Dynamic Systems and Applications 15 (2006) 155-172

FUNCTIONAL PERTURBATIONS OF NONOSCILLATORY SECOND ORDER DIFFERENCE EQUATIONS

WILLIAM F. TRENCH

Trinity University, San Antonio, Texas, 78212 USA Mailing address: 95 Pine Lane, Woodland Park, Colorado 80863, USA Email: wtrench@trinity.edu

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. We consider functional perturbations of the nonoscillatory equation

Δ(r_{n−1}Δx_{n−1}) + p_{n}x_{n} = 0.

Let S_{k} be the set of all real sequences of the form Y = {y_{n}}_{n=k}^{∞}. For each n > k, let f_{n}(Y) denote a real-valued functional of Y ∈ S_{k}. We give sufficient conditions on {f_{n}(Y)}_{n=k+1}^{∞} for the equation

Δ(r_{n−1}Δy_{n−1}) + p_{n}y_{n} = f_{n}(Y), n > k,

to have a solution Ŷ ∈ S_{k} that behaves in a precisely defined way like a given solution X̂ of the unperturbed equation as n → ∞. A specific family of nonlinear functional perturbations is considered in detail.

AMS (MOS) Subject Classification. 39A11.

1. INTRODUCTION

Throughout this paper all quantities are real. We consider functional perturbations of the equation

(1.1)  Δ(r_{n−1}Δx_{n−1}) + p_{n}x_{n} = 0,

under the standing assumption that {r_{n}}^{∞} and {p_{n}}^{∞} are infinite sequences of real numbers with r_{n} > 0, and (1.1) is nonoscillatory. In this case there is an integer k and sequences X_{1} = {x_{1n}}_{n=k}^{∞} and X_{2} = {x_{2n}}_{n=k}^{∞} of positive numbers that satisfy (1.1) for n > k, such that

(1.2)  r_{n}(x_{1n}x_{2,n+1} − x_{1,n+1}x_{2n}) = 1, n ≥ k,

and

(1.3)  lim_{n→∞} \frac{x_{2n}}{x_{1n}} = ∞.

Received October 27, 2004               1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

156 WILLIAM F. TRENCH

The solutions X_{1} and X_{2} are said to be recessive (or principal) and dominant (or nonprincipal), respectively, and X_{1} is unique up to a positive constant multiplier [3]. Let

ρ_{n} = \frac{x_{2n}}{x_{1n}} .

From (1.2) and (1.3),

(1.4)  Δρ_{n} = \frac{1}{r_{n}x_{1n}x_{1,n+1}} > 0 and lim_{n→∞} ρ_{n} = ∞.

Let S_{k} be the set of real sequences Y = {y_{n}}_{n=k}^{∞}. For each n > k, let f_{n}(Y) denote a real-valued functional of Y. We give sufficient conditions on {f_{n}(Y)}_{n=k+1}^{∞} for the functional equation

(1.5)  Δ(r_{n−1}Δy_{n−1}) + p_{n}y_{n} = f_{n}(Y), n > k,

to have a solution Ŷ ∈ S_{k} that behaves in a precisely defined way like a given solution X̂ of (1.1) as n → ∞. The case where f_{n}(Y) = f_{n}y_{n} has previously been considered in [1, 4, 5]. Some nonlinear and nonhomogeneous perturbations were also considered in [1].

2. PRELIMINARY CONSIDERATIONS

We impose the metric

d(X, Y) = ∑_{n=k}^{∞} \frac{1}{2^{n−k}} \frac{|x_{n} − y_{n}|}{1 + |x_{n} − y_{n}|}

on S_{k}. If {Y_{ν}} is a sequence in S_{k}, then lim_{ν→∞} Y_{ν} = Y with respect to this metric if and only if lim_{ν→∞} y_{νn} = y_{n} for every n ≥ k. Moreover, S_{k} is complete; i.e., (S_{k}, d) is a Frechét space.

We use the following lemmas to obtain our results. The first is a special case of the Schauder–Tychonoff theorem [2, Corollary 0.1, p 405].

Lemma 2.1. If U is a closed, convex, and compact subset of a Frechét space and T is a continuous mapping of U into itself, then T has a fixed point in T (U).

Lemma 2.2. Suppose that ∑_{n=k}^{∞} u_{m}x_{2m} converges, and let

(2.1)  ψ_{n} = sup_{ℓ≥n} |∑_{m=ℓ}^{∞} u_{m}x_{2m}| , n ≥ k.

Then ∑_{m=k}^{∞} u_{m}x_{1m} also converges,

(2.2)  |∑_{m=n}^{∞} u_{m}x_{1m}| ≤ \frac{2ψ_{n}}{ρ_{n}} , n ≥ k,

and

(2.3)  | ∑_{m=n+1}^{∞}(x_{2m} − x_{1m}ρ_{n})u_{m}| ≤ ψ_{n+1}, n ≥ k.

Proof. Define g_{n} = ∑_{m=n}^{∞} u_{m}x_{2m}. If k ≤ n < N, summation by parts yields

∑_{m=n}^{N} u_{m}x_{1m} = ∑_{m=n}^{N} \frac{u_{m}x_{2m}}{ρ_{m}} = ∑_{m=n}^{N} \frac{g_{m} − g_{m+1}}{ρ_{m}}

= \frac{g_{n}}{ρ_{n}} − \frac{g_{N+1}}{ρ_{N}} + ∑_{m=n+1}^{N} g_{m} (\frac{1}{ρ_{m}} − \frac{1}{ρ_{m−1}} ) . (2.4)

From (1.4) and (2.1),

(2.5)  ∑_{m=n+1}^{∞} |g_{m}| |\frac{1}{ρ_{m}} − \frac{1}{ρ_{m−1}} | ≤ \frac{ψ_{n+1}}{ρ_{n}} , n ≥ k,

and lim_{N→∞} g_{N+1}/ρ_{N} = 0. Therefore (2.4) implies that ∑_{m=n}^{∞} u_{m}x_{1m} converges and satisfies (2.2).

If k ≤ n ≤ N, summation by parts yields

∑_{m=n+1}^{N}(x_{2m} − x_{1m}ρ_{n})u_{m} = ∑_{m=n+1}^{N} (1 − \frac{ρ_{n}}{ρ_{m}} ) (g_{m} − g_{m+1})

= − (1 − \frac{ρ_{n}}{ρ_{N}} ) g_{N+1} + ρ_{n} ∑_{m=n+1}^{N} (\frac{1}{ρ_{m−1}} − \frac{1}{ρ_{m}} ) g_{m.}

Letting N → ∞ and invoking (2.5) yields (2.3).

The following elementary lemma motivates the choices of transformations of S_{k} whose fixed points are solutions of (1.5).

Lemma 2.3. If ∑_{n=k+1}^{∞} q_{n}x_{1n} converges, then

z_{n} = −x_{1n} ∑_{ℓ=k}^{n−1}(Δρ_{ℓ}) ∑_{m=ℓ+1}^{∞} q_{m}x_{1m}, n ≥ k,

is a solution of

(2.6)  Δ(r_{n−1}Δz_{n−1}) + p_{n}z_{n} = q_{n}, n > k.

If ∑_{n=k+1}^{∞} q_{n}x_{2n} converges, then

z_{n} = ∑_{m=n+1}^{∞}(x_{2m}x_{1n} − x_{1m}x_{2n})q_{m}, n ≥ k,

is a solution of (2.6).

3. GENERAL RESULTS

Theorem 3.1. Let X̂ be a given solution of (1.1) for n ≥ k, and let {σ_{n}}_{n=k+1}^{∞} be a sequence of positive numbers. Let U_{1} be the set of all Y ∈ S_{k} such that

(3.1)  |\frac{y_{n} − x̂_{n}}{x_{1n}} | ≤ σ_{n+1} and |Δ (\frac{y_{n} − x̂_{n}}{x_{1n}} )| ≤ 2σ_{n+1} \frac{Δρ_{n}}{ρ_{n+1}} , n ≥ k.

Suppose that (i) {f_{n}(Y)}_{n=k+1}^{∞} is defined if Y ∈ U_{1}; (ii) If {Y_{ν}} is a sequence in U_{1} such that lim_{ν→∞} Y_{ν} = Y, then lim_{ν→∞} f_{n}(Y_{ν}) =

f_{n}(Y), n > k;

(iii) ∑_{m=k+1}^{∞} x_{2m}f_{m}(Y) converges for every Y ∈ U_{1}, and there is a sequence {ϕ_{n}}_{n=k+1}^{∞} such that

(3.2)  sup_{ℓ≥n} |∑_{m=ℓ}^{∞} x_{2m}f_{m}(Y)| ≤ ϕ_{n}, n > k, Y ∈ U_{1},
(3.3)  ϕ_{n} ≤ σ_{n}, n > k,

and lim_{n→∞} ϕ_{n} = 0. Then (1.5) has a solution Ŷ ∈ S_{k} such that

(3.4)  |\frac{ŷ_{n} − x̂_{n}}{x_{1n}} | ≤ ϕ_{n+1} and |Δ (\frac{ŷ_{n} − x̂_{n}}{x_{1n}} )| ≤ 2ϕ_{n+1} \frac{Δρ_{n}}{ρ_{n+1}} , n ≥ k.

Proof. U_{1} is closed, convex, and compact. From (iii) and Lemma 2.2 with u_{m} = f_{m}(Y), we can define T_{1} on U_{1} by

(T_{1}(Y))_{n} = x̂_{n} + ∑_{m=n+1}^{∞}(x_{2m}x_{1n} − x_{1m}x_{2n})f_{m}(Y), n ≥ k.

Then

(3.5)  \frac{(T_{1}(Y))_{n} − x̂_{n}}{x_{1n}} = ∑_{m=n+1}^{∞}(x_{2m} − x_{1m}ρ_{n})f_{m}(Y),

and

Δ (\frac{(T_{1}(Y))_{n} − x̂_{n}}{x_{1n}} ) = −(Δρ_{n}) ∑_{m=n+1}^{∞} x_{1m}f_{m}(Y),

so (3.2) and Lemma 2.2, with u_{m} = f_{m}(Y), imply that

(3.6)  |\frac{(T_{1}(Y))_{n} − x̂_{n}}{x_{1n}} | ≤ ϕ_{n+1}, n ≥ k,

and

(3.7)  |Δ (\frac{(T_{1}(Y))_{n} − x̂_{n}}{x_{1n}} )| ≤ 2ϕ_{n+1} \frac{Δρ_{n}}{ρ_{n+1}} , n ≥ k.

Hence, (3.1) and (3.3) imply that T_{1}(U_{1}) ⊂ U_{1}.

FUNCTIONAL PERTURBATIONS 159

Now suppose that {Y_{ν}}_{ν=1}^{∞} is a sequence in U_{1} such that lim_{ν→∞} Y_{ν} = Y. We want to show that lim_{ν→∞} T_{1}(Y_{ν}) = T_{1}(Y); i.e., that

(3.8)  lim_{ν→∞}(T_{1}(Y_{ν}))_{n} = (T_{1}(Y))_{n}, n ≥ k.

If ϵ > 0, choose N > k + 1 such that ϕ_{n} < ϵ/4 if n > N. Then (3.2) implies that

(3.9)  |∑_{m=n}^{∞}(f_{m}(Y_{ν}) − f_{m}(Y))x_{2m}| ≤ ϵ/2, ν ≥ 1, n > N.

With N now fixed, (ii) implies that there is a ν_{0} so that

∑_{m=k+1}^{N} |f_{m}(Y_{ν}) − f_{m}(Y)|x_{2m} < ϵ/2, ν ≥ ν_{0}.

This and (3.9) imply that

(3.10)  |∑_{m=n}^{∞}(f_{m}(Y_{ν}) − f_{m}(Y))x_{2m}| < ϵ, n > k, ν ≥ ν_{0}.

Therefore, (3.5) and Lemma 2.2 with u_{m} = f_{m}(Y_{ν}) − f_{m}(Y) imply that

|(T_{1}(Y_{ν}))_{n} − (T_{1}(Y))_{n}| = | ∑_{m=n+1}^{∞}(x_{2m} − x_{1m}ρ_{n})(f_{m}(Y_{ν}) − f_{m}(Y))| ≤ ϵx_{1n},

n ≥ k, ν ≥ ν_{0}, which implies (3.8). Now Lemma 2.1 implies that there is a Ŷ ∈ U_{1} such T_{1}(Ŷ) = Ŷ; i.e.,

ŷ_{n} = x̂_{n} + ∑_{m=n+1}^{∞}(x_{2m}x_{1n} − x_{1m}x_{2n})f_{m}(Ŷ), n ≥ k,

and Lemma 2.3 implies that Ŷ satisfies (1.5). Setting Y = Ŷ = T (Ŷ) in (3.6) and (3.7) verifies (3.4).

Theorem 3.2. Let X̂ be a given solution of (1.1) for n ≥ k, and let {σ_{n}}_{n=k+1}^{∞} be a positive sequence. Let U_{2} be the set of all Y ∈ S such that

(3.11)  |\frac{y_{n} − x̂_{n}}{x_{2n}} | ≤ σ_{n+1} and |Δ (\frac{y_{n} − x̂_{n}}{x_{2n}} )| ≤ 2σ_{n+1} \frac{Δρ_{n}}{ρ_{n+1}} , n ≥ k.

Suppose that (i) {f_{n}(Y)}_{n=k+1}^{∞} is defined if Y ∈ U_{2}; (ii) If {Y_{ν}} is a sequence in U_{2} such that lim_{ν→∞} Y_{ν} = Y, then lim_{ν→∞} f_{n}(Y_{ν}) =

f_{n}(Y), n > k;

(iii) ∑_{m=k+1}^{∞} x_{1m}f_{m}(Y) converges for every Y ∈ U_{2}, and there is a sequence {ϕ_{n}}_{n=k+1}^{∞} such that

(3.12)  sup_{ℓ≥n} |∑_{m=ℓ}^{∞} x_{1m}f_{m}(Y)| ≤ ϕ_{n}, n > k, Y ∈ U_{2},

160 WILLIAM F. TRENCH

lim_{n→∞} ϕ_{n} = 0, and

(3.13)  max(ϕ̂_{n}, ϕ_{n+1}) ≤ σ_{n+1}, n ≥ k,

where

(3.14)  ϕ̂_{n} = \frac{1}{ρ_{n}} ∑_{ℓ=k}^{n−1} ϕ_{ℓ+1}Δρ_{ℓ}.

Then (1.5) has a solution Ŷ ∈ S_{k} such that

(3.15)  |\frac{ŷ_{n} − x̂_{n}}{x_{2n}} | ≤ ϕ̂_{n} and |Δ (\frac{ŷ_{n} − x̂_{n}}{x_{2n}} )| ≤ (ϕ̂_{n} + ϕ_{n+1})\frac{Δρ_{n}}{ρ_{n+1}} , n ≥ k.

Proof. U_{2} is closed, convex, and compact. From (iii), we can define T_{2} on U_{2} by

(3.16)  (T_{2}(Y))_{n} = x̂_{n} − x_{1n} ∑_{ℓ=k}^{n−1}(Δρ_{ℓ}) ∑_{m=ℓ+1}^{∞} f_{m}(Y)x_{1m}, n ≥ k.

Then

\frac{(T_{2}(Y))_{n} − x̂_{n}}{x_{2n}} = −\frac{1}{ρ_{n}} ∑_{ℓ=k}^{n−1}(Δρ_{ℓ}) ∑_{m=ℓ+1}^{∞} f_{m}(Y)x_{1m}

and

Δ (\frac{(T_{2}(Y))_{n} − x̂_{n}}{x_{2n}} ) =\frac{Δρ_{n}}{ρ_{n}ρ_{n+1}} ∑_{ℓ=k}^{n−1}(Δρ_{ℓ}) ∑_{m=ℓ+1}^{∞} f_{m}(Y)x_{1m}

− \frac{Δρ_{n}}{ρ_{n+1}} ∑_{m=n+1}^{∞} f_{m}(Y)x_{1m}.

Therefore, (3.12) and (3.14) imply that

(3.17) |\frac{(T_{2}(Y))_{n} − x̂_{n}}{x_{2n}} | ≤ ϕ̂_{n}, n ≥ k,

and

(3.18) |Δ (\frac{(T_{2}(Y))_{n} − x̂_{n}}{x_{2n}} )| ≤ (ϕ̂_{n} + ϕ_{n+1})\frac{Δρ_{n}}{ρ_{n+1}} , n ≥ k.

Now (3.11) and (3.13) imply that T_{2}(U_{2}) ⊂ U_{2}.

Now suppose that {Y_{ν}}_{ν=1}^{∞} is a sequence in U_{2} such that lim_{ν→∞} Y_{ν} = Y. We want to show that lim_{ν→∞} T_{2}(Y_{ν}) = T_{2}(Y); i.e. that,

(3.19) lim_{ν→∞} T_{2}(Y_{ν})_{n} = T_{2}(Y)_{n}, n ≥ k.

Let ϵ > 0. The argument used to obtain (3.10) shows that there is an integer ν_{0} such that

|∑_{m=n}^{∞}(f_{m}(Y_{ν}) − f_{m}(Y))x_{1m}| < ϵ, n > k, ν ≥ ν_{0}.

Therefore, from (3.16),

|(T_{2}Y_{ν})_{n} − (T_{2}Y)_{n}| ≤ ϵx_{1n} ∑_{ℓ=k}^{n−1} Δρ_{ℓ} = ϵx_{1n}(ρ_{n} − ρ_{k}), n ≥ k, ν ≥ ν_{0}.

This implies (3.19). Now Lemma 2.1 implies that there is a Ŷ ∈ U_{2} such that T_{2}(Ŷ) = Ŷ; i.e.,

ŷ_{n} = x̂_{n} − x_{1n} ∑_{ℓ=k}^{n−1}(Δρ_{ℓ}) ∑_{m=ℓ+1}^{∞} f_{m}(Ŷ)x_{1m}, n ≥ k,

and Lemma 2.3 implies that Ŷ satisfies (1.5). Setting Y = Ŷ = T_{2}(Ŷ) in (3.17) and (3.18) verifies (3.15).

Note that lim_{n→∞} ϕ̂_{n} = 0. To see this, suppose ϵ > 0. Since lim_{n→∞} ϕ_{n} = 0, there is an integer N such that ϕ_{m} < ϵ if m > N. Therefore, (3.14) implies that

ϕ̂_{n} < \frac{1}{ρ_{n}} ∑_{ℓ=k}^{N}(Δρ_{ℓ})ϕ_{ℓ+1} + ϵ, n > N + 1.

Hence, lim sup_{n→∞} ϕ̂_{n} ≤ ϵ. Since ϵ is arbitrary, it follows that lim_{n→∞} ϕ̂_{n} = 0.

4. A SPECIFIC RESULT

Now we consider the equation

(4.1)  Δ(r_{n−1}Δy_{n−1}) + p_{n}y_{n} = ρ_{n}^{−α}a_{n} ∑_{ℓ=k}^{n} b_{ℓ}y_{ℓ}^{γ}, n > k.

We will need the following lemmas.

Lemma 4.1 (Abel-Dini Theorem). If s > 0 then

∑^{∞} \frac{Δρ_{n−1}}{ρ_{n}^{s+1}} < ∞.

Lemma 4.2. If ρ_{n+1} = O(ρ_{n}) and s, t are real, then

(4.2)  ∑_{m=n}^{∞} ρ_{m}^{s}Δρ_{m}^{t} = O(ρ_{n}^{s+t}) if s + t < 0,

and

(4.3)  ∑_{m=k+1}^{n} ρ_{m}^{s}Δρ_{m−1}^{t} = \begin{cases}O(ρ_{n}^{s+t}) & if s + t > 0, \\ O(1) & if s + t < 0. \end{cases}

Proof. For (4.2), it suffices to show that ρ_{m}^{s}Δρ_{m}^{t} = O (Δρ_{m}^{s+t}). By the mean value theorem,

ρ_{m}^{s}Δρ_{m}^{t} = tρ_{m}^{s}u_{m}^{t−1}Δρ_{m} and Δρ_{m}^{s+t} = (s + t)v_{m}^{s+t−1}Δρ_{m},

where ρ_{m} < u_{m}, v_{m} < ρ_{m+1}. Therefore

\frac{ρ_{m}^{s}Δρ_{m}^{t}}{Δρ_{m}^{s+t}} = \frac{t}{s + t} (\frac{ρ_{m}}{v_{m}} )^{s} (\frac{u_{m}}{v_{m}} )^{t−1} ,

which is bounded for all m.

The proof of (4.3) is similar.

Theorem 4.3. Suppose that γ(≠ 0, 1) is real, c > 0, and

(4.4)  A_{n} = ∑_{ℓ=k}^{n−1} a_{ℓ}x_{1ℓ}, B_{n} = ∑_{ℓ=k}^{n−1} b_{ℓ}x_{1ℓ}^{γ}, C_{n} = ∑_{ℓ=k}^{n−1} A_{ℓ}b_{ℓ}x_{1ℓ}^{γ} n ≥ k,

are bounded. Suppose also that one of the following hypotheses hold:

H_{1} : i = r = 1, j = 2, α > 1, and λ = α − 1.

H_{2} : i = 1, j = r = 2, α > max(1, γ + 1), α ≠ 2γ, λ = α − max(γ + 1, 1),
(4.5)  ρ_{n}^{−λ} = O(Δρ_{n−1}),

and

(4.6)  ρ_{n+1} = O(ρ_{n}).

H_{3} : i = r = 2, j = 1, 0 < α − max(γ, 0) < 1, α ≠ 2γ, (4.6) holds, and

(4.7)  λ = α − max(γ, 0).

Then (4.1) has a solution Ŷ ∈ S_{k} such that

\frac{ŷ_{n} − cx_{rn}}{x_{in}} = O(ρ_{n+1}^{−λ}) and Δ (\frac{ŷ_{n} − cx_{rn}}{x_{in}} ) = O (\frac{Δρ_{n}}{ρ_{n+1}^{λ+1}} ) ,

if c^{γ−1} is sufficiently small.

Proof. Let 0 < θ < 1 and let U be the set of all Y ∈ S_{k} such that

(4.8)  |\frac{y_{n} − cx_{rn}}{x_{in}} | ≤ θc\frac{ρ_{k}^{λ+r−i}}{ρ_{n+1}^{λ}}

and

(4.9)  |Δ (\frac{y_{n} − cx_{rn}}{x_{in}} )| ≤ 2θcρ_{k}^{λ+r−i} \frac{Δρ_{n}}{ρ_{n+1}^{λ+1}} , n ≥ k.

To avoid annoying repetition, we invoke the boundedness of {A_{n}}, {B_{n}} and {C_{n}} repeatedly without stating that we are doing so. We also use summation by parts repeatedly without stating this explicitly, and without verifying in advance that the infinite series that arise from this are convergent. They are, in fact, all absolutely convergent, as our subsequent estimates will show. To avoid unnecesary and annoying subscripts, we will use the symbol M (as in |u_{n}| ≤ M|v_{n}|) as a generic constant throughout this proof; thus, the value of M may be different in each appearance. It is to be understood throughout that M is independent of Y ∈ U and c. If g_{n} is a

FUNCTIONAL PERTURBATIONS 163

functional of Y, we write g_{n}(Y) = O(c^{γ}v_{n}) to indicate that |g_{n}(Y)| ≤ Mc^{γ}|v_{n}|, n ≥ k, for all Y ∈ U and c > 0. Finally, whenever we invoke Lemma 4.2, we are also invoking (4.6).

Under all three hypotheses,

(4.10)  λ = \begin{cases}α − j + 1 − (r − 1)γ & if γ > 0, \\ α − j + 1 & if γ < 0. \end{cases}

It is to be understood throughout the proof that whenever we state that a quantity is O(ρ_{n}^{−λ}) or O(c^{γ}ρ_{n}^{−λ}), we are invoking (4.10).

It is convenient to work with the variable z_{n} = y_{n}/x_{in}. Then (4.8) and (4.9) become

(4.11)  |z_{n} − cρ_{n}^{r−i}| ≤ θc\frac{ρ_{k}^{λ+r−i}}{ρ_{n+1}^{λ}}

and

(4.12)  |Δ(z_{n} − cρ_{n}^{r−i})| ≤ 2θcρ_{k}^{λ+r−i} \frac{Δρ_{n}}{ρ_{n+1}^{λ+1}} , n ≥ k.

Note that if Y ∈ U and

|ζ_{n} − cρ_{n}^{r−i}| ≤ |z_{n} − cρ_{n}^{r−i}|, n ≥ k,

then

(4.13)  c(1 − θ)ρ_{n}^{r−i} ≤ ζ_{n} ≤ c(1 + θ)ρ_{n}^{r−i}, n ≥ k.

From (1.5) and (4.1),

f_{n}(Y) = ρ_{n}^{−α}a_{n} ∑_{ℓ=k}^{n} b_{ℓ}y_{ℓ}^{γ}, n > k.

We will show that

(4.14)  ∑_{m=n}^{∞} x_{jm}f_{m}(Y) = O(c^{γ}ρ_{n}^{−λ})

under all three hypotheses. Then we will apply Theorems 3.1 and 3.2 obtain the conclusions.

To establish (4.14), we write

∑_{m=n}^{∞} x_{jm}f_{m}(Y) = c^{γ}I_{n} + Z_{n}(Y),

where

(4.15)  I_{n} = ∑_{m=n}^{∞} x_{jm}f_{m}(X_{r}) = ∑_{m=n}^{∞} ρ_{m}^{−α}a_{m}x_{jm} ∑_{ℓ=k}^{n} b_{ℓ}x_{rℓ}^{γ}

164 WILLIAM F. TRENCH

and

(4.16)  Z_{n}(Y) = ∑_{m=n}^{∞} x_{jm}f_{m}(Y) − c^{γ}I_{n} = ∑_{m=n}^{∞} ρ_{m}^{−α}a_{m}x_{jm} ∑_{ℓ=k}^{m} (y_{ℓ}^{γ} − c^{γ}x_{rℓ}^{γ}) b_{ℓ}.

It suffices to show that

(4.17)  I_{n} = O(ρ_{n}^{−λ})

and

(4.18)  Z_{n}(Y) = O(c^{γ}ρ_{n}^{−λ}), Y ∈ U, c > 0.

We begin with (4.17). Recall that x_{jm} = ρ_{m}^{j−1}x_{1m} and x_{rℓ} = ρ_{ℓ}^{r−1}x_{1m}. From (4.4) and (4.15),

(4.19)  I_{n} = ∑_{m=n}^{∞} ρ_{m}^{j−α−1}a_{m}x_{1m} ∑_{ℓ=k}^{m} ρ_{ℓ}^{(r−1)γ}b_{ℓ}x_{1ℓ}^{γ} = ∑_{m=n}^{∞} ρ_{m}^{j−α−1}h_{m}ΔA_{m},

with

(4.20)  h_{m} = ∑_{ℓ=k}^{m} ρ_{ℓ}^{(r−1)γ}ΔB_{ℓ}.

If r = 1 then h_{m} = B_{m+1} = O(1). If r = 2 then

h_{m} = ρ_{m}^{γ}B_{m+1} − ∑_{ℓ=k+1}^{m} B_{ℓ}Δ (ρ_{ℓ−1}^{γ}) = \begin{cases}O(ρ_{m}^{γ}) & if γ > 0, \\ O(1) & if γ < 0. \end{cases}

In either case,

(4.21)  h_{m} = \begin{cases}O(ρ_{m}^{(r−1)γ}) & if γ > 0, \\ O(1) & if γ < 0. \end{cases}

From the second equality in (4.19),

I_{n} = −E_{n} − F_{n} − G_{n},

where

(4.22)  E_{n} = ρ_{n}^{j−α−1}h_{n}A_{n},
(4.23)  F_{n} = ∑_{m=n+1}^{∞} A_{m}ρ_{m}^{j−α−1}Δh_{m−1},

and

(4.24)  G_{n} = ∑_{m=n}^{∞} A_{m+1}h_{m}Δ (ρ_{m}^{j−α−1}) .

FUNCTIONAL PERTURBATIONS 165

From (4.21) and (4.22), E_{n} = O(ρ_{n}^{−λ}). From (4.4), (4.20), and (4.23),

F_{n} = ∑_{m=n+1}^{∞} A_{m}ρ_{m}^{j−α−1+(r−1)γ}b_{m}x_{1m}^{γ} = ∑_{m=n+1}^{∞} ρ_{m}^{j−α−1+(r−1)γ}ΔC_{m}

= −ρ_{n+1}^{j−α−1+(r−1)γ}C_{n+1} − ∑_{m=n+2}^{∞} C_{m}Δ (ρ_{m−1}^{j−α−1+(r−1)γ}) = O(ρ_{n}^{−λ}).

From (4.21) and (4.24), G_{n} = O(ρ_{n}^{−λ}) if r = 1 or γ < 0, while if r = 2 and γ > 0, then

|G_{n}| ≤ M ∑_{m=n}^{∞} ρ_{m}^{(r−1)γ}|Δρ_{m}^{j−α−1}| = O(ρ_{n}^{−λ})

by Lemma 4.2. We have now verified (4.14) under all three hypotheses.

Turning to (4.18), substituting

y_{ℓ} = z_{ℓ}x_{iℓ} = ρ_{ℓ}^{i−1}z_{ℓ}x_{1ℓ} and x_{rℓ} = ρ_{ℓ}^{(r−1)}x_{1ℓ}

into (4.16) and invoking (4.4) yields

(4.25)  Z_{n}(Y) = ∑_{m=n}^{∞} ρ_{m}^{j−α−1}g_{m}ΔA_{m},

where

(4.26)  g_{m} = ∑_{ℓ=k}^{m} (z_{ℓ}^{γ} − c^{γ}ρ_{ℓ}^{(r−i)γ}) ρ_{ℓ}^{(i−1)γ}ΔB_{ℓ} = P_{m} − Q_{m} − R_{m},

with

(4.27)  P_{m} = B_{m+1}(z_{m}^{γ} − c^{γ}ρ_{m}^{(r−i)γ})ρ_{m}^{(i−1)γ},
(4.28)  Q_{m} = ∑_{ℓ=k+1}^{m} B_{ℓ}ρ_{ℓ}^{(i−1)γ}Δ(z_{ℓ−1}^{γ} − c^{γ}ρ_{ℓ−1}^{(r−i)γ}),

and

(4.29)  R_{m} = ∑_{ℓ=k+1}^{m} B_{ℓ}(z_{ℓ−1}^{γ} − c^{γ}ρ_{ℓ−1}^{(r−i)γ})Δρ_{ℓ−1}^{(i−1)γ}.

From (4.25),

Z_{n}(Y) = −S_{n} − T_{n} − U_{n},

with

(4.30)  S_{n} = ρ_{n}^{j−α−1}g_{n}A_{n},
(4.31)  T_{n} = ∑_{m=n}^{∞} A_{m+1}g_{m}Δρ_{m}^{j−α−1},

and

U_{n} = ∑_{m=n+1}^{∞} A_{m}ρ_{m}^{j−α−1}Δg_{m−1}

= ∑_{m=n+1}^{∞} ρ_{m}^{j−α−1+(i−1)γ}(z_{m}^{γ} − c^{γ}ρ_{m}^{(r−i)γ})ΔC_{m} (from (4.4) and (4.26))

= −V_{n} − W_{n} − X_{n},

with

(4.32)  V_{n} = ρ_{n+1}^{j−α−1+(i−1)γ}(z_{n+1}^{γ} − c^{γ}ρ_{n+1}^{(r−i)γ})C_{n+1},
(4.33)  W_{n} = ∑_{m=n+2}^{∞} C_{m}ρ_{m}^{j−α−1+(i−1)γ}Δ(z_{m−1}^{γ} − c^{γ}ρ_{m−1}^{(r−i)γ}),

and

(4.34)  X_{n} = ∑_{m=n+1}^{∞} C_{m+1}(z_{m}^{γ} − c^{γ}ρ_{m}^{(r−i)γ})Δρ_{m}^{j−α−1+(i−1)γ}.

To complete the proof, we must show that S_{n}, T_{n}, V_{n}, W_{n} and X_{n} are O(c^{γ}ρ_{n}^{−λ}) under all three hypotheses. We consider the three cases separately.

Case 1: Suppose H_{1} holds.

From (4.11), (4.12), (4.13), and the mean value theorem,

(4.35)  z_{n}^{γ} − c^{γ} = O(c^{γ}ρ_{n+1}^{−λ})

and

(4.36)  Δz_{n}^{γ} = O (c^{γ} \frac{Δρ_{n}}{ρ_{n+1}^{λ+1}} ) .

From (4.27) and (4.35), P_{m} = O(c^{γ}ρ_{m+1}^{−λ}). By Lemma 4.1,

(4.37)  ∑^{∞} \frac{Δρ_{n}}{ρ_{n+1}^{λ+1}} < ∞,

so (4.28) and (4.36) imply that Q_{m} = 0(c^{γ}). From (4.29), R_{m} = 0. Hence, from (4.26), g_{m} = O(c^{γ}). Therefore, from (4.30) and (4.31), S_{n} = O(c^{γ}ρ_{n}^{−λ}) and T_{n} = O(c^{γ}ρ_{n}^{−λ}). Moreover, (4.32) and (4.35) imply that V_{n} = O(c^{γ}ρ_{n+1}^{−2λ}). From (4.33), (4.36), and (4.37),

|W_{n}| ≤ Mc^{γ} ∑_{m=n+2}^{∞} ρ_{m}^{−λ} \frac{Δρ_{m−1}}{ρ_{m}^{λ+1}} = O(c^{γ}ρ_{n+2}^{−λ}),

by Lemma 4.1. Finally, from (4.34) and (4.35),

|X_{n}| ≤ Mc^{γ} ∑_{m=n+2}^{∞} ρ_{m}^{−λ}|Δρ_{m−1}^{−λ}| = O(c^{γ}ρ_{n+1}^{−2λ}).

This concludes the proof of (4.18) under the hypothesis H_{1}.

FUNCTIONAL PERTURBATIONS 167

Case 2. Suppose H_{2} holds.

From (4.11), (4.13), and the mean value theorem,

(4.38)  z_{m}^{γ} − c^{γ}ρ_{m}^{γ} = O(c^{γ}ρ_{m}^{γ−λ−1}).

Therefore, from (4.27),

(4.39)  P_{m} = O(c^{γ}ρ_{m}^{γ−λ−1}).

To deal with Q_{m} (see (4.28) with i = 1 and r = 2), we denote

(4.40)  Φ_{ℓ} = Δ(z_{ℓ−1}^{γ} − c^{γ}ρ_{ℓ−1}^{γ}).

By the mean value theorem,

z_{ℓ}^{γ} − c^{γ}ρ_{ℓ}^{γ} = γζ_{ℓ}^{γ−1}(z_{ℓ} − cρ_{ℓ}) and z_{ℓ−1}^{γ} − c^{γ}ρ_{ℓ−1}^{γ} = γζ_{ℓ−1}^{γ−1}(z_{ℓ−1} − cρ_{ℓ−1}), where

(4.41)  |ζ_{ℓ} − cρ_{ℓ}| ≤ |z_{ℓ} − cρ_{ℓ}|, |ζ_{ℓ−1} − cρ_{ℓ−1}| ≤ |z_{ℓ−1} − cρ_{ℓ−1}|,

and, from (4.13),

(4.42)  c(1 − θ)ρ_{ℓ} ≤ ζ_{ℓ}, ζ_{ℓ−1} ≤ c(1 + θ)ρ_{ℓ}.

Now we can write

Φ_{ℓ} = γ (ζ_{ℓ}^{γ−1} − ζ_{ℓ−1}^{γ−1}) (z_{ℓ} − cρ_{ℓ}) + γζ_{ℓ−1}^{γ−1}Δ(z_{ℓ−1} − cρ_{ℓ−1}),

and so, by the mean value theorem,

(4.43)  |Φ_{ℓ}| ≤ |γ(γ − 1)|û_{ℓ}^{γ−2}|Δζ_{ℓ−1}|(z_{ℓ} − cρ_{ℓ}) + |γ|ζ_{ℓ−1}^{γ−1} |Δ(z_{ℓ−1} − cρ_{ℓ−1})| ,

where

(4.44)  c(1 − θ)ρ_{ℓ} ≤ û_{ℓ} ≤ c(1 + θ)ρ_{ℓ}.

However,

|Δζ_{ℓ−1}| ≤ |ζ_{ℓ} − cρ_{ℓ}| + cΔρ_{ℓ−1} + |ζ_{ℓ−1} − cρ_{ℓ−1}|
≤ |z_{ℓ} − cρ_{ℓ}| + cΔρ_{ℓ−1} + |z_{ℓ−1} − cρ_{ℓ−1}| from (4.41)
= cΔρ_{ℓ−1} + O(cρ_{ℓ}^{−λ}) = O(cΔρ_{ℓ−1})

from (4.5) and (4.11). Now (4.11), (4.12), (4.40), (4.42), (4.43), and (4.44) imply that

(4.45)  Δ(z_{ℓ−1}^{γ} − c^{γ}ρ_{ℓ−1}^{γ}) = O (c^{γ} \frac{Δρ_{ℓ−1}}{ρ_{ℓ}^{λ−γ+2}} ) .

From this and (4.28),

|Q_{m}| ≤ Mc^{γ} ∑_{ℓ=k+1}^{m} \frac{Δρ_{ℓ−1}}{ρ_{ℓ}^{λ−γ+2}} = \begin{cases}O(c^{γ}) & if γ < α/2, \\ O(c^{γ−λ−1}) & if γ > α/2, \end{cases}

by Lemmas 4.1 and 4.2. (Note that if γ > 0 then γ − λ − 1 = 2γ − α.) Since R_{m} = 0 (see (4.29) with i = 1), this, (4.26), and (4.39) imply that

(4.46)  g_{m} = \begin{cases}O(c^{γ}) & if γ < α/2, \\ O(c^{γ}ρ_{m}^{γ−λ−1}) & if γ > α/2. \end{cases}

From this and (4.30),

S_{n} = \begin{cases}O(c^{γ}ρ_{n}^{1−α}) & if γ < α/2, \\ O(c^{γ}ρ_{n}^{γ−α−λ}) & if γ > α/2; \end{cases}

hence, S_{n} = O(c^{γ}ρ_{n}^{−λ}). From (4.31) and (4.46),

|T_{n}| ≤ Mc^{γ} ∑_{m=n}^{∞} |Δρ_{m}^{1−α}| = O(c^{γ}ρ_{n}^{1−α})

if γ < α/2, and

|T_{n}| ≤ Mc^{γ} ∑_{m=n}^{∞} ρ_{m}^{γ−λ−1}|Δρ_{m}^{1−α}| = O(c^{γ}ρ_{n}^{γ−α−λ})

if γ > α/2, by Lemma 4.2; hence, T_{n} = O(c^{γ}ρ_{n}^{−λ}).

From (4.32) and (4.38), V_{n} = O(c^{γ}ρ_{n+1}^{γ−α−λ}). From (4.33), (4.45), and Lemma 4.1,

|W_{n}| ≤ Mc^{γ} ∑_{m=n+2}^{∞} ρ_{m}^{1−α} \frac{Δρ_{m−1}}{ρ_{m}^{λ−γ+2}} ≤ Mc^{γ}ρ_{n+2}^{−λ} ∑_{m=n+1}^{∞} \frac{Δρ_{m−1}}{ρ_{m}^{α−γ+1}} = O(c^{γ}ρ_{n+2}^{−λ}).

Finally, from (4.34) and (4.38),

|X_{n}| ≤ Mc^{γ} ∑_{m=n+1}^{∞} ρ_{m}^{γ−λ−1} |Δρ_{m}^{1−α}| = O(c^{γ}ρ_{n+1}^{γ−α−λ}),

by Lemma 4.2.

Case 3. Suppose that H_{3} holds. We first note that since α ≠ 2γ, γ ≠ λ.

Now (4.35) and (4.36) hold. From (4.27) and (4.35), P_{m} = O(c^{γ}ρ_{m}^{γ−λ}). From (4.28) and (4.36),

|Q_{m}| ≤ Mc^{γ} ∑_{ℓ=k+1}^{m} ρ_{ℓ}^{γ−λ−1}Δρ_{ℓ−1} = \begin{cases}O(c^{γ}ρ_{m}^{γ−λ}) & if γ > λ, \\ O(c^{γ}) & if γ < λ, \end{cases}

by Lemma 4.2. From (4.29), (4.35), and Lemma 4.2,

|R_{m}| ≤ Mc^{γ} ∑_{ℓ=k+1}^{m} ρ_{ℓ}^{−λ}Δρ_{ℓ−1}^{γ} = \begin{cases}O(c^{γ}ρ_{m}^{γ−λ}) & if γ > λ, \\ O(c^{γ}) & if γ < λ. \end{cases}

Therefore, from (4.26),

(4.47)  g_{m} = \begin{cases}O(c^{γ}ρ_{m}^{γ−λ}) & if γ > λ, \\ O(c^{γ}) & if γ < λ. \end{cases}

From this and (4.30),

S_{n} = \begin{cases}O(c^{γ}ρ_{n}^{−2λ}) & if γ > λ, \\ O(c^{γ}ρ_{n}^{−α}) & if γ < λ; \end{cases}

hence, S_{n} = O(c^{γ}ρ_{n}^{−λ}). From (4.31) and (4.47), T_{n} = O(ρ_{n}^{−α}) if γ < λ, and

|T_{n}| ≤ Mc^{γ} ∑_{m=n}^{∞} ρ_{m}^{γ−λ}Δρ_{m}^{−α} = O(c^{γ}ρ_{n}^{−2λ})

if γ > λ, by (4.7) and Lemma 4.2. In either case, T_{n} = O(c^{γ}ρ_{n}^{−λ}). From (4.32) and (4.35), V_{n} = O(c^{γ}ρ_{n+1}^{−2λ}). From (4.33) and (4.36),

|W_{n}| ≤ c^{γ} ∑_{m=n+2}^{∞} ρ_{m}^{γ−α−λ−1}Δρ_{m−1} = O(c^{γ}ρ_{n+2}^{−2λ}),

by Lemma 4.2. Finally, from (4.34) and (4.35),

|X_{n}| ≤ Mc^{γ} ∑_{m=n+1}^{∞} ρ_{m}^{−λ}Δρ_{m}^{γ−α} = O(c^{γ}ρ_{n+1}^{−2λ}),

by Lemma 4.2.

To complete the proof, we apply Theorems 3.1 and 3.2 with σ_{n} = θcρ_{k}^{λ+r−i}ρ_{n}^{−λ}. (If i = 1 as in H_{1} and H_{2}, compare (4.8) and (4.9) with (3.1); if i = 2 as in H_{3}, compare (4.8) and (4.9) with (3.11).) Under all three hypotheses, we have shown that if (4.8) and (4.9) hold, then

∑_{m=n}^{∞} x_{jm}f_{m}(Y) = O(c^{γ}ρ_{n}^{−λ}).

Thus, under H_{1} and H_{2} (where j = 2), (3.2) holds with ϕ_{n} = Mc^{γ}ρ_{n}^{−λ}, so ϕ_{n}/σ_{n} < 1 for n > k if c^{γ−1} is sufficiently small. Hence, Theorem 3.1 implies the conclusion. Under H_{3} (where j = 1), (3.12) holds with ϕ_{n} = Mc^{γ}ρ_{n}^{−λ}. Therefore, since λ < 1 in this case, (3.14) and Lemma 4.2 imply that

ϕ̂_{n} ≤ \frac{Mc^{γ}}{ρ_{n}} ∑_{ℓ=k}^{n−1} ρ_{ℓ+1}^{−λ}Δρ_{ℓ} ≤ M_{1}c^{γ}ρ_{n}^{−λ}.

Hence, (3.13) holds if c^{γ−1} is sufficiently small, so Theorem 3.2 implies the conclusion.

5. AN EXAMPLE

It is clear that {A_{n}}_{n=k}^{∞}, {B_{n}}_{n=k}^{∞}, and {C_{n}}_{n=k}^{∞} in (4.4) are bounded if, for example, {A_{n}}_{n=k}^{∞} is bounded and ∑^{∞} |b_{ℓ}|x_{1ℓ}^{γ} < ∞. However, they may be bounded even if ∑^{∞} a_{ℓ}x_{1ℓ}, ∑^{∞} b_{ℓ}x_{1ℓ}^{γ}, and ∑^{∞} A_{ℓ}b_{ℓ}x_{1ℓ}^{γ} all diverge. For example, if

a_{ℓ} = \frac{a cos(ℓt_{1} + s_{1})}{x_{1ℓ}} and b_{ℓ} = \frac{cos(ℓt_{2} + s_{2})}{x_{1ℓ}^{γ}}

170 WILLIAM F. TRENCH

where a ≠ 0, s_{1} and s_{2} are arbitrary,

0 < t_{1}, t_{2} < 2π, t_{1} ≠ t_{2}, and t_{1} + t_{2} ≠ 2π,

then {A_{n}}_{n=k}^{∞}, {B_{n}}_{n=k}^{∞}, and {C_{n}}_{n=k}^{∞} are bounded but divergent.

For a specific example, consider the equation

(5.1)  Δ (\frac{Δy_{n−1}}{n(n − 1)}) + \frac{2y_{n}}{n^{2}(n^{2} − 1)} = a_{n}n^{−α} ∑_{ℓ=1}^{n} b_{ℓ}y_{ℓ}^{γ}, n > 1,

where

a_{ℓ} = \frac{a cos(ℓt_{1} + s_{1})}{ℓ}, b_{ℓ} = \frac{cos(ℓt_{2} + s_{2})}{ℓ^{γ}}, ℓ ≥ 1,

and the constants are as above. The solutions of the unperturbed equation

Δ (\frac{Δx_{n−1}}{n(n − 1)}) + \frac{2x_{n}}{n^{2}(n^{2} − 1)} = 0, n > 1,

that satisfy (1.2) are x_{1n} = n and x_{2n} = n^{2}. Note that Δx_{1n} = 1 and Δx_{2n} = 2n + 1. Theorem 4.3 implies the following assertions for sufficiently small c^{γ−1}:

(i) If α > 1 then (5.1) has a solution Ŷ = {ŷ_{n}}_{n=1}^{∞} such that

\frac{ŷ_{n} − cn}{n} = O(n^{1−α}) and Δ (\frac{ŷ_{n} − cn}{n}) = O(n^{−α}),

which implies that

ŷ_{n} = [c + O(n^{1−α})] n and Δŷ_{n} = c + O(n^{1−α}).

(ii) If α > max(γ + 1, 1), α ≠ 2γ, and λ = α − max(γ + 1, 1), then (5.1) has a solution Ŷ = {ŷ_{n}}_{n=1}^{∞} such that

\frac{ŷ_{n} − cn^{2}}{n} = O(n^{−λ}) and Δ (\frac{ŷ_{n} − cn^{2}}{n} ) = O(n^{−λ−1}),

which implies that ŷ_{n} = [c + O(n^{−λ−1})] n^{2} and Δŷ_{n} = [c + O(n^{−λ−1})] (2n + 1).

(iii) If 0 < α − max(γ, 0) < 1, α ≠ 2γ, and λ = α − max(γ, 0), then (5.1) has a solution Ŷ = {ŷ_{n}}_{n=1}^{∞} such that

\frac{ŷ_{n} − cn^{2}}{n^{2}} = O(n^{−λ}) and Δ (\frac{ŷ_{n} − cn^{2}}{n^{2}} ) = O(n^{−λ−1}),

which implies that ŷ_{n} = [c + O(n^{−λ})] n^{2} and Δŷ_{n} = [c + O(n^{−λ})] (2n + 1).

REFERENCES

[1] S. Chen, A problem of Hartman and Wintner: approximation for discrete perturbations, Comput. Math. Appl. 4 (2001), 655-669.

[2] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York, London, Sydney, 1964.

[3] W. T. Patula, Growth and oscillation properties of second order linear difference equations, SIAM J. Math. Anal. 10 (1979), 55-61.

[4] W. F. Trench, Linear perturbations of a nonoscillatory second order difference equation, J. Math. Anal. Appl. 255 (2001), 627-635.

[5] W. F. Trench, Linear perturbations of a nonoscillatory second order difference equation II, J. Math. Anal. Appl. 273 (2002), 548-557.