Existence of Bounded Oscillatory Solutions to Second Order Forced Neutral Dynamic Equations with Time Delay on Time Scales

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Existence of Bounded Oscillatory Solutions to Second Order Forced Neutral Dynamic Equations with Time Delay on Time Scales

JINJUAN FAN Shandong Normal University School of Mathematical Sciences 88 Culture East Road, 250014 Ji’nan

CHINA fjj18@126.com

XIAODONG LU Shandong Normal University School of Mathematical Sciences 88 Culture East Road, 250014 Ji’nan

CHINA 2801108032@qq.com

Abstract: In this paper, we study the existence of oscillatory solutions to a class of second order forced neutral dynamic equations with time delay on time scales and give several suff cient conditions for oscillation of bounded solutions.

Key–Words:time scales; time delay; neutral; dynamic equations; oscillatory solution

1 Introduction

The theory of oscillation of neutral differential equations is rich and has wide applications, and there are much research and abundant achievements on the theory of oscillation of differential equations. In the past ten years, oscillation and nonoscillation of dynamic equations on time scales have aroused wide attention of scholars at home and abroad, and the theory of time scales is also gradually improved^[1−2]. However, the research results of oscillation of dynamic equations with time delay on time scales is relatively little.

Agarwal et al.^[3]studied the following second order neutral differential equation

x^∆∆(t) + p(t)x(τ (t)) = 0, t ∈ T,

and gave several suff cient conditions for oscillation of the above equation.

Erbe et al.^[4] considered second order nonlinear dynamic equation with time delay on time scales

[r(t)x^∆(t)]^∆+ p(t)f (x(τ (t))) = 0, t ∈ T, and obtained some new suff cient conditions for en- suring that every solution oscillates or converges to zero by Riccati transformation techniques.

Sahiner^[5]studied the oscillation of solution to the following second order neutral dynamic equation with time delay on time scales

[r(t)((x(t)+p(t)(x(t−τ ))^∆)^γ]^∆+f (t, x(t−δ)) = 0, t ∈ T.

Saker^[6] studied the following second order nonlinear neutral dynamic equation with time delay on time scales

[r(t)((x(t)+p(t)(x(τ (t))^∆)^γ]^∆+f (t, x(δ(t))) = 0, t ∈ T,

and gained several suff cient conditions for oscillation of the above equation.

Based on the above results, we study the existence of oscillatory solution to more general neutral equations by adding forced term, and give several suf- f cient conditions for oscillation of bounded solutions.

In this paper, we consider the existence of bounded oscillatory solutions to the forced second order neutral dynamic equation with time delay on time s- cales as follows

[r(t)z^∆(t)]^∆+

m

X

i=1

fi(t, x(δi(t))) = h(t), t ∈ [t0, ∞)T, (1) where z(t) = x(t)+p(t)x(τ(t)), and T is a unbound- ed time scale.

2 Preliminaries and Lemmas

A time scale is a nonempty closed subset in the set of real numbers R. Assuming T is a time scale, when t < sup T, we def ne forward jump operator σ : T → T by σ(t) = inf{s ∈ T : s > t}; when t > inf T, we def ne back jump operator ρ : T → T by ρ(t) = sup{s ∈ T : s < t}. If σ(t) > t, we say that t ∈ T is right-scattered; while if σ(t) = t, we say that t ∈ T is right-dense. If ρ(t) < t, we say that t ∈ T is left- scattered; while if ρ(t) = t, we say that t ∈ T is left-dense. Def ning the set T^κas following: If T has a maximum left-scattered point M, then T^κ = T − {M }. Otherwise T^κ = T. A function u : T → R is right-dense continuous or rd-continuous provided it is continuous at right-dense points in T, and its left-sided limits exist(f nite) at left-dense points in T, denoted

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Jinjun Fan, Xiaodong Lu

E-ISSN: 2224-266X 26 Volume 17, 2018

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by u ∈ Crd(T). Assuming f : T → R, t ∈ T^κ. If there is a constant α, for any ε > 0, and there is a neighborhood UT(= UT T) of t, such that

|f (σ(t)) − f (s) − α[σ(t) − s]| ≤ ε|σ(t) − s|, s ∈ UT, then we say that f is ∆ differentiable in t, and the derivative is α, which is def ned as f^∆(t). Similarly, we can def ne n order ∆ derivative f^∆ⁿ = (f^∆ⁿ⁻¹)^∆. Other concept and calculation on time scales can be seen in [1-2,7].

We say that a nontrivial solution to Eq. (1) is an oscillatory solution if it is f nally not positive or negative, otherwise we say it a nonoscillatory solution. If all solutions to Eq. (1) are oscillatory solutions, then we say that the equation is oscillatory.

In this paper, we give the following assumptions:

(H₁) r ∈ Crd([t₀, ∞)T, (0, ∞)), p ∈ Crd([t₀, ∞)T, [0, ∞)),

h ∈ Crd([t₀, ∞)T, R), h(t)is bounded, Z +∞

t0

| h(t) | ∆t < +∞, and ∃d > 0, s.t. r(t) ≤ d, ∀t ∈ [t0, ∞)T;

(H₂) τ, δ_i ∈ Crd([t₀, ∞)T, [t₀, ∞)T), τ(t) ≤ t, δi(t) ≤ t, ∀t ∈ [t0, ∞)T, and lim

t→+∞τ (t) = lim

t→+∞δ_i(t) = +∞, i = 1, 2, · · ·, m;

(H₃) f_i ∈ C([t0, ∞)T× R, R), fi(t, x)does not increase about x,f_i(t, x)

x > 0(x 6= 0), ∀t ∈ [t0, ∞)T, and lim

t→+∞| fi(t, λ) |= +∞, i = 1, 2, · · ·, m, λ is an arbitrary constant which has nothing to do with t.

3 Main Results

Theorem 1 Assuming that (H1) ∼ (H3)hold, and

m

X

i=1

Z +∞

t0

| fi(t, c) | ∆t = +∞,

where c is an arbitrary constant which has nothing to do with t. Then all bounded solutions to Eq. (1) oscillate.

Proof: Supposing that x(t) is a f nal bounded positive solution of Eq. (1)(f nal bounded negative solution can be proved in the same way), then there exists suff cient large t1 ∈ [t0, ∞)T, when t ∈ [t1, ∞)T, x(t) > 0, x(δ_i(t)) > 0(i = 1, 2, · · ·, m), x(τ (t)) >

0, z(t) > 0. Because x(t) is bounded, there exists M > 0, x(t) ≤ M for arbitrary t ∈ [t1, ∞)T. And

because lim

t→+∞δ_i(t) = +∞, i = 1, 2, · · ·, m, there exists t2 ∈ [t1, ∞)T, for arbitrary t ∈ [t2, ∞)T, 0 <

x(δi(t)) ≤ M, i = 1, 2, · · ·, m.

By Eq. (1) and (H3), when t ∈ [t2, ∞)T,

[r(t)z^∆(t)]^∆= −

m

X

i=1

f_i(t, x(δ_i(t))) + h(t)

≤ −

m

X

i=1

f_i(t, M )+ | h(t) | . (2) By (H1), (H₃), when t → +∞, the right side tends to

−∞, so [r(t)z(t)^∆]^∆is f nally negative.

Integrating both sides of Eq. (2) from t2to t

r(t)z^∆(t) ≤ r(t2)z^∆(t₂) −

m

X

i=1

Z t t2

f_i(s, M )∆s

+ Z t

t2

| h(s) | ∆s.

According to (H1), (H₃), when t → +∞, the right side tends to −∞, so r(t)z^∆(t) is f nally negative. Also [r(t)z^∆(t)]^∆ is f nally negative. There- fore, there exists t3 ∈ [t2, ∞)T, M₁ < 0, such that r(t₃)z^∆(t₃) = M₁holds and when t ∈ [t3, ∞)T,

r(t)z^∆(t) ≤ r(t3)z^∆(t₃) = M₁. So z^∆(t) ≤ M₁

r(t)≤ M₁

d , integrating from t3to t, z(t) ≤ z(t3) +M₁

d (t − t3) → −∞(t → +∞), and this contradicts that z(t) is f nally positive. Thus all bounded solutions to Eq. (1) oscillate. The proof

is completed. 2

Theorem 2 Assuming that (H1) ∼ (H3) hold, 0 ≤ p(t) ≤ 1, and there exist

λ > 0, Q_i ∈ Crd([t₀, ∞)T, R⁺), F_i ∈ Crd(R, R), s.t.

| fi(t, x) |≥ Qi(t) | Fi(x) |,F_i(x) x ≥ λ, i = 1, 2, · · ·, m and

m

X

i=1

Z _+∞

t0

Q_i(t)[1 − p(δi(t))]∆t = +∞.

Then all bounded solutions to Eq. (1) oscillate.

Proof: Supposing that x(t) is a f nal bounded positive solution to Eq. (1) (f nal bounded negative solution can be proved in the same way), then there exists

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suff cient large t1 ∈ [t0, ∞)T, when t ∈ [t1, ∞)T, x(t) > 0, x(δ_i(t)) > 0, i = 1, 2, · · ·, m, x(τ (t)) >

0, z(t) > 0, z(t) ≥ x(t). Because x(t) is bounded, there exists M > 0, x(t) ≤ M for arbitrary t ∈ [t1, ∞)T. And because lim

t→+∞δ_i(t) = +∞, i = 1, 2, · · ·, m, there exists t2 ∈ [t1, ∞)T, for arbitrary t ∈ [t2, ∞)T, 0 < x(δ_i(t)) ≤ M, i = 1, 2, · · ·, m.

From Eq. (1) and (H3), when t ∈ [t2, ∞)T, we have

[r(t)z(t)^∆]^∆= −

m

X

i=1

f_i(t, x(δ_i(t))) + h(t)

≤ −

m

X

i=1

f_i(t, M )+ | h(t) | . From (H1), (H₃), when t → +∞, the right side tends to −∞, so [r(t)z^∆(t)]^∆is f nally negative and r(t)z^∆(t)is f nally positive or negative.

If r(t)z^∆(t)is f nally negative, by [r(t)z^∆(t)]^∆ being f nally negative, then there exists t3 ∈ [t₂, ∞)T, M₁ < 0, such that r(t3)z^∆(t₃) = M₁ holds and when t ∈ [t3, ∞)T,

r(t)z^∆(t) ≤ r(t3)z^∆(t₃) = M₁. So z^∆(t) ≤ M₁

r(t) ≤ M₁

d , integrating from t3to t, z(t) ≤ z(t3) +M₁

d (t − t3) → −∞(t → +∞), and this contradicts that z(t) is f nally positive.

If r(t)z^∆(t)is f nally positive, then z^∆(t)is also f nally positive. And because z(t) is f nally positive,

lim

t→+∞δ_i(t) = +∞, i = 1, 2, · · ·, m, then there exist T1 ∈ [t1, ∞)T, M₂ > 0, when t ∈ [T1, ∞)T, z(δi(t)) ≥ M2, i = 1, 2, · · ·, m. From Eq. (1), one obtains

[r(t)z^∆(t)]^∆= −

m

X

i=1

fi(t, x(δi(t))) + h(t)

≤ −

m

X

i=1

Q_i(t)F_i(x(δ_i(t))) + h(t)

≤ −λ

m

X

i=1

Qi(t)x(δi(t)) + h(t)

= −λ

m

X

i=1

Q_i(t)[z(δ_i(t)) − p(δi(t))x(τ (δ_i(t)))] + h(t).

For z(t) ≥ x(t), z^∆(t)being f nally positive, there exists T2 ∈ [t1, ∞)T, when t ∈ [T2, ∞)T, 0 <

x(τ (δ_i(t))) ≤ z(τ (δi(t))) ≤ z(δi(t)), i = 1, 2, ···, m.

Letting T3 = max{T1, T₂}, then when t ∈ [T3, ∞)T,

[r(t)z^∆(t)]^∆

≤ −λ

m

X

i=1

Q_i(t)[1 − p(δi(t))]z(δ_i(t)) + h(t)

≤ −λM2 m

X

i=1

Qi(t)[1 − p(δi(t))]+ | h(t) |, integrating from T3 to t, we have

r(t)z^∆(t)

≤ r(T3)z^∆(T₃)

−λM2 m

X

i=1

Z t T₃

Q_i(s)[1 − p(δi(s))]∆s

+ Z t

T₃ | h(s) | ∆s

→ −∞ (t → +∞),

and this contradicts that z^∆(t) is f nally positive.

Therefore all bounded solutions to Eq. (1) oscillate.

The proof is completed. 2

4 Conclusion

In this paper, we study the existence of bounded oscillatory solutions to a class of second order neutral dynamic equations by adding forced term on time scales, and give several suff cient conditions for oscillation of bounded solutions.

Acknowledgements: The research was supported by NNSF of China(No.11471196) and NSF of Shandong Province(No. ZR2013AM005).

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[1] R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications. Results Math.35, 1999, pp. 3–22.

[2] R. P. Agarwal, M. Bohner and D. O’Regan, Dy- namic equations on time scales: a survey. J.

Comput. Appl. Math., 141(1-2), 2002, pp. 1-26.

[3] R. P. Agarwal, M. Bohner and S. H. Saker, Os- cillation of second order delay dynamic equa- tions,Canadian Applied Mathematics Quarterly, 13(1), 2005, pp. 1-18.

[4] L. H. Erbe, S. Peterson and S. H. Saker, Oscil- lation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl., 333(1), 2007, pp. 505-522.

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[5] Y. Sahiner, Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales, Advances in Difference Equations, Vol.2006, 2006, pp. 1-9.

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