Nonoscillatory Solutions of Second-Order Differential Equations without Monotonicity Assumptions

Abstract

The continuability, boundedness, monotonicity, and asymptotic properties of nonoscillatory solutions for a class of second-order nonlinear differential equations [𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥^′(𝑡))]^′=𝑞(𝑡)𝑔(𝑥(𝑡)) are discussed without monotonicity assumption for function g. It is proved that all solutions can be extended to infinity, are eventually monotonic, and can be classified into disjoint classes that are fully characterized in terms of several integral conditions. Moreover, necessary and sufficient conditions for the existence of solutions in each class and for the boundedness of all solutions are established.

1. Introduction

This paper studies the continuability, boundedness, monotonicity, and asymptotic properties of nonoscillatory solutions for a class of second-order nonlinear differential equations 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)^′=𝑞(𝑡)𝑔(𝑥(𝑡)),𝑡≥𝑎. (1.1) Some special cases of (1.1) such as half-linear equation 𝑝(𝑡)Φ_𝑝𝑥′(𝑡)^′=𝑞(𝑡)Φ_𝑝(𝑥(𝑡)), (1.2) where Φ_𝑝(𝑟)=|𝑟|^𝑝−2𝑟,  𝑝>1, is the so-called 𝑝-Laplacian operator, Emden-Fowler equation 𝑝(𝑡)Φ_𝑝𝑥′(𝑡)^′=𝑞(𝑡)Φ_𝛽(𝑥(𝑡)), (1.3) and differential equation 𝑝(𝑡)ℎ(𝑥(𝑡))𝑥′(𝑡)^′=𝑞(𝑡)𝑔(𝑥(𝑡)) (1.4) have been extensively discussed in the literature; see, for example, [1–15] and references cited therein. Equation (1.1) with general nonlinear function 𝑓(𝑟) is investigated in [16–18]. It is worth to point out that 𝑔(𝑟) is assumed to be monotonic in most cited papers, but [2, 6] explain that this assumption does not hold in some applications. The aim of this paper is to investigate the continuability, boundedness, monotonicity, and asymptotic properties of nonoscillatory solutions of (1.1) without monotonic assumption for 𝑔. Some techniques and ideas have been used by the authors in [17].

By solution of (1.1), we mean a differentiable function 𝑥 such that 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥′(𝑡)) is differentiable and satisfies (1.1) on the maximum existence interval [𝑎,𝛼_𝑥), 𝛼_𝑥≤∞. A solution 𝑥 of (1.1) is said to be eventually monotonic if there exists a 𝑡_𝑥≥𝑎 such that 𝑥 is monotonic on [𝑡_𝑥,𝛼_𝑥). In this paper, we consider only solutions that are not eventually identically equal to zero.

Throughout the paper, we always assume that

(H) 𝑝(𝑡),𝑞(𝑡)∶[𝑎,∞)→ℝarecontinuousand𝑝(𝑡)>0and𝑞(𝑡)>0;

 ℎ(𝑟)∶ℝ→ℝiscontinuousandℎ(𝑟)>0;

 𝑔(𝑟)∶ℝ→ℝiscontinuousand𝑟𝑔(𝑟)>0for𝑟≠0;

 𝑓(𝑟)∶ℝ→ℝiscontinuous,increasing,and𝑟𝑓(𝑟)>0for𝑟≠0.

(H1) There exists a constant 𝑀₁>0 such that 𝑓⁻¹(𝑢𝑣)≤𝑀₁𝑓⁻¹(𝑢)𝑓⁻¹(𝑣),∀𝑢,𝑣∈ℝ. (1.5)

Remark 1.1. (H1) holds for 𝑝-Laplacian operator; indeed, 𝑓⁻¹(𝑢𝑣)=𝑓⁻¹(𝑢)𝑓⁻¹(𝑣). (1.6) However, there are nonlinear functions 𝑓 that satisfy (H1) but not (1.6); see [17].

The paper is organized as follows: Section 1 briefly addresses the background and the motivation of the paper. Continuability, classification, and boundedness of solutions are discussed in Section 2. Sections 3 and 4 deal with the existence of class A and class B solutions, respectively. Finally, several remarks are provided in Section 5 to compare our results with existing ones.

2. Continuability, Classification, and Boundedness of Solutions

In this section we discuss continuability, classification, and boundedness of solutions of (1.1). First of all, we cite a result from [17] that will be used later on.

Lemma 2.1. If 𝑥 is a solution of (1.1) with maximal existence interval [𝑎,𝛼_𝑥), 𝛼_𝑥≤∞, then 𝑥 is eventually monotonic. Moreover, if 𝑥 is bounded on all finite subinterval of [𝑎,𝛼_𝑥), then 𝛼_𝑥=∞.

Remark 2.2. From Lemma 2.1 all solutions of (1.1) except eventually trivial solutions can be classified into two classes 𝐴=𝑥isdefinedon𝑎,𝛼_𝑥∶𝑥(𝑡)𝑥′(𝑡)>0inaleftneighborhoodof𝛼_𝑥,𝐵=𝑥isdefinedon[𝑎,∞)∶𝑥(𝑡)𝑥′(𝑡)<0for𝑡≥𝑎. (2.1)

Next theorem establishes the continuability for all solutions of (1.1), in other words, all solutions can be extended to [𝑎,∞).

Theorem 2.3. Assume the following assmputions hold.(H2) There exists a real number 𝑚>0 and a continuous function 𝐺(𝑟)∶ℝ→ℝ such that 𝐺(𝑟) is increasing and |𝑔(𝑟)|≤|𝐺(𝑟)| for |𝑟|≥𝑚, and 𝑟𝐺(𝑟)>0 for 𝑟≠0; (H3) There exists a real number 𝑟₀>0 such that ^∞_𝑟0𝑑𝑟𝑓⁻¹(𝑧(𝑟))=∞,^−𝑟0_−∞𝑑𝑟𝑓⁻¹(𝑧(𝑟))=−∞, (2.2) where 𝑧(𝑟)=𝐺(𝑟)/ℎ(𝑟). Then all solutions of (1.1) can be extended to [𝑎,∞).

Proof. The proof is similar to that of Theorem 2.3 [17]. We point out that as in the proof of Theorem 2.3 [17], for a class A solution 𝑥, we have 𝑓(𝑡)≤𝐺(𝑥(𝑡𝑥′))𝑝(𝑡)ℎ(𝑥(𝑡𝑝(𝑑)ℎ(𝑥(𝑑𝑓(𝑑)))))𝑥′𝐺(𝑥(𝑑+))^𝑡_𝑑𝑞(𝑠)𝑑𝑠_{𝑥𝑡𝑥(𝑡)1},𝑑𝑟𝑓⁻¹(𝑧(𝑟))≤𝑀²₁𝑓⁻¹(𝑘)^𝑡_𝑡1𝑓⁻¹1𝑝(𝑠)^𝑠_𝑑𝑞(𝜎)𝑑𝜎𝑑𝑠. (2.3)

Remark 2.4. The function 𝑔(𝑟)=𝑟+sin𝑟 is not monotonic. Clearly, |𝑔(𝑟)|≤2|𝑟|, so 𝑔 is bounded by an increasing function 𝐺(𝑟)=2𝑟. Therefore, the existing results which require the monotonic condition for 𝑔 would not apply, but Theorem 2.3 does.

From Remark 2.2 and Theorem 2.3, all solutions of (1.1) can be classified further into four disjoint classes 𝐴_𝑏=𝑥∈𝐴∶lim_𝑡→∞|𝑥(𝑡,𝐴)|=ℓ<∞_∞=𝑥∈𝐴∶lim_𝑡→∞|𝑥(𝑡,𝐵)|=∞_𝑏=𝑥∈𝐵∶lim_𝑡→∞𝑥(𝑡),𝐵=ℓ≠0₀=𝑥∈𝐵∶lim_𝑡→∞𝑥(𝑡)=0. (2.4)

We will show that the existence of solutions in each class and the boundedness of all solutions are fully characterized by means of convergence or divergence of the following integrals: 𝐽₁=^∞_𝑎𝑓⁻¹1𝑝(𝑡)^𝑡_𝑎𝑞(𝑠)𝐽𝑑𝑠𝑑𝑡,₂=^∞_𝑎𝑓⁻¹−1𝑝(𝑡)^𝑡_𝑎𝑞(𝑠)𝐽𝑑𝑠𝑑𝑡,₃=^∞_𝑎𝑓⁻¹1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝐽𝑑𝑠𝑑𝑡,₄=^∞_𝑎𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝐽𝑑𝑠𝑑𝑡,₅=^∞_𝑎𝑓⁻¹1𝑝(𝑡)𝑑𝑡. (2.5)

Theorem 2.5. Let (H2) and (H3) hold. Then all positive (negative) solutions of (1.1) are bounded if and only if 𝐽₁<∞(𝐽₂>−∞).

Proof. We consider positive solutions only since the case of negative solutions can be handled similarly.
Necessity. Let 𝑥 be a positive bounded class A solution. Then 𝑥(𝑡)>0 and 𝑥′(𝑡)>0 for 𝑡≥𝑏>𝑎 and lim_𝑡→∞𝑥(𝑡)=𝑙∈(0,∞). By the Extreme Value Theorem, we have 𝐿₁∶=min_{𝑥(𝑏)≤𝑟≤𝑙}𝑔(𝑟)>0. Hence 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)=𝑝(𝑏)ℎ(𝑥(𝑏))𝑓𝑥′(𝑏)+^𝑡_𝑏𝑞(𝑠)𝑔(𝑥(𝑠))𝑑𝑠≥𝐿₁^𝑡_𝑏𝑞(𝑠)𝑑𝑠. (2.6) Since 𝑥 is continuous and bounded and ℎ(𝑟) is continuous, then ℎ(𝑥(𝑡)) is bounded. Let ℎ(𝑥(𝑡))≤𝐾 for 𝑡∈[𝑎,∞). Then 𝐾𝑝(𝑡)𝑓𝑥′(𝑡)≥𝑝(𝑡)ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)≥𝐿₁^𝑡_𝑏𝑞(𝑠)𝐾𝑑𝑠,𝐿₁𝑓(𝑡)≥1𝑥′𝑝(𝑡)^𝑡_𝑏𝑞(𝑠)𝑑𝑠. (2.7) By (H1), we have 𝑓⁻¹1𝑝(𝑡)^𝑡_𝑏𝑞(𝑠)𝑑𝑠≤𝑓⁻¹𝐾𝐿₁𝑓𝑥^′(𝑡)≤𝑀₁𝑓⁻¹𝐾𝐿₁𝑥^′(𝑡). (2.8) Integrating from 𝑏 to 𝑡 and letting 𝑡→∞, we have 𝐽₁=^∞_𝑏𝑓⁻¹1𝑝(𝑡)^𝑡_𝑏𝑞(𝑠)𝑑𝑠𝑑𝑡≤𝑀₁𝑓⁻¹𝐾𝐿₁(𝑙−𝑥(𝑏))<∞. (2.9)
Sufficiency. We will prove by contradiction. Let 𝑥 be a unbounded class A solution. Then 𝑥(𝑡)>0 and 𝑥′(𝑡)>0 on [𝑏,∞), and there exists a real number 𝑑≥𝑏 such that 𝑥(𝑡)≥𝑚 for 𝑑≤𝑡<∞. Similar to the proof of Theorem 2.3, we have _𝑥𝑡^𝑥(𝑡)₁𝑑𝑟𝑓⁻¹(𝑧(𝑟))≤𝑀²₁𝑓⁻¹(𝑘)^𝑡_𝑡1𝑓⁻¹1𝑝(𝑠)^𝑠_𝑑𝑞(𝜎)𝑑𝜎𝑑𝑠. (2.10) Letting 𝑡→∞ and noting that 𝑥(∞)=∞, we have ^∞_𝑥𝑡1𝑑𝑟𝑓⁻¹(𝑧(𝑟))≤𝑀²₁𝑓⁻¹(𝑘)^∞_𝑡1𝑓⁻¹1𝑝(𝑠)^𝑠_𝑏𝑞(𝜎)𝑑𝜎𝑑𝑠≤𝑀²₁𝑓⁻¹(𝑘)𝐽₁<∞, (2.11) a contradiction to (H3). Therefore, 𝑥 is bounded.

Corollary 2.6. Let (H2) and (H3) hold. If (1.1) has a positive (negative) bounded class A solution, then all positive (negative) solutions are bounded. On the other hand, if (1.1) has an unbounded positive (negative) class A solution, then all positive (negative) solutions are unbounded.

3. Class A Solutions

In this section, we consider the existence of class 𝐴_𝑏 and class 𝐴_∞ solutions of (1.1). The necessary and sufficient conditions for the existence of class 𝐴_𝑏 solutions and the sufficient conditions for the existence of class 𝐴_∞ solutions are provided.

Theorem 3.1. Equation (1.1) has both positive and negative class A solutions.

Proof. Similar to the proof of Theorem 3.1 in [17].

Theorem 3.2. Equation (1.1) has a positive (negative) 𝐴_𝑏 solution if and only if 𝐽₁<∞(𝐽₂>−∞).

Proof. Necessity. Without loss of generality, we assume that 𝑥 is a positive 𝐴_𝑏 solution. In this case, there exists a 𝑏≥𝑎 such that 𝑥(𝑡)>0 and 𝑥′(𝑡)>0 for 𝑡≥𝑏. Note that 𝑥(∞)∶=lim_𝑡→∞𝑥(𝑡)<∞, we have 𝑚₁∶=min_{𝑥(𝑏)≤𝑟≤𝑥(∞)}𝑔(𝑟)𝑐>0,₁∶=max_{𝑏≤𝑡<∞}ℎ(𝑥(𝑡))≤max_{𝑥(𝑏)≤𝑟≤𝑥(∞)}ℎ(𝑟)<∞. (3.1) Then 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)=𝑝(𝑏)ℎ(𝑥(𝑏))𝑓𝑥′(𝑏)+^𝑡_𝑏𝑞(𝑠)𝑔(𝑥(𝑠))𝑑𝑠≥𝑚₁^𝑡_𝑏𝑞(𝑠)𝑑𝑠, (3.2) and hence 1𝑝(𝑡)^𝑡_𝑏𝑞(𝑠)1𝑑𝑠≤𝑚₁ℎ(𝑥(𝑡))𝑓𝑥′(𝑡). (3.3) Taking 𝑓⁻¹ on both sides and applying (H1) imply that 𝑓⁻¹1𝑝(𝑡)^𝑡_𝑏𝑞(𝑠)𝑑𝑠≤𝑓⁻¹1𝑚₁ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)≤𝑀₁𝑓⁻¹𝑐₁𝑚₁𝑥′(𝑡). (3.4) Therefore 𝐽₁=^∞_𝑏𝑓⁻¹1𝑝(𝑡)^𝑡_𝑏𝑞(𝑠)𝑑𝑠𝑑𝑠≤𝑀₁𝑓⁻¹𝑐₁𝑚₁(𝑥(∞))−(𝑥(𝑏))<∞. (3.5)
Sufficiency. Define 𝑚₂=max_1≤𝑟≤2𝑔(𝑟)>0,𝑐₂=min_1≤𝑟≤2ℎ(𝑟)>0. (3.6) Since 𝐽₁<∞, we may select a 𝑑≥𝑎 such that ^∞_𝑑𝑓⁻¹1𝑝(𝑡)^𝑡_𝑑𝑞(𝑠)1𝑑𝑠𝑑𝑡≤𝑀₁𝑓⁻¹𝑚₂/𝑐₂. (3.7) Let 𝐶𝐵[𝑑,∞) be the Banach space of all bounded and continuous functions defined on [𝑑,∞) endowed with the supremum norm, and let 𝑋={𝑥∈𝐶𝐵[𝑑,∞)∶1≤𝑥(𝑡)≤2,𝑡≥𝑑}. Clearly, 𝑋 is a bounded convex subset of 𝐶𝐵[𝑑,∞). Define a mapping 𝐹₁∶𝑋→𝐶𝐵[𝑑,∞) by 𝐹₁𝑥(𝑡)=1+^𝑡_𝑑𝑓⁻¹1𝑝(𝑠)ℎ(𝑥(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔(𝑥(𝜎))𝑑𝜎𝑑𝑠. (3.8) In order to apply Schauder's fixed-point theorem to show that 𝐹₁ has a fixed point in 𝑋, we need to prove that 𝐹₁ maps into 𝑋 and is continuous, and 𝐹₁(𝑋) is precompact in 𝐶𝐵[𝑑,∞).
Let 𝑥∈𝑋. Considering (3.7), we have 1≤𝐹₁𝑥(𝑡)≤1+𝑀₁𝑓⁻¹𝑚₂𝑐₂^𝑡_𝑑𝑓⁻¹1𝑝(𝑠)^𝑠_𝑑𝑞(𝜎)𝑑𝜎𝑑𝑠≤1+𝑀₁𝑓⁻¹𝑚₂𝑐₂^∞_𝑑𝑓⁻¹1𝑝(𝑡)^𝑡_𝑑𝑞(𝑠)𝑑𝑠𝑑𝑡≤2. (3.9) Hence, 𝐹₁ maps 𝑋 into 𝑋.
Now, we show that if 𝑥_𝑛, 𝑥^∗∈𝑋 and ‖𝑥_𝑛−𝑥^∗‖→0 as 𝑛→∞, then ‖𝐹₁𝑥_𝑛−𝐹₁𝑥^∗‖→0. Indeed, for any fixed 𝑠∈[𝑑,∞), since 𝑥_𝑛(𝑠)→𝑥^∗(𝑠) as 𝑛→∞, we have 𝑓⁻¹1𝑝(𝑠)ℎ𝑥_𝑛(𝑠)^𝑠_𝑑𝑞(𝜎)𝑔𝑥_𝑛(𝜎)𝑑𝜎−𝑓⁻¹1𝑝(𝑠)ℎ(𝑥^∗(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔𝑥^∗(𝜎)𝑑𝜎⟶0as𝑛⟶∞. (3.10) Note that 𝑓⁻¹1𝑝(𝑠)ℎ𝑥_𝑛(𝑠)^𝑠_𝑑𝑞(𝜎)𝑔𝑥_𝑛(𝜎)𝑑𝜎−𝑓⁻¹1𝑝(𝑠)ℎ(𝑥^∗(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔𝑥^∗(𝜎)≤𝑓𝑑𝜎₋₁1𝑝(𝑠)ℎ𝑥_𝑛(𝑠)^𝑠_𝑑𝑞(𝜎)𝑔𝑥_𝑛(𝜎)+𝑓𝑑𝜎₋₁1𝑝(𝑠)ℎ(𝑥^∗(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔𝑥^∗(𝜎)𝑑𝜎≤2𝑀₁𝑓⁻¹𝑚₂𝑐₂𝑓⁻¹1𝑝(𝑠)^𝑠_𝑑𝑞(𝜎)𝑑𝜎∶=𝐹(𝑠), (3.11) and that ^∞_𝑑𝐹(𝑠)𝑑𝑠=^∞_𝑑2𝑀₁𝑓⁻¹𝑚₂𝑐₂𝑓⁻¹1𝑝(𝑠)^𝑠_𝑑𝑞(𝜎)𝑑𝜎𝑑𝑠=2𝑀₁𝑓⁻¹𝑚₂𝑐₂𝐽₁. (3.12) By Lebesgue’s dominated convergence theorem and considering (3.11) and (3.12) we have 𝐹₁𝑥_𝑛−𝐹₁𝑥^∗≤sup_{𝑏≤𝑡<∞}^𝑡_𝑑𝑓⁻¹1𝑝(𝑠)ℎ𝑥_𝑛(𝑠)^𝑠_𝑑𝑞(𝜎)𝑔𝑥_𝑛(𝜎)𝑑𝜎−𝑓⁻¹1𝑝(𝑠)ℎ(𝑥^∗(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔𝑥^∗(𝜎)≤𝑑𝜎𝑑𝑠_∞𝑑𝑓⁻¹1𝑝(𝑠)ℎ𝑥_𝑛(𝑠)^𝑠_𝑑𝑞(𝜎)𝑔𝑥_𝑛(𝜎)𝑑𝜎−𝑓⁻¹1𝑝(𝑠)ℎ(𝑥^∗(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔𝑥^∗(𝜎)𝑑𝜎𝑑𝑠⟶0 (3.13) as 𝑛→∞. Therefore, 𝐹₁ is continuous in 𝑋.
Finally, we show the precompactness of 𝐹₁(𝑋) in 𝐶𝐵[𝑑,∞), which means that for any sequence 𝑥_𝑛∈𝑋, 𝐹₁𝑥_𝑛 has a convergent subsequence in 𝐶𝐵[𝑑,∞). This can be proved by showing that 𝐹₁𝑥_𝑛 has a convergent subsequence in 𝐶[𝑏₁,𝑏₂] for any compact subinterval [𝑏₁,𝑏₂] of [𝑏,∞) as well as the diagonal rule. In fact, 𝐹₁𝑥_𝑛 is uniformly bounded on [𝑏₁,𝑏₂]. Since 𝐹₁𝑥_𝑛^′(𝑡)=𝑓⁻¹1𝑝(𝑡)ℎ𝑥_𝑛(𝑡)^𝑡_𝑑𝑞(𝑠)𝑔𝑥_𝑛(𝑠)𝑑𝑠≤𝑀₁𝑓⁻¹𝑚₂𝑐₂𝑓⁻¹1𝑝(𝑡)^𝑡_𝑑𝑞(𝑡)𝑑𝑠. (3.14) By the Mean Value Theorem, we have 𝐹₁𝑥_𝑛𝑡₁−𝐹₁𝑥_𝑛𝑡₂=𝐹₁𝑥_𝑛′(𝜉)𝑡₁−𝑡₂≤𝑀₁𝑓⁻¹𝑚₂𝑐₂max_{𝑏1≤𝑡≤𝑏2}𝑓⁻¹1𝑝(𝑡)^𝑡_𝑑𝑞(𝑠)𝑑𝑠𝑡₁−𝑡₂. (3.15) Then 𝐹₁𝑥_𝑛 is uniformly bounded and equicontinuous in 𝐶[𝑏₁,𝑏₂]. So 𝐹₁𝑥_𝑛 has a convergent subsequence in 𝐶[𝑏₁,𝑏₂] by Arzelà-Ascoli Theorem.
Now all conditions of Schauder's fixed-point theorem are satisfied, so 𝐹₁ has a fixed point 𝑥 in 𝑋, that is, 𝑥(𝑡)=1+^𝑡_𝑑𝑓⁻¹1𝑝(𝑠)ℎ(𝑥(𝑠))^𝑠_𝑑𝑞(𝜎)𝑔(𝑥(𝜎))𝑑𝜎𝑑𝑠. (3.16) It is easy to verify that [𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥^′(𝑡))]′=𝑞(𝑡)𝑔(𝑥(𝑡)). Hence, 𝑥 is a positive 𝐴_𝑏 solution of (1.1). The proof is complete.

Theorem 3.3. Let (H2) and (H3) hold. Then(a)𝐴_∞=∅ if and only if 𝐽₁<∞ and 𝐽₂>−∞.(b)Equation (1.1) has a positive (negative) 𝐴_∞ solution if 𝐽₁=∞(𝐽₂=−∞).

Proof. By Theorem 2.5 all solutions of (1.1) are bounded if and only if 𝐽₁<∞ and 𝐽₂>−∞, so part (a) follows.
If 𝐽₁=∞, there is no positive 𝐴_𝑏 solution of (1.1) from Theorem 3.2. Therefore, Theorem 3.1 guarantees the existence of a positive 𝐴_∞ solution of (1.1). Similarly, there exists a negative 𝐴_∞ solution of (1.1) if 𝐽₂=−∞.

4. Class B Solutions

In this section the existence of class 𝐵, 𝐵_𝑏, and 𝐵₀ solutions are discussed. We assume that (1.1) has a unique solution for any initial conditions 𝑥(𝑎)=𝑥₀≠0 and 𝑥′(𝑎)=𝑥₁.

Theorem 4.1. Assume the following assumptions hold.(H2a) There exists a continuous function 𝐺(𝑟)∶ℝ→ℝ such that G is increasing, 𝑟𝐺(𝑟)>0 for 𝑟≠0 and |𝑔(𝑟)|≤|𝐺(𝑟)|;(H4) There exists 𝑟₀>0 such that ^±𝑟0₀𝑑𝑟𝑓⁻¹(𝑧(𝑟))=∞. (4.1)Then (1.1) has (a)both positive and negative solutions in class B; (b)no solution which is eventually identically equal to zero.

Proof. (a) We prove that class 𝐵 has a positive solution, the case of having a negative solution is similar. Assume 𝑥₀>0. The solution of (1.1) with initial conditions 𝑥(𝑎)=𝑥₀ and 𝑥′(𝑎)=𝑐, denoted by 𝑥(𝑡)∶=𝑥(𝑡,𝑐), has the form 𝑥(𝑡)=𝑥₀+^𝑡_𝑎𝑓⁻¹𝑝(𝑎)ℎ𝑥₀𝑓(𝑐)𝑝(𝑠)ℎ(𝑥(𝑠+1))𝑝(𝑠)ℎ(𝑥(𝑠))^𝑠_𝑎𝑞(𝜎)𝑔(𝑥(𝜎))𝑑𝜎𝑑𝑠. (4.2) Define two sets 𝑈 and 𝐿 as 𝑈=𝑐∈ℝ∶thereexistssome𝑡≥𝑎suchthat𝑥^′𝑡,𝑐>0,𝐿=𝑐∈ℝ∶thereexistssome𝑡≥𝑎suchthat𝑥𝑡,𝑐<0. (4.3) Then 𝑈∩𝐿=∅. Clearly, 𝑈≠∅. We claim that 𝑈 is open. Indeed, if 𝑐₀∈𝑈, there exists 𝑡>𝑎 such that 𝑥′(𝑡,𝑐₀)>0. For any 𝑐∈ℝ, we have 𝑝𝑡ℎ𝑥𝑡,𝑐₀𝑓𝑥′𝑡,𝑐₀−𝑝𝑡ℎ𝑥𝑡,𝑐𝑓𝑥′(𝑎)ℎ𝑥𝑡,𝑐=𝑝₀𝑓𝑐₀−𝑝(𝑎)ℎ𝑥₀𝑓(𝑐)+^𝑡_𝑎𝑞(𝑠)𝑔𝑥𝑠,𝑐₀−𝑔(𝑥(𝑠,𝑐))𝑑𝑠. (4.4) Since (1.1) has a unique solution for any initial conditions 𝑥(𝑎)≠0, 𝑥′(𝑎), this solution is continuously dependent on initial data. If 𝑐→𝑐₀, we have 𝑔(𝑥(𝑠,𝑐))−𝑔(𝑥(𝑠,𝑐₀))→0 uniformly for 𝑠 on [𝑎,𝑡]. Hence, 𝑥′(𝑡,𝑐)>0 for all 𝑐 that are close to 𝑐₀, this proves the openness of 𝑈.
Next we show that 𝐿≠∅. Define 𝑀₂∶=min_{0≤𝑟≤𝑥0}ℎ(𝑟)>0,𝑀₃∶=min_{𝑎≤𝑡≤𝑎+1}𝑝(𝑡)>0. (4.5) Let 𝑐<𝑓⁻¹𝑀₂𝑀₃𝑓⁻¹−𝑥₀−𝐺𝑥₀∫_𝑎^𝑎+1𝑞(𝑠)𝑑𝑠𝑝(𝑎)ℎ𝑥₀<0. (4.6) If there exists 𝑏∈(𝑎,𝑎+1] such that 𝑥(𝑏,𝑐)<0, then 𝑐∈𝐿 and 𝐿≠∅. Otherwise, 𝑥(𝑡,𝑐)≥0 on [𝑎,𝑎+1]. In this case, we claim 𝑥′(𝑡,𝑐)<0 on [𝑎,𝑎+1]. If this is not true, since 𝑥′(𝑎,𝑐)=𝑐<0, there exists 𝑡₁∈(𝑎,𝑎+1] such that 𝑥′(𝑡₁,𝑐)=0 and 𝑥′(𝑡,𝑐)<0 for 𝑡∈[𝑎,𝑡₁). Taking into account (4.6) we have 0=𝑝(𝑡)ℎ𝑥𝑡₁,𝑐𝑓𝑥′𝑡₁(𝑎)ℎ𝑥,𝑐=𝑝₀𝑓(𝑐)+^𝑡1_𝑎𝑞(𝑠)𝑔(𝑥((𝑎)ℎ𝑥𝑠,𝑐))𝑑𝑠≤𝑝₀𝑓(𝑐)+𝐺𝑥_0𝑎^𝑎+1𝑞(𝑠)𝑑𝑠<0. (4.7) This is a contradiction and hence 𝑥′(𝑡,𝑐)<0 on [𝑎,𝑎+1]. Notice that 𝑥(𝑎+1,𝑐)=𝑥₀+_𝑎^𝑎+1𝑓⁻¹𝑝(𝑎)ℎ𝑥₀𝑓(𝑐)𝑝(𝑡)ℎ(𝑥(𝑡+1))𝑝(𝑡)ℎ(𝑥(𝑡))^𝑡_𝑎𝑞(𝑠)𝑔(𝑥(𝑠))𝑑𝑠𝑑𝑡≤𝑥₀+_𝑎^𝑎+1𝑓⁻¹𝑝(𝑎)ℎ𝑥₀𝑓(𝑐)+𝐺𝑥₀∫_𝑎^𝑎+1𝑞(𝑠)𝑑𝑠𝑀₂𝑀₃𝑑𝑡<0, (4.8) we know 𝑐∈𝐿. Clearly, 𝐿 is open, then ℝ−(𝑈∪𝐿)≠∅. Take 𝑐∈ℝ−(𝑈∪𝐿), 𝑥(𝑡,𝑐) is a nonincreasing nonnegative solution on [𝑎,∞). We will show that 𝑥(𝑡,𝑐)>0 on [𝑎,∞). If not, there exists 𝑡₀>𝑎 such that 𝑥(𝑡₀)=0 and 𝑥(𝑡)=0 for 𝑡≥𝑡₀ and 𝑥′(𝑡₀)=0. Note that for 𝑡∈[𝑎,𝑡₀] we have 𝑥^′(𝑡)=𝑓⁻¹−1𝑝(𝑡)ℎ(𝑥(𝑡))^𝑡0_𝑡𝑞(𝑠)𝑔(𝑥(𝑠))𝑑𝑠≥𝑓⁻¹−𝐺(𝑥(𝑡))𝑝(𝑡)ℎ(𝑥(𝑡))^𝑡0_𝑡𝑞(𝑠)𝑑𝑠≥𝑀₁𝑓⁻¹(𝑧(𝑥(𝑡)))𝑓⁻¹−1𝑝(𝑡)^𝑡0_𝑡𝑞(𝑠)𝑑𝑠. (4.9) Dividing both sides by 𝑓⁻¹(𝑧(𝑥(𝑡))) and integrating from 𝑎 to 𝑡₀, we have 𝑥′(^𝑡0_𝑎𝑡)𝑓⁻¹(𝑧(𝑥(𝑡)))𝑑𝑡≥𝑀₁^𝑡0_𝑎𝑓⁻¹−1𝑝(𝑡)^𝑡0_𝑡𝑞(𝑠)𝑑𝑠𝑑𝑡. (4.10) That is ^𝑥0₀1𝑓⁻¹(𝑧(𝑟))𝑑𝑟≤−𝑀₁^𝑡0_𝑎𝑓⁻¹−1𝑝(𝑡)^𝑡0_𝑡𝑞(𝑠)𝑑𝑠𝑑𝑡<∞, (4.11) a contradiction to (H4). Therefore, 𝑥(𝑡)>0 for 𝑡≥𝑎 and 𝑥∈𝐵.
The proof of part (b) follows from the end part of the proof of part (a).

Theorem 4.2. Equation (1.1) has a positive (negative) 𝐵_𝑏 solution if and only if 𝐽₄>−∞(𝐽₃<∞).

Proof. Necessity. We assume that 𝑥 is a positive 𝐵_𝑏 solution. The case of negative 𝐵_𝑏 solution is similar. In this case, we have 𝑥(𝑡)>0 and 𝑥′(𝑡)<0 for 𝑡≥𝑎. Let 𝑚₁=min_{𝑥(∞)(𝑎)≤𝑟≤𝑥}𝑔(𝑟)>0,𝑐₁=max_{𝑥(∞)≤𝑟≤𝑥(𝑎)}ℎ(𝑟)>0 (4.12) and note that 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥′(𝑡))<0, (𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥′(𝑡)))′>0. Then lim_𝑡→∞𝑝(𝑡)ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)=𝐵≤0. (4.13) Integrating both sides of (1.1) from 𝑡 to ∞ implies that 𝑚₁^∞_𝑡𝑞(𝑠)d𝑠≤^∞_𝑡𝑞(𝑠)𝑔(𝑥(𝑠𝑝(𝑡)ℎ(𝑥(𝑡𝑓(𝑡)(𝑡)ℎ(𝑥(𝑡𝑓(𝑡.))𝑑𝑠=𝐵−))𝑥′≤−𝑝))𝑥′) (4.14) Hence, 𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝑑𝑠≥𝑀₁𝑓⁻¹𝑐₁𝑚₁𝑥′(𝑡). (4.15) Again, integrating both sides of the above inequality we have 𝐽₄=^∞_𝑎𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝑑𝑠𝑑𝑡≥𝑀₁𝑓⁻¹𝑐₁𝑚₁(𝑥(∞)−𝑥(𝑎))>−∞. (4.16)
Sufficiency. Let 𝑚₂=max_1≤𝑟≤2𝑔(𝑟)>0,𝑐₂=min_1≤𝑟≤2ℎ(𝑟)>0. (4.17) Since 𝐽₄>−∞ we choose 𝑑>𝑎 such that ^∞_𝑑𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)1𝑑𝑠𝑑𝑡≥−𝑀₁𝑓⁻¹𝑚₂/𝑐₂. (4.18) Let 𝑋 and 𝐶𝐵[𝑑,∞) as defined in Theorem 3.2. Define 𝐹₂∶𝑋→𝐶𝐵[𝑑,∞) by 𝐹₂𝑥(𝑡)=1−^∞_𝑡𝑓⁻¹−1𝑝(𝑠)ℎ(𝑥(𝑠))^∞_𝑠𝑞(𝜎)𝑔(𝑥(𝜎))𝑑𝜎𝑑𝑠. (4.19) For any 𝑥∈𝑋, we have 1≤𝐹₂𝑥(𝑡)≤1−^∞_𝑑𝑀₁𝑓⁻¹𝑚₂𝑐₂𝑓⁻¹−1𝑝(𝑠)^∞_𝑠𝑞(𝜎)𝑑𝜎𝑑𝑠≤2. (4.20) This proves that 𝐹₂ maps 𝑋 into 𝑋. Similar to the proof of Theorem 3.2, we are able to show that 𝐹₂ is continuous in 𝑋, and 𝐹₂(𝑋) is precompact in 𝐶𝐵[𝑑,∞). Then 𝐹₂ has a fixed-point 𝑥(𝑡) in 𝑋 by Schauder's fixed-point theorem, that is, 𝑥(𝑡)=1−^∞_𝑡𝑓⁻¹−1𝑝(𝑠)ℎ(𝑥(𝑠))^∞_𝑠𝑞(𝜎)𝑔(𝑥(𝜎))𝑑𝜎𝑑𝑠. (4.21) It is easy to verify that 𝑥(𝑡) is a positive 𝐵_𝑏 solution of (1.1). The proof is complete.

Theorem 4.3. Let (H2a) and (H4) hold and let 𝐽₅=∞. Then (1.1) has a positive (negative) 𝐵₀ solution if and only if 𝐽₄=−∞ (𝐽₄=∞).

Proof. We prove the assertion for positive solutions without loss of generality.
Necessity. Assume 𝑥(𝑡) is a positive 𝐵₀ solution. Then 𝑥(𝑡)>0 and 𝑥′(𝑡)<0 for 𝑡≥𝑎, 𝑥(∞)=0, and lim_𝑡→∞𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥′(𝑡))=𝐿∈(−∞,0]. We claim 𝐿=0. In fact, if 𝐿<0, since 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥′(𝑡)) is negative and increasing on [𝑎,∞), then 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓(𝑥′(𝑡))≤𝐿 and 𝑥′(𝑡)≤𝑓⁻¹𝐿𝑐₁𝑝(𝑡)≤𝑀₁𝑓⁻¹𝐿𝑐₁𝑓⁻¹1𝑝(𝑡), (4.22) where 𝑐₁=max_{0≤𝑟≤𝑥(𝑎)}ℎ(𝑟)>0.
Integrating both sides from 𝑎 to ∞ and noting that 𝑥(∞)=0, we have 𝑥(𝑎)≥−𝑀₁𝑓⁻¹𝐿𝑐₁^∞_𝑎𝑓⁻¹1𝑝(𝑡)𝑑𝑡, (4.23) a contradiction to 𝐽₅=∞ and hence 𝐿=0.
Integrating both sides of (1.1) from 𝑡 to ∞ we have 𝑝(𝑡)ℎ(𝑥(𝑡))𝑓𝑥′(𝑡)=−^∞_𝑡𝑞(𝑠)𝑔(𝑥(𝑠))𝑑𝑠. (4.24) Then 𝑥^′(𝑡)=𝑓⁻¹−1𝑝(𝑡)ℎ(𝑥(𝑡))^∞_𝑡𝑞(𝑠)𝑔(𝑥(𝑠))𝑑𝑠≥𝑀₁𝑓⁻¹(𝑧(𝑥(𝑡)))𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝑑𝑠. (4.25) Hence, 𝑥′(𝑡)𝑓⁻¹(𝑧(𝑥(𝑡)))≥𝑀₁𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝑑𝑠. (4.26) Integrating both sides of the above inequality from 𝑎 to ∞ implies that ₀^𝑥(𝑎)𝑑𝑟𝑓⁻¹(𝑧(𝑟))≤−𝑀₁^∞_𝑎𝑓⁻¹−1𝑝(𝑡)^∞_𝑡𝑞(𝑠)𝑑𝑠𝑑𝑡. (4.27) Therefore, 𝐽₄=−∞ from (H4).
Sufficiency. By Theorem 4.1 (1.1) has a positive class B solution 𝑥, either 𝑥∈𝐵_𝑏 or 𝑥∈𝐵₀. Note that 𝐽₄=−∞ implies that 𝑥∉𝐵_𝑏 from Theorem 4.2. So 𝑥∈𝐵₀. The proof is complete.

5. Remarks

In this section, we present several remarks about comparison of our results with the existing ones in the literature.

Theorems 2.3 and 2.5 improve [10, Theorem 1] since (H3) reduces to (iii) of [10] if 𝑓(𝑟)=𝑟 and the differentiability of 𝑝(⋅) and ℎ(⋅) is not required. Theorems 2.3, 2.5, and 4.2 complement and generalize [2, Theorem 8]. Moreover, under (H2), Theorems 2.3, 2.5, and 4.2 improve [2, Theorem 8] since (H3) improves (22) of [2]; see the discussion in [16]. Theorem 2.5 generalizes [13, Theorem 3.9]. Theorems 2.3, 3.1, and 4.2 generalize [16, Theorem 1]. Theorem 3.2 generalizes [2, Theorem 3], [16, Theorem 3], and [18, Theorem 2.1]. Theorem 3.3 generalizes [18, Theorem 2.2]. Theorem 4.1 generalizes [13, Theorem 2.1] and improves [3, Theorem 6] under (H2a) since (hp) in [3] is replaced by a weaker condition (H4). Theorem 4.2 generalizes [2, Theorem 1], [16, Theorem 5], and [18, Theorem 3.1]. Theorem 4.3 generalize [16, Theorem 6] and [18, Theorem 3.2].