Sufficient conditions for the existence of non-oscillatory solutions are established by using Krasnoselskii fixed point theorem

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF NON-OSCILLATORY SOLUTIONS FOR A HIGHER-ORDER NONLINEAR NEUTRAL DIFFERENCE

EQUATION

ZHENYU GUO, MIN LIU

Abstract. This article concerns the solvability of the higher-order nonlinear neutral delay difference equation

∆

“

akn. . .∆`

a2n∆(a1n∆(xn+bnxn−d))´” +

s

X

j=1

pjnfj(xn−r_jn) =qn, wheren≥n0≥0,d, k, j, sare positive integers, fj:R→Randxfj(x)≥0 forx6= 0. Sufficient conditions for the existence of non-oscillatory solutions are established by using Krasnoselskii fixed point theorem. Five theorems are stated according to the range of the sequence{bn}.

1. Introduction and preliminaries

Interest in the solvability of difference equations has increased lately, as inferred from the number of related publications; see for example the references in this article and their references. Authors have examined various types difference equations, as follows:

∆(an∆xn) +pnxg(n)= 0, n≥0, in [14], (1.1)

∆(an∆xn) =qnxn+1, ∆(an∆xn) =qnf(xn+1), n≥0, in [11], (1.2)

∆²(xn+px_n−m) +pnx_n−k−qnx_n−l= 0, n≥n0, in [6], (1.3)

∆²(xn+px_n−k) +f(n, xn) = 0, n≥1, in [10], (1.4)

∆²(xn−pxn−τ) =

m

X

i=1

qifi(xn−σ_i), n≥n0, in [9], (1.5)

∆(an∆(xn+bx_n−τ)) +f(n, x_n−d1n, x_n−d2n, . . . , x_n−dkn) =cn,

n≥n₀, in [8], (1.6)

∆^m(xn+cxn−k) +pnxn−r= 0, n≥n0, in [15], (1.7)

∆^m(xn+cnx_n−k) +pnf(x_n−r) = 0, n≥n0, in [3, 4, 12, 13], (1.8)

2000Mathematics Subject Classification. 34K15, 34C10.

Key words and phrases. Nonoscillatory solution; neutral difference equation;

Krasnoselskii fixed point theorem.

c

2010 Texas State University - San Marcos.

Submitted July 30, 2010. Published October 14, 2010.

1

∆^m(x_n+cx_n−k) +

u

X

s=1

p^s_nf_s(x_n−rs) =q_n, n≥n₀, in [16], (1.9)

∆^m(xn+cxn−k) +pnxn−r−qnxn−l= 0, n≥n0, in [17]. (1.10) Motivated by the above publications, we investigate the higher-order nonlinear neutral difference equation

∆

a_kn. . .∆ a_2n∆(a_1n∆(x_n+b_nx_n−d)) +

s

X

j=1

p_jnf_j(x_n−rjn) =q_n, (1.11) where n ≥ n0 ≥ 0, d, k, j, s are positive integers, {ain}_n≥n0 (i = 1,2, . . . , k), {bn}_n≥n0, {pjn}_n≥n0 (1 ≤ j ≤ s) and {qn}_n≥n0 are sequences of real numbers, rjn∈Z(1≤j ≤s, n0≤n),fj:R→Randxfj(x)≥0 for x6= 0 (j= 1,2, . . . , s).

Clearly, difference equations (1.1)–(1.10) are special cases of (1.11), for which we use Krasnoselskii fixed point theorem to obtain non-oscillatory solutions.

Lemma 1.1(Krasnoselskii Fixed Point Theorem). LetΩbe a bounded closed con- vex subset of a Banach space X and T1, T2 : S → X satisfy T1x+T2y ∈ Ω for each x, y ∈ Ω. If T1 is a contraction mapping and T2 is a completely continuous mapping, then the equation T1x+T2x=xhas at least one solution inΩ.

As usual, the forward difference ∆ is defined as ∆x_n = x_n+1−x_n, and for a positive integermthe higher-order difference is defined as

∆^mxn= ∆(∆^m−1xn), ∆⁰xn =xn.

In this article, R = (−∞,+∞), N is the set of positive integers, Z is the sets of all integers, α = inf{n−rjn : 1 ≤ j ≤ s, n0 ≤ n}, β = min{n0 −d, α}, lim_n→∞(n−rjn) = +∞, 1 ≤ j ≤ s, l_β^∞ denotes the set of real-valued bounded sequences x = {xn}_n≥β. It is well known that l_β^∞ is a Banach space under the supremum normkxk= sup_n≥β|xn|.

ForN > M >0, let A(M, N) =

x={xn}n≥β∈l^∞_β :M ≤x_n≤N, n≥β . Obviously,A(M, N) is a bounded closed and convex subset ofl_β^∞. Put

b= lim sup

n→∞

bn and b= lim inf

n→∞ bn.

Definition 1.2 ([5]). A set Ω of sequences in l^∞_β is uniformly Cauchy (or equi- Cauchy) if for everyε >0, there exists an integer N₀such that

|xi−xj|< ε, wheneveri, j > N0 for anyx=xk in Ω.

Lemma 1.3(Discrete Arzela-Ascoli’s theorem [5]). A bounded, uniformly Cauchy subsetΩof l_β^∞ is relatively compact.

By a solution of (1.11), we mean a sequence {xn}n≥β with a positive integer N0 ≥n0+d+|α|such that (1.11) is satisfied for alln≥N0. As is customary, a solution of (1.11) is said to be oscillatory about zero, or simply oscillatory, if the termsx_nof the sequence{xn}n≥β are neither eventually all positive nor eventually all negative. Otherwise, the solution is called non-oscillatory.

2. Existence of non-oscillatory solutions

In this section, we will give five sufficient conditions of the existence of non- oscillatory solutions of (1.11).

Theorem 2.1. If there exist constants M andN with N > M >0 and such that

|bn| ≤b < N−M

2N , eventually, (2.1)

∞

X

t=n₀

max 1

|ait|,|pjt|,|qt|: 1≤i≤k,1≤j≤s <+∞, (2.2) then (1.11) has a non-oscillatory solution inA(M, N).

Proof. Choose L ∈ (M +bN, N −bN). By (2.1) and (2.2), an integer N0 >

n0+d+|α|can be chosen such that

|bn| ≤b < N−M

2N , ∀n≥N0, (2.3)

and

∞

X

t₁=N₀

∞

X

t₂=t₁

· · ·

∞

X

t_k=t_k−1

∞

X

t=t_k

F

Ps j=1pjt

+|qt|

Qk i=1a_iti

≤min{L−bN −M, N−bN −L}, (2.4) where F = max_M≤x≤N{f_j(x) : 1 ≤ j ≤ s}. Define two mappings T₁, T₂ : A(M, N)→X by

(T1x)n =

(L−bnx_n−d, n≥N0,

(T₁x)_N0, β≤n < N₀, (2.5)

(T₂x)_n=











(−1)^kP∞ t₁=n

P∞ t₂=t₁. . . P∞

tk=tk−1

P∞ t=tk

Ps

j=1p_jtf_j(xt−rjt)−q_t Qk

i=1a_iti , n≥N0,

(T₂x)_N0, β≤n < N₀,

(2.6)

for allx∈A(M, N).

(i) Note that T1x+T2y ∈A(M, N) for all x, y ∈ A(M, N). In fact, for every x, y∈A(M, N) andn≥N0, by (2.4), we have

(T₁x+T₂y)_n≥L−bN−

∞

X

t1=n

∞

X

t2=t1

· · ·

∞

X

tk=tk−1

∞

X

t=tk

Ps

j=1pjtfj(y_t−rjt)−qt

Qk

i=1aiti

≥L−bN−

∞

X

t1=N0

∞

X

t₂=t₁

· · ·

∞

X

t_k=tk−1

∞

X

t=t_k

F

Ps j=1pjt

+|qt|

Qk i=1ait_i

≥M

and

(T₁x+T₂y)_n≤L+bN+

∞

X

t₁=N₀

∞

X

t2=t1

· · ·

∞

X

tk=tk−1

∞

X

t=tk

F Ps

j=1pjt

+|qt|

Qk

i=1ait_i

≤N.

That is, (T₁x+T₂y)(A(M, N))⊆A(M, N).

(ii) W show that T1 is a contraction mapping on A(M, N). For any x, y ∈ A(M, N) andn≥N0, it is easy to derive that

(T1x)n−(T1y)n

≤ |bnkx_n−d−y_n−d| ≤bkx−yk, which implies

kT₁x−T₁yk ≤bkx−yk.

Thenb < ^N_2N^−M <1 ensures that T1 is a contraction mapping onA(M, N).

(iii) We show thatT2is completely continuous. First, we showT2 that is contin- uous. Letx^(u)={x^(u)n } ∈A(M, N) be a sequence such that x^(u)n →x_n as u→ ∞.

SinceA(M, N) is closed,x={x_n} ∈A(M, N). Then, forn≥N₀, T2x^(u)_n −T2xn

≤

∞

X

t₁=N₀

∞

X

t₂=t₁

· · ·

∞

X

t_k=t_k−1

∞

X

t=t_k

Ps j=1pjt

fj(x^(u)_t−rjt)−fj(xt−rjt)|

Qk

i=1a_iti

.

Since Ps

j=1pjt

fj(x^(u)_t−rjt)−fj(x_t−rjt)|

Qk i=1ait_i

≤ Ps

j=1pjt

|fj(x^(u)_t−rjt)|+|fj(x_t−rjt)|

Qk i=1ait_i

≤2F

Ps j=1pjt

Qk i=1ait_i

and|fj(x^(u)_t−rjt)−fj(x_t−rjt)| →0 asu→ ∞forj= 1,2, . . . , s, it follows from (2.2) and the Lebesgue dominated convergence theorem that lim_u→∞kT₂x^(u)−T₂xk= 0, which means thatT2is continuous.

Next, we show thatT2A(M, N) is relatively compact. By (2.2), for any ε >0, takeN1≥N0 large enough,

∞

X

t1=N1

∞

X

t₂=t₁

· · ·

∞

X

t_k=tk−1

∞

X

t=t_k

F Ps

j=1p_jt +|qt|

Qk

i=1ait_i

<ε

2. (2.7)

Then, for anyx={x_n} ∈A(M, N) andn₁, n₂≥N₁, (2.7) ensures that T2xn₁−T2xn₂

≤

∞

X

t₁=n₁

∞

X

t₂=t₁

· · ·

∞

X

t_k=tk−1

∞

X

t=t_k

Ps

j=1pjtfj(yt−r_jt)−qt

Qk i=1ait_i

+

∞

X

t1=n2

∞

X

t2=t1

· · ·

∞

X

tk=tk−1

∞

X

t=tk

Ps

j=1pjtfj(y_t−rjt)−qt

Qk

i=1ait_i

≤

∞

X

t₁=N₁

∞

X

t₂=t₁

· · ·

∞

X

t_k=t_k−1

∞

X

t=t_k

F

Ps j=1pjt

+|qt|

Qk i=1a_iti

+

∞

X

t1=N1

∞

X

t2=t1

· · ·

∞

X

t_k=tk−1

∞

X

t=t_k

F Ps

j=1p_jt +|q_t|

Qk

i=1ait_i

< ε 2 +ε

2 =ε,

which impliesT2A(M, N) begin uniformly Cauchy. Therefore, by Lemma 1.3, the set T2A(M, N) is relatively compact. By Lemma 1.1, there exists x = {xn} ∈ A(M, N) such thatT1x+T2x=x, which is a bounded non-oscillatory solution to

(1.11). This completes the proof.

Theorem 2.2. If (2.2)holds,

bn≥0 eventually, 0≤b≤b <1, (2.8) and there exist constants M and N with N > ^2−b

1−bM > 0 then (1.11) has a non- oscillatory solution in A(M, N).

Proof. Choose L ∈ (M + ^1+b₂ N, N +₂^bM). By (2.2) and (2.8), an integer N0 >

n0+d+|α|can be chosen such that b

2 ≤bn≤ 1 +b

2 , ∀n≥N0 (2.9)

and ∞

X

t1=N0

∞

X

t₂=t₁

· · ·

∞

X

t_k=tk−1

∞

X

t=t_k

F

Ps j=1pjt

+|qt|

Qk i=1ait_i

≤minn

L−M−1 +b

2 N, N−L+b 2Mo

,

(2.10)

where F = max_M≤x≤N{fj(x) : 1≤ j ≤s}. Then define T1, T2 : A(M, N) →X as (2.5) and (2.6). The rest proof is similar to that of Theorem 2.1, and it is

omitted.

Theorem 2.3. If (2.2)holds,

bn ≤0eventually, −1< b≤b≤0, (2.11) and there exist constants M andN with N > ^2+b_1+bM >0, then (1.11) has a non- oscillatory solution in A(M, N).

Proof. ChooseL∈(^2+b₂ M,^1+b₂ N). By (2.2) and (2.11), an integerN0> n0+d+|α|

can be chosen such that

b−1

2 ≤bn≤ b

2, ∀n≥N0, (2.12)

and ∞

X

t₁=N₀

∞

X

t2=t1

· · ·

∞

X

tk=tk−1

∞

X

t=tk

F Ps

j=1pjt

+|qt|

Qk i=1a_iti

≤minn

L−2 +b

2 M,1 +b

2 N−Lo ,

(2.13)

whereF = maxM≤x≤N{fj(x) : 1≤j≤s}. Then defineT1, T2:A(M, N)→X by (2.5) and (2.6). The rest proof is similar to that of Theorem 2.1, and is omitted.

Theorem 2.4. If (2.2)holds,

bn>1 eventually, 1< b, andb < b²<+∞, (2.14) and there exist constants M and N with N > ^b(b

2−b)

b(b²−b)M > 0, then (1.11) has a non-oscillatory solution inA(M, N).

Proof. Takeε∈(0, b−1) sufficiently small satisfying

1< b−ε < b+ε <(b−ε)² (2.15) and

(b+ε)(b−ε)²−(b+ε)²

N > (b+ε)²(b−ε)−(b−ε)²

M. (2.16)

ChooseL∈ (b+ε)M+^b+ε_b−εN,(b−ε)N+^b−ε

b+εM

. By (2.2) and (2.15), an integer N₀> n₀+d+|α| can be chosen such that

b−ε < b_n< b+ε, ∀b≥N₀ (2.17) and

∞

X

t₁=N₀

∞

X

t2=t1

· · ·

∞

X

tk=tk−1

∞

X

t=tk

F Ps

j=1pjt

+|qt|

Qk i=1a_iti

≤minnb−ε

b+εL−(b−ε)M−N,b−ε

b+εM+ (b−ε)N−Lo ,

(2.18)

where F = max_M≤x≤N{f_j(x) : 1 ≤ j ≤ s}. Define two mappings T₁, T₂ : A(M, N)→X by

(T1x)n= ( _L

b_n+d−^x_b^n+d

n+d, n≥N0,

(T1x)N₀, β ≤n < N0, (2.19)

T2x)n=











(−1)^k bn+d

P∞ t₁=n

P∞ t₂=t₁. . . P∞

t_k=tk−1

P∞ t=t_k

Ps

j=1p_jtf_j(x_t−rjt)−qt

Qk

i=1a_iti , n≥N₀,

(T2x)N₀, β≤n < N0,

(2.20)

for all x ∈ A(M, N). The rest proof is similar to that in Theorem 2.1, and is

omitted.

Theorem 2.5. If (2.2)holds,

b_n<−1 eventually, −∞< b, b <−1 (2.21) and there exist constants M andN with N > ^1+b

1+bM >0, then (1.11) has a non- oscillatory solution in A(M, N).

Proof. Take∈ 0,−(1 +b)

sufficiently small satisfying

b− < b+ <−1 (2.22)

and

(1 +b+)N <(1 +b−)M. (2.23) Choose L ∈ (1 +b+)N,(1 +b−)M

. By (2.2) and (2.22), an integer N0 >

n0+d+|α|can be chosen such that

b− < bn < b+, ∀n≥N0, (2.24) and

∞

X

t₁=N₀

∞

X

t2=t1

· · ·

∞

X

tk=tk−1

∞

X

t=tk

F Ps

j=1pjt

+|qt|

Qk

i=1ait_i

≤minn

b++b+ b−

M−b+

b−L, L−(1 +b+)No ,

(2.25)

where F = maxM≤x≤N{fj(x) : 1≤ j ≤s}. Then define T1, T2 : A(M, N) →X as (2.19) and (2.20). The rest proof is similar to that in Theorem 2.1, and is

omitted.

Remark 2.6. Theorems 2.1–2.5 extend the results in Cheng [6, Theorem 1], Liu, Xu and Kang [8, Theorems 2.3-2.7], Zhou and Huang [16, Theorems 1-5] and cor- responding theorems in [3, 4, 9, 10, 11, 12, 13, 14, 15].

Acknowledgments. The authors are grateful to the anonymous referees for their careful reading, editing, and valuable comments and suggestions.

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Zhenyu Guo

School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China E-mail address:[email protected]

Min Liu

School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China E-mail address:min [email protected]