Existence of Unbounded Solutions for a Third-Order Boundary Value Problem on Infinite Intervals

Abstract

We generalize the unbounded upper and lower solution method to a third-order ordinary differential equation on the half line subject to the Sturm-Liouville boundary conditions. By using such techniques and the Schäuder fixed point theorem, some criteria are presented for the existence of solutions and positive ones to the problem discussed.

1. Introduction

Boundary value problems on infinite intervals, arising from the study of radially symmetric solutions of nonlinear elliptic equation [1], have received much attention in recent years. Because the infinite interval is noncompact, the discussion about BVPs on the half-line is more complicated. There have been many existence results for some boundary value problems of differential equations on the half line. The main methods are the extension of continuous solutions on the corresponding finite intervals under a diagonalization process, fixed point theorems in special Banach space or in special Fréchet space; see [1–12] and the references therein.

The method of upper and lower solutions is a powerful technique to deal with the existence of boundary value problems (BVPs). In many cases, when given one pair of well-ordered lower and upper solution, nonlinear BVPs always have at least one solution in the closed interval. To obtain this kind of result, we can employ topological degree theory, the monotone iterative technique, or critical theory. For details, we refer the reader to see [1–4, 7, 9, 12–14] and therein.

When the method of upper and lower solution is applied to the infinite interval problems, diagonalization process is always used; see [2, 3, 7]. For example, in [3], Agarwal and O’Regan discussed a Sturm-Liouville boundary value problem of second-order differential equation: 1𝑝(𝑡)𝑝(𝑡)𝑦^−𝑎₀(𝑡)^=𝑞(𝑡)𝑓(𝑡,𝑦(𝑡)),𝑡∈(0,+∞),𝑦𝑦((0𝑡))+bo𝑏u₀nlidmed_𝑡→0+𝑝o(𝑡n)𝑦[^0(,𝑡)+=∞𝑐)₀,,oorrlliimm_{𝑡→0+𝑡→+∞}𝑝(𝑡)𝑦^(𝑡)=0,𝑦(𝑡)=0,(1.1) where 𝑎₀>0,𝑏₀≥0. General existence criteria were obtained to guarantee the existence of bounded solutions. The methods used therein were based on a diagonalization arguments and existence results of appropriate boundary value problems on finite intervals.

In [12], Yan et al. investigated the boundary value problem 𝑦𝑎^𝑦((𝑡0))+−Φ𝑏𝑦(𝑡_)𝑓𝑡,𝑦(𝑡),𝑦^(𝑡)=0,𝑡∈(0,+∞),(0)=𝑦₀≥0,lim_𝑡→+∞𝑦^(𝑡)=𝑘>0,(1.2) where 𝑎>0,𝑏>0. By using the upper and lower solutions method and the fixed point theorem, the authors presented sufficient conditions for the existence of unbounded positive solutions. In [9], Lian and the coauthors discussed further the existence of the unbounded solutions.

There are many results of third-order boundary value problems on finite interval; see [14, 15] and the references therein. However, there has been few papers concerned with the upper and lower solutions technique for the boundary value problems of third-order differential equation on infinite intervals. In this paper, we aim to investigate a general Sturm-Liouville boundary value problem for third-order differential equation on the half line 𝑢𝑢^((𝑡0))+=𝜙𝐴(,𝑡)𝑓𝑢𝑡,_𝑢(𝑡),𝑢^(𝑡),𝑢^(𝑡)=0,𝑡∈(0,+∞),(0)−𝑎𝑢^(0)=𝐵,𝑢^(+∞)=𝐶,(1.3) where 𝜙∶(0,+∞)→(0,+∞), 𝑓∶[0,+∞)×ℝ³→ℝ are continuous, 𝑎>0, 𝐴,𝐵,𝐶∈ℝ. The methods mainly depend on the unbounded upper and lower solutions method and topological degree theory. The nonlinear is admitted to involve in the high-order derivatives under the considerations of the Nagumo condition. The solutions obtained can be unbounded in this paper. The results obtained in this paper generalize those in [4].

2. Preliminaries

We present here some definitions and lemmas which are essential in the proof of the main results.

Definition 2.1. A function 𝛼∈𝐶²[0,+∞)∩𝐶³(0,+∞) is called a lower solution of BVP (1.3) if 𝛼𝛼^((𝑡0))+≤𝜙𝐴(,𝑡)𝑓𝛼𝑡,_𝛼(𝑡),𝛼^(𝑡),𝛼^(𝑡)≥0,𝑡∈(0,+∞),(0)−𝑎𝛼^(0)≤𝐵,𝛼^(+∞)<𝐶.(2.1) Similarly, a function 𝛽∈𝐶²[0,+∞)∩𝐶³(0,+∞) is called an upper solution of BVP (1.3) if 𝛽𝛽^((𝑡0))+≥𝜙𝐴(,𝑡)𝑓𝛽𝑡,_𝛽(𝑡),𝛽^(𝑡),𝛽^(𝑡)≤0,𝑡∈(0,+∞),(0)−𝑎𝛽^(0)≥𝐵,𝛽^(+∞)>𝐶.(2.2)

Definition 2.2. Given a positive function 𝜙∈𝐶(0,+∞) and a pair of functions 𝛼,𝛽∈𝐶¹[0,+∞) satisfying 𝛼(0)≤𝛽(0) and 𝛼′(𝑡)≤𝛽′(𝑡),𝑡∈[0,+∞); a function 𝑓∶[0,+∞)×𝑅³→𝑅 is said to satisfy the Nagumo condition with respect to the pair of functions 𝛼,𝛽, if there exist positive functions 𝜓,ℎ∈𝐶[0,+∞) satisfying ∫^0+∞𝜓(𝑠)𝜙(𝑠)𝑑𝑠<+∞,∫^+∞𝑠/ℎ(𝑠)𝑑𝑠=+∞ such that ||𝑓(𝑡,𝑥,𝑦,𝑧)||≤𝜓(𝑡)ℎ(|𝑧|)(2.3) holds for all 0≤𝑡<+∞,𝛼(𝑡)≤𝑥≤𝛽(𝑡),𝛼′(𝑡)≤𝑦≤𝛽′(𝑡), and 𝑧∈ℝ.
Let 𝑣₀(𝑡)=1+𝑡²,𝑣₁(𝑡)=1+𝑡,𝑣₂(𝑡)=1 and consider the space 𝑋 defined by 𝑋=𝑥∈𝐶²[0,+∞),lim_𝑡→+∞𝑥^(𝑖)(𝑡)𝑣_𝑖(𝑡)exist,𝑖=0,1,2,(2.4) with the norm ‖𝑥‖=max{‖𝑥‖₀,‖𝑥′‖₁,‖𝑥^‖₂}, where ‖𝑥‖_𝑖=sup_{𝑡∈[0,+∞)}|𝑥(𝑡)/𝑣_𝑖(𝑡)|. By the standard arguments, we can prove that (𝑋,‖⋅‖) is a Banach space.

Lemma 2.3. If 𝑒∈𝐿¹[0,+∞), then the BVP of third-order linear differential equation 𝑢(0)=𝐴,𝑢^(𝑡𝑢)_+𝑒(𝑡)=0,𝑡∈(0,+∞),(0)−𝑎𝑢^(0)=𝐵,𝑢^(+∞)=𝐶,(2.5) has a unique solution in 𝑋. Moreover this solution can be expressed as 𝑢(𝑡)=𝑝(𝑡)+^0+∞𝐺(𝑡,𝑠)𝑒(𝑠)𝑑𝑠,(2.6) where 𝑝(𝑡)=𝐴+(𝑎𝐶+𝐵)𝑡+𝐶2𝑡²𝐺(𝑡,𝑠)=⎧⎪⎨⎪⎩𝑎𝑡+𝑠𝑡−1,2𝑠²1,0≤𝑠≤𝑡<+∞,2𝑡²+𝑎𝑡,0≤𝑡≤𝑠<+∞.(2.7)

Proof. It is easy to verify that (2.6) satisfies BVP (2.5). Now we show the uniqueness. Suppose 𝑢 is a solution of (2.5). Let 𝑣=𝑢′, then we have 𝑣𝑣(0^)(𝑡−)+𝑎𝑣𝑒_(𝑡)=0,𝑡∈(0,+∞),(0)=𝐵,𝑣^(+∞)=𝐶.(2.8) By a direct calculation, we obtain the general solution of the above equation: 𝑣(𝑡)=𝑐₁+𝑐₂𝑡+^𝑡0^𝜏+∞=𝑐₁𝑒(𝑠)𝑑𝑠𝑑𝜏+𝑐₂𝑡+^𝑡0𝑠𝑒(𝑠)𝑑𝑠+𝑡^𝑡+∞𝑒(𝑠)𝑑𝑠.(2.9) Substituting this to the boundary condition, we arrive at 𝑐₁=𝑎𝐶+𝐵𝑐+𝑎^0+∞𝑒(𝑠)𝑑𝑠,₂=𝐶.(2.10) Therefore, (2.8) has a unique solution 𝑣(𝑡)=𝑎𝐶+𝐵+𝐶𝑡+^0+∞𝑔(𝑡,𝑠)𝑒(𝑠)𝑑𝑠,(2.11) where 𝑔(𝑡,𝑠)=𝑎𝑎++𝑠𝑡,,00≤≤𝑠𝑡≤≤𝑠𝑡<<++∞∞,.(2.12) Furthermore, 𝑢′=𝑣,𝑢(0)=𝐴, so 𝑢(𝑡)==𝑢𝐴(+0)+_𝑡0^𝑡0𝑣(𝜏)𝑑𝜏𝑎𝐶+𝐵+𝐶𝜏+^0+∞=𝑝(𝑡)+_0+∞𝑔(𝜏,𝑠)𝑒(𝑠)𝑑𝑠𝑑𝜏^𝑡0=𝑝(𝑡)+_0+∞𝑔(𝜏,𝑠)𝑑𝜏𝑒(𝑠)𝑑𝑠𝐺(𝑡,𝑠)𝑒(𝑠)𝑑𝑠.(2.13) The proof is complete.

Theorem 2.4 (see [1]). Let 𝑀⊂𝐶_∞={𝑥∈𝐶[0,+∞),lim_𝑡→+∞𝑥(𝑡)exists}. Then 𝑀 is relatively compact if the following conditions hold:(a)all functions from 𝑀 are uniformly bounded;(b)all functions from 𝑀 are equicontinuous on any compact interval of [0,+∞);(c)all functions from 𝑀 are equiconvergent at infinity; that is, for any given 𝜖>0, there exists a 𝑇=𝑇(𝜖)>0 such that |𝑓(𝑡)−𝑓(+∞)|<𝜖, for all 𝑡>𝑇 and 𝑓∈𝑀.
From the above results, we can obtain the following general criteria for the relative compactness of subsets in 𝐶[0,+∞).

Theorem 2.5. Given 𝑛+1 continuous functions 𝜌_𝑖 satisfying 𝜌_𝑖≥𝜀>0,𝑖=0,1,…,𝑛 with 𝜀 a positive constant. Let 𝑀⊂𝐶^𝑛∞={𝑥∈𝐶^𝑛[0,+∞),lim_𝑡→+∞𝜌_𝑖(𝑡)𝑥^(𝑖)(𝑡)exists,𝑖=0,1,2,…,𝑛}. Then 𝑀 is relatively compact if the following conditions hold:(a)all functions from 𝑀 are uniformly bounded;(b)the functions from {𝑦_𝑖∶𝑦_𝑖=𝜌_𝑖𝑥^(𝑖),𝑥∈𝑀} are equicontinuous on any compact interval of [0,+∞), 𝑖=0,1,2,…,𝑛;(c)the functions from {𝑦_𝑖∶𝑦_𝑖=𝜌_𝑖𝑥^(𝑖),𝑥∈𝑀} are equiconvergent at infinity, 𝑖=0,1,2,…,𝑛.

Proof. Set 𝑀_𝑖={𝑦_𝑖∶𝑦_𝑖=𝜌_𝑖𝑥^(𝑖),𝑥^(𝑖)∈𝑀}, then 𝑀_𝑖⊂𝐶_∞,𝑖=0,1,2…,𝑛. From conditions (a)–(c), we have 𝑀_𝑖 is relatively compact in 𝐶_∞. Therefore, for any sequence {𝑦_𝑖,𝑚}^∞𝑚=1⊂𝑀_𝑖, it has a convergent subsequence. Without loss of generality, we denote it this sequence. Then there exists 𝑦_𝑖,0∈𝑀_𝑖 such that 𝑦_𝑖,𝑚=𝜌_𝑖𝑥^𝑚(𝑖)⟶𝑦_𝑖,0,𝑚⟶+∞,𝑖=0,1,2,…,𝑛.(2.14)
Set 𝑥_𝑖,0=(1/𝜌_𝑖)𝑦_𝑖,0, then 𝑥^𝑚(𝑖)→𝑥_𝑖,0,𝑖=0,1,2,…,𝑛. Noticing that all functions from 𝑀 are uniformly continuous, we can obtain that 𝑥_𝑖,0=𝑥^(𝑖)0,0,𝑖=1,2,…,𝑛. So 𝑀 is relatively compact.

3. Main Results

In this section, we present the existence criteria for the existence of solutions and positive solutions of BVP (1.3). We first cite conditions (H₁) and (H₂) here.(H₁):(1) BVP (1.3) has a pair of upper and lower solutions 𝛽,𝛼 in 𝑋 with 𝛼′(𝑡)≤𝛽′(𝑡),𝑡∈[0,+∞);(2)𝑓∈𝐶([0,+∞)×ℝ³,ℝ) satisfies the Nagumo condition with respect to 𝛼 and 𝛽.(H₂): For any 0≤𝑡<+∞,𝛼′(𝑡)≤𝑦≤𝛽′(𝑡) and 𝑧∈ℝ, it holds 𝑓(𝑡,𝛼(𝑡),𝑦,𝑧)≤𝑓(𝑡,𝑥,𝑦,𝑧)≤𝑓(𝑡,𝛽(𝑡),𝑦,𝑧),as𝛼(𝑡)≤𝑥≤𝛽(𝑡).(3.1)

Lemma 3.1. Suppose condition (H₁) holds. And suppose further that the following condition holds:(H₃) there exists a constant 𝛾>1 such that sup_{0≤𝑡<+∞}(1+𝑡)^𝛾𝜙(𝑡)𝜓(𝑡)<+∞.
If 𝑢 is a solution of (1.3) satisfying 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡),𝛼^(𝑡)≤𝑢^(𝑡)≤𝛽^(𝑡),𝑡∈[0,+∞),(3.2) then there exists a constant 𝑅>0 (without relations to 𝑢) such that ‖𝑢^‖₂≤𝑅.

Proof. Let 𝛿>0 and 𝑅>𝐶, 𝜂≥maxsup_{𝑡∈[𝛿,+∞)}𝛽^(𝑡)−𝛼^(0)𝑡,sup_{𝑡∈[𝛿,+∞)}𝛽^(0)−𝛼^(𝑡)𝑡(3.3) such that ^𝑅𝜂𝑠ℎ(𝑠)𝑑𝑠≥𝑀sup_{𝑡∈[0,+∞)}𝛽^(𝑡)(1+𝑡)^𝛾−inf_{𝑡∈[0,+∞)}𝛼^(𝑡)(1+𝑡)^𝛾+𝛾𝛾−1⋅sup_{𝑡∈[0,+∞)}𝛽^(𝑡)1+𝑡,(3.4) where 𝐶 is the nonhomogeneous boundary value, 𝑀=sup_{0≤𝑡<+∞}(1+𝑡)^𝛾𝜙(𝑡)𝜓(𝑡), then |𝑢^(𝑡)|≤𝑅,𝑡∈[0,+∞). If it is untrue, we have the following three cases.

Case 1. Consider ||𝑢^(𝑡)||>𝜂,∀𝑡∈[0,+∞).(3.5) Without loss of generality, we suppose 𝑢^(𝑡)>𝜂,𝑡∈[0,+∞). While for any 𝑇≥𝛿, 𝛽^(𝑇)−𝛼^(0)𝑇≥𝑢^(𝑇)−𝑢^(0)𝑇=𝑇1^𝑇0𝑢^(𝑠)𝑑𝑠>𝜂≥𝛽^(𝑇)−𝛼^(0)𝑇,(3.6) which is a contraction. So there must exist 𝑡₀∈[0,+∞) such that |𝑢^(𝑡₀)|≤𝜂.

Case 2. Consider ||𝑢^(𝑡)||≤𝜂,∀𝑡∈[0,+∞).(3.7) Just take 𝑅=𝜂 and we can complete the proof.

Case 3. There exists [𝑡₁,𝑡₂]⊂[0,+∞) such that |𝑢^(𝑡₁)|=𝜂,|𝑢^(𝑡)|>𝜂,𝑡∈(𝑡₁,𝑡₂] or |𝑢^(𝑡₂)|=𝜂,|𝑢^(𝑡)|>𝜂,𝑡∈[𝑡₁,𝑡₂).
Suppose that 𝑢^(𝑡₁)=𝜂,𝑢^(𝑡)>𝜂,𝑡∈(𝑡₁,𝑡₂]. Obviously, ^{𝑢′′(𝑡2)𝑢}_{′′(𝑡1)}𝑠ℎ(𝑠)𝑑𝑠=^𝑡2_𝑡1𝑢^(𝑠)ℎ(𝑢^(𝑠))𝑢^=(𝑠)𝑑𝑠_𝑡2𝑡1−𝜙(𝑠)𝑓𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑢^(𝑠)≤ℎ(𝑢^(𝑠))𝑑𝑠_𝑡2𝑡1𝑢^(𝑠)𝜙(𝑠)𝜓(𝑠)𝑑𝑠≤𝑀^𝑡2_𝑡1𝑢^(𝑠)(1+𝑠)^𝛾=𝑀_𝑡2𝑡1𝑑𝑠(1𝑢+′(𝑠𝑠))^𝛾^𝑑𝑠+^𝑡2_𝑡1𝛾𝑢′(𝑠)(1+𝑠)^1+𝛾≤𝑀sup_{𝑡∈[0,+∞)}𝑑𝑠𝛽′(𝑡)(1+𝑡)^𝛾−inf_{𝑡∈[0,+∞)}𝛼′(𝑡)(1+𝑡)^𝛾+sup_{𝑡∈[0,+∞)}𝛽′(𝑡)1+𝑡^0+∞𝛾(1+𝑠)^𝛾≤𝑑𝑠_𝑅𝜂𝑠ℎ(𝑠)𝑑𝑠(3.8) concludes that 𝑢^(𝑡₂)≤𝑅. For 𝑡₁ and 𝑡₂ are arbitrary, we have 𝑢^(𝑡)≤max{𝑅,𝜂}=𝑅,𝑡∈[0,+∞).
Similarly if 𝑢^(𝑡₁)=−𝜂,𝑢^(𝑡)<−𝜂,𝑡∈(𝑡₁,𝑡₂], we can also obtain that 𝑢^(𝑡)>−𝑅,𝑡∈[0,+∞).
Thus there exists 𝑅>0, just related with 𝛼,𝛽, and ℎ, such that ‖𝑢^‖₂≤𝑅.

Remark 3.2. Condition (H₃) is necessary for an a priori estimation of 𝑢^ in Lemma 3.1. Because the upper and lower solutions are in 𝑋, 𝛽′(𝑡) and 𝛼′(𝑡) are at most linearly increasing, especially at infinity. Otherwise, sup_{𝑡∈[0,+∞)}𝛼′(𝑡) and sup_{𝑡∈[0,+∞)}𝛽′(𝑡) may be equal to infinity.

Theorem 3.3. Suppose 𝜙∈𝐿¹[0,+∞) and the conditions (H₁)–(H₃) hold. Then BVP (1.3) has at least one solution 𝑢∈𝐶²[0,+∞)∩𝐶³(0,+∞) such that 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡),𝛼^(𝑡)≤𝑢^(𝑡)≤𝛽^(𝑡),𝑡∈[0,+∞).(3.9)

Proof. Let 𝑅>0 be the same definition in Lemma 3.1 and consider the boundary value problem 𝑢^𝑢((𝑡0))+=𝜙𝐴(𝑡,)𝑓^∗𝑢𝑡_,𝑢(𝑡),𝑢^(𝑡),𝑢^(𝑡)=0,𝑡∈(0,+∞),(0)−𝑎𝑢^(0)=𝐵,𝑢^(+∞)=𝐶,(3.10) where 𝑓^∗(𝑡,𝑥,𝑦,𝑧)=⎧⎪⎪⎨⎪⎪⎩𝐹^𝑅𝑡,𝑥,𝛼^(𝑡),𝑧+𝑦−𝛼^(𝑡)1+||𝑦−𝛼^(𝑡)||,𝑦<𝛼^𝐹(𝑡),_𝑅𝐹(𝑡,𝑥,𝑦,𝑧),𝛼^(𝑡)≤𝑦≤𝛽^(𝑡),_𝑅𝑡,𝑥,𝛽^(𝑡),𝑧−𝑦−𝛽^(𝑡)1+||𝑦−𝛽^(𝑡)||,𝑦>𝛽^𝐹(𝑡),_𝑅(𝑡,𝑥,𝑦,𝑧)=⎧⎪⎨⎪⎩𝑓^𝑅𝑓(𝑡,𝛼(𝑡),𝑦,𝑧),𝑥<𝛼(𝑡),_𝑅𝑓(𝑡,𝑥,𝑦,𝑧),𝛼(𝑡)≤𝑥≤𝛽(𝑡),_𝑅𝑓(𝑡,𝛽(𝑡),𝑦,𝑧),𝑥>𝛽(𝑡),_𝑅(𝑡,𝑥,𝑦,𝑧)=⎧⎪⎨⎪⎩𝑓𝑓𝑓(((𝑡𝑡𝑡,,,𝑥𝑥𝑥,,,𝑦𝑦𝑦,,,−𝑧𝑅)𝑅),,),𝑧−𝑧𝑅<>≤−𝑅𝑅.𝑧,≤𝑅,(3.11) Firstly we prove that BVP (3.10) has at least one solution 𝑢. To this end, define the operator 𝑇∶𝑋→𝑋 by (𝑇𝑢)(𝑡)=𝑝(𝑡)+^0+∞𝐺(𝑡,𝑠)𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠.(3.12) By Lemma 2.3, we can see that the fixed points of 𝑇 coincide with the solutions of BVP (3.10). So it is enough to prove that 𝑇 has a fixed point.
We claim that 𝑇∶𝑋→𝑋 is completely continuous.
(1)𝑇∶𝑋→𝑋 is well defined. For any 𝑢∈𝑋,‖𝑢‖<+∞ and it holds ^0+∞𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠≤^0+∞𝜙(𝑠)𝐻₀𝜓(𝑠)+1𝑑𝑠<+∞,(3.13) where 𝐻₀=max_{0≤𝑠≤‖𝑢‖}ℎ(𝑠). By the Lebesgue-dominated convergence theorem, we have lim_𝑡→+∞(𝑇𝑢)(𝑡)𝑣₀(𝑡)=lim^𝑡→+∞𝑣𝑙₀((𝑡𝑡))+^0+∞𝐺(𝑡,𝑠)𝑣₀(𝑡)𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^=𝐶(𝑠),𝑢^(𝑠)𝑑𝑠2+12^0+∞lim_𝑡→+∞𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠<+∞,(𝑇𝑢)^(𝑡)𝑣₁(𝑡)=lim^𝑡→+∞(𝑎𝐶+𝑣₁𝐵(𝑡))+𝐶𝑡+^0+∞𝑔(𝑡,𝑠)𝑣₁(𝑡)𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^=𝐶+_0+∞(𝑠),𝑢^(𝑠)𝑑𝑠lim_𝑡→+∞𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠<+∞,(𝑇𝑢)^(𝑡)=lim_𝑡→+∞𝐶+^𝑡+∞=𝐶<+∞,𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠(3.14) so 𝑇𝑢∈𝑋.
(2)𝑇∶𝑋→𝑋 is continuous. For any convergent sequence 𝑢_𝑛→𝑢 in 𝑋, there exists 𝑟₁>0 such that sup_𝑛∈𝑁‖𝑢_𝑛‖≤𝑟₁. Similarly, we have ‖‖𝑇𝑢_𝑛−𝑇𝑢‖‖=≤ma_0+∞x‖‖𝑇𝑢_𝑛−𝑇𝑢‖‖₀,‖‖𝑇𝑢_𝑛^−(𝑇𝑢)′‖‖₁,‖‖𝑇𝑢_𝑛^−(𝑇𝑢)^‖‖₂maxsup_{0≤𝑡<+∞}||||𝐺(𝑡,𝑠)𝑣₀(𝑡)||||,sup_{0≤𝑡<+∞}||||𝑔(𝑡,𝑠)⋅𝜙(𝑠)||𝑓_∗𝑣₁(𝑡)||||,1≤𝑠,𝑢_𝑛(𝑠),𝑢^𝑛(𝑠),𝑢^𝑛(𝑠)−𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)||𝑑𝑠_0+∞⟶0,𝜙a(s𝑠)𝑛||𝑓⟶^∗𝑠,+𝑢_𝑛∞(𝑠,),𝑢^𝑛(𝑠),𝑢^𝑛(𝑠)−𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)||𝑑𝑠(3.15) so 𝑇∶𝑋→𝑋 is continuous.
(3)𝑇∶𝑋→𝑋 is compact. Let 𝐵 be any bounded subset of 𝑋, then there exists 𝑟>0 such that ‖𝑢‖≤𝑟,forall𝑢∈𝐵. For any 𝑢∈𝐵, one has ‖𝑇𝑢‖=≤ma_0+∞x‖𝑇𝑢‖₀,‖(𝑇𝑢)′‖₁,‖‖(𝑇𝑢)^‖‖₂≤𝜙(𝑠)||𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)||𝑑𝑠_0+∞𝜙(𝑠)𝐻_𝑟𝜓(𝑠)+1𝑑𝑠<+∞,(3.16) where 𝐻_𝑟=max_{0≤𝑠≤𝑟}ℎ(𝑠), so 𝑇𝐵 is uniformly bounded. Meanwhile, for any 𝑇>0, if 𝑡₁,𝑡₂∈[0,𝑇], we have ||||𝑇𝑢𝑡₁𝑣₀𝑡₁−𝑇𝑢𝑡²𝑣₀𝑡₂||||=|||||^0+∞𝐺𝑡¹,𝑠𝑣₀𝑡₁−𝐺𝑡²,𝑠𝑣₀𝑡₂𝜙(𝑠)𝑓^∗≤𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠_0+∞|||||||||𝐺𝑡₁,𝑠𝑣₀𝑡₁−𝐺𝑡²,𝑠𝑣₀𝑡₂||||𝜙(𝑠)𝐻^𝑟⟶0,as𝑡₁𝜓(𝑠)+1𝑑𝑠||||(𝑇𝑢)_⟶𝑡₂,𝑡₁𝑣₁𝑡₁−(𝑇𝑢)^𝑡₂𝑣₁𝑡₁||||=|||||^0+∞𝑔𝑡¹,𝑠𝑣₁𝑡₁−𝑔𝑡²,𝑠𝑣₁𝑡₂𝜙(𝑠)𝑓^∗≤𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠_0+∞|||||||||𝑔𝑡₁,𝑠𝑣₁𝑡₁−𝑔𝑡²,𝑠𝑣₁𝑡₂||||𝜙(𝑠)𝐻^𝑟⟶0,as𝑡₁𝜓(𝑠)+1𝑑𝑠||||(𝑇𝑢)_⟶𝑡₂,𝑡₁𝑣₂𝑡₁−(𝑇𝑢)^𝑡₂𝑣₂𝑡₁||||=||||^𝑡2𝑡₁≤𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠_𝑡2𝑡1||||⟶0,𝜙(𝑠a)s𝑡𝐻_𝑟1𝜓(𝑠)+1𝑑𝑠⟶𝑡₂;(3.17) that is, 𝑇𝐵 is equicontinuous. From Theorem 2.5, we can see that if 𝑇𝐵 is equiconvergent at infinity, then 𝑇𝐵 is relatively compact. In fact, ||||𝑇𝑢(𝑡)𝑣₀(𝑡)−lim^𝑡→+∞𝑇𝑢(𝑡)=||||𝑙(𝑡)𝑣₀(𝑡)||||𝑣₀(𝑡)−𝐶2+^0+∞𝐺𝑣₀(𝑡(,𝑡𝑠))−21𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^≤||||𝑙(𝑡)(𝑠),𝑢^(𝑠)𝑑𝑠||||𝑣₀(𝑡)−𝐶2||||+^0+∞||||𝐺(𝑡,𝑠)𝑣₀(𝑡)−21||||𝜙(𝑠)𝐻^𝑟||||(𝑇⟶𝑢)_0,as𝑡⟶+∞,𝜓(𝑠)+1𝑑𝑠(𝑡)𝑣₁(𝑡)−lim^𝑡→+∞(𝑇𝑢)^(𝑡)=||||(𝑎𝐶+𝐵)+𝐶𝑣𝑡₁(𝑡)||||𝑣₁(𝑡)−𝐶+^0+∞𝑔𝑣(₁𝑡(,𝑡𝑠))−1𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^≤||||(𝑎𝐶+𝐵)+𝐶𝑡(𝑠),𝑢^(𝑠)𝑑𝑠||||𝑣₁(𝑡)−𝐶||||+^0+∞||||𝑔(𝑡,𝑠)𝑣₁(𝑡)−1||||𝜙(𝑠)𝐻^𝑟||||(𝑇⟶𝑢)_0,as𝑡⟶+∞,𝜓(𝑠)+1𝑑𝑠(𝑡)=𝑣₂||||(𝑡)_𝑡+∞||||≤𝜙(𝑠)𝑓^∗𝑠,𝑢(𝑠),𝑢^(𝑠),𝑢^(𝑠)𝑑𝑠_𝑡+∞||||⟶0,𝜙a(s𝑠)𝑡⟶𝐻_𝑟𝜓+(𝑠)∞+.1𝑑𝑠(3.18) Then we can obtain that 𝑇∶𝑋→𝑋 is completely continuous.
By the Schäuder fixed point theorem, 𝑇 has at least one fixed point 𝑢∈𝑋. Next we will prove 𝑢 satisfying 𝛼′(𝑡)≤𝑢′(𝑡)≤𝛽′(𝑡),𝑡∈[0,+∞). If 𝑢′(𝑡)≤𝛽′(𝑡),𝑡∈[0,+∞) does not hold, then, sup_{0≤𝑡<+∞}𝑢^(𝑡)−𝛽^(𝑡)>0.(3.19) Because 𝑢^(+∞)−𝛽^(+∞)<0, so there are two cases.

Case 1. Consider lim_𝑡→0+𝑢^(𝑡)−𝛽^(𝑡)=sup_{0≤𝑡<+∞}𝑢^(𝑡)−𝛽^(𝑡)>0.(3.20) Easily, 𝑢^(0⁺)−𝛽^(0⁺)≤0. While by the boundary condition, we have 𝑎𝑢^(0)−𝛽^(0)≥𝑢^(0)−𝛽^(0)>0,(3.21) which is a contraction.

Case 2. There exists 𝑡^∗∈(0,+∞) such that 𝑢^𝑡^∗−𝛽^𝑡^∗=sup_{0≤𝑡<+∞}𝑢^(𝑡)−𝛽^(𝑡)>0.(3.22) So, 𝑢^(𝑡^∗)−𝛽^(𝑡^∗)=0,𝑢^(𝑡^∗)−𝛽^(𝑡^∗)≤0. Unfortunately, 𝑢^𝑡^∗−𝛽^𝑡^∗≥=𝜙𝜙𝑡𝑡^∗_∗𝑓𝑡^∗,𝛽𝑡^∗,𝛽^𝑡^∗,𝛽^𝑡^∗−𝑓^∗𝑡^∗,𝑢𝑡^∗,𝑢^𝑡^∗,𝑢^𝑡^∗+𝜙𝑡_∗𝑓𝑡^∗,𝛽𝑡^∗,𝛽^𝑡^∗,𝛽^𝑡^∗−𝑓^∗𝑡^∗,𝑢𝑡^∗,𝛽^𝑡^∗,𝛽^𝑡^∗𝑢^𝑡^∗−𝛽^𝑡^∗≥𝜙𝑡_∗1+||𝑢^(𝑡^∗)−𝛽^(𝑡^∗)||𝑢^𝑡^∗−𝛽^𝑡^∗>0.1+||𝑢^(𝑡^∗)−𝛽^(𝑡^∗)||(3.23)
Consequently, 𝑢′(𝑡)≤𝛽′(𝑡) holds for all 𝑡∈[0,+∞). Similarly, we can show that 𝛼′(𝑡)≤𝑢′(𝑡) for all 𝑡∈[0,+∞). Noticing that 𝛼(0)≤𝐴≤𝛽(0), from the inequality 𝛼′(𝑡)≤𝑢′(𝑡)≤𝛽′(𝑡), we can obtain that 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡). Lemma 3.1 guarantee that ‖𝑢^‖_∞𝑅. So, 𝑢^(𝑡)=−𝑓^∗𝑡,𝑢(𝑡),𝑢^(𝑡),𝑢^(𝑡)=−𝑓𝑡,𝑢(𝑡),𝑢^(𝑡),𝑢^(𝑡);(3.24) that is, 𝑢 is a solution of BVP (1.3).

Remark 3.4. For finite interval problem, it is sharp to define the lower and upper solutions satisfying 𝛼^(𝑏)≤𝐶 and 𝛽^(𝑏)≥𝐶; see [15].
If 𝑓∶[0,+∞)⁴→[0,+∞), we can establish a criteria for the existence of positive solutions.

Theorem 3.5. Let 𝑓∶[0,+∞)⁴→[0,+∞) be continuous and 𝜙∈𝐿¹[0,+∞). Suppose the condition (H₂) holds and the following conditions hold.(P₁) BVP (1.3) has a pair of positive upper and lower solutions 𝛼,𝛽∈𝑋 satisfying 𝛼^(𝑡)≤𝛽^(𝑡),𝑡∈[0,+∞).(3.25)(P₂) For any 𝑟>0, there exists 𝜑_𝑟 satisfying ∫^0+∞𝜙(𝑠)𝜑_𝑟(𝑠)𝑑𝑠<+∞ such that 𝑓(𝑡,𝑥,𝑦,𝑧)≤𝜑_𝑟(𝑡)(3.26) holds for all 𝑡∈[0,+∞),𝛼(𝑡)≤𝑥≤𝛽(𝑡),𝛼′(𝑡)≤𝑦≤𝛽′(𝑡), and 0≤𝑧≤𝑟.
Then BVP (1.3) with 𝐴,𝐵,𝐶≥0 has at least one solution such that 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡),𝛼^(𝑡)≤𝑢^(𝑡)≤𝛽^(𝑡),𝑡∈[0,+∞).(3.27)

Proof. Choose 𝑅=(1/𝑎)(𝐵+𝛽′(0)) and consider the boundary value problem (3.10) except 𝑓_𝑅 substituting by 𝑓_𝑅(𝑡,𝑥,𝑦,𝑧)=⎧⎪⎨⎪⎩𝑓𝑓𝑓(((𝑡𝑡𝑡,,,𝑥𝑥𝑥,,,𝑦𝑦𝑦,,,𝑧0𝑅))),,,𝑧𝑧0<>≤0𝑅𝑧,.≤𝑅,(3.28) Similarly, we can obtain that (3.10) has at least one solution 𝑢 satisfying 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡) and 𝛼′(𝑡)≤𝑢′(𝑡)≤𝛽′(𝑡),𝑡∈[0,+∞). Because 𝑢^(𝑡)=−𝜙(𝑡)𝑓^∗𝑡,𝑢(𝑡),𝑢^(𝑡),𝑢^(𝑡)≤0(3.29) and 𝑢^(+∞)=𝐶≥0, we have 0≤𝑢^(𝑡)≤𝑢^(0)=1𝑎𝐵+𝑢^(0)≤𝑅.(3.30) Consequently, the solution 𝑢 is a positive solution of (1.3).

Funding

This research is supported by the National Natural Science Foundation of China (nos. 11101385 and 60974145) and by the Fundamental Research Funds for the Central Universities.