Existence of Positive Solutions for Multi-Point Boundary Value Problems on Infinite Intervals in Banach Spaces

Abstract

We investigate the positive solutions of a class of second-order nonlinear singular differential equations with multi-point boundary value conditions on an infinite interval in Banach spaces. The tools we used are the cone theory and Mönch fixed point theorem and a monotone iterative technique. An example is also given to demonstrate the applications of our results, which include and extend some existing results.

1. Introduction

Let 𝐸 be a real Banach space and let 𝑃 be a cone in 𝐸. 𝑃 is said to be normal if there exists a positive constant 𝑁 such that 𝜃≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖. 𝑃 is said to be regular (fully regular) if 𝑢₁≤𝑢₂≤⋯≤𝑢_𝑛≤𝑦⋯,  sup_𝑛‖𝑢_𝑛‖<∞ implies that their exists 𝑥∈𝐸 such that lim_𝑛→∞‖𝑢_𝑛−𝑥‖=0. Let 𝑃₊=𝑃⧵{𝜃}. In what follows, we always assume that 𝑥^∗₀∈𝑃₊. Let 𝑃_0𝜆={𝑢∈𝑃∶𝑢≥𝜆𝑥^∗₀}(𝜆>0). Obviously, 𝑃_0𝜆⊂𝑃₊ for any 𝜆>0. When 𝜆=1, we write 𝑃₀=𝑃₀₁, that is, 𝑃₀={𝑢∈𝑃∶𝑢≥𝑥^∗₀}.

In this paper, we will consider the following boundary value problems (BVPs) for multipoint singular differential equations of mixed type on an unbounded domain in a real Banach space (𝐸,‖⋅‖)𝑢^′′(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),(𝑇𝑢(𝑡)),(𝑆𝑢(𝑡))=𝜃,𝑡∈𝐽₊,𝑎𝑢(0)−𝑏𝑢^′(0)−^𝑛_𝑖=1𝑘_𝑖𝑢𝜉_𝑖=𝜃,lim_𝑡→+∞𝑢^′(𝑡)=𝑦_∞, (1.1) where 𝐽=[0,+∞), 𝐽₊=(0,+∞), 𝑎>0,𝑏≥0, 𝑘_𝑖≥0,0<𝜉₁<𝜉₂<⋯<𝜉_𝑛<+∞, 𝑓∈𝐶[𝐽₊×𝑃_0𝜆×𝑃_0𝜆×𝑃×𝑃,𝑃],  𝜆>0,𝑦_∞≥𝑥^∗₀, (𝑇𝑢)(𝑡)=^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,(𝑆𝑢)(𝑡)=^∞₀𝐻(𝑡,𝑠)𝑢(𝑠)𝑑𝑠, (1.2)𝐾∈𝐶[𝐷,𝐽],𝐷={(𝑡,𝑠)∈𝐽×𝐽∶𝑡≥𝑠},𝐻∈𝐶[𝐽×𝐽,𝐽]. Here the nonlinear term 𝑓 may be singular at 𝑡=0 and 𝑥₀,𝑥₁,𝑥₂,𝑥₃=𝜃. By singularity, we mean that ‖𝑓(𝑡,𝑥₀,𝑥₁,𝑥₂,𝑥₃)‖→∞ as 𝑡→0⁺ or 𝑥_𝑖→𝜃⁺(𝑖=0,1,2,3).

Second-order boundary value problems (BVPs) on infinite intervals, arising from the study of radially symmetric solutions of nonlinear elliptic equation and models of gas pressure in a semi-infinite porous medium, have received much attention. We can see papers [1, 2] and the references therein. In a recent paper, Liu [3] investigated the existence of solutions of the following second-order two-point boundary value problems on the half-line:𝑥^′′(𝑡)((𝑡(),𝑥(0)+𝑓𝑡,𝑥))=0,𝑡∈0,+∞=0,𝑥^′(∞)=𝑦_∞>0. (1.3) Lian and Ge [4] studied the solvability of the three-point BVP 𝑢^′′(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)(),𝑢(0)𝜂=0,𝑡∈0,+∞=𝛼𝑢,lim_𝑡→+∞𝑢^′(𝑡)=0, (1.4) where 𝛼≠1,𝜂∈(0,+∞). With the help of the established Green function and the Leray-Schauder continuation theorem, suitable conditions imposed on 𝑓 are presented for the existence of solutions. Yan et al. [5] established the results of existence and multiplicity of positive solutions to the BVP on the half-line 𝑢^′′(𝑡)+Φ(𝑡)𝑓𝑡,𝑢,𝑢^′(),(0)=0,𝑡∈0,+∞𝑎𝑢−𝑏𝑢^′(0)=𝑢₀≥0,lim_𝑡→+∞𝑢^′(𝑡)=𝑘>0, (1.5) by using the lower and upper solutions technique. Zhang [6] researched the problem 𝑥^′′(𝑡)+𝑞(𝑡)𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐽₊,𝑥(0)=^𝑚−2_𝑖=1𝛼_𝑖𝑥𝜉_𝑖,𝑥^′(∞)=𝑦_∞, (1.6) by using the fixed point theorem and the monotone iterative technique.

We note that these works [3–6] are all in real space. To the best of our knowledge, very few literatures are available for the computation of positive solutions for mutlipoint BVP on the half-line in Banach space. There are two papers we should present here. Liu [7] discussed the existence of solutions of the following second-order two-point BVP on infinite intervals in a Banach space 𝐸: 𝑥^′′(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)𝑥(0),𝑡∈𝐽,=𝑥₀,𝑥^′(∞)=𝑦_∞, (1.7) where 𝑓∈𝐶[𝐽×𝐸×𝐸],𝐽=[0,+∞),𝑥^′(∞)=lim_𝑡→∞𝑥^′(𝑡). The main tool is the Sadovskii’s fixed point theorem. Zhang [8] concerned the existence of solutions of the following singular problems 𝑥^′′(𝑡)+𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)=𝜃,𝑡∈𝐽₊,𝑥(0)=^𝑚−2_𝑖=1𝛼_𝑖𝑥𝜉_𝑖,𝑥^′(∞)=𝑦_∞, (1.8) where 𝐽₊=(0,+∞). The nonlinear term 𝑓 may be singular at 𝑡=0,𝑥=𝜃 and/or 𝑥^′=𝜃. They used the Mönch fixed point theorem.

Motivated by the above papers, we also use the Mönch fixed point theorem to give the existence of a positive solutions of the more general BVP (1.1) for integrodifferential equations on infinite intervals in a Banach spaces. The main features of the present paper are as follows. Firstly, comparing with [3–6], the space in this paper is Banach space. The equation we discussed here is more general than those of [3–8] because the function 𝑓 of (1.1) has new terms 𝑇𝑢, 𝑆𝑢 and the boundary value conditions are more complicated. Moreover, the singularity of nonlinear term 𝑓 in this paper is more complex than [2, 7, 9–11]. Furthermore, an iterative sequence for the solution under some normal type conditions is established which makes it very important and convenient in applications. [3–7, 9–11] did not obtain this kind of result.

The rest of this paper is organized as follows. In Section 2, we give several important Lemmas. The main theorems are formulated and proved in Section 3, followed by an example in Section 4 to demonstrate the application of our results.

2. The Preliminary and Several Lemmas

Let 𝐹𝐶[𝐽,𝐸]=𝑢∈𝐶[𝐽,𝐸]∶sup_𝑡∈𝐽‖𝑢(𝑡)‖𝑒^𝑡<∞,𝐷𝐶¹[𝐽,𝐸]=𝑢∈𝐶¹[𝐽,𝐸]∶sup_𝑡∈𝐽‖𝑢(𝑡)‖𝑒^𝑡<∞,sup_𝑡∈𝐽𝑢^′(𝑡)𝑒^𝑡<∞. (2.1) Evidently, 𝐶¹[𝐽,𝐸]⊂𝐶[𝐽,𝐸],𝐷𝐶¹[𝐽,𝐸]⊂𝐹𝐶[𝐽,𝐸]. It is easy to see that 𝐹𝐶[𝐽,𝐸] is a Banach space with norm ‖𝑢‖_𝐹=sup_𝑡∈𝐽‖𝑢(𝑡)‖/𝑒^𝑡 and 𝐷𝐶¹[𝐽,𝐸] is also a Banach space with norm ‖𝑢‖_𝐷=max{‖𝑢‖_𝐹,‖𝑢^′‖_𝐶}, where ‖𝑢^′‖_𝐶=sup_𝑡∈𝐽‖𝑢^′(𝑡)‖/𝑒^𝑡.

Let 𝑃(𝐹)={[](𝑡)},𝑃(𝐷)=𝑢∈𝐹𝐶𝐽,𝐸∶𝑢≥𝜃,∀𝑡∈𝐽𝑢∈𝐷𝐶¹[𝐽,𝐸]∶𝑢(𝑡)≥𝜃,𝑢^′(𝑡)≥𝜃,∀𝑡∈𝐽. (2.2) It is clear that 𝑃(𝐹),𝑃(𝐷) are cones in 𝐹𝐶[𝐽,𝐸] and 𝐷𝐶¹[𝐽,𝐸], respectively. A map 𝑢∈𝐷𝐶¹[𝐽,𝐸]⋂𝐶²[𝐽₊,𝐸] is called a positive solutions of the BVP (1.1) if 𝑢∈𝑃(𝐷) and satisfies (1.1).

Let 𝛼,𝛼_𝐷 denote the Kuratowski measure of noncompactness in 𝐸 and 𝐷𝐶¹[𝐽,𝐸], respectively. For details on the definition and properties of the measure of noncompactness, the reader is referred to [12, 13].

Some conditions to be used throughout the rest of the paper are listed below.(𝐻₁)lim_𝑡→∞𝑒^−𝑡𝑡₀𝐾(𝑡,𝑠)𝑒^𝑠𝑑𝑠=0,^∞₀𝐻(𝑡,𝑠)𝑒^𝑠𝑑𝑠<∞,∀𝑡∈𝐽,lim_𝑡→∞𝑒^−𝑡∞₀𝐻(𝑡,𝑠)𝑒^𝑠𝑑𝑠=0,lim_{𝑡′→𝑡}^∞₀𝐻𝑡^′,𝑠−𝐻(𝑡,𝑠)𝑒^𝑠𝑑𝑠=0,∀𝑡,𝑡^′∈𝐽. (2.3) In this case, let 𝑘^∗=sup_𝑡∈𝐽𝑒^−𝑡𝑡₀𝐾(𝑡,𝑠)𝑒^𝑠𝑑𝑠,ℎ^∗=sup_𝑡∈𝐽𝑒^−𝑡∞₀𝐻(𝑡,𝑠)𝑒^𝑠𝑑𝑠. (2.4)(𝐻₂)𝑓∈𝐶[𝐽₊×𝑃_0𝜆×𝑃_0𝜆×𝑃×𝑃,𝑃], for any 𝜆>0 and there exist 𝑚,𝑝,𝑞∈𝐿[𝐽₊,𝐽] and 𝑔∈𝐶[𝐽₊×𝐽₊×𝐽×𝐽,𝐽] such that 𝑓𝑡,𝑢₀,𝑢₁,𝑢₂,𝑢₃≤𝑚(𝑡)+𝑝(𝑡)𝑔𝑢₀,𝑢₁,𝑢₂,𝑢₃,∀𝑡∈𝐽₊,𝑢₀,𝑢₁∈𝑃₀,𝑢₂,𝑢₃𝑓∈𝑃,𝑡,𝑢₀,𝑢₁,𝑢₂,𝑢₃𝑞(𝑡)𝑢₀+𝑢₁+𝑢₂+𝑢₃𝑢→0₀,𝑢₁∈𝑃₀,𝑢₂,𝑢₃∈𝑃,𝑢₀+𝑢₁+𝑢₂+𝑢₃→∞ (2.5) uniformly for 𝑡∈𝐽₊, and 𝑚^∗=^∞₀𝑚(𝑡)𝑑𝑡<∞,𝑝^∗=^∞₀𝑝(𝑡)𝑑𝑡<∞,𝑞^∗=^∞₀𝑞(𝑡)𝑒^𝑡𝑑𝑡<∞. (2.6)(𝐻₃) For any 𝑡∈𝐽₊, 𝑅>0 and countable sets 𝑉_𝑖⊂𝐷𝐶¹[𝐽,𝑃^∗_0𝑅](𝑖=0,1),𝑉_𝑖⊂𝐷𝐶¹[𝐽,𝑃^∗_𝑅](𝑖=2,3), there exist 𝐿_𝑖∈𝐿[𝐽,𝐽](𝑖=0,1,2,3) such that 𝛼𝑓𝑡,𝑉₀(𝑡),𝑉₁(𝑡),𝑉₂(𝑡),𝑉₃(𝑡)≤³_𝑖=0𝐿_𝑖(𝑡)𝛼𝑉_𝑖(𝑡,𝐿_∗)=^∞₀𝐿₀(𝑠)+𝐿₁(𝑠)+𝐿₂(𝑠)𝑘^∗+𝐿₃(𝑠)ℎ^∗𝑒^𝑠1𝑑𝑠<4, (2.7) where 𝑃^∗_0𝑅={𝑢∈𝑃∶𝑢≥𝑥^∗₀,‖𝑢‖≤𝑅}  (𝑖=0,1) and 𝑃^∗_𝑅={𝑢∈𝑃∶‖𝑢‖≤𝑅}.(𝐻₄)𝑡∈𝐽₊,𝑥^∗₀≤𝑢_𝑖≤𝑢_𝑖(𝑖=0,1),𝜃≤𝑢₂≤𝑢₂,𝜃≤𝑢₃≤𝑢₃ imply 𝑓𝑡,𝑢₀,𝑢₁,𝑢₂,𝑢₃≤𝑓𝑡,𝑢₀,𝑢₁,𝑢₂,𝑢₃. (2.8)

Lemma 2.1 (Lemma 1 see, [9]). If condition (𝐻₁) is satisfied, then the operators T and S defined by (1.2) are bounded linear operators from 𝐹𝐶[𝐽,𝐸] into 𝐹𝐶[𝐽,𝐸] and ‖𝑇‖≤𝑘^∗,‖𝑆‖≤ℎ^∗,𝑇(𝑃(𝐹))⊂𝑃(𝐹),𝑆(𝑃(𝐹))⊂𝑃(𝐹). (2.9)

In what follows, we write 𝑄=𝑢∈𝐷𝐶¹[𝐽,𝑃]∶𝑢^(𝑖)(𝑡)≥𝑥^∗₀,∀𝑡∈𝐽,𝑖=0,1. (2.10) Evidently, 𝑄 is a closed convex set in 𝐷𝐶¹[𝐽,𝐸]. We will reduce the BVP (1.1) to an integral equation in 𝐸. To this end, we first consider the operator 𝐴 defined by (𝑡)=1𝐴𝑢)(Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑠),(𝑠)+𝑇𝑢)(𝑆𝑢)(𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏𝑑𝑠,𝑢∈𝑄, (2.11) where Δ=𝑎−^𝑛_𝑖=1𝑘_𝑖>0,𝛿=𝑏+^𝑛_𝑖=1𝑘_𝑖𝜉_𝑖,Δ≤𝛿. (2.12)

Lemma 2.2. If conditions (𝐻₁) and (𝐻₂) hold, then the operator 𝐴 defined by (2.11) is a continuous operator from 𝑄 into 𝑄.

Proof. Firstly, we will show 𝐴(𝑄)⊂𝑄. Let 𝜀₀=12(1+𝛿/Δ)𝑞^∗2+𝑘^∗+ℎ^∗,𝑥𝑟=^∗0𝑁>0. (2.13) By virtue of condition (𝐻₂), there exists a 𝑅>𝑟 such that 𝑓𝑡,𝑥,𝑦,𝑧,𝑤≤𝜀₀𝑞(𝑡)‖𝑥‖+𝑦+‖𝑧‖+‖𝑤‖,∀𝑡∈𝐽₊,𝑥,𝑦∈𝑃₀‖𝑥‖+𝑦+‖𝑧‖+‖𝑤‖𝑓(𝑡)(𝑡),𝑧,𝑤∈𝑃,>𝑅,𝑡,𝑥,𝑦,𝑧,𝑤≤𝑚+𝑀𝑝,∀𝑡∈𝐽₊,𝑥,𝑦∈𝑃₀,𝑧,𝑤∈𝑃,‖𝑥‖+𝑦+‖𝑧‖+‖𝑤‖≤𝑅, (2.14) where 𝑀=max𝑔𝑢₀,𝑢₁,𝑢₂,𝑢₃∶𝑟≤𝑢_𝑖≤𝑅(𝑖=0,1),0≤𝑢_𝑖≤𝑅(𝑖=2,3). (2.15) Hence, 𝑓𝑡,𝑥,𝑦,𝑧,𝑤≤𝜀₀𝑞(𝑡)‖𝑥‖+𝑦+‖𝑧‖+‖𝑤‖+𝑚(𝑡)+𝑀𝑝(𝑡),∀𝑡∈𝐽₊,𝑥,𝑦∈𝑃₀,𝑧,𝑤∈𝑃. (2.16)
Let 𝑢∈𝑄. We have by (2.16) and Lemma 2.1𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)≤𝜀₀𝑞(𝑡)‖𝑢(𝑡)‖+𝑢^′(𝑡)+‖(𝑇𝑢)(𝑡)‖+‖(𝑆𝑢)(𝑡)‖+𝑚(𝑡)+𝑀𝑝(𝑡)≤𝜀₀𝑞(𝑡)𝑒^𝑡‖𝑢‖_𝐹+𝑢^′_𝐶+𝑘^∗‖𝑢‖_𝐹+ℎ^∗‖𝑢‖_𝐹+𝑚(𝑡)+𝑀𝑝(𝑡)≤𝜀₀𝑞(𝑡)𝑒^𝑡2+𝑘^∗+ℎ^∗‖𝑢‖_𝐷+𝑚(𝑡)+𝑀𝑝(𝑡),∀𝑡∈𝐽₊, (2.17) which together with condition (𝐻₂) implies the convergence of the infinite integral ^∞₀𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)𝑑𝑡. (2.18) Thus, we have ^𝑡₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝜏),(𝜏)≤𝑇𝑢)(𝑆𝑢)(𝑑𝜏𝑑𝑠_𝑡0∞𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝜏),(𝜏)≤𝑇𝑢)(𝑆𝑢)(𝑑𝜏𝑑𝑠_∞0𝑡0𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝑠𝑑𝜏≤𝑡^∞₀𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏,∀𝑡∈𝐽₊, (2.19) (2.11), (2.19), and (𝐻₂) tell us that 𝑡≤1‖(𝐴𝑢)()‖Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑠),(𝑠)+𝑇𝑢)(𝑆𝑢)(𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝜏),(𝜏)𝑦𝑇𝑢)(𝑆𝑢)(𝑑𝜏𝑑𝑠+𝑡_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝜏),(𝜏≤𝛿𝑇𝑢)(𝑆𝑢)()𝑑𝜏𝑑𝑠Δ𝑦_∞+𝑡𝑦_∞+𝑏Δ^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑠),(𝑠)+∑𝑇𝑢)(𝑆𝑢)(𝑑𝑠_𝑛𝑖=1𝑘_𝑖𝜉_𝑖Δ^∞₀𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏+𝑡^∞₀𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏. (2.20) Therefore, ‖(𝐴𝑢)(𝑡)‖𝑒^𝑡≤𝛿Δ𝑦_∞+𝑦_∞+𝑏Δ^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑠),(𝑠)+∑𝑇𝑢)(𝑆𝑢)(𝑑𝑠_𝑛𝑖=1𝑘_𝑖𝜉_𝑖Δ^∞₀𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝜏),(𝜏)+𝑇𝑢)(𝑆𝑢)(𝑑𝜏_∞0𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝜏),(𝜏)≤𝛿𝑇𝑢)(𝑆𝑢)(𝑑𝜏1+Δ^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑠),(𝑠)𝛿𝑇𝑢)(𝑆𝑢)(𝑑𝑠+Δ+1𝑦_∞≤𝛿1+Δ𝜀₀𝑞^∗2+𝑘^∗+ℎ^∗‖𝑢‖_𝐷+𝑚^∗+𝑀𝑝^∗+𝛿Δ+1𝑦_∞≤12‖𝑢‖_𝐷+𝛿1+Δ𝑚^∗+𝑀𝑝^∗+𝛿Δ+1𝑦_∞. (2.21) Differentiating (2.11), we get (𝐴𝑢)^′(𝑡)=^∞_𝑡𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠)𝑑𝑠+𝑦_∞. (2.22) Hence, (𝐴𝑢)^′(𝑡)𝑒^𝑡≤^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠)𝑑𝑠+𝑦_∞≤𝜀₀𝑞^∗2+𝑘^∗+ℎ^∗‖𝑢‖_𝐷+𝑚^∗+𝑀𝑝^∗+𝑦_∞≤12‖𝑢‖_𝐷+𝑚^∗+𝑀𝑝^∗+𝑦_∞,∀𝑡∈𝐽. (2.23) It follows from (2.21) and (2.23) that ‖(𝐴𝑢)(𝑡)‖_𝐷≤12‖𝑢‖_𝐷+𝛿1+Δ𝑚^∗+𝑀𝑝^∗+𝛿Δ+1𝑦_∞. (2.24) So, 𝐴𝑢∈𝐷𝐶¹[𝐽,𝐸]. On the other hand, it can be easily seen that (𝑡)≥𝛿𝐴𝑢)(Δ𝑦_∞≥𝑦_∞≥𝑥^∗₀,(𝐴𝑢)^′(𝑡)≥𝑦_∞≥𝑥^∗₀,∀𝑡∈𝐽. (2.25) So, 𝐴𝑢∈𝑄. Thus, 𝐴 maps 𝑄 into 𝑄 and (2.24) holds.
Secondly, we will show that 𝐴 is continuous. Let 𝑢_𝑚,𝑢∈𝑄,‖𝑢_𝑚−𝑢‖_𝐷→0  (𝑚→∞). Then 𝑟=sup_𝑚‖𝑢_𝑚‖_𝐷<∞ and ‖𝑢‖_𝐷≤𝑟. Similar to (2.21) and (2.23), it is easy to say 𝐴𝑢_𝑚−𝐴𝑢_𝐷≤𝛿1+Δ^∞₀𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)−𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠)𝑑𝑠. (2.26) So we get 𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)→𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠)as𝑚→∞,∀𝑡∈𝐽₊. (2.27) Then we know from (2.17) that 𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)−𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢𝑠)𝜀)(≤2₀𝑞(𝑡)𝑒^𝑡2+𝑘^∗+ℎ^∗(𝑡)(𝑡(𝑡)𝐽𝑟+𝑚+𝑀𝑝)∶=𝜎∈𝐿₊,𝐽,𝑚=1,2,3,…,∀𝑡∈𝐽₊. (2.28) It follows from (2.27) and (2.28) and the dominated convergence theorem that lim_𝑚→∞^∞₀𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)−𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠)=0. (2.29) It follows from (2.26) and (2.29) that ‖𝐴𝑢_𝑚−𝐴𝑢‖_𝐷→0 as 𝑚→∞, and the continuity of 𝐴 is proved.

Lemma 2.3. Let conditions (𝐻₁) and (𝐻₂) be satisfied. Then 𝑢∈𝑄∩𝐶²[𝐽₊,𝐸] is a solution of the BVP (1.1) if and only if 𝑢∈𝑄 is a solution of the following integral equation: 𝑢(𝑡)=1Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑠),(𝑠)+𝑇𝑢)(𝑆𝑢)(𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏𝑑𝑠. (2.30)

Proof. Integrating the differential equation in (1.1) from 𝑡 to ∞, one has 𝑢^′(𝑡)=𝑦_∞+^∞_𝑡𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠),(𝑇𝑢)(𝑠),(𝑆𝑢)(𝑠)𝑑𝑠. (2.31) Then, integrating (2.31) from 0 to 𝑡, we have 𝑢(𝑡)=𝑢(0)+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏)𝑑𝜏𝑑𝑠. (2.32) By Lemma 2.2, we know that ∫^𝑡₀∫^∞_𝑠𝑓(𝜏,𝑢(𝜏),𝑢^′(𝜏),(𝑇𝑢)(𝜏),(𝑆𝑢)(𝜏))𝑑𝜏𝑑𝑠 is convergent. Since 𝑎𝑢(0)−𝑏𝑢^′(0)−∑^𝑛_𝑖=1𝑘_𝑖𝑢(𝜉_𝑖)=𝜃, we can compute those coefficients 𝑢(0),  𝑢^′(0),  𝑢(𝜉_𝑖) and then obtain (2.30).
Conversely, if 𝑢 is a solution of integral equation (2.30), then direct differentiation gives the proof.

Lemma 2.4 (Mönch Fixed Point Theorem [10]). Let 𝑄 be a closed convex set of 𝐸 and 𝑢∈𝑄. Assume that the continuous operator 𝐴∶𝑄→𝑄 has the following property: 𝑉⊂𝑐𝑜({𝑢}⋃𝐹(𝑉))⇒𝑉 is relatively compact. Then 𝐹 has a fixed point in 𝑄.

Lemma 2.5. Let 𝐻 be a bounded set in 𝐷𝐶¹[𝐽,𝐸]. Suppose that 𝐻^′(𝑡) is equicontinuous on any finite subinterval [0,𝑐]  (𝑐>0) of 𝐽 and 𝑒^−𝑡‖𝑢^(𝑖)‖→0 as 𝑡→∞ uniformly for 𝑢∈𝐻(𝑖=0,1). Then 𝛼_𝐷(𝐻)=maxsup_𝑡∈𝐽𝑒^−𝑡𝛼(𝐻(𝑡)),sup_𝑡∈𝐽𝑒^−𝑡𝛼𝐻⁽¹⁾(𝑡), (2.33) where 𝐻^(𝑖)(𝑡)={(𝑥_𝑚)^(𝑖)(𝑡)∶𝑥_𝑚∈𝐻,𝑚=1,2,3,…}(𝑖=0,1),𝛼 and 𝛼_𝐷 denote the Kuratowski measure of noncompactness of bounded sets in 𝐸 and 𝐷𝐶¹[𝐽,𝐸], respectively.

Proof. The proof is similar to [11, Lemma 7], we omit it.

Lemma 2.6. If condition (𝐻₄) is satisfied, then 𝑥,𝑦∈𝑄,𝑥^(𝑖)≤𝑦^(𝑖),𝑡∈𝐽(𝑖=0,1) imply that (𝐴𝑥)^(𝑖)≤(𝐴𝑦)^(𝑖),𝑡∈𝐽(𝑖=0,1).

Proof. It is easy to see that this lemma follows from (2.11) and (2.22) and condition (𝐻₄).

3. Main Results

In the following, we will give the main results of this paper.

Theorem 3.1. Assume that (𝐻₁)-(𝐻₃) hold, then the BVP (1.1) has a positive solution 𝑢∈𝐷𝐶¹[𝐽,𝐸]⋂𝐶²[𝐽₊,𝐸] satisfying (𝑢)^(𝑖)(𝑡)≥𝑥^∗₀  for 𝑡∈𝐽(𝑖=0,1).

Proof. By Lemma 2.2, the operator 𝐴 defined by (2.11) is a continuous operator from 𝑄 into 𝑄. And, by Lemma 2.3, we only need to show that 𝐴 has a fixed point 𝑢 in 𝑄.
Choose 𝛿𝑅>21+Δ𝑚^∗+𝑀𝑝^∗+𝛿Δ+1𝑦_∞, (3.1) and let 𝑄₁={𝑢∈𝑄∶‖𝑢‖_𝐷≤𝑅}. Obviously, 𝑄₁ is a bounded closed convex set in the space 𝐷𝐶¹[𝐽,𝐸]. It is easy to see that 𝑄₁ is not empty since (𝛿/Δ+1)𝑦_∞∈𝑄₁. It follows from (2.24) and (3.1) that 𝑢∈𝑄₁ implies 𝐴𝑢∈𝑄₁, that is, 𝐴 maps 𝑄₁ into 𝑄₁.
Now, we are in position to show that 𝐴(𝑄₁) is relatively compact. Let 𝑉={𝑢_𝑚∶𝑚=1,2,…}⊂𝑄₁ satisfying 𝑉⊂𝑐𝑜({𝑢}⋃{(𝐴𝑉)}) for some 𝑢∈𝑄₁. Then ‖𝑢_𝑚‖_𝐷≤𝑅. So we have by (2.11) and (2.22) that 𝐴𝑢_𝑚(𝑡)=1Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)+𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑢_𝑚(𝜏),𝑢^′_𝑚(𝜏),𝑇𝑢_𝑚(𝜏),𝑆𝑢_𝑚(𝜏)𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑢_𝑚(𝜏),𝑢^′_𝑚(𝜏),𝑇𝑢_𝑚(𝜏),𝑆𝑢_𝑚(𝜏=1)𝑑𝜏𝑑𝑠Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)+𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑢_𝑚(𝜏),𝑢^′_𝑚(𝜏),𝑇𝑢_𝑚(𝜏),𝑆𝑢_𝑚(𝜏)𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀𝑠𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)+𝑑𝑠_∞𝑡𝑡𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)𝑑𝑠,𝐴𝑢_𝑚^′(𝑡)=𝑦_∞+^∞_𝑡𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)𝑑𝑠. (3.2) So for any 𝑡₁,𝑡₂∈[0,𝑐](𝑐>0),𝑡₁<𝑡₂, we have that 𝐴𝑢_𝑚^′𝑡₂−𝐴𝑢_𝑚^′𝑡₁≤^𝑡2_𝑡1𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)𝑑𝑠≤𝜀₀2+𝑘^∗+ℎ^∗𝑅^𝑡2_𝑡1𝑞(𝑠)𝑒^𝑠𝑑𝑠+^𝑡2_𝑡1𝑚(𝑠)+𝑀𝑝(𝑠)𝑑𝑠. (3.3) (3.3) implies that {(𝐴𝑢_𝑚)^′(𝑡)}  (𝑚=1,2,3,…) is equicontinuous on any finite subinterval of 𝐽.
On the other hand, by (2.17), (2.20), (3.2) we can obtain 𝐴𝑢_𝑚(𝑡≤1)2𝑢_𝑚𝐷+𝛿𝑡+Δ𝑚^∗+𝑀𝑝^∗+𝛿Δ+𝑡𝑦_∞,∀𝑡∈𝐽₊,𝐴𝑢_𝑚^′(𝑡≤1)2𝑢_𝑚𝐷+𝑚^∗+𝑀𝑝^∗+𝑦_∞,∀𝑡∈𝐽₊, (3.4) which implies that 𝑒^−𝑡‖(𝐴𝑢_𝑚)^(𝑖)(𝑡)‖→0 as 𝑡→∞(𝑖=0,1) uniformly for 𝑚=1,2,3,…. Hence, by Lemma 2.5 we have 𝛼_𝐷(𝐴𝑉)=maxsup_𝑡∈𝐽𝑒^−𝑡𝛼((𝐴𝑉)(𝑡)),sup_𝑡∈𝐽𝑒^−𝑡𝛼(𝐴𝑉)⁽¹⁾(𝑡), (3.5) where 𝐴𝑉={𝐴𝑢_𝑚∶𝑢_𝑚∈𝑉,𝑚=1,2,3,…}, and (𝐴𝑉)⁽¹⁾={(𝐴𝑢_𝑚)⁽¹⁾(𝑡)∶𝑢_𝑚∈𝑉,𝑚=1,2,3,…}.
It follows from (2.18) that the infinite integral  ∫^∞₀‖𝑓(𝑡,𝑢_𝑚(𝑡),𝑢^′_𝑚(𝑡),(𝑇𝑢_𝑚)(𝑡), (𝑆𝑢_𝑚)(𝑡))‖𝑑𝑡  is  convergence uniformly for 𝑚=1,2,3,…. So, for any 𝜀>0, we can choose a large 𝑇>0 such that ^∞_𝑇𝑓𝑡,𝑢_𝑚(𝑡),𝑢^′_𝑚(𝑡),𝑇𝑢_𝑚(𝑡),𝑆𝑢_𝑚(𝑡)𝑑𝑡<𝜀 (3.6) holds for any 𝑚. Then, by [6, Theoerm 1.2.3], (3.6), and (𝐻₃), we obtain 𝑒^−𝑡𝛼𝑡𝑡((𝐴𝑉)())≤2𝑒^𝑡𝑇_𝑡𝛼𝑓𝑠,𝑉(𝑠),𝑉^′(𝑠),(𝑠),(𝑠)𝑠𝑇𝑉)(𝑆𝑉)(𝑑𝑠+2𝜀+2𝑒^𝑡𝑡₀𝛼𝑓𝑠,𝑉(𝑠),𝑉^′(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)𝑑𝑠≤4^∞₀𝛼𝑓𝑠,𝑉(𝑠),𝑉^′(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)𝑑𝑠+2𝜀≤4𝛼_𝐷(𝑉)^∞₀𝐿₀(𝑠)+𝐿₁(𝑠)+𝐿₂(𝑠)𝑘^∗+𝐿₃(𝑠)ℎ^∗𝑒^𝑠𝑒𝑑𝑠+2𝜀,_−𝑡𝛼(𝐴𝑉)⁽¹⁾(𝑡)≤2^∞₀𝛼𝑓𝑠,𝑉(𝑠),𝑉^′(𝑠),(𝑇𝑉)(𝑠),(𝑆𝑉)(𝑠)𝑑𝑠+2𝜀≤2𝛼_𝐷(𝑉)^∞₀𝐿₀(𝑠)+𝐿₁(𝑠)+𝐿₂(𝑠)𝑘^∗+𝐿₃(𝑠)ℎ^∗𝑒^𝑠𝑑𝑠+2𝜀. (3.7) By (3.7) and noting the fact that 𝜀>0 is arbitrary, we see that 𝑎_𝐷(𝐴𝑉)≤4𝐿^∗𝑎_𝐷(𝑉)≤𝑎_𝐷(𝑉). (3.8) On the other hand, 𝛼_𝐷(𝑉)≤𝛼_𝐷{𝑐𝑜({𝑢}⋃(𝐴𝑉))}=𝛼_𝐷(𝐴𝑉). Then 𝛼_𝐷(𝑉)=0, that is, 𝑉 is relatively compact in 𝐷𝐶¹[𝐽,𝐸].
Hence, the Mönch fixed point theorem implies that 𝐴 has a fixed point 𝑢 in 𝑄₁ and this theorem is proved.

Theorem 3.2. Let cone 𝑃 be normal and let conditions (𝐻₁)–(𝐻₄) be satisfied. Then the BVP (1.1) has a positive solution 𝑦∈𝑄⋂[𝐽₊,𝐸] which is minimal in the sense that 𝑢^(𝑖)(𝑡)≥𝑦^(𝑖)(𝑡),𝑡∈𝐽  (𝑖=0,1) for any positive solution 𝑢∈𝑄⋂[𝐽₊,𝐸]. Moreover, ‖𝑦‖_𝐷≤2𝛾+‖𝑥₀‖_𝐷, where 𝛿𝛾=1+Δ𝑚^∗+𝑀𝑝^∗+𝛿Δ+1𝑦_∞ (3.9) and there exists a monotone iterative sequence {𝑢_𝑚} such that 𝑢_𝑚^(𝑖)→𝑦^(𝑖) as 𝑚→∞(𝑖=0,1) uniformly on 𝐽 and 𝑢_𝑚^′′(𝑡)→𝑦^′′(𝑡) as 𝑚→∞ for any 𝑡∈𝐽₊, where 𝑢₀(𝑡)=1Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑥^∗₀,𝑥^∗₀,𝜃,𝜃𝑑𝑠+^𝑛_𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑥^∗₀,𝑥^∗₀,𝜃,𝜃𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑥^∗₀,𝑥^∗₀𝑢,𝜃,𝜃𝑑𝜏𝑑𝑠,_𝑚(𝑡)=1Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑢_𝑚−1(𝑠),𝑢^′_𝑚−1(𝑠),𝑇𝑢_𝑚−1(𝑠),𝑆𝑢_𝑚−1𝑠)+(𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑢_𝑚−1(𝜏),𝑢^′_𝑚−1(𝜏),𝑇𝑢_𝑚−1(𝜏),𝑆𝑢_𝑚−1(𝜏)𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑢_𝑚−1(𝜏),𝑢^′_𝑚−1(𝜏),𝑇𝑢_𝑚−1(𝜏),𝑆𝑢_𝑚−1(𝜏)𝑑𝜏𝑑𝑠. (3.10)(3.11)

Proof. From (3.10), we can find that 𝑢₀∈𝐶[𝐽,𝐸] and 𝑢^′₀(𝑡)=^∞_𝑡𝑓𝑠,𝑥^∗₀,𝑥^∗₀,𝜃,𝜃𝑑𝑠+𝑦_∞. (3.12) By (3.10) and (3.12), we have 𝑢₀^(𝑖)≥𝑦_∞≥𝑥^∗₀  (𝑖=0,1) and 𝑢₀(𝑡≤1)Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑥^∗₀,𝑥^∗₀,𝜃,𝜃𝑑𝑠+^𝑛_𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑥^∗₀,𝑥^∗₀𝑦,𝜃,𝜃𝑑𝜏𝑑𝑠+𝑡_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑥^∗₀,𝑥^∗₀≤𝛿,𝜃,𝜃𝑑𝜏𝑑𝑠Δ𝑦_∞+𝑡𝑦_∞+𝑏Δ^∞₀𝑓𝑠,𝑥^∗₀,𝑥^∗₀+∑,𝜃,𝜃𝑑𝑠_𝑛𝑖=1𝑘_𝑖𝜉_𝑖Δ^∞₀𝑓𝜏,𝑥^∗₀,𝑥^∗₀,𝜃,𝜃𝑑𝜏+𝑡^∞₀𝑓𝜏,𝑥^∗₀,𝑥^∗₀𝑢,𝜃,𝜃𝑑𝜏,_′0(𝑡)≤^∞_𝑡𝑓𝑠,𝑥^∗₀,𝑥^∗₀,𝜃,𝜃𝑑𝑠+𝑦_∞. (3.13) (2.18) implies that 𝑒^−𝑡‖𝑢₀^(𝑖)‖→0 as 𝑡→∞(𝑖=0,1). Hence, ‖𝑢₀‖_𝐹, ‖𝑢₀‖_𝐶<∞(𝑖=0,1), which implies 𝑢₀∈𝐷𝐶¹[𝐽,𝐸]. From (2.11) and (3.11) we get 𝑢_𝑚(𝑡)=𝐴𝑢_𝑚−1(𝑡),∀𝑡∈𝐽,𝑚=1,2,3,…. (3.14) By Lemma 2.2, we get 𝑢_𝑚∈𝑄 and 𝑢_𝑚(𝑡)_𝐷=𝐴𝑢_𝑚−1(𝑡)_𝐷≤12𝑢_𝑚−1𝐷+𝛾,𝑚=1,2,3,…, (3.15) By Lemma 2.6 and (3.15), we have 𝑢^∗₀≤𝑢₀^(𝑖)(𝑡)≤𝑢₁^(𝑖)(𝑡)≤⋯≤𝑢_𝑚^(𝑖)(𝑡)≤⋯,∀𝑡∈𝐽(𝑖=0,1). (3.16) It follows from (3.15), by induction, that 𝑢_𝑚(𝑡)_𝐷1≤𝛾+21𝛾+⋯2^𝑚−11𝛾+2^𝑚𝑢_0𝐷≤2𝛾+𝑢_0𝐷,𝑚=1,2,3,…. (3.17) Let 𝐾={𝑢∈𝑄∶‖𝑢‖≤2𝛾+‖𝑢₀‖_𝐷}. So, 𝐾 is a bounded closed convex set in the space 𝐷𝐶¹[𝐽,𝐸] and operator 𝐴 maps 𝐾 into 𝐾. Clearly, 𝐾 is not empty since 𝑢₀∈𝐾.
Let 𝑊={𝑢_𝑚∶𝑚=0,1,2,…} and 𝐴𝑊={𝐴𝑢_𝑚∶𝑚=0,1,2,…}. Obviously, 𝑊⊂𝐾 and 𝑊={𝑢₀}⋃(𝐴𝑊). Similarly, as the proof in Theorem 3.1, we can obtain 𝛼_𝐷(𝑊)=0, that is, 𝑊 is relatively compact in 𝐷𝐶¹[𝐽,𝐸]. So there exist a 𝑦∈𝐷𝐶¹[𝐽,𝐸] and a subsequence {𝑢_𝑚𝑗∶𝑗=1,2,3,…}⊂𝑊 such that {𝑢_𝑚^(𝑖)_𝑗∶𝑗=1,2,3,…} converges to 𝑦^(𝑖)(𝑡) uniformly on 𝐽(𝑖=0,1). Since that 𝑃 is normal and {𝑢_𝑚^(𝑖)(𝑡)∶𝑚=1,2,3,…} is nondecreasing on account of (3.16), it is easy to see that the entire sequence {𝑢_𝑚^(𝑖)(𝑡)∶𝑚=1,2,3,…} converges to 𝑦^(𝑖)(𝑡) uniformly on 𝐽(𝑖=0,1). By 𝑢_𝑚∈𝐾 and 𝐾 is a closed convex set in space 𝐷𝐶¹[𝐽,𝐸], we have 𝑦∈𝐾. Let 𝑡∈𝐽 be arbitrarily fixed. It is clear that 𝐻(𝑡,𝑠)𝑢_𝑚(𝑠)−𝑦(𝑠)→0,as𝑚→∞,∀𝑠∈𝐽 (3.18) and, by (3.17), we have 𝐻(𝑡,𝑠)𝑢_𝑚(𝑠)−𝑦(𝑠)≤22𝛾+𝑢_0𝐷𝐻(𝑡,𝑠)𝑒^𝑠=𝜌(𝑠)∈𝐿[𝐽,𝐽],∀𝑠∈𝐽(𝑚=1,2,…). (3.19) It follows from (3.18) and (3.19) and the dominated convergence theorem that 𝑆𝑢_𝑚(𝑠)−𝑆𝑦(𝑠)≤^∞₀𝐻(𝑡,𝑠)𝑢_𝑚(𝑠)−𝑦(𝑠)𝑑𝑠→0(𝑚→∞), (3.20) which implies that (𝑆𝑢_𝑚)(𝑡)→(𝑆𝑦)(𝑡), as 𝑚→∞ for any 𝑡∈𝐽. Hence, 𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)→𝑓𝑠,𝑦(𝑠),𝑦^′(𝑠),𝑇𝑦(𝑠),𝑆𝑦(𝑠)as𝑚→∞,∀𝑠∈𝐽₊. (3.21) By virtue of (3.17) and Lemma 2.1, we have 𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠)−𝑓𝑠,𝑦(𝑠),𝑦^′(𝑠),(𝑠),(𝑠)𝜀𝑇𝑦𝑆𝑦≤2₀𝑞(𝑠)𝑒^𝑠2+𝑘^∗+ℎ^∗𝑢_𝑚𝐷(𝑠)(𝑠𝜀+𝑚+𝑀𝑝)≤2₀𝑞(𝑠)𝑒^𝑠2+𝑘^∗+ℎ^∗2𝛾+𝑢_0𝐷+𝑚(𝑠)+𝑀𝑝(𝑠). (3.22) Now, noting (3.21) and (3.22) and taking limit as 𝑚→∞ in (3.11), we obtain 𝑦(𝑡)=1Δ𝛿𝑦_∞+𝑏^∞₀𝑓𝑠,𝑦(𝑠),𝑦^′(𝑠),(𝑠),(𝑠)+𝑇𝑦𝑆𝑦𝑑𝑠_𝑛𝑖=1𝑘_𝑖^𝜉𝑖₀^∞_𝑠𝑓𝜏,𝑦(𝜏),𝑦^′(𝜏),𝑇𝑦(𝜏),𝑆𝑦(𝜏)𝑑𝜏𝑑𝑠+𝑡𝑦_∞+^𝑡₀^∞_𝑠𝑓𝜏,𝑦(𝜏),𝑦^′(𝜏),𝑇𝑦(𝜏),𝑆𝑦(𝜏)𝑑𝜏𝑑𝑠, (3.23) which follows from Lemma 2.3 that 𝑦∈𝐾⋂𝐶²[𝐽₊,𝐸] and 𝑦(𝑡) is positive solution of the BVP (1.1). Differentiating (3.11) twice, we get 𝑢_𝑚^′′=−𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠),𝑇𝑢_𝑚(𝑠),𝑆𝑢_𝑚(𝑠),∀𝑠∈𝐽₊,𝑚=1,2,3,…. (3.24) Hence, by (3.21), we obtain lim_𝑚→∞𝑢_𝑚^′′=−𝑓𝑠,𝑦(𝑠),𝑦^′(𝑠),𝑇𝑦(𝑠),𝑆𝑦(𝑠),∀𝑠∈𝐽₊. (3.25)
Let 𝑢(𝑡) be any other positive solution of the BVP (1.1). By Lemma 2.3, we have 𝑢(𝑡)∈𝑄 and 𝑢(𝑡)=(𝐴𝑢)(𝑡), for 𝑡∈𝐽. It is clear that 𝑢^(𝑖)(𝑡)≥𝑥^∗₀>0 for any 𝑡∈𝐽(𝑖=0,1). So, by Lemma 2.6, we can have 𝑢^(𝑖)(𝑡)≥𝑢₀^(𝑖) for any 𝑡∈𝐽(𝑖=0,1). Assume that 𝑢^(𝑖)≥𝑢^(𝑖)_𝑚−1(𝑡) for any 𝑡∈𝐽,𝑚≥1(𝑖=0,1). Then, it follows from Lemma 2.6 that (𝐴𝑢)^(𝑖)≥(𝐴𝑢_𝑚−1)^(𝑖)(𝑡) for any 𝑡∈𝐽,(𝑖=0,1), that is, 𝑢^(𝑖)(𝑡)≥𝑢_𝑚^(𝑖)(𝑡) for any 𝑡∈𝐽,(𝑖=0,1). Hence, by induction, we get 𝑢^(𝑖)(𝑡)≥𝑢_𝑚^(𝑖)(𝑡),∀𝑡∈𝐽,(𝑖=0,1;𝑚=0,1,2,3,…). (3.26) Now, taking limits as 𝑚→∞ in (3.26), we get 𝑢^(𝑖)(𝑡)≥𝑦^(𝑖)(𝑡) for 𝑡∈𝐽(𝑖=0,1), and this completes the proof.

Theorem 3.3. Let cone 𝑃 be fully regular and let conditions (𝐻₁),(𝐻₂),(𝐻₄) be satisfied. Then the conclusion of Theorem 3.2 holds also.

Proof. The proof is almost the same as that of Theorem 3.2. The only difference is that, instead of using condition (𝐻₃), the conclusion 𝛼_𝐷(𝑊)=0 was implied directly by (3.16) and (3.17) and full regularity of 𝑃 and Lemma 2.5.

4. An Example

Example 4.1. Consider the following infinite system of scalar second-order multipoint singular integrodifferential equation: −𝑢_𝑛^′′(𝑡)=𝑒^−2𝑡𝑛²√𝑡1+𝑢_𝑛+𝑢^′_2𝑛+12𝑛²𝑢_𝑛(𝑡)+18𝑛³𝑢^′_2𝑛(𝑡)^1/2+𝑒^−3𝑡√𝑛𝑡^𝑡0𝑒^{−𝑠(𝑡+1)}𝑢_𝑛(𝑠)𝑑𝑠(1+𝑡𝑠+𝑠)^31/3+𝑒^−4𝑡𝑛√𝑡(1+𝑡)^∞0𝑒^−2𝑠cos²(𝑡−𝑠)𝑢_2𝑛(𝑠)𝑑𝑠^1/5,∀𝑡∈𝐽₊,80𝑢_𝑛(0)−3𝑢^′_𝑛(0)−⁷_𝑖=1𝑖𝑢_𝑛𝑖+34=0,lim_𝑡→+∞𝑢^′_𝑛(𝑡)=1𝑛². (4.1)

Conclusion. Infinite system (4.2) has a minimal positive solution 𝑢_𝑛(𝑡) satisfying 𝑢_𝑛(𝑡),𝑢^′_𝑛(𝑡)≥1/𝑛² for 0≤𝑡≤+∞(𝑛=1,2,3,…), and this minimal solution can be obtained by taking limits from some iterative sequences.

Proof. Let 𝐸=𝑙¹={𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛,…)∶∑^∞_𝑛=1|𝑢_𝑛|<∞} with the norm ‖𝑢‖=∑^∞_𝑛=1|𝑢_𝑛|. Choose 𝑃={𝑢=(𝑢_𝑛)∈𝑙¹∶𝑢_𝑛≥0,𝑛=1,2,…}. It is easy to see that 𝐸 is weakly sequence complete, and 𝑃 is a normal cone in 𝐸. Thus, 𝑃 is fully regular.
Now we consider the infinite system (4.2), which can be regarded as the BVP (1.1) with 𝑎=80, 𝑏=3,  𝑘_𝑖=𝑖,  𝜉_𝑖=(𝑖+3)/4  (𝑖=1,2,…,7), 𝑦_∞=(1,1/4,1/9,…). So we have Δ=80−⁷_𝑖=1𝑖=52,𝛿=3+⁷_𝑖=1𝑖𝑖+34=59,Δ≤𝛿. (4.2) In this situation, 𝑢=𝑢₁,𝑢₂,…,𝑢_𝑛,…,𝑣=𝑣₁,𝑣₂,…,𝑣_𝑛,𝑤,…𝑤=₁,𝑤₂,…,𝑤_𝑛,…,𝑧=𝑧₁,𝑧₂,…,𝑧_𝑛,𝐾()=𝑒,…𝑡,𝑠^{−𝑠(𝑡+1)}(1+𝑡𝑠+𝑠)³,𝐻(𝑡,𝑠)=𝑒^−2𝑠cos²(),𝑓𝑡−𝑠𝑓=₁,𝑓₂,…,𝑓_𝑛,…, (4.3) in which 𝑓_𝑛()=𝑒𝑡,𝑢,𝑣,𝑤,𝑧^−2𝑡𝑛²√𝑡1+𝑢_𝑛+𝑣_2𝑛+12𝑛²𝑢_𝑛(𝑡)+18𝑛³𝑣_2𝑛(𝑡)^1/2+𝑒^−3𝑡√𝑤𝑛𝑡^𝑛1/3+𝑒^−4𝑡𝑛√𝑡()𝑧1+𝑡^1/52𝑛. (4.4) Let 𝑥^∗₀=𝑦_∞=(1,1/4,1/9,…). Then 𝑃_0𝜆={𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛,…)∶𝑢_𝑛≥𝜆/𝑛²,𝑛=1,2,…} for any 𝜆>0 and the condition (𝐻₁) holds for 𝑘^∗<1/2 and ℎ^∗<1. It is clear that 𝑓∈𝐶[𝐽₊×𝑃_0𝜆×𝑃_0𝜆×𝑃×𝑃,𝑃] for any 𝜆>0. By (4.4) we get 𝑓(≤𝑒𝑡,𝑢,𝑣,𝑤,𝑧)^−2𝑡√𝑡114+‖𝑢‖+‖𝑣‖^1/2+‖𝑤‖^1/3+‖𝑧‖^1/5. (4.5) So, the condition (𝐻₂) is satisfied for 𝑚(𝑡)=0, 𝑝(𝑡)=𝑞(𝑡)=𝑒^−2𝑡/√𝑡 and 𝑔𝑢₀,𝑢₁,𝑢₂,𝑢₃=114+𝑢₀+𝑢₁^1/2+𝑢₂^1/3+𝑢₃^1/5. (4.6) It is easy to see that (𝐻₄) holds. Thus, our conclusion follows from Theorem 3.3 immediately.

Remark 4.2. It is easy to see that the Example 4.1 cannot be solved by [3–11, 13].

Acknowledgments

The authors are grateful to the anonymous referees for their helpful suggestions and comments. Z. Hao acknowledges support from NSFC (10771117). Ph.D. Programs Foundation of Ministry of Education of China (20093705120002) and NSF of Shandong Province of China (Y2008A24). China Postdoctoral Science Foundation (20090451290), ShanDong Province Postdoctoral Foundation (200801001), and Foundation of Qufu Normal University (BSQD07026).