Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 48, 1-19; http://www.math.u-szeged.hu/ejqtde/

OSCILLATORY BEHAVIOUR OF A CLASS OF NONLINEAR SECOND ORDER MIXED

DIFFERENCE EQUATIONS

A.K.Tripathy Department of Mathematics Kakatiya Institute of Technology and Science Warangal-506015, INDIA e-Mail: arun tripathy70@rediffmail.com

Abstract

In this paper oscillatory and asymptotic behaviour of solutions of a class of nonlinear second order neutral difference equations with positive and negative coefficients of the form

(E) Δ(r(n)Δ(y(n)+p(n)y(n−m)))+f(n)H_{1}(y(n−k_{1}))−g(n)H_{2}(y(n−k_{2})) = q(n) and

Δ(r(n)Δ(y(n) + p(n)y(n − m))) + f(n)H_{1}(y(n − k_{1})) − g(n)H_{2}(y(n − k_{2})) = 0

are studied under the assumptions

∑_{n=0}^{∞} \frac{1}{r(n)} < ∞

and

∑_{n=0}^{∞} \frac{1}{r(n)} = ∞

for various ranges of p(n). Using discrete Krasnoselskii’s fixed point theorem sufficient conditions are obtained for existence of positive bounded solutions of (E).

Mathematics Subject classification (2000): 39 A 10, 39 A 12 Keywords: Oscillation, Nonoscillation, Neutral Difference Equations, Existence of Positive Solutions.

1 Introduction

In recent years there has been extensive research activity concerning the oscillation and nonoscillation of solutions of differential and difference equations. In particular, much attention has been given to nonlinear neutral delay equations with positive and negative coefficients for existence of positive bounded solutions. We refer the papers [2, 3, 6, 7, 4, 13, 14] for comprehensive treatment of this theory. But very little work [11, 12] is available on the study of oscillatory and asymptotic behaviour of solutions of such equations which is due to the technical difficulties arising in its analysis.

The object of this work is to study the oscillation properties of a class of non-linear neutral difference equations with positive and negative coefficients of the form

Δ(r(n)Δ(y(n) + p(n)y(n − m))) + f(n)H_{1}(y(n − k_{1})) − g(n)H_{2}(y(n − k_{2})) = q(n), (1)

where Δ is the forward difference operator defined by Δy(n) = y(n+1)−y(n), r, p, f, g and q are real valued functions such that r(n) > 0, f(n) > 0 and g(n) ≥ 0 for all n, m > 0, k_{i} ≥ 0 are integers and H_{i} ∈ C(R, R) is a nondecreasing function such that xH_{i}(x) > 0, x ≠ 0 for i = 1, 2 under the assumptions

(A_{0})  ∑_{s=0}^{∞} \frac{1}{r(s)} ∑_{t=s}^{∞} g(t) < ∞
(A_{1})  ∑_{n=0}^{∞} \frac{1}{r(n)} < ∞

and

(A_{2})  ∑_{n=0}^{∞} \frac{1}{r(n)} = ∞

The corresponding unforced equation

Δ(r(n)Δ(y(n) + p(n)y(n − m))) + f(n)H_{1}(y(n − k_{1})) − g(n)H_{2}(y(n − k_{2})) = 0 (2)

is also studied under the assumptions (A_{1}) and (A_{2}) for various ranges of p(n).

This work is motivated by the recent paper [5], where the authors Li et al. have studied the existence of nonoscillatory solution of

Δ^{m}(y(n) + p(n)y(τ(n))) + f_{1}(n, y(σ_{1}(n))) − f_{2}(n, y(σ_{2}(n))) = g(n)  (3)

and its associated unforced equation

Δ^{m}(y(n) + p(n)y(τ(n))) + f_{1}(n, y(σ_{1}(n))) − f_{2}(n, y(σ_{2}(n))) = 0  (4)

under various ranges of p(n). If r(n) ≡ 1 and m = 2, then Eqns. (1) and (2) are particular cases of Eqns. (3) and (4) respectively. However, for m = 2, Eqns. (1) and (2) can not be treated as the particular cases of (3) and (4) in view of (A_{1}) and (A_{2}). Hence study of (1) and (2) is very much interesting. Necessary and sufficient conditions for oscillation of (1)/(2) are investigated in this paper.

By a solution of Eqn.(1) (see for e.g [9], [10]) we mean a real valued function y(n) defined on N(−ρ) = {−ρ, −ρ + 1, ⋯} which satisfies (1) for n ≥ 0, where ρ = max{m, k_{1}, k_{2}}. If

y(n) = A_{n}, n = −ρ, −ρ + 1, ⋯ 0,  (5)

are given, then (1) admits a unique solution satisfying the initial condition (5). Recall that a solution y(n) of (1) is oscillatory, if for any given integer N > 0, there exists an n ≥ N such that y(n)y(n + 1) ≤ 0 for n ≥ N; otherwise it is called nonoscillatory.

2 Preliminary Results

This section deals with the results which play an important role in establishing the present work.

Lemma 2.1 Assume that (A_{1}) hold. Let u(n) be an eventually positive real valued function such that Δ(r(n)Δu(n) ≤ 0 but ≢ 0 for all large n and r(n) > 0. Then the following hold :

i) if Δu(n) > 0, then there exists a constant C > 0 such that u(n) ≥ CR(n), for all large n;

ii) if Δu(n) < 0, then u(n) ≥ −r(n)Δu(n) R(n), where R(n) = ∑_{s=n}^{∞} \frac{1}{r(s)}.

Proof (i) Since R(0) < ∞, R(n) → 0 as n → ∞ and u(n) is nondecreasing, we can find a constant C > 0 such that u(n) ≥ C R(n) for all large n.

(ii) For s ≥ n + 1 > n, r(s)Δu(s) ≤ r(n)Δu(n) and hence

∑_{t=n}^{s−1} Δu(t) < r(n)Δu(n) ∑_{t=n}^{s−1} \frac{1}{r(t)}

implies that

u(s) < u(n) + r(n)Δu(n) ∑_{t=n}^{s−1} \frac{1}{r(t)}.

Thus 0 < u(s) < u(n) + r(n) Δu(n) ∑_{t=n}^{s−1} \frac{1}{r(t)} implies that u(n) ≥ −r(n) Δu(n) R(n) for all

large n.

Lemma 2.2 Assume that (A_{2}) hold. Let u(n) and Δu(n) be eventually positive real valued functions for n ≥ M + 1 > M ≥ 0. Then u(n) ≥ (n − M − 1)Δu(n) = β(n)Δu(n) for n ≥ M + 1, where β(n) = \frac{(n−M−1)}{r(n)} and M > 0 is an integer. Lemma 2.3 [15, 8] Let p, y, z be real valued functions such that z(n) = y(n) + p(n)y(n − m), n ≥ m ≥ 0, y(n) > 0 for n ≥ n_{1} > m, lim inf_{n→∞} y(n) = 0 and lim_{n→∞} z(n) = L exists. Let p(n) satisfy one of the following conditions

(i) 0 ≤ p(n) ≤ b_{1} < 1

(ii) 1 < b_{2} ≤ p(n) ≤ b_{3},
(iii) b_{4} ≤ p(n) ≤ 0,

where b_{i} is a constant, 1 ≤ i ≤ 4. Then L = 0. Lemma 2.4 [1] Let K be a closed bounded and convex subset of ℓ^{∞}, the Banach space consisting of all bounded real sequences. Suppose Γ is a continuous map such that Γ(K) ⊂ K and suppose further that Γ(K) is uniformly Cauchy. Then Γ has a fixed point in K.

3 Oscillation Results

This section deals with the sufficient conditions for oscillation of solutions of Eq.(1) and Eq.(2) under the assumptions (A_{0}), (A_{1}) and (A_{2}). We need the following conditions for our use in the sequel. (A_{3}) For u > 0and v > 0, there exists λ > 0 such that H_{1}(u) + H_{1}(v) ≥ λ H_{1}(u + v), (A_{4}) H_{1}(uv) = H_{1}(u)H_{1}(v), H_{2}(uv) = H_{2}(u)H_{2}(v) for u, v ∈ R. (A_{5}) F (n) = min{f(n), f(n − m)}, n ≥ m. (A_{6}) H_{1}(x) is sublinear and ∫_{0}^{±C} \frac{dx}{H_{1}(x)} < ∞. (A_{7}) There exists a real valued function Q(n) such that Q(n) changes sign with

−∞ < lim inf_{n→∞} Q(n) < 0 < lim sup_{n→∞} Q(n) < ∞ and Δ(r(n)ΔQ(n)) = q(n).

(A_{8}) Q^{+}(n) = max{Q(n), 0} and Q^{−}(n) = max{−Q(n), 0}. Remark The prototype of H_{1} satisfying (A_{3}) and (A_{4}) is

H_{1}(u) = (a + b |u|^{λ}) |u|^{μ} sgn u, where a ≥ 0, b > 0, λ ≥ 0 and μ ≥ 0 such that a + b = 1.

Remark Indeed, H_{1}(1) H_{1}(1) = H_{1}(1) due to (A_{4}). Hence H_{1}(1) = 1. Further, H_{1}(−1) H_{1}(−1) = H_{1}(1) = 1 gives to (H_{1}(−1))^{2} = 1. Because H_{1}(−1) < 0, then H_{1}(−1) = −1. Consequently, H_{1}(−u) = H_{1}(−1) H_{1}(u) = −H_{1}(u) for every u ∈ R.

Remark If y(n) is a solution of (1)/(2), then x(n) = −y(n) is also a solution of (1)/(2) provided that H_{1} satisfies (A_{4}).

Theorem 3.1 Let 0 ≤ p(n) ≤ p < ∞. If (A_{0}), (A_{1}), (A_{3}), (A_{4}), (A_{5}), (A_{7}), (A_{8}), and

(A_{9})  ∑_{n=k_{1}}^{∞} F (n) H_{1}(R(n − k_{1})) = ∞

hold, then (1) is oscillatory.

Proof Suppose on the contrary that y(n) is a nonoscillatory solution of (1) such that y(n) > 0 for n ≥ n_{0}. Setting

K(n) = ∑_{s=n}^{∞} \frac{1}{r(s)} ∑_{t=s}^{∞} g(t)H_{2}(y(t − k_{2}))

w(n) = y(n) + p(n) y(n − m) − K(n) = z(n) − K(n),  (6)

let

U(n) = w(n) − Q(n)  (7)

for n ≥ n_{0}. We note that K(n) > 0, ΔK(n) < 0 and lim_{n→∞} K(n) = 0. Using (6) and (7), Eq.(1) becomes

Δ(r(n)ΔU(n)) = −f(n) H_{1}(y(n − k_{1})) ≤ 0, ≢ 0  (8)

for n ≥ n_{1} > n_{0} + ρ. Accordingly, ΔU(n) and U(n) are monotonic functions. Assume that ΔU(n) < 0 for n ≥ n_{1}. If U(n) < 0 for n ≥ n_{2} > n_{1}, then z(n) < K(n) + Q(n) and hence

lim inf_{n→∞} z(n) ≤ lim inf_{n→∞}(k(n) + Q(n))
≤ lim sup_{n→∞} k(n) + lim inf_{n→∞} Q(n)
= lim_{n→∞} k(n) + lim inf_{n→∞} Q(n)
< 0,

a contradiction to the fact that z(n) > 0. Thus U(n) > 0 for n ≥ n_{2}. Using Lemma 2.1 (ii) with u(n) is replaced by U(n), we get U(n) ≥ −r(n)(ΔU(n))R(n) and hence

z(n) ≥ −r(n)R(n)ΔU(n) + K(n) + Q^{+}(n)
> −r(n)R(n)ΔU(n)

for n ≥ n_{2}. Further, r(n)ΔU(n) is non-increasing. So we can find a constant −α > 0 and n_{3} > n_{2} such that −r(n)ΔU(n) ≥ −α for n ≥ n_{3}. Using Eq.(1) and (7) we obtain

0 = Δ(r(n)ΔU(n)) + H_{1}(p)Δ(r(n − m)ΔU(n − m))
+ f(n)H_{1}(y(n − k_{1})) + H_{1}(p)f(n − m)H_{1}(y(n − m − k_{1})),

that is,

0 ≥ Δ(r(n)ΔU(n)) + H_{1}(p)Δ(r(n − m)ΔU(n − m))
+ λF (n)H_{1}(z(n − k_{1}))  (9)

due to (A_{3}), (A_{4}) and (A_{5}). Hence (9) becomes

λH_{1}(−α)F (n)H_{1}(R(n − k_{1})) ≤ −Δ(r(n)ΔU(n)) − H_{1}(p)Δ(r(n − m)ΔU(n − m)),

for n ≥ n_{4} > n_{3} + k_{1}. Since lim_{n→∞} U(n) exists, we claim that y(n) is bounded. If not, there exists {n^{j}} ⊂ {n} such that U(n) = z(n) − K(n) − Q(n) > y(n) − K(n) − Q(n) implies that U(n^{j}) − K(n^{j}) − Q(n^{j}) → ∞ as j → ∞ and n^{j} → ∞, a contradiction. Consequently, lim_{n→∞}(r(n)ΔU(n)) exists. Summing the last inequality from n_{4} to ∞, we obtain

∑_{n=n_{4}}^{∞} F (n)H_{1}(R(n − k_{1})) < ∞,

a contradiction to our hypothesis (A_{9}).

Let ΔU(n) > 0 for n ≥ n_{1}. The argument for the case U(n) < 0 is the same. Consider, U(n) > 0 for n ≥ n_{2} > n_{1}. By Lemma 2.1(i), it follows that U(n) ≥ C R(n), that is,

z(n) ≥ C R(n) + k(n) + Q^{+}(n) > C R(n),

for n ≥ n_{2}. Using (9) and proceeding as above we get a contradiction to our hypothesis (A_{9}).

If y(n) < 0 for n ≥ n_{0}, then we set x(n) = −y(n) to obtain x(n) > 0 for n ≥ n_{0} and

Δ(r(n)Δ(x(n) + p(n)x(n − m))) + f(n)H_{1}(x(n − k_{1})) − g(n)H_{2}(x(n − k_{2})) = q̃(n),

where q̃(n) = −q(n). If Q̃(n) = −Q(n), then Δ(r(n)ΔQ̃(n)) = −q(n) = q̃(n) and Q̃(n) changes sign. Further, Q̃^{+}(n) = Q^{−}(n) and Q̃^{−}(n) = Q^{+}(n). Proceeding as above we obtain a contradiction. Thus the proof of the theorem is complete.

Theorem 3.2 Let −1 < p ≤ p(n) ≤ 0. If (A_{0}), (A_{1}), (A_{4}), (A_{7}). (A_{8}) and

(A_{10})  ∑_{n=0}^{∞} f(n)H_{1}(R(n − k_{1})) = ∞

hold, then every solution of (1) oscillates.

Proof Let y(n) be a nonoscillatory solution of (1) such that y(n) > 0 for n ≥ n_{0}. The case y(n) < 0 for n ≥ n_{0} is similar. Setting as in (6) and (7), we get (8). Hence ΔU(n) is a monotonic function on [n_{1}, ∞), n_{1} > n_{0} + ρ. Let ΔU(n) < 0 for n ≥ n_{1}. Accordingly, U(n) is a monotonic function and lim_{n→∞} U(n) = lim_{n→∞}(z(n)−Q(n)) implies that z(n)−Q(n) < 0 when U(n) < 0, that is, z(n) < Q(n) for n ≥ n_{2} > n_{1}. If z(n) > 0, then 0 < z(n) < Q(n), which is absurd. Hence z(n) < 0 for n ≥ n_{2}, that is, y(n) < y(n − m), n ≥ n_{2} implies that y(n) is bounded on [n_{3}, ∞), n_{3} > n_{2} +m. Consequently, U(n) is bounded and lim_{n→∞}(r(n)ΔU(n)) exists. Using the fact that py(n−m) < z(n) < −Q^{−}(n) and Q(n) is bounded, we may conclude that lim inf_{n→∞} y(n) ≠ 0. On the other hand when (A_{10}) holds and since R(n) → 0 as n → ∞, then it follows that lim inf_{n→∞} y(n) = 0, a contradiction. Thus U(n) > 0 for n ≥ n_{2} > n_{1}. Using Lemma 2.1(ii), we have U(n) ≥ −r(n)R(n)ΔU(n) and hence for n ≥ n_{2}

z(n) ≥ −r(n)R(n)ΔU(n) + K(n) + Q^{+}(n),

that is,

y(n) ≥ −r(n)R(n)ΔU(n) + K(n) + Q^{+}(n)
> −r(n)R(n)ΔU(n).

Further, r(n)ΔU(n) is non-increasing. So we can find a constant C_{1} > 0 and n_{3} > n_{2} such that −r(n)ΔU(n) ≥ C_{1} for n ≥ n_{3}. Hence y(n) > −C_{1}R(n) for n ≥ n_{3}. Summing (8) from n_{4} to ∞, we get

H_{1}(−C_{1}) ∑_{n=n_{4}}^{∞} f(n)H_{1}(R(n − k_{1})) < − ∑_{n=n_{4}}^{∞} Δ(r(n)ΔU(n)),

n_{4} > n_{3} + k_{1}. Since lim_{n→∞} U(n) exists, then it follows from Theorem 3.1 that y(n) is bounded. Consequently, lim_{n→∞}(r(n)ΔU(n)) exists and the last inequality becomes

∑_{n=n_{4}}^{∞} f(n)H_{1}(R(n − k_{1})) < ∞,

a contradiction to (A_{10}).

Let ΔU(n) > 0 for n ≥ n_{1}. Then lim_{n→∞}(r(n)ΔU(n)) exists. Similar contradictions can be obtained for U(n) > 0 and U(n) < 0 for n ≥ n_{2} > n_{1}. The case y(n) < 0 for n ≥ n_{0} is similar. Hence the proof of the theorem is complete.

Theorem 3.3 Let −∞ < p ≤ p(n) ≤ −1. If all the conditions of Theorem 3.2 are satisfied, then every bounded solution of (1) oscillates.

Proof The proof follows from Theorem 3.2.

Theorem 3.4 If 0 ≤ p(n) ≤ p < ∞ and (A_{0}), (A_{2}) - (A_{5}), (A_{7}), (A_{8}) and

(A_{11})  ∑_{n=0}^{∞} F (n) = ∞

hold, then (1) is oscillatory.

Proof Proceeding as in the proof of Theorem 3.1, we assume that ΔU(n) < 0 for n ≥ n_{1}. Accordingly, U(n) < 0 for n ≥ n_{2} > n_{1} due to (A_{2}). Using the same type of reasoning as in the proof of Theorem 3.1, U(n) < 0 is a contradiction. Hence ΔU(n) > 0 for n ≥ n_{1}. Ultimately, U(n) > 0, n ≥ n_{2} > n_{1}. Since U(n) is nondecreasing, there exists a constant α > 0 and n_{3} > n_{2} such that U(n) ≥ α, n ≥ n_{3}. Therefore,

z(n) ≥ α + K(n) + F (n) ≥ α + K(n) + F^{+}(n) > α,

for n ≥ n_{3}. Summing (9) from n_{4} to ∞ and using the last inequality, we obtain

∑_{n=n_{4}}^{∞} F (n) < ∞, n_{4} > n_{3} + k_{1},

a contradiction to our assumption (A_{11}). This completes the proof of the theorem.

Theorem 3.5 Let 0 ≤ p(n) ≤ p < ∞ and m ≤ k_{1}. If (A_{0}), (A_{2}) - (A_{8}) and

(A_{12})  ∑_{n=k_{1}}^{∞} F (n)β(n − k_{1}) = ∞

hold, then every solution of (1) oscillates.

Proof Following the proof of the Theorem 3.4, we may consider the case ΔU(n) > 0 and U(n) > 0 for n ≥ n_{1}. Applying Lemma 2.2, inequality (9) yields

λF (n)H_{1}(β(n − k_{1})) ≤ − [H_{1}(r(n − k_{1})ΔU(n − k_{1}))]^{−1}Δ(r(n)ΔU(n))
− H_{1}(p)[H_{1}(r(n − k_{1})ΔU(n − k_{1}))]^{−1}Δ(r(n − m)ΔU(n − m))

for n ≥ n_{2} > n_{1}. Consequently,

λF (n)H_{1}(β(n − k_{1})) ≤ \frac{−Δ(r(n)ΔU(n))}{H_{1}(r(n)ΔU(n))} − \frac{H_{1}(p)Δ(r(n − m)ΔU(n − m))}{H_{1}(r(n − m)ΔU(n − m))}

for n ≥ n_{2}. Hence for r(n)ΔU(n) < x < r(n + 1)ΔU(n + 1) and r(n − m)ΔU(n − m) < y < r(n + 1 − m)ΔU(n + 1 − m), the last inequality becomes

λF (n)H_{1}(β(n − k_{1})) ≤ − ∫_{a(n)}^{a(n+1)}\frac{dx}{H_{1}(r(n)ΔU(n))}

− H_{1}(p) ∫_{a(n−m)}^{a(n+1−m)}\frac{dy}{H_{1}(r(n − m)ΔU(n − m))}

< − ∫_{a(n)}^{a(n+1)}\frac{dx}{H_{1}(x)} − H_{1}(p) ∫_{a(n−m)}^{a(n+1−m)}\frac{dy}{H_{1}(y)}

where a(n) = r(n)ΔU(n). Because lim_{n→∞}(r(n)ΔU(n)) exits, then for n ≥ n_{3} > n_{2},

λ lim_{t→∞} ∑_{n=n_{3}}^{t} F (n)H_{1}(β(n − k_{1})) < − lim_{t→∞} ∑_{n=n_{3}}^{t} ∫_{a(n)}^{a(n+1)}\frac{dx}{H_{1}(x)}

− H_{1}(p) lim_{t→∞} ∑_{n=n_{3}}^{t} ∫_{a(n−m)}^{a(n+1−m)}\frac{dy}{H_{1}(y)}

= − lim_{t→∞} [ ∫_{a(n_{3})}^{a(t+1)}\frac{dx}{H_{1}(x)} + H_{1}(p) ∫_{a(n_{3}−m)}^{a(t+1−m)}\frac{dy}{H_{1}(y)}]
< ∞ ,

a contradiction to (A_{12}). Thus the proof of the theorem is complete.

Theorem 3.6 If −1 < p ≤ p(n) ≤ 0 and (A_{0}), (A_{2}), (A_{4}), (A_{7}), (A_{8}) and

(A_{13})  ∑_{n=0}^{∞} f(n) = ∞

hold, then (1) is oscillatory.

Proof Let y(n) be a nonoscillatory solution of (1) such that y(n) > 0 for n ≥ n_{0}. The case y(n) < 0 for n ≥ n_{0} is similar. Proceeding as in the proof of Theorem 3.4, U(n) < 0 for n ≥ n_{2} > n_{1} when ΔU(n) < 0 for n ≥ n_{1}. Using the same type of reasoning as in the proof of Theorem 3.2, U(n) < 0 for n ≥ n_{2} yields a contradiction. Hence ΔU(n) > 0 for n ≥ n_{1}. Consequently, lim_{n→∞}(r(n)ΔU(n)) exists. Since U(n) is monotonic, then either U(n) > 0 or U(n) < 0 for n ≥ n_{2} > n_{1}. The contrdiction is similar, if U(n) < 0 for n ≥ n_{2}. Assume that U(n) > 0 for n ≥ n_{2}. Then z(n) − K(n) > Q(n) for n ≥ n_{2}. If z(n) < 0 then Q(n) < 0 for n ≥ n_{2}, which is absurd. Let z(n) > 0 for n ≥ n_{3} > n_{2}. Following to the proof of the Theorem 3.4, we have a contradiction due to (A_{13}). This completes the proof of the theorem.

Theorem 3.7 Suppose that −∞ < p ≤ p(n) ≤ −1. Let all the conditions of Theorem 3.6 hold. Then every bounded solution of (1) is oscillatory.

Proof The proof of the theorem follows from the proof of the Theorem 3.6 and hence the detail is omitted.

Theorem 3.8 Let 0 ≤ p(n) ≤ p < ∞. Assume that (A_{0}), (A_{1}), (A_{3}) - (A_{5}) and (A_{9}) hold. Then every solution of (2) either oscillates or tends to zero as n → ∞.

Proof Let y(n) be a nonoscillatory solution of (2) such that y(n) > 0 for n ≥ n_{0}. Setting as in (6), we write Eq.(2) as follows :

Δ(r(n)Δw(n)) = −f(n)H_{1}(y(n − k_{1})) ≤ 0, ≢ 0  (10)

for n ≥ n_{1} > n_{0} + ρ. Hence (r(n)Δw(n)) is a monotonic function on [n_{1}, ∞). We may suppose that Δw(n) < 0 for n ≥ n_{1}. If w(n) < 0, then y(n) ≤ z(n) ≤ k(n), n ≥ n_{1}. Because k(n) is bounded, there exists a constant γ > 0 such that y(n) ≤ γ for n ≥ n_{2} > n_{1}. Ultimately, w(n) is bounded and lim_{n→∞} w(n) exists. This is a contradiction to the fact that lim_{n→∞} w(n) = lim_{n→∞} z(n) ≠ 0 implies that z(n) < 0 for n ≥ n_{2} > n_{1}. Assume that w(n) > 0 for n ≥ n_{1}. Successive summations of the inequality Δ(r(n)Δw(n)) ≤ 0 from n_{1} to n, we can find a constant η > 0 such that w(n) ≤ η for n ≥ n_{2} > n_{1}. Thus y(n) ≤ z(n) ≤ K(n) + η implies y(n) is bounded. Therefore lim_{n→∞}(r(n)Δw(n)) exists. Using Lemma 2.1(ii) with U(n) replaced by w(n), we get w(n) ≥ −r(n)R(n)Δw(n) and hence z(n) ≥ −r(n)R(n)Δw(n) for n ≥ n_{2}. Repeated application of Eq.(2) and use of (A_{3}) and (A_{4}) yields

0 = Δ(r(n)Δw(n)) + H_{1}(p)Δ(r(n − m)Δw(n − m))
+ f(n)H_{1}(y(n − k_{1})) + H_{1}(p)f(n − m)H_{1}(y(n − m − k_{1})),

that is,

0 ≥ Δ(r(n)Δw(n))  +  H_{1}(p)Δ(r(n − m)Δw(n − m))
+ λF (n)H_{1}(z(n − k_{1}))
≥ Δ(r(n)Δw(n)) + H_{1}(p)Δ(r(n − m)Δw(n − m))
+  λF (n)H_{1}(−r(n − k_{1})R(n − k_{1})Δw(n − k_{1})))
= Δ(r(n)Δw(n)) + H_{1}(p)Δ(r(n − m)Δw(n − m))
+  λF (n)H_{1}(R(n − k_{1}))H_{1}(−r(n − k_{1})Δw(n − k_{1})) (11)

due to (A_{5}) for n ≥ n_{3} > n_{2} + k_{1}. Since −r(n)Δw(n) is nondecreasing, there exists a constant C_{1} > 0 and n_{4} > n_{3} such that −r(n)Δw(n) ≥ C_{1} for n ≥ n_{4}. Accordingly, the last inequality becomes

λF (n)H_{1}(C_{1})H_{1}(R(n − k_{1})) ≤ −Δ(r(n)Δw(n)) − H_{1}(p)Δ(r(n − m)Δw(n − m)) for n ≥ n_{5} > n_{4} + k_{1} which on summation from n_{5} to ∞, we get

∑_{n=n_{5}}^{∞} F (n)H_{1}(R(n − k_{1})) < ∞,

a contradiction to (A_{9}).

Next, we suppose that Δw(n) > 0 for n ≥ n_{1}. If w(n) < 0, then lim_{n→∞} w(n) exists and 0 ≠ lim_{n→∞} w(n) = lim_{n→∞} z(n) will imply that z(n) < 0, a contradiction to the fact that z(n) > 0. Let lim_{n→∞} w(n) = 0. Accordingly, lim_{n→∞} z(n) = 0 which provides lim_{n→∞} y(n) = 0 due to y(n) ≤ z(n) for n ≥ n_{2} > n_{1}. Consider, w(n) > 0 for n ≥ n_{2} > n_{1}. Lemma 2.1(i) can be applied here and it follows that w(n) ≥ C R(n), that is, z(n) ≥ w(n) ≥ C R(n) for n ≥ n_{2}. Consequently, (11) yields

λF (n)H_{1}(C)H_{1}(R(n − k_{1})) ≤ −Δ(r(n)Δw(n)) − H_{1}(p)Δ(r(n − m)Δw(n − m))

for n ≥ n_{2} + k_{1}. Summing the above inequality from n_{3} to ∞, we get

∑_{n=n_{3}}^{∞} F (n)H_{1}(R(n − k_{1})) < ∞, n_{3} > n_{2} + 2k_{1}

a contradiction.

If y(n) < 0 for n ≥ n_{0} > 0, then we set x(n) = −y(n) to obtain x(n) > 0 for n ≥ n_{0}. Above procedure can be applied for x(n) > 0 and hence we have the contradiction to (A_{9}). This completes the proof of the theorem.

Theorem 3.9 Let−1 < p ≤ p(n) ≤ 0. If (A_{0}), (A_{1}), (A_{4}) and (A_{10}) hold, then every solution of (2) either oscillates or tends to zero as n → ∞.

Proof Let y(n) be a nonoscillatory solution of (2) such that y(n) > 0 for n ≥ n_{0} ≥ 0. Setting as in (6), we get (10) for n ≥ n_{0} + ρ. Accordingly, Δw(n) is a monotonic function on [n_{1}, ∞) for which either w(n) > 0 or w(n) < 0 for n ≥ n_{2} > n_{1}. Consider Δw(n) < 0 and w(n) < 0 for n ≥ n_{2}. Then 0 ≠ lim_{n→∞} w(n) = lim_{n→∞} z(n) yields that z(n) < 0 for n ≥ n_{2}. Hence y(n) < y(n − m) for n ≥ n_{3} > n_{2}, that is, y(n) is a bounded real valued function on [n_{3}, ∞). Consequently, w(n) is bounded and lim_{n→∞}(r(n)Δw(n)) exists. Further, w(n) is monotonic implies that lim_{n→∞} w(n) = L, L ∈ (−∞, 0), that is, lim_{n→∞} z(n) = L. We claim that lim inf_{n→∞} y(n) = 0. If not, there exists γ > 0 and n_{4} > n_{3} such that y(n) ≥ γ or n ≥ n_{4}. Summing (10) from n_{4} to ∞, we get

∑_{n=n_{4}}^{∞} F (n) < ∞,

a contradiction to the fact that R(n) → 0 as n → ∞ and (A_{10}) implies that (A_{13}) hold. So our claim holds and by the Lemma 2.3, L = 0. We note that

0 = lim_{n→∞} z(n) = lim sup_{n→∞} (y(n) + p(n) y(n − m))
≥ lim sup_{n→∞} [y(n) + py(n − m)]
≥ lim sup_{n→∞} y(n) + lim inf_{n→∞} (py(n − m))
= (1 + p) lim sup_{n→∞} y(n)

implies lim sup_{n→∞} y(n) = 0 and hence lim_{n→∞} y(n) = 0. Next, we suppose that w(n) > 0 for

n ≥ n_{2}. Let lim_{n→∞} w(n) = a, a ∈ [0, ∞). If y(n) is unbounded, then there exists {n_{j}^{1}}_{j=1}^{∞} such that n_{j}^{1} → ∞ and y(n_{j}^{1}) → ∞ as j → ∞ and

y(n_{j}^{1}) = max {y(n) : n_{2} ≤ n ≤ n_{j}^{1}} .

Hence

w(n_{j}^{1}) ≥ y(n_{j}^{1}) + p y(n_{j}^{1} − m) − k(n_{j}^{1})
≥ (1 + p)y(n_{j}^{1}) − k(n_{j}^{1})

yields that w(n_{j}^{1}) → ∞ as j → ∞, a contradiction to the fact that lim_{n→∞} w(n) exists. Thus y(n) is bounded and hence lim_{n→∞}(r(n)Δw(n) exists. Using Lemma 2.1(ii) with u(n) replaced by w(n), we get w(n) ≥ −r(n)R(n)Δw(n) and

y(n) ≥ w(n) ≥ −r(n)R(n)Δw(n), n ≥ n_{3} > n_{2}.

Consequently, (10) becomes

f(n)H_{1}(R(n − k_{1}))H_{1}(−r(n − k_{1})Δw(n − k_{1})) ≤ −Δ(r(n)Δw(n))

for n ≥ n_{4} > n_{3} + k_{1}. Due to nonincreasing (r(n)Δw(n)), we can find a constant b > 0 and n_{5} > n_{4} + k_{1} such that r(n − k_{1})Δw(n − k_{1}) ≤ −b for n ≥ n_{5}. Summing the last inequality from n_{5} to ∞ , we get

∑_{n=n_{5}}^{∞} f(n)H_{1}(R(n − k_{1})) < ∞,

a contradiction to (A_{10}).

Assume that Δw(n) > 0 for n ≥ n_{1}. We have two cases, w(n) > 0 and w(n) < 0. If the former holds, then by the Lemma 2.1(i)

y(n) ≥ w(n) ≥ C R(n), n ≥ n_{2} > n_{1}

and accordingly, Eq.(10) can be written as

f(n)H_{1}(C R(n − k_{1})) ≤ −Δ(r(n)Δw(n)),

for n ≥ n_{3} > n_{2} + k_{1}. Summing the above inequality from n_{3} to ∞, we get

∑_{n=n_{3}}^{∞} f(n)H_{1}(R(n − k_{1})) < ∞,

a contradiction. Suppose the later holds. Then lim_{n→∞} w(n) exists and 0 ≠ lim_{n→∞} w(n) = lim_{n→∞} z(n) implies that z(n) < 0 for n ≥ n_{2} > n_{1}. Consquently, y(n) is a bounded real valued function on [n_{3}, ∞), n_{3} > n_{2} + ρ. Using the same type of reasoning as above, we obtained lim_{n→∞} y(n) = 0. If 0 = lim_{n→∞} w(n) = lim_{n→∞} z(n), we claim that y(n) is bounded. Otherwise there is a contradiction to the fact that w(n_{j}^{1}) → ∞ as j → ∞. Proceeding as above, we obtained

lim_{n→∞} y(n) = 0.

The case y(n) < 0 for n ≥ n_{0} ≥ 0 is similar. Hence the theorem is proved.

Theorem 3.10 Let −∞ < p_{1} ≤ p(n) ≤ p_{2} < −1. If (A_{0}), (A_{1}), (A_{4}) and (A_{10}) hold, then every bounded solution of (2) either oscillates or tends to zero as n → ∞.

Proof Let y(n) be a bounded nonoscillatory solution of (2) such that y(n) > 0 for n ≥ n_{0} ≥ 0. Then setting as in (6), we get (10) for n ≥ n_{1} > 0 + ρ. From (10) it follows that Δw(n) > 0 or Δw(n) < 0 for n ≥ n_{1}. Consider the case Δw(n) < 0 for n ≥ n_{1}. Proceeding as in the proof of the Theorem 3.9, we obtain L = 0. Thus

0 = lim_{n→∞} z(n) = lim inf_{n→∞} [y(n) + p(n) y(n − m)]
≤ lim inf_{n→∞} [y(n) + p_{2} y(n − m)]
≤ lim sup_{n→∞} y(n) + lim inf_{n→∞} (p_{2} y(n − m))
≤ lim sup_{n→∞} y(n) + p_{2} lim sup_{n→∞} y(n − m)
= (1 + p_{2}) lim sup_{n→∞} y(n)

implies that lim_{n→∞} y(n) = 0, since (1 + p_{2}) < 0. Rest of the proof can be followed from the proof of the Theorem 3.9. Hence the proof of the theorem is complete.

Theorem 3.11 Let 0 ≤ p(n) ≤ p < ∞. If (A_{0}), (A_{2}) - (A_{5}), and (A_{11}) hold, then a solution of (2) either oscillates or tends to zero as n → ∞.

Proof Let y(n) be a non-oscillatory solution of (2) such that y(n) > 0 for n ≥ n_{0} ≥ 0. The case y(n) < 0 for n ≥ n_{0} can similarly dealt with. Proceeding as in the proof of the Theorem 3.8, we assume that Δw(n) < 0 for n ≥ n_{1}. Accordingly, w(n) < 0 for n ≥ n_{2} > n_{1} due to (A_{2}). Using the same type of reasoning as in the proof of Theorem 3.8, we obtain a contradiction. Hence Δw(n) > 0 for n ≥ n_{1}. If w(n) < 0, then lim_{n→∞} w(n) exists for which there is a contradiction when 0 ≠ lim_{n→∞} w(n) = lim_{n→∞} z(n). Let lim_{n→∞} w(n) = 0. Following the proof of Theorem 3.8 we get, lim_{n→∞} y(n) = 0. Assume that w(n) > 0 for n ≥ n_{1}. Hence there exists a constant α > 0 such that w(n) ≥ α for n ≥ n_{2} > n_{1}, that is, z(n) ≥ w(n) ≥ α for n ≥ n_{2}. Accordingly, (10) yields that

∑_{n=n_{3}}^{∞} F (n) < ∞, n_{3} > n_{2} + k_{1},

a contradiction to our hypothesis. This completes the proof of the theorem.

Theorem 3.12 Let 0 ≤ p(n) ≤ p < ∞ and m ≤ k_{1}. If (A_{0}), (A_{2})-(A_{6}) and (A_{12}) hold, then every solution of (2) either oscillates or tends to zero as n → ∞.

Proof Proceeding as in the proof of the Theorem 3.11, we only consider the case Δw(n) > 0 and w(n) > 0 for n ≥ n_{1}. From (11) it follows that

0 ≥ Δ(r(n)Δw(n)) + H_{1}(p)Δ(r(n − m)Δw(n − m))
+ λF (n)H_{1}(β(n − k_{1}))H_{1}(r(n − k_{1})Δw(n − k_{1}))

due to Lemma 2.2, for n ≥ n_{2} > n_{1}. Hence

λF (n)H_{1}(β(n − k_{1})) ≤ − [H_{1}(r(n − k_{1})Δw(n − k_{1}))]^{−1} Δ(r(n)Δw(n))
− H_{1}(p) [H_{1}(r(n − k_{1})Δw(n − k_{1}))]^{−1} Δ(r(n − m)Δw(n − m)).

Rest of the proof follows from the Theorem 3.5 and hence the details are omitted.

Remark In Theorem 3.11, H_{1} could be linear, sublinear or superlinear. However, if we restrict m and k_{1}, H_{1} could be sublinear in Theorem 3.12 due to (A_{6}).

Theorem 3.13 Assume that −1 < p ≤ p(n) ≤ 0. If (A_{0}), (A_{2}), (A_{4}) and (A_{13}) hold, then a solution of (2) either oscillates or tends to zero as n → ∞.

Proof Proceeding as in the proof of the Theorem 3.11. We consider the case Δw(n) < 0 and w(n) < 0 for n ≥ n_{2} > n_{1}. Accordingly, w(n) is a monotonic function on [n_{2}, ∞) and 0 ≠ lim_{n→∞} w(n) = lim_{n→∞} z(n) exists. It follows from the Theorem 3.9 that lim_{n→∞} y(n) = 0.

Let Δw(n) > 0 for n ≥ n_{1}. If w(n) < 0 for n ≥ n_{2} > n_{1}, then use the same arguments as in Theorem 3.9 to obtain lim_{n→∞} y(n) = 0. Suppose that w(n) > 0 for n ≥ n_{2} > n_{1}. Then there exists a constant γ > 0 and n_{3} > n_{2} such that w(n) ≥ γ, n ≥ n_{3}. Consequently, y(n) ≥ w(n) ≥ γ for n ≥ n_{3}. Summing (10) from n_{3} + k_{1} to ∞, we obtain a contradiction to (A_{13}). Hence the proof of the theorem is complete.

Theorem 3.14 Let −∞ < p_{1} ≤ p(t) ≤ p_{2} < −1. If (A_{0}), (A_{2}), (A_{4}) and (A_{13}) hold, then every bounded solution of (2) either oscillates or tends to zero as n → ∞.

Proof The proof of the theorem can be followed from the Theorems 3.13 and 3.10.

Example Consider

Δ[e^{n}Δ(y(n) + p(n) y(n − 1))] + f(n) y(n − 2) − g(n) y^{3}(n − 4) = (−1)^{n}e^{n}, n ≥ 0, (12)

where p(n) = [2 + (−1)^{n}], f(n) = (2e + 3)e^{n} + e^{−n}, g(n) = e^{−n}. Indeed, F (n) = (2e + 3)e^{n−1} + e^{−(n−1)} and R(n) = \frac{e}{e−1}e^{−n}. If we choose Q(n) = [2(1 + e)]^{−1}(−1)^{n}, then q(n) = Δ(e^{n}ΔQ(n)). Clearly, (A_{9}) is satisfied. Theorem 3.1 can be applied to (12). Consequently, every solution of (12) oscillates. In particular, y(n) = (−1)^{n} is one of such solution.

Example Consider

Δ[e^{−n}Δ(y(n) + p(n) y(n − 1))] + f(n) y^{3}(n − 1) − g(n) y(n − 2) = 0,  (13)

where 0 ≥ p(n) = −\frac{1}{2}e^{−n} > −1, f(n) = \frac{(e+1)(e^{2}+1)}{e^{6}}e^{n} and g(n) = \frac{(e^{2}+1)(e^{3}+1)}{2e^{6}}e^{−2n}. Indeed, all the conditions of Theorem 3.13 are satisfied. Hence every solution of (13) either oscillates or tends to zero as n → ∞. In particular, y(n) = (−1)^{n}e^{−n} is such a solution of (13).

4 Existence Theorems

Theorem 4.1 Let 0 ≤ p(n) ≤ b_{1} < 1. Suppose that q(n) satisfies (A_{7}). If

(A_{14})  ∑_{n=0}^{∞} A(n + 1)f(n) < ∞ ;  ∑_{n=0}^{∞} A(n + 1)g(n) < ∞,

where A(n) = ∑_{S=M_{0}}^{n−1} \frac{1}{r(s)} , then Eq.(1) admits a positive bounded solution.

Proof It is possible to find M_{0} large enough such that

∑_{n=M_{0}}^{∞} A(n + 1)f(n) < \frac{1 − b_{1}}{10 H_{1}(1)},  ∑_{n=M_{0}}^{∞} A(n + 1)g(n) < \frac{1 − b_{1}}{20 H_{1}(1)}

Let Q(n) be such that −\frac{(1−b_{1})}{20} ≤ Q(n) ≤ \frac{1−b_{1}}{10} for n ≥ M_{0}. We can choose m_{1} > m_{2} ≥ M_{2} > M_{1} > M_{0} such that |Q(m_{1}) − Q(m_{2})| < \frac{1−b_{1}}{3} and

∑_{n=m_{2}}^{m_{1}−1} A(n + 1)f(n) < \frac{1 − b_{1}}{6 H_{1}(1)},  ∑_{n=m_{2}}^{m_{1}−1} A(n + 1)g(n) < \frac{1 − b_{1}}{6 H_{1}(1)}

Let X = ℓ_{M_{0}}^{∞} be the Banach space of all bounded real valued functions x(n), n ≥ M_{0} with the sup norm defined by

∥x∥ = sup{|x(n)| : n ≥ M_{0}}.

Define a set S ⊂ X as follows:

S = \{x ∈ X : \frac{1 − b_{1}}{10} ≤ x(n) ≤ 1, n ≥ M_{0}\} .

Then S is a closed bounded and convex subset of X. Define two maps T_{1} and T_{2} on S as follows :

(T_{1} x)(n) = \{ \begin{matrix}(T_{1} x)(M_{1}) , M_{0} ≤ n ≤ M_{1} \\ −p(n)x(n − m) + \frac{1+4b_{1}}{5} , n ≥ M_{1} \end{matrix}

(T_{2} x)(n) = \{ \begin{matrix}(T_{2} x)(M_{1}) , M_{0} ≤ n ≤ M_{1} \\ Q(n) + A(n) ∑_{s=n}^{∞}[f(s)H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))] \\ + ∑_{s=M_{0}}^{n−1} A(s + 1)[f(s)H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))], \\ n ≥ M_{1}. \end{matrix}

Clearly,

(T_{1} x)(n) + (T_{2} x)(n) ≤ \frac{1 + 4b_{1}}{5} + \frac{1 − b_{1}}{10} + H_{1}(1) ∑_{s=n}^{∞} A(s)f(s)
+ H_{1}(1) ∑_{s=M_{0}}^{n−1} A(s + 1)f(s)
≤ \frac{1 + 4b_{1}}{5} + \frac{1 − b_{1}}{10} + H_{1}(1) ∑_{s=M_{0}}^{∞} A(s + 1)f(s) < 1

and

(T_{1} x)(n) + (T_{2} x)(n) ≥ −b_{1} + \frac{1 + 4b_{1}}{5} − \frac{1 − b_{1}}{20} − \frac{1 − b_{1}}{20} = \frac{1 − b_{1}}{10}

implies that T_{1} x + T_{2} x ∈ S. Since 0 ≤ p(n) ≤ b_{1} < 1, it is easy to check that T_{1} is a contraction mapping.

Next, we show that T_{2} is continuous. Let {x_{j}(n)} be a sequence in S such that, ∥x_{j} − x∥ = 0 as j → ∞. Since S is a closed set, then x1 − b_{1}S and

|(T_{2} x_{j})(n) − (T_{2}x)(n)| ≤ ∑_{s=M_{0}}^{∞} A(s + 1)|f(s){H_{1}(x_{j}(s − k_{1}))H_{1}(x(s − k_{1}))}
− g(s){H_{2}(x_{j}(S − k_{2})) − H_{2}(x(s − k_{2}))}|.

As H_{i} is continuous, then lim_{j→∞} ∥T_{2} x_{j} − T_{2} x∥ = 0. We know that T_{2} is uniformly bounded,

there exists M_{2} > 0 such that m_{1} > m_{2} ≥ M_{2} and for all x(n) ϵ S,

|T_{2}x(m_{1}) − T_{2}x(m_{2})| ≤ |Q(m_{1}) − Q(m_{2})|
+ | ∑_{s=m_{2}}^{m_{1}−1} A(s + 1)[f(s) H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))]|

+ | ∑_{s=M_{2}}^{m_{1}−1} A(s + 1) [f(s) H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))]|

< \frac{1 − b_{1}}{3} + 2 (\frac{1 − b_{1}}{6} + \frac{1 − b_{1}}{6} ) = 1 − b_{1}.

Hence by the Lemma 2.4, T_{2} has a fixed point. Consequently, it follows from the discrete Krasnoselskii’s fixed point theorem that T_{2} x + T_{1} x has a fixed point in S, that is

x(n) = \frac{1 + 4b_{1}}{5} − p(n)x(n − m) + Q(n)
+ A(n) ∑_{s=n}^{∞}[f(s) H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))]
+ ∑_{s=M_{0}}^{n−1} A(s + 1) [f(s) H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))] .

Theorem 4.2 Let 1 < b_{1} ≤ p(n) ≤ b_{2} < \frac{1}{2} b_{1}^{2}. Suppose that q(n) satisfies (A_{7}). If (A_{14}) holds, then Eq.(1) admits a positive bounded solution.

Proof It is possible to find M_{0} large enough such that

H_{1}(1) ∑_{n=M_{0}}^{∞} A(n + 1)f(n) < \frac{b_{1} − 1}{8b_{1}} + \frac{b_{1} − 1}{16b_{2}}, H_{1}(1) ∑_{n=M_{0}}^{∞} A(n + 1)g(n) < \frac{b_{1} − 1}{16b_{2}}.

Let Q(n) be such that −\frac{(b_{1}−1)}{16b_{1} b_{2}} ≤ Q(n) ≤ \frac{b_{1}−1}{8b_{1}^{2}} + \frac{b_{1}−1}{16b_{1} b_{2}} for n ≥ M_{0}. We can choose m_{1} > m_{2} ≥ M_{2} > M_{1} > M_{0} such that |Q(m_{1}) − Q(m_{2})| < \frac{1−b_{1}}{3b_{1}} and

∑_{n=m_{2}+m}^{m_{1}+m−1} A(n + 1)f(n) < \frac{1 − b_{1}}{6b_{1} H_{1}(1)}, ∑_{n=m_{2}+m}^{m_{1}+m−1} A(n + 1)g(n) < \frac{1 − b_{1}}{6b_{1} H_{1}(1)}.

∥x∥ = sup{|x(n)| : n ≥ M_{0}}.

Define a set S ⊂ X as follows :

S = \{x ∈ X : \frac{b_{1} − 1}{8b_{1} b_{2}} ≤ x(n) ≤ 1, n ≥ M_{0}\} .

Clearly, S is a closed bounded and convex subset of X. Define two maps T_{1} and T_{2} on S as follows :

(T_{1} x)(n) = \{ \begin{matrix}(T_{1} x)(M_{1}) , M_{0} ≤ n ≤ M_{1} \\ −\frac{x(n+m)}{p(n+m)} + \frac{2b_{1}^{2}+b_{1}−1}{4b_{1} p(n+m)}, n ≥ M_{1} \end{matrix}

(T_{2} x)(n) = \{ \begin{matrix}(T_{2} x)(M_{1}) , M_{0} ≤ n ≤ M_{1} \\ \frac{Q(n+m)}{p(n+m)} + \frac{A(n+m)}{p(n+m)} ∑_{s=n+m}^{∞}[f(s)H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))] \\ +\frac{1}{p(n+m)} ∑_{s=M_{0}}^{n+m−1} A(s + 1)[f(s)H_{1}(x(s − k_{1})) − g(s)H_{2}(x(s − k_{2}))], \\ n ≥ M_{1}. \end{matrix}

It is easy to verify that T_{1} is a contraction mapping and T_{1} x + T_{2} x ∈ S.

Rest of the analysis can be followed from the Theorem 4.1. Hence the proof of the theorem is complete.

Remark In other ranges of p(n) except p(n) = ±1, the discrete Krasnoselskii’s fixed point theorem can be applied for existence of positive solutions of (1) under the suitable mappings T_{1} and T_{2}. The following theorems are stated without proof.

Theorem 4.3 Let−1 < b_{1} ≤ p(n) ≤ 0. Suppose that q(n) satisfies (A_{7}). If (A_{14}) holds, then Eq.(1) admits a positive bounded solution.

Theorem 4.4 Let −∞ < b_{1} ≤ p(n) ≤ b_{2} < −1. Assume that q(n) satisfied (A_{7}). If (A_{14}) holds, then Eq.(1) admits a positive bounded solution.

Acknowledgment : The author is thankful to the Referee for his kind helpful suggestions and necessary correction.

5 References

1. S. Z. Chen, L. Erbe; Oscillation results for second order scalar and matrix difference equations, Computers. Math. Applic. 28(1-3) (1994), 55-70.

2. M. M. A. EL-Sheikh; Oscillation and nonoscillation criteria for second order nonlinear differential equations, J. Math. Anal. Appl. 179(1993), 14-27.

3. Y. Jun, G. X. Ping; Positive solution of a class of neutral delay difference equations, Acta Mathematica Sinica 44 (2001), 409-416.

4. M. R. S. Kulenovic, S. Hadziomerspahic; Existence of nonoscillatory solution of second order linear neutral delay equation, J. Math. Anal. Appl. 228(1998), 436-448.

5. Q. Li, H. Liang, W. Dong, Z. Zhang; Existence of nonoscillatory solutions to higher order mixed difference equations, Elect. J. Diff. eqns. 2007(2007), 1-7.

6. K. Y. Liu, Z. Q. Zhang; Existence of positive solutions in neutral differential and difference equations with delays, Acta. Sci. Nat. Univ. Norm. Human 21(1998), 12-18.

7. A. Lomtatidze; Oscillation and nonoscillation criteria for second order linear differential equations, Geor. Math. J. 4 (1997), 129-138.

8. N. Parhi, A. K. Tripathy; Oscillation of solutions of forced nonlinear neutral difference equations of higher order, Proceedings of the VIII Ramanujan Symposium on Recent Developments in Nonlinear Systems, 2002.

9. A. K. Tripathy; Oscillation of fourth order nonlinear neutral difference equations-I, Math. Slovaca, 58(2008), No.2, 221-240.

10. A. K. Tripathy; Oscillation of fourth order nonlinear neutral difference equations-II, Math. Slovaca. 58(2008), No.5, 581-604.

11. A. K. Tripathy, K. V. V. S. Rao; Oscillation properties of a class of nonlinear differential equations of neutral type (to appear in Diff. Equ. Dyn. systems).

12. A. K. Tripathy; Oscillation properties of a class of neutral differential equations with positive and negative coefficients (to appear in Fasciculli Mathematici).

13. E. Wahlen; Positive solutions of second order differential equations, Nonlinear Analysis, 58(2004), 359-366.

14. Y. Yu, H. Wang; Nonoscillatory solutions of second order nonlinear neutral delay equations, J. Math. Anal. Appl. 311(2005), 445-456.

15. A. Zafer, Y. Yalcin, Y. S. Yilmaz; Positive solutions and oscillation of higher order neutral difference equations, Arch. Math. (BRNO), Tomus, 36(2000), 623636, CDDE 2000 issue.

(Received October 23, 2009)