Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations

Zeqing Liu; Ming Jia; Jeong Sheok Ume; Shin Min Kang

doi:10.1186/1029-242X-2014-302

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Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations

Zeqing Liu¹,
Ming Jia²,
Jeong Sheok Ume³Email author and
Shin Min Kang⁴

Journal of Inequalities and Applications20142014:302

https://doi.org/10.1186/1029-242X-2014-302

Received: 20 February 2014

Accepted: 8 July 2014

Published: 19 August 2014

Abstract

This paper deals with the solvability of the higher-order nonlinear neutral delay differential equation $\frac{d^{n}}{d t^{n}} [x (t) + p (t) x (t - τ)] + {(- 1)}^{n + 1} \sum_{i = 1}^{m} q_{i} (t) x (α_{i} (t)) + {(- 1)}^{n + 1} f (t, x (β_{1} (t)), \dots, x (β_{l} (t))) = r (t)$ , $t \geq t_{0}$ , where $τ > 0$ , $n, m, l \in N$ , $p, r, q_{i}, α_{i}, β_{j} \in C ([t_{0}, + \infty), R)$ , and $f \in C ([t_{0}, + \infty) \times R^{l}, R)$ satisfying ${lim}_{t \to + \infty} α_{i} (t) = {lim}_{t \to + \infty} β_{j} (t) = + \infty$ , $i \in {1, 2, \dots, m}$ , $j \in {1, 2, \dots, l}$ . 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.

Keywords

higher-order nonlinear neutral delay differential equation uncountably many bounded nonoscillatory solutions Krasnoselskii fixed point theorem Schauder fixed point theorem

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}{d t} [x (t) + c (t) x (t - τ)] + P (t) x (t - α) - Q (t) x (t - β) = 0, t \geq t_{0}

(1.1)

and

\frac{d}{d t} [x (t) + c (t) x (t - τ)] + \sum_{i = 1}^{m} A_{i} (t) x (t - σ_{i}) - \sum_{i = m + 1}^{n} A_{i} (t) x (t - σ_{i}) = 0, t \geq t_{0},

(1.2)

where

τ > 0

,

α, β, σ_{i} \in R^{+}

,

c \in C ([t_{0}, + \infty), R)

and

P, Q, A_{i} \in C ([t_{0}, + \infty), 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}{d t} [x (t) + c x (t - τ)] = g_{1} (t, x (t)) + g_{2} (t, x (t - α (t))) + f (t),

(1.3)

where c, τ are constants,

c \neq \pm 1

,

f, α \in C (R, R)

,

g_{1}, g_{2} \in C (R^{2}, 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) + c x (t - τ)] + P (t) x (t - σ) - Q (t) x (t - δ) = 0, t \geq t_{0},

(1.4)

where

c \in R ∖ {\pm 1}

,

σ, δ \in R^{+}

and

P, Q \in C ([t_{0}, + \infty), 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 - τ)] + f (t) x (t - α) = 0, t \geq t_{0},

(1.5)

and gave conditions for each type of nonoscillatory solutions to exist, where n is an odd number,

τ > 0

,

α \in R

, and

f \in C (R^{+}, 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) + c x (t - τ)] + {(- 1)}^{n + 1} [P (t) x (t - σ) - Q (t) x (t - δ)] = 0, t \geq t_{0},

(1.6)

where

c \in R ∖ {\pm 1}

,

τ, σ, δ \in R^{+}

and

P, Q \in C ([t_{0}, + \infty), 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) + c x (t - τ)] + {(- 1)}^{n} [P (t) x (f (t)) - Q (t) x (g (t))] = 0, t \geq t_{0},

(1.7)

where

c \in R ∖ {- 1}

,

τ \in R^{+}

,

P, Q \in C ([t_{0}, + \infty), R^{+})

,

f, g \in C ([t_{0}, + \infty), R)

, and

{lim}_{t \to + \infty} f (t) = {lim}_{t \to + \infty} g (t) = + \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) + c x (t - τ)] + \sum_{i = 1}^{m} q_{i} (t) x (t - α_{i} (t)) = 0, t \geq 0,

(1.8)

where

c \in [0, 1)

,

τ > 0

,

q_{i} \in C (R^{+}, R)

, and

α_{i} \in C (R^{+}, 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 - τ)] = q (t) {| x (t - σ) |}^{α - 1} x (t - σ), t \geq 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,

α \geq 1

,

τ > 0

,

σ > 0

,

c, q \in C ([t_{0}, + \infty), 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 - τ)] + \sum_{i = 1}^{m} q_{i} (t) f (x (t - σ_{i})) = g (t), t \geq t_{0},

(1.10)

where

τ, σ_{i} \in R^{+}

,

c, q_{i}, g \in C ([t_{0}, + \infty), R)

for

i \in {1, 2, \dots, m}

and

f \in C (R, 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) + c x (t - τ)] + {(- 1)}^{n + 1} f (t, x (t - σ_{1}), x (t - σ_{2}), \dots, x (t - σ_{k})) = g (t), t \geq t_{0},

(1.11)

where $c \in R ∖ {- 1}$ , $τ > 0$ , $σ_{i} \in R^{+}$ for $i \in {1, 2, \dots, k}$ , $f \in C ([t_{0}, + \infty) \times R^{k}, R)$ , and $g \in C ([t_{0}, + \infty), R^{+})$ .

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{matrix} \frac{d^{n}}{d t^{n}} [x (t) + p (t) x (t - τ)] + {(- 1)}^{n + 1} \sum_{i = 1}^{m} q_{i} (t) x (α_{i} (t)) \\ + {(- 1)}^{n + 1} f (t, x (β_{1} (t)), \dots, x (β_{l} (t))) = r (t), t \geq t_{0}, \end{matrix}

(1.12)

where

τ > 0

,

n, m, l \in N

,

p, r, q_{i}, α_{i}, β_{j} \in C ([t_{0}, + \infty), R)

, and

f \in C ([t_{0}, + \infty) \times R^{l}, R)

satisfying

lim_{t \to + \infty} α_{i} (t) = lim_{t \to + \infty} β_{j} (t) = + \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,

R^{+}

and ℕ denote the sets of all real numbers, nonnegative numbers and positive integers, respectively, and

γ = min {t_{0} - τ, inf {α_{i} (t), β_{j} (t) : t \in [t_{0}, + \infty), i \in {1, 2, \dots, m}, j \in {1, 2, \dots, l}}} .

Let

C B ([γ, + \infty), R)

stand for the Banach space of all continuous and bounded functions on

[γ, + \infty)

with norm

∥ x ∥ = {sup}_{t \geq γ} | x (t) |

for all

x \in C B ([γ, + \infty), R)

and

A (N, M) = {x \in C B ([γ, + \infty), R) : N \leq x (t) \leq M, t \geq γ},

where $M, N \in R$ with $M > N$ . It is easy to see that $A (N, M)$ is a nonempty bounded closed convex subset of $C B ([γ, + \infty), R)$ .

By a solution of (1.12), we mean a function $x \in C ([γ, + \infty), R)$ for some $T \geq t_{0}$ , such that $x (t) + p (t) x (t - τ)$ is n times continuously differentiable on $[T, + \infty)$ and (1.12) holds for $t \geq T$ . 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 $U x + S y \in X$ for every pair $x, y \in X$ . If U is a contraction and S is completely continuous, then the equation $U x + S y = x$ 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), \forall t \in [T, + \infty) .

Theorem 2.1 Assume that there exist

h \in C ([t_{0}, + \infty), R^{+})

and constants M, N,

p_{0}

, and c satisfying

| f (t, u_{1}, \dots, u_{l}) | \leq h (t), \forall (t, u_{1}, \dots, u_{l}) \in [t_{0}, + \infty) \times {[N, M]}^{l};

(2.1)

\int_{t_{0}}^{+ \infty} s^{n - 1} max {\sum_{i = 1}^{m} | q_{i} (s) |, | r (s) |, h (s)} d s < + \infty;

(2.2)

0 < N < (1 + p_{0}) M, - 1 < p_{0} \leq p (t) \leq 0, \forall t \geq c \geq 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

θ \in (0, 1)

and

T > | t_{0} | + τ + | c | + | γ | + n

sufficiently large satisfying

θ = | p_{0} | + \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} \sum_{i = 1}^{m} | q_{i} (s) | d s

(2.4)

and

\frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \leq min {(p_{0} + 1) M - L, L - N} .

(2.5)

Define two mappings

U_{L}

and

S_{L} : A (N, M) \to C B ([γ, + \infty), R)

by

\begin{aligned} (U_{L} x) (t) = {\begin{matrix} L - p (t) x (t - τ) + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} \\ \times (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s, & t \geq T, \\ (U_{L} x) (T), & γ \leq t < T; \end{matrix} \\ (S_{L} x) (t) = {\begin{matrix} \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s, & t \geq T, \\ (S_{L} x) (T), & γ \leq t < T, \end{matrix} \end{aligned}

(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), ∥ U_{L} x - U_{L} y ∥ \leq θ ∥ x - y ∥, \forall x, y \in A (N, M) .

(2.7)

Let

x, y \in A (N, M)

and

t \geq T

. By (2.3)-(2.6), we get

\begin{matrix} (U_{L} x) (t) + (S_{L} y) (t) \\ = L - p (t) x (t - τ) + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) d s \\ \leq L + | p (t) | | x (t) | \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | \\ + | f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) |) d s \\ \leq L - p_{0} M + \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \leq L - p_{0} M + min {(p_{0} + 1) M - L, L - N} \\ \leq M, \\ (U_{L} x) (t) + (S_{L} y) (t) \\ \geq L - \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | \\ + | f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) |) d s \\ \geq L - \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \geq L - min {(p_{0} + 1) M - L, L - N} \\ \geq N \end{matrix}

and

\begin{matrix} | (U_{L} x) (t) - (U_{L} y) (t) | \\ = | L - p (t) x (t - τ) + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ - L + p (t) y (t - τ) - \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) y (α_{i} (s)) + {(- 1)}^{n} r (s)) d s | \\ \leq - p_{0} ∥ x - y ∥ + \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} \sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) - y (α_{i} (s)) | d s \\ \leq θ ∥ x - y ∥, \end{matrix}

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 N}

be an arbitrary sequence in

A (N, M)

with

lim_{k \to \infty} x_{k} = x \in C B ([γ, + \infty), R) .

(2.8)

Since

A (N, M)

is a closed subset of

C B ([γ, + \infty), R)

, it follows that

x \in A (N, M)

. Put

\begin{matrix} G_{k} (s) = | f (s, x_{k} (β_{1} (s)), \dots, x_{k} (β_{l} (s))) - f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) |, \\ \forall (s, k) \in [T, + \infty) \times N . \end{matrix}

From (2.8) and the continuity of f and

β_{j}

for

j \in {1, 2, \dots, l}

we infer that

lim_{k \to \infty} G_{k} (s) = 0, \forall s \in [T, + \infty),

which together with (2.6) and the Lebesgue dominated convergence theorem yields for any

t \in [T, + \infty)

\begin{matrix} | (S_{L} x_{k}) (t) - (S_{L} x) (t) | \\ \leq \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} | f (s, x_{k} (β_{1} (s)), \dots, x_{k} (β_{l} (s))) - f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s \\ \leq \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} G_{k} (s) d s \\ \to 0 as k \to \infty, \end{matrix}

which gives

lim_{k \to \infty} ∥ S_{L} x_{k} - S_{L} x ∥ = lim_{k \to \infty} sup_{t \geq γ} | (S_{L} x_{k}) (t) - (S_{L} x) (t) | = lim_{k \to \infty} sup_{t \geq T} | (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 \geq T

\begin{aligned} | (S_{L} x) (t) | & = | \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s | \\ \leq \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} h (s) d s \\ \leq min {(p_{0} + 1) M - L, L - N} \\ \leq M, \end{aligned}

which yields

∥ S_{L} x ∥ \leq M, \forall x \in A (N, M) .

(2.9)

That is,

S_{L} (A (N, M))

is uniformly bounded in

[γ, + \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

[γ, + \infty)

. Let

ε > 0

be given. Equation (2.2) ensures that there exists

T^{*} > T

satisfying

\frac{2}{(n - 1)!} \int_{T^{*}}^{+ \infty} s^{n - 1} h (s) d s < ε .

(2.10)

Put

δ = min {\frac{ε}{1 + M}, \frac{ε}{1 + max {h (t) : t \in [T, T^{*}]}}} .

(2.11)

Now we consider the following possible cases:

(i)

$t_{2} > t_{1} \geq T^{*}$ with $| t_{2} - t_{1} | < δ$ . By (2.1), (2.6), and (2.10) we have
$\begin{matrix} | (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \\ = \frac{1}{(n - 1)!} | \int_{t_{2}}^{+ \infty} {(s - t_{2})}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s \\ - \int_{t_{1}}^{+ \infty} {(s - t_{1})}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | \\ \leq \frac{2}{(n - 1)!} \int_{T^{*}}^{+ \infty} s^{n - 1} h (s) d s \\ < ε, \forall x \in A (N, M) . \end{matrix}$

(2.12)
(ii)

$T \leq t_{1} < t_{2} \leq T^{*}$ with $| t_{2} - t_{1} | < δ$ and $n = 1$ . By means of (2.1), (2.6), and (2.11) we infer that
$\begin{matrix} | (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \\ = | \int_{t_{2}}^{+ \infty} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s - \int_{t_{1}}^{+ \infty} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s | \\ \leq \int_{t_{1}}^{t_{2}} | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s \\ \leq \int_{t_{1}}^{t_{2}} h (s) d s \\ < ε, \forall x \in A (N, M) . \end{matrix}$

(2.13)
(iii)

$T \leq t_{1} < t_{2} \leq T^{*}$ with $| t_{2} - t_{1} | < δ$ and $n \geq 2$ . In light of (2.1), (2.5), and (2.6), we get
$\begin{aligned} | \frac{d}{d t} (S_{L} x) (t) | & = \frac{1}{(n - 1)!} | \frac{d}{d t} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s | \\ = | \frac{1}{(n - 2)!} \int_{t}^{+ \infty} {(s - t)}^{n - 2} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s | \\ \leq \frac{1}{(n - 2)!} \int_{T}^{+ \infty} s^{n - 2} h (s) d s \\ \leq \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} h (s) d s \\ \leq min {(p_{0} + 1) M - L, L - N} \\ \leq M, \forall x \in A (N, M), t \in [T, T^{*}], \end{aligned}$

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

| (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \leq M | t_{1} - t_{2} | < ε, \forall x \in A (N, M) .

(2.14)

(iv)

$γ \leq t_{1} < t_{2} \leq T$ with $| t_{2} - t_{1} | < δ$ . Clearly (2.6) means that
$| (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | = 0 < ε, \forall x \in A (N, M) .$

(2.15)

It follows from (2.12)-(2.15) that

S_{L} (A (N, M))

is equicontinuous in

[γ, + \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 - τ) + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s, \forall t \geq T, \end{array}

which implies that

\begin{matrix} \frac{d^{n}}{d t^{n}} [x (t) + p (t) x (t - τ)] \\ = {(- 1)}^{n} (\sum_{i = 1}^{m} q_{i} (t) x (α_{i} (t)) + {(- 1)}^{n} r (t)) + {(- 1)}^{n} f (t, x (β_{1} (t)), \dots, x (β_{l} (t))), \forall t \geq T, \end{matrix}

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} \neq L_{2}

. As in the above proof we can deduce that for each

k \in {1, 2}

, there exist constants

θ_{k} \in (0, 1)

,

T_{k} > | t_{0} | + τ + | c | + | γ | + n

, and mappings

U_{L_{k}}, S_{L_{k}} : A (N, M) \to C B ([γ, + \infty), R)

satisfying (2.4)-(2.6), where θ, T, L,

U_{L}

and

S_{L}

are replaced by

θ_{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} \neq z_{2}

. In view of (2.2) there exists

T_{3} > max {T_{1}, T_{2}}

satisfying

\frac{1}{(n - 1)!} \int_{T_{3}}^{+ \infty} s^{n - 1} h (s) d s < \frac{| L_{1} - L_{2} |}{4} .

(2.16)

Note that (2.6) means that for

t \geq T_{3}

,

\begin{aligned} z_{1} (t) = L_{1} - p (t) z_{1} (t - τ) \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) z_{1} (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, z_{1} (β_{1} (s)), \dots, z_{1} (β_{l} (s))) d s, \\ z_{2} (t) = L_{2} - p (t) z_{2} (t - τ) \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) z_{2} (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, z_{2} (β_{1} (s)), \dots, z_{2} (β_{l} (s))) d s . \end{aligned}

(2.17)

It follows from (2.1), (2.4), and (2.17) that for

t \geq T_{3}

\begin{matrix} | z_{1} (t) - z_{2} (t) | \\ = | L_{1} - L_{2} - p (t) (z_{1} (t - τ) - z_{2} (t - τ)) \\ + \frac{1}{(n - 1)!} (\int_{t}^{+ \infty} {(s - t)}^{n - 1} \sum_{i = 1}^{m} q_{i} (s) (z_{1} (α_{i} (s)) - z_{2} (α_{i} (s))) d s \\ + \int_{t}^{+ \infty} {(s - t)}^{n - 1} (f (s, z_{1} (α_{1} (s)), \dots, z_{1} (α_{l} (s))) \\ - f (s, z_{2} (β_{1} (s)), \dots, z_{2} (β_{l} (s)))) d s) | \\ \geq | L_{1} - L_{2} | - (- p_{0} + \frac{1}{(n - 1)!} \int_{T_{3}}^{+ \infty} s^{n - 1} \sum_{i = 1}^{m} | q_{i} (s) | d s) ∥ z_{1} - z_{2} ∥ \\ - \frac{2}{(n - 1)!} \int_{T_{3}}^{+ \infty} s^{n - 1} h (s) d s \\ \geq | L_{1} - L_{2} | - θ ∥ z_{1} - z_{2} ∥ - \frac{2}{(n - 1)!} \int_{T_{3}}^{+ \infty} s^{n - 1} h (s) d s, \end{matrix}

which together with (2.16) implies that

\begin{aligned} ∥ z_{1} - z_{2} ∥ & \geq \frac{1}{1 + θ} (| L_{1} - L_{2} | - \frac{2}{(n - 1)!} \int_{T_{3}}^{+ \infty} s^{n - 1} h (s) d s) \\ > \frac{1}{1 + θ} (| L_{1} - L_{2} | - \frac{| L_{1} - L_{2} |}{2}) \\ = \frac{| L_{1} - L_{2} |}{2 (1 + θ)} \\ > 0, \end{aligned}

which yields $z_{1} \neq z_{2}$ . This completes the proof. □

Theorem 2.2 Assume that there exist

h \in C ([t_{0}, + \infty), R^{+})

and constants M, N,

p_{0}

and c satisfying (2.1), (2.2), and

0 < N < (1 - p_{0}) M, 0 \leq p (t) \leq p_{0} < 1, \forall t \geq c \geq 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

θ \in (0, 1)

and

T > | t_{0} | + τ + | c | + | γ | + n

satisfying (2.4) and

\frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \leq min {M - L, L - p_{0} M - N} .

(2.19)

Define two mappings $U_{L}$ and $S_{L} : A (N, M) \to C B ([t_{0}, + \infty), R)$ by (2.6).

Let

x, y \in A (N, M)

and

t \geq T

. In terms of (2.6), (2.18), and (2.19), we arrive at

\begin{matrix} (U_{L} x) (t) + (S_{L} y) (t) \\ = L - p (t) x (t - τ) + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) d s \\ \leq L + \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | \\ + | f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) |) d s \\ \leq L + \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \leq L + min {M - L, L - p_{0} M - N} \\ \leq M, \\ (U_{L} x) (t) + (S_{L} y) (t) \\ \geq L - | p (t) | | x (t - τ) | \\ - \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | \\ + | f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) |) d s \\ \geq L - p_{0} M - \frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \geq L - p_{0} M - min {M - L, L - p_{0} M - N} \\ \geq N, \end{matrix}

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}, + \infty), R^{+})

and constants M, N,

p_{0}

,

p_{1}

, and c satisfying (2.1), (2.2), and

M > N max {1, \frac{p_{0} (p_{1}^{2} - p_{0})}{p_{1} (p_{0}^{2} - p_{1})}} > 0, p_{0}^{2} > p_{1} \geq p (t) \geq p_{0} > 1, \forall t \geq c \geq 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

θ \in (0, 1)

and

T > | t_{0} | + τ + | c | + | γ | + n

satisfying

θ = \frac{1}{| p_{0} |} (1 + \frac{1}{(n - 1)!}) \int_{T}^{+ \infty} s^{n - 1} \sum_{i = 1}^{m} | q_{i} (s) | d s

(2.21)

and

\frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \leq 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 C B ([t_{0}, + \infty), R)

by

\begin{aligned} (U_{L} x) (t) = {\begin{matrix} \frac{L}{p (t + τ)} - \frac{x (t + τ)}{p (t + τ)} + \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} \\ \times (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s, & t \geq T, \\ (U_{L} x) (T), & γ \leq t < T; \end{matrix} \\ (S_{L} x) (t) = {\begin{matrix} \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s, & t \geq T, \\ (S_{L} x) (T), & γ \leq t < T . \end{matrix} \end{aligned}

(2.23)

We show that (2.7) holds. In fact, for every

x, y \in A (N, M)

and

t \geq T

, by (2.1) and (2.20)-(2.23), we get

\begin{matrix} (U_{L} x) (t) + (S_{L} y) (t) \\ = \frac{L}{p (t + τ)} - \frac{x (t - τ)}{p (t + τ)} + \frac{1}{p (t + τ) (n - 1)!} \\ \times \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) d s \\ \leq \frac{L}{p_{0}} - \frac{N}{p_{1}} + \frac{1}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \leq \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, \\ (U_{L} x) (t) + (S_{L} y) (t) \\ \geq \frac{L}{p_{1}} - \frac{M}{p_{0}} - \frac{1}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \geq \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} \geq N \end{matrix}

and

\begin{matrix} | (U_{L} x) (t) - (U_{L} y) (t) | \\ = | \frac{L}{p (t + τ)} - \frac{x (t - τ)}{p (t + τ)} + \frac{1}{p (t + τ) (n - 1)!} \\ \times \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ - \frac{L}{p (t + τ)} + \frac{y (t - τ)}{p (t + τ)} - \frac{1}{p (t + τ) (n - 1)!} \\ \times \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) y (α_{i} (s)) + {(- 1)}^{n} r (s)) d s | \\ \leq \frac{1}{p_{0}} ∥ x - y ∥ + \frac{∥ x - y ∥}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} \sum_{i = 1}^{m} | q_{i} (s) | d s \\ = θ ∥ x - y ∥, \end{matrix}

which means that we have (2.7).

Next we show that

S_{L} (A (N, M))

is equicontinuous in

[γ, + \infty)

. For any given

ε > 0

, (2.2) guarantees that (2.10) holds for some sufficiently large

T^{*} > T

. Set

\bar{f} = max {| f (t, u_{1}, \dots, u_{l}) | : t \in [T, T^{*} + τ], 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

δ_{0} > 0

such that

| p (t_{1} + τ) - p (t_{2} + τ) | < \frac{ε}{4 (M + N)}

(2.25)

whenever

t_{1}, t_{2} \in [T, T^{*}]

with

| t_{1} - t_{2} | < δ_{0}

. Put

δ = min {δ_{0}, \frac{ε}{4 n (M + N)}, \frac{p_{0} ε}{4 (1 + \bar{f}) {(T^{*} - T)}^{n - 1}}} .

(2.26)

We have to consider the following possible cases:

(i)

$t_{2} > t_{1} \geq T^{*}$ with $| t_{2} - t_{1} | < δ$ . It follows from (2.1), (2.10), (2.20), (2.23), and (2.26) that
$\begin{matrix} | (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \\ = | \frac{1}{p (t_{2} + τ) (n - 1)!} \int_{t_{2} + τ}^{+ \infty} {(s - t_{2} - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s \\ - \frac{1}{p (t_{1} + τ) (n - 1)!} \int_{t_{1} + τ}^{+ \infty} {(s - t_{1} - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | \\ \leq \frac{2}{p_{0} (n - 1)!} \int_{T^{*}}^{+ \infty} s^{n - 1} h (s) d s \\ < ε, \forall x \in A (N, M) . \end{matrix}$

(2.27)
(ii)

$T \leq t_{1} < t_{2} \leq T^{*}$ with $| t_{2} - t_{1} | < δ$ . For each $s \in (t_{2} + τ, + \infty)$ , it follows from the mean value theorem that there exists $ξ \in (s - t_{2} - τ, s - t_{1} - τ)$ satisfying
${(s - t_{2} - τ)}^{n - 1} - {(s - t_{1} - τ)}^{n - 1} = (n - 1) ξ^{n - 2} (t_{1} - t_{2}),$

which together with (2.1), (2.20), (2.22), and (2.26) yields for each

n \geq 2

\begin{matrix} \frac{1}{p (t_{2} + τ) (n - 1)!} \int_{t_{2} + τ}^{+ \infty} | {(s - t_{2} - τ)}^{n - 1} - {(s - t_{1} - τ)}^{n - 1} | | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s \\ \leq \frac{| t_{2} - t_{1} |}{p_{0} (n - 2)!} \int_{t_{2} + τ}^{+ \infty} ξ^{n - 2} h (s) d s \\ \leq \frac{(n - 1) | t_{2} - t_{1} |}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} h (s) d s \\ \leq \frac{n - 1}{p_{0}} \cdot \frac{ε}{4 n (M + N)} min {p_{0} M - L + \frac{p_{0}}{p_{1}} N, \frac{p_{0}}{p_{1}} L - M - p_{0} N} \\ < \frac{ε}{4}, \forall x \in A (N, M), \end{matrix}

which implies that for each

n \in N

\begin{matrix} \frac{1}{p (t_{2} + τ) (n - 1)!} \int_{t_{2} + τ}^{+ \infty} | {(s - t_{2} - τ)}^{n - 1} - {(s - t_{1} - τ)}^{n - 1} | | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s \\ < \frac{ε}{4}, \forall x \in A (N, M) . \end{matrix}

(2.28)

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

\begin{matrix} | (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \\ = \frac{1}{(n - 1)!} | \frac{1}{p (t_{2} + τ)} \int_{t_{2} + τ}^{+ \infty} {(s - t_{2} - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s \\ - \frac{1}{p (t_{1} + τ)} \int_{t_{1} + τ}^{+ \infty} {(s - t_{1} - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s | \\ \leq \frac{1}{(n - 1)!} (\frac{1}{p (t_{2} + τ)} | \int_{t_{2} + τ}^{+ \infty} {(s - t_{2} - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s \\ - \int_{t_{1} + τ}^{+ \infty} {(s - t_{1} - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s | \\ + | \frac{1}{p (t_{2} + τ)} - \frac{1}{p (t_{1} + τ)} | \int_{t_{1} + τ}^{+ \infty} {(s - t_{1} - τ)}^{n - 1} | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s) \\ \leq \frac{1}{p (t_{2} + τ) (n - 1)!} (\int_{t_{2} + τ}^{+ \infty} | {(s - t_{2} - τ)}^{n - 1} - {(s - t_{1} - τ)}^{n - 1} | \\ \times | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s \\ + \int_{t_{1} + τ}^{t_{2} + τ} | {(s - t_{1} - τ)}^{n - 1} | | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s \\ + \frac{| p (t_{2} + τ) - p (t_{1} + τ) |}{p (t_{1} + τ)} \int_{t_{1} + τ}^{+ \infty} {(s - t_{1} - τ)}^{n - 1} | f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) | d s) \\ \leq \frac{ε}{4} + \frac{{(T^{*} - T)}^{n - 1} \bar{f}}{p_{0} (n - 1)!} | t_{2} - t_{1} | + \frac{| p (t_{2} + τ) - p (t_{1} + τ) |}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} h (s) d s \\ \leq \frac{ε}{4} + \frac{{(T^{*} - T)}^{n - 1} \bar{f}}{p_{0} (n - 1)!} \cdot \frac{p_{0} ε}{4 {(T^{*} - T)}^{n - 1} (1 + \bar{f})} \\ + \frac{ε}{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} \\ < \frac{ε}{4} + \frac{ε}{4} + \frac{ε}{4} \\ < ε, \forall x \in A (N, M) . \end{matrix}

(2.29)

(iii)

$γ \leq t_{1} < t_{2} \leq T$ with $| t_{2} - t_{1} | < δ$ . Obviously, (2.23) guarantees that
$| (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | = 0 < ε, \forall x \in A (N, M) .$

(2.30)

Using (2.27), (2.29), and (2.30), we conclude that

S_{L} (A (N, M))

is equicontinuous in

[γ, + \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 + τ)} - \frac{x (t + τ)}{p (t + τ)} \\ + \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s, \forall t \geq T + τ, \end{array}

which yields

\begin{matrix} x (t + τ) + p (t + τ) x (t) \\ = L + \frac{1}{(n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{(n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s, \forall t \geq T + τ, \end{matrix}

which implies that

\begin{array}{rcl} \frac{d^{n}}{d t^{n}} [x (t) + p (t) x (t - τ)] & = & {(- 1)}^{n} (\sum_{i = 1}^{m} q_{i} (t) x (α_{i} (t)) + {(- 1)}^{n} r (t)) \\ + {(- 1)}^{n} f (t, x (β_{1} (t)), \dots, x (β_{l} (t))), \forall t \geq 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} \neq L_{2}

. As in the above proof, we infer that for each

k \in {1, 2}

, there exist constants

θ_{k} \in (0, 1)

,

T_{k} > | t_{0} | + τ + | c | + | γ | + n

, and mappings

U_{L_{k}}, S_{L_{k}} : A (N, M) \to C B ([γ, + \infty), R)

satisfying (2.21)-(2.23), where θ, T, L,

U_{L}

,

S_{L}

are replaced by

θ_{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}}^{+ \infty} s^{n - 1} h (s) d s < \frac{p_{0} | L_{1} - L_{2} |}{4 p_{1}} .

(2.31)

It follows from (2.21), (2.23), and (2.31) that for

t \geq T_{3} + τ

\begin{matrix} | z_{1} (t) - z_{2} (t) | \\ = | \frac{L_{1} - L_{2}}{p (t + τ)} - \frac{z_{1} (t + τ) - z_{2} (t + τ)}{p (t + τ)} \\ + \frac{1}{p (t + τ) (n - 1)!} (\int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} \sum_{i = 1}^{m} q_{i} (s) (z_{1} (α (s)) - z_{2} (α (s))) d s \\ + \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (f (s, z_{1} (β_{1} (s)), \dots, z_{1} (β_{l} (s))) \\ - f (s, z_{2} (β_{1} (s)), \dots, z_{2} (β_{l} (s)))) d s) | \\ \geq \frac{| L_{1} - L_{2} |}{p_{1}} - \frac{∥ z_{1} - z_{2} ∥}{p_{0}} (1 + \frac{1}{(n - 1)!}) \int_{T_{3}}^{+ \infty} s^{n - 1} \sum_{i = 1}^{m} | q_{i} (s) | d s \\ - \frac{2}{p_{0} (n - 1)!} \int_{T_{3}}^{+ \infty} s^{n - 1} h (s) d s \\ \geq \frac{| L_{1} - L_{2} |}{p_{1}} - θ ∥ z_{1} - z_{2} ∥ - \frac{2}{p_{0}} \cdot \frac{p_{0} | L_{1} - L_{2} |}{4 p_{1}} \\ = \frac{| L_{1} - L_{2} |}{2 p_{1}} - θ ∥ z_{1} - z_{2} ∥, \end{matrix}

which yields

∥ z_{1} - z_{2} ∥ \geq \frac{| L_{1} - L_{2} |}{2 p_{1} (1 + θ)} > 0,

that is, $x_{1} \neq x_{2}$ . This completes the proof. □

Theorem 2.4 Assume that there exist

h \in C ([t_{0}, + \infty), R^{+})

and constants M, N,

p_{0}

,

p_{1}

, and c satisfying (2.1), (2.2), and

\begin{aligned} M > N > 0, p_{0} N (1 + \frac{1}{p_{1}}) > p_{1} M (1 + \frac{1}{p_{0}}), \\ p_{1} \leq p (t) \leq p_{0} < - 1, \forall t \geq c \geq t_{0} . \end{aligned}

(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

θ \in (0, 1)

and

T > τ + | t_{0} | + | c | + | γ |

satisfying (2.21) and

\frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \leq 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 $S_{L} : A (N, M) \to C B ([t_{0}, + \infty), R)$ 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 \geq T

\begin{matrix} (U_{L} x) (t) + (S_{L} y) (t) \\ = \frac{L}{p (t + τ)} - \frac{x (t - τ)}{p (t + τ)} + \frac{1}{p (t + τ) (n - 1)!} \\ \times \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) d s \\ \leq \frac{L}{p_{1}} - \frac{M}{p_{0}} - \frac{1}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \leq \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} \\ \leq M \end{matrix}

and

\begin{matrix} (U_{L} x) (t) + (S_{L} y) (t) \\ = \frac{L}{p (t + τ)} - \frac{x (t - τ)}{p (t + τ)} + \frac{1}{p (t + τ) (n - 1)!} \\ \times \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s)) d s \\ + \frac{1}{p (t + τ) (n - 1)!} \int_{t + τ}^{+ \infty} {(s - t - τ)}^{n - 1} f (s, y (β_{1} (s)), \dots, y (β_{l} (s))) d s \\ \geq \frac{L}{p_{0}} - \frac{N}{p_{1}} + \frac{1}{p_{0} (n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \\ \geq \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} \\ \geq N, \end{matrix}

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}, + \infty), R^{+})

and constants M, N, and c satisfying (2.1), (2.2), and

0 < N < M, p (t) \equiv 1, \forall t \geq c \geq 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 > τ + | t_{0} | + | c | + | γ |

satisfying

\int_{T}^{+ \infty} H (s) d s \leq min {M - L, L - N} .

(2.35)

Define a mapping

S_{L} : A (N, M) \to C B ([γ, + \infty), R)

by

(S_{L} x) (t) = {\begin{matrix} L + \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) \\ + f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) d s), & t \geq T, \\ (S_{L} x) (T), & γ \leq t < T . \end{matrix}

(2.36)

For every

x \in A (N, M)

and

t \geq T

, by (2.1), (2.35), and (2.36), we deduce that

\begin{matrix} (S_{L} x) (t) = L + \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (t, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s \\ \leq L + \int_{t}^{+ \infty} H (s) d s \\ \leq L + min {M - L, L - N} \\ \leq M, \\ (S_{L} x) (t) \geq L - \int_{t}^{+ \infty} H (s) d s \\ \geq L - min {M - L, L - N} \\ \geq N, \end{matrix}

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

Let

{x_{k}}_{k \in 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} (α_{i} (s)) - x (α_{i} (s)) | + | f (s, x_{k} (β_{1} (s)), \dots, x_{k} (β_{l} (s))) \\ - f (s, x (β_{1} (s)), \dots, x (β_{l} (s))) |, \forall (s, k) \in [T, + \infty) \times N . \end{array}

(2.37)

Using (2.8), (2.37), and the continuity of f,

q_{i}

,

α_{i}

, and

β_{j}

for

i \in {1, 2, \dots, m}

and

j \in {1, 2, \dots, l}

, we obtain

{lim}_{t \to + \infty} G_{k} (s) = 0

for all

s \in [T, + \infty)

. In light of (2.36) and the Lebesgue dominated convergence theorem, we conclude that for any

t \geq T

\begin{matrix} | (S_{L} x_{k}) (t) - (S_{L} x) (t) | \\ \leq \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} (\sum_{i = 1}^{m} | q_{i} (s) | | x_{k} (α_{i} (s)) - x (α_{i} (s)) | \\ + | f (s, x_{k} (β_{1} (t)), \dots, x_{k} (β_{l} (t))) - f (s, x (β_{1} (t)), \dots, x (β_{l} (s))) |) d s \\ \leq \int_{T}^{+ \infty} G_{k} (s) d s \to 0 as k \to \infty, \end{matrix}

which means that

lim_{k \to \infty} ∥ S_{L} x_{k} - S_{L} x ∥ = lim_{k \to \infty} sup_{t \geq T} | (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^{*} > T

large enough such that

\int_{T^{*}}^{+ \infty} H (s) d s < \frac{ε}{4} .

(2.38)

Set

δ = \frac{ε}{8 K B} and B = 1 + max {H (s) : s \in [T, T^{*} + 2 K τ]},

(2.39)

where

K \in N

satisfies

\begin{aligned} either T + 2 (K - 1) τ < T^{*} and T + (2 K - 1) τ \geq T^{*} or \\ T + (2 K - 1) τ < T^{*} and T + 2 K τ \geq T^{*} . \end{aligned}

(2.40)

We consider the following possible cases:

(i)

$t_{2} > t_{1} \geq T^{*}$ with $| t_{2} - t_{1} | < δ$ . From (2.1), (2.36), and (2.38), we conclude immediately that
$\begin{matrix} | (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \\ \leq \sum_{a = 1}^{\infty} \int_{t_{2} + (2 a - 1) τ}^{t_{2} + 2 a τ} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | + | f (t, x (β_{1} (s)), \dots, x (β_{l} (s))) |) d s \\ + \sum_{a = 1}^{\infty} \int_{t_{1} + (2 a - 1) τ}^{t_{1} + 2 a τ} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | + | f (t, x (β_{1} (s)), \dots, x (β_{l} (s))) |) d s \\ \leq 2 \int_{T^{*}}^{+ \infty} H (s) d s \\ < \frac{ε}{2}, \forall x \in A (N, M) . \end{matrix}$

(2.41)
(ii)

$T \leq t_{1} < t_{2} \leq T^{*}$ with $| t_{1} - t_{2} | < δ$ . In terms of (2.1) and (2.36)-(2.40), we deduce that
$\begin{matrix} | (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | \\ \leq | \sum_{a = 1}^{K} \int_{t_{2} + (2 a - 1) τ}^{t_{2} + 2 a τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (t, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s \\ + \sum_{a = 1}^{K} \int_{t_{1} + 2 a τ}^{t_{2} + (2 a - 1) τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (t, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s \\ - \sum_{a = 1}^{K} \int_{t_{1} + 2 a τ}^{t_{2} + (2 a - 1) τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (t, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s \\ - \sum_{a = 1}^{K} \int_{t_{1} + (2 a - 1) τ}^{t_{1} + 2 a τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (t, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s | \\ + \sum_{a = K + 1}^{\infty} \int_{t_{2} + (2 a - 1) τ}^{t_{2} + 2 a τ} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | + | f (t, x (β_{1} (s)), \dots, x (β_{l} (s))) |) d s \\ + \sum_{a = K + 1}^{\infty} \int_{t_{1} + (2 a - 1) τ}^{t_{1} + 2 a τ} (\sum_{i = 1}^{m} | q_{i} (s) | | x (α_{i} (s)) | + | r (s) | + | f (t, x (β_{1} (s)), \dots, x (β_{l} (s))) |) d s \\ \leq \sum_{a = 1}^{K} \int_{t_{1} + 2 a τ}^{t_{2} + 2 a τ} H (s) d s + \sum_{a = 1}^{K} \int_{t_{1} + (2 a - 1) τ}^{t_{2} + (2 a - 1) τ} H (s) d s + 2 \int_{T^{*}}^{+ \infty} H (s) d s \\ \leq 2 K B | t_{2} - t_{1} | + \frac{ε}{2} \\ < ε, \forall x \in A (N, M) . \end{matrix}$

(2.42)
(iii)

$γ \leq t_{1} < t_{2} \leq T$ with $| t_{1} - t_{2} | < δ$ . Equation (2.36) means that
$| (S_{L} x) (t_{2}) - (S_{L} x) (t_{1}) | = 0 < ε, \forall x \in A (N, M) .$

(2.43)

It follows from (2.41)-(2.43) that

S_{L} (A (N, M))

is equicontinuous in

[γ, + \infty)

. Thus Lemma 1.2 means that

S_{L}

has a fixed point

x \in A (N, M)

, that is, for any

t \geq T + τ

x (t) = L + \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (s, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s

and

x (t - τ) = L + \sum_{a = 1}^{\infty} \int_{t + 2 (a - 1) τ}^{t + (2 a - 1) τ} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (s, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s,

which give for any

t \geq T + τ

x (t) + x (t - τ) = 2 L + \int_{t}^{+ \infty} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) - r (s) + f (s, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s,

which implies that

\frac{d}{d t} [x (t) + x (t - τ)] + \sum_{i = 1}^{m} q_{i} (t) x (α_{i} (t)) + f (t, x (β_{1} (t)), \dots, x (β_{l} (t))) = r (t), t \geq T + τ,

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 $n \in N ∖ {1}$ . Assume that there exist $h \in C ([t_{0}, + \infty), R^{+})$ 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 > τ + | t_{0} | + | c | + | γ |

satisfying

\frac{1}{(n - 1)!} \int_{T}^{+ \infty} s^{n - 1} H (s) d s \leq min {M - L, L - N} .

(2.44)

Define a mapping

S_{L} : A (N, M) \to C B ([γ, + \infty), R)

by

(S_{L} x) (t) = {\begin{matrix} L + \frac{1}{(n - 2)!} \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} d u \int_{u}^{+ \infty} {(s - u)}^{n - 2} \\ \times (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s) \\ + f (s, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s, & t \geq T, \\ (S_{L} x) (T), & γ \leq t < T, \end{matrix}

(2.45)

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

Let

x \in A (N, M)

and

t \geq T

. By (2.44) and (2.45), we get

\begin{array}{rcl} | (S_{L} x) (t) - L | & = & \frac{1}{(n - 2)!} | \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} d u \int_{u}^{+ \infty} {(s - u)}^{n - 2} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) \\ + {(- 1)}^{n} r (s) + f (s, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s | \\ \leq & \frac{1}{(n - 2)!} \int_{t}^{+ \infty} d u \int_{u}^{+ \infty} {(s - u)}^{n - 2} H (s) d s \\ = & \frac{1}{(n - 2)!} \int_{t}^{+ \infty} d s \int_{t}^{s} {(s - u)}^{n - 2} H (s) d u \\ = & \frac{1}{(n - 1)!} \int_{t}^{+ \infty} {(s - t)}^{n - 1} H (s) d s \\ \leq & 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 $[γ, + \infty)$ .

Let

{x_{k}}_{k \in N} \subset A (N, M)

satisfy (2.8) for some

x \in A (N, M)

and

{G_{k}}_{k \in N}

be defined by (2.37). Using (2.8), (2.37), (2.45), the continuity of f and

β_{j}

for

j \in {1, 2, \dots, l}

, and the Lebesgue dominated convergence theorem, we conclude that for any

t \geq T

\begin{matrix} | (S_{L} x_{k}) (t) - (S_{L} x) (t) | \\ \leq \frac{1}{(n - 2)!} \sum_{a = 1}^{\infty} \int_{t + (2 a - 1) τ}^{t + 2 a τ} d u \int_{u}^{+ \infty} {(s - u)}^{n - 2} (\sum_{i = 1}^{m} | q_{i} (s) | | x_{k} (α_{i} (s)) - x (α_{i} (s)) | \\ + | f (s, x_{k} (β_{1} (t)), \dots, x_{k} (β_{l} (t))) - f (s, x (β_{1} (t)), \dots, x (β_{l} (s))) |) d s \\ \leq \frac{1}{(n - 2)!} \int_{t}^{+ \infty} d u \int_{u}^{+ \infty} {(s - u)}^{n - 2} G_{k} (s) d s \\ \leq \frac{1}{(n - 1)!} \int_{T}^{+ \infty} {(s - T)}^{n - 1} G_{k} (s) d s \\ \to 0 as k \to \infty, \end{matrix}

which yields

lim_{k \to \infty} ∥ S_{L} x_{k} - S_{L} x ∥ = lim_{k \to \infty} sup_{t \geq T} | (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

[γ, + \infty)

. Let

ε > 0

. Equation (2.34) ensures that there exists

T^{*} > T

large enough satisfying

\frac{1}{(n - 1)!} \int_{T^{*}}^{+ \infty} s^{n - 1} H (s) d s < \frac{ε}{16} .

(2.46)

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

δ = \frac{ε}{4 [n K min {M - L, L - N} + B {(T^{*} + 2 K τ)}^{n - 1}]} .

(2.47)

Now we have to consider the following possible cases:

(i)

$t_{2} > t_{1} \geq T^{*}$ with $| t_{1} - t_{2} | < δ$ . In terms of (2.45)-(2.47), we know that
$\begin{matrix} | (S_{L} x) (t_{1}) - (S_{L} x) (t_{2}) | \\ = \frac{1}{(n - 2)!} | \sum_{a = 1}^{\infty} \int_{t_{1} + (2 a - 1) τ}^{t_{1} + 2 a τ} d u \int_{u}^{+ \infty} {(s - u)}^{n - 2} (\sum_{i = 1}^{m} q_{i} (s) x (α_{i} (s)) + {(- 1)}^{n} r (s) \\ + f (s, x (β_{1} (s)), \dots, x (β_{l} (s)))) d s \\ - \sum_{a = 1}^{\infty} \int_{t_{2} + (2 a - 1) τ}^{t_{2} + 2 a τ} d u \int_{u}^{+ \infty} (s - u) \end{matrix}$