Oscillation criteria of Second-Order non-linear dynamic equations with integro forcing term on Time scales Текст научной статьи по специальности «Математика»

MSC 34N05, 39A99, 34C10, 34K11

DOI: 10.14529/mmp 170103

OSCILLATION CRITERIA OF SECOND-ORDER NON-LINEAR DYNAMIC EQUATIONS WITH INTEGRO FORCING TERM ON TIME SCALES

S.S. Negi, School of Basic Sciences, Indian Institute of Technology Mandi, India, [email protected],

S. Abbas, School of Basic Sciences, Indian Institute of Technology Mandi, India, [email protected],

M. Malik, School of Basic Sciences, Indian Institute of Technology Mandi, India, [email protected]

This paper is concerned with the oscillatory properties of second order non-linear dynamic equation with integro forcing term on an arbitrary time scales. We reduce our original dynamic equation into an alternate equation by introducing a function of forward jump operator. To study oscillations we establish some crucial Lemmas and employ generalized Riccati transformation technique which transforms our second order dynamic equation into the first order dynamic equation on an arbitrary time scales. These results also guarantee that the solution of our equation oscillates. Furthermore, we establish the Kamenev-type oscillation criteria of our system. At the end, we consider a second order dynamic equation on time scales with deviating argument and compare it with our result which gives the sufficient conditions of oscillation of it.

Keywords: time scale; dynamic equation; Riccati transformation technique; oscillation.

Introduction and Preliminaries

The oscillation theory of differential equation and difference equation has been receiving a lot of attention in the past few decades. It is a very important concept in the qualitative behaviour of the solutions of both types of equations. Many researchers have taken an interest in studying the oscillation and non-oscillation criteria of the solution of both differential and difference equations [1-3]. The problem of obtaining the sufficient conditions for oscillation of both types of equations takes much time. Therefore, it is necessary to find a single way for both equations.

The time scale theory which removes this ambiguity has been first introduced by a German mathematician Stefan Hilger in his Ph.D. dissertation (1988) [4]. The significance of this theory is that it does not only avoid the dual analysis but also harmonize both continuous and discrete calculus.

Time scale is a non-empty closed subset of the real numbers (i.e., R), e.g., set of natural numbers (N), integers (Z), Cantor set, set of harmonic numbers etc. In this way, the results do not only relate to the set of real numbers or set of integers but also pertain to more general times scales. The three most popular examples of calculus on time scales are differential calculus (T = R), difference calculus (T = Z), and quantum calculus (T = qZ[j{0},q> 1), for more details see [5,6] and references therein. It has many applications in various fields, e.g., population dynamics, economics, neural networks, quantum physics and social science etc. [7,8].

We briefly recall some basic definitions, useful Theorems, Lemmas, assumptions and basic facts of time scales etc.

Definition 1. [5, 6] For t G T, we define a forward jump operator a : T ^ T by a(t) := inf{s G T : s > t}, if t < a(t) and t < supT as well as t = a(t), then t is right-scattered and right dense respectively. A backward jump operator p : T ^ T by p(t) := sup{s G T : s < t}, if t > p(t) and t > inf T as well as t = p(t), then t is left-scattered and left dense respectively. The graininess operator j : T ^ [0, to) is defined by j(t) = a(t) — t.

Remark 1. We put inf 0 = supT (i.e., a(t) = t if T has a maximum t), sup0 = inf T p(t) = t T t

Definition 2. [5,6] A function f : T ^ R is called rd-continuous provided it is continuous

T

T, denoted by Crd = Crd(T) = Crd(T, R). We define some notations as follows:

+Cd(T) = {q : q(t) is positive rd-continuous function and qA(t) G Crd(T)} . +Cd+(T) = {q : q(t), qA(t) are positive rd-continuous functions} .

Definition 3. [5, 6] A function G : T ^ R is called an anti-derivative of g : T ^ R, provided GA (t) = g(t) Vt G T. Then Va, b G T, the Cauchy integral is defined by

Define

{

/ g(s)A(s) = G(b) — G(a).

J a

T — {l}, if T has a left-scattered maximum l,

T — ,

T = T, otherwise.

Definition 4. [5,6] For a function f : T ^ R and t G TK, we define fA(t), to be a number (provided it exists) with the property that given any t > 0, there exists a neighborhood A = (t — 8,t + i) P| T for some 5 > 0 such that

\[f(a(t)) — f(r)] — fA(t)[a(t) — r]| < t\a(t) — r\ Vr G A.

Thus, we call f A(t) the delta or Hilger derivative of f at t. f is also called differentiable t

Theorem 1. [5,6] For the functions g, f : T ^ R and t G TK, we have the following:

1. If f is differentiable at t, then f is continuous at t;

2. If f is continuous at t and t is right-scattered, then f has a delta derivative at t and

fA(t) = «aam_m;

j(t)

3. If t is right-dense, then f is differentiable at t if the limit

f A(t) = lim f (t) — f (s)

s vt t — s

exists and has a finite value; ft

f ° = f (a(t)) = f (t)+ j(t)f A(t);

b

5. If f and g both are differentiable at t, then the product fg : T ^ R is differentiable at t and

(fg)A(t) = f A(t)g(t) + f (a(t))gA(t) = f (t)gA(t) + f A(t)g(a(t));

6. If g(t)g(a(t)) = 0 with g(t) = 0, then ft) is differentiable at t and

(g г«

\ f A(t)g(t) - f (t)gA(t) gj g(t)g(a(t)) '

Definition 5. [5,6] A function q : T ^ R is called regressive if 1 + ¡i(t)q(t) = 0, Wt E T. We denote the collections of all functions h : T ^ R which are rd-continuous and regressive by R and R+ = {q eR : 1 + ¡i(t)p(t) > 0 for a 111 E T}.

Definition 6. [5,6] (Time scale version of exponential function). If q eR, then we define the exponential function by

eq(t, s) = exp ^^ ^{t) (q(T)) A^j , Wt E T, s E TK,

where nh(z) is the cylinder transformation, which is defined by

Vh(z) = { , H h =0,

Vh(Z) = \ z, if h = 0. Definition 7. [5,6] If q ER, then the first order linear dynamic equation

yA(t) = q(t)y(t) (1)

is called regressive.

Theorem 2. [5,6] Suppose that (1) is regressive and fix t0 E T. Then eq(.,t0) is a solution of the initial value problem

yA(t) = q(t)y(t), y(t0) = l (2)

T.

Theorem 3. [5,6] If (1) is regressive, then eq(.,t0) is the only solution of (2). Theorem 4. [5,6] Ifp,q E R, then

1. e0(t, s) = 1 and ep(t, t) = 1;

2. ep(a(t),s) = (1 + fi(t)p(t))ep(t, s);

3. eQp(t, s) = eJt,s) ;

4- ep(t,s)eq (t,s) = ep®q (t,s);

Вестник ЮУрГУ. Серия «Математическое моделирование 27

и программирование» (Вестник ЮУрГУ ММП). 2017. Т. 10, № 1. С. 35-47

5 i 1 \A — _ p(t)

ep(.,s) J el(,,s)>

6. If q e R+, then eq(t, s) > 0 for all t e T.

In this paper, we study the oscillation criteria of the second order non-linear dynamic

T

yAA(t)+ pyA(t) — B(t)y(t) + HA(t,y(t), f J(t - s)H(s,y(s))A sV (3)

where B, J : T ^ R are the function of t, and the feeing terms H : T x R2 ^ R and H : T x R ^ R. For the alternate form of equation (3), we multiply (3) by a function r° (t). We have

t

(t)yAA(t)+pra (t)yA(t) — r* (t)B (t)y(t)+ra (t)H M t,y(t), I J (t - s)H(s,y(s))A s

(t,y(t),f_ J(t - s)H(s,y(s))A

From above relation, we obtain

r*(t)yAA(t) + rA(t)yA(t) -rA(t)yA(t) + Pr*(t)yA(t)

"V

t

— r"(t)B(t)y(t) + r°(t)HA (ty(t),j_ J(t - s)H(s,y(s))As^j

which is equivalent to

(r(t)yA(t))A + (Pr*(t) - rA(t))yA(t) —

— B(t)r*(t)y(t) + r*(t)HA ^t, y(t)J_ J(t - s)H(s, y(s))A^ .

(4)

We study oscillation criteria of (4). A non-trivial y(t) such that y(t) £ ([ty, to)t), r(t)yA(t) £ CAd([ty, to)t) for certain ty > t\ and satisfying (4) for ty < t is called non-oscillatory if it is eventually positive or eventually negative, otherwise it is called oscillatory. In other words, it is said to be oscillatory if it has an arbitrarily large number of zeros, i.e., there exists a sequence {sn} such th at lim sn = to as well as y(sn) = 0, Wn. A dynamic

equation is called oscillatory if every solution is oscillatory, and non-oscillatory otherwise. Our attention is restricted to those solutions of (4) which exist on some [ty, to)t and satisfy sup{|y(t)| : t > t*} > 0 for any ty < t*.

In the past few years, there have been many research activities concerning the oscillation of solutions of various forced second-order dynamic equations on time scales. The oscillation criteria of the second order linear and non-linear dynamic equations on time scales have been studied by many researchers, and we have plenty of essential papers, articles etc. [9-14]. In particular: In (2003), Erbe, et. al. [13] considered linear damped dynamic equation

xAA(t) + p(t)xA(t) + q(t)x(t) = 0,

and non-linear equation

xAA(t) + p(t)xAa (t) + q(t)(foxa )(t) = 0, (5)

to establish the oscillation criteria. In (2007), Saker, et. al. [14] gerneralized (5) as follows: (r(t)xA(t))A + p(t)xAa (t) + q(t)(foxa )(t) = 0, (6)

and studied an oscillation criteria. Finally, in (2008), Erbe, et. al. [15] have considered a non-linear damped delay dynamic equation by introducing a constant term where 3 is a quotient of odd positive integers and studied the sufficient conditions for oscillation:

{r(t)(xA(t)f )A + p(t)(xA (t)f + q(t)f (x(t (t))) = 0, (7)

which extend and improve the results [13,14].

In (2004), Bohner, et. al. [10] considered a second-order perturbed dynamic equation:

(r(t)(xA(t))Y )A + F (t,xa (t)) = G(t, x(t),xA(t)), (8)

where y is a positive odd integer. In (2006), Agarwal, et. al. [9] modified equation (8)

(r(t)(xA(t))Y )A + F (t,x (t)) = G(t, x(t),xA(t)),

and both have established the sufficient conditions for oscillation. In (2010), Chen, et. al. [11] considered a dynamic equation with damping on time scale

((xA(t))Y )A + p(t)(xA(t))Y + q(t)f (x(a(t)) = 0,

and studied the oscillation criteria as well as established the Kamenev type and the Philos-type oscillation criteria of it. Throughout this paper we denote [a, œ) fl T = [a, œ)T, where sup(T) = œ.

This paper is organized as follows: In this section, we establish some necessary Lemmas. In the next section, we use Eiccati transformation technique to establish the sufficient conditions for oscillation of (4) and also establish the Kamenev type oscillation criteria. Finally, we consider another second order dynamic equation with deviating argument and establish the same for oscillation.

It will be convenient to make the following notations:

(t) - rA(t) D(t *) = e Q(t) (t *), m(t) = (p(t) - B(t)V(t),

r(t) l-M(t)Q(t)

R _ (óA(t) - ó*(t)Q(t)m) B _ ют A m _( pa(t,s*)s(t) \ Bi(t)- m ,B2(t)_ щш' r(t)(ó*(t))2D(t,s*))

(ó_A(t)_ D*(t,s*)ó(t) л v ó*(t) D(t, s*)ó*(t)Q[t)) -

(t) D(t, s*)5°(t) To establish our results we use the following assumptions:

(Ui) B : [t\, ^ R is a negative rd-continuous function and p : [t\, ^ R is a positive rd-continuous function;

(U2) B : [t\, ^ R is an rd-continuous function and p : [t\, ^ R is a positive

p(t) - B(t) > 0;

(U3) Q : [ti, to)t ^ R is a positive rd-continuous function such that 1 — ¡i(t)Q(t) > 0, and r(t) e +CArd(T);

(U4) H : T x R2 is a A-derivative finction (w.r.t first variable) such that ri(t)HA(t,ri(t),£(t)) < 0, Vrj(t) e R - {0}, V£(t) e R, Vt e T;

(U5) \HA(t,v(t),at))\> p(t)\v(t)\, V'n(t) e R - {0}, vat) e R, t e T;

, ™ 1

(U6) J e-Q(t)(t,s*)A t =

Now we establish two Lemmas, which will be used in the next section.

Lemma 1. Let y(t) be an eventually positive solution of (4)- Assume that (U3) — (U6) and either (U\) or (U2) holds. Then there exists s* > ti such that

y(t) > 0, yA(t) > 0 and (D(t,s*)r(t)yA(t))A < 0 Vs* < t. (9)

Proof. Since y(t) is an eventually positive solution of (4). Take t2 e [ti, to)t such that y(t) > 0 on [t2, to)t. In view of equation (4) and (Ui) — (U5), we have

(r(t)yA(t))A + Q(t)r(t)y (t) < —(p(t) — B(t))r°(t)y(t) = —M(t)y(t) < 0 (10)

on [t2, to)t. Now, by dividing equation (10) by a non-negative function 1 — ¡i(t)Q(t), we get

(r(t)yA(t))A + Q(t) A,t) < 0

1—nwm + i — ,<m(t)mv [ > <

or

(D(t,s*)r(t)yA(t))A < 0.

Then D(t, s*)r(t)yA(t) is an eventually decreasing function and thus it is eventually of one sign, i.e. it is either eventually positive or eventually negative. We assert that D(t, s*)r(t)yA(t) is eventually non-negative. Suppose it is eventually negative, then there exist t3 > t2 and a const ant Ki < 0 such that

D(t, s*)r(t)yA(t) < Ki < 0.

Integrating the above relation from t3 to t, we obtain

r* 1

y(t) < 1 rmx^fx=

f* 1

= y(t3) + Ki —— e-Q(x)(x, s*)A x ^ as t ^

Jt3 r(x)

which is a contradiction since y(t) > 0 for all t2 < t. Hence, D(t, s*)r(t)yA(t) is eventually non-negative, i.e., yA(t) is eventually positive function or there exists s* e T such that yA(t) > 0 (or a 11 t e [s*, to)t-Thus, we have

y(t) > 0, yA(t) > 0 Mid (D(t, s*)r(t)yA(t))A < 0, Vs* < t. ^

Lemma 2. Let Lemma I hold. Futhermore, assume rA(t) > 0. Then

Щ < < 1 (И)

w va (t) ' 1 ;

t-s*

whereat)-- —, +Kt). Proof. From (9), we obtain

y(t) > 0, yA(t) > 0, yAA(t) < 0. (12)

Also from (12), we have

y(t) = y(t) - y(s*) = SyA(V)AV > yA(t)(t — s*). (13)

From (13) and y°(t) = y(t) + ¡i(t)yA(t), we get

3(t) < -^tt < 1. (14)

v ; " ya (t) - K '

□

1. Main Results

In this section, we give some new oscillation criteria for equation (4).

Theorem 5. Assume that (U3) — (U6) and either (U{) or (U2) hold. Furthermore, there exists S(t) E ^^(T) (not necessary rd-eontinuity of SA) such that for all sufficiently large

lim sup /

t^-œ J s*

m (от - Al(0

or

t

lim sup

M(t№) -

4A2(£ )

r(C) {¿A(C) - )Q(C))

m)

А С = œ (15)

А С = œ. (16)

Then, every solution of equation (4) is oscillatory.

Proof. Suppose to the contrary that y(t) is a non-oscillatory solution of equation (4). We may assume without loss of generality that y(t) is eventually positive. By Lemma 1, we

haVe y(t) > 0, yA(t) > 0 and (D(t,s*)r(t)yA(t))A < 0 Vs* < t.

Now define a function w(t) by the Riccati substitution

w(t) = S(t) r(t%(t), t > s* > t1. (17)

Then, w(t) > 0 and

wAt) = (m = Ж + i) •

From (17)- we ^ w(t) S(t)yA(t) \ +. ... aM)a (m \

_w (t) = T& [s (t) - —ytr) +(r(t)y (t)) щ) •_

Вестник ЮУрГУ. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2017. Т. 10, № 1. С. 35-47

S*

t

2

From (10) and (17), we obtain

W(t) (a^ 6(t)yA(t)

wA(t) <

6° (t)

(

6A(t) -

- (M(t)y(t) + Q(t)(r(t)y

A

-M(t)6(t) - Q(t)w(t) +

y(t)

6A(t)wa (t) wa (t)w(t)

D)

(t) r(t)6a(t) '

Since D(t, s*)r(t)yA(t) is eventually decreasing function, then

(t, s*)(r(t)yA(t)f < D(t, s*)r (t)yA (t), as t < a(t). From (17), we obtain

D° (t,s*)ya (t)5(t)W (t) U~ D(t, s*)y(t)5a(t) *

y(t)

y(t) < ya(t).

From (19) and (20), we have

Da(t, s*)5(t)wa(t)

w(t) >

D(t, s*)6°(t)

or

D° (t,s*)6(t)w° (t) -Q(t)w(t) <---Q(t)■

D(t, s*)6°(t)

From (19) and (21), we obtain

uA(t) < -m(t)6(t) - D°(t)Qt) +6A(t)w°(t)

(w° (t))2D° (t,s*)6(t) r(t)D(t, s*)(6a(t))2

D(t, s*)6a(t)

6° (t)

-M(t)6(t) -

-M(t)6(t) + Ai(t)wa(t) - (w°(t))2Â2(t) ] 2

w° wy/m -

+ AUt)

4A2(t)'

Hence

wA (t) < -

M(t)6(t) -

Aj(t) 4A2(t)

Integrating (23) from s* to t, we obtain

m(06(0 -

Am

4A2(£ )

A £ < w(s*) - w(t) < w(s*) < œ,

(18)

(19)

(20)

(21)

(22)

(23)

(24)

t

complete due to asumption (15).

t

Now we prove the result using (16). From equations (19), (20) and Da(t,s*) > D(t,s*), we obtain

w(t) >

Similarly, from (23) and (25), we obtain

8(t)wa (t) (t) '

(25)

M (СЖ) -

r(t ) {¿a(o - НсШ ))-

rn)

< -wA(t)

(26)

Integrating equation (16) from s* to t, we get

M(С)8(С) -

r(Q {5A(С) - 5(QQ(0Y ЩС)

А С < w(s*) - w(t) < w(s*) < ю, (27)

t,

due to assumption (16). Thus, the whole proof is complete.

□

As an immediate consequence of Theorem (5), we have the following corollaries for different values of 8 (t).

For 8(t) = constant (say, C > 0), t > ti7 we have the following result.

Corollary 1. Assume that (U3) — (U6) and either (Ui) or (U2) hold. Furthermore, assume

lim sup

t^œ J s

tr r(0D° (Os*)Q2(£ )

M (С) -

4D(Ç,s*)

А С = ю

or

lim sup

t^œ J s

tvM С) - гшт.

А С = ю.

Then, every solution of equation (4) is oscillatory.

For 8(t) = t2, we have following result. Corollary 2. Assume that (U3) — (U6) and either (Ui) or (U2) hold. Furthermore, assume

)2

d° (c,s*)e

or

lim sup

t^œ J s*

lim sup

t

m(С)С2 -

2 (w«)2 D^s*)^))2

Q(0

)

(

D (С,.з*)С2

rm^ovD^s*)

>

А С = ю

m(С)С2 -

2 г(с)(с+а(о - eQ(of

4С2

А С = ю.

Then, every solution of equation (4) is oscillatory.

We add one more condition, i.e. rA(t) > 0 or r(t) E +CA+(T) in Theorem (5) to obtain some new oscillation criteria of equation (4).

t

t

t

Theorem 6. Assume that (U3) — (U6) and either (U1) or (U2) hold. Moreover, rA(t) > 0 or r(t) E +CAd+(T) and there exists 8(t) E +Cfd(T)(not necessary rd-continuity of 5A) such that for all sufficiently large s*,

lim sup

m (SW (S )5(S) —

4B2(S )J

A S = w.

(28)

Then, every solution of equation (4) is oscillatory.

Proof. In view of Theorem (5) and equation (17), we obtain

(*) +**(t) (* f

By using equation (10), Lemma 2 and after manipulation, we obtain

wA(t) < —M(t)5a(t)5(t) + Bi(t)w(t) — B2(t)w2(t),

which is equivalent to

wA(t) < —M(t)5a(t)5(t) —

—J2fe

+

B2(t)

4B2(t)

s* t

m (S W (S)5(S) —

B2(S) 4b2 (S)

A S < w(s*) — w(t) < w(s*) < w,

(29)

□

t

Now as a special case if 5(t) is positive constant, we obtain a result:

Corollary 3. Assume that (U3) — (U6) and either (U1) or (U2) hold. Moreo ver, rA(t) > 0 or r(t) E +CA/(T) and

lim sup <5 (t)

t

m (S) —

r(S )Q2(S)

4

A S = w.

(30)

Then, every solution of equation (4) is oscillatory.

Now we establish Kamenev-type oscillation criteria for (4). We need the following result of [16] ((t — s)m)As < —m(t — a(s))m-1 < 0 for m > 1 and a(s) < t to establish our results.

Theorem 7. Assume that (U3) — (U6) and either (U1) or (U2) holds. Furthermore, there exists 5(t) E +CAd(T) (not necessary rd-continuity of 5A) such that for K > 1 and for all sufficiently large s*,

1 -r......... A?(S) -

limsup^ (t — S)

t J s*

M(S)W(S) —

4A2(S)

A S = w

(31)

or

t

2

t

t

1 ft

limsuP^ (t - С?

t^œ t Js*

M(С)6(С) -

г(С) (sA(C) - ¿(C)Q(C))'

4*(С)

А С = ю.

Then, every solution of (4) is oscillatory. Proof. In view of Theorem 5, from (23), we have

wA(t) < -

M (t)5(t) -

A2(t)

4A2(t)

Thus

(t - С)^А(С)А с < - I (t - С)

M(С)5(С) -

АКС) 4А2(С )J

А С.

We know

(32)

(33)

(t - cfwAm с = -(t - s*fw(s*) - I ((t - с Лa w^o^ с

(34)

and using the remark (3.3) in [16], we have

((t — C< — N(t — a(Of-i < 0, ) < t Mid N > 1. From equations (33), (34) and (35), we obtain

-It* (t - С)Л M (С )S(c ) -

Af(Q

4A2(0

(35)

А С > ft* (t - C)*wAm С = -(t - s*)*w(s*)-

- ((t - C)*)A w(a(C ))А С >-(t - s* )*w(s*)

or

(t - С)

M(С)S(C) -

Thus

1

limsuP^ (t - c)

t^œ t Js*

H

M(С)S(C) -

Aj(C) A(C)

A2(C )

А С < (t - s*)*w(s*).

4A2(C)

А С < limsup ( 1--) w(s*) < ю.

t^œ V t у

H

Which contradicts assumption (31). To prove the result using (32), we apply the above process for equation (26), then we get a contradiction since we have (32). Hence, the proof is complete.

□

Theorem 8. Assume that (U3) — (U6) and either (U\) or (U2) holds. Moreover; either rA(t) > 0 or r(t) E +CA+(T) and there exists 8(t) E +CAd(T) (not necessary rd-continuity of 8A) such that for H > 1 and for all sufficiently large s*,

1t

limsuP^ (t - c)

t^œ t Js*

M (С )s° (С)Э(С) -

B2 (С) 4B2(C)

А С = ю.

(36)

Then, every solution of equation (4) is oscillatory. Proof of the above Theorem is same as proof of Theorem 8.

t

t

t

t

t

t

Remark 2. The result of Theorem 8 and Theorem 9 holds for any 8(t) E + C^(T) (not necessary rd-continuity of óA). Thus, by these Theorems, we can immediately obtain corollaries with different choice of ó(t).

Let us consider a second order non-linear dynamic equation with deviating argument on an arbitrary time scale T :

where 3 > 0,wl(t,y(t)) = bl(t,y(b2(t, ••• ,y(bm0 (t,y(t))) •••))),B is a function oft, and the forcing terms W and bi,i = 1, 2, • • • , m0.

Remark 3. If we replace the integro forcing term H J(t — s)H(s, y(s))A s

(in 3) by the deviating argument W(t,y(t),y(wl(t,y(t)))) (in 37), then in the similar manner we can obtain oscillation criteria of (37). Thus, all the above Theorems and corollaries can be achieved for the second order dynamic equation (37), i.e., all the results will remain the same for it.

References

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yAA(t) + 33yA(t) = B(t)y(t)+ WA(t,y(t),y(wi (t,y(t)))),

(37)

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Received November 11, 2016

УДК 517.9 Б01: 10.14529/ттр170103

КРИТЕРИИ КОЛЕБАНИЙ НЕЛИНЕЙНЫХ ДИНАМИЧЕСКИХ УРАВНЕНИЙ ВТОРОГО ПОРЯДКА С ИНТЕГРАЛЬНОЙ СОСТАВЛЯЮЩЕЙ НА ВРЕМЕННЫХ МАСШТАБАХ

Ш.С. Неги, С. Аббас, М. Малик

В статье рассматриваются колебательные свойства нелинейного динамического уравнения второго порядка с интегральной составляющей на произвольном промежутке времени. При помощи введения оператора сдвига исходное динамическое уравнение редуцируется к альтернативному уравнению. Для изучения колебаний мы представим некоторые важные леммы и будем использовать обобщенное преобразование Риккати, которое переводит динамическое уравнение второго порядка в динамическое уравнение первого порядке на произвольном промежутке времени. Полученные результаты также гарантируют, что решение исходного уравнения осциллирует. Кроме того, мы устанавливаем критерий колебаний Каменева для нашей системы. В итоге, мы рассмотрим динамическое уравнение второго порядка на временных масштабах с отклоняющимся аргументом и сравним его с результатом, который дает достаточные условия его колебания.

Ключевые слова: масштаб времени; динамические уравнения; преобразование Риккати; колебание.

Шекхар Сингх Неги, Школа фундаментальных наук, Индийский институт технологий в Манди (г. Манди, Индия), [email protected].

Сайед Аббас, Школа фундаментальных наук, Индийский институт технологии в Манди, (г. Манди, Индия), [email protected].

Муслим Малик, Школа фундаментальных наук, Индийский институт технологий в Манди, (г. Манди, Индия), [email protected].

Поступила в редакцию 11 ноября 2016 г.