# Solvability of a Second Order Nonlinear Neutral Delay Difference Equation

Authors: Zeqing Liu; Liangshi Zhao; Jeong Sheok Ume; Shin Min Kang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper studies the second-order nonlinear neutral delay difference equationΔ[anΔ(xn+bnxn−τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn, n≥n0. By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence
results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation,
respectively. Ten examples are given to illustrate the results presented in this paper.


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## Body

## 1. Introduction

We are concerned with the second-order nonlinear neutral delay difference equation of the form(1.1)Δ[anΔ(xn+bnxn-τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥n0,
where τ,k∈ℕ, n0∈ℕ0, {an}n∈ℕn0, {bn}n∈ℕn0 and {cn}n∈ℕn0 are real sequences with an≠0 for each n∈ℕn0, f,g∈C(ℕn0×ℝk,ℝ), and fl,gl:ℕn0→ℤ with(1.2)lim⁡n→∞fln=lim⁡n→∞gln=+∞,l∈{1,2,…,k}.
Note that a few special cases of (1.1) were studied in [1–9]. In particular, González and Jiménez-Melado [3] used a fixed-point theorem derived from the theory of measures of noncompactness to investigate the existence of solutions for the second-order difference equation(1.3)Δ(qnΔxn)+fn(xn)=0,n≥0.
By applying the Leray-Schauder nonlinear alternative theorem for condensing operators, Agarwal et al. [1] studied the existence of a nonoscillatory solution for the second-order neutral delay difference equation(1.4)Δ(anΔ(xn+pxn-τ))+F(n+1,xn+1-σ)=0,n≥0,
where p∈ℝ∖{±1}. Using the Banach contraction principle, Cheng [5] discussed the existence of a positive solution for the second-order neutral delay difference equation with positive and negative coefficients(1.5)Δ2(xn+pxn-m)+pnxn-k-qnxn-l=0,n≥n0,
where p∈ℝ∖{-1}, Liu et al. [6] and Liu et al. [7] extended the results due to cheng [5] and got the existence of uncountably many bounded nonoscillatory solutions for (1.1) and the second-order nonlinear neutral delay difference equation(1.6)Δ[anΔ(xn+bxn-τ)]+f(n,xn-d1n,xn-d2n,…,xn-dkn)=cn,n≥n0,
with respect to b∈ℝ, where f is Lipschitz continuous, respectively.The purpose of this paper is to establish the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions, respectively, for (1.1) by using the Krasnoselskii fixed point theorem, Schauder fixed point theorem, and a few new techniques. The results obtained in this paper improve essentially the corresponding results in [5–7] by removing the Lipschitz continuity condition. Ten nontrivial examples are given to reveal the superiority and applications of our results.


## 2. Preliminaries

Throughout this paper, we assume thatΔ is the forward difference operator defined by Δxn=xn+1-xn,Δ2xn=Δ(Δxn),ℝ=(-∞,+∞),ℝ+=[0,+∞),ℝ-=(-∞,0), and ℤ, ℕ, and ℕ0 stand for the sets of all integers, positive integers, and nonnegative integers, respectively,(2.1)Nn0={n:n∈N0withn≥n0},n0∈N0,β=min⁡{n0-τ,inf⁡{fln,gln:1≤l≤k,n∈Nn0}},Zβ={n:n∈Zwithn≥β}.
Let lβ∞ denote the Banach space of all bounded sequences in ℤβ with norm(2.2)‖x‖=sup⁡n∈Zβ|xn|forx={xn}n∈Zβ∈lβ∞,B(d,D)={x={xn}n∈Zβ∈lβ∞:‖x-d‖≤D}ford={d}n∈Zβ∈lβ∞,D>0
represent the closed ball centered at d and with radius D in lβ∞.By a solution of (1.1), we mean a sequence {xn}n∈Zβ with a positive integer T≥n0+τ+|β| such that (1.1) is satisfied for all n≥T. As is customary, a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory.Lemma 2.1 ([2]).
A bounded, uniformly Cauchy subsetY of lβ∞ is relatively compact.Lemma 2.2 (Krasnoselskii fixed point theorem [10]).
LetY be a nonempty bounded closed convex subset of a Banach space X and S,G:Y→X mappings such that Sx+Gy∈Y for every pair x,y∈Y. If S is a contraction and G is completely continuous, then
(2.3)Sx+Gx=x
has a solution in Y.Lemma 2.3 (Schauder fixed point theorem [10]).
LetY be a nonempty closed convex subset of a Banach space X and S:Y→Y a continuous mapping such that S(Y) is a relatively compact subset of X. Then, S has a fixed point in Y.Lemma 2.4.
Letτ∈ℕ,n0∈ℕ0 and {an}n∈ℕn0 be a nonnegative sequence. Then,
(2.4)∑i=0∞∑s=n0+iτ∞as<+∞⟺∑s=n0∞sas<+∞.
Moreover, if ∑i=0∞∑s=n0+iτ∞as<+∞, then
(2.5)∑i=0∞∑s=n0+iτ∞as≤∑s=n0∞(1+sτ)as<+∞.Proof.
For eacht∈ℝ, let [t] stand for the largest integer not exceeding t. It follows that
(2.6)∑i=0∞∑s=n0+iτ∞as=∑s=n0∞as+∑s=n0+τ∞as+∑s=n0+2τ∞as+⋯=∑s=n0∞(1+[s-n0τ])as≤∑s=n0∞(1+sτ)as,(2.7)lim⁡s→∞1+[s-n0/τ]s/τ=1.
Combining (2.6) and (2.7), we infer that (2.4) holds. Assume that ∑i=0∞∑s=n0+iτ∞as<+∞. In view of (2.4), we get that ∑s=n0∞sas<+∞, which gives that ∑s=n0∞as<+∞. It follows that
(2.8)∑s=n0∞(1+sτ)as<+∞.
This completes the proof.


## 3. Existence of Uncountably Many Bounded Positive Solutions

Now, we use the Krasnoselskii fixed point theorem to prove the existence of uncountably many bounded nonoscillatory, positive, and negative solutions of (1.1) under various conditions relative to the sequence {bn}n∈ℕβ⊂ℝ.Theorem 3.1.
Assume that there existn1∈ℕn0,d∈ℝ,D,b∈ℝ+∖{0} and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying
(3.1)1<|d|D<1-bb,|bn|≤b,∀n≥n1,(3.2)|f(n,u1,…,uk)|≤Fn,|g(n,u1,…,uk)|≤Gn,∀(n,ul)∈Nn0×[d-D,d+D],1≤l≤k,(3.3)∑i=n0+1∞1|ai|max⁡{Fi,∑j=n0i-1max⁡{Gj,|cj|}}<+∞.
Then, (1.1) has uncountably many bounded nonoscillatory solutions in B(d,D).Proof.
LetL∈(d-(1-b)D+b|d|,d+(1-b)D-b|d|). It follows from (3.3) that there exists T≥1+n0+n1+τ+|β| satisfying
(3.4)∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤(1-b)D-b|d|-|L-d|.
Define two mappings SL and GL:B(d,D)→lβ∞ by
(3.5)SLxn={L-bnxn-τ,n≥T,SLxT,β≤n<T,(3.6)GLxn={∑i=n∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,GLxT,β≤n<T,
for any x={xn}n∈ℤβ∈B(d,D).
Now, we assert that(3.7)SLx+GLy∈B(d,D),∀x,y∈B(d,D),(3.8)‖SLx-SLy‖≤b‖x-y‖,∀x,y∈B(d,D),(3.9)‖GLx‖≤D,∀x∈B(d,D).
It follows from (3.1), (3.2), and (3.4)–(3.6) that for any x={xn}n∈ℤβ,y={yn}n∈ℤβ∈B(d,D), and n≥T,
(3.10)|SLxn+GLyn-d|=|{f(i,yf1i,…,yfki)+∑j=n0i-1[g(j,yg1j,…,ygkj)-cj]}L-d-bnxn-τ+∑i=n∞1ai×{f(i,yf1i,…,yfki)+∑j=n0i-1[g(j,yg1j,…,ygkj)-cj]}|≤|L-d|+b(|d|+D)+∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤|L-d|+b(|d|+D)+(1-b)D-b|d|-|L-d|=D,|SLxn-SLyn|=|bn(xn-τ-yn-τ)|≤b‖x-y‖,|GLxn|≤∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤(1-b)D-b|d|-|L-d|≤D,
which imply that (3.7)–(3.9) hold.
Next, we prove thatGL is continuous and GL(B(d,D)) is uniformly Cauchy. It follows from (3.3) that for each ɛ>0, there exists M>T satisfying
(3.11)∑i=M∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]<ɛ4.
Let xν={xnν}n∈ℤβ and x={xn}n∈ℤβ∈B(d,D) satisfy that
(3.12)lim⁡ν→∞xν=x.
In view of (3.12) and the continuity of f and g, we know that there exists V∈ℕ such that
(3.13)∑i=TM-11|ai|[∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|]<ɛ2,∀ν≥V.
Combining (3.6), (3.11), and (3.13), we obtain that
(3.14)‖GLxν-GLx‖≤∑i=T∞1|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}≤∑i=TM-11|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}+2∑i=M∞1|ai|(Fi+∑j=n0i-1Gj)<ɛ,∀ν≥V,
which means that GL is continuous in B(d,D).
In view of (3.6) and (3.11), we obtain that for any x={xn}n∈ℤβ∈B(d,D) and t,h≥M(3.15)|GLxt-GLxh|≤|∑i=t∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|+|∑i=h∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|≤2∑i=M∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]<ɛ,
which implies that GL(B(d,D)) is uniformly Cauchy, which together with (3.9) and Lemma 2.1 yields that GL(B(d,D)) is relatively compact. Consequently, GL is completely continuous in B(d,D). Thus, (3.7), (3.8), and Lemma 2.2 ensure that the mapping SL+GL has a fixed point x={xn}n∈ℤβ∈B(d,D), which together with (3.5) and (3.6) implies that
(3.16)xn=L-bnxn-τ+∑i=n∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,
which yields that
(3.17)Δ[anΔ(xn+bnxn-τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥T.
That is, x={xn}n∈ℤβ is a bounded nonoscillatory solution of (1.1) in B(d,D).
LetL1,L2∈(d-(1-b)D+b|d|,d+(1-b)D-b|d|), and L1≠L2. Similarly, we can prove that for each l∈{1,2}, there exist a constant Tl≥1+n0+n1+τ+|β| and two mappings SLl and GLl:B(d,D)→lβ∞ satisfying (3.4)–(3.6), where T,L,SL, and GL are replaced by Tl,Ll,SLl, and GLl, respectively, and SLl+GLl has a fixed point zl∈B(d,D), which is a bounded nonoscillatory solution of (1.1); that is,
(3.18)znl=Ll-bnzn-τl+∑i=n∞1ai{f(i,zf1il,…,zfkil)+∑j=n0i-1[g(j,zg1jl,…,zgkjl)-cj]},∀n≥Tl,l∈{1,2}.
Note that (3.3) implies that there exists T3>max{T1,T2} satisfying
(3.19)∑i=T3∞1|ai|(Fi+∑j=n0i-1Gj)<|L1-L2|4.
Using (3.2), (3.18), and (3.19), we get that for any n≥T3,
(3.20)|zn1-zn2+bn(zn-τ1-zn-τ2)|=|L1-L2+∑i=n∞1ai{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|[f(j,zf1j1,…,zfkj1)-f(j,zf1j2,…,zfkj2)]+∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)]}|≥|L1-L2|-2∑i=T3∞1|ai|(Fi+∑j=n0i-1Gj)>|L1-L2|2>0,
that is, z1≠z2. Therefore, (1.1) possesses uncountably many bounded nonoscillatory solutions in B(d,D). This completes the proof.Theorem 3.2.
Assume that there existn1∈ℕn0,d∈ℝ,D,b∈ℝ+∖{0}, and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.21)|d|D<1-bb,|bn|≤b,∀n≥n1.
Then, (1.1) has uncountably many bounded solutions in B(d,D).The proof of Theorem3.2 is analogous to that of Theorem 3.1 and hence is omitted.Theorem 3.3.
Assume that there existn1∈ℕn0,d∈ℝ,D,b*,b*∈ℝ+∖{0}, and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.22)|d|D<b*-1b*+1,1<b*≤|bn|≤b*,∀n≥n1.
Then, (1.1) has uncountably many bounded solutions in B(d,D).Proof.
LetL∈(-(b*-1)D+(b*+1)|d|,(b*-1)D-(b*+1)|d|). It follows from (3.3) that there exists T≥1+n0+n1+τ+|β| satisfying
(3.23)∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤(b*-1)D-(b*+1)|d|-|L|.
Define two mappings SL and GL:B(d,D)→lβ∞ by
(3.24)SLxn={Lbn+τ-xn+τbn+τ,n≥T,SLxT,β≤n<T,(3.25)GLxn={1bn+τ∑i=n+τ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,GLxT,β≤n<T,
for any x={xn}n∈ℤβ∈B(d,D).
Now, we assert that (3.7), (3.9), and the below
(3.26)‖SLx-SLy‖≤1b*‖x-y‖,∀x,y∈B(d,D)
hold. It follows from (3.2) and (3.22)–(3.25) that for any x={xn}n∈ℤβ,y={yn}n∈ℤβ∈B(d,D), and n≥T,
(3.27)|SLxn+GLyn-d|=|{f(i,yf1i,…,yfki)+∑j=n0i-1[g(j,yg1j,…,ygkj)-cj]}Lbn+τ-d-xn+τbn+τ+1bn+τ∑i=n+τ∞1ai×{f(i,yf1i,…,yfki)+∑j=n0i-1[g(j,yg1j,…,ygkj)-cj]}|≤1b*|L-bn+τd|+|d|+Db*+1b*∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤1b*(|L|+b*|d|)+|d|+Db*+1b*[(b*-1)D-(b*+1)|d|-|L|]≤D,|SLxn-SLyn|=|1bn+τ(xn+τ-yn+τ)|≤1b*‖x-y‖,|GLxn|≤1b*∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤1b*[(b*-1)D-(b*+1)|d|-|L|]≤D,
which imply (3.7), (3.9), and (3.26).
Next, we show thatGL is continuous and GL(B(d,D)) is uniformly Cauchy. It follows from (3.3) that for each ɛ>0, there exists M>T satisfying (3.11). Let xν={xnν}n∈ℤβ and x={xn}n∈ℤβ∈B(d,D) with (3.12). It follows from (3.12) and the continuity of f and g that there exists V∈ℕ satisfying (3.13). In light of (3.11), (3.13), and (3.25), we deduce that
(3.28)‖GLxν-GLx‖≤1b*∑i=T+τ∞1|ai|[∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|]≤1b*∑i=TM-11|ai|[∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|]+2b*∑i=M∞1|ai|(Fi+∑j=n0i-1Gj)<ɛb*,∀ν≥V,
which yields that GL is continuous in B(d,D).
Using (3.1) and (3.25), we get that for any x={xn}n∈ℤβ∈B(d,D) and t,h≥M(3.29)|GLxt-GLxh|≤|1bt+τ∑i=t+τ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|+|1bh+τ∑i=h+τ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|≤2b*∑i=M∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]<ɛb*,
which means that GL(B(d,D)) is uniformly Cauchy, which together with (3.9) and Lemma 2.1 yields that GL(B(d,D)) is relatively compact. Consequently, GL is completely continuous in B(d,D). Thus, (3.22), (3.26), and Lemma 2.2 ensure that the mapping SL+GL has a fixed point x={xn}n∈ℤβ∈B(d,D); that is,
(3.30)xn=Lbn+τ-xn+τbn+τ+1bn+τ∑i=n+τ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,
which gives that
(3.31)Δ[anΔ(xn+bnxn-τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥T+τ.
That is, x={xn}n∈ℤβ is a bounded solution of (1.1) in B(d,D).
LetL1,L2∈(-(b*-1)D+(b*+1)|d|,(b*-1)D-(b*+1)|d|) and L1≠L2. Similarly, we conclude that for each l∈{1,2}, there exist a constant Tl≥1+n0+n1+τ+|β| and two mappings SLl and GLl:B(d,D)→lβ∞ satisfying (3.23)–(3.25), where T,L,SL, and GL are replaced by Tl,Ll,SLl, and GLl, respectively, and SLl+GLl has a fixed point zl∈B(d,D), which is a bounded solution of (1.1); that is,
(3.32)znl=Llbn+τ-zn+τlbn+τ+1bn+τ∑i=n+τ∞1ai{f(i,zf1il,…,zfkil)+∑j=n0i-1[g(j,zg1jl,…,zgkjl)-cj]},
for all n≥Tl and l∈{1,2}. Note that (3.3) implies that there exists T3>max{T1,T2} satisfying (3.19). By means of (3.2), (3.19), and (3.32), we infer that for any n≥T3,
(3.33)|zn1-zn2+zn+τl-zn+τ2bn+τ|=|L1-L2bn+τ+1bn+τ∑i=n+τ∞1ai{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|[f(j,zf1j1,…,zfkj1)-f(j,zf1j2,…,zfkj2)]+∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)]}|≥|L1-L2|b*-2b*∑i=T3∞1|ai|(Fi+∑j=n0i-1Gj)>|L1-L2|2b*>0,
that is, z1≠z2. Therefore, (1.1) possesses uncountably many bounded solutions in B(d,D). This completes the proof.Similar to the proofs of Theorems3.1 and 3.3, we have the following results.Theorem 3.4.
Assume that there existn1∈ℕn0,d∈ℝ,D,b*,b*∈ℝ+∖{0} and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.34)|d|>D,(b*2b*+b*b*2-b*2-b*2)D>(b*2-b*2-b*2b*+b*b*2)|d|,1<b*≤bn≤b*,∀n≥n1.
Then, (1.1) has uncountably many bounded nonoscillatory solutions in B(d,D).Theorem 3.5.
Assume that there existn1∈ℕn0,d,D∈ℝ+∖{0},b*,b*∈ℝ- and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.35)d>D,D(2+b*+b*)<d(b*-b*),b*≤bn≤b*<-1,∀n≥n1.
Then, (1.1) has uncountably many bounded positive solutions in B(d,D).Theorem 3.6.
Assume that there existn1∈ℕn0,D∈ℝ+∖{0},d,b*∈ℝ-∖{0},b*∈ℝ- and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.36)-d>D,d(b*-b*)>D(2+b*+b*),b*≤bn≤b*,∀n≥n1.
Then, (1.1) has uncountably many bounded negative solutions in B(d,D).Theorem 3.7.
Assume that there existn1∈ℕn0,d∈ℝ∖{0},b*,D∈ℝ+∖{0} and two negative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.37)1<|d|D<2-b*b*,0≤bn≤b*,∀n≥n1.
Then, (1.1) has uncountably many bounded nonoscillatory solutions in B(d,D).Theorem 3.8.
Assume that there existn1∈ℕn0,d∈ℝ∖{0},b*∈ℝ-∖{0},D∈ℝ+∖{0} and two negative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.38)1<|d|D<b*+2-b*,b*≤bn≤0,∀n≥n1.
Then, (1.1) has uncountably many bounded nonoscillatory solutions in B(d,D).Next, we investigate the existence of uncountably bounded nonoscillatory solutions for (1.1) with the help of the Schauder fixed point theorem under the conditions of bn=±1.Theorem 3.9.
Assume that there existn1∈ℕn0,d∈ℝ,D∈ℝ+∖{0} and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2), (3.3), and
(3.39)|d|>D,bn=1,∀n≥n1.
Then, (1.1) has uncountably many bounded nonoscillatory solutions in B(d,D).Proof.
LetL∈(d-D,d+D). It follows from (3.3) that there exists T≥1+n0+n1+τ+|β| satisfying
(3.40)∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤D-|L-d|.
Define a mapping SL:B(d,D)→lβ∞ by
(3.41)SLxn={L+∑s=1∞∑i=n+(2s-1)τn+2sτ-11ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,SLxT,β≤n<T,
for any x={xn}n∈ℤβ∈B(d,D).
Now, we prove that(3.42)SLx∈B(d,D),‖SLx‖≤|L|+D,∀x∈B(d,D).
It follows from (3.2) and (3.39)–(3.41) that for any x={xn}n∈ℤβ∈B(d,D) and n≥T,
(3.43)|SLxn-d|=|L-d+∑s=1∞∑i=n+(2s-1)τn+2sτ-11ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|≤|L-d|+∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤|L-d|+D-|L-d|=D,|SLxn|≤|L|+∑i=T∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤|L|+D-|L-d|≤|L|+D,
which imply (3.42).
Next, we prove thatSL is continuous and SL(B(d,D)) is uniformly Cauchy. It follows from (3.3) that for each ɛ>0, there exists M>T satisfying (3.11). Let xν={xnν}n∈ℤβ and x={xn}n∈ℤβ∈B(d,D) satisfying (3.12). It follows from (3.12) and the continuity of f and g that there exists V∈ℕ satisfying (3.13). Combining (3.11), (3.13), and (3.41), we infer that
(3.44)‖SLxν-SLx‖≤sup⁡n≥T{+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}∑s=1∞∑i=n+(2s-1)τn+2sτ-11|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}}≤∑i=T∞1|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}≤∑i=TM-11|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}+2∑i=M∞1|ai|(Fi+∑j=n0i-1Gj)<ɛ,∀ν≥V,
which implies that SL is continuous in B(d,D).
By means of (3.11) and (3.41), we get that for any x={xn}n∈ℤβ∈B(d,D) and t,h≥M(3.45)|SLxt-SLxh|≤|∑s=1∞∑i=t+(2s-1)τt+2sτ-11ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|+|∑s=1∞∑i=h+(2s-1)τh+2sτ-11ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|≤2∑i=M∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]<ɛ,
which means that SL(B(d,D)) is uniformly Cauchy, which together with (3.42) and Lemma 2.1 yields that SL(B(d,D)) is relatively compact. It follows from Lemma 2.3 that the mapping SL has a fixed point x={xn}n∈ℤβ∈B(d,D); that is,
(3.46)xn=L+∑s=1∞∑i=n+(2s-1)τn+2sτ-11ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,
which give that
(3.47)Δ[anΔ(xn+xn-τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥T+τ.
That is, x={xn}n∈ℤβ∈B(d,D) is a bounded nonoscillatory solution of (1.1).
LetL1,L2∈(d-D,d+D) and L1≠L2. Similarly, we infer that for each l∈{1,2}, there exist a constant Tl≥1+n0+n1+τ+|β| and a mapping SLl:B(d,D)→lβ∞ satisfying (3.41), where L,T,andSL are replaced by Tl,Ll, and SLl, respectively, and SLl has a fixed point zl∈B(d,D), which is a bounded nonoscillatory solution of (1.1); that is,
(3.48)znl=Ll+∑s=1∞∑i=n+(2s-1)τn+2sτ-11ai{f(i,zf1il,…,zfkil)+∑j=n0i-1[g(j,zg1jl,…,zgkjl)-cj]},n≥Tl,
for l∈{1,2}. Note that (3.3) implies that there exists T3>max{T1,T2} satisfying (3.19). Using (3.2), (3.19), and (3.48), we conclude that for any n≥T3(3.49)|zn1-zn2|=|eftdot?>+∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)]}L1-L2+∑s=1∞∑i=n+(2s-1)τn+2sτ-11ai{∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)][f(i,zf1i1,…,zfki1)-f(i,zf1i2,…,zfki2)]+∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)]}|≥|L1-L2|-2∑s=1∞∑i=n+(2s-1)τn+2sτ-11|ai|(Fi+∑j=n0i-1Gj)≥|L1-L2|-2∑i=T3∞1|ai|(Fi+∑j=n0i-1Gj)>|L1-L2|2>0,
which gives that z1≠z2. Therefore, (1.1) possesses uncountably many bounded nonoscillatory solutions in B(d,D). This completes the proof.Theorem 3.10.
Assume that there existn1∈ℕn0,d∈ℝ,D∈ℝ+∖{0} and two nonnegative sequences {Fn}n∈ℕn0 and {Gn}n∈ℕn0 satisfying (3.2),
(3.50)|d|>D,bn=-1,∀n≥n1,(3.51)∑s=1∞∑i=n0+sτ∞1|ai|max⁡{Fi,∑j=n0i-1max⁡{Gj,|cj|}}<+∞.
Then, (1.1) has uncountably many bounded nonoscillatory solutions in B(d,D).Proof.
LetL∈(d-D,d+D). It follows from (3.41) that there exists T≥1+n0+n1+τ+|β| satisfying
(3.52)∑s=1∞∑i=T+sτ∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤D-|L-d|.
Define a mapping SL:B(d,D)→lβ∞ by
(3.53)SLxn={L-∑s=1∞∑i=n+sτ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥TSLxT,β≤n<T,
for any x={xn}n∈ℤβ∈B(d,D). It follows from (3.2), (3.52), and (3.53) that for any x={xn}n∈ℤβ∈B(d,D) and n≥T(3.54)|SLxn-d|=|L-d-∑s=1∞∑i=n+sτ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|≤|L-d|+∑s=1∞∑i=T+sτ∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤|L-d|+D-|L-d|=D|SLxn|≤|L|+∑s=1∞∑i=T+sτ∞1|ai|[∑j=n0i-1(Gj+|cj|)+Fi]≤|L|+D,
which imply (3.42).
Next, we show thatSL is continuous and SL(B(d,D)) is uniformly Cauchy. It follows from (3.51) and Lemma 2.4 that for each ɛ>0, there exists M>1+T+τ satisfying
(3.55)∑i=M∞(1+iτ)1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]<ɛ4.
Let xν={xnν}n∈ℤβ and x={xn}n∈ℤβ∈B(d,D) satisfying (3.12). By means of (3.12) and the continuity of f and g, we deduce that there exists V∈ℕ satisfying
(3.56)∑i=T+τM-1(1+iτ)1|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}<ɛ2,∀ν≥V.
In light of (3.2), (3.53)–(3.56) and Lemma 2.4, we conclude that
(3.57)‖SLxν-SLx‖≤∑s=1∞∑i=T+sτ∞1|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}≤∑i=T+τ∞(1+iτ)1|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}≤∑i=T+τM-1(1+iτ)1|ai|{∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)||f(i,xf1iν,…,xfkiν)-f(i,xf1i,…,xfki)|+∑j=n0i-1|g(j,xg1jν,…,xgkjν)-g(j,xg1j,…,xgkj)|}+2∑i=M∞(1+iτ)1|ai|(Fi+∑j=n0i-1Gj)<ɛ,∀ν≥V,
which implies that SL is continuous in B(d,D).
By virtue of (3.53), (3.55), and Lemma 2.4, we get that for any x={xn}n∈ℤβ∈B(d,D) and t,h≥M,
(3.58)|SLxt-SLxh|≤|∑s=1∞∑i=t+sτ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|+|∑s=1∞∑i=h+sτ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]}|≤2∑s=1∞∑i=M+sτ∞1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]≤2∑i=M∞(1+iτ)1|ai|[Fi+∑j=n0i-1(Gj+|cj|)]<ɛ,
which means that SL(B(d,D)) is uniformly Cauchy, which together with (3.42) and Lemma 2.1 yields that SL(B(d,R)) is relatively compact. It follows from Lemma 2.3 that the mapping SL has a fixed point x={xn}n∈ℤβ∈B(d,R); that is,
(3.59)xn=L-∑s=1∞∑i=n+sτ∞1ai{f(i,xf1i,…,xfki)+∑j=n0i-1[g(j,xg1j,…,xgkj)-cj]},n≥T,
which means that
(3.60)Δ[anΔ(xn-xn-τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥T+τ.
That is, x={xn}n∈ℤβ is a bounded nonoscillatory solution of (1.1) in B(d,D).
LetL1,L2∈(d-D,d+D) and L1≠L2. Similarly, we conclude that for each l∈{1,2}, there exist a positive integer Tl≥1+n0+n1+τ+|β| and a mapping SLl:B(d,D)→lβ∞ satisfying (3.53), where T,L,andSL are replaced by Tl,Ll, and SLl, respectively, and SLl has a fixed point zl∈B(d,D), which is a bounded nonoscillatory solution of (1.1); that is,
(3.61)znl=Ll-∑s=1∞∑i=n+sτ∞1ai{f(i,zf1il,…,zfkil)+∑j=n0i-1[g(j,zg1jl,…,zgkjl)-cj]},n≥T,
for l∈{1,2}. Note that (3.41) implies that there exists T3>max{T1,T2} satisfying
(3.62)∑s=1∞∑i=T3+sτ∞1|ai|(Fi+∑j=n0i-1Gj)<|L1-L2|4,
which together with (3.2), (3.53), and (3.61) gives that
(3.63)|zn1-zn2|=|+∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)]}L1-L2-∑s=1∞∑i=n+sτ∞1ai{∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)][f(j,zf1j1,…,zfkj1)-f(j,zf1j2,…,zfkj2)]+∑j=n0i-1[g(j,zg1j1,…,zgkj1)-g(j,zg1j2,…,zgkj2)]}|≥|L1-L2|-2∑s=1∞∑i=T3+sτ∞1|ai|(Fi+∑j=n0i-1Gj)>|L1-L2|2>0,∀n≥T3,
that is, z1≠z2. Therefore, (1.1) possesses uncountably many bounded nonoscillatory solutions in B(d,D). This completes the proof.Remark 3.11.
Theorems3.1 and 3.4–3.10 generalize Theorem  1 in [5] and Theorems 2.1–2.7 in [6, 7], respectively. The examples in Section 4 reveal that Theorems 3.1 and 3.4–3.10 extend authentically the corresponding results in [5–7].


## 4. Examples and Applications

Now, we construct ten examples to explain the advantage and applications of the results presented in Section3. Note that Theorem  1 in [5] and Theorem   2.1–2.7 in [6, 7] are invalid for Examples 4.1–4.10, respectively.Example 4.1.
Consider the second-order nonlinear neutral delay difference equation(4.1)Δ[(n6ln⁡n)Δ(xn+(-1)n(n-1)3n+1xn-τ)+5n3x2n+1n+x3n-152]+n2xn3+1x2n2+3=(n-1)2,n≥2,
where n0=2 and τ∈ℕ are fixed. Let n1=6, k=2, d=±6, D=5, b=1/3, β=min{2-τ,-9}, and
(4.2)an=n6ln⁡n,bn=(-1)n(n-1)3n+1,cn=(n-1)2,f(n,u,v)=5un3n+v2,f1n=2n+1,f2n=3n-15,g(n,u,v)=uvn2,g1n=n3+1,g2n=2n2+3,Fn=55n2,Gn=121n2,(n,u,v)∈Nn0×[d-D,d+D]2.
It is easy to show that (3.1)–(3.3) hold. It follows from Theorem 3.1 that (4.1) possesses uncountably many bounded nonoscillatory solutions in B(d,D).Example 4.2.
Consider the second-order nonlinear neutral delay difference equation(4.3)Δ[n2(1-2n)3Δ(xn+(-1)n-1(3+sin⁡n)4+sin⁡nxn-τ)+3n2xn2x5n]+2nx2n-91+n3|x3n3|=n2,n≥2,
where n0=2 and τ∈ℕ are fixed. Let n1=5, k=2, d=±2, D=11, b=5/6, β=min{2-τ,-5}, and
(4.4)an=n2(1-2n)3,bn=(-1)n-1(3+sin⁡⁡n)4+sin⁡⁡n,cn=n2,f(n,u,v)=3n2uv,f1n=n2,f2n=5n,g(n,u,v)=2nu1+n3|v|,g1n=2n-9,g2n=3n3,Fn=507n2,Gn=26n,(n,u,v)∈Nn0×[d-D,d+D]2.
It is clear that (3.2), (3.3), and (3.21) hold. It follows from Theorem 3.2 that (4.3) possesses uncountably many bounded solutions in B(d,D).Example 4.3.
Consider the second-order nonlinear neutral delay difference equation(4.5)Δ[(n8sin⁡(-1)nn)Δ(xn+(-1)n-1(4n3+1)n3+2n+2xn-τ)+4n2xn+131+nxn2-12]+n4x2n2xn-33=n3,n≥1,
where n0=1 and τ∈ℕ are fixed. Let n1=10, k=2, d=±1, D=5, b*=3, b*=4, β=min{1-τ,-2}, and
(4.6)an=n8sin⁡(-1)nn,bn=(-1)n-1(4n3+1)n3+2n+2,cn=n3,f(n,u,v)=4nu31+nv2,f1n=n+1,f2n=n2-1,g(n,u,v)=n4u2v3,g1n=2n,g2n=n-3,Fn=864n2,Gn=7776n4,(n,u,v)∈Nn0×[d-D,d+D]2.
It is easy to see that (3.2), (3.3), and (3.22) hold. It follows from Theorem 3.3 that (4.5) possesses uncountably many bounded solutions in B(d,D).Example 4.4.
Consider the second-order nonlinear neutral delay difference equation(4.7)Δ[n5(sin⁡1n)-1/(ln⁡n)Δ(xn+3n2+1n2+n+2xn-τ)+nxn2+321+ncos⁡2x2n]+n4-x3nn2+xn+12=(-1)nn-1,n≥2,
where n0=2 and τ∈ℕ are fixed. Let n1=5, k=2, d=±4, D=3, b*=2, b*=3, β=2-τ, and
(4.8)an=n5(sin⁡1n)-1/(ln⁡n),bn=3n2+1n2+n+2,cn=(-1)nn-1,f(n,u,v)=nu21+ncos⁡2v,f1n=n2+3,f2n=2n,g(n,u,v)=n4-un2+v2,g1n=3n,g2n=n+1,Fn=49n,Gn=n2+7,(n,u,v)∈Nn0×[d-D,d+D]2.
It is easy to show that (3.2), (3.3), and (3.34) hold. It follows from Theorem 3.4 that (4.7) has uncountably many bounded nonoscillatory solutions in B(d,D).Example 4.5.
Consider the second-order nonlinear neutral delay difference equation(4.9)Δ[n6(1+11+n)3nΔ(xn-6-2ln⁡(1+n2)5+ln⁡(1+n2)xn-τ)+nx2n+1xn-94]+n2x3nx4n3=2n2,n≥0,
where n0=0 and τ∈ℕ are fixed. Let n1=10, k=2, d=9, D=7, b*=-2, b*=-17/15, β=min{-τ,-9}, and
(4.10)an=n6(1+11+n)3n,bn=-6-2ln⁡(1+n2)5+ln⁡(1+n2),cn=2n2,f(n,u,v)=nuv4,f1n=2n+1,f2n=n+2,g(n,u,v)=n2uv3,g1n=3n,g2n=4n,Fn=1048576n,Gn=65536n2,(n,u,v)∈Nn0×[d-D,d+D]2.
It is easy to show that (3.2), (3.3), and (3.35) hold. It follows from Theorem 3.5 that (4.9) has uncountably many bounded positive solutions in B(d,D).Example 4.6.
Consider the second-order nonlinear neutral delay difference equation(4.11)Δ[(-1)n-1n17(n-4)5Δ(xn+3-2n45+n+n4xn-τ)+n18-n7x3n-192ln⁡(3+n5|x2n|)]+n2xn+521+x2n+32=n15,n≥5,
where n0=5 and τ∈ℕ are fixed. Let n1=5, k=2, d=-9, D=7, b*=-2, b*=-1715, β=min{5-τ,-4}, and
(4.12)an=(-1)n-1n17(n-4)5,bn=3-2n45+n+n4,cn=n15,f(n,u,v)=n18-n7u2ln⁡(3+n5|v|),f1n=3n-18,f2n=2n,g(n,u,v)=n2u21+v2,g1n=n+5,g2n=2n+3,Fn=n18+256n7,Gn=256n2,(n,u,v)∈Nn0×[d-D,d+D]2.
It is easy to show that (3.2), (3.3), and (3.36) hold. It follows from Theorem 3.6 that (4.11) has uncountably many bounded negative solutions in B(d,D).Example 4.7.
Consider the second-order nonlinear neutral delay difference equation(4.13)Δ[n8ln⁡(cos⁡πn)Δ(xn+5n2-2n+1706n2+n+1xn-τ)-n2x2nn+x4n2]+n2-x2n-331+n|xn-6|=n2(2-n),n≥3,
where n0=3 and τ∈ℕ are fixed. Let n1=60, k=2, d=±7, D=6, b*=5/6, β=min{3-τ,-3}, and
(4.14)an=n8ln⁡(cos⁡πn),bn=5n2-2n+1706n2+n+1,cn=n2(2-n),f(n,u,v)=-n2un+v2,f1n=2n,f2n=4n,g(n,u,v)=n2-u31+n|v|,g1n=2n-3,g2n=n-6,Fn=13n,Gn=2197+n2,(n,u,v)∈Nn0×[d-D,d+D]2.
It is clear (3.2), (3.3), and (3.37) hold. It follows from Theorem 3.7 that (4.13) has uncountably many bounded nonoscillatory solutions in B(d,D).Example 4.8.
Consider the second-order nonlinear neutral delay difference equation(4.15)Δ[n6Δ(xn+1-8n43+9n4xn-τ)+n3xn+1-(n2+1)xn-321+nxn-32]+n2+x2n+532+n2|x3n-1|=(-1)nn3,n≥4,
where n0=4 and τ∈ℕ are fixed. Let n1=4, k=2, d=±7, D=6, b*=-8/9, β=min{4-τ,1}, and
(4.16)an=n6,bn=1-8n43+9n4,cn=(-1)nn3,f(n,u,v)=n3u-(n2+1)v21+nv2,f1n=n+1,f2n=n-3,g(n,u,v)=n2+u32+n2|v|,g1n=2n+5,g2n=3n-1,Fn=13n3+169(n2+1),Gn=n2+2197,(n,u,v)∈Nn0×[d-D,d+D]2.
It is clear (3.2), (3.3), and (3.38) hold. It follows from Theorem 3.8 that (4.15) has uncountably many bounded nonoscillatory solutions in B(d,D).Example 4.9.
Consider the second-order nonlinear neutral delay difference equation(4.17)Δ[n6(1+1n)nΔ(xn+xn-τ)+(n-1)2-nx3n+1n2ln⁡(3+nx6n2)]+1-n3+nx7n21+n+n5|x3n5x7n3|=(-1)nn2n3+1,n≥1,
where n0=1 and τ∈ℕ are fixed. Let n1=1, k=2, d=±6, D=2, β=1-τ, and
(4.18)an=n6(1+1n)n,cn=(-1)nn2n3+1,f(n,u,v)=(n-1)2-nun2ln⁡(3+nv2),f1n=3n+1,f2n=6n,g(n,u,v)=1-n3+nu21+n+n5|v5u3|,g1n=7n,g2n=3n,Fn=1+8n,Gn=1+64n+n3,(n,u,v)∈Nn0×[d-D,d+D]2.
It is clear (3.2), (3.3), and (3.39) hold. It follows from Theorem 3.9 that (4.17) has uncountably many bounded nonoscillatory solutions in B(d,D).Example 4.10.
Consider the second-order nonlinear neutral delay difference equation(4.19)Δ[n4(1-2n)(2n-3)2Δ(xn-xn-τ)+5nx3n3-n+22+n3|x5n5+3|]+n2x3n3+4x4n3-5=n2,n≥2,
where n0=2 and τ∈ℕ are fixed. Let n1=2, k=2, d=±10, D=6, β=2-τ, and
(4.20)an=n4(1-2n)(2n-3)2,cn=n2,f(n,u,v)=5nu2+n3|v|,f1n=3n3-n+2,f2n=5n5+3,g(n,u,v)=uvn2,g1n=3n3+4,g2n=4n3-5,Fn=40n,Gn=256n2,(n,u,v)∈Nn0×[d-D,d+D]2.
It is clear (3.2), (3.50), and (3.51) hold. It follows from Theorem 3.10 that (4.19) possesses uncountably bounded nonoscillatory solutions in B(d,D).



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*Source: 328914-2011-12-26.xml*