Bounded solutions of k-dimensional system of nonlinear difference equations of neutral type

Bounded solutions of k-dimensional system of nonlinear difference equations of neutral type

Małgorzata Migda

, Ewa Schmeidel

and Małgorzata Zdanowicz

1Poznan University of Technology, Piotrowo 3A, 60-965 Pozna ´n, Poland

2University of Bialystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland Received 3 August 2015, appeared 21 November 2015

Communicated by Josef Diblík

Abstract.Thek-dimensional system of neutral type nonlinear difference equations with delays in the following form







∆

x_i(n) +p_i(n)x_i(n−τ_i)=a_i(n)f_i(x_i+1(n−σ_i)) +g_i(n)_,

∆

x_k(n) +p_k(n)x_k(n−τ_k)=a_k(n)f_k(x₁(n−σ_k)) +g_k(n)_,

where i = _{1, . . . ,}k−1, is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded solutions of the above system with various(p_i(n)),i=1, . . . ,k,k≥2.

Keywords: system of difference equation,k-dimensional, neutral type, nonoscillatory solutions, boundedness, existence.

2010 Mathematics Subject Classification: 39A10, 39A22.

1 Introduction

In this paper we consider a nonlinear difference system ofk(k ≥2)equations of the form







∆

x_i(n) +p_i(n)x_i(n−τ_i)=a_i(n)f_i(x_i+1(n−σ_i)) +g_i(n),

∆

x_k(n) +p_k(n)x_k(n−τ_k)=a_k(n)f_k(x₁(n−σ_k)) +g_k(n),

(1.1)

where n ∈ _N0, i = 1, . . . ,k−1, ∆ is the forward difference operator defined by ∆u(n) = u(n+1)−u(n). Here R is a set of real numbers, N = {0, 1, 2, . . .} and σ_i,τ_i ∈ _N for i = 1, . . . ,k. Byn₀we denote max{τ₁, . . . ,τ_k,σ₁, . . . ,σ_k}, andN0= {n₀,n₀+1, . . .}. Moreovera_i = (a_i(n)),g_i = (g_i(n)),p_i = (p_i(n)) fori = 1, . . . ,k are given sequences of real numbers, x_i = (x_i(n))fori=1, . . . ,kare unknown real sequences and functions f_i: R→R. Throughout this paper X denotes an unknown vector (x₁, . . . ,x_k) and X(n) denotes (x₁(n), . . . ,x_k(n)) ∈ _R^k. For the elements ofR^k the symbol| · |stands for the maximum norm.

BCorresponding author. Email: eschmeidel@math.uwb.edu.pl

By B we denote the Banach space of all bounded sequences in R^k with the supremum norm, i.e.

B= (

X: N→_R^k :kXk=sup

n∈_N

|X(n)|<_∞ )

, and byBthe following subset ofB

B={X= (x₁, . . . ,x_k)∈ B:x_i is nonnegative or nonpositive fori=1, . . . ,k}.

A sequence of real numbers is said to be nonoscillatory if it is either eventually positive or eventually negative. By a solution of system (1.1) we mean a vectorXsuch that its compo- nents, i.e. x₁, . . . ,x_k, satisfy the system (1.1) for sufficiently largen. The solutionX of system (1.1) is called nonoscillatory if all its components are nonoscillatory. The solutionXof system (1.1) is called bounded if all its components are bounded.

Any higher-order nonlinear neutral difference equation could be rewritten as k-dimen- sional system of difference equations with one equation of neutral type but not vice-versa.

Higher-order nonlinear neutral difference equations have been studied by many authors, see for example [2–4,8–10,13–23], and the references cited therein. The theorems presented here generalize and improve the results obtained for three dimensional system in [13].

The following definition and theorems will be used in the sequel.

Definition 1.1(Uniformly Cauchy subset, [6]). A setΩof sequences inl^∞is uniformly Cauchy if for everyε > 0, there exists an integer n such that|X(i)−X(j)| < ε whenever i,j > n for anyX∈_Ω.

Lemma 1.2(Arzelà–Ascoli theorem, [1]). A bounded and uniformly Cauchy subset of l^∞is relatively compact.

Theorem 1.3(Krasnoselskii’s fixed point theorem, [7]). LetΩbe a bounded closed convex subset of a Banach space and let F, T be maps such that Fx+Ty ∈ _Ωfor every pair x,y ∈ Ω. If F is a contraction and T is completely continuous, then the equation Fx+Tx= x has a solution inΩ.

Theorem 1.4 (Schauder’s fixed point theorem, [5]). Let Ω be a nonempty, compact and convex subset of a Banach space and let T: Ω→_Ωbe continuous. Then T has a fixed point in M.

2 Main results

In this section, using the Krasnoselskii’s fixed point theorem and Schauder’s fixed point the- orem, we establish sufficient conditions for the existence of nonoscillatory bounded solutions of system (1.1).

Theorem 2.1. Assume that for i=1, . . . ,k

∑

|a_i(n)|<_∞, (2.1)

∑

|g_i(n)|<_∞, (2.2)

f_i: R→_Ris a continuous function (2.3)

and for any closed subset J ⊂_R

max

i=1,...,k

sup

t∈J

{|f_i(t)|}>0. (2.4)

Assume also that for each i= 1, . . . ,k the terms of sequence p_i are of the same sign for n ∈ _N0. If for each i=1, . . . ,k there exists a positive real constant c_pi such that

0≤ p_i(n)≤c_pi <1, n∈_N0, (2.5) or

−1<−c_pi ≤ p_i(n)≤0, n∈_N0, (2.6) then system(1.1)has a bounded nonoscillatory solution.

Proof. For the fixed positive real numberr we define a set Ω1 =

X∈ B: 1

8(1−cp_i)r≤ |x_i(n)| ≤r, i=1, . . . ,k, n∈_N

.

ClearlyΩ1 is a bounded closed convex subset of the Banach space B. Since condition (2.3) is satisfied, we can take

M_f = max

i=1,...,k

|f_i(t)|:|t| ∈ 1

8(1−c_pi)r,r

. From (2.1) and (2.2), there exists suchn₁ ∈_N0that

∑

|a_i(n)| ≤ (1−cp_i)r 8M_f ,

∑

|g_i(n)| ≤ (1−cp_i)r

4 .

Let I₁,I2,I3,I₄be subsets of the set{1, . . . ,k}and moreover, I_i∩I_j =_∅fori6= j,i,j=1, 2, 3, 4 and I₁∪I₂∪I₃∪I₄ ={1, . . . ,k}.

We consider four cases (i)

(0≤ p_i(n)≤ c_pi <1,

x_i(n)>0, fori∈ I₁, n ≥n₁, (ii)

(−1<−c_pi ≤ p_i(n)≤0, x_i(n)<0, fori∈ I₂, n ≥n₁, (iii)

(0≤ p_i(n)≤ c_pi <1,

x_i(n)<0, fori∈ I₃, n ≥n₁, (iv)

(−1<−c_pi ≤ p_i(n)≤0, x_i(n)>0, fori∈ I₄, n ≥n₁.

Next, we define the maps F,T: Ω1→ Bwhere

F =





 F₁

... F_k





, T=





 T₁

... T_k





,

(F_iX)(n) =











(F_iX)(n₁) _fori=_{1, . . . ,}k, 0≤n <n₁,

−p_i(n)x_i(n−τ_i) +⁽¹⁺₂^cpi)r fori∈ I₁∪I2, n≥n₁,

−p_i(n)x_i(n−τ_i) +⁽¹⁻₂^cpi)r fori∈ I₃∪I₄, n≥n₁,

(2.7)

and fori=1, . . . ,k−1 (T_iX)(n) =







(T_iX)(n₁) for 0≤n<n₁,

− _∑^∞

s=na_i(s) f_i(x_i+1(s−σ_i))− _∑^∞

s=ng_i(s) forn≥n₁, (2.8) and

(T_kX)(n) =







(T_kX)(n₁) for 0≤n <n₁,

− _∑^∞

s=n

a_k(s) f_k(x₁(s−σ_k))− _∑^∞

s=n

g_k(s) _forn≥n₁. (2.9) We will show that F and T satisfy the assumptions of Theorem 1.3. First we prove that if X, ¯X∈_Ω1, thenFX+TX¯ ∈_Ω1.

Forn ≥n₁,i∈ I₁∪I₂andi6=kwe have

(F_iX)(n) + (T_iX¯)(n) = −p_i(n)x_i(n−τ_i) +(1+c_pi)r 2

−

∑

s=n

a_i(s) f_i(x¯_i+₁(s−σ_i))−

∑

s=n

g_i(s)

≤ (1+c_pi)r

2 +

∑

|a_i(s)| |f_i(x¯_i+1(s−σ_i))|+

∑

|g_i(s)|

≤ ¹ 2r+ ¹

2c_pir+M_f ·(1−cp_i)r

8M_f + (1−cp_i)r 4

= ⁷ 8r+ ¹

8c_pir≤r.

Moreover,

(F_iX)(n) + (T_iX¯)(n) = −p_i(n)x_i(n−τ_i) + (1+c_pi)r 2

−

∑

s=n

a_i(s) f_i(x¯_i+₁(s−σ_i))−

∑

s=n

g_i(s)

≥ − |p_i(n)||x_i(n−τ_i)|+ (1+c_pi)r 2

−

∑

s=n

|a_i(s)| |f_i(x¯_i+₁(s−σ_i))| −

∑

s=n

|g_i(s)|

≥ −c_pir+ ¹ 2r+¹

2c_pir−M_f ·(1−c_pi)r

8M_f − (1−c_pi)r 4

= ¹

8(1−c_pi)r.

Forn≥n₁andi∈ I₃∪I₄, andi6=k we have

(F_iX)(n) + (T_iX¯)(n) = −p_i(n)x_i(n−τ_i) + (1−c_pi)r 2

−

∑

a_i(s) f_i(x¯_i+1(s−σ_i))−

∑

g_i(s)

≤ |p_i(n)||x_i(n−τ_i)|+ (1−c_pi)r 2 +

∑

|a_i(s)| |f_i(x¯_i+1(s−σ_i))|+

∑

|g_i(s)|

≤c_pir+ ¹ 2r−¹

2c_pir+M_f · (1−c_pi)r

8M_f +(1−c_pi)r 4

= ¹

8c_pir+⁷ 8r ≤r.

On the other hand,

(F_iX)(n) + (T_iX¯)(n) = −p_i(n)x_i(n−τ_i) + (1−c_pi)r 2

−

∑

a_i(s) f_i(x¯_i+1(s−σ_i))−

∑

g_i(s)

≥ (1−c_pi)r

2 −

∑

s=n

|a_i(s)| |f_i(x¯_i+₁(s−σ_i))| −

∑

s=n

|g_i(s)|

≥ ¹ 2r−¹

2c_pir−M_f · (1−c_pi)r

8M_f −(1−c_pi)r 4

= ¹

8(1−c_pi)r.

Fori=kthere is a different definition of the mappingT_k, but all estimations are analogous, and hence omitted.

The task is now to prove thatFis a contraction mapping. It is easy to see that

|(F_iX)(n)−(F_iX¯)(n)| ≤ |p_i(n)| |x_i(n−τ_i)−x¯_i(n−τ_i)|

≤cp_i|x_i(n−τ_i)−x¯_i(n−τ_i)|, for any X, ¯X∈_Ω1,i=_{1, . . . ,}kandn≥n₁. Hence

kFX−FX¯k ≤ max

i=1,...,k{c_pi} · kX−X^¯k, where, by (2.5) and (2.6), there is 0<max_i=1,...,k{c_pi}<1.

The next step is to show continuity ofT. LetX_j = (x_1j, . . . ,x_kj)∈_Ω1for j∈_Nand fori= 1, . . . ,kthere isx_ij(n)→x_i(n)asj→_{∞. SinceΩ1}is closed, we haveX= (x₁, . . . ,x_k)∈_Ω1. By (2.1), (2.3), (2.8) and Lebesgue’s dominated convergence theorem we obtain fori=1, . . . ,k−1

(T_iX_j)(n)−(T_iX)(n) ≤

∑

s=n

|a_i(s)| |f_i(x_i+1j(s−σ_i))− f_i(x_i+1(s−σ_i))| →0 if j→_∞, wheren ∈N. Analogously we conclude fori= k. Therefore

k(TX_j)−(TX)k →_{0 if} j→_∞,

and we see thatTis a continuous mapping.

In order to prove that T is completely continuous we can use Lemma1.2. Hence we have to show thatTΩ1is uniformly Cauchy (see Definition1.1). We show transformations for any T_i,i=1, . . . ,k−1. Similar arguments apply toT_k.

Let X ∈ _Ω1. We conclude from the assumptions (2.1), (2.2) and (2.3) that for any given ε>0 there exists an integern₂ >n₁ such that forn≥n₂ we have

∑

|a_i(s)| |f_i(x_i+1(s−σ_i))|+

∑

|g_i(s)|< ^ε 2. Hence, forn₄>n₃≥n₂, we obtain

|(T_iX)(n₄)−(T_iX)(n₃)|=

∑

a_i(s) f_i(x_i+1(s−σ_i)) +

∑

g_i(s)

−

∑

s=n₃

a_i(s) f_i(x_i+1(s−σ_i))−

∑

s=n₃

g_i(s)

<ε.

ThereforeTΩ1is uniformly Cauchy.

By Theorem1.3, there existsXsuch that(FX)(n) + (TX)(n) =X(n).

Finally, we verify that Xsatisfies system (1.1) forn≥ n₁. As(F_iX)(n) + (T_iX)(n) = x_i(n), i=1, . . . ,k, we have fori∈ I₁∪I₂ andi6=k

−p_i(n)x_i(n−τ_i) + (1+cp_i)r

2 −

∑

a_i(s) f_i(x_i+1(s−σ_i))−

∑

g_i(s) = x_i(n),

∆(x_i(n) +p_i(n)x_i(n−τ_i)) =−_∆

∑

s=n

a_i(s) f_i(x_i+₁(s−σ_i))−_∆

∑

s=n

g_i(s)_,

∆(x_i(n) +p_i(n)x_i(n−τ_i)) = a_i(n)f_i(x_i+1(n−σ_i)) +g_i(n). (2.10) Similarly, we get (2.10) fori∈ I₃∪I₄ andi6=k. In all cases, fori=k, the reasoning is also the same as above. The proof is complete.

Note that forp_i(n)≡0,i=1, . . . ,k, system (1.1) is not of the neutral type, but Theorem2.1 is still true.

Example 2.2. Consider a difference system































∆ x₁(n) + _2n¹ x₁(n−1)= ^{5n4−21n3+22n2+4n−8}

4n⁶−16n⁵+10n⁴+16n³−14n² x₂(n−2) + ¹

n²,

∆ x2(n)− _2n¹ x2(n−2)= ²ⁿ⁵_2n⁻¹⁷ⁿ_8−4n⁴⁺₇⁴³ⁿ_−6n³⁻₆₊⁴⁸ⁿ_8n5²⁺₊²⁷ⁿ_8n4⁻⁷ x³₃(n−1)− _n¹₂,

∆ x₃(n) + _2n¹ x₃(n−1)= ^3n3−3n2+2

4n⁵−2n⁴−6n³ x₄(n−1) + _n¹₃,

∆ x₄(n)− _2n¹ x₄(n−1)= ^2n2−5n+3

n⁴+n³ x²₁(n−1).

All assumptions of Theorem2.1are satisfied. The system above has the bounded (but not unique) solutionX= 1+_n¹, −2+ ¹

n²

, −1− ¹_n, 2− ¹_nforn≥3.

Theorem 2.3. Assume that conditions(2.1), (2.2),(2.3) and(2.4) are satisfied. If there exist positive real numbersc˜p_i, i =1, . . . ,k that

1< c˜_pi ≤ p_i(n)_, n∈_{N0, (2.11)} or

p_i(n)≤ −c˜p_i <−1, n∈_N0, (2.12) then system(1.1)has a bounded nonoscillatory solution.

Proof. We define a subsetΩ2of Bin the following way Ω2 =

X∈ B: 1

8(c˜_pi −1)r≤ |x_i(n)| ≤c˜_pir, i=1, . . . ,k, n∈_N

.

whereris a fixed positive real number. ObviouslyΩ2 is a bounded, closed and convex subset of B. Let us set

M˜ _f = max

i=1,...,k

|f_i(t)|:|t| ∈ 1

8(c˜p_i −1)r, ˜cp_ir

. From assumptions (2.1) and (2.2), we conclude that there existsn₅ ∈_N0that

∑

|a_i(n)| ≤ (c˜_pi −1)r 8 ˜M_f ,

∑

|g_i(n)| ≤ (c˜_pi −1)r

4 .

Let ˜I₁, ˜I₂, ˜I₃, ˜I₄ be such subsets of the set{1, . . . ,k}that ˜I_i∩I^˜_j =_∅ fori6= j,i,j= 1, 2, 3, 4 and I˜₁∪I^˜₂∪I^˜₃∪I^˜₄= {1, . . . ,k}.

Since we seek for the nonoscillatory solution, we consider the following cases (i)

(1<c˜p_i ≤ p_i(n),

x_i(n)>_{0, for}i∈ I^˜₁, n ≥n₅, (ii)

(p_i(n)≤ −c˜_pi <−1,

x_i(n)<0, fori∈ I^˜₂, n ≥n₅, (iii)

(1<c˜p_i ≤ p_i(n),

x_i(n)<0, fori∈ I^˜₃, n ≥n₅, (iv)

(p_i(n)≤ −c˜_pi <−1,

x_i(n)>0, fori∈ I^˜₄, n ≥n₅. We define the maps F,T: Ω2→ Bin the following way

(F_iX)(n) =











(F_iX)(n₅) for i=1, . . . ,k, 0≤n<n₅,

−^x_p^i(n+τi)

i(n+τ_i)+ ⁽¹⁺₂^c˜pi)r fori∈ I^˜₁∪I^˜₂, n≥n₅,

−^xi(n+τi)

pi(n+τ_i)+ ^(c˜pi−₂^1)r fori∈ I^˜₃∪I^˜₄, n≥n₅,

(2.13)

and fori=1, . . . ,k−1 (T_iX)(n) =







(T_iX)(n₅) for 0≤n<n₅,

−_p ¹

i(n+τi)

∑∞ s=n+τi

a_i(s) f_i(x_i+1(s−σ_i))− _p ¹

i(n+τi)

∑∞ s=n+τi

g_i(s) _forn≥ n₅, (2.14) and

(T_kX)(n) =







(T_kX)(n₅) for 0≤n< n₅,

−_p ¹

k(n+τ_k)

∑∞ s=n+τ_k

a_k(s)f_k(x₁(s−σ_k))− _p ¹

k(n+τ_k)

∑∞ s=n+τi

g_k(s) forn≥n5. (2.15) Let X, ¯X ∈ _Ω2,n ≥ n₅. Then alsoFX+TX¯ ∈ _Ω2. We will present all transformations for thei-th components ofFandT, wherei=1, . . . ,k−1. We have fori∈ I^˜₁∪I^˜₂

(_FiX)(_n) + (_TiX^¯)(_n) = − ^xi(n+τ_i)

p_i(n+τ_i)+(1+c˜_pi)r 2

− ¹

p_i(n+τ_i)

∑

a_i(s) f_i(x_i+1(s−σ_i))

− ¹

p_i(n+τ_i)

∑

g_i(s)

≤ (1+c˜_pi)r

2 + ¹

|p_i(n+τ_i)|

∑

|a_i(s)| |f_i(x_i+1(s−σ_i))|

+ ¹

|p_i(n+τ_i)|

∑

|g_i(s)|

≤ ¹

2c˜p_ir+ ¹

2r+M^˜ _f ·(c˜p_i−1)r

8 ˜M_f + (c˜p_i−1)r 4

= ⁷

8c˜_pir+ ¹

8r≤c˜_pir.

On the other hand,

(F_iX)(n) + (T_iX¯)(n) = − ^xi(n+τ_i)

p_i(n+τ_i)+(1+c˜_pi)r 2

− ¹

p_i(n+τ_i)

∑

a_i(s) f_i(x_i+1(s−σ_i))

− ¹

p_i(n+τ_i)

∑

g_i(s)

≥ −|x_i(n+τ_i)|

|p_i(n+τ_i)|+(1+c˜_pi)r 2

− ¹

|p_i(n+τ_i)|

∑

|a_i(s)| |f_i(x_i+1(s−σ_i))|

− ¹

|p_i(n+τ_i)|

∑

|g_i(s)|

≥ −r+ ¹

2c˜_pir+ ¹

2r−M^˜_f ·(c˜_pi−1)r

8 ˜M_f − (c˜_pi −1)r 4

= ¹

8(c˜_pi −1)r.

Next we have fori∈ I^˜₃∪I^˜₄

(F_iX)(n) + (T_iX¯)(n) = − ^xi(n+τ_i)

p_i(n+τ_i)+ (c˜p_i−1)r 2

− ¹

p_i(n+τ_i)

∑

a_i(s) f_i(x¯_i+1(s−σ_i))

− ¹

p_i(n+τ_i)

∑

g_i(s)

≤ |x_i(n+τ_i)|

|p_i(n+τ_i)| +(c˜p_i −1)r 2

+ ¹

|p_i(n+τ_i)|

∑

|a_i(s)| |f_i(x¯_i+₁(s−σ_i))|

+ ¹

|p_i(n+τ_i)|

∑

|g_i(s)|

≤r+¹

2c˜p_ir−¹

2r+M^˜ _f ·(c˜p_i −1)r

8 ˜M_f +(c˜p_i −1)r 4

= ⁷

8c˜p_ir+¹

8r ≤c˜p_ir.

Moreover,

(F_iX)(n) + (T_iX¯)(n) = − ^xi(n+τ_i)

p_i(n+τ_i)+(c˜_pi −1)r 2

− ¹

p_i(n+τ_i)

∑

a_i(s) f_i(x¯_i+1(s−σ_i))

− ¹

p_i(n+τ_i)

∑

g_i(s)

≥ (c˜p_i −1)r 2

− ¹

|p_i(n+τ_i)|

∑

|a_i(s)| |f_i(x¯_i+1(s−σ_i))|

− ¹

|p_i(n+τ_i)|

∑

|g_i(s)|

≥ ¹

2c˜_pir− ¹

2r−M^˜ _f ·(c˜_pi−1)r

8 ˜M_f − (c˜_pi−1)r 4

= ¹

8(c˜p_i −1)r.

To see that Fis a contraction mapping let us observe that fori=1, . . . ,k

|(F_iX)(n)−(F_iX¯)(n)| ≤ ¹

|p_i(n+τ_i)||x_i(n+τ_i)−x¯_i(n+τ_i)|

≤ ¹

˜

c_pi |x_i(n+τ_i)−x¯_i(n+τ_i)|. Hence

kFX−FX¯k ≤ ¹

min_i=1,...,k{c˜_pi}kX−X^¯k,

but _min ¹

i=1,...,k{c˜_pi} <1 by (2.11) and (2.12).

The proof of the continuity of the mapping T can be performed exactly in the same way as previously.

By virtue of Theorem1.3, there existsXthat(_FX)(_n) + (_TX)(_n) = _X(_n). Finally, we show that X satisfies system (1.1) forn ≥ n₅. Let (F_iX)(n) + (T_iX)(n) = x_i(n)for i= 1, . . . ,k. We show all transformations only for i ∈ I^˜₁∪I^˜₂ and i 6= k, because for the other cases they are analogous. Since

x_i(n) = − ^xi(n+τ_i)

p_i(n+τ_i)+(1+c˜_pi)r

2 − ¹

p_i(n+τ_i)

∑

a_i(s) f_i(x_i+₁(s−σ_i)

− ¹

p_i(n+τ_i)

∑

g_i(s), then we have

∆

x_i(n) + ^xi(n+τ_i) p_i(n+τ_i)

= −_∆ ¹

p_i(n+τ_i)

∑

a_i(s) f_i(x_i+1(s−σ_i))

!

−_∆ ¹

p_i(n+τ_i)

∑

g_i(s)

! . Therefore

1

p_i(n+τ_i+1)^∆

x_i(n+τ_i) +p_i(n+τ_i)x_i(n)+

∆ 1 p_i(n+τ_i)

x_i(n+τ_i) +p_i(n+τ_i)x_i(n)

=− ¹

p_i(n+τ_i+1)^∆

∑

a_i(s) f_i(x_i+₁(s−σ_i))

!

− ¹

p_i(n+τ_i+1)^∆

∑

g_i(s)

!

−

∆ 1 p_i(n+τ_i)

_∞

s=

∑

a_i(s) f_i(x_i+₁(s−σ_i))

!

−

∆ 1 p_i(n+τ_i)

_∞

s=

∑

g_i(s)

! . It is easy to notice that

−_∆

∑

a_i(s) f_i(x_i+1(s−σ_i))

!

= a_i(n+τ_i) f_i(x_i+1(n+τ_i−σ_i)), and

−_∆

∑

g_i(s)

!

=g_i(n+τ_i). Then

∆

x_i(n+τ_i) +p_i(n+τ_i)x_i(n)= a_i(n+τ_i) f_i(x_i+1(n+τ_i−σ_i)) +g_i(n+τ_i). Now we can transform the last equation into

∆

x_i(n) +p_i(n)x_i(n−τ_i)= a_i(n) f_i(x_i+1(n−σ_i)) +g_i(n)_. The proof is complete.

Example 2.4. Now, let us consider a difference system































∆ x₁(n) + 2+ ₂¹_n x₁(n−2) =−^{13·8n−1+3·4n−1}

16ⁿ−4·8ⁿ+4ⁿ⁺¹ x²₂(n−2) +₂¹_n,

∆ x2(n) + −1−₂¹n

x2(n−2)= ⁻₂³_·8^·4nⁿ+⁻4⁶·4^·2nn x3(n−1)−₂¹n,

∆ x₃(n) + 1+ ₂¹n

x₃(n−1) = ⁴₆^·_·⁴₈ⁿn⁺+³4^··²4ⁿⁿ x₄(n−1),

∆ x₄(n) + −1−₂¹_n x₄(n−1)= _8·16n⁴₊^·8₃₂ⁿ⁺_·8³_n^·4₊ⁿ_32·4n x₁²(n−2). All assumptions of Theorem2.3are satisfied. The sequence

X=

2+ ¹ 2ⁿ

,

−2+ ¹ 2ⁿ

,

−1− ¹ 2ⁿ

,

3+ ¹

2ⁿ

forn≥2 is the bounded solution of the above system.

Now we can formulate the theorem that join both Theorem2.1and Theorem2.3.

Let I₅,I₆,I₇,I₈ be subsets of the set{1, . . . ,k}such that I_i∩I_j = _∅ fori6= j, i,j =5, 6, 7, 8 and I5∪I6∪I7∪I8 ={1, . . . ,k}.

Theorem 2.5. Let assumptions(2.1), (2.2), (2.3) and(2.4)hold. If there exist positive real numbers cp_i, i∈ I₅∪I₆ andc˜p_i, i∈ I₇∪I₈that satisfy the inequalities

0≤ p_i(n)≤c_pi <1, for i∈ I₅, n∈_N0,

−1<−c_pi ≤ p_i(n)≤0, for i∈ I₆, n∈_N0, 1< c˜p_i ≤ p_i(n), for i∈ I₇, n∈_N0, p_i(n)≤ −c˜p_i <−1, for i∈ I8, n∈_N0, then system(1.1)has a bounded nonoscillatory solution.

Proof. For the fixed positive real numberr we define the set Ω3 =

X∈ B: 1

8(1−cp_i)r≤ |x_i(n)| ≤r, i∈ I₅∪I₆, 1

8(c˜_pi −1)r≤ |x_i(n)| ≤c˜_pir, i∈ I₇∪I₈, n∈_N

. Ω3 is bounded closed convex subset of the Banach spaceB.

Letn₆ =max{c₁,c₅}. From assumptions (2.1) and (2.2) we have

∑

|a_i(n)| ≤ (₁−c_pi)r

8M_f , i∈ I5∪I6,

∑

|g_i(n)| ≤ (1−c_pi)r

4 , i∈ I₅∪I₆,

∑

|a_i(n)| ≤ (c˜_pi −1)r

8 ˜M_f , i∈ I₇∪I₈,

∑

|g_i(n)| ≤ (c˜p_i −1)r

4 , i∈ I₇∪I₈,

where

M_f = max

i∈I5∪I6

|f_i(t)|:|t| ∈ 1

8(1−c_pi)r,r

, M˜_f = max

i∈I7∪I8

|f_i(t)|:|t| ∈ 1

8(c˜_pi −1)r, ˜c_pir

.

We can now proceed analogously as in the proof of Theorem 2.1 and Theorem 2.3. Re- peating reasoning in these proofs we define forn ≥ n₆ the maps F,T: Ω3 → Bby formulas (2.7)–(2.8) fori ∈ I₅∪I₆ and (2.13)–(2.15) fori∈ I₇∪I₈. The rest of the proof also runs as in Theorem2.1 and Theorem2.3.

In the next theorem we consider the case p_i(n)≡1,i =1, . . . ,kand get even better result than in the previous theorems.

Theorem 2.6. Assume that conditions (2.1), (2.2), (2.3) and (2.4) are satisfied. If p_i(_n) ≡ 1, i = 1, . . . ,k then for any real constants d₁, . . . ,d_k there exists a solution X of system (1.1) that lim_n→_∞X(n) = (d₁, . . . ,d_k).

Proof. Let d_i ∈ _R, i = 1, . . . ,k and let ε be any positive real number. There exists a constant M>0 such that

|f_i(t)| ≤ M for t ∈[d_i−ε,d_i+ε], i=1, . . . ,k.

Let us denote

S_ai(n) =

∑

|a_i(j)|, S_gi(n) =

∑

|g_i(j)|, i=_{1, . . . ,}k.

By (2.1) and (2.2) there exists such an indexn7≥n0 that forn≥n7we have S_ai(n)≤ ^ε

2M, and S_gi(n)≤ ^ε

2, i=1, . . . ,k.

We define a subsetΩ5ofB by

Ω5 =X∈ B:X(0) =· · ·= X(n7−1) =Dand|X(n)−D| ≤ M|S_A(n)|+|S_G(n)|forn≥n7 , where D= (d₁, . . . ,d_k), S_A = (S_a1, . . . ,S_ak), S_G = (S_g1, . . . ,S_gk). It is easy to check, thatΩ5 is the convex subset ofB. It can be also shown thatΩ5 is compact (see, for example, the proof of Theorem 1 in [12] or Lemma 4.7 in [11]). Now, forn≥0, we define a map

T: Ω5→ B, as follows, fori=_{1, . . . ,}k−₁

(T_iX)(n) =































d_i, for n<n₇, d_i− _∑^∞

j=₁

n+2jτ_i−1

∑

s=n+(2j−1)τ_i

a_i(s)f_i(x_i+1(s−σ_i))− _∑^∞

j=₁

n+2jτ_i−1

∑

s=n+(2j−1)τ_i

g_i(s), forn≥n₇ andτ_i >0,

d_i−¹_{2 ∑}^∞

s=_na_i(s)f_i(x_i+1(s−σ_i))−¹_{2 ∑}^∞

s=_ng_i(s), forn≥n₇ andτ_i =0,