Existence of Positive Solutions of Neutral Differential Equations

Abstract

The paper contains some suffcient conditions for the existence of positive solutions which are bounded below and above by positive functions for the nonlinear neutral differential equations of higher order. These equations can also support the existence of positive solutions approaching zero at infinity.

1. Introduction

This paper is concerned with the existence of a positive solution of the neutral differential equations of the form:𝑑^𝑛𝑑𝑡^𝑛[𝑥(𝑡)−𝑎(𝑡)𝑥(𝑡−𝜏)]=(−1)^𝑛+1𝑝(𝑡)𝑓(𝑥(𝑡−𝜎)),𝑡≥𝑡₀, (1.1) where 𝑛>0 is an integer, 𝜏>0, 𝜎≥0, 𝑎∈𝐶([𝑡₀,∞),(0,∞)), 𝑝∈𝐶(𝑅,(0,∞)), 𝑓∈𝐶(𝑅,𝑅), 𝑓 is a nondecreasing function and 𝑥𝑓(𝑥)>0, 𝑥≠0.

By a solution of (1.1) we mean a function 𝑥∈𝐶([𝑡₁−𝜏,∞),𝑅) for some 𝑡₁≥𝑡₀, such that 𝑥(𝑡)−𝑎(𝑡)𝑥(𝑡−𝜏) is 𝑛-times continuously differentiable on [𝑡₁,∞) and such that (1.1) is satisfied for 𝑡≥𝑡₁.

The problem of the existence of solutions of neutral differential equations has been studied and discussed by several authors in the recent years. For related results we refer the reader to [1–17] and the references cited therein. However, there is no conception which guarantees the existence of positive solutions which are bounded below and above by positive functions. Maybe it is due to the technical difficulties arising in the analysis of the problem. In this paper we presented some conception. The method also supports the existence of positive solutions which approaching zero at infinity. Some examples illustrating the results.

The existence and asymptotic behavior of solutions of the nonlinear neutral differential equations and systems have been also solved in [1–7, 12, 15].

As much as we know for (1.1) in the literature, there is no result for the existence of solutions which are bounded by positive functions. Only the existence of solutions which are bounded by constants is treated and discussed, for example, in [10, 15, 17]. It seems that conditions of theorems are rather complicate, but cannot be simpler due to Corollaries 2.4, 2.8, and 3.3.

The following fixed point theorem will be used to prove the main results in the next section.

Lemma 1.1 (see [7, 10, 12] Krasnoselskii’s fixed point theorem). Let 𝑋 be a Banach space, let Ω be a bounded closed convex subset of 𝑋, and let 𝑆₁,𝑆₂ be maps of Ω into 𝑋 such that 𝑆₁𝑥+𝑆₂𝑦∈Ω for every pair 𝑥,𝑦∈Ω. If 𝑆₁ is a contractive and 𝑆₂ is completely continuous then the equation:𝑆₁𝑥+𝑆₂𝑥=𝑥 (1.2) has a solution in Ω.

2. The Existence of Positive Solution

In this section, we will consider the existence of a positive solution for (1.1) which is bounded by two positive functions. We will use the notation 𝑚=max{𝜏,𝜎}.

Theorem 2.1. Suppose that there exist bounded functions 𝑢,𝑣∈𝐶¹([𝑡₀,∞),(0,∞)), constant 𝑐>0, and 𝑡₁≥𝑡₀+𝑚 such that 𝑢(𝑡)≤𝑣(𝑡),𝑡≥𝑡₀,𝑣(𝑡)−𝑣𝑡₁−𝑢(𝑡)+𝑢𝑡₁≥0,𝑡₀≤𝑡≤𝑡₁1,𝑢()𝑢(𝑡)+1𝑡−𝜏(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓(𝑣((𝑡)≤1𝑠−𝜎))𝑑𝑠≤𝑎𝑣()𝑣(𝑡)+1𝑡−𝜏(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓(𝑢(𝑠−𝜎))𝑑𝑠≤𝑐<1,𝑡≥𝑡₁. (2.1)(2.2)(2.3) Then (1.1) has a positive solution which is bounded by the functions 𝑢, 𝑣.

Proof. Let 𝐶([𝑡₀,∞),𝑅) be the set of all continuous bounded functions with the norm ||𝑥||=sup_𝑡≥𝑡0|𝑥(𝑡)|. Then 𝐶([𝑡₀,∞),𝑅) is a Banach space. We define a closed, bounded, and convex subset Ω of 𝐶([𝑡₀,∞),𝑅) as follows: Ω=𝑥=𝑥(𝑡)∈𝐶𝑡₀,∞,𝑅∶𝑢(𝑡)≤𝑥(𝑡)≤𝑣(𝑡),𝑡≥𝑡₀. (2.4) We now define two maps 𝑆₁ and 𝑆₂∶Ω→𝐶([𝑡₀,∞),𝑅) as follows: 𝑆₁𝑥𝑡)=𝑎(𝑡)𝑥()(𝑡−𝜏,𝑡≥𝑡₁,𝑆₁𝑥𝑡₁,𝑡₀≤𝑡≤𝑡₁,𝑆₂𝑥(𝑡)=−1(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠,𝑡≥𝑡₁,𝑆₂𝑥𝑡₁+𝑣(𝑡)−𝑣𝑡₁,𝑡₀≤𝑡≤𝑡₁. (2.5) We will show that for any 𝑥,𝑦∈Ω we have 𝑆₁𝑥+𝑆₂𝑦∈Ω. For every 𝑥,𝑦∈Ω and 𝑡≥𝑡₁ we obtain 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)(𝑡)𝑣()−1≤𝑎𝑡−𝜏(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓(𝑢(𝑠−𝜎))𝑑𝑠≤𝑣(𝑡). (2.6) For 𝑡∈[𝑡₀,𝑡₁] we have 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)=𝑆₁𝑥𝑡₁+𝑆₂𝑦𝑡₁+𝑣(𝑡)−𝑣𝑡₁≤𝑣𝑡₁+𝑣(𝑡)−𝑣𝑡₁=𝑣(𝑡). (2.7) Furthermore for 𝑡≥𝑡₁ we get 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)(𝑡)𝑢()−1≥𝑎𝑡−𝜏(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≥𝑢(𝑡). (2.8) Finally let 𝑡∈[𝑡₀,𝑡₁] and with regard to (2.2) we get 𝑣(𝑡)−𝑣𝑡₁+𝑢𝑡₁≥𝑢(𝑡),𝑡₀≤𝑡≤𝑡₁. (2.9) Then for 𝑡∈[𝑡₀,𝑡₁] and any 𝑥,𝑦∈Ω we get 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)=𝑆₁𝑥𝑡₁+𝑆₂𝑦𝑡₁+𝑣(𝑡)−𝑣𝑡₁≥𝑢𝑡₁+𝑣(𝑡)−𝑣𝑡₁≥𝑢(𝑡). (2.10) Thus, we have proved that 𝑆₁𝑥+𝑆₂𝑦∈Ω for any 𝑥,𝑦∈Ω.
We will show that 𝑆₁ is a contraction mapping on Ω. For 𝑥,𝑦∈Ω and 𝑡≥𝑡₁ we have𝑆₁𝑥(𝑡)−𝑆₁𝑦(𝑡)=|𝑎(𝑡)|𝑥(𝑡−𝜏)−𝑦(𝑡−𝜏)≤𝑐𝑥−𝑦. (2.11) This implies that 𝑆₁𝑥−𝑆₁𝑦≤𝑐𝑥−𝑦. (2.12) Also for 𝑡∈[𝑡₀,𝑡₁] the inequality above is valid. We conclude that 𝑆₁ is a contraction mapping on Ω.
We now show that 𝑆₂ is completely continuous. First we will show that 𝑆₂ is continuous. Let 𝑥_𝑘=𝑥_𝑘(𝑡)∈Ω be such that 𝑥_𝑘(𝑡)→𝑥(𝑡) as 𝑘→∞. Because Ω is closed, 𝑥=𝑥(𝑡)∈Ω. For 𝑡≥𝑡₁ we have𝑆₂𝑥_𝑘(𝑡)−𝑆₂𝑥(𝑡≤1)(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓𝑥_𝑘()(𝑥(≤1𝑠−𝜎−𝑓𝑠−𝜎))𝑑𝑠(𝑛−1)!^∞_𝑡1𝑠−𝑡₁^𝑛−1𝑝(𝑠)𝑓𝑥_𝑘(𝑠−𝜎)−𝑓(𝑥(𝑠−𝜎))𝑑𝑠. (2.13) According to (2.8) we get ^∞_𝑡1𝑠−𝑡₁^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠<∞. (2.14) Since |𝑓(𝑥_𝑘(𝑠−𝜎))−𝑓(𝑥(𝑠−𝜎))|→0 as 𝑘→∞, by applying the Lebesgue dominated convergence theorem we obtain that lim_𝑘→∞𝑆₂𝑥_𝑘(𝑡)−𝑆₂𝑥(𝑡)=0. (2.15) This means that 𝑆₂ is continuous.
We now show that 𝑆₂Ω is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞). The uniform boundedness follows from the definition of Ω. For the equicontinuity we only need to show, according to Levitan result [8], that for any given 𝜀>0 the interval [𝑡₀,∞) can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have change of amplitude less than 𝜀. With regard to the condition (2.14), for 𝑥∈Ω and any 𝜀>0 we take 𝑡^∗≥𝑡₁ large enough so that1(𝑛−1)!^∞_𝑡^∗𝑠−𝑡₁^𝑛−1𝑝(𝑠)𝑓(𝑥(𝜀𝑠−𝜎))𝑑𝑠<2. (2.16) Then for 𝑥∈Ω,𝑇₂>𝑇₁≥𝑡^∗ we have 𝑆₂𝑥𝑇₂−𝑆₂𝑥𝑇₁≤1(𝑛−1)!^∞_𝑇2𝑠−𝑡₁^𝑛−1𝑝(𝑠)𝑓(𝑥(1𝑠−𝜎))𝑑𝑠+(𝑛−1)!^∞_𝑇1𝑠−𝑡₁^𝑛−1𝑝(𝑠)𝑓(𝑥(<𝜀𝑠−𝜎))𝑑𝑠2+𝜀2=𝜀. (2.17) For 𝑥∈Ω, 𝑡₁≤𝑇₁<𝑇₂≤𝑡^∗ and 𝑛≥2 we get 𝑆₂𝑥𝑇₂−𝑆₂𝑥𝑇₁=1(𝑛−1)!^∞_𝑇1𝑠−𝑇₁^𝑛−1𝑝(𝑠)𝑓(𝑥(−𝑠−𝜎))𝑑𝑠_∞𝑇2𝑠−𝑇₂^𝑛−1𝑝(𝑠)𝑓(𝑥(=1𝑠−𝜎))𝑑𝑠(𝑛−1)!^𝑇2_𝑇1𝑠−𝑇₁^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠+^∞_𝑇2𝑠−𝑇₁^𝑛−1𝑝(𝑠)𝑓(𝑥(−𝑠−𝜎))𝑑𝑠_∞𝑇2𝑠−𝑇₂^𝑛−1𝑝(𝑠)𝑓(𝑥(≤1𝑠−𝜎))𝑑𝑠(𝑛−1)!^𝑇2_𝑇1𝑠^𝑛−1𝑝(𝑠)𝑓(𝑥(+1𝑠−𝜎))𝑑𝑠(𝑛−1)!^∞_𝑇2𝑠−𝑇₁^𝑛−1−𝑠−𝑇₂^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠≤max_{𝑡1≤𝑠≤𝑡∗}1()!𝑠𝑛−1^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑇₂−𝑇₁+1(𝑛−1)!^∞_𝑇2𝑠−𝑇₁−𝑠−𝑇₂×𝑠−𝑇₁^𝑛−2+𝑠−𝑇₁^𝑛−3𝑠−𝑇₂+⋯+𝑠−𝑇₁𝑠−𝑇₂^𝑛−3+𝑠−𝑇₂^𝑛−2𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠≤max_{𝑡1≤𝑠≤𝑡∗}1()!𝑠𝑛−1^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑇₂−𝑇₁+1(𝑛−2)!^∞_𝑇2𝑇₂−𝑇₁𝑠−𝑇₁^𝑛−2𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠. (2.18) With regard to the condition (2.14) we have that 1(𝑛−2)!^∞_𝑇2𝑠−𝑇₁^𝑛−2𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠<𝐵,𝐵>0. (2.19) Then we obtain 𝑆₂𝑥𝑇₂−𝑆₂𝑥𝑇₁<max_{𝑡1≤𝑠≤𝑡∗}1()!𝑠𝑛−1^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))+𝐵𝑇₂−𝑇₁. (2.20) Thus there exists a 𝛿₁=𝜀/(𝑀+𝐵), where 𝑀=max_{𝑡1≤𝑠≤𝑡∗}1()!𝑠𝑛−1^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎)), (2.21) such that 𝑆₂𝑥𝑇₂−𝑆₂𝑥𝑇₁<𝜀if0<𝑇₂−𝑇₁<𝛿₁. (2.22) For 𝑛=1 we proceed by the similar way as above. Finally for any 𝑥∈Ω, 𝑡₀≤𝑇₁<𝑇₂≤𝑡₁ there exists a 𝛿₂>0 such that 𝑆₂𝑥𝑇₂−𝑆₂𝑥𝑇₁=𝑣𝑇₁−𝑣𝑇₂=^𝑇2_𝑇1𝑣^′(𝑠)𝑑𝑠≤max_{𝑡0≤𝑠≤𝑡1}𝑣^′(𝑠)𝑇₂−𝑇₁<𝜀if0<𝑇₂−𝑇₁<𝛿₂. (2.23) Then {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞) and hence 𝑆₂Ω is relatively compact subset of 𝐶([𝑡₀,∞),𝑅). By Lemma 1.1 there is an 𝑥₀∈Ω such that 𝑆₁𝑥₀+𝑆₂𝑥₀=𝑥₀. We conclude that 𝑥₀(𝑡) is a positive solution of (1.1). The proof is complete.

Corollary 2.2. Suppose that all conditions of Theorem 2.1 are satisfied andlim_𝑡→∞𝑣(𝑡)=0. (2.24) Then (1.1) has a positive solution which tends to zero.

Corollary 2.3. Suppose that there exist bounded functions 𝑢,𝑣∈𝐶¹([𝑡₀,∞),(0,∞)), constant 𝑐>0 and 𝑡₁≥𝑡₀+𝑚 such that (2.1), (2.3) hold and𝑣^′(𝑡)−𝑢^′(𝑡)≤0,𝑡₀≤𝑡≤𝑡₁. (2.25) Then (1.1) has a positive solution which is bounded by the functions 𝑢,𝑣.

Proof. We only need to prove that condition (2.25) implies (2.2). Let 𝑡∈[𝑡₀,𝑡₁] and set 𝐻(𝑡)=𝑣(𝑡)−𝑣𝑡₁−𝑢(𝑡)+𝑢𝑡₁. (2.26) Then with regard to (2.25), it follows that 𝐻′(𝑡)=𝑣′(𝑡)−𝑢′(𝑡)≤0, 𝑡₀≤𝑡≤𝑡₁. Since 𝐻(𝑡₁)=0 and 𝐻′(𝑡)≤0 for 𝑡∈[𝑡₀,𝑡₁], this implies that 𝐻(𝑡)=𝑣(𝑡)−𝑣𝑡₁−𝑢(𝑡)+𝑢𝑡₁≥0,𝑡₀≤𝑡≤𝑡₁. (2.27) Thus all conditions of Theorem 2.1 are satisfied.

Corollary 2.4. Suppose that there exists a bounded function 𝑣∈𝐶¹([𝑡₀,∞),(0,∞)), constant 𝑐>0 and 𝑡₁≥𝑡₀+𝑚 such that𝑎(𝑡)=1𝑣()𝑣(𝑡)+1𝑡−𝜏(𝑛−1)!^∞_𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≤𝑐<1,𝑡≥𝑡₁. (2.28) Then (1.1) has a solution 𝑥(𝑡)=𝑣(𝑡),𝑡≥𝑡₁.

Proof. We put 𝑢(𝑡)=𝑣(𝑡) and apply Theorem 2.1.

Theorem 2.5. Suppose that 𝑝 is bounded and there exist bounded functions 𝑢,𝑣∈𝐶¹([𝑡₀,∞),(0,∞)), constant 𝑐>0 and 𝑡₁≥𝑡₀+𝑚 such that (2.1), (2.2) hold and 1𝑢()𝑢(𝑡)−1𝑡−𝜏(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑢((𝑡)≤1𝑠−𝜎))𝑑𝑠≤𝑎𝑣()𝑣(𝑡)−1𝑡−𝜏(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≤𝑐<1,𝑡≥𝑡₁, (2.29) if 𝑛 is odd, 1𝑢()𝑢(𝑡)+1𝑡−𝜏(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑣((𝑡)≤1𝑠−𝜎))𝑑𝑠≤𝑎𝑣()𝑣(𝑡)+1𝑡−𝜏(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑢(𝑠−𝜎))𝑑𝑠≤𝑐<1,𝑡≥𝑡₁, (2.30) if 𝑛 is even, and ^𝑡_𝑡1(𝑡−𝑠)^𝑛−2𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≤𝐾,𝑡≥𝑡₁,𝐾>0,𝑛≥2. (2.31) Then (1.1) has a positive solution which is bounded by the functions 𝑢, 𝑣.

Proof. Let 𝐶([𝑡₀,∞),𝑅) be the set as in the proof of Theorem 2.1. We define a closed, bounded, and convex subset Ω of 𝐶([𝑡₀,∞),𝑅) as in the proof of Theorem 2.1. We define two maps 𝑆₁ and 𝑆₂∶Ω→𝐶([𝑡₀,∞),𝑅) as follows: 𝑆₁𝑥𝑡)=𝑎(𝑡)𝑥()(𝑡−𝜏,𝑡≥𝑡₁,𝑆₁𝑥𝑡₁,𝑡₀≤𝑡≤𝑡₁,𝑆₂𝑥(𝑡)=(−1)^𝑛+1(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑥(𝑠−𝜎))𝑑𝑠,𝑡≥𝑡₁,𝑆₂𝑥𝑡₁+𝑣(𝑡)−𝑣𝑡₁,𝑡₀≤𝑡≤𝑡₁. (2.32) We shall show that for any 𝑥,𝑦∈Ω we have 𝑆₁𝑥+𝑆₂𝑦∈Ω. For 𝑛 odd, every 𝑥,𝑦∈Ω and 𝑡≥𝑡₁ we obtain 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)(𝑡)𝑣()+1≤𝑎𝑡−𝜏(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≤𝑣(𝑡). (2.33) For 𝑡∈[𝑡₀,𝑡₁], we have 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)=𝑆₁𝑥𝑡₁+𝑆₂𝑦𝑡₁+𝑣(𝑡)−𝑣𝑡₁≤𝑣𝑡₁+𝑣(𝑡)−𝑣𝑡₁=𝑣(𝑡). (2.34) Furthermore for 𝑡≥𝑡₁, we get 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)(𝑡)𝑢()+1≥𝑎𝑡−𝜏(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑢(𝑠−𝜎))𝑑𝑠≥𝑢(𝑡). (2.35) Let 𝑡∈[𝑡₀,𝑡₁] and according to (2.2) we have 𝑣(𝑡)−𝑣𝑡₁+𝑢𝑡₁≥𝑢(𝑡). (2.36) Then for 𝑡∈[𝑡₀,𝑡₁] and any 𝑥,𝑦∈Ω we get 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)=𝑆₁𝑥𝑡₁+𝑆₂𝑦𝑡₁+𝑣(𝑡)−𝑣𝑡₁≥𝑢𝑡₁+𝑣(𝑡)−𝑣𝑡₁≥𝑢(𝑡). (2.37) Thus we have proved that 𝑆₁𝑥+𝑆₂𝑦∈Ω for any 𝑥,𝑦∈Ω.
For 𝑛 even by the similar way as above we can prove that 𝑆₁𝑥+𝑆₂𝑦∈Ω for any 𝑥,𝑦∈Ω.
As in the proof of Theorem 2.1, we can show that 𝑆₁ is a contraction mapping on Ω.
We now show that 𝑆₂ is completely continuous. First, we will show that 𝑆₂ is continuous. Let 𝑥_𝑘=𝑥_𝑘(𝑡)∈Ω be such that 𝑥_𝑘(𝑡)→𝑥(𝑡) as 𝑘→∞. Because Ω is closed, 𝑥=𝑥(𝑡)∈Ω. For 𝑡≥𝑡₁ we have 𝑆₂𝑥_𝑘(𝑡)−𝑆₂𝑥(𝑡≤1)(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓𝑥_𝑘(𝑠−𝜎)−𝑓(𝑥(𝑠−𝜎))𝑑𝑠. (2.38) According to (2.33) there exists a positive constant 𝑀 such that ^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≤𝑀for𝑡≥𝑡₁. (2.39) The inequality above also holds for 𝑛 even.
Since |𝑓(𝑥_𝑘(𝑠−𝜎))−𝑓(𝑥(𝑠−𝜎))|→0 as 𝑘→∞, by applying the Lebesgue dominated convergence theorem we obtain thatlim_𝑘→∞𝑆₂𝑥_𝑘(𝑡)−𝑆₂𝑥(𝑡)=0. (2.40) This means that 𝑆₂ is continuous.
We now show that 𝑆₂Ω is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞). The uniform boundedness follows from the definition of Ω. For 𝑛≥2 and with regard to (2.31) we have𝑑𝑆𝑑𝑡²𝑥(𝑡=1)(𝑛−2)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−2𝑝(𝑠)𝑓(𝑥(≤1𝑠−𝜎))𝑑𝑠(𝑛−2)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−2𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))≤𝑀₁, (2.41) and for 𝑛=1 we obtain 𝑑𝑆𝑑𝑡²𝑥(𝑡)=𝑝(𝑡)𝑓(𝑣(𝑡−𝜎))≤𝑀₂, (2.42) for 𝑡≥𝑡₁, 𝑀₂>0 and |(𝑑/𝑑𝑡)(𝑆₂𝑥)(𝑡)|=|𝑣′(𝑡)|≤𝑀₃ for 𝑡₀≤𝑡≤𝑡₁, 𝑀₃>0, which shows the equicontinuity of the family 𝑆₂Ω, (cf. [7, page 265]). Hence 𝑆₂Ω is relatively compact and therefore 𝑆₂ is completely continuous. By Lemma 1.1, there is 𝑥₀∈Ω such that 𝑆₁𝑥₀+𝑆₂𝑥₀=𝑥₀. Thus 𝑥₀(𝑡) is a positive solution of (1.1). The proof is complete.

Corollary 2.6. Suppose that all conditions of Theorem 2.5 are satisfied andlim_𝑡→∞𝑣(𝑡)=0. (2.43) Then (1.1) has a positive solution which tends to zero.

Corollary 2.7. Suppose that 𝑝 is bounded and there exist bounded functions 𝑢,𝑣∈𝐶¹([𝑡₀,∞),(0,∞)), constant 𝑐>0 and 𝑡₁≥𝑡₀+𝑚 such that (2.1), (2.29), (2.30), (2.31) hold and𝑣^′(𝑡)−𝑢^′(𝑡)≤0,𝑡₀≤𝑡≤𝑡₁. (2.44) Then (1.1) has a positive solution which is bounded by the functions 𝑢,𝑣.

Proof. The proof is similar to that of Corollary 2.3 and we omit it.

Corollary 2.8. Suppose that 𝑝 is bounded and there exists a bounded function 𝑣∈𝐶¹([𝑡₀,∞),(0,∞)), constant 𝑐>0 and 𝑡₁≥𝑡₀+𝑚 such that (2.31) holds and𝑎(𝑡)=1𝑣()𝑣(𝑡)+()𝑡−𝜏−1^𝑛(𝑛−1)!^𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝑝(𝑠)𝑓(𝑣(𝑠−𝜎))𝑑𝑠≤𝑐<1,𝑡≥𝑡₁. (2.45) Then (1.1) has a solution 𝑥(𝑡)=𝑣(𝑡),𝑡≥𝑡₁.

Proof. We put 𝑢(𝑡)=𝑣(𝑡) and apply Theorem 2.5.

3. Applications and Examples

In this section, we give some applications of the theorems above.

Theorem 3.1. Suppose that 0<𝑘₁≤𝑘₂ and there exist 𝛾≥0, 𝑐>0, 𝑡₁≥𝑡₀+𝑚 such that 𝑘₁𝑘₂exp𝑘₂−𝑘₁^𝑡0_𝑡0−𝛾𝑝(𝑡)𝑑𝑡≥1,exp−𝑘₂^𝑡_𝑡−𝜏𝑝(𝑠)+1𝑑𝑠()!𝑘𝑛−1exp_2𝑡^𝑡−𝜏_0−𝛾𝑝(𝑠)×𝑑𝑠_∞𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓exp−𝑘_1𝑡^𝑠−𝜎_0−𝛾𝑝(𝜉)𝑑𝜉𝑑𝑠≤𝑎(𝑡)≤exp−𝑘₁^𝑡_𝑡−𝜏𝑝(𝑠)+1𝑑𝑠()!𝑘𝑛−1exp_1𝑡^𝑡−𝜏_0−𝛾𝑝(𝑠)×𝑑𝑠_∞𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓exp−𝑘_2𝑡^𝑠−𝜎_0−𝛾𝑝(𝜉)𝑑𝜉𝑑𝑠≤𝑐<1,𝑡≥𝑡₁. (3.1)(3.2) Then (1.1) has a positive solution which is bounded by two exponential functions.

Proof. We set 𝑢(𝑡)=exp−𝑘₂^𝑡_𝑡0−𝛾𝑝(𝑠)𝑑𝑠,𝑣(𝑡)=exp−𝑘₁^𝑡_𝑡0−𝛾𝑝(𝑠)𝑑𝑠,𝑡≥𝑡₀. (3.3) We will show that the conditions of Corollary 2.3 are satisfied. With regard to (3.1) for 𝑡∈[𝑡₀,𝑡₁] we get 𝑣^′(𝑡)−𝑢^′(𝑡)=−𝑘₁𝑝(𝑡)𝑣(𝑡)+𝑘₂𝑝(𝑡)𝑢(𝑡)=𝑝(𝑡)𝑣(𝑡)−𝑘₁+𝑘₂𝑢(𝑡)exp𝑘₁^𝑡_𝑡0−𝛾𝑝(𝑠)(𝑡)𝑣(𝑡𝑑𝑠=𝑝)−𝑘₁+𝑘₂exp𝑘₁−𝑘₂^𝑡_𝑡0−𝛾𝑝(𝑠)(𝑡)𝑣(𝑡𝑑𝑠≤𝑝)−𝑘₁+𝑘₂exp𝑘₁−𝑘₂^𝑡0_𝑡0−𝛾𝑝(𝑠)𝑑𝑠≤0. (3.4) Other conditions of Corollary 2.3 are also satisfied. The proof is complete.

Corollary 3.2. Suppose that all conditions of Theorem 3.1 are satisfied and^∞_𝑡0𝑝(𝑡)𝑑𝑡=∞. (3.5) Then (1.1) has a positive solution which tends to zero.

Corollary 3.3. Suppose that 𝑘>0,𝑐>0,𝑡₁≥𝑡₀+𝑚 and 𝑎(𝑡)=exp−𝑘^𝑡_𝑡−𝜏𝑝(𝑠)+1𝑑𝑠()!𝑘𝑛−1exp_𝑡^𝑡−𝜏₀𝑝(𝑠)×𝑑𝑠_∞𝑡(𝑠−𝑡)^𝑛−1𝑝(𝑠)𝑓exp−𝑘_𝑡^𝑠−𝜎₀𝑝(𝜉)𝑑𝜉𝑑𝑠≤𝑐<1,𝑡≥𝑡₁. (3.6) Then (1.1) has a solution: 𝑥(𝑡)=exp−𝑘^𝑡_𝑡0𝑝(𝑠)𝑑𝑠,𝑡≥𝑡₁. (3.7)

Proof. We put 𝑘₁=𝑘₂=𝑘,𝛾=0 and apply Theorem 3.1.

Example 3.4. Consider the nonlinear neutral differential equation: [𝑥(𝑡)−𝑎(𝑡)𝑥(𝑡−2)]^′=𝑝𝑥³(𝑡−1),𝑡≥𝑡₀, (3.8) where 𝑝∈(0,∞). We will show that the conditions of Theorem 3.1 are satisfied. The condition (3.1) has a form: 𝑘₁𝑘₂exp𝑘₂−𝑘₁𝑝𝛾≥1, (3.9)0<𝑘₁≤𝑘₂,𝛾≥0. For function 𝑎(𝑡), we obtain exp−2𝑝𝑘₂+13𝑘₁exp𝑝𝑘₂𝛾−𝑡₀−2−3𝑘₁𝛾−𝑡₀−1+𝑘₂−3𝑘₁𝑡≤𝑎(𝑡)≤exp−2𝑝𝑘₁+13𝑘₂exp𝑝𝑘₁𝛾−𝑡₀−2−3𝑘₂𝛾−𝑡₀−1+𝑘₁−3𝑘₂𝑡,𝑡≥𝑡₀. (3.10) For 𝑝=1,𝑘₁=1,𝑘₂=2,𝛾=1,𝑡₀=1, the condition (3.9) is satisfied and 𝑒⁻⁴+1𝑒3𝑒^−𝑡≤𝑎(𝑡)≤𝑒⁻²+𝑒⁴6𝑒^−5𝑡,𝑡≥𝑡₁≥3. (3.11) If the function 𝑎(𝑡) satisfies (3.11), then (3.8) has a solution which is bounded by the functions 𝑢(𝑡)=exp(−2𝑡),𝑣(𝑡)=exp(−𝑡),𝑡≥3.

Example 3.5. Consider the nonlinear differential equation: [𝑥(𝑡)−𝑎(𝑡)𝑥(𝑡−𝜋)]^′=𝑝(𝑡)𝑓(𝑥(𝑡−𝜋)),𝑡≥0, (3.12) where 𝑓(𝑥)=√𝑥,𝑥>0,𝑝(𝑡)=0.8exp(𝜋−𝑡+0.05cos𝑡),𝑡≥0, and 𝑒^0.1cos𝑡𝑒^0.1cos𝑡+0.8𝑒^𝜋−𝑡−1≤𝑎(𝑡)≤𝑒^0.1cos𝑡𝑒^0.1cos𝑡+0.8√𝑏𝑒^𝜋−𝑡−1<1, (3.13) for 𝑡≥𝜋,𝑏∈[1,2]. Set 𝑢(𝑡)=𝑒^0.1cos𝑡,𝑣(𝑡)=𝑏𝑒^0.1cos𝑡,𝑡≥0. (3.14) Then we have 𝑣^′(𝑡)−𝑢^′(𝑡)=−0.1𝑏𝑒^0.1cos𝑡sin𝑡+0.1𝑒^0.1cos𝑡()𝑒sin𝑡=−0.1𝑏−1^0.1cos𝑡sin𝑡≤0for𝑡∈[0,𝜋]. (3.15) By Corollary 2.7, (3.12) has a solution which is bounded by the functions 𝑒^0.1cos𝑡 and 𝑏𝑒^0.1cos𝑡,𝑡≥𝜋. If 𝑎(𝑡)=𝑒^0.1cos𝑡𝑒^0.1cos𝑡+0.8𝑒^𝜋−𝑡−1for𝑡≥𝜋, (3.16) then (3.12) has the positive periodic solution 𝑥(𝑡)=𝑢(𝑡)=𝑒^0.1cos𝑡,𝑡≥𝜋.

Acknowledgment

The research was supported by the Grants 1/0090/09 and 1/1260/12 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic.