑡∈[0,1]𝐓}¹₀𝐺(𝑡,𝑠)|𝑢₂(𝑠)−𝑢₁(𝑠)|∇𝑠≤𝜔𝐴‖𝑢₂−𝑢₁‖𝐴⁻¹=𝜔‖𝑢₂−𝑢₁‖. (3.13) Hence, we can deduce that ‖𝑢_𝑛+2−𝑢_𝑛+1‖=‖𝑇𝑢_𝑛+1−𝑇𝑢_𝑛‖≤𝜔^𝑛‖𝑇0−0‖=𝜔^𝑛‖𝑢‖𝑇0‖,_𝑛+𝑘+2−𝑢_𝑛+1‖≤(𝜔^𝑛+𝑘+𝜔^{𝑛+𝑘−1}+⋯+𝜔^𝑛𝜔)‖𝑇0‖<^𝑛1−𝜔‖𝑇0‖. (3.14) It implies that ‖𝑇^𝑛+10−𝑢^∗𝜔‖≤^𝑛1−𝜔‖𝑇0‖. (3.15) The proof is complete.

4. Existence of 𝑛 positive solutions

Theorem 4.1. Assume (𝐻₁)-(𝐻₃) hold, and there exist 2𝑛 positive numbers 𝑎₁,…,𝑎_𝑛,𝑏₁,…,𝑏_𝑛 with 𝑏₁<𝑎₁<𝑏₂<𝑎₂<⋯<𝑏_𝑛<𝑎_𝑛 such that(𝐸₁)max{𝑓(𝑡,𝑎_𝑖)∶𝑡∈[0,1]_𝐓}≤𝑎_𝑖𝐴, min{𝑓(𝑡,Ψ𝑏_𝑖)∶𝑡∈[𝜉₁,1]_𝐓}≥𝑏_𝑖𝐵,𝑖=1,2,…,𝑛; (𝐸₂)𝑓(𝑡,𝑢₁)≤𝑓(𝑡,𝑢₂) for any 𝑡∈[0,1]_𝐓,0≤𝑢₁≤𝑢₂≤𝑎_𝑛. Then the BVP (1.1) has 𝑛 positive solutions 𝑢^∗_𝑖,𝑖=1,2,…,𝑛 such that 𝑏_𝑖≤‖𝑢^∗_𝑖‖≤𝑎_𝑖 and lim_𝑛→∞𝑇^𝑛𝑢_𝑖=𝑢^∗_𝑖, that is, lim_𝑛→∞sup_{𝑡∈[0,1]𝐓}|𝑇^𝑛𝑢_𝑖(𝑡)−𝑢^∗_𝑖(𝑡)|=0, (4.1) where 𝑢_𝑖(𝑡)≡𝑎_𝑖,𝑡∈[0,1]_𝐓,𝑖=1,2,…,𝑛.

Corollary 4.2. Assume that (𝐻₁)-(𝐻₃) and (𝐶^′₁)-(𝐶^′₂) hold, and the following condition is satisfied (𝐸^′) there exist 2(𝑛−1) positive numbers 𝑎₁<𝑏₂<𝑎₂<⋯<𝑏_𝑛−1<𝑎_𝑛−1<𝑏_𝑛 such that max{𝑓(𝑡,𝑎_𝑖)∶𝑡∈[0,1]_𝐓}<𝑎_𝑖𝐴,𝑖=1,…,𝑛−1,min{𝑓(𝑡,Ψ𝑏_𝑖)∶𝑡∈[𝜉_𝑖,1]_𝐓}>𝑏_𝑖𝐵,𝑖=2,…,𝑛. (4.2)Then the BVP (1.1) has 𝑛 positive solutions 𝑢^∗_𝑖,𝑖=1,2,…,𝑛, and there exists a positive number 𝑎_𝑛with 𝑎_𝑛>𝑏_𝑛such that lim_𝑛→∞𝑇^𝑛𝑢_𝑖=𝑢^∗_𝑖, where 𝑢_𝑖(𝑡)≡𝑎_𝑖,𝑡∈[0,1]_𝐓,𝑖=1,2,…,𝑛.

5. Examples

Example 5.1. Let 𝐓=[0,1/3]⋃[1/2,1]. Considering the following BVP: 𝑢^Δ∇(𝑡)+𝑓(𝑡,𝑢)=0,𝑡∈[0,1]_𝐓,1𝑢(0)=0,𝑢(1)=81𝑢(31)+61𝑢(2), (5.1) where 𝑓(𝑡,𝑢)=(200/109)𝑢²+1, it is easy to check that 𝑓(𝑡,0)=1>0, for any 𝑡∈[0,1]_𝐓.
Further calculations give us
7𝑑=8,𝐴^′=[¹₀𝜃(𝑠)∇𝑠]⁻¹8=[7¹₀𝑠(1−𝑠)∇𝑠]⁻¹8={7[₀^1/3𝑠(1−𝑠)𝑑𝑠+^1/2_𝜌(1/2)𝑠(1−𝑠)∇𝑠+¹_1/2𝑠(1−𝑠)𝑑𝑠]}⁻¹=567.109 (5.2) Choose 𝑎=1, it is easy to check that 𝑓(𝑡,⋅)∶[0,1]_𝐓→[0,+∞) is nondecreasing for any 𝑡∈[0,1]_𝐓 and max_{𝑡∈[0,1]𝐓}𝑓(𝑡,1)=200109+1≤1⋅567.109 (5.3) Let 𝑢₀(𝑡)≡0, for 𝑛=0,1,2,…, we have 𝑢_𝑛+1(𝑡)=−^𝑡₀(𝑡−𝑠)(200𝑢109^𝑛8(𝑠)+1)∇𝑠+7𝑡¹₀(1−𝑠)(200𝑢109^𝑛−8(𝑠)+1)∇𝑠71𝑡[8₀^1/3(13−𝑠)(200𝑢109^𝑛1(𝑠)+1)∇𝑠+6₀^1/2(12−𝑠)(200𝑢109^𝑛(𝑠)+1)∇𝑠]. (5.4) By Theorem 3.4, the BVP (5.1) has one positive solution 𝑢^∗ such that 0<‖𝑢^∗‖≤1 and 𝑇^𝑛0→𝑢^∗. On the other hand, for any 0≤𝑢₁,𝑢₂≤1, we have |𝑓(𝑢₁)−𝑓(𝑢₂)|=200109|𝑢²₁−𝑢²₂|≤400109|𝑢₁−𝑢₂|=567⋅109400567|𝑢₁−𝑢₂|=400𝐴567^′|𝑢₁−𝑢₂|. (5.5) Then, ‖𝑇^𝑛+10−𝑢^∗‖≤(400/567)^𝑛1−400/567‖𝑇0‖=567(167400)567^𝑛‖𝑇0‖. (5.6) The first and second terms of this scheme are as follows: 𝑢₀𝑢(𝑡)=0,₁−𝑡(𝑡)=2²+199𝑡1378,𝑡∈[0,3−𝑡],₂2+199𝑡+1378172,𝑡∈[2,1]. (5.7) Now, we compute the third term of this scheme.
For 𝑡∈[0,1/3],
𝑢₂(𝑡)=175𝑡327⁴−9950𝑡61803³−𝑡²2+(6720175+70084602199378)𝑡. (5.8) For 𝑡∈[1/2,1], 𝑢₂(𝑡)=175𝑡327⁴−9950𝑡61803³−503𝑡²981+(6720175+70084602100097185409)𝑡. (5.9)

Example 5.2. Let 𝐓={0}⋃{1/3^𝑛∶𝑛∈𝐍_𝟎}. Considering the BVP on 𝐓,𝑢^Δ∇+𝑢(𝑡)=0,𝑡∈[0,1]_𝐓,𝑢^Δ1(0)=0,𝑢(1)=31𝑢(91)+91𝑢(3). (5.10) By direct computation, we can get 5𝑑=9,𝐴^′=916,𝐵^′=8185,Ψ=.54 (5.11) Choose 𝑎=100,𝑏=1/1875, it is easy to see that the nonlinear term 𝑓(𝑡,𝑢)=𝑓(𝑢)=𝑢(𝑡) possesses the following properties(a)𝑓∶[0,1]_𝐓×[0,+∞)→[0,+∞) is continuous; (b)𝑓(𝑡,𝑢₁)≤𝑓(𝑡,𝑢₂) for any 𝑡∈[0,1]_𝐓 and 0≤𝑢₁≤𝑢₂≤100; (c)max{𝑓(𝑡,100)∶𝑡∈[0,1]_𝐓}=√100≤𝑎𝐴^′=100×9/16, min{𝑓(𝑡,Ψ𝑏)∶𝑡∈[𝜉₁,1]_𝐓}=5/54×1/1875>𝑏𝐵^′=1/1875×81/8. By Theorem 3.1, the BVP (5.10) has one positive solution 𝑢^∗ such that 1/1875≤‖𝑢^∗‖≤100 and lim_𝑛→∞𝑇^𝑛𝑢=𝑢^∗, where 𝑢(𝑡)≡100, 𝑡∈[0,1]_𝐓. Let 𝑢₀(𝑡)≡100,𝑡∈[0,1]_𝐓. For 𝑛=0,1,2,…, we have 𝑢_𝑛+1=¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢_𝑛(𝑠))∇𝑠=−^𝑡₀(𝑡−𝑠)𝑢_𝑛9(𝑠)∇𝑠+5¹₀(1−𝑠)𝑢_𝑛−3(𝑠)∇𝑠5₀^1/9(19−𝑠)𝑢_𝑛1(𝑠)∇𝑠−5₀^1/3(13−𝑠)𝑢_𝑛(𝑠)∇𝑠. (5.12)

Remark 5.3. By Theorems  3.1, 3.2, and  3.3 in [12, 16, 17], the existence of positive solutions for the BVP (5.1) can be obtained, however, we cannot give a way to find the solutions which will be useful from an application viewpoint. Therefore, Theorem 3.1 improves and extends the main results of [12, 16, 17]. On the other hand, in Example 5.2, since 𝑓(0)=0, we cannot obtain the above mentioned results by use of Theorem 3.4, thus, Theorems 3.1 and 3.4 do not contain each other.

Acknowledgment

This project supported by the Youth Science Foundation of Shanxi Province (2009021001-2).