Computation of Positive Solutions for Nonlinear Impulsive Integral Boundary Value Problems with 𝑝-Laplacian on Infinite Intervals

Abstract

This paper deals with the existence and iteration of positive solutions for nonlinear second-order impulsive integral boundary value problems with 𝑝-Laplacian on infinite intervals. Our approach is based on the monotone iterative technique.

1. Introduction

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years. It has been extensively applied to biology, biologic medicine, optimum control in economics, chemical technology, population dynamics, and so on. It is much richer because all the structure of its emergence has deep physical background and realistic mathematical model and coincides with many phenomena in nature. For an introduction of the basic theory of impulsive differential equations in 𝑅^𝑛, the reader is referred to see Lakshmikantham et al. [1, 2], Samoĭlenko and Perestyuk [3], and the references therein.

Boundary value problems on infinite intervals arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium; see [4–7], for example. In a recent paper [8], by means of a fixed-point theorem due to Avery and Peterson, Li and Nieto obtained some new results on the existence of multiple positive solutions for the following multipoint boundary value problem with a finite number of impulsive times on an infinite interval: 𝑢^′′(𝑡)+𝑞(𝑡)𝑓(𝑡,𝑢(𝑡))=0,∀0<𝑡<∞,𝑡≠𝑡_𝑘𝑡,𝑘=1,2,…,𝑝,Δ𝑢_𝑘=𝐼_𝑘𝑢𝑡_𝑘𝑢(0)=,𝑘=1,2,…,𝑝,_{𝑚−2𝑖=1}𝛼_𝑖𝑢𝜉_𝑖,𝑢^′(∞)=0, (1) where 𝑓∈𝐶([0,+∞)×[0,+∞),[0,+∞)), 𝐼_𝑘∈𝐶([0,+∞),[0,+∞)), 𝑢^′(∞)=lim_𝑡→∞𝑢^′(𝑡), 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<+∞, 0<𝑡₁<𝑡₂<⋯<𝑡_𝑝<+∞, and 𝑞∈𝐶([0,+∞),[0,+∞)).

Boundary value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. Moreover, boundary value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems. They include two-point, three-point, and multipoint boundary value problems as special cases; see [9–14]. For boundary value problems with other integral boundary conditions and comments on their importance, we refer the reader to the papers [11–20] and the references therein.

There are relatively few papers available for integral boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times up to now. In [21], Zhang et al. investigated the existence of minimal nonnegative solution for the following second-order impulsive differential equation −𝑥^′′(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)=0,𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,Δ𝑥^′_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘𝑥(0),𝑘=1,2,…,=₀^+∞𝑔(𝑡)𝑥(𝑡)d𝑡,𝑥^′(∞)=0, (2) where 𝑓∈𝐶(𝐽×𝐽×𝐽,𝐽), 𝐼_𝑘∈𝐶(𝐽,𝐽), 𝐼_𝑘∈𝐶(𝐽,𝐽), 𝐽=[0,+∞), 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, 𝑡_𝑘→∞, and 𝑔(𝑡)∈𝐿[𝐽,𝐽] with ∫₀^+∞𝑔(𝑡)d𝑡<1. Δ𝑥|_{𝑡=𝑡𝑘} denotes the jump of 𝑥(𝑡) at 𝑡=𝑡_𝑘, that is, Δ𝑥|_{𝑡=𝑡𝑘}=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘, (3) where 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) represent the right-hand limit and left-hand limit of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. Δ𝑥^′|_{𝑡=𝑡𝑘} has a similar meaning to 𝑥^′(𝑡).

In the past few years, the existence and the multiplicity of bounded or unbounded positive solutions to nonlinear differential equations on infinite intervals have been studied by several different techniques; we refer the reader to [5–8, 21–29] and the references therein. However, most of these papers only considered the existence of positive solutions under various boundary value conditions. Seeing such a fact, a natural question which arises is “how can we find the solutions when they are known to exist?” More recently, Ma et al. [30] and Sun et al. [31, 32] established iterative schemes for approximating the solutions for some boundary value problems defined on finite intervals by virtue of the iterative technique.

However, to the author's knowledge, the corresponding theory for impulsive integral boundary value problems with 𝑝-Laplacian operator and infinite impulsive times on infinite intervals has not been considered till now. Motivated by previous papers, the purpose of this paper is to obtain the existence of positive solutions and establish a corresponding iterative scheme for the following impulsive integral boundary value problem of second-order differential equation with 𝑝-Laplacian on an infinite interval 𝜑_𝑝𝑥^′(𝑡)^′+𝑞(𝑡)𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)=0,𝑡∈𝐽^′₊,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘𝑥(0)=,𝑘=1,2,…,_0+∞𝑔(𝑡)𝑥(𝑡)d𝑡,𝑥^′(∞)=𝑥_∞, (4) where 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, 𝐽=[0,+∞), 𝐽₊=(0,+∞), 𝐽^′₊=𝐽₊⧵{𝑡₁,𝑡₂,…,𝑡_𝑘,…}, 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, 𝑡_𝑘→∞, and 𝑔(𝑡)∈𝐿[𝐽,𝐽] with ∫₀^+∞𝑔(𝑡)d𝑡<1, ∫₀^+∞𝑡𝑔(𝑡)d𝑡<+∞, and 0≤𝑥^′(∞)=lim_𝑡→+∞𝑥^′(𝑡).

It is clear that 𝜑_𝑝()≤2𝑠+𝑡_𝑝−1𝜑_𝑝(𝑠)+𝜑_𝑝(𝑡)𝜑,𝑝≥2,𝑠,𝑡>0,_𝑝(𝑠)+𝜑_𝑝(𝑡),1<𝑝<2,𝑠,𝑡>0,𝜑_𝑝⁻¹()≤2𝑠+𝑡_1/(𝑝−1)𝜑_𝑝⁻¹(𝑠)+𝜑_𝑝⁻¹(𝑡)𝜑,𝑝≥2,𝑠,𝑡>0,_𝑝−1(𝑠)+𝜑_𝑝⁻¹(𝑡),1<𝑝<2,𝑠,𝑡>0. (5)(6) Throughout this paper, we adopt the following assumptions.(H₁)𝑓(𝑡,𝑢,𝑣)∈𝐶(𝐽×𝐽×𝐽,𝐽),𝑓(𝑡,0,0)≢0 on any subinterval of 𝐽, and when 𝑢,𝑣 are bounded, 𝑓(𝑡,(1+𝑡)𝑢,𝑣) is bounded on 𝐽. (H₂)𝑞(𝑡) is a nonnegative measurable function defined in 𝐽₊ and 𝑞(𝑡) does not identically vanish on any subinterval of 𝐽₊, and 0<₀^+∞𝑞(𝑡)d𝑡<+∞,0<₀^+∞𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)d𝜏d𝑠<+∞. (7)(H₃)𝐼_𝑘∈𝐶(𝐽,𝐽), and there exist 𝑎_𝑘≥0,𝑏_𝑘≥0 such that 0≤𝐼_𝑘(𝑥)≤𝑎_𝑘+𝑏_𝑘𝑥,for(),𝑎𝑥∈𝐽𝑘=1,2,3,…_∗=^∞_𝑘=1𝑎_𝑘<+∞,𝑏^∗=^∞_𝑘=1𝑏_𝑘1+𝑡_𝑘<+∞, (8)with 𝑏^∗<(1/3)(1−∫₀^+∞𝑔(𝑡)d𝑡).

If 𝑝=2,𝐼_𝑘=0(𝑘=1,2,…), 𝑔(𝑡)≡0, 𝑥^′(∞)=0, then BVP (4) reduces to the following two-point boundary value problem: −𝑥^′′(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)𝑥(0)=0,𝑡∈𝐽,=0,𝑥^′(∞)=0, (9) which has been studied in [23].

Compared with [8, 21], the main features of the present paper are as follows. Firstly, second-order differential operator is replaced by a more general 𝑝-Laplacian operator. Secondly, in this paper, 𝑥_∞ in boundary value conditions may not be zero which will bring about computational difficulties. Thirdly, by applying monotone iterative techniques, we construct successive iterative schemes starting off with simple known functions. It is worth pointing out that the first terms of our iterative schemes are simple functions. Therefore, the iterative schemes are significant and feasible.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is presented to illustrate the main results.

2. Preliminaries and Several Lemmas

Definition 1. Let 𝐸 be a real Banach space. A nonempty closed set 𝑃⊂𝐸 is said to be a cone provided that (1)𝑎𝑢+𝑏𝑣∈𝑃 for all 𝑢,𝑣∈𝑃 and all 𝑎≥0,𝑏≥0,(2)𝑢,−𝑢∈𝑃 implies that 𝑢=0.

Definition 2. A map 𝛼∶𝑃→[0,+∞) is said to be concave on 𝑃, if 𝛼(𝑡𝑢+(1−𝑡)𝑣)≥𝑡𝛼(𝑢)+(1−𝑡)𝛼(𝑣) for all 𝑢,𝑣∈𝑃 and 𝑡∈[0,1].
Let 𝑃𝐶[𝐽,𝑅]={𝑥∶𝑥 is a map from 𝐽 into 𝑅 such that 𝑥(𝑡) is continuous at 𝑡≠𝑡_𝑘, left continous at 𝑡=𝑡_𝑘 and 𝑥(𝑡⁺_𝑘) exists for 𝑘=1,2,…,}, 𝑃𝐶¹[𝐽,𝑅]={𝑥∈𝑃𝐶[𝐽,𝑅]∶𝑥^′(𝑡) exists and is continuous at 𝑡≠𝑡_𝑘, left continous at 𝑡=𝑡_𝑘 and 𝑥^′(𝑡⁺_𝑘) exists for 𝑘=1,2,…,}[]=[]𝐹𝑃𝐶𝐽,𝑅𝑥∈𝑃𝐶𝐽,𝑅∶sup_𝑡∈𝐽|𝑥(𝑡)|,1+𝑡<∞𝐸=𝐷𝑃𝐶¹[]=𝐽,𝑅𝑥∈𝑃𝐶¹[𝐽,𝑅]∶sup_𝑡∈𝐽|𝑥(𝑡)|1+𝑡<∞,sup_𝑡∈𝐽𝑥^′(𝑡)<∞. (10) Obviously, 𝐷𝑃𝐶¹[𝐽,𝑅]⊂𝐹𝑃𝐶[𝐽,𝑅]. It is clear that 𝐹𝑃𝐶[𝐽,𝑅] is a Banach space with the norm ‖𝑥‖_𝐹=sup_𝑡∈𝐽|𝑥(𝑡)|,1+𝑡 (11) and 𝐷𝑃𝐶¹[𝐽,𝑅] is also a Banach space with the norm ‖𝑥‖_𝐷=max‖𝑥‖_𝐹,𝑥^′_𝐵, (12) where ‖𝑥^′‖_𝐵=sup_𝑡∈𝐽|𝑥^′(𝑡)|. Let 𝐽₀=[0,𝑡₁], 𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1](𝑘=1,2,3,…). Define a cone 𝑃⊂𝐸 by 𝑃=𝑥∈𝐸∶𝑥isconcaveandnondecreasing𝑥(𝑡)on𝐽,≥0,𝑥^′(𝑡)≥0,𝑡∈𝐽. (13)

Remark 3. If 𝑥 satisfies (4), then (𝜑_𝑝(𝑥^′(𝑡)))^′=−𝑞(𝑡)𝑓(𝑡,𝑥(𝑡),𝑥^′(𝑡))≤0, and 𝑡∈[0,+∞) which implies that 𝜑_𝑝(𝑥^′(𝑡)) is nonincreasing on 𝐽; that is, 𝑥^′(𝑡) is also nonincreasing on 𝐽. Thus, 𝑥 is concave on [0,+∞). Moreover, if 𝑥^′(∞)=𝑥_∞≥0, then 𝑥^′(𝑡)≥0, 𝑡∈[0,+∞), and so 𝑥 is monotone increasing on [0,+∞).

Lemma 4. Let conditions (H₁)-(H₃) hold. Then, 𝑥∈𝑃 with (𝜑_𝑝(𝑥^′(𝑡)))^′∈𝐶(0,+∞) is a solution of BVP (4) if and only if 𝑥∈𝐶[0,+∞) is a fixed point of the following operator equation: (𝑡)=1𝐴𝑥)(1−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘. (14)

Proof. Suppose that 𝑥∈𝑃 with (𝜑_𝑝(𝑥^′(𝑡)))^′∈𝐶(0,+∞) is a solution of BVP (4). For 𝑡∈𝐽, integrating (4) from 𝑡 to +∞, we have _𝑡^+∞𝜑_𝑝𝑥^′(𝜏)^′d𝜏=−_𝑡^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏. (15) That is 𝜑_𝑝𝑥^′(∞)−𝜑_𝑝𝑥^′(𝑡)=−_𝑡^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏, (16) which implies that 𝑥^′(𝑡)=𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞. (17) If 𝑡₁<𝑡≤𝑡₂, integrating (17) from 0 to 𝑡₁, we get that 𝑥𝑡₁(0)=−𝑥_𝑡10𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠. (18) Integrating (17) from 𝑡₁ to 𝑡, we obtain 𝑥(𝑡)−𝑥𝑡⁺₁=^𝑡_𝑡1𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠. (19) Adding (18) and (19) together, we have 𝑥(𝑡)(0)+=𝑥_𝑡0𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+𝐼₁𝑥𝑡₁,𝑡₁<𝑡≤𝑡₂. (20) Repeating previous process, we get that 𝑥(𝑡)(0)+=𝑥_𝑡0𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘=₀^+∞𝑔(𝑡)𝑥(𝑡)d𝑡+^𝑡₀𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘. (21) It follows that ₀^+∞𝑔(𝑡)𝑥(𝑡)d𝑡=11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡. (22) Substituting (22) into (21), we get that 𝑥(𝑡)=11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘. (23)
For 𝑥∈𝑃, there exists 𝑟₀ such that ‖𝑥‖_𝐷<𝑟₀. Set 𝐵_𝑟0=sup{𝑓(𝑡,(1+𝑡)𝑢,𝑣)∣(𝑡,𝑢,𝑣)∈𝐽×[0,𝑟₀]×[0,𝑟₀]}, and we have by (H₁) and (H₃) that ₀^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠≤₀^+∞𝑞(𝑠)d𝑠⋅𝐵_𝑟0,_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘≤^∞_𝑘=1𝐼_𝑘𝑥𝑡_𝑘≤𝑎^∗+𝑏^∗𝑟₀<+∞. (24) By (6), (24), we have 𝑥(𝑡)=11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘≤11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑡𝑔(𝑡)d𝑡⋅𝜑_𝑝⁻¹×₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+𝑡𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑟₀≤2^1/(𝑝−1)11−∫₀^+∞𝑔(𝑡)d𝑡⋅₀^+∞𝑡𝑔(𝑡)d×𝜑𝑡+𝑡_{𝑝−10+∞}𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑟₀. (25) It follows from (24) and (25) that the right term in (23) is well defined. Thus, we have proved that 𝑥 is a fixed point of the operator 𝐴 defined by (14).
Conversely, suppose that 𝑥∈𝐶[0,+∞) is a fixed point of the operator equation (14). Evidently, Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘(𝑘=1,2,…). (26) Direct differentiation of (14) implies that, for 𝑡≠𝑡_𝑘, 𝑥^′(𝑡)=𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠+𝜑_𝑝𝑥_∞,Δ𝑥^′_{𝑡=𝑡𝑘}(),𝜑=0𝑘=1,2,…_𝑝𝑥^′(𝑡)^′=−𝑞(𝑡)𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡), (27) which means that (𝜑_𝑝(𝑥^′(𝑡)))^′∈𝐶(𝐽^′). It is easy to verify that 𝑥(0)=∫₀^+∞𝑔(𝑡)𝑥(𝑡)d𝑡,  𝑥^′(∞)=𝑥_∞. The proof of Lemma 4 is complete.

To obtain the complete continuity of 𝐴, the following lemma is still needed.

Lemma 5 (see [33, 34]). Let 𝑊 be a bounded subset of 𝑃. Then, 𝑃 is relatively compact in 𝐸 if {𝑊(𝑡)/(1+𝑡)} and {𝑊^′(𝑡)} are both equicontinuous on any finite subinterval 𝐽_𝑘∩[0,𝑇](𝑘=1,2,…) for any 𝑇>0, and for any 𝜀>0, there exists 𝑁>0 such that 𝑥𝑡^′1+𝑡^′−𝑥𝑡^′′1+𝑡^′′<𝜀,𝑥^′𝑡^′−𝑥^′𝑡^′′<𝜀,∀𝑡^′,𝑡^′′≥𝑁, (28) uniformly with respect to 𝑥∈𝑊 as 𝑡^′,𝑡^′′≥𝑁, where 𝑊(𝑡)={𝑥(𝑡)∣𝑥∈𝑊}, 𝑊^′(𝑡)={𝑥^′(𝑡)∣𝑥∈𝑊}, 𝑡∈[0,+∞).

This lemma is a simple improvement of the Corduneanu theorem in [33, 34].

Lemma 6. Let (H₁)-(H₃) hold. Then 𝐴∶𝑃→𝑃 is completely continuous.

Proof. For any 𝑥∈𝑃, by (14), we have 𝜑_𝑝(𝐴𝑥)^′(𝑡)=_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠+𝜑_𝑝𝑥_∞,𝜑_𝑝(𝐴𝑥)^′(𝑡)^′=−𝑞(𝑡)𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡). (29) It follows from (14), (29), and (H₁) that (𝐴𝑥)(𝑡)≥0, (𝐴𝑥)^′(𝑡)≥𝑥_∞≥0, (𝐴𝑥)^′′(𝑡)≤0, that is, 𝐴(𝑃)⊂𝑃. Now, we prove that 𝐴 is continuous and compact respectively. Let 𝑥_𝑛∈𝑃, 𝑥_𝑛→𝑥 as 𝑛→∞, then there exists 𝑟₀ such that sup_{𝑛∈𝑁⧵{0}}‖𝑥_𝑛‖<𝑟₀. Let 𝐵_𝑟0=sup{𝑓(𝑡,(1+𝑡)𝑢,𝑣)∣(𝑡,𝑢,𝑣)∈𝐽×[0,𝑟₀]×[0,𝑟₀]}. By (H₁) and (H₂), we have ₀^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)−𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏≤2𝐵_𝑟0⋅₀^+∞𝑞(𝑠)d𝑠<+∞. (30) It follows from (30) and dominated convergence theorem that ₀^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)−𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏→₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏, (31) which implies that 𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠−𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠→0,𝑛→∞. (32) By (30)–(32), (H₃) and dominated convergence theorem, we get that 𝐴𝑥_𝑛−𝐴𝑥_𝐹=sup_𝑡∈𝐽1×11+𝑡1−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠d𝑡−₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠d𝑡+₀^+∞𝑔(𝑡)⋅_{𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡_𝑘−𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)d𝜏+𝜑_𝑝𝑥_∞−𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡_𝑘−𝐼_𝑘𝑥𝑡_𝑘≤11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑡𝑔(𝑡)d𝑡⋅𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)d𝜏+𝜑_𝑝𝑥_∞−𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥_𝑛(𝜏),𝑥^′_𝑛(𝜏)d𝜏+𝜑_𝑝𝑥_∞−𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)⋅_{𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡_𝑘−𝐼_𝑘𝑥𝑡_𝑘d𝑡+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡_𝑘−𝐼_𝑘𝑥𝑡_𝑘→0(𝑛→∞),𝐴𝑥_𝑛^′−(𝐴𝑥)^′_𝐵=sup_𝑡∈𝐽𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥_𝑛(𝑠),𝑥^′_𝑛(𝑠)d𝑠+𝜑_𝑝𝑥_∞−𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠+𝜑_𝑝𝑥_∞→0(𝑛→∞). (33)It follows from (33) that 𝐴 is continuous.
Let Ω⊂𝑃 be any bounded subset. Then, there exists 𝑟>0 such that ‖𝑥‖_𝐷≤𝑟 for any 𝑥∈Ω. Obviously, ‖𝐴𝑥‖_𝐹=sup_𝑡∈𝐽1×11+𝑡1−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘≤11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑡𝑔(𝑡)d𝑡⋅𝜑_𝑝⁻¹×₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑟≤2^1/(𝑝−1)11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d×𝜑𝑡+1_𝑝−1𝐵_𝑟⋅𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏+𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑟,(𝐴𝑥)^′_𝐵=sup_𝑡∈𝐽𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠+𝜑_𝑝𝑥_∞≤2^1/(𝑝−1)𝜑_𝑝⁻¹₀^+∞𝑞(𝑠)d𝑠⋅𝜑_𝑝⁻¹𝐵_𝑟+𝑥_∞. (34)From (34), (H₂), and (H₃), we know that 𝐴Ω is bounded.
For any 𝑇>0,𝑥∈Ω,𝑡^′, 𝑡^′′∈𝐽_𝑘∩[0,𝑇] with 𝑡^′<𝑡^′′, by the absolute continuity of the integral, we have (𝐴𝑥)𝑡^′1+𝑡^′−(𝐴𝑥)𝑡^′′1+𝑡^′′≤11+𝑡^′′1−∫₀^+∞𝑔(𝑡)d𝑡⋅₀^+∞𝑔(𝑡)d𝑡⋅^𝑡′′_𝑡′𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)d𝑡⋅^𝑡′₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠⋅11+𝑡^′′−11+𝑡^′+11+𝑡^′′×^𝑡′′_𝑡′𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+^𝑡′₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠⋅11+𝑡^′′−11+𝑡^′+𝑎^∗+𝑏^∗𝑟1−∫₀^+∞𝑔(𝑡)d𝑡⋅11+𝑡^′′−11+𝑡^′≤2^1/(𝑝−1)1−∫₀^+∞𝑔(𝑡)d𝑡×^𝑡′′_𝑡′𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏⋅𝜑_𝑝⁻¹𝐵_𝑟+𝑥_∞d𝑠+^𝑡′₀𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏⋅𝜑_𝑝⁻¹𝐵_𝑟+𝑥_∞d𝑠⋅11+𝑡^′′−11+𝑡^′+𝑎^∗+𝑏^∗𝑟1−∫₀^+∞𝑔(𝑡)d𝑡⋅11+𝑡^′′−11+𝑡^′→0uniformlyas𝑡^′→𝑡^′′,𝜑_𝑝(𝐴𝑥)^′𝑡^′−𝜑_𝑝(𝐴𝑥)^′𝑡^′′=^𝑡′′_𝑡′𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠≤𝐵_𝑟⋅^𝑡′′_𝑡′𝑞(𝑠)d𝑠→0uniformlyas𝑡^′→𝑡^′′. (35) Thus, we have proved that 𝐴Ω is equicontinuous on any 𝐽_𝑘∩[0,𝑇].
Next, we prove that for any 𝜀>0,𝑥∈Ω, there exits sufficiently large 𝑁>0 such that (𝐴𝑥)𝑡^′1+𝑡^′−(𝐴𝑥)𝑡^′′1+𝑡^′′)<𝜀,(𝐴𝑥^′𝑡^′−(𝐴𝑥)^′𝑡^′′<𝜀,∀𝑡^′,𝑡^′′≥𝑁. (36) For any 𝑥∈Ω, we have lim_𝑡→+∞1⋅11+𝑡1−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘≤lim_𝑡→+∞1⋅𝑎1+𝑡^∗+𝑏^∗𝑟1−∫₀^+∞𝑔(𝑡)d𝑡=0,lim_𝑡→+∞1⋅1+𝑡_0+∞𝑔(𝑡)⋅^𝑡₀𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠d𝑡≤lim_𝑡→+∞121+𝑡^1/(𝑝−1)×11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d𝑡×𝜑_𝑝⁻¹𝐵_𝑟⋅𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏+𝑥_∞=0,lim_𝑡→+∞1⋅1+𝑡_𝑡0𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠=lim_𝑡→+∞𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞=𝑥_∞. (37) It follows from (37) that lim_𝑡→∞(𝐴𝑥)(𝑡)1+𝑡=lim_𝑡→∞1×11+𝑡1−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘=𝑥_∞. (38) On the other hand, we arrive at lim_𝑡→∞(𝐴𝑥)^′(𝑡)=lim_𝑡→∞𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠+𝜑_𝑝𝑥_∞=𝑥_∞. (39) Thus, (36) can be easily derived from (38) and (39). So, by Lemma 5, we know that 𝐴Ω is relatively compact. Thus, we have proved that 𝐴∶𝑃→𝑃 is completely continuous.

3. Main Results

For notational convenience, we denote that 𝑚=2^1/(𝑝−1)11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d𝑡+1⋅𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏,𝑚^′=11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d𝑡+1⋅𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏,𝑛=2^1/(𝑝−1)11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d𝑡+1𝑥_∞,𝑛^′=11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d𝑡+1𝑥_∞,𝑎Λ=max^∗1−∫₀^+∞𝑔(𝑡)d𝑡−3𝑏^∗,Λ,𝑛_′𝑎=max^∗1−∫₀^+∞𝑔(𝑡)d𝑡−3𝑏^∗,𝑛^′. (40)(41)(42)(43)(44)

Theorem 7. Assume that (H₁)-(H₃) hold, and there exists 𝑑>3Λ,𝑎𝑠𝑝≥2,3Λ^′,𝑎𝑠1<𝑝<2 (45) such that (A₁)  𝑓(𝑡,𝑥₁,𝑦₁)≤𝑓(𝑡,𝑥₂,𝑦₂) for any 0≤𝑡<+∞,0≤𝑥₁≤𝑥₂≤𝑑,0≤𝑦₁≤𝑦₂≤𝑑,(A₂)     𝑓(())≤𝜑𝑡,1+𝑡𝑢,𝑣^𝑝𝑑𝜑3𝑚,𝑎𝑠𝑝≥2,_𝑝𝑑3𝑚^′()∈[)×[]×[],,𝑎𝑠1<𝑝<2,𝑡,𝑢,𝑣0,+∞0,𝑑0,𝑑 (46)(A₃)  𝐼_𝑘(𝑥₁)≤𝐼_𝑘(𝑥₂)(𝑘=1,2,…,), for any 0≤𝑥₁≤𝑥₂.
Then, the boundary value problem (4) admits positive, nondecreasing on [0,+∞) and concave solutions 𝑤^∗ and 𝑣^∗ such that 0<‖𝑤^∗‖_𝐷≤𝑑, and lim_𝑛→∞𝑤_𝑛=lim_𝑛→∞𝐴^𝑛𝑤₀=𝑤^∗, where 𝑤₀(𝑡)=𝑑+𝑑𝑡,𝑡∈𝐽, (47) and 0<‖𝑣^∗‖_𝐷≤𝑑, lim_𝑛→∞𝑣_𝑛=lim_𝑛→∞𝐴^𝑛𝑣₀=𝑣^∗, where 𝑣₀(𝑡)=0,𝑡∈𝐽.

Proof. We only prove the case that 𝑝≥2. Another case can be proved in a similar way. By Lemma 6, we know that 𝐴∶𝑃→𝑃 is completely continuous. From the definition of 𝐴, (A₁), and (A₃), we can easily get that 𝐴𝑥₁≤𝐴𝑥₂ for any 𝑥₁,𝑥₂∈𝑃 with 𝑥₁≤𝑥₂, 𝑥^′₁≤𝑥^′₂. Denote that 𝑃_𝑑=𝑥∈𝑃∣‖𝑥‖_𝐷≤𝑑. (48) In what follows, we first prove that 𝐴∶𝑃_𝑑→𝑃_𝑑. If 𝑥∈𝑃_𝑑, then ‖𝑥‖_𝐷≤𝑑. By (6), (40), (42), (44), (H₃), (A₂), and (A₃), we get that ‖𝐴𝑥‖_𝐹=sup_𝑡∈𝐽1×11+𝑡1−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘≤11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑡𝑔(𝑡)d𝑡⋅𝜑_𝑝⁻¹×₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑑≤2^1/(𝑝−1)11−∫₀^+∞𝑔(𝑡)d𝑡^0+∞𝑡𝑔(𝑡)d×𝜑𝑡+1_𝑝−1𝜑_𝑝𝑑3𝑚⋅𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏+𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑑≤𝑑3+𝑑3+𝑑3=𝑑,(𝐴𝑥)^′_𝐵=sup_𝑡∈𝐽𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑥(𝑠),𝑥^′(𝑠)d𝑠+𝜑_𝑝𝑥_∞≤2^1/(𝑝−1)𝜑_𝑝⁻¹₀^+∞𝑞(𝑠)d𝑠𝜑_𝑝⁻¹𝜑_𝑝𝑑3𝑚+𝑥_∞≤𝑑. (49)(50)Thus, we get that ‖𝐴𝑥‖_𝐷≤𝑑. Hence, we have proved that 𝐴∶𝑃_𝑑→𝑃_𝑑.
Let 𝑤₀(𝑡)=𝑑+𝑑𝑡, 0≤𝑡<+∞, then 𝑤₀(𝑡)∈𝑃_𝑑. Let 𝑤₁=𝐴𝑤₀, 𝑤₂=𝐴²𝑤₀, then by Lemma 6, we have that 𝑤₁∈𝑃_𝑑 and 𝑤₂∈𝑃_𝑑. Denote that 𝑤_𝑛+1=𝐴𝑤_𝑛=𝐴^𝑛+1𝑤₀,𝑛=0,1,2,…. (51) Since 𝐴∶𝑃_𝑑→𝑃_𝑑, we have that 𝑤_𝑛∈𝐴𝑃_𝑑⊂𝑃_𝑑,𝑛=1,2,3,…. (52) It follows from the complete continuity of 𝐴 that {𝑤_𝑛}^∞_𝑛=1 is a sequentially compact set. We assert that {𝑤_𝑛}^∞_𝑛=1 has a convergent subsequence {𝑤_𝑛𝑘}^∞_𝑘=1, and there exists 𝑤^∗∈𝑃_𝑑 such that 𝑤_𝑛𝑘→𝑤^∗.
By (51), (A₁)-(A₃), we get that 𝑤₁(𝑡)=11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑔(𝑡)×^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑤₀(𝜏),𝑤^′₀(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑤₀𝑡_𝑘d𝑡+^𝑡₀𝜑_𝑝⁻¹×_𝑠^+∞𝑞(𝜏)𝑓𝜏,𝑤₀(𝜏),𝑤^′₀(𝜏)d𝜏+𝜑_𝑝𝑥_∞d𝑠+_{𝑡𝑘<𝑡}𝐼_𝑘𝑤₀𝑡_𝑘≤11−∫₀^+∞𝑔(𝑡)d𝑡×₀^+∞𝑡𝑔(𝑡)d𝑡⋅𝜑_𝑝⁻¹×₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+𝑡𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)d𝜏+𝜑_𝑝𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑑≤2^1/(𝑝−1)11−∫₀^+∞𝑔(𝑡)d𝑡⋅₀^+∞𝑡𝑔(𝑡)d𝑡×𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏𝜑_𝑝⁻¹𝜑_𝑝𝑑3𝑚+𝑥_∞+2^1/(𝑝−1)𝑡𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏𝜑_𝑝⁻¹𝜑_𝑝𝑑3𝑚+𝑥_∞+11−∫₀^+∞𝑔(𝑡)d𝑡𝑎^∗+𝑏^∗𝑑≤𝑑+𝑑𝑡=𝑤₀(𝑡),𝑤^′₁(𝑡)=𝐴𝑤₀^′(𝑡)=𝜑_𝑝⁻¹_𝑡^+∞𝑞(𝑠)𝑓𝑠,𝑤₀(𝑠),𝑤^′₀(𝑠)d𝑠+𝜑_𝑝𝑥_∞≤2^1/(𝑝−1)𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏𝜑_𝑝⁻¹𝜑_𝑝𝑑3𝑚+𝑥_∞≤𝑑=𝑤^′₀(𝑡),0≤𝑡<+∞. (53)So, by (53) (A₁)-(A₃) we have 𝑤₂(𝑡)=𝐴𝑤₁(𝑡)≤𝐴𝑤₀(𝑡)=𝑤₁(𝑡)𝑤,0≤𝑡<+∞,_′2(𝑡)=𝐴𝑤₁^′(𝑡)≤𝐴𝑤₀^′(𝑡)=𝑤₁^′(𝑡),0≤𝑡<+∞. (54) By induction, we get that 𝑤_𝑛+1(𝑡)≤𝑤_𝑛(𝑡),𝑤^′_𝑛+1(𝑡)≤𝑤^′_𝑛(𝑡),0≤𝑡<+∞,𝑛=0,1,2,…. (55) Hence, we claim that 𝑤_𝑛→𝑤^∗ as 𝑛→∞. Applying the continuity of 𝐴 and 𝑤_𝑛+1=𝐴𝑤_𝑛, we get that 𝐴𝑤^∗=𝑤^∗.
Let 𝑣₀(𝑡)=0, 0≤𝑡<+∞, then 𝑣₀(𝑡)∈𝑃_𝑑. Let 𝑣₁=𝐴𝑣₀,𝑣₂=𝐴²𝑣₀. By Lemma 6, we have that 𝑣₁∈𝑃_𝑑 and 𝑣₂∈𝑃_𝑑. Denote 𝑣_𝑛+1=𝐴𝑣_𝑛=𝐴^𝑛+1𝑣₀,𝑛=0,1,2,…. (56) Since 𝐴∶𝑃_𝑑→𝑃_𝑑, we have that 𝑣_𝑛∈𝐴(𝑃_𝑑)⊂𝑃_𝑑,𝑛=1,2,3,…. It follows from the complete continuity of 𝐴 that {𝑣_𝑛}^∞_𝑛=1 is a sequentially compact set. And, we assert that {𝑣_𝑛}^∞_𝑛=1 has a convergent subsequence {𝑣_𝑛𝑘}^∞_𝑘=1 and there exists 𝑣^∗∈𝑃_𝑑 such that 𝑣_𝑛𝑘→𝑣^∗.
Since 𝑣₁=𝐴𝑣₀∈𝑃_𝑑, we have 𝑣₁(𝑡)=𝐴𝑣₀(𝑡)=(𝑡)𝑣𝐴0)(≥0,0≤𝑡<+∞,_′1(𝑡)=𝐴𝑣₀^′(𝑡)=(𝐴0)^′(𝑡)≥0=𝑣^′₀(𝑡),0≤𝑡<+∞. (57) By (A₁)-(A₃), we have 𝑣₂(𝑡)=𝐴𝑣₁(𝑡)≥(𝐴0)(𝑡)=𝑣₁(𝑡)𝑣,0≤𝑡<+∞,_′2(𝑡)=𝐴𝑣₁^′(𝑡)≥(𝐴0)^′(𝑡)=𝑣^′₁(𝑡),0≤𝑡<+∞. (58) By induction, we get that 𝑣_𝑛+1(𝑡)≥𝑣_𝑛(𝑡),𝑣^′_𝑛+1(𝑡)≥𝑣^′_𝑛(𝑡),0≤𝑡<+∞,𝑛=0,1,2,…. (59) Hence, we claim that 𝑣_𝑛→𝑣^∗ as 𝑛→∞. Applying the continuity of 𝐴 and 𝑣_𝑛+1=𝐴𝑣_𝑛, we get that 𝐴𝑣^∗=𝑣^∗.
Since 𝑓(𝑡,0,0)≢0,0≤𝑡<∞, then the zero function is not the solution of BVP (4). Thus, 𝑣^∗ is a positive solution of BVP (4). By Lemma 4 we know that 𝑤^∗ and 𝑣^∗ are positive, nondecreasing on [0,+∞) and concave solutions of the BVP (4).
We can easily get that Theorem 7 holds for 1<𝑝<2 in a similar manner.

Remark 8. The iterative schemes in Theorem 7 are 𝑤₀(𝑡)=𝑑+𝑑𝑡, 𝑤_𝑛+1=𝐴𝑤_𝑛=𝐴^𝑛+1𝑤₀, 𝑛=0,1,2,… and 𝑣₀(𝑡)=0, 𝑣_𝑛+1=𝐴𝑣_𝑛=𝐴^𝑛+1𝑣₀, 𝑛=0,1,2,…. They start off with a known simple linear function and the zero function respectively. This is convenient in application.

Theorem 9. Assume that (𝐻₁)-(𝐻₃) hold, and there exist 𝑑_𝑛>𝑑_𝑛−1>⋯>𝑑₁>3Λ,𝑎𝑠𝑝≥2,3Λ^′,𝑎𝑠1<𝑝<2 (60) such that (A^′₁)𝑓(𝑡,𝑥₁,𝑦₁)≤𝑓(𝑡,𝑥₂,𝑦₂) for any 0≤𝑡<+∞, 0≤𝑥₁≤𝑥₂, 0≤𝑦₁≤𝑦₂. (A^′₂)  𝑓(())≤𝜑𝑡,1+𝑡𝑢,𝑣^𝑝𝑑_𝑘𝜑3𝑚,𝑎𝑠𝑝≥2,_𝑝𝑑_𝑘3𝑚^′()∈[)×,𝑎𝑠1<𝑝<2𝑡,𝑢,𝑣0,+∞0,𝑑_𝑘×0,𝑑_𝑘,𝑘=1,2,…,𝑛. (61)(A^′₃)𝐼_𝑘(𝑥₁)≤𝐼_𝑘(𝑥₂)(𝑘=1,2,…,), for any 0≤𝑥₁≤𝑥₂.
Then, the boundary value problem (4) admits positive nondecreasing on [0,+∞) and concave solutions 𝑤^∗_𝑘 and 𝑣^∗_𝑘, such that 0<‖𝑤^∗_𝑘‖_𝐷≤𝑑_𝑘, and lim_𝑛→∞𝑤_𝑘𝑛=lim_𝑛→∞𝐴^𝑛𝑤_𝑘0=𝑤^∗_𝑘, where 𝑤₀(𝑡)=𝑑_𝑘+𝑑_𝑘𝑡,𝑡∈𝐽, (62) and 0<‖𝑣^∗_𝑘‖_𝐷≤𝑑_𝑘, lim_𝑛→∞𝑣_𝑘𝑛=lim_𝑛→∞𝐴^𝑛𝑣_𝑘0=𝑣^∗_𝑘, where 𝑣₀(𝑡)=0, 𝑡∈𝐽.

Remark 10. It is easy to see that 𝑤^∗ and 𝑣^∗ in Theorem 7 may coincide, and then the boundary value problem (4) has only one solution in 𝑃. Similarly, positive solutions 𝑤^∗_𝑘 and 𝑣^∗_𝑘 may also coincide.

4. An Example

Example 11. Consider the following impulsive integral boundary value problem: 𝑥^′𝑥^′′+𝑒^−6𝑡𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)=0,𝑡∈𝐽₊,Δ𝑥|_𝑡=𝑘=1912^𝑘+2𝑥(𝑘)+12^𝑘+1(1+𝑥(𝑘))^1/6,𝑥(0)=₀^+∞1(1+𝑡)³𝑥(𝑡)d𝑡,𝑥^′(∞)=√23, (63) where 𝑓()=1𝑡,𝑢,𝑣|(+164sin101𝑡+20)|𝑢721+𝑡³+1𝑣10120,𝑢≤2,|(+164sin101𝑡+20)|2721+𝑡³+1𝑣1020,𝑢≥2. (64) It is clear that conditions (H₁), (A₁), and (A₃) hold for 𝑝=3, 𝑞(𝑡)=𝑒^−6𝑡,  𝑔(𝑡)=1/(1+𝑡)³. By direct computation, we obtain that ₀^+∞𝑞(𝑡)d1𝑡=6,₀^+∞𝜑_𝑝⁻¹_𝑠^+∞𝑞(𝜏)d𝜏d√𝑠=3,18 (65) which implies that (H₂) holds.
Obviously, 𝐼_𝑘∈𝐶(𝐽,𝐽). Using a simple inequality (1+𝑢)^𝛼≤1+𝛼𝑢,∀𝑢≥0,0<𝛼<1, (66) we get that 𝐼_𝑘(𝑥(𝑘≤1))912^𝑘+2𝑥(𝑘)+12^𝑘+111+6𝑥(𝑘≤1)9⋅12^𝑘+1+2⋅1272^𝑘+1𝑥(𝑘). (67) Thus, (H₃) holds for 𝑎_𝑘=(1/9)⋅(1/2^𝑘+1), 𝑏_𝑘=(2/27)⋅(1/2^𝑘+1). Considering that ₀^+∞𝑔(𝑡)d𝑡=₀^+∞1(1+𝑡)³d1𝑡=2_0+∞,𝑡𝑔(𝑡)d𝑡=₀^+∞𝑡(1+𝑡)³d1𝑡=2,𝜑_𝑝⁻¹₀^+∞𝑞(𝜏)d𝜏=𝜑_𝑝⁻¹16=√66, (68) we can obtain that 𝑎^∗=^∞_𝑘=1𝑎_𝑘=1,𝑏18_∗=227^∞_𝑘=11+𝑘2^𝑘+1=227^∞_𝑘=112^𝑘+1+^∞_𝑘=1𝑘2^𝑘+1=19,𝑚=√3,𝑛=2,Λ=2. (69) Take 𝑑=8. In this case, we have 𝜑_𝑝𝑑3𝑚=𝜑_𝑝83√3=64.27 (70) On the other hand, nonlinear term 𝑓 satisfies 𝑓(())≤1𝑡,1+𝑡𝑢,𝑣+1649+1=252401[)[],14400,𝑡∈0,+∞,𝑢,𝑣∈0,8 (71) which means that (A₂) holds. Thus, we have checked that all the conditions of Theorem 7 are satisfied. Therefore, the conclusion of Theorem 7 holds.

Acknowledgments

This paper is supported financially by the Foundation for Outstanding Middle-Aged and Young Scientists of Shandong Province (BS2010SF004), the Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA53), and the National Natural Science Foundation of China (10971179).