Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations

Abstract

This paper deals with the solvability of the higher-order nonlinear neutral delay differential equation $\frac{d^n}{d{t}^n}[x(t)+p(t)x(t-\tau )]+{(-1)}^{n+1}{\sum }_{i=1}^m{q}_i(t)x({\alpha }_i(t))+{(-1)}^{n+1}f(t,x({\beta }_1(t)),\dots ,x({\beta }_l(t)))=r(t)$, $t\ge t_0$, where $\tau >0$, $n,m,l\in \mathbb{N}$, $p,r,q_i,{\alpha }_i,{\beta }_j\in C([t_0,+\mathrm{\infty }),\mathbb{R})$, and $f\in C([t_0,+\mathrm{\infty })\times {\mathbb{R}}^l,\mathbb{R})$ satisfying ${lim}_{t\to +\mathrm{\infty }}{\alpha }_i(t)={lim}_{t\to +\mathrm{\infty }}{\beta }_j(t)=+\mathrm{\infty }$, $i\in \{1,2,\dots ,m\}$, $j\in \{1,2,\dots ,l\}$. With respect to various ranges of the function p, we investigate the existence of uncountably many bounded nonoscillatory solutions for the equation. The main tools used in this paper are the Krasnoselskii and Schauder fixed point theorems together with some new techniques. Six nontrivial examples are given to illustrate the superiority of the results presented in this paper.

MSC:39A10, 39A20, 39A22.

1 Introduction and preliminaries

In the past two decades, the oscillation, nonoscillation, and existence of solutions for some kinds of neutral delay differential equations have been extensively studied by many authors. See, for example, [1–16] and the references cited therein.

Recently, Zhang et al. [14] and Öcalan [10] got several existence results of a nonoscillatory solution or positive solution for the first-order neutral delay differential equations

$$\frac{d}{dt}[x(t)+c(t)x(t-\tau )]+P(t)x(t-\alpha )-Q(t)x(t-\beta )=0,t\ge t_0$$

(1.1)

and

$$\frac{d}{dt}[x(t)+c(t)x(t-\tau )]+\sum _{i=1}^m{A}_i(t)x(t-{\sigma }_i)-\sum _{i=m+1}^n{A}_i(t)x(t-{\sigma }_i)=0,t\ge t_0,$$

(1.2)

where $\tau >0$, $\alpha ,\beta ,{\sigma }_i\in {\mathbb{R}}^{+}$, $c\in C([t_0,+\mathrm{\infty }),\mathbb{R})$ and $P,Q,A_i\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ for $i\in \{1,2,\dots ,n\}$. Shen and Debnath [12] obtained some sufficient conditions for the oscillations of (1.1) and Luo and Shen [9] established a few oscillation and nonoscillation criteria for (1.2). Liu and Huang [6] used the coincidence degree theory to get the existence and uniqueness results of a T-periodic solution for the first-order neutral functional differential equation with a deviating argument of the form

$$\frac{d}{dt}[x(t)+cx(t-\tau )]=g_1(t,x(t))+g_2(t,x(t-\alpha (t)))+f(t),$$

(1.3)

where c, τ are constants, $c\ne \pm 1$, $f,\alpha \in C(\mathbb{R},\mathbb{R})$, $g_1,g_2\in C({\mathbb{R}}^2,\mathbb{R})$, f, α are T-periodic and $g_1$, $g_2$ are T-periodic in the first argument. Using the Banach fixed point theorem, Kulenović and Hadžiomerspahić [4] studied the existence of a nonoscillatory solution for the second-order neutral delay differential equation with positive and negative coefficients

$$\frac{d^2}{d{t}^2}[x(t)+cx(t-\tau )]+P(t)x(t-\sigma )-Q(t)x(t-\delta )=0,t\ge t_0,$$

(1.4)

where $c\in \mathbb{R}\setminus \{\pm 1\}$, $\sigma ,\delta \in {\mathbb{R}}^{+}$ and $P,Q\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$. Kong et al. [3] established a complete classification of nonoscillatory solutions for the higher-order neutral differential equation

$$\frac{d^n}{d{t}^n}[x(t)-x(t-\tau )]+f(t)x(t-\alpha )=0,t\ge t_0,$$

(1.5)

and gave conditions for each type of nonoscillatory solutions to exist, where n is an odd number, $\tau >0$, $\alpha \in \mathbb{R}$, and $f\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$. Zhou and Zhang [16] extended the results in [5] to higher-order neutral functional differential equation with positive and negative coefficients

$$\frac{d^n}{d{t}^n}[x(t)+cx(t-\tau )]+{(-1)}^{n+1}[P(t)x(t-\sigma )-Q(t)x(t-\delta )]=0,t\ge t_0,$$

(1.6)

where $c\in \mathbb{R}\setminus \{\pm 1\}$, $\tau ,\sigma ,\delta \in {\mathbb{R}}^{+}$ and $P,Q\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$. Liu et al. [8] investigated the higher-order neutral delay differential equation with positive and negative coefficients

$$\frac{d^n}{d{t}^n}[x(t)+cx(t-\tau )]+{(-1)}^n[P(t)x(f(t))-Q(t)x(g(t))]=0,t\ge t_0,$$

(1.7)

where $c\in \mathbb{R}\setminus \{-1\}$, $\tau \in {\mathbb{R}}^{+}$, $P,Q\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$, $f,g\in C([t_0,+\mathrm{\infty }),\mathbb{R})$, and ${lim}_{t\to +\mathrm{\infty }}f(t)={lim}_{t\to +\mathrm{\infty }}g(t)=+\mathrm{\infty }$. Utilizing the Banach fixed point theorem, they obtained the existence of bounded nonoscillatory solutions for (1.7), suggested some algorithms for approximating these bounded nonoscillatory solutions, and discussed the convergence and stability of iteration sequences generated by the algorithms. Parhi [11] discussed the oscillation of solutions for the higher-order neutral delay linear differential equation

$$\frac{d^n}{d{t}^n}[x(t)+cx(t-\tau )]+\sum _{i=1}^m{q}_i(t)x(t-{\alpha }_i(t))=0,t\ge 0,$$

(1.8)

where $c\in [0,1)$, $\tau >0$, $q_i\in C({\mathbb{R}}^{+},\mathbb{R})$, and ${\alpha }_i\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ for $i\in \{1,2,\dots ,m\}$. Li et al. [5] considered the following higher-order neutral delay differential equation with unstable type:

$$\frac{d^n}{d{t}^n}[x(t)+c(t)x(t-\tau )]=q(t){|x(t-\sigma )|}^{\alpha -1}x(t-\sigma ),t\ge 0,$$

(1.9)

proved bounded oscillation and nonoscillation criteria and the existence of an unbounded positive solution for (1.9), where n is an even integer, $\alpha \ge 1$, $\tau >0$, $\sigma >0$, $c,q\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$. Zhou and Zhang [15] used the Krasnoselskii and Schauder fixed point theorems to prove the existence of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation

$$\frac{d^n}{d{t}^n}[x(t)+c(t)x(t-\tau )]+\sum _{i=1}^m{q}_i(t)f(x(t-{\sigma }_i))=g(t),t\ge t_0,$$

(1.10)

where $\tau ,{\sigma }_i\in {\mathbb{R}}^{+}$, $c,q_i,g\in C([t_0,+\mathrm{\infty }),\mathbb{R})$ for $i\in \{1,2,\dots ,m\}$ and $f\in C(\mathbb{R},\mathbb{R})$. Liu et al. [7] got the existence of infinitely many nonoscillatory solutions for the n th-order neutral delay differential equation

$$\frac{d^n}{d{t}^n}[x(t)+cx(t-\tau )]+{(-1)}^{n+1}f(t,x(t-{\sigma }_1),x(t-{\sigma }_2),\dots ,x(t-{\sigma }_k))=g(t),t\ge t_0,$$

(1.11)

where $c\in \mathbb{R}\setminus \{-1\}$, $\tau >0$, ${\sigma }_i\in {\mathbb{R}}^{+}$ for $i\in \{1,2,\dots ,k\}$, $f\in C([t_0,+\mathrm{\infty })\times {\mathbb{R}}^k,\mathbb{R})$, and $g\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$.

However, to the best of our knowledge, there exist no results for the existence of solutions of the higher-order nonlinear neutral delay differential equation

$$\begin{array}{c}\frac{d^n}{d{t}^n}[x(t)+p(t)x(t-\tau )]+{(-1)}^{n+1}\sum _{i=1}^m{q}_i(t)x({\alpha }_i(t))\hfill \\ +{(-1)}^{n+1}f(t,x({\beta }_1(t)),\dots ,x({\beta }_l(t)))=r(t),t\ge t_0,\hfill \end{array}$$

(1.12)

where $\tau >0$, $n,m,l\in \mathbb{N}$, $p,r,q_i,{\alpha }_i,{\beta }_j\in C([t_0,+\mathrm{\infty }),\mathbb{R})$, and $f\in C([t_0,+\mathrm{\infty })\times {\mathbb{R}}^l,\mathbb{R})$ satisfying

$$\underset{t\to +\mathrm{\infty }}{lim}{\alpha }_i(t)=\underset{t\to +\mathrm{\infty }}{lim}{\beta }_j(t)=+\mathrm{\infty },i\in \{1,2,\dots ,m\},j\in \{1,2,\dots ,l\}.$$

It is clear that (1.12) includes (1.1)-(1.11) as special cases. The purpose of this paper is to study the solvability of (1.12) under various ranges of the function p. Utilizing the Krasnoselskii and Schauder fixed point theorems and some new techniques, we study sufficient conditions of the existence of uncountably many bounded nonoscillatory solutions for (1.12) relative to various ranges of the function p. The results presented in this paper extend, improve, and unify the corresponding results in [4] and [13–16]. Six nontrivial examples are also given to illustrate the importance of the results obtained in this paper.

Throughout this paper, we assume that ℝ, ${\mathbb{R}}^{+}$ and ℕ denote the sets of all real numbers, nonnegative numbers and positive integers, respectively, and

$$\gamma =min\{t_0-\tau ,inf\{{\alpha }_i(t),{\beta }_j(t):t\in [t_0,+\mathrm{\infty }),i\in \{1,2,\dots ,m\},j\in \{1,2,\dots ,l\}\}\}.$$

Let $CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ stand for the Banach space of all continuous and bounded functions on $[\gamma ,+\mathrm{\infty })$ with norm $\parallel x\parallel ={sup}_{t\ge \gamma }|x(t)|$ for all $x\in CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ and

$$A(N,M)=\{x\in CB([\gamma ,+\mathrm{\infty }),\mathbb{R}):N\le x(t)\le M,t\ge \gamma \},$$

where $M,N\in \mathbb{R}$ with $M>N$. It is easy to see that $A(N,M)$ is a nonempty bounded closed convex subset of $CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$.

By a solution of (1.12), we mean a function $x\in C([\gamma ,+\mathrm{\infty }),\mathbb{R})$ for some $T\ge t_0$, such that $x(t)+p(t)x(t-\tau )$ is n times continuously differentiable on $[T,+\mathrm{\infty })$ and (1.12) holds for $t\ge T$. As is customary, a solution of (1.12) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise.

The following lemmas are well known.

Lamma 1.1 (Krasnoselskii fixed point theorem [1])

Let X be a nonempty bounded closed convex subset of a Banach space E and let U, S be maps of X into E such that $Ux+Sy\in X$ for every pair $x,y\in X$. If U is a contraction and S is completely continuous, then the equation $Ux+Sy=x$ has a solution in X.

Lamma 1.2 (Schauder fixed point theorem [1])

Let X be a nonempty closed convex subset of a Banach space E. Let $S:X\to X$ be a continuous mapping such that SX is a relatively compact subset of X. Then S has at least one fixed point in X.

2 The existence of uncountably many bounded nonoscillatory solutions

Now we investigate sufficient conditions for the existence of uncountably many bounded nonoscillatory solutions of (1.12) under various ranges of the function p. The proofs of the results presented in this section are based on the Krasnoselskii and Schauder fixed point theorems and a few new and key techniques, one of which is to construct the mappings $U_L$ and $S_L$ satisfying the conditions in the cited fixed point theorems for each constant L, which belongs to certain interval. Let

$$H(t)=M\sum _{i=1}^m|q_i(t)|+|r(t)|+h(t),\mathrm{\forall }t\in [T,+\mathrm{\infty }).$$

Theorem 2.1 Assume that there exist $h\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ and constants M, N, $p_0$, and c satisfying

$$|f(t,u_1,\dots ,u_l)|\le h(t),\mathrm{\forall }(t,u_1,\dots ,u_l)\in [t_0,+\mathrm{\infty })\times {[N,M]}^l;$$

(2.1)

$${\int }_{t_0}^{+\mathrm{\infty }}{s}^{n-1}max\left\{\sum _{i=1}^m\right|q_i(s)|,|r(s)|,h(s)\}ds<+\mathrm{\infty };$$

(2.2)

$$0<N<(1+p_0)M,-1<p_0\le p(t)\le 0,\mathrm{\forall }t\ge c\ge t_0.$$

(2.3)

Then (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$.

Proof Let $L\in (N,(1+p_0)M)$. It follows from (2.2) and (2.3) that there exist constants $\theta \in (0,1)$ and $T>|t_0|+\tau +|c|+|\gamma |+n$ sufficiently large satisfying

$$\theta =|p_0|+\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}\sum _{i=1}^m|q_i(s)|ds$$

(2.4)

and

$$\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\le min\{(p_0+1)M-L,L-N\}.$$

(2.5)

Define two mappings $U_L$ and $S_L:A(N,M)\to CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ by

$$\begin{array}{r}(U_{L}x)(t)=\{\begin{array}{c}L-p(t)x(t-\tau )+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}\hfill \\ \times ({\sum }_{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds,\hfill & t\ge T,\hfill \\ (U_{L}x)(T),\hfill & \gamma \le t<T;\hfill \end{array}\\ (S_{L}x)(t)=\{\begin{array}{cc}\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds,\hfill & t\ge T,\hfill \\ (S_{L}x)(T),\hfill & \gamma \le t<T,\hfill \end{array}\end{array}$$

(2.6)

for each $x\in A(N,M)$.

First of all we show that

$$U_{L}x+S_{L}y\in A(N,M),\parallel U_{L}x-U_{L}y\parallel \le \theta \parallel x-y\parallel ,\mathrm{\forall }x,y\in A(N,M).$$

(2.7)

Let $x,y\in A(N,M)$ and $t\ge T$. By (2.3)-(2.6), we get

$$\begin{array}{c}(U_{L}x)(t)+(S_{L}y)(t)\hfill \\ =L-p(t)x(t-\tau )+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ +\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))ds\hfill \\ \le L+|p(t)||x(t)|\hfill \\ +\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|\hfill \\ +\left|f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))\right|)ds\hfill \\ \le L-p_{0}M+\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \le L-p_{0}M+min\{(p_0+1)M-L,L-N\}\hfill \\ \le M,\hfill \\ (U_{L}x)(t)+(S_{L}y)(t)\hfill \\ \ge L-\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|\hfill \\ +\left|f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))\right|)ds\hfill \\ \ge L-\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \ge L-min\{(p_0+1)M-L,L-N\}\hfill \\ \ge N\hfill \end{array}$$

and

$$\begin{array}{c}|(U_{L}x)(t)-(U_{L}y)(t)|\hfill \\ =|L-p(t)x(t-\tau )+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ -L+p(t)y(t-\tau )-\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)y({\alpha }_i(s))+{(-1)}^{n}r(s))ds|\hfill \\ \le -p_0\parallel x-y\parallel +\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}\sum _{i=1}^m|q_i(s)||x({\alpha }_i(s))-y({\alpha }_i(s))|ds\hfill \\ \le \theta \parallel x-y\parallel ,\hfill \end{array}$$

which imply (2.7).

Second, we show that $S_L$ is continuous in $A(N,M)$ and $S_L(A(N,M))$ is relatively compact. Let ${\{x_k\}}_{k\in \mathbb{N}}$ be an arbitrary sequence in $A(N,M)$ with

$$\underset{k\to \mathrm{\infty }}{lim}{x}_k=x\in CB([\gamma ,+\mathrm{\infty }),\mathbb{R}).$$

(2.8)

Since $A(N,M)$ is a closed subset of $CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$, it follows that $x\in A(N,M)$. Put

$$\begin{array}{c}{G}_k(s)=|f(s,x_k({\beta }_1(s)),\dots ,x_k({\beta }_l(s)))-f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))|,\hfill \\ \mathrm{\forall }(s,k)\in [T,+\mathrm{\infty })\times \mathbb{N}.\hfill \end{array}$$

From (2.8) and the continuity of f and ${\beta }_j$ for $j\in \{1,2,\dots ,l\}$ we infer that

$$\underset{k\to \mathrm{\infty }}{lim}{G}_k(s)=0,\mathrm{\forall }s\in [T,+\mathrm{\infty }),$$

which together with (2.6) and the Lebesgue dominated convergence theorem yields for any $t\in [T,+\mathrm{\infty })$

$$\begin{array}{c}|(S_L{x}_k)(t)-(S_{L}x)(t)|\hfill \\ \le \frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}|f(s,x_k({\beta }_1(s)),\dots ,x_k({\beta }_l(s)))-f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))|ds\hfill \\ \le \frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}{G}_k(s)ds\hfill \\ \to 0\text{as }k\to \mathrm{\infty },\hfill \end{array}$$

which gives

$$\underset{k\to \mathrm{\infty }}{lim}\parallel S_L{x}_k-S_{L}x\parallel =\underset{k\to \mathrm{\infty }}{lim}\underset{t\ge \gamma }{sup}|(S_L{x}_k)(t)-(S_{L}x)(t)|=\underset{k\to \mathrm{\infty }}{lim}\underset{t\ge T}{sup}|(S_L{x}_k)(t)-(S_{L}x)(t)|=0,$$

which implies that $S_L$ is continuous in $A(N,M)$.

Using (2.1), (2.5), and (2.6), we conclude that for any $x\in A(N,M)$ and $t\ge T$

$$\begin{array}{rl}|(S_{L}x)(t)|& =\left|\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\right|\\ \le \frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\\ \le min\{(p_0+1)M-L,L-N\}\\ \le M,\end{array}$$

which yields

$$\parallel S_{L}x\parallel \le M,\mathrm{\forall }x\in A(N,M).$$

(2.9)

That is, $S_L(A(N,M))$ is uniformly bounded in $[\gamma ,+\mathrm{\infty })$. In order to prove that $S_L(A(N,M))$ is relatively compact, we have to prove that $S_L(A(N,M))$ is equicontinuous in $[\gamma ,+\mathrm{\infty })$. Let $\epsilon >0$ be given. Equation (2.2) ensures that there exists $T^{\ast }>T$ satisfying

$$\frac{2}{(n-1)!}{\int }_{T^{\ast }}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds<\epsilon .$$

(2.10)

Put

$$\delta =min\{\frac{\epsilon }{1+M},\frac{\epsilon }{1+max\{h(t):t\in [T,T^{\ast }]\}}\}.$$

(2.11)

Now we consider the following possible cases:

• (i) $t_2>t_1\ge T^{\ast }$ with $|t_2-t_1|<\delta$. By (2.1), (2.6), and (2.10) we have

  $$\begin{array}{c}|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\hfill \\ =\frac{1}{(n-1)!}|{\int }_{t_2}^{+\mathrm{\infty }}{(s-t_2)}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\hfill \\ -{\int }_{t_1}^{+\mathrm{\infty }}{(s-t_1)}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))|\hfill \\ \le \frac{2}{(n-1)!}{\int }_{T^{\ast }}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\hfill \\ <\epsilon ,\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

  (2.12)

• (ii) $T\le t_1<t_2\le T^{\ast }$ with $|t_2-t_1|<\delta$ and $n=1$. By means of (2.1), (2.6), and (2.11) we infer that

  $$\begin{array}{c}|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\hfill \\ =|{\int }_{t_2}^{+\mathrm{\infty }}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds-{\int }_{t_1}^{+\mathrm{\infty }}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds|\hfill \\ \le {\int }_{t_1}^{t_2}\left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds\hfill \\ \le {\int }_{t_1}^{t_2}h(s)ds\hfill \\ <\epsilon ,\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

  (2.13)

• (iii) $T\le t_1<t_2\le T^{\ast }$ with $|t_2-t_1|<\delta$ and $n\ge 2$. In light of (2.1), (2.5), and (2.6), we get

  $$\begin{array}{rl}|\frac{d}{dt}(S_{L}x)(t)|& =\frac{1}{(n-1)!}\left|\frac{d}{dt}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\right|\\ =\left|\frac{1}{(n-2)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-2}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\right|\\ \le \frac{1}{(n-2)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-2}h(s)ds\\ \le \frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\\ \le min\{(p_0+1)M-L,L-N\}\\ \le M,\mathrm{\forall }x\in A(N,M),t\in [T,T^{\ast }],\end{array}$$

which together with the mean value theorem and (2.11) yields

$$|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\le M|t_1-t_2|<\epsilon ,\mathrm{\forall }x\in A(N,M).$$

(2.14)

• (iv) $\gamma \le t_1<t_2\le T$ with $|t_2-t_1|<\delta$. Clearly (2.6) means that

  $$|(S_{L}x)(t_2)-(S_{L}x)(t_1)|=0<\epsilon ,\mathrm{\forall }x\in A(N,M).$$

  (2.15)

It follows from (2.12)-(2.15) that $S_L(A(N,M))$ is equicontinuous in $[\gamma ,+\mathrm{\infty })$. Thus Lemma 1.1 means that there exists $x\in A(N,M)$ such that $U_{L}x+S_{L}x=x$. That is,

$$\begin{array}{rcl}x(t)& =& L-p(t)x(t-\tau )+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\\ +\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds,\mathrm{\forall }t\ge T,\end{array}$$

which implies that

$$\begin{array}{c}\frac{d^n}{d{t}^n}[x(t)+p(t)x(t-\tau )]\hfill \\ ={(-1)}^n(\sum _{i=1}^m{q}_i(t)x({\alpha }_i(t))+{(-1)}^{n}r(t))+{(-1)}^{n}f(t,x({\beta }_1(t)),\dots ,x({\beta }_l(t))),\mathrm{\forall }t\ge T,\hfill \end{array}$$

that is, $x(t)$ is a bounded nonoscillatory solution of (1.12) in $A(N,M)$.

Finally, we show (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$. Let $L_1,L_2\in (N,(1+p_0)M)$ with $L_1\ne L_2$. As in the above proof we can deduce that for each $k\in \{1,2\}$, there exist constants ${\theta }_k\in (0,1)$, $T_k>|t_0|+\tau +|c|+|\gamma |+n$, and mappings $U_{L_k},S_{L_k}:A(N,M)\to CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ satisfying (2.4)-(2.6), where θ, T, L, $U_L$ and $S_L$ are replaced by ${\theta }_k$, $T_k$, $L_k$, $U_{L_k}$, $S_{L_k}$, respectively, and $U_{L_k}+S_{L_k}$ has a fixed point $z_k\in A(N,M)$. That is, $z_1$ and $z_2$ are also bounded nonoscillatory solutions of (1.12) in $A(N,M)$. We now need to show that $z_1\ne z_2$. In view of (2.2) there exists $T_3>max\{T_1,T_2\}$ satisfying

$$\frac{1}{(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds<\frac{|L_1-L_2|}{4}.$$

(2.16)

Note that (2.6) means that for $t\ge T_3$,

$$\begin{array}{r}{z}_1(t)=L_1-p(t)z_1(t-\tau )\\ \phantom{z_1(t)=}+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)z_1({\alpha }_i(s))+{(-1)}^{n}r(s))ds\\ \phantom{z_1(t)=}+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,z_1({\beta }_1(s)),\dots ,z_1({\beta }_l(s)))ds,\\ z_2(t)=L_2-p(t)z_2(t-\tau )\\ \phantom{z_2(t)=}+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)z_2({\alpha }_i(s))+{(-1)}^{n}r(s))ds\\ \phantom{z_2(t)=}+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,z_2({\beta }_1(s)),\dots ,z_2({\beta }_l(s)))ds.\end{array}$$

(2.17)

It follows from (2.1), (2.4), and (2.17) that for $t\ge T_3$

$$\begin{array}{c}|z_1(t)-z_2(t)|\hfill \\ =|L_1-L_2-p(t)(z_1(t-\tau )-z_2(t-\tau ))\hfill \\ +\frac{1}{(n-1)!}({\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}\sum _{i=1}^m{q}_i(s)(z_1({\alpha }_i(s))-z_2({\alpha }_i(s)))ds\hfill \\ +{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(f(s,z_1({\alpha }_1(s)),\dots ,z_1({\alpha }_l(s)))\hfill \\ -f(s,z_2({\beta }_1(s)),\dots ,z_2({\beta }_l(s)))\left)ds\right)|\hfill \\ \ge |L_1-L_2|-(-p_0+\frac{1}{(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}\sum _{i=1}^m|q_i(s)\left|ds\right)\parallel z_1-z_2\parallel \hfill \\ -\frac{2}{(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\hfill \\ \ge |L_1-L_2|-\theta \parallel z_1-z_2\parallel -\frac{2}{(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds,\hfill \end{array}$$

which together with (2.16) implies that

$$\begin{array}{rl}\parallel z_1-z_2\parallel & \ge \frac{1}{1+\theta }(|L_1-L_2|-\frac{2}{(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds)\\ >\frac{1}{1+\theta }(|L_1-L_2|-\frac{|L_1-L_2|}{2})\\ =\frac{|L_1-L_2|}{2(1+\theta )}\\ >0,\end{array}$$

which yields $z_1\ne z_2$. This completes the proof. □

Theorem 2.2 Assume that there exist $h\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ and constants M, N, $p_0$ and c satisfying (2.1), (2.2), and

$$0<N<(1-p_0)M,0\le p(t)\le p_0<1,\mathrm{\forall }t\ge c\ge t_0.$$

(2.18)

Then (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$.

Proof Let $L\in (N,(1-p_0)M)$. By (2.2) and (2.18), we choose constants $\theta \in (0,1)$ and $T>|t_0|+\tau +|c|+|\gamma |+n$ satisfying (2.4) and

$$\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\le min\{M-L,L-p_{0}M-N\}.$$

(2.19)

Define two mappings $U_L$ and $S_L:A(N,M)\to CB([t_0,+\mathrm{\infty }),\mathbb{R})$ by (2.6).

Let $x,y\in A(N,M)$ and $t\ge T$. In terms of (2.6), (2.18), and (2.19), we arrive at

$$\begin{array}{c}(U_{L}x)(t)+(S_{L}y)(t)\hfill \\ =L-p(t)x(t-\tau )+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ +\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))ds\hfill \\ \le L+\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|\hfill \\ +\left|f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))\right|)ds\hfill \\ \le L+\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \le L+min\{M-L,L-p_{0}M-N\}\hfill \\ \le M,\hfill \\ (U_{L}x)(t)+(S_{L}y)(t)\hfill \\ \ge L-|p(t)||x(t-\tau )|\hfill \\ -\frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|\hfill \\ +\left|f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))\right|)ds\hfill \\ \ge L-p_{0}M-\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \ge L-p_{0}M-min\{M-L,L-p_{0}M-N\}\hfill \\ \ge N,\hfill \end{array}$$

which yields $U_{L}x+S_{L}y\in A(N,M)$ for any $x,y\in A(N,M)$. The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof. □

Theorem 2.3 Assume that there exist $h\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ and constants M, N, $p_0$, $p_1$, and c satisfying (2.1), (2.2), and

$$M>Nmax\{1,\frac{p_0(p_1^2-p_0)}{p_1(p_0^2-p_1)}\}>0,p_0^2>p_1\ge p(t)\ge p_0>1,\mathrm{\forall }t\ge c\ge t_0.$$

(2.20)

Then (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$.

Proof Let $L\in (\frac{p_1}{p_0}M+p_{1}N,p_{0}M+\frac{p_0}{p_1}N)$. It follows from (2.2) and (2.20) that there exist constants $\theta \in (0,1)$ and $T>|t_0|+\tau +|c|+|\gamma |+n$ satisfying

$$\theta =\frac{1}{|p_0|}(1+\frac{1}{(n-1)!}){\int }_T^{+\mathrm{\infty }}{s}^{n-1}\sum _{i=1}^m|q_i(s)|ds$$

(2.21)

and

$$\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\le min\{p_{0}M-L+\frac{p_0}{p_1}N,\frac{p_0}{p_1}L-M-p_{0}N\}.$$

(2.22)

Define two mappings $U_L$ and $S_L:A(N,M)\to CB([t_0,+\mathrm{\infty }),\mathbb{R})$ by

$$\begin{array}{r}(U_{L}x)(t)=\{\begin{array}{c}\frac{L}{p(t+\tau )}-\frac{x(t+\tau )}{p(t+\tau )}+\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}\hfill \\ \times ({\sum }_{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds,\hfill & t\ge T,\hfill \\ (U_{L}x)(T),\hfill & \gamma \le t<T;\hfill \end{array}\\ (S_{L}x)(t)=\{\begin{array}{cc}\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds,\hfill & t\ge T,\hfill \\ (S_{L}x)(T),\hfill & \gamma \le t<T.\hfill \end{array}\end{array}$$

(2.23)

We show that (2.7) holds. In fact, for every $x,y\in A(N,M)$ and $t\ge T$, by (2.1) and (2.20)-(2.23), we get

$$\begin{array}{c}(U_{L}x)(t)+(S_{L}y)(t)\hfill \\ =\frac{L}{p(t+\tau )}-\frac{x(t-\tau )}{p(t+\tau )}+\frac{1}{p(t+\tau )(n-1)!}\hfill \\ \times {\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ +\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))ds\hfill \\ \le \frac{L}{p_0}-\frac{N}{p_1}+\frac{1}{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \le \frac{L}{p_0}-\frac{N}{p_1}+\frac{1}{p_0}min\{p_{0}M-L+\frac{p_0}{p_1}N,\frac{p_0}{p_1}L-M-p_{0}N\}=M,\hfill \\ (U_{L}x)(t)+(S_{L}y)(t)\hfill \\ \ge \frac{L}{p_1}-\frac{M}{p_0}-\frac{1}{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \ge \frac{L}{p_1}-\frac{M}{p_0}-\frac{1}{p_0}min\{p_{0}M-L+\frac{p_0}{p_1}N,\frac{p_0}{p_1}L-M-p_{0}N\}\ge N\hfill \end{array}$$

and

$$\begin{array}{c}|(U_{L}x)(t)-(U_{L}y)(t)|\hfill \\ =|\frac{L}{p(t+\tau )}-\frac{x(t-\tau )}{p(t+\tau )}+\frac{1}{p(t+\tau )(n-1)!}\hfill \\ \times {\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ -\frac{L}{p(t+\tau )}+\frac{y(t-\tau )}{p(t+\tau )}-\frac{1}{p(t+\tau )(n-1)!}\hfill \\ \times {\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)y({\alpha }_i(s))+{(-1)}^{n}r(s))ds|\hfill \\ \le \frac{1}{p_0}\parallel x-y\parallel +\frac{\parallel x-y\parallel }{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}\sum _{i=1}^m|q_i(s)|ds\hfill \\ =\theta \parallel x-y\parallel ,\hfill \end{array}$$

which means that we have (2.7).

Next we show that $S_L(A(N,M))$ is equicontinuous in $[\gamma ,+\mathrm{\infty })$. For any given $\epsilon >0$, (2.2) guarantees that (2.10) holds for some sufficiently large $T^{\ast }>T$. Set

$$\overline{f}=max\left\{\right|f(t,u_1,\dots ,u_l)|:t\in [T,T^{\ast }+\tau ],u_j\in [N,M],j\in \{1,2,\dots ,l\}\}.$$

(2.24)

It follows from the uniform continuity of p in $[N,M]$ that there exists ${\delta }_0>0$ such that

$$|p(t_1+\tau )-p(t_2+\tau )|<\frac{\epsilon }{4(M+N)}$$

(2.25)

whenever $t_1,t_2\in [T,T^{\ast }]$ with $|t_1-t_2|<{\delta }_0$. Put

$$\delta =min\{{\delta }_0,\frac{\epsilon }{4n(M+N)},\frac{p_0\epsilon }{4(1+\overline{f}){(T^{\ast }-T)}^{n-1}}\}.$$

(2.26)

We have to consider the following possible cases:

• (i) $t_2>t_1\ge T^{\ast }$ with $|t_2-t_1|<\delta$. It follows from (2.1), (2.10), (2.20), (2.23), and (2.26) that

  $$\begin{array}{c}|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\hfill \\ =|\frac{1}{p(t_2+\tau )(n-1)!}{\int }_{t_2+\tau }^{+\mathrm{\infty }}{(s-t_2-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\hfill \\ -\frac{1}{p(t_1+\tau )(n-1)!}{\int }_{t_1+\tau }^{+\mathrm{\infty }}{(s-t_1-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))|\hfill \\ \le \frac{2}{p_0(n-1)!}{\int }_{T^{\ast }}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\hfill \\ <\epsilon ,\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

  (2.27)

• (ii) $T\le t_1<t_2\le T^{\ast }$ with $|t_2-t_1|<\delta$. For each $s\in (t_2+\tau ,+\mathrm{\infty })$, it follows from the mean value theorem that there exists $\xi \in (s-t_2-\tau ,s-t_1-\tau )$ satisfying

  $${(s-t_2-\tau )}^{n-1}-{(s-t_1-\tau )}^{n-1}=(n-1){\xi }^{n-2}(t_1-t_2),$$

which together with (2.1), (2.20), (2.22), and (2.26) yields for each $n\ge 2$

$$\begin{array}{c}\frac{1}{p(t_2+\tau )(n-1)!}{\int }_{t_2+\tau }^{+\mathrm{\infty }}|{(s-t_2-\tau )}^{n-1}-{(s-t_1-\tau )}^{n-1}|\left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds\hfill \\ \le \frac{|t_2-t_1|}{p_0(n-2)!}{\int }_{t_2+\tau }^{+\mathrm{\infty }}{\xi }^{n-2}h(s)ds\hfill \\ \le \frac{(n-1)|t_2-t_1|}{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\hfill \\ \le \frac{n-1}{p_0}\cdot \frac{\epsilon }{4n(M+N)}min\{p_{0}M-L+\frac{p_0}{p_1}N,\frac{p_0}{p_1}L-M-p_{0}N\}\hfill \\ <\frac{\epsilon }{4},\mathrm{\forall }x\in A(N,M),\hfill \end{array}$$

which implies that for each $n\in \mathbb{N}$

$$\begin{array}{c}\frac{1}{p(t_2+\tau )(n-1)!}{\int }_{t_2+\tau }^{+\mathrm{\infty }}|{(s-t_2-\tau )}^{n-1}-{(s-t_1-\tau )}^{n-1}|\left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds\hfill \\ <\frac{\epsilon }{4},\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

(2.28)

By means of (2.1), (2.20), (2.22)-(2.26), and (2.28), we get

$$\begin{array}{c}|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\hfill \\ =\frac{1}{(n-1)!}|\frac{1}{p(t_2+\tau )}{\int }_{t_2+\tau }^{+\mathrm{\infty }}{(s-t_2-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\hfill \\ -\frac{1}{p(t_1+\tau )}{\int }_{t_1+\tau }^{+\mathrm{\infty }}{(s-t_1-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds|\hfill \\ \le \frac{1}{(n-1)!}\left(\frac{1}{p(t_2+\tau )}\right|{\int }_{t_2+\tau }^{+\mathrm{\infty }}{(s-t_2-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds\hfill \\ -{\int }_{t_1+\tau }^{+\mathrm{\infty }}{(s-t_1-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds|\hfill \\ +|\frac{1}{p(t_2+\tau )}-\frac{1}{p(t_1+\tau )}|{\int }_{t_1+\tau }^{+\mathrm{\infty }}{(s-t_1-\tau )}^{n-1}\left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds)\hfill \\ \le \frac{1}{p(t_2+\tau )(n-1)!}\left({\int }_{t_2+\tau }^{+\mathrm{\infty }}\right|{(s-t_2-\tau )}^{n-1}-{(s-t_1-\tau )}^{n-1}|\hfill \\ \times \left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds\hfill \\ +{\int }_{t_1+\tau }^{t_2+\tau }\left|{(s-t_1-\tau )}^{n-1}\right|\left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds\hfill \\ +\frac{|p(t_2+\tau )-p(t_1+\tau )|}{p(t_1+\tau )}{\int }_{t_1+\tau }^{+\mathrm{\infty }}{(s-t_1-\tau )}^{n-1}\left|f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\right|ds)\hfill \\ \le \frac{\epsilon }{4}+\frac{{(T^{\ast }-T)}^{n-1}\overline{f}}{p_0(n-1)!}|t_2-t_1|+\frac{|p(t_2+\tau )-p(t_1+\tau )|}{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\hfill \\ \le \frac{\epsilon }{4}+\frac{{(T^{\ast }-T)}^{n-1}\overline{f}}{p_0(n-1)!}\cdot \frac{p_0\epsilon }{4{(T^{\ast }-T)}^{n-1}(1+\overline{f})}\hfill \\ +\frac{\epsilon }{4(M+N)p_0(n-1)!}min\{p_{0}M-L+\frac{p_0}{p_1}N,\frac{p_0}{p_1}L-M-p_{0}N\}\hfill \\ <\frac{\epsilon }{4}+\frac{\epsilon }{4}+\frac{\epsilon }{4}\hfill \\ <\epsilon ,\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

(2.29)

• (iii) $\gamma \le t_1<t_2\le T$ with $|t_2-t_1|<\delta$. Obviously, (2.23) guarantees that

  $$|(S_{L}x)(t_2)-(S_{L}x)(t_1)|=0<\epsilon ,\mathrm{\forall }x\in A(N,M).$$

  (2.30)

Using (2.27), (2.29), and (2.30), we conclude that $S_L(A(N,M))$ is equicontinuous in $[\gamma ,+\mathrm{\infty })$. As in the proof of Theorem 2.1, we prove similarly that $S_L$ is continuous in $A(N,M)$ and $S_L(A(N,M))$ is uniformly bounded. It follows that $S_L(A(N,M))$ is relatively compact. Consequently, Lemma 1.1 shows that there is $x\in A(N,M)$ such that $U_{L}x+S_{L}x=x$. That is,

$$\begin{array}{rcl}x(t)& =& \frac{L}{p(t+\tau )}-\frac{x(t+\tau )}{p(t+\tau )}\\ +\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\\ +\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds,\mathrm{\forall }t\ge T+\tau ,\end{array}$$

which yields

$$\begin{array}{c}x(t+\tau )+p(t+\tau )x(t)\hfill \\ =L+\frac{1}{(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ +\frac{1}{(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds,\mathrm{\forall }t\ge T+\tau ,\hfill \end{array}$$

which implies that

$$\begin{array}{rcl}\frac{d^n}{d{t}^n}[x(t)+p(t)x(t-\tau )]& =& {(-1)}^n(\sum _{i=1}^m{q}_i(t)x({\alpha }_i(t))+{(-1)}^{n}r(t))\\ +{(-1)}^{n}f(t,x({\beta }_1(t)),\dots ,x({\beta }_l(t))),\mathrm{\forall }t\ge T,\end{array}$$

that is, $x(t)$ is a bounded nonoscillatory solution of (1.12) in $A(N,M)$.

Finally we show (1.12) has uncountably many bounded nonoscillatory solutions. Let $L_1,L_2\in (\frac{p_1}{p_0}M+p_{1}N,p_{0}M+\frac{p_0}{p_1}N)$ with $L_1\ne L_2$. As in the above proof, we infer that for each $k\in \{1,2\}$, there exist constants ${\theta }_k\in (0,1)$, $T_k>|t_0|+\tau +|c|+|\gamma |+n$, and mappings $U_{L_k},S_{L_k}:A(N,M)\to CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ satisfying (2.21)-(2.23), where θ, T, L, $U_L$, $S_L$ are replaced by ${\theta }_k$, $T_k$, $L_k$, $U_{L_k}$, $S_{L_k}$, respectively, and (1.12) possesses a bounded nonoscillatory solution $z_k\in A(N,M)$. In terms of (2.2), we select $T_3>max\{T_1,T_2\}$ satisfying

$$\frac{1}{(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds<\frac{p_0|L_1-L_2|}{4p_1}.$$

(2.31)

It follows from (2.21), (2.23), and (2.31) that for $t\ge T_3+\tau$

$$\begin{array}{c}|z_1(t)-z_2(t)|\hfill \\ =|\frac{L_1-L_2}{p(t+\tau )}-\frac{z_1(t+\tau )-z_2(t+\tau )}{p(t+\tau )}\hfill \\ +\frac{1}{p(t+\tau )(n-1)!}({\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}\sum _{i=1}^m{q}_i(s)(z_1(\alpha (s))-z_2(\alpha (s)))ds\hfill \\ +{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(f(s,z_1({\beta }_1(s)),\dots ,z_1({\beta }_l(s)))\hfill \\ -f(s,z_2({\beta }_1(s)),\dots ,z_2({\beta }_l(s)))\left)ds\right)|\hfill \\ \ge \frac{|L_1-L_2|}{p_1}-\frac{\parallel z_1-z_2\parallel }{p_0}(1+\frac{1}{(n-1)!}){\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}\sum _{i=1}^m|q_i(s)|ds\hfill \\ -\frac{2}{p_0(n-1)!}{\int }_{T_3}^{+\mathrm{\infty }}{s}^{n-1}h(s)ds\hfill \\ \ge \frac{|L_1-L_2|}{p_1}-\theta \parallel z_1-z_2\parallel -\frac{2}{p_0}\cdot \frac{p_0|L_1-L_2|}{4p_1}\hfill \\ =\frac{|L_1-L_2|}{2p_1}-\theta \parallel z_1-z_2\parallel ,\hfill \end{array}$$

which yields

$$\parallel z_1-z_2\parallel \ge \frac{|L_1-L_2|}{2p_1(1+\theta )}>0,$$

that is, $x_1\ne x_2$. This completes the proof. □

Theorem 2.4 Assume that there exist $h\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ and constants M, N, $p_0$, $p_1$, and c satisfying (2.1), (2.2), and

$$\begin{array}{r}M>N>0,p_{0}N(1+\frac{1}{p_1})>p_{1}M(1+\frac{1}{p_0}),\\ p_1\le p(t)\le p_0<-1,\mathrm{\forall }t\ge c\ge t_0.\end{array}$$

(2.32)

Then (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$.

Proof Let $L\in (p_{1}M(1+\frac{1}{p_0}),p_{0}N(1+\frac{1}{p_1}))$. It follows from (2.2) and (2.32) that there exist constants $\theta \in (0,1)$ and $T>\tau +|t_0|+|c|+|\gamma |$ satisfying (2.21) and

$$\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\le min\{-M(p_0+1)+\frac{p_{0}L}{p_1},p_{0}N(1+\frac{1}{p_1})-L\}.$$

(2.33)

Let the mappings $U_L$ and $S_L:A(N,M)\to CB([t_0,+\mathrm{\infty }),\mathbb{R})$ be defined by (2.23).

Note that (2.1), (2.21), (2.23), and (2.33) imply that for each $x,y\in A(N,M)$, and $t\ge T$

$$\begin{array}{c}(U_{L}x)(t)+(S_{L}y)(t)\hfill \\ =\frac{L}{p(t+\tau )}-\frac{x(t-\tau )}{p(t+\tau )}+\frac{1}{p(t+\tau )(n-1)!}\hfill \\ \times {\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ +\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))ds\hfill \\ \le \frac{L}{p_1}-\frac{M}{p_0}-\frac{1}{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \le \frac{L}{p_1}-\frac{M}{p_0}-\frac{1}{p_0}min\{-M(p_0+1)+\frac{p_{0}L}{p_1},p_{0}N(1+\frac{1}{p_1})-L\}\hfill \\ \le M\hfill \end{array}$$

and

$$\begin{array}{c}(U_{L}x)(t)+(S_{L}y)(t)\hfill \\ =\frac{L}{p(t+\tau )}-\frac{x(t-\tau )}{p(t+\tau )}+\frac{1}{p(t+\tau )(n-1)!}\hfill \\ \times {\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s))ds\hfill \\ +\frac{1}{p(t+\tau )(n-1)!}{\int }_{t+\tau }^{+\mathrm{\infty }}{(s-t-\tau )}^{n-1}f(s,y({\beta }_1(s)),\dots ,y({\beta }_l(s)))ds\hfill \\ \ge \frac{L}{p_0}-\frac{N}{p_1}+\frac{1}{p_0(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\hfill \\ \ge \frac{L}{p_0}-\frac{N}{p_1}+\frac{1}{p_0}min\{-M(p_0+1)+\frac{p_{0}L}{p_1},p_{0}N(1+\frac{1}{p_1})-L\}\hfill \\ \ge N,\hfill \end{array}$$

which yields $U_{L}x+S_{L}y\in A(N,M)$ for any $x,y\in A(N,M)$. The rest of the proof is similar to that of Theorem 2.3 and is omitted. This completes the proof. □

Theorem 2.5 Let $n=1$. Assume that there exist $h\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ and constants M, N, and c satisfying (2.1), (2.2), and

$$0<N<M,p(t)\equiv 1,\mathrm{\forall }t\ge c\ge t_0.$$

(2.34)

Then (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$.

Proof Let $L\in (N,M)$. It follows from (2.2) and (2.34) that there exists a constant $T>\tau +|t_0|+|c|+|\gamma |$ satisfying

$${\int }_T^{+\mathrm{\infty }}H(s)ds\le min\{M-L,L-N\}.$$

(2.35)

Define a mapping $S_L:A(N,M)\to CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ by

$$(S_{L}x)(t)=\{\begin{array}{c}L+{\sum }_{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }({\sum }_{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)\hfill \\ +f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))ds),\hfill & t\ge T,\hfill \\ (S_{L}x)(T),\hfill & \gamma \le t<T.\hfill \end{array}$$

(2.36)

For every $x\in A(N,M)$ and $t\ge T$, by (2.1), (2.35), and (2.36), we deduce that

$$\begin{array}{c}(S_{L}x)(t)=L+\sum _{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds\hfill \\ \phantom{(S_{L}x)(t)}\le L+{\int }_t^{+\mathrm{\infty }}H(s)ds\hfill \\ \phantom{(S_{L}x)(t)}\le L+min\{M-L,L-N\}\hfill \\ \phantom{(S_{L}x)(t)}\le M,\hfill \\ (S_{L}x)(t)\ge L-{\int }_t^{+\mathrm{\infty }}H(s)ds\hfill \\ \phantom{(S_{L}x)(t)}\ge L-min\{M-L,L-N\}\hfill \\ \phantom{(S_{L}x)(t)}\ge N,\hfill \end{array}$$

which yield $S(A(N,M))\subseteq A(N,M)$ and hence $S(A(N,M))$ is uniformly bounded in $[\gamma ,+\mathrm{\infty })$.

Let ${\{x_k\}}_{k\in \mathbb{N}}$ be a sequence in $A(N,M)$ and $x\in A(N,M)$ satisfying (2.8) and let

$$\begin{array}{rcl}{G}_k(s)& =& \sum _{i=1}^m|q_i(s)||x_k({\alpha }_i(s))-x({\alpha }_i(s))|+|f(s,x_k({\beta }_1(s)),\dots ,x_k({\beta }_l(s)))\\ -f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))|,\mathrm{\forall }(s,k)\in [T,+\mathrm{\infty })\times \mathbb{N}.\end{array}$$

(2.37)

Using (2.8), (2.37), and the continuity of f, $q_i$, ${\alpha }_i$, and ${\beta }_j$ for $i\in \{1,2,\dots ,m\}$ and $j\in \{1,2,\dots ,l\}$, we obtain ${lim}_{t\to +\mathrm{\infty }}{G}_k(s)=0$ for all $s\in [T,+\mathrm{\infty })$. In light of (2.36) and the Lebesgue dominated convergence theorem, we conclude that for any $t\ge T$

$$\begin{array}{c}|(S_L{x}_k)(t)-(S_{L}x)(t)|\hfill \\ \le \sum _{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x_k({\alpha }_i(s))-x({\alpha }_i(s))|\hfill \\ +|f(s,x_k({\beta }_1(t)),\dots ,x_k({\beta }_l(t)))-f(s,x({\beta }_1(t)),\dots ,x({\beta }_l(s)))|)ds\hfill \\ \le {\int }_T^{+\mathrm{\infty }}{G}_k(s)ds\to 0\text{as }k\to \mathrm{\infty },\hfill \end{array}$$

which means that

$$\underset{k\to \mathrm{\infty }}{lim}\parallel S_L{x}_k-S_{L}x\parallel =\underset{k\to \mathrm{\infty }}{lim}\underset{t\ge T}{sup}|(S_L{x}_k)(t)-(S_{L}x)(t)|=0,$$

which implies that $S_L$ is continuous in $A(N,M)$.

Let ε be an arbitrary positive number. It follows from (2.2) that there exists $T^{\ast }>T$ large enough such that

$${\int }_{T^{\ast }}^{+\mathrm{\infty }}H(s)ds<\frac{\epsilon }{4}.$$

(2.38)

Set

$$\delta =\frac{\epsilon }{8KB}\text{and}B=1+max\{H(s):s\in [T,T^{\ast }+2K\tau ]\},$$

(2.39)

where $K\in \mathbb{N}$ satisfies

$$\begin{array}{r}\text{either}T+2(K-1)\tau <T^{\ast }\text{and}T+(2K-1)\tau \ge T^{\ast }\text{or}\\ T+(2K-1)\tau <T^{\ast }\text{and}T+2K\tau \ge T^{\ast }.\end{array}$$

(2.40)

We consider the following possible cases:

• (i) $t_2>t_1\ge T^{\ast }$ with $|t_2-t_1|<\delta$. From (2.1), (2.36), and (2.38), we conclude immediately that

  $$\begin{array}{c}|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\hfill \\ \le \sum _{a=1}^{\mathrm{\infty }}{\int }_{t_2+(2a-1)\tau }^{t_2+2a\tau }\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|+|f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\left|\right)ds\hfill \\ +\sum _{a=1}^{\mathrm{\infty }}{\int }_{t_1+(2a-1)\tau }^{t_1+2a\tau }\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|+|f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\left|\right)ds\hfill \\ \le 2{\int }_{T^{\ast }}^{+\mathrm{\infty }}H(s)ds\hfill \\ <\frac{\epsilon }{2},\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

  (2.41)

• (ii) $T\le t_1<t_2\le T^{\ast }$ with $|t_1-t_2|<\delta$. In terms of (2.1) and (2.36)-(2.40), we deduce that

  $$\begin{array}{c}|(S_{L}x)(t_2)-(S_{L}x)(t_1)|\hfill \\ \le |\sum _{a=1}^K{\int }_{t_2+(2a-1)\tau }^{t_2+2a\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds\hfill \\ +\sum _{a=1}^K{\int }_{t_1+2a\tau }^{t_2+(2a-1)\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds\hfill \\ -\sum _{a=1}^K{\int }_{t_1+2a\tau }^{t_2+(2a-1)\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds\hfill \\ -\sum _{a=1}^K{\int }_{t_1+(2a-1)\tau }^{t_1+2a\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds|\hfill \\ +\sum _{a=K+1}^{\mathrm{\infty }}{\int }_{t_2+(2a-1)\tau }^{t_2+2a\tau }\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|+|f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\left|\right)ds\hfill \\ +\sum _{a=K+1}^{\mathrm{\infty }}{\int }_{t_1+(2a-1)\tau }^{t_1+2a\tau }\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x({\alpha }_i(s))|+|r(s)|+|f(t,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\left|\right)ds\hfill \\ \le \sum _{a=1}^K{\int }_{t_1+2a\tau }^{t_2+2a\tau }H(s)ds+\sum _{a=1}^K{\int }_{t_1+(2a-1)\tau }^{t_2+(2a-1)\tau }H(s)ds+2{\int }_{T^{\ast }}^{+\mathrm{\infty }}H(s)ds\hfill \\ \le 2KB|t_2-t_1|+\frac{\epsilon }{2}\hfill \\ <\epsilon ,\mathrm{\forall }x\in A(N,M).\hfill \end{array}$$

  (2.42)

• (iii) $\gamma \le t_1<t_2\le T$ with $|t_1-t_2|<\delta$. Equation (2.36) means that

  $$|(S_{L}x)(t_2)-(S_{L}x)(t_1)|=0<\epsilon ,\mathrm{\forall }x\in A(N,M).$$

  (2.43)

It follows from (2.41)-(2.43) that $S_L(A(N,M))$ is equicontinuous in $[\gamma ,+\mathrm{\infty })$. Thus Lemma 1.2 means that $S_L$ has a fixed point $x\in A(N,M)$, that is, for any $t\ge T+\tau$

$$x(t)=L+\sum _{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds$$

and

$$x(t-\tau )=L+\sum _{a=1}^{\mathrm{\infty }}{\int }_{t+2(a-1)\tau }^{t+(2a-1)\tau }(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds,$$

which give for any $t\ge T+\tau$

$$x(t)+x(t-\tau )=2L+{\int }_t^{+\mathrm{\infty }}(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))-r(s)+f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds,$$

which implies that

$$\frac{d}{dt}[x(t)+x(t-\tau )]+\sum _{i=1}^m{q}_i(t)x({\alpha }_i(t))+f(t,x({\beta }_1(t)),\dots ,x({\beta }_l(t)))=r(t),t\ge T+\tau ,$$

that is, $x\in A(N,M)$ is a bounded nonoscillatory solution of (1.12). The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof. □

Theorem 2.6 Let $n\in \mathbb{N}\setminus \{1\}$. Assume that there exist $h\in C([t_0,+\mathrm{\infty }),{\mathbb{R}}^{+})$ and constants M, N, and c satisfying (2.1), (2.2), and (2.34). Then (1.12) has uncountably many bounded nonoscillatory solutions in $A(N,M)$.

Proof Let $L\in (N,M)$. It follows from (2.2) and (2.34) that there exists a constant $T>\tau +|t_0|+|c|+|\gamma |$ satisfying

$$\frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{s}^{n-1}H(s)ds\le min\{M-L,L-N\}.$$

(2.44)

Define a mapping $S_L:A(N,M)\to CB([\gamma ,+\mathrm{\infty }),\mathbb{R})$ by

$$(S_{L}x)(t)=\{\begin{array}{c}L+\frac{1}{(n-2)!}{\sum }_{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }du{\int }_u^{+\mathrm{\infty }}{(s-u)}^{n-2}\hfill \\ \times ({\sum }_{i=1}^m{q}_i(s)x({\alpha }_i(s))+{(-1)}^{n}r(s)\hfill \\ +f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s))))ds,\hfill & t\ge T,\hfill \\ (S_{L}x)(T),\hfill & \gamma \le t<T,\hfill \end{array}$$

(2.45)

for each $x\in A(N,M)$.

Let $x\in A(N,M)$ and $t\ge T$. By (2.44) and (2.45), we get

$$\begin{array}{rcl}|(S_{L}x)(t)-L|& =& \frac{1}{(n-2)!}\left|\sum _{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }du{\int }_u^{+\mathrm{\infty }}{(s-u)}^{n-2}\right(\sum _{i=1}^m{q}_i(s)x({\alpha }_i(s))\\ +{(-1)}^{n}r(s)+f(s,x({\beta }_1(s)),\dots ,x({\beta }_l(s)))\left)ds\right|\\ \le & \frac{1}{(n-2)!}{\int }_t^{+\mathrm{\infty }}du{\int }_u^{+\mathrm{\infty }}{(s-u)}^{n-2}H(s)ds\\ =& \frac{1}{(n-2)!}{\int }_t^{+\mathrm{\infty }}ds{\int }_t^s{(s-u)}^{n-2}H(s)du\\ =& \frac{1}{(n-1)!}{\int }_t^{+\mathrm{\infty }}{(s-t)}^{n-1}H(s)ds\\ \le & min\{M-L,L-N\},\end{array}$$

which gives that $S(A(N,M))\subseteq A(N,M)$ and $S(A(N,M))$ is uniformly bounded in $[\gamma ,+\mathrm{\infty })$.

Let ${\{x_k\}}_{k\in \mathbb{N}}\subset A(N,M)$ satisfy (2.8) for some $x\in A(N,M)$ and ${\{G_k\}}_{k\in \mathbb{N}}$ be defined by (2.37). Using (2.8), (2.37), (2.45), the continuity of f and ${\beta }_j$ for $j\in \{1,2,\dots ,l\}$, and the Lebesgue dominated convergence theorem, we conclude that for any $t\ge T$

$$\begin{array}{c}|(S_L{x}_k)(t)-(S_{L}x)(t)|\hfill \\ \le \frac{1}{(n-2)!}\sum _{a=1}^{\mathrm{\infty }}{\int }_{t+(2a-1)\tau }^{t+2a\tau }du{\int }_u^{+\mathrm{\infty }}{(s-u)}^{n-2}\left(\sum _{i=1}^m\right|q_i(s)\left|\right|x_k({\alpha }_i(s))-x({\alpha }_i(s))|\hfill \\ +|f(s,x_k({\beta }_1(t)),\dots ,x_k({\beta }_l(t)))-f(s,x({\beta }_1(t)),\dots ,x({\beta }_l(s)))|)ds\hfill \\ \le \frac{1}{(n-2)!}{\int }_t^{+\mathrm{\infty }}du{\int }_u^{+\mathrm{\infty }}{(s-u)}^{n-2}{G}_k(s)ds\hfill \\ \le \frac{1}{(n-1)!}{\int }_T^{+\mathrm{\infty }}{(s-T)}^{n-1}{G}_k(s)ds\hfill \\ \to 0\text{as }k\to \mathrm{\infty },\hfill \end{array}$$

which yields

$$\underset{k\to \mathrm{\infty }}{lim}\parallel S_L{x}_k-S_{L}x\parallel =\underset{k\to \mathrm{\infty }}{lim}\underset{t\ge T}{sup}|(S_L{x}_k)(t)-(S_{L}x)(t)|=0,$$

that is, $S_L$ is continuous in $A(N,M)$.

Next we show that $S_L(A(N,M))$ is equicontinuous in $[\gamma ,+\mathrm{\infty })$. Let $\epsilon >0$. Equation (2.34) ensures that there exists $T^{\ast }>T$ large enough satisfying

$$\frac{1}{(n-1)!}{\int }_{T^{\ast }}^{+\mathrm{\infty }}{s}^{n-1}H(s)ds<\frac{\epsilon }{16}.$$

(2.46)

Let B and K be defined by (2.39) and (2.40), respectively. Put

$$\delta =\frac{\epsilon }{4[nKmin\{M-L,L-N\}+B{(T^{\ast }+2K\tau )}^{n-1}]}.$$

(2.47)

Now we have to consider the following possible cases:

• (i) $t_2>t_1\ge T^{\ast }$ with $|t_1-t_2$