Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 25, 1–8; http://www.math.u-szeged.hu/ejqtde/

BOUNDED NONOSCILLATORY SOLUTIONS OF NEUTRAL TYPE DIFFERENCE SYSTEMS

E. Thandapani ^{1}, R.Karunakaran ^{2} and I.M.Arockiasamy ^{3}

1 Ramanujan Institute for Advanced Study in Mathematics University of Madras, Chepauk, Chennai-600 005, India e-mail: ethandapani@yahoo.co.in 2 Department of Mathematics, Periyar University Salem-636 011, Tamil Nadu, India 3 Department of Mathematics, St. Paul’s Hr. Sec. School Salem-636 007, Tamil Nadu, India

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract This paper deals with the existence of a bounded nonoscillatory solution of nonlinear neutral type difference systems. Examples are provided to illustrate the main results.

Key words and phrases: Neutral difference system, bounded nonoscillatory solution, fixed point theorem. AMS (MOS) Subject Classifications: 39A10, 39A12

1 Introduction

Consider nonlinear neutral type difference systems of the form

\begin{matrix}Δ^{2} (x_{1} (n) − a_{1} (n) x_{1} (n − k_{1})) = p_{1} (n) f_{1} (x_{2} (n − ℓ_{2})) + e_{1} (n) \\ Δ^{2} (x_{2} (n) − a_{2} (n) x_{2} (n − k_{2})) = p_{2} (n) f_{2} (x_{1} (n − ℓ_{1})) + e_{2} (n) \end{matrix}  (1)

where Δ is a forward difference operator defined by Δx (n) = x (n + 1) − x (n), n ∈ ℕ (n_{0}) = {n_{0}, n_{0} + 1, ⋯}, n_{0} ≥ 0, k_{i}, ℓ_{i} are non negative integers {a_{i} (n)}, {p_{i} (n)}, {e_{i} (n)} are real sequences and f_{i} : ℝ → ℝ is continuous function for i = 1, 2.

Let θ = max {k_{1}, k_{2}, ℓ_{1}, ℓ_{2}}. By a solution of the system (1) we mean a function X = ({x_{1} (n)} , {x_{2} (n)}), defined for all n ≥ n_{0} − θ , and which satisfies the system (1) for all n ≥ n_{0}.

In this paper, we obtain some sufficient conditions for the existence of a nonoscillatory bounded solutions of the system (1) via fixed point theorems and some new techniques. Here we allow the sequences {p_{i} (n)} and {e_{i} (n)} , i = 1, 2, to be oscillatory. The existence of nonoscillatory solutions of neutral type difference equations has been treated in [1, 2, 6, 12] and in papers cited therein.

2 Main Results

In this section, we obtain sufficient conditions for the existence of bounded nonoscillatory solutions of the system (1). We consider the following cases:

(i) 0 < a_{i} (n) < 1, (ii) 1 < a_{i} ≡ a_{i} (n) < ∞, i = 1, 2, and their combination.

In this sequel we use the following fixed point theorem.

Lemma 2.1 [Krasnoselskii’s fixed point theorem] Let B be a Banach space. Let S be a bounded, closed, convex subset of B and let F, T be maps of S into B such that F x + T y ∈ S for all x, y ∈ S. If F is contractive and T is completely continuous, then the equation

F x + T x = x

has a solution in S.

The details of Lemma 2.1 can be found in [1] and [4] .

Theorem 2.1 Assume that 0 < a_{i} (n) ≤ a_{i} < 1,

∑_{n=n_{0}}^{∞} n |p_{i} (n)| < ∞  (2)

and

| ∑_{n=n_{0}}^{∞} ne_{i} (n)| < ∞, i = 1, 2.  (3)

Then the system (1) has a bounded nonoscillatory solution.

Proof. In view of conditions (2) and (3), one can choose N ∈ ℕ (n_{0}) sufficiently large that

∑_{n=N}^{∞} n |p_{i} (n)| ≤ \frac{1 − a_{i}}{5M_{i}} , |∑_{n=N}^{∞} ne_{i} (n)| ≤ \frac{1 − a_{i}}{10} , i = 1, 2,

where M_{i} = max \{|f_{i} (y_{3−i} (n))| : \frac{2−2a_{i}}{5} ≤ y_{3−i} (n) ≤ 1\} , i = 1, 2.

Let B (n_{0}) be the set of all real valued sequences X (n) = ({x_{1} (n)} , {x_{2} (n)}) with the norm ∥x (n)∥ = sup_{n≥n_{0}} (|x_{1} (n)| , |x_{2} (n)|) < ∞. Then B (n_{0}) is a Banach space. We

define a closed, bounded and convex subset S of B (n_{0}) as follows:

S = \{x (n) = (\{x_{1} (n)\} , \{x_{2} (n)\}) ∈ B (n_{0}) : \frac{2 − 2a_{i}}{5} ≤ x_{i} (n) ≤ 1, i = 1, 2, n ≥ n_{0}\} .

Let maps F = (F_{1}, F_{2}) and T = (T_{1}, T_{2}) : S → B (n_{0}) be defined by

(F_{i}x) (n) = \{ \begin{matrix}a_{i} (n) x_{i} (n − k_{i}) + \frac{7 − 7a_{i}}{10} , n ≥ N, \\ (F_{i}x) (N) , n_{0} ≤ n ≤ T, i = 1, 2, \end{matrix}

and

(T_{i}x) (n) = \{ \begin{matrix}∑_{s=n}^{∞} (s − n + 1) (p_{i} (s) f_{i} (x_{3−i} (s − ℓ_{3−i}) + e_{i} (s))) , n ≥ N, \\ (T_{i}x) (N) , n_{0} ≤ n ≤ T, i = 1, 2. \end{matrix}

First we show that if x, y ∈ S, then F x + T y ∈ S. Hence, for x = {x (n)} and y = {y (n)} ∈ S and n ≥ N, we have

(F_{i}x) (n) + (T_{i}y) (n) ≤ a_{i} (n) x_{i} (n − k_{i}) + \frac{7 − 7a_{i}}{10} + ∑_{s=n}^{∞} (s − n + 1) |p_{i} (s)|

|f_{i} (y_{3−i} (s − ℓ_{3−i}))| + |∑_{s=n}^{∞} (s − n + 1) e_{i} (s)|

≤ a_{i} + \frac{7 − 7a_{i}}{10} + M_{i} ∑_{s=n}^{∞} s |p_{i} (s)| + |∑_{s=n}^{∞}se_{i} (s)|

≤ a_{i} + \frac{7 − 7a_{i}}{10} + \frac{1 − a_{i}}{5} + \frac{1 − a_{i}}{10} = 1, i = 1, 2.

Further we obtain

(F_{i}x) (n) + (T_{i}y) (n)

≥ \frac{7 − 7a_{i}}{10} − (∑^{∞} ; s=n (s − n + 1) |p_{i} (s)| |f_{i} (y_{3−i} (s − ℓ_{3−i}))| + |∑^{∞} ; s=n (s − n + 1) e_{i} (s)|)

≥ \frac{7 − 7a_{i}}{10} − \frac{1 − a_{i}}{5} − \frac{1 − a_{i}}{10} = \frac{2 − 2a_{i}}{5} , i = 1, 2.

Hence F x + T y ∈ S for any x, y ∈ S, that is, F S ∪ T S ⊂ S. Next we show that F is a contraction on S. In fact for x, y ∈ S and n ≥ N, we have

|(F_{i}x) (n) − (F_{i}y) (n)| ≤ a_{i} (n) |x_{i} (n − k_{i}) − y_{i} (n − k_{i})|
≤ max (a_{i}) |x_{i} (n − k_{i}) − y_{i} (n − k_{i})| , i = 1, 2.

This implies that

∥F x − F y∥ ≤ |a| ∥x − y∥ .

Since 0 < |a| < 1, a = (a_{1}, a_{2}) , we conclude that F is a contraction mapping on S. Next we show that T is completely continuous. For this, first we show that T is continuous. Let x_{k} = ({x_{1k} (n)} , {x_{2k} (n)}) ∈ S and x_{ik} (n) → x_{i} (n) as k → ∞, i = 1, 2. Since S is closed, x = (x_{1} (n) , x_{2} (n)) ∈ S. For n ≥ N, we obtain

|(T_{i}x_{k})(n) − (T_{i}x)(n)|

≤ ∑_{s=n}^{∞} (s − n + 1) |p_{i} (s)| |f_{i} (x_{3−ik} (s − ℓ_{3−i})) − f_{i} (x_{3−i} (s − ℓ_{3−i}))|

≤ ∑_{s=n}^{∞}s |p_{i} (s)| |f_{i} (x_{3−ik} (s − ℓ_{3−i})) − f_{i} (x_{3−i} (s − ℓ_{3−i}))| , i = 1, 2.

Since |f_{i} (x_{3−ik} (s − ℓ_{3−i})) − f_{i} (x_{3−i} (s − ℓ_{3−i}))| → 0 as k → ∞, for i = 1, 2, we have T is continuous.

Next we show that T S is relatively compact. Using the result [[3], Theorem 3.3], we need only to show that T S is uniformly Cauchy. Let x = ({x_{1} (n)} , {x_{2} (n)}) be in S. From (2) and (3) it follows that for ϵ > 0, there exists N^{∗} > N such that for n ≥ N^{∗},

∑_{s=n}^{∞} (s − n + 1) |p_{i} (s)| |f_{i} (x_{3−i} (s − ℓ_{3−i}))| + |∑_{s=n}^{∞} (s − n + 1) e_{i} (s)| < \frac{ε}{2}, i = 1, 2.

Then for n_{2} > n_{2} ≥ N^{∗}, we have

|(T_{i}x) (n_{2}) − (T_{i}x) (n_{1})| < ε.

Thus T S is uniformly Cauchy. Hence it is relatively compact.

Thus by Lemma 2.1, there is a x_{0} (n) = ({x_{01} (n)} , {x_{02} (n)}) ∈ S such that F x_{0} + T x_{o} = x_{0}. We see that {x_{0} (n)} is a bounded nonoscillatory solution of the system (1). The proof is now complete.

Theorem 2.2 Suppose that 1 < a_{i} ≡ a_{i} (n) < ∞ and conditions (2) and (3) hold. Then the system (1) has a bounded nonoscillatory solution.

Proof. In view of conditions (2) and (3), we can choose N > n_{0} sufficiently large that

∑_{s=N+k_{i}}^{∞} s|p_{i}(s)| ≤ \frac{a_{i} − 1}{3D_{i}}, | ∑_{s=N+k_{i}}^{∞} se_{i} (s)| ≤ \frac{a_{i} − 1}{6}

where D_{i} = max_{a_{i}−1≤x_{3−i}^{(n)}≤2a_{i}} {|f_{i} (x_{3−i}^{(n)})|} , i = 1, 2.

Let B (n_{0}) be the Banach space defined in the proof of Theorem 2.1. We define a closed,bounded and convex subset S of B(n_{0}) as follows:

S = {x = ({x_{1} (n)} , {x_{2} (n)}) ∈ B (n_{0}) : a_{i} − 1 ≤ x_{i} (n) ≤ 2a_{i}, i = 1, 2, n ≥ n_{0}} .

Let maps F = (F_{1}, F_{2}) and T = (T_{1}, T_{2}) : S → B (n_{0}) be defined by

(F_{i}x) (n) = \{ \begin{matrix}\frac{3a_{i} − 3}{2} + \frac{1}{a_{i} (n)}x_{i} (n + k_{i}) , n ≥ N, \\ (F_{i}x) (N) , n_{0} ≤ n ≤ N, i = 1, 2. \end{matrix}

(T_{i}x) (n) = \{ \begin{matrix}−\frac{1}{a_{i} (n)} ∑_{s=n+k_{i}}^{∞} (s − n − k_{i}) (p_{i} (s) f_{i} (x_{3−i} (s − ℓ_{3−i})) + e_{i} (s)) , n ≥ N, \\ (T_{i}x) (N) , n_{0} ≤ n ≤ N, i = 1, 2. \end{matrix}

Now we show that F is a contractive mapping on S. For any x, y ∈ S and n ≥ N, we obtain

|(F_{i}x) (n) − (F_{i}y) (n)| ≤ \frac{1}{a_{i} (n)} |x_{i} (n + k_{i}) − y_{i} (n + k_{i})|

≤ \frac{1}{a_{i}} ∥x − y∥ ≤ max \{\frac{1}{a_{i}} \} ∥x − y∥ , i = 1, 2.

This implies that

∥F x − F y∥ ≤ max \{\frac{1}{a_{i}} , i = 1, 2\} ∥x − y∥ .

Since 0 < max \{\frac{1}{a_{i}} , i = 1, 2\} < 1 we conclude that F is a contraction mapping on S.

Next we show that for any x, y ∈ S, F x + T y ∈ S. For every x, y ∈ S and n ≥ N, we have

(F_{i}x) (n) + (T_{i}y) (n) ≤ \frac{3a_{i} − 3}{2} + \frac{1}{a_{i} (n)}x_{i} (n + k_{i})

+ \frac{1}{a_{i} (n)} ∑_{s=n+k_{i}}^{∞} (s − k_{i} − n + 1) |p_{i} (s)| |f_{i} (y_{3−i} (s − ℓ_{3−i}))|

+ | ∑_{s=n+k_{i}}^{∞} (s − k_{i} − n + 1) e_{i} (s)|

≤ \frac{3a_{i} − 3}{2} + 2 + (D_{i} ∑_{s=N+k_{i}}^{∞} s |p_{i} (s)| + | ∑_{s=N+k_{i}}^{∞} se_{i} (s)|)

≤ \frac{3a_{i} − 3}{2} + 2 + \frac{a_{i} − 1}{3} + \frac{a_{i} − 1}{6} = 2a_{i}, i = 1, 2.

Further, we obtain

(F_{i}x) (n) − (T_{i}y) (n) ≥ \frac{3a_{i} − 3}{2} + \frac{1}{a_{i} (n)}x_{i} (n + k_{i})

− \frac{1}{a_{i} (n)} [ ∑_{s=n+k_{i}}^{∞} (s − k_{i} − n + 1) |p_{i} (s)| |f_{i} (y_{3−i} (s − ℓ_{3−i}))|

+ | ∑_{s=n+k_{i}}^{∞} (s − k_{i} − n + 1) e_{i} (s)|]

≥ \frac{3a_{i} − 3}{2} − (D_{i} ∑_{s=N+k_{i}}^{∞} s |p_{i} (s)| + | ∑_{s=N+k_{i}}^{∞} se_{i} (s)|)

≥ \frac{3a_{i} − 3}{2} − \frac{a_{i} − 1}{3} − \frac{a_{i} − 1}{6} = a_{i} − 1, i = 1, 2.

Hence

a_{i} − 1 ≤ (F_{i}x) (n) + (T_{i}y) (n) ≤ 2a_{i}, i = 1, 2 for n ≥ n_{0}.

Thus we proved that F x + T y ∈ S for any x, y ∈ S.

Proceeding similarly as in the proof of Theorem 2.1, we obtain that the mapping T is completely continuous. By Lemma 2.1, there is a x_{0} ∈ S such that F x_{0} + T x_{0} = x_{0}. We see that {x_{0} (n)} is a non oscillatory bounded solution of the system (1). This completes the proof of Theorem 2.2.

Theorem 2.3 Suppose that 0 < a_{1} (n) ≤ a_{1} < 1, 1 < a_{2} ≡ a_{2} (n) < ∞ and conditions (2) and (3) hold. Then the difference system (1) has a bounded nonoscillatory solution.

Proof. In view of conditions (2) and (3), we can choose a N > n_{0} sufficiently large that

∑_{n=N}^{∞} n |p_{1} (n)| ≤ \frac{1 − a_{1}}{5M_{1}} , |∑_{n=N}^{∞} ne_{1} (n)| ≤ \frac{1 − a_{1}}{10}

and

∑_{n=N+k_{2}}^{∞} n |p_{2} (n)| ≤ \frac{a_{2} − 1}{3D_{2}}, | ∑_{n=N+k_{2}}^{∞} ne_{2} (n)| ≤ \frac{a_{2} − 1}{6}.

Let B (n_{0}) be the Banach space defined as in Theorem 2.1. We define a closed, bounded and convex subset S of B (n_{0}) as follows:

S = \{x = (\{x_{1} (n)\} , \{x_{2} (n)\}) ∈ B (n_{0}) : \frac{2 − 2a_{1}}{5} ≤ x_{1} (n) ≤ 1,
a_{2} − 1 ≤ x_{2} (n) ≤ 2a_{2}, n ≥ n_{0}}.

Let maps F = (F_{1}, F_{2}) and T = (T_{1}, T_{2}) : S → B (n_{0}) be defined by

(F_{1}x) (n) = \{ \begin{matrix}a_{1} (n) x_{1} (n − k_{1}) + \frac{7 − 7a_{1}}{10} , n ≥ N, \\ (F_{1}x) (N) , n_{0} ≤ n ≤ N, \end{matrix}

(T_{1}x) (n) = \{ \begin{matrix}∑_{s=n}^{∞} (s − n + 1) (p_{i} (s) f_{1} (x_{2} (s − k_{2})) + e_{1} (s)) , n ≥ N, \\ (T_{1}x) (N) , n_{0} ≤ n ≤ N, \end{matrix}

and

(F_{2}x) (n) = \{ \begin{matrix}\frac{1}{a_{2} (n)}x_{2} (n + k_{2}) + \frac{3a_{2} − 3}{2}, n ≥ N, \\ (F_{2}x) (N) , n_{0} ≤ n ≤ N, \end{matrix}

(T_{2}x) (n) = \{ \begin{matrix}− ∑_{s=n+k_{2}}^{∞} (s − n − k_{2} + 1) (p_{2} (s) f_{2} (x_{1} (s − k_{1})) + e_{2} (s)) , n ≥ N, \\ (T_{2}x) (N) , n_{0} ≤ n ≤ N. \end{matrix}

As in the proof of Theorems 2.1 and 2.2 one can show that F_{1}, F_{2} are contractive mappings on S. It is easy to show that for any x, y ∈ S. It is easy to show that for any x, y ∈ S, F_{1}x + T_{1}y ∈ S and also F_{2}x + T_{2}y ∈ S. Proceeding as in the proof of Theorem 2.1, we obtain that the mappings T_{1}, T_{2} are completely continuous. By Lemma 2.1, there are x_{01}, x_{02} ∈ S such that F_{1}x_{01} + T_{1}x_{01} = x_{01}, F_{2}x_{02} + T_{2}x_{02} = x_{02}. We see that x_{0} (n) = ({x_{01} (n)} , {x_{02} (n)}) is a nonoscillatory bounded solution of the difference system (1). The proof is now complete.

In the following we provide some examples to illustrate the results.

Example 2.1 Consider the difference system

\begin{matrix}Δ^{2} (x_{1} (n) − \frac{1}{2}x_{1} (n − 1)) = \frac{1}{3^{n}} x_{2}^{3} (n − 1) − \frac{2^{7}}{3^{4n}} \\ Δ^{2} (x_{2} (n) − \frac{1}{2}x_{2} (n − 1)) = \frac{1}{2^{n}} x_{1}^{3} (n − 1) − \frac{8}{2^{4n}} , n ≥ 1. \end{matrix}  (4)

It is easy to see that all conditions of Theorem 2.1 are satisfied, and hence the system

(4) has a bounded nonoscillatory solution. In fact x (n) = (\{\frac{1}{2^{n}} \} , \{\frac{1}{3^{n}} \}) is one

such solution of the system (4).

Example 2.2 Consider the difference system

\begin{matrix}Δ^{2} (x_{1} (n) − 2x_{1} (n − 1)) = \frac{−12}{n(n+2)(n+3)}x_{2} (n − 1) − \frac{2}{(n+1)(n+2)(n+3)} \\ Δ^{2} (x_{2} (n) − 2x_{2} (n − 2)) = \frac{−(26n+48)}{(n+1)(n+2)(n+3)(n+4)}x_{1} (n − 1) − \frac{2n}{(n+1)(n+2)(n+3)(n+4)}, n ≥ 1. \end{matrix}

(5)

By Theorem 2.2, the system (5) has a bounded nonoscillatory solution. In fact x (n) =

(\{\frac{1}{n + 1}\} , \{\frac{1}{n + 2}\}) is one such solution of the system (5).

Example 2.3 Consider the difference system

\begin{matrix}Δ^{2} (x_{1} (n) − \frac{1}{2}x_{1} (n − 1)) = −\frac{1}{n^{3}} x_{2}^{3} (n − 1) + \frac{1}{n^{6}} \\ Δ^{2} (x_{2} (n) − 2x_{2} (n − 1)) = \frac{1}{2^{n+1}} x_{1} (n − 1) − \frac{2 (n + 6)}{n (n + 1) (n + 2) (n + 3)} − \frac{1}{4^{n}} . \end{matrix} (6)

It is easy to see that all conditions of Theorem 2.3 are satisfied, and hence the system

(6) has a bounded nonoscillatory solution. In fact x (n) = (\{\frac{1}{2^{n}} \} , \{\frac{1}{n + 1}\}) is one

such solution of the system.

Remark 2.1 It is easy to see that the difference system (1) includes different types of (ordinary, delay, neutral) fourth order difference equations, and hence the results obtained in this paper generalize many of the existing results for the fourth order difference equations, see for example [5, 7, 8, 9, 10, 11] and the references cited therein.

References

[1] R. P. Agarwal, Difference Equations and Inequalities, Mercel Dekker, New York, 2000.

[2] R. P. Agarwal and S. R. Grace, Oscillation of higher order nonlinear difference equations of neutral type, Appl. Math. Lett. 12 (1999), 77-83.

[3] S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear difference equation, Nonlinear Anal. 20 (1993), 193-203.

[4] L. H. Erbe, Q. K. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.

[5] J. R. Graef and E. Thandapani, Oscillatory and asymptotic behavior of fourth order nonlinear delay difference equations, Fasc. Math. 31 (2001), 23-36.

[6] N. Parhi and A. K. Tripathy, Oscillation of forced nonlinear neutral delay difference equations of first order, Czech. Math. J. 53 (2003), 83-101.

[7] E. Thadapani and I. M. Arockiasamy, Some oscillation and nonoscillation theorems for fourth order difference equations, ZAA 19 (2000), 863-872.

[8] E. Thadapani and I. M. Arockiasamy, Oscillatory and asymptotic behavior of fourth order nonlinear neutral delay difference equations, Indian J. Pure Appl. Math. 32 (2001), 109-123.

[9] E. Thandapani and I. M. Arockiasamy, Fourth order nonlinear oscillations of difference equations, Comput. Math. Appl. 42 (2001) 357-368.

[10] E. Thandapani, S. Pandian, R. Dhanasekaran and J. R. Graef, Asymptotic results for a class of fourth order quasilinear difference equations, J. Difference Equ. Appl. 13 (2007), 1085-1103.

[11] E. Thandapani and B. Selvaraj, Oscillatory and nonoscillatory behavior of fourth order quasilinear difference equations, Far East J. Appl. Math. 17 (2004), 287-307.

[12] Y. Zhou and Y. Q. Huang, Existence for nonoscillatory solutions of higher order nonlinear neutral difference equations, J. Math. Anal. Appl. 280 (2003), 63-76.