Solvability of a Higher Order Nonlinear Neutral Delay Difference Equation

(1)

doi:10.1155/2010/767620

Research Article

Solvability of a Higher-Order Nonlinear Neutral

Delay Difference Equation

Min Liu and Zhenyu Guo

School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China

Correspondence should be addressed to Zhenyu Guo,[email protected]

Received 19 March 2010; Revised 10 July 2010; Accepted 5 September 2010

Academic Editor: S. Grace

Copyrightq2010 M. Liu and Z. Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay diﬀerence equationΔakn· · ·Δa2nΔa1nΔxnbnxn−dfn, xn−r1n, xn−r2n, . . . , xn−rsn 0,n≥n0, wheren0 ≥0,d >0,k >0, ands >0 are integers,{ain}_n≥n₀ i1,2, . . . , kand{bn}_n≥n₀are real sequences,s_j₁{rjn}n≥n0 ⊆ Z, andf : {n : n ≥ n0} ×R

s _→ _R_{is a mapping, is studied. Some} suﬃcient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence{bn}n≥n0.

Moreover, these suﬃcient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

1. Introduction and Preliminaries

Recently, the interest in the study of the solvability of diﬀerence equations has been increasing

see 1–17and references cited therein. Some authors have paied their attention to various diﬀerence equations. For example,

ΔanΔxn pnxgn0, n≥0 1.1

see 14,

(2)

see 11,

Δ2_x

npxn−m

pnxn−k−qnxn−l0, n≥n0 1.3

see 6,

Δ2_x

npxn−k

fn, xn 0, n≥1 1.4

see 10,

Δ2_x

n−pxn−τ

m

i1

qifixn−σi, n≥n0 1.5

see 9,

ΔanΔxnbxn−τ fn, xn−d1n, xn−d2n, . . . , xn−dkn cn, n≥n0 1.6

see 8,

Δm_x

ncxn−k pnxn−r 0, n≥n0 1.7

see 15,

Δm_x

ncnxn−k pnfxn−r 0, n≥n0 1.8

see 3,4,12,13,

Δm_x

ncxn−k u

s1

p_nsfsxn−rs qn, n≥n0 1.9

see 16,

Δm_x

ncxn−k pnxn−r−qnxn−l0, n≥n0 1.10

see 17.

Motivated and inspired by the papers mentioned above, in this paper, we investigate the following higher-order nonlinear neutral delay diﬀerence equation:

Δakn· · ·Δa2nΔa1nΔxnbnxn−d fn, xn−r1n, xn−r2n, . . . , xn−rsn 0, n≥n0, 1.11

wheren0≥0,d >0,k >0, ands >0 are integers,{ain}n≥n0i1,2, . . . , kand{bn}n≥n0are real

sequences,s_j₁{rjn}n≥n0⊆Z, andf :{n:n≥n0} ×R

(3)

equations1.1–1.10are special cases of1.11. By using Schauder fixed point theorem and Krasnoselskii fixed point theorem, the existence of bounded nonoscillatory solutions of1.11

is established.

Lemma 1.1Schauder fixed point theorem. LetΩbe a nonempty closed convex subset of a Banach spaceX. LetT :Ω → Ωbe a continuous mapping such thatTΩis a relatively compact subset ofX. ThenT has at least one fixed point inΩ.

Lemma 1.2 Krasnoselskii fixed point theorem. LetΩ be a bounded closed convex subset of a Banach spaceX, and letT1, T2 : Ω → X satisfy T1xT2y ∈ Ω for each x, y ∈ Ω. If T1 is a

contraction mapping andT2is a completely continuous mapping, then the equationT1xT2xxhas

at least one solution inΩ.

The forward diﬀerence Δ is defined as usual, that is,Δxn xn1 −xn. The higher-order

diﬀerence for a positive integermis defined asΔm_x

n ΔΔm−1xn,Δ0xn xn. Throughout this

paper, assume thatR −∞,∞,NandZstand for the sets of all positive integers and integers, respectively,α inf{n−rjn : 1 ≤j ≤s, n ≥n0},β min{n0−d, α},limn→ ∞n−rjn ∞, 1≤j ≤s, andl_β∞denotes the set of real sequences defined on the set of positive integers lager thanβ where any individual sequence is bounded with respect to the usual supremum normxsup_n_≥_β|xn|

forx{xn}n≥β∈l∞β. It is well known thatl∞β is a Banach space under the supremum norm. A subset

Ωof a Banach spaceXis relatively compact if every sequence inΩhas a subsequence converging to an element ofX.

Definition 1.3see 5. A setΩof sequences inl∞_β is uniformly Cauchyor equi-Cauchyif, for everyε >0, there exists an integerN0such that

_x_i₋_x_j_{< ε,} _1.12

wheneveri, j > N0for anyx{xk}k≥βinΩ.

Lemma 1.4discrete Arzela-Ascoli’s theorem 5. A bounded, uniformly Cauchy subsetΩofl∞_β is relatively compact.

Let

AM, N x{xn}n≥β∈l∞β :M≤xn≤N,∀n≥β

forN > M >0. 1.13

Obviously,AM, Nis a bounded closed and convex subset ofl∞_β. Put

blim sup

n→ ∞ bn, blim infn→ ∞ bn. 1.14

By a solution of 1.11, we mean a sequence {xn}n≥β with a positive integerN0 ≥

(4)

2. Existence of Nonoscillatory Solutions

In this section, a few suﬃcient conditions of the existence of bounded nonoscillatory solutions of1.11are given.

Theorem 2.1. Assume that there exist constants M and N with N > M > 0 and sequences {ain}n≥n0 1≤i≤k,{bn}n≥n0,{hn}n≥n0, and{qn}n≥n0such that, forn≥n0,

bn ≡ −1, eventually, 2.1

fn, u1, u2, . . . , us−fn, v1, v2, . . . , vs≤hnmax{|ui−vi|:ui, vi∈ M, N,1≤i≤s},

2.2

_{fn, u}₁, u2, . . . , us≤qn, ui∈ M, N, 1≤i≤s, 2.3

∞

tn0

max 1

|ait|

, ht, qt: 1≤i≤k

<∞. 2.4

Then1.11has a bounded nonoscillatory solution inAM, N.

Proof. Choose L ∈ M, N. By2.1, 2.4, and the definition of convergence of series, an integerN0> n0d|α|can be chosen such that

bn≡ −1, ∀n≥N0, 2.5

∞

j1

∞

t1N0jd

∞

t2t1 · · · ∞

tktk−1

∞

ttk qt

k

i1aiti

≤min{L−M, N−L}. 2.6

Define a mappingTL:AM, N → Xby

TLxn

⎧ ⎪ ⎨ ⎪ ⎩

L−−1k

∞

j1

∞

t1njd

∞

t2t1 · · · ∞

tktk−1

∞

ttk

ft, xt−r1t, xt−r2t, . . . , xt−rst k

i1aiti

, n≥N0,

TLxN0, β≤n < N0

2.7

for allx∈AM, N.

(5)

In fact, for everyx∈AM, Nandn≥N0, it follows from2.3and2.6that

This inequality and2.4imply thatTLis continuous.

(6)

(7)

which derives that

by which it follows that

Δa1nΔxn−xn−d −1k−1

(8)

Remark 2.2. The conditions of Theorem 2.1 ensure the 1.11 has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions. In fact, let L1, L2 ∈ M, N with L1/L2. For L1 and L2, as the preceding proof

which are bounded nonoscillatory solutions of1.11inAM, N. For the sake of proving that1.11possesses uncountably many bounded nonoscillatory solutions inAM, N, it is only needed to show thatx /y. In fact, by2.7, we know that, forn≥N4,

Theorem 2.3. Assume that there exist constants M and N with N > M > 0 and sequences {ain}n≥n0 1≤i≤k,{bn}n≥n0,{hn}n≥n0,{qn}n≥n0, satisfying2.2–2.4and

bn≡1, eventually. 2.18

(9)

Proof. ChooseL∈M, N. By2.18and2.4, an integerN0> n0d|α|can be chosen such

Theorem 2.1. It is claimed that the fixed pointxis a bounded nonoscillatory solution of1.11. In fact, forn≥N0d,

by which it follows that

xnxn−d2L −1k

The rest of the proof is similar to that inTheorem 2.1. This completes the proof.

Theorem 2.4. Assume that there exist constantsb,M, andN with N > M > 0and sequences {ain}n≥n01≤i≤k,{bn}n≥n0,{hn}n≥n0,{qn}n≥n0, satisfying2.2–2.4and

|bn| ≤b <

N−M

2N , eventually. 2.23

(10)

(11)

Theorem 2.5. Assume that there exist constantsMandNwithN >2−b/1−bM >0and sequences{ain}n≥n01≤i≤k,{bn}n≥n0,{hn}n≥n0,{qn}n≥n0, satisfying2.2–2.4and

bn≥0, eventually, and 0≤b≤b <1. 2.29

Proof. ChooseL ∈ M 1b/2N, N b/2M. By 2.29and 2.4, an integerN0 >

n0d|α|can be chosen such that

b

2 ≤bn≤ 1b

2 , ∀n≥N0,

∞

t1N0

∞

t2t1 · · · ∞

tktk−1

∞

ttk qt

k

i1aiti ≤min

L−M−1b

2 N, N−L

b

2M

.

2.30

Define two mappingsT1L, T2L:AM, N → Xas2.25. The rest of the proof is analogous to that inTheorem 2.4. This completes the proof.

Similar to the proof ofTheorem 2.5, we have the following theorem.

Theorem 2.6. Assume that there exist constantsMandNwithN >2b/1bM >0and sequences{ain}n≥n01≤i≤k,{bn}n≥n0,{hn}n≥n0,{qn}n≥n0, satisfying2.2–2.4and

bn≤0, eventually, and −1< b≤b≤0. 2.31

Theorem 2.7. Assume that there exist constantsMandNwithN >bb2−b/bb2−bM >0

and sequences{ain}n≥n0 1≤i≤k,{bn}n≥n0,{hn}n≥n0,{qn}n≥n0, satisfying2.2–2.4and

bn>1, eventually, 1< bandb < b2<∞. 2.32

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

1< b−ε < bε <b−ε2 ,

bεb−ε2−bε2

N >bε2b−ε−b−ε2

M.

(12)

ChooseL∈bεM bε/b−εN,b−εN b−ε/bεM. By2.33, an integer

N0> n0d|α|can be chosen such that

b−ε < bn< bε, ∀b≥N0,

∞

t1N0

∞

t2t1 · · · ∞

tktk−1

∞

ttk qt

k

i1aiti ≤min

b−ε

bεL−

b−εM−N,b−ε bεM

b−εN−L

.

2.34

Define two mappingsT1L, T2L:AM, N → Xby

T1Lxn

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

L bnd −

xnd

bnd

, n≥N0,

T1LxN0, β≤n < N0,

T2Lxn

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

−1k

bnd

∞

t1n

∞

t2t1 · · · ∞

tktk−1

∞

ttk

ft, xt−r1t, xt−r2t, . . . , xt−rst k

i1aiti

, n≥N0,

T2LxN0, β≤n < N0

2.35

for allx∈AM, N. The rest of the proof is analogous to that inTheorem 2.4. This completes the proof.

Similar to the proof ofTheorem 2.7, we have

Theorem 2.8. Assume that there exist constantsMandNwithN >1b/1bM >0and sequences{ain}n≥n0 1≤i≤k,{bn}n≥n0,{hn}n≥n0,{qn}n≥n0, satisfying2.2–2.4and

bn <−1, eventually, −∞< bandb <−1. 2.36

Remark 2.9. Similar toRemark 2.2, we can also prove that the conditions of Theorems2.3–2.8

ensure that 1.11has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

Remark 2.10. Theorems 2.1–2.8 extend and improve Theorem 1 of Cheng 6, Theorems 2.1–2.7 of Liu et al. 8, and corresponding theorems in 3,4,9–17.

3. Examples

In this section, two examples are presented to illustrate the advantage of the above results.

Example 3.1. Consider the following fourth-order nonlinear neutral delay diﬀerence equation:

(13)

ChooseM1 andN2. It is easy to verify that the conditions ofTheorem 2.1are satisfied. ThereforeTheorem 2.1ensures that3.1has a nonoscillatory solution inA1,2. However, the results in 3,4,6,8–17are not applicable for3.1.

Example 3.2. Consider the following third-order nonlinear neutral delay diﬀerence equation:

Δ

2n−nΔn2−n1Δ

xn2

n₋₁

3n xn−4

sin2xn−2

n2 −

cos3xn−3

n3 0, n≥5,

3.2

where

a1nn2−n1, a2n2n−n, bn 2 n₋₁

3n ,

fn, u1, u2 sin2u1

n2 −

cos3u2

n3 , hnqn 2

n2.

3.3

ChooseM1 andN5. It can be verified that the assumptions ofTheorem 2.5are fulfilled. It follows fromTheorem 2.5that3.2has a nonoscillatory solution inA1,5. However, the results in 3,4,6,8–17are unapplicable for3.2.

Acknowledgment

The authors are grateful to the editor and the referee for their kind help, careful reading and editing, valuable comments and suggestions.

References

1 R. P. Agarwal, Diﬀerence Equations and Inequalities: Theory, Methods, and Application, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.

2 R. P. Agarwal, S. R. Grace, and D. O’Regan,Oscillation Theory for Diﬀerence and Functional Diﬀerential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

3 R. P. Agarwal, E. Thandapani, and P. J. Y. Wong, “Oscillations of higher-order neutral diﬀerence equations,”Applied Mathematics Letters, vol. 10, no. 1, pp. 71–78, 1997.

4 R. P. Agarwal and S. R. Grace, “Oscillation of higher-order nonlinear diﬀerence equations of neutral type,”Applied Mathematics Letters, vol. 12, no. 8, pp. 77–83, 1999.

5 S. S. Cheng and W. T. Patula, “An existence theorem for a nonlinear diﬀerence equation,”Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 3, pp. 193–203, 1993.

6 J. Cheng, “Existence of a nonoscillatory solution of a second-order linear neutral diﬀerence equation,” Applied Mathematics Letters, vol. 20, no. 8, pp. 892–899, 2007.

7 I. Gy˝ori and G. Ladas, Oscillation Theory of Delay Diﬀerential Equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, NY, USA, 1991.

8 Z. Liu, Y. Xu, and S. M. Kang, “Global solvability for a second order nonlinear neutral delay diﬀerence equation,”Computers & Mathematics with Applications, vol. 57, no. 4, pp. 587–595, 2009.

9 Q. Meng and J. Yan, “Bounded oscillation for second-order nonlinear neutral diﬀerence equations in critical and non-critical states,”Journal of Computational and Applied Mathematics, vol. 211, no. 2, pp. 156–172, 2008.

(14)

11 E. Thandapani, M. M. S. Manuel, J. R. Graef, and P. W. Spikes, “Monotone properties of certain classes of solutions of second-order diﬀerence equations,”Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 291–297, 1998.

12 F. J. Yang and J. C. Liu, “Positive solution of even-order nonlinear neutral diﬀerence equations with variable delay,”Journal of Systems Science and Mathematical Sciences, vol. 22, no. 1, pp. 85–89, 2002. 13 B. G. Zhang and B. Yang, “Oscillation in higher-order nonlinear diﬀerence equations,”Chinese Annals

of Mathematics, vol. 20, no. 1, pp. 71–80, 1999.

14 Z. Zhang and Q. Li, “Oscillation theorems for second-order advanced functional diﬀerence equations,”Computers & Mathematics with Applications, vol. 36, no. 6, pp. 11–18, 1998.

15 Y. Zhou, “Existence of nonoscillatory solutions of higher-order neutral diﬀerence equations with general coeﬃcients,”Applied Mathematics Letters, vol. 15, no. 7, pp. 785–791, 2002.