Oscillation criteria for two-dimensional difference and differential systems of first-order linear difference equations are generalized and extended to nonlinear dynamic equations on arbitrary time scales

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

OSCILLATION AND NONOSCILLATION CRITERIA FOR TWO-DIMENSIONAL TIME-SCALE SYSTEMS OF FIRST-ORDER NONLINEAR DYNAMIC EQUATIONS

DOUGLAS R. ANDERSON

Abstract. Oscillation criteria for two-dimensional difference and differential systems of first-order linear difference equations are generalized and extended to nonlinear dynamic equations on arbitrary time scales. This unifies and extends under one theory previous linear results from discrete and continuous systems. An example is given illustrating that a key theorem is sharp on all time scales.

1. prelude

Jiang and Tang [14] establish sufficient conditions for the oscillation of the linear two-dimensional difference system

∆xn=pnyn, ∆y_n−1=−qnxn, n∈Z, (1.1) where {pn}, {qn} are nonnegative real sequences and ∆ is the forward difference operator given via ∆x_n =x_n+1−x_n; see also Li [16]. The system (1.1) may be viewed as a discrete analogue of the differential system

x⁰(t) =p(t)y(t), y⁰(t) =−q(t)x(t), t∈R, (1.2) investigated by Lomtatidze and Partsvania [17].

Oscillation questions in difference and differential equations are an interesting and important area of study in modern mathematics. Furthermore, within the past two decades, these two related but distinct areas have begun to be combined under a powerful, more robust and general theory titled dynamic equations on time scales, a theory introduced by Hilger [13]. We wish to generalize (1.1) and (1.2) to the nonlinear time-scale system of the form

x^∆(t) =p(t)f y(t)

, y^∆(t) =−q(t)g x(t)

, t∈T, (1.3) where T is an arbitrary time scale (any nonempty closed set of real numbers) unbounded above, with the special cases of T = Z and T = R yielding systems

2000Mathematics Subject Classification. 34B10, 39A10.

Key words and phrases. Nonoscillation; nonlinear system; time scales.

c

2009 Texas State University - San Marcos.

Submitted January 20, 2009. Published January 29, 2009.

1

related to (1.1) and (1.2), respectively, as important corollaries. In this general time-scale setting, ∆ represents the delta (or Hilger) derivative [4, Definition 1.10],

z^∆(t) := lim

s→t

z(σ(t))−z(s) σ(t)−s = lim

s→t

z^σ(t)−z(s) σ(t)−s ,

where σ(t) := inf{s ∈ T : s > t} is the forward jump operator, µ(t) := σ(t)−t is the forward graininess function, andz◦σis abbreviated asz^σ. In particular, if T=R, thenσ(t) =tandx^∆=x⁰, while ifT=hZfor anyh >0, thenσ(t) =t+h and

x^∆(t) = x(t+h)−x(t)

h .

A function f : T→ Ris right-dense continuous provided it is continuous at each right-dense point t∈T(a point where σ(t) =t) and has a left-sided limit at each left-dense pointt∈T. The set of right-dense continuous functions onTis denoted by Crd(T). It can be shown that any right-dense continuous function f has an antiderivative (a function Φ :T→Rwith the property Φ^∆(t) =f(t) for allt∈T).

Then the Cauchy delta integral off is defined by Z t1

t₀

f(t)∆t= Φ(t₁)−Φ(t₀),

where Φ is an antiderivative off onT. For example, ifT=Z, then Z t₁

t₀

f(t)∆t=

t1−1

X

t=t0

f(t), and ifT=R, then

Z t₁ t₀

f(t)∆t= Z t₁

t₀

f(t)dt.

Throughout we assume that t0 < t1 are points in T, and define the time-scale interval [t0, t1]_T = {t ∈ T : t0 ≤ t ≤ t1}. Other time-scale intervals are defined similarly.

Time scales and time-scale notation are introduced well in the fundamental texts by Bohner and Peterson [4, 5]. For related oscillation and nonoscillation results for dynamic equations on time scales, please see some of the many recent papers in this area, including Akin-Bohner, Bohner, and Saker [1], Bohner, Erbe, and Peterson [3], Bohner and Saker [6, 7], Bohner and Tisdell [8], Erbe and Peterson [9], Erbe, Peterson, and Saker [10, 11, 12], and Saker [18]. Recent papers on extensions of second-order self-adjoint equations to dynamic systems on time scales include Anderson and Hall [2], and Xu and Xu [19].

2. preliminary results on oscillation

LetTbe a time scale that is unbounded above, and lett0∈T. In (1.3), assume p: T → Ris right-dense continuous with p > 0 on [t0,∞)_T, and q: T → R is a right-dense continuous function satisfyingq≥0 on [t0,∞)_Twithqnonzero and not eventually zero; note that pand qare delta integrable. Moreover, we assume that f, g: R→Rare nondecreasing continuous functions that satisfy zf(z), zg(z)>0 forz6= 0, and that there exist positive real numbersFandGsuch thatf(y)/y≥F andg(x)/x≥G.

A solution (x, y) of (1.3) is oscillatory if both component functions x and y are oscillatory, that is to say neither eventually positive nor eventually negative;

otherwise, the solution is nonoscillatory. The nonlinear dynamic system (1.3) is oscillatory if all its solutions are oscillatory.

Lemma 2.1. The component functionsxandy of a nonoscillatory solution(x, y) of (1.3)are themselves nonoscillatory.

Proof. Assume to the contrary thatxoscillates buty is eventually positive. Then x^∆ = pf(y) > 0 eventually, so that x(t) > 0 or x(t) < 0 for all large t ∈ T, a contradiction. The case where y is eventually negative is similar. Likewise, assuming that y oscillates while x is eventually positive or eventually negative

leads to comparable contradictions.

Lemma 2.2. If Z ∞

t0

p(r)∆r=∞ and Z ∞

t0

q(s)∆s=∞, (2.1)

then each solution of nonlinear system (1.3)is oscillatory.

Proof. Let (x, y) be a nonoscillatory solution of (1.3). First assume thatx >0; then y^∆=−qg(x)≤0, and in view of Lemma 2.1,ymust be of constant sign eventually.

Ify(t₁)<0 for somet₁∈[t₀,∞)_T, theny <0 on [t₁,∞)_Tand x^∆=pf(y)<0 on [t₁,∞)_T; after delta integrating from t₁ tot, we have

x(t) =x(t1) + Z t

t1

p(r)f y(r)

∆r. (2.2)

Since y is negative and nonincreasing, and yf(y) > 0 with f nondecreasing, we know f(y)< 0, and by the first assumption in (2.1) the right-hand side of (2.2) tends to −∞, a contradiction of x > 0. Consequently, y > 0 with y^∆ ≤ 0 on [t0,∞)_T, and x^∆ >0 on [t0,∞)_T by the first equation of (1.3). Thus there exists a constant c > 0 and t1 ∈ [t0,∞)_T such that x(t) ≥ c for t ∈ [t1,∞)_T. Delta integrating the second equation of (1.3), we obtain

g(c) Z ∞

t₁

q(s)∆s≤y(t₁)<∞,

and this contradicts the second assumption in (2.1). Similar contradictions are

reached forx <0.

Lemma 2.3. If Z ∞

t0

p(r)∆r <∞ and Z ∞

t0

q(s)∆s <∞, (2.3)

then nonlinear system (1.3)is nonoscillatory.

Proof. Suppose that (2.3) holds. Then there existst₁∈[t₀,∞)_T such that Z ∞

t1

p(r)f

1 +g(2) Z ∞

r

q(s)∆s

∆r <1. (2.4)

LetB= C_rd(T) be the Banach space of right-dense continuous functions onT, with normkxk= sup_t≥t

1,t∈T|x(t)|and the usual pointwise ordering≤. Define a subset S ofBas follows:

S={x∈ B: 1≤x(t)≤2, t∈[t₁,∞)_T}.

For any subsetQof S, we have that infQ ∈ S and supQ ∈ S. LetL:S → B be the functional given via

(Lx)(t) = 1 + Z t

t₁

p(r)f 1 +

Z ∞ r

q(s)g x(s)

∆s

∆r, t∈[t1,∞)_T. By the assumptions on x∈ S and pand q and the fact that f and g are nonde- creasing, (Lx)(t)≥1 for allt∈[t1,∞)_T, and

(Lx)(t)≤1 + Z t

t1

p(r)f 1 +

Z ∞ r

q(s)g(2)∆s

∆r≤2 by (2.4). Moreover,

(Lx)^∆(t) =p(t)f 1 +

Z ∞ t

q(s)g x(s)

∆s

>0, (2.5)

ensuring thatL :S → S is increasing. By Knaster’s fixed-point theorem [15], we can conclude that there exists anx∈ S such that x=Lx. If we let

y(t) = 1 + Z ∞

t

q(s)g x(s)

∆s, t∈[t1,∞)_T using the fixed pointx∈ S, then we have

x^∆(t) = (Lx)^∆(t) =p(t)f y(t)

and y^∆(t) =−q(t)g x(t)

fort∈[t1,∞)_Tby using (2.5). Thus (x, y) is a nonoscillatory solution of (1.3).

In view of Lemmas 2.2 and 2.3, respectively, we could assume that either Z ∞

t₀

p(r)∆r=∞ and Z ∞

t₀

q(s)∆s <∞, or (2.6) Z ∞

t0

p(r)∆r <∞ and Z ∞

t0

q(s)∆s=∞; (2.7)

in fact, we will focus on (2.6). Moreover, in preparation for what follows, we introduce the following notation. Let

P(t) :=

Z t t₀

p(r)∆r. (2.8)

Lemma 2.4. Assume that (2.6)holds, P is given by (2.8), andλ∈[0,1)is a real number. If

t→∞lim

µ(t)p(t)

P(t) = 0,

equivalently, lim

t→∞

P^σ(t) P(t) = 1

(2.9) then given >0 there exists at1≡t1()∈(t0,∞)_T such that for anyt∈[t1,∞)_T,

Z ∞ t

P^λ∆

(r)2

p(r)P^λ(r) ∆r≤ λ²

1−λ(1 +)^2−λP^λ−1(t), and (2.10) Z ∞

t

p(r)

P^2−λ(r)∆r≤(1 +)^2−λ

1−λ P^λ−1(t). (2.11)

Proof. Forr∈(t₀,∞)_T, by the chain rule [4, Theorem 1.90] we have P^λ∆

(r) =







P^λ(σ(r))−P^λ(r)

µ(r) :µ(r)>0, λp(r)P^λ−1(r) :µ(r) = 0.

By [4, Theorem 1.16 (iv)],µP^∆=P^σ−P, so thatµp=P^σ−PonT. Ifr∈(t0,∞)_T is a right-scattered point, thenµ(r)>0 and, suppressing ther,

P^λ∆2

pP^λ = p

µ²p²P^λ

P^σλ

−P^λ2

= p P^λ

P^σλ

−P^λ P^σ−P

²

MVT= p

P^λ λξ^λ−12

, ξ∈ P(r), P^σ(r)

R

≤ pλ²

P^λP^2λ−2, λ−1<0

=λ²pP^λ−2.

Ifr∈(t0,∞)_Tis a right-dense point, thenµ(r) = 0 and P^λ∆2

pP^λ =

λpP^λ−12

pP^λ =λ²pP^λ−2. It follows that in either case,

h P^λ∆

(r)i²

p(r)P^λ(r) ≤λ²p(r)P^λ−2(r), r∈(t₀,∞)_T. (2.12) Similarly, ifr∈(t₀,∞)_T is a right-scattered point, then once again µ(r)>0 and, suppressing ther,

− P^λ−1∆

= −p µp

P^σλ−1

−P^λ−1

=−p P^σλ−1

−P^λ−1 P^σ−P

MVT= p(1−λ)η^λ−2, η∈ P(r), P^σ(r)

R

≥p(1−λ) P^σλ−2

.

Ifris a right-dense point, thenP^σ =P,µ(r) = 0, andp(1−λ)P^λ−2=− P^λ−1∆ . Summarizing, in either case we have

− P^λ−1∆

≥p(1−λ) P^σλ−2

, r∈(t0,∞)_T. (2.13) Combining (2.12) and (2.13), we see that

P^λ∆ (r)² p(r)P^λ(r) ≤ λ²

1−λ P(r)

P^σ(r) λ−2

− P^λ−1∆ (r)

.

By (2.9), given > 0 there exists a t1 ∈ [t0,∞)_T such that P^σ/P ≤ (1 +) on [t1,∞)_T. Consequently, for any t∈[t1,∞)_T,

Z ∞ t

P^λ∆

(r)2

p(r)P^λ(r) ∆r≤ λ²

1−λ(1 +)^2−λ Z ∞

t

− P^λ−1∆

(r)

∆r

(2.6),(2.8)

= λ²

1−λ(1 +)^2−λP^λ−1(t),

which is (2.10). Moreover, again for anyr∈[t1,∞)_T, p(r)

P^2−λ(r) = p(r) P^2−λ(σ(r))

P^2−λ(σ(r))

P^2−λ(r) ≤(1 +)^2−λ p(r) P^2−λ(σ(r))

(2.13)

≤ (1 +)^2−λ

λ−1 P^λ−1∆ (r).

(2.14)

Delta integrating (2.14) fromt to infinity, we obtain Z ∞

t

p(r)

P^2−λ(r)∆r≤(1 +)^2−λ λ−1

Z ∞ t

P^λ−1∆

(r)∆r(2.6),(2.8)

= (1 +)^2−λ

1−λ P^λ−1(t),

which is (2.11).

Note that ifT=R, then (2.9) is automatically satisfied, asµ(t)≡0.

Lemma 2.5. Assume that (2.6) holds, that P is given by (2.8), and that (2.9) holds. If for some real number λ <1 we have

Z ∞ t₁

q(r)P^λ(r)∆r=∞ for t1≥σ(t0), (2.15) then nonlinear system (1.3)is oscillatory.

Proof. By Lemma 2.3, we can focus onλ∈(0,1). Assume that (x, y) is a nonoscil- latory solution of nonlinear system (1.3), and assume that x >0 on [t0,∞)_T; the case wherex <0 on [t₀,∞)_T is similar and consequently omitted. As in the proof of Lemma 2.2, y >0 with y^∆≤0 and x^∆ >0 on [t₀,∞)_T. Let w:=y/x. Then w >0, and suppressing the argument, we have by the delta quotient rule and (1.3) that on [t₀,∞)_T,

w^∆= x^σy^∆−y^σx^∆

xx^σ =−qg(x)

x −pww^σf(y)

y ≤ −qG−pww^σF <0. (2.16) In fact this gives us

w^∆≤ −qG−p(w^σ)²F, (2.17) and from the previous line we obtain on [t0,∞)_Tthat

1 w

^∆

= −w^∆

ww^σ ≥ qG+pww^σF ww^σ ≥pF; delta integrating fromt0 totwe see that

1>1− w(t)

w(t0) ≥F w(t) Z t

t₀

p(r)∆r=F w(t)P(t)≥0, t∈[t₀,∞)_T. (2.18)

Again by the mean value theorem, P^λ∆

≤ λpP^λ−1 for λ∈ (0,1). Multiplying (2.17) byP^λand delta integrating fromt₁≥σ(t₀) tot we obtain

G Z t

t1

q(r)P^λ(r)∆r≤ − Z t

t1

P^λ(r)w^∆(r)∆r−F Z t

t1

p(r)P^λ(r)(w^σ)²(r)∆r

parts

= −P^λ(t)w(t) +P^λ(t₁)w(t₁) + Z t

t1

P^λ∆

(r)w^σ(r)∆r

−F Z t

t₁

p(r)P^λ(r)(w^σ)²(r)∆r

≤ −P^λ(t)w(t) +P^λ(t1)w(t1) + Z t

t₁

λp(r)P^λ−1(r)w^σ(r)∆r

−F Z t

t₁

p(r)P^λ(r)(w^σ)²(r)∆r

=−P^λ(t)w(t) +P^λ(t₁)w(t₁) +

Z t t₁

p(r)P^λ−2(r)

P(r)w^σ(r)

λ−F P(r)w^σ(r)

∆r.

(2.19) Since by (2.18) we have

0≤F P(t)w^σ(t)≤F P(t)w(t)<1, t∈[t0,∞)_T, (2.20) there exists a positive real numberksuch that

P(r)w^σ(r) λ−F P(r)w^σ(r) < k.

As a result we have lim_t→∞−P^λ(t)w(t) = 0 by (2.18) for 0< λ <1, and

Z t

t1

p(r)P^λ−2(r)

P(r)w^σ(r) λ−F P(r)w^σ(r)

∆r < k

Z ∞ t1

p(r)P^λ−2(r)∆r

(2.11)

≤ k(1 +)^2−λ

1−λ P^λ−1(t1) for allt∈[t1,∞)_T. Therefore,

Z ∞ t₁

q(r)P^λ(r)∆r <∞,

a contradiction of (2.15).

Due to (2.6) and the establishment of Lemma 2.5, we will henceforth restrict our analysis to the case

Z ∞ t0

p(r)∆r=∞, and Z ∞

t1

q(r)P^λ(r)∆r <∞ for λ <1, t1≥σ(t0).

(2.21) We also adopt the following notation. Set

g(t, λ) :=G

(P^1−λ(t)R∞

t q(r)P^λ(r)∆r :λ <1, P^1−λ(t)Rt

t₀q(r)P^λ(r)∆r :λ >1.

In either case, take

g_∗(λ) := lim inf

t→∞ g(t, λ) and g^∗(λ) := lim sup

t→∞

g(t, λ).

Lemma 2.6. Assume that (2.21) holds, that P is given by (2.8), and that (2.9) holds. If(x, y)is a nonoscillatory solution of nonlinear system (1.3), then

lim inf

t→∞ w(t)P(t)≥ 1 2F

1−p

1−4F g∗(0)

, (2.22)

lim sup

t→∞

w(t)P(t)≤ 1 2F

1 +p

1−4F g∗(2)

, (2.23)

where again w:=y/x.

Proof. By (2.18), we can introduce the constants r:= lim inf

t→∞ w(t)P(t), R:= lim sup

t→∞

w(t)P(t), (2.24) and by (2.21), we must have

t→∞lim w(t) = 0. (2.25)

From (2.16) we have w^∆ ≤ −qG−pww^σF; delta integrate this fromt to ∞, use (2.25), and multiply byP to see that

w(t)P(t)≥GP(t) Z ∞

t

q(τ)∆τ+F P(t) Z ∞

t

p(τ)w(τ)w^σ(τ)∆τ (2.26) holds fort∈[t1,∞)_T. From (2.24) this yields

r≥g_∗(0). (2.27)

This time multiply (2.17) byP²and delta integrate from t1 tot to get G

Z t t1

q(τ)P²(τ)∆τ ≤ − Z t

t1

P²(τ)w^∆(τ)∆τ−F Z t

t1

p(τ)P²(τ) w^σ2 (τ)∆τ

=−P²(t)w(t) +P²(t₁)w(t₁) + Z t

t₁

(P²)^∆(τ)w^σ(τ)∆τ

−F Z t

t₁

p(τ)P²(τ) w^σ2

(τ)∆τ

=−P²(t)w(t) +P²(t1)w(t1) + Z t

t₁

µ(τ)p²(τ)w^σ(τ)∆τ +

Z t t1

p(τ)P(τ)w^σ(τ)[2−F P(τ)w^σ(τ)]∆τ fort∈[t1,∞)_T, which leads to

w(t)P(t)

≤ −GP⁻¹(t) Z t

t₁

q(τ)P²(τ)∆τ+P⁻¹(t) Z t

t₁

µ(τ)p²(τ)w^σ(τ)∆τ +P⁻¹(t)P²(t₁)w(t₁) +P⁻¹(t)

Z t t₁

p(τ)P(τ)w^σ(τ)[2−F P(τ)w^σ(τ)]∆τ.

(2.28)

Using (2.20), 0<(1−F P w^σ)², leading toF P w^σ[2−F P w^σ]<1. Thus for large t∈T,

P⁻¹(t) Z t

t₁

p(τ)P(τ)w^σ(τ) [2−P(τ)w^σ(τ)] ∆τ ≤1/F.

Applying L’Hˆopital’s rule [4, Theorem 1.120], (2.20) again, and (2.9) we have 0≤ lim

t→∞

Rt

t1µ(τ)p²(τ)w^σ(τ)∆τ

P(t) = lim

t→∞µ(t)p(t)w^σ(t)≤ lim

t→∞

µ(t)p(t) P(t) = 0.

Altogether then, inequality (2.28) implies that

R≤1/F −g∗(2). (2.29)

If g_∗(0) = 0 = g_∗(2), then estimates (2.22) and (2.23) follow directly from (2.27) and (2.29), respectively. Thus we pick a real number∈ 0,min{g_∗(0), g_∗(2)}

and t2∈[t1,∞)_T such that fort∈[t2,∞)_T,

r− < w(t)P(t)< R+, w(t)P(t)≥GP(t) Z ∞

t

q(τ)∆τ > g_∗(0)−, GP⁻¹(t)

Z t t₀

q(τ)P²(τ)∆τ > g∗(2)−. From (2.26) and L’Hˆopital’s rule we have fort∈[t2,∞)_T that

w(t)P(t)≥g∗(0)−+F(r−)².

Multiply (2.16) byP² and delta integrate fromt1to tto see that this leads to w(t)P(t)≤ −GP⁻¹(t)

Z t t1

q(τ)P²(τ)∆τ+P⁻¹(t) Z t

t1

µ(τ)p²(τ)w^σ(τ)∆τ +P⁻¹(t)P²(t₁)w(t₁) +P⁻¹(t)

Z t t1

p(τ)P(τ)w^σ(τ)[2−F w(τ)P(τ)]∆τ.

(2.30) From (2.30) we have fort∈[t2,∞)_T that

w(t)P(t)≤ P²(t₁)w(t₁) +Rt

t1µ(τ)p²(τ)w^σ(τ)∆τ

P(t) −g_∗(2) ++ (R+)(2−F(R+)), sinceF w^σP ≤F wP <1. These two inequalities lead to

r≥g_∗(0) +F r², R≤R(2−F R)−g_∗(2). (2.31) Consequently,

r≥ 1 2F

1−p

1−4F g_∗(0)

, R≤ 1 2F

1 +p

1−4F g_∗(2) ,

and the lemma is proven.

3. main oscillation results

We use the lemmas obtained previously to prove our main results.

Theorem 3.1. Assume that (2.21) holds, thatP is given by (2.8), and that (2.9) holds. If

g∗(0) = lim inf

t→∞ P(t) Z ∞

t

q(τ)∆τ > 1

4F, or (3.1)

g_∗(2) = lim inf

t→∞

1 P(t)

Z t t₀

q(τ)P²(τ)∆τ > 1

4F, (3.2)

then every solution of nonlinear system (1.3)is oscillatory.

Proof. Suppose to the contrary that (x, y) is a nonoscillatory solution of (1.3) with x(t)>0 for t∈[t0,∞)_T. Let

r:= lim inf

t→∞ w(t)P(t), R:= lim sup

t→∞

w(t)P(t),

where w = y/x. By Lemma 2.6 and its proof (in particular (2.31)) and simple calculus, we have

g_∗(0)≤r−F r²≤ 1

4F and g_∗(2)≤R−F R²≤ 1 4F,

a contradiction of both (3.1) and (3.2). The case withx(t)<0 fort∈[t₀,∞)_T is

similar.

Theorem 3.2. Assume that (2.21) holds, thatP is given by (2.8), and that (2.9) holds. Let g_∗(2)≤1/(4F), and assume there exists a real number λ∈[0,1) such that

g^∗(λ)> λ²

4F(1−λ)+ 1 2F

1 +p

1−4F g_∗(2)

. (3.3)

Then every solution of nonlinear system (1.3)is oscillatory.

Proof. Suppose to the contrary that (x, y) is a nonoscillatory solution of (1.3) with x(t)>0 for t∈[t0,∞)_T. By (2.17) we have

Gq(t)≤ −w^∆(t)−F p(t)(w^σ)²(t), t∈[t0,∞)_T,

wherew=y/x; multiply this byP^λand delta integrate from tto infinity to get G

Z ∞ t

q(τ)P^λ(τ)∆τ≤ − Z ∞

t

w^∆(τ)P^λ(τ)∆τ−F Z ∞

t

p(τ)(w^σ)²(τ)P^λ(τ)∆τ

=P^λ(t)w(t) + Z ∞

t

P^λ∆

(τ)w^σ(τ)∆τ

−F Z ∞

t

p(τ)P^λ(τ)(w^σ)²(τ)∆τ

=P^λ(t)w(t) + 1 4F

Z ∞ t

(P^λ)^∆2

(τ) p(τ)P^λ(τ) ∆τ

− Z ∞

t

pF p(τ)P^λ/2(τ)w^σ(τ)− P^λ∆ (τ) 2p

F p(τ)P^λ/2(τ) 2

∆τ

≤P^λ(t)w(t) + 1 4F

Z ∞ t

(P^λ)^∆2

(τ) p(τ)P^λ(τ) ∆τ.

It follows that P^1−λ(t)G

Z ∞ t

q(τ)P^λ(τ)∆τ < P(t)w(t) +P^1−λ(t) 4F

Z ∞ t

(P^λ)^∆2

(τ)

p(τ)P^λ(τ) ∆τ. (3.4) By (2.10), (2.23), and (3.4),

g^∗(λ)≤ 1 2F

1 +p

1−4F g∗(2)

+ λ²

4F(1−λ),

a contradiction of (3.3). Similarly ifx(t)<0 fort∈[t0,∞)_T.

Corollary 3.3. Assume that (2.21)holds, thatP is given by (2.8), and that (2.9) holds. Ifg_∗(2)≤1/(4F)andg^∗(0)>_2F¹

1 +p

1−4F g_∗(2)

, then every solution of nonlinear system (1.3)is oscillatory.

Theorem 3.4. Assume that (2.21) holds, thatP is given by (2.8), and that (2.9) holds. Letg∗(0), g∗(2)≤1/(4F), and assume there exists a real number λ∈[0,1) such that

g_∗(0)>λ(2−λ)

4F , and (3.5)

g^∗(λ)> g_∗(0) 1−λ+ 1

2F

p1−4F g_∗(0) +p

1−4F g_∗(2)

. (3.6)

Then every solution of nonlinear system (1.3)is oscillatory.

Proof. Suppose to the contrary that (x, y) is a nonoscillatory solution of (1.3) with x(t)>0 for t ∈ [t0,∞)_T; the case with x(t) <0 for t ∈[t0,∞)_T is omitted. Let r= lim inf_t→∞w(t)P(t) andR= lim sup_t→∞w(t)P(t), where w=y/x. By (2.22) and (2.23),

r≥m:= 1 2F

1−p

1−4F g_∗(0)

, R≤M := 1 2F

1 +p

1−4F g_∗(2) . (3.7) Using (3.5) and (3.7) we find that m > λ/(2F), whence given ∈ 0, m−_2F^λ

, there exists at₁∈[t₀,∞)_T such that

m− < w(t)P(t)< M+, t∈[t₁,∞)_T. (3.8) Similar to what we did in (2.19), multiply (2.17) byP^λ and delta integrate fromt to infinity to get

G Z ∞

t

q(τ)P^λ(τ)∆τ

≤w(t)P^λ(t) + Z ∞

t

p(τ)P^λ−2(τ)

λw^σ(τ)P(τ)−F P(τ)w^σ(τ)²

∆τ;

this leads to P^1−λ(t)G

Z ∞ t

q(τ)P^λ(τ)∆τ ≤w(t)P(t) +P^1−λ(t) Z ∞

t

p(τ)P^λ−2(τ)

×

λw^σ(τ)P(τ)−F P(τ)w^σ(τ)²

∆τ.

(3.9) Since the functionγ(z) :=λz−F z² is decreasing over the real interval [_2F^λ ,∞), it follows from (3.8), (3.9), and Lemma 2.4 that

P^1−λ(t)G Z ∞

t

q(τ)P^λ(τ)∆τ

< M++ (m−)(λ−F(m−))P^1−λ(t) Z ∞

t

p(τ)P^λ−2(τ)∆τ

< M++(m−)(λ−F(m−))(1 +)^2−λ

1−λ .

This in tandem with (3.7) yields g^∗(λ)≤M +m(λ−F m)

1−λ = g_∗(0) 1−λ+ 1

2F

p1−4F g_∗(0) +p

1−4F g_∗(2) ,

a contradiction of (3.6).

Corollary 3.5. Assume that (2.21)holds, thatP is given by (2.8), and that (2.9) holds. Let0< g^∗(0)≤1/(4F)andg∗(2)≤1/(4F). If

g^∗(0)> g∗(0) + 1 2F

p1−4F g∗(0) +p

1−4F g∗(2) , then every solution of nonlinear system (1.3)is oscillatory.

4. example

We illustrate Theorem 3.1 with the following example.

Example 4.1. LetTbe an arbitrary time scale unbounded above, and let pand F be positive constants. Then the linear system

x^∆(t) =pF y(t), y^∆(t) = −1

tσ(t)x(t), t∈[t0,∞)_T (4.1) fort0 >0, is nonoscillatory for 0< p≤1/(4F) and oscillatory for p >1/(4F). In other words, the inequality in (3.1) is sharp on all time scales.

Proof. Note that p(t)≡p, f(y) =F y, q(t) = _tσ(t)¹ , and g(x) =x. Thus we have P(t) =p(t−t0),f(y)/y=F, andG≡1, so that

g_∗(0) = lim inf

t→∞ GP(t) Z ∞

t

q(r)∆r= lim inf

t→∞

p(t−t₀) t =p.

By Theorem 3.1 and (3.1), any solution (x, y) of (4.1) oscillates if p > 1/(4F).

Converting (4.1) to a second-order dynamic equation forx, we arrive at a Cauchy- Euler equation [5, Section 2.3] of the form

tσ(t)x^∆∆(t) +pF x(t) = 0, with general solution

x(t) =Ae1+√ 1−4F p 2t

(t, t₀) +Be1−√ 1−4F p 2t

(t, t₀), (4.2) where we have used a linear combination involving the time-scale exponential func- tion [4, Section 2.2]. From elementary analysis and Euler’s formula we know that x is nonoscillatory for p ≤ 1/(4F) and oscillatory for p > 1/(4F), showing in particular that the 1/(4F) in (3.1) is sharp for all time scalesT. Remark 4.2. In Example 4.1 we can identify the exponential functions that occur in (4.2) for specific time scales [5, Example 2.19]. Lettingλ= ¹⁺

√1−4F p

2 , we get that

T=R:e1+√ 1−4F p 2t

(t, t0) =t t₀

λ

, T=q^Z:e1+√

1−4F p 2t

(t, t0) =t t0

^logq[1+(q−1)λ]

, T=Z:e1+√

1−4F p 2t

(t, t0) = Γ(t+λ)Γ(t₀) Γ(t)Γ(t0+λ), where Γ is the gamma function.

References

[1] E. Akin-Bohner, M. Bohner and S. H. Saker, Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations,Electron. Trans. Numer. Anal.27 (2007), 1—12.

[2] D. R. Anderson and W. R. Hall, Oscillation criteria for two-dimensional systems of first-order linear dynamic equations on time scales,Involve(2009), to appear.

[3] M. Bohner, L. Erbe, and A. Peterson, Oscillation for nonlinear second order dynamic equations on a time scale,J. Math. Anal. Appl.301 (2005), no. 2, 491—507.

[4] M. Bohner and A. Peterson,Dynamic Equations on Time Scales, An Introduction with Ap- plications, Birkh¨auser, Boston, 2001.

[5] M. Bohner and A. Peterson, editors, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.

[6] M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales,Rocky Mountain J. Math.34 (2004), no. 4, 1239—1254.

[7] M. Bohner and S. H. Saker, Oscillation of second order half-linear dynamic equations on discrete time scales,Int. J. Difference Equ.1 (2006), no. 2, 205—218.

[8] M. Bohner and C. C. Tisdell, Oscillation and nonoscillation of forced second order dynamic equations,Pacific J. Math.230 (2007), no. 1, 59—71.

[9] L. Erbe and A. Peterson, Boundedness and oscillation for nonlinear dynamic equations on a time scale,Proc. Amer. Math. Soc.132 (2004) 735–744.

[10] L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales,J. London Math. Soc.67 (2003) 701–714.

[11] L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations,J. Math. Anal. Appl.333:1 (2007) 505—522.

[12] L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for a forced second-order nonlinear dynamic equation,J. Difference Eq. Appl.14:10 (2008) 997—1009.

[13] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math.18 (1990) 18–56.

[14] J. C. Jiang and X. H. Tang, Oscillation criteria for two-dimensional difference systems of first order linear difference equations,Comput. Math. Appl.54 (2007) 808—818.

[15] B. Knaster, Un th´eor`eme sur les fonctions d’ensembles, Ann. Soc. Polon. Math. 6 (1928) 133—134.

[16] W. T. Li, Classification schemes for nonoscillatory solutions of two-dimensional nonlinear difference systems,Comput. Math. Appl.42 (2001) 341—355.

[17] A. Lomtatidze and N. Partsvania, Oscillation and nonoscillation criteria for two-dimensional systems of first order linear ordinary differential equations,Georgian Math. J.6 (1999) 285—

298.

[18] S. H. Saker, Oscillation of nonlinear dynamic equations on time scales,Appl. Math. Comput.

148 (2004) 81–91.

[19] Y. J. Xu and Z. T. Xu, Oscillation criteria for two-dimensional dynamic systems on time scales,J. Computational Appl. Math.225:1 (2009) 9—19.

Douglas R. Anderson

Concordia College, Department of Mathematics and Computer Science, Moorhead, MN 56562, USA

E-mail address:[email protected]