EXISTENCE AND EXPONENTIAL STABILITY OF POSITIVE PERIODIC SOLUTIONS FOR SECOND-ORDER DYNAMIC EQUATIONS

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 42-53

DOI: 10.15826/umj.2020.1.004

EXISTENCE AND EXPONENTIAL STABILITY OF POSITIVE PERIODIC SOLUTIONS FOR SECOND-ORDER DYNAMIC EQUATIONS

Faycal Bouchelaghem

High School of Management Sciences Annaba, Annaba 23000, Algeria

fyb1500@gmail.com

Abdelouaheb Ardjouni

Department of Mathematics and Informatics, Souk Ahras University, P.O. Box 1553, Souk Ahras, 41000, Algeria abd_ardjouni(Syahoo.fr

Ahcene Djoudi

Department of Mathematics, Annaba University, P.O. Box 12, Annaba 23000, Algeria adjoudi@yahoo.com

Abstract: In this article, we establish the existence of positive periodic solutions for second-order dynamic equations on time scales. The main method used here is the Schauder fixed point theorem. The exponential stability of positive periodic solutions is also studied. The results obtained here extend some results in the literature. An example is also given to illustrate this work.

Time scales theory was initiated by Stefan Hilger in 1988 as a means of unifying theories from discrete analysis and continuous analysis. Difference equations are defined on discrete sets while differential equations are defined on an interval of the set of real numbers. However, dynamic equations on time scales are very important in the physical applications because they are either difference equations, differential equations or a combination of both. This means that dynamic equations are defined on discrete, connected or combination of both types of sets. Hence, the theory of time scales provides an extension of difference analysis and differential analysis, see [6, 7, 15, 17] and the references therein.

Delay dynamic equations arise in many applications of different fields of science and engineering. For example, these equations appear in applied sciences, physics, chemistry, biology, medicine, etc. In particular, qualitative analysis such as positivity, periodicity and stability of solutions of dynamic equations on time scales has received the attention of many authors, see [1-17] and the references therein.

Let T be a periodic time scale such that t0 € T. In this paper, we are interested in the positivity, periodicity and exponential stability of solutions of second-order dynamic equations. Inspired and

motivated by the references in this paper, we consider the following second-order dynamic equation

xAA (t) + a (xa)A (t) + q (t) x3 (t) - r (t) xa (t) = 0, t > to, (1.1)

with xA(t0) + axa(t0) = 0 and x(t0) = 1. Throughout this paper we assume that a > 0, q, r € Crd([t0, to) n T, R), a, ft € (0, to). To prove the positivity and periodicity of solutions of (1.1), we convert (1.1) into an equivalent integral equation and then employ the Schauder fixed point theorem. The sufficient conditions for the exponential stability of positive solutions are also considered. In the special case T = R, Dorociakova, Michalkova, Olach and Saga in [13] show the existence and the exponential stability of positive solutions of (1.1). Then, the results presented in this paper extend the main results in [13].

The rest of this work is organized as follows. In Section 2, we present some basic concepts concerning the calculus on time scales that will be used to show our main results. We give some properties of the exponential function on a time scale as well as the Schauder fixed point theorem. We refer the reader to the monograph [18] for more details on the Schauder theorem. In Section 3, we prove our main results for the existence of positive periodic solutions by using the Schauder theorem, and we give an example to illustrate our existence results. In Section 4, we study the exponential stability of a positive periodic solution of (1.1). In Section 5, we establish new sufficient conditions for the existence and the exponential stability for a pipe-tank flow configuration.

The theory of dynamic equations is a fairly new branch in mathematics (see [1-10, 14-17]). Dynamic equations extend and unify the difference and differential equations. We assume that most readers are familiar with the basic concepts of the dynamic equations on time scales and for more details we refer to the books [6, 7, 17].

Definition 1 [6]. A time scale T is an arbitrary nonempty closed subset of R.

The definition of periodic time scales was introduced by Kaufmann and Raffoul [16]. The following two definitions are found in [16].

Definition 2. A time scale T is said to be periodic provided there exists a T > 0 such that if t € T then t ± T € T. For T = R, the period of the time scale is the smallest positive T.

Example 1 [16]. The following time scales are periodic.

Remark 1 [16]. All periodic time scales are unbounded above and below.

Definition 3. Let T = R be a periodic time scale with period T. The function f : T ^ R is said to be periodic with period w provided there exists a natural number n such that w = nT, f (t ± w) = f (t) for all t € T and w is the smallest number such that f (t ± w) = f (t). If T = R, f is said to be periodic with period w > 0 provided w is the smallest positive number such that f (t ± w) = f (t) for all t € T.

Definition 4 [6]. Let T be a time scale. The forward jump operator a : T — T is defined by

Remark 2 [16]. Let T be a periodic time scale with period T. Then, the forward jump operator a satisfies a(t ± nT) = a(t) ± nT. Hence, ^(i ± nT) = a(t ± nT) - (t ± nT) = a(t) - t = ^(i). So, ^ is a periodic function with period T.

Definition 5 [6]. We say that the function f : T — R is regulated if its right-sided limits exist at all right-dense points in T and its left-sided limits exist at all left-dense points in T.

Definition 6 [6]. We say that the function f : T — R is rd-continuous if it is continuous at every right-dense point t € T and its left-sided limits exist, and is finite at every left-dense point t € T. We denote the set of rd-continuous functions f : T — R by

We denote the set of differentiable functions f : T — R and whose derivative is rd-continuous by

Definition 7 [6]. The delta derivative fA (t) of a function f : T — R at a point t € Tk = T\\ {sup T} exists provided that for any given e > 0, there exists a neighborhood U of t such that

We say that the function fA : Tk — R is the delta derivative of f on Tk.

Definition 8 [6]. A function p : T — R is said to be regressive if 1 + ^(t)p(t) = 0 for all t € T. We denote the set of all rd-continuous and regressive functions p : T — R by R = R(T, R). We define the set R+ of all rd-continuous and positively regressive functions by

Theorem 1 [6]. Suppose f : T — R is a regulated function. Then there exists a function F which is pre-differentiable with region of differentiation D such that

FA (t) = f (t) for all t € D.

Definition 9 [6]. Suppose f : T — R is a regulated function. We say that the function F as in Theorem 1 is a pre-antiderivative of f. The indefinite integral of a regulated function f is defined

where F is a pre-antiderivative of f and C is an arbitrary constant. The Cauchy integral is defined

a (t) = inf {s € T : s > t} for all t € T, while the graininess function ^ : T — [0, to) is defined by

^(t) = a(t) -1 for all t € T.

Crd = CVd(T) = CVd(T, R).

Cld = Cld(T) = CUT, R).

(f (*(t)) - f (s)) - f A(t)(v(t) - s)| Ki) - s| for all s € U.

R+ = R+(T, R) = {p € R : 1 + ß(t)p(t) > 0, Vi € T}.

We say that a function F : T ^ R is an antiderivative of f : T ^ R if

FA (t) = f (t) for all t € Tk.

Theorem 2 [6]. Every rd-continuous function has an antiderivative.

Definition 10 [6]. For p € R, we define the generalized exponential function ep as the unique solution of the initial value problem

xA(t) = p(t)x(t), x(s) = 1, where s € T.

We give an explicit formula for ep(t, s) by

ep(t, s) = exp ^ J ^(v) (p(v))Av^ Vs, t € T,

,, , J log(1 + ^} if MO, I p if p = 0,

with log is the principal logarithm function.

Lemma 1 [6]. For p,q € R, we define the functions p ® q and Qp by (P ® q) (t) = p (t) + q (t) + p (t) p (t) q (t) Vt € Tk,

which are elements of R.

Lemma 2 [6]. Let p,q €R. Then

(i) e0(t,s) = 1 and ep(t,t) = 1,

(ii) ep(a(t),s) = (1 + p(t)p(t))ep(t,s),

(iii) -i-pe^f,*).

(iv) e.p(t,s) = = eQp(s,t),

(v) ep(t, s)ep(s, r) = ep(t, r),

(vi)& 1 p(i)

eP (&,s)J ep (&,s)

Lemma 3 [1]. If p € R+, then

The following Schauder fixed point theorem plays important role to prove the existence results in the next section.

Theorem 3 [18, Schauder&s fixed point theorem]. Suppose that Q is a bounded closed convex nonempty subset of a Banach space X. Let S : Q ^ Q be a completely continuous mapping. Then S has a fixed point in Q.

Next theorem guarantee the existence of positive w-periodic solutions of (1.1).

Theorem 4. Assume that there exist positive constants m and M, and a rd-continuous function k € Crd([t0, to) n T, R) such that

- k € R+,

/t+ui

Us) [Ö (a - k(s))]As = 0, t > to,

k(t)ee(o-k) ((t),to) = [ \\r(s) exp (af [e (a - k(v))] Av

Jto1 K J to

-q(s) exp [ß jf £Mv)[e(a - k(v))]AvJJ As, t > to.

Then (1.1) has a positive w -periodic solution.

Proof. Let X = BCrd([to, to) n T, R) be the Banach space of all bounded rd-continuous functions endowed with the supremum norm ||x|| = supt>t0 |x(t)|. Consider the bounded closed convex nonempty subset Q of X as follows

Q = {x € X : x(t + u) = x(t), t > to, m < x(t) < M, t > to,

f \\r (s) xa (s) - q (s) xß (s)l As = k(t), t > to),

and define the operator S : Q ^ X as follows

(Sx)(t) = exp ( f Us)

e a xa(s)

[r (v) xa (v) - q (v) xß (v)] Av

for t > t0. We will prove that SQ C Q. By using (3.1), for every x € Q and t > t0 we obtain

(Sx)(t) = exp ( f Us)

Also for x € Ü and t > to we have

e a xa (s)

[r (v) xa (v) - q (v) xß (v)] Av

= ee(o-k)(t,to) < M.

(Sx)(t) = exp ( f Us)

e a xCT (s)

[r (v) xa (v) - q (v) xß (v) ] Av

ee(o-k)(t,to) >

From (3.3), for every x € Q and t > to we obtain

k(t)(Sx)f(t)= k(t)exp( / eM(s)

&to ra(t)

xf(s) to

= k(t) exp ( jf {^(s) [0 (a — k(s))] As) = k(t)ee(o-k) (a (t), to)

r (s) exp (a J £ß(v) [0 (a — k(v))] Av) — q (s) exp (ß J (ß(v) [0 (a — k(v))] Av

r (s) (Sx)a (s) — q (s) (Sx)ß (s)

Finally we will prove that for x € Q, t > t0 the function Sx is w-periodic. By using (3.2), for x € Q and t > to we get

/ rt+u (Sx)(t + W)=expi /

[r (v) xa (v) — q (v) xß (v)] Av

exp I / £^(s) t0

/•t+u

xf (s) J to 1 fs

x° (s) J to

[r (v) xa (v) — q (v) xß (v)] Av

[r (v) xa (v) — q (v) xß (v)] Av

= (Sx)(t) exp ^ J Q (a - k(s)) ]As) = (Sx)(t).

So Sx is w-periodic on [t0, to) n T. Hence, SQ C Q.

Now, we need to prove that the mapping S is completely continuous. So we will show that the mapping S is continuous. Let xi € Q be such that xi —> x € Q as i —> to. For t > t0, we have

|(Sxi)(t) — (Sx)(t)| =

exp( / {Ms) t0

[r (v) x" (v) — q (v) xß (v)] Av

exp( I £p(s) t0

[r (v) xa (v) — q (v) xß (v)] Av

By applying the Lebesgue dominated convergence theorem we obtain that

lim ||Sxi - Sx|| =0. i—^^

Therefore S is continuous.

Next, we are going to prove that SQ is relatively compact by applying the Arzela-Ascoli theorem. The uniform boundedness of SQ follows from the definition of Q. For t > t0 and x € Q we have

| (Sx)A(t) | = |-(a--^jT [r (v) xa (v) - q (v) x&3 (v) ] Av)

[r (v) xa (v) — q (v) x^ (v) ] Av

x exp I / £^(s) t0

xf (s) J to 1 ^

x exH / {M(s) t0

xf (t) J to 1

/ [r (v) xa (v) — q (v) xß (v)] Av t0

[r (v) xa (v) — q (v) xß (v)] Av

xf (s) j to

|0 (a — k(t))| ee(o-k)(t, to) < Mi, Mi > 0,

which implies that the family SQ is equicontinuous. By using the Arzela-Ascoli theorem SQ is relatively compact. Therefore, S is completely continuous. By Theorem 3 there is an x0 € Q such that Sx0 = x0. We see that x0 is a positive u-periodic solution of (1.1). The proof is complete. □

To illustrate the applications of Theorem 4 we give the following example. Example 2. Consider the dynamic equation on T = nZ then t) = n,

xAA (t) + a (xa)A (t)+ q (t) xfi (t) - r (t) xa (t) = 0, t > t0. (3.4)

We take t0 € T which

gOOS to _ 1 gOOS t _ 1

a = --> 0, k(t) = a-~-, w = 47T, a, (3 € (0, oo).

Then for the conditions (3.1), (3.2) and u = 4n we obtain

/t+w r t+4n 1

Us) [e(a - fc(s))] As = j — log [1 + fjL(s) (9 (a - k(s)))]As

r t+4n 1

(a fc(s)) Jti(s) + l] As = - £+47T ^log[l + fj,(t) (a- k(s))] As

-1 + ß(s) (a — k(s))

/•t+4n

— cos (s) As = i cos(s) n 2

t+4n

= 0, t > to,

ee(a-fc) (t, to) = exp / {^(s) [ e (a — k(s)) ] A

= exp / —-logh+Aí(s)(H(a-fc(s)))lAs = e(cosí-cosío)/2, í > í0./to Ms)

We take m = e- 1and M = e, then

Also, we put

-cosí _ --cosí . 1 .

r (t) = —-—-e-f(cost-cost0) a+i e-3Cost0)

p 2 ^ _ p 2 ^ ß -i

q^-j _ ____g tj(cOSÍ COS¿o)g &o COSÍO

, ecos t — 1. /• CT(t)

k(t)ee{a_k)(a (t), to) = (a - —--J exp jf [© (a - fc(s))] As

_ 2_W-|(cos(í)+cosío) _ ,le|(cos(í)-cosío) ¿ > ¿

V nJ n & _ &

Jo r (s) exp (a Jo £Mv) [0 (a — k(v))] Av) — q (s) exp (ß jt £Mv) [0 (a — k(v))] Av) As

n V n J n2

_ |-a j_Ae-|(cosi+cosio) _ (cost-costo) £ >

All conditions of Theorem 4 are satisfied. Thus (3.4) has a positive w = 4n-periodic solution

x(t) = e.e{a_k)(t,to) = eM^t-^, t > to,

with x(to) = 1 and

xA(t0) + axa(t0) = e-îcost0 i —--J + ae"cosio = 0.

In this section, we will prove the exponential stability of a positive w-periodic solution of (1.1). Let xi be the positive w-periodic solution of (1.1) with the initial condition xi (t0) = 1 and xA (t0) + axl (t0) = 0. Let x be the another positive w-periodic solution of (1.1) with the initial condition x (t0) = c1 > 0, c1 = 1 and xA (t0) + axa (t0) = 0. Let

y(t) = x (t) - xi (t), t > t0.

After integration of (1.1), we obtain

rt ft rt

xAA (s) As + a / (xa)A (s) As + / |q (s) x^ (s) - r (s) xa ( s It 0 Jt 0 Jt 0

[ xAA (s) As + af (xf )A (s) As + / [q (s) xß (s) — r (s) xa (s) ]As = 0,

J to J to J to

(t) — xA (to) + axf (t) — axf (to) = / [r (s) xa (s) — q (s) xß (s) ] As.

xA (t) + axf

In a similar way one can easily show that

(t) + axf (t) = / [r (s) xa (s) — q (s) xß (s) ]As.

xA (t) + axf (t) = [ [r (s) xa (s) — q (s) xß (s) ] As.

Therefore

yA (t) = xA (t) — xA (t) = —a [xf (t) — xf (t)]

+ / [r (s) (xa (s) — xa (s)) — q (s) (xß (s) — xß (s) )] As

This implies

yA(t) = —ayf (t)+ / [r (s) (xa (s) — xa (s)) — q (s) (xß (s) — xß (s) )]As.

By using the mean value theorem, we get

yA(t) = -ay°(t) + / r(s)ax&a-1(s)(x(s) - xi(s^ - q(s)ftx^-1(s)(x(s) - xi(s))

= -aya(t)+ / r (s) axT1 (s) - q (s) ftx3 1 (s) y(s)As, t > t0,

for x*,x0 € [x,x1] or x*,x0 € [x1,x].

For m < x (t) < M, we suppose that the function

f (t,x (t)) = - axCT (t)+ [ \\r (s) xa (s) - q (s) x13 (s)l As, t > t0,

is Lipschitzian in second argument.

Definition 11. Assume that x1 is the positive u-periodic solution of (1.1). If there exist positive constants Kxi and X for every positive u-periodic solution x of (1.1) such that

|x (t) - x1 (t)| <KxieeA (t, 0) Vt>t0, then x1 is said to be exponentially stable.

In the next theorem, we prove the exponential stability of the positive periodic solution x1 of (1.1).

Theorem 5. Assume that q,r € Crd([t0, to) n T, (t0, to)) and there exist positive constants m and M, and a function k € Crd([t0, to) nT, R) such that (3.1)-(3.3) hold. Let a > 0, 0 < a < ft < 1 and there exist constants m*,M* € (0, to) such that m* < m, M* > M and

ama-1r (t) - ftM33-1q (t) < 0 for t > t0.

Then (1.1) has a positive u-periodic solution which is exponentially stable.

Proof. Conditions (3.1)-(3.3) imply that (1.1) has a positive u-periodic solution x1. Let x be a positive u-periodic solution of (1.1) such that m* < x(t) < M*,

xA (t0) + axa (t0) - xA (t0) - axl (t0) = 0.

We prove that there exists X € (0, to) such that

|x (t) - x1 (t)| <KxieeA (t, 0), t > t0,

where Kxi = eA (t0,0) |y(t0>| + 1. We define the Lyapunov function

L(t) = |y(t)| e\\ (t, 0), t > t0, X € (0, a) .

For t > t0, we assume that L(t) < Kxi. On the other hand there exists t* > t0 such that L(t*) = Kxi and L(t) < Kxi for t € [t0,t*). By calculation of the upper left delta derivative of L(t) along the solution of (4.1), we get

- f t r i

(L(t))A~ < -a y (t)| e\\ (t, 0) + e\\ (t, 0W r (s) axa-1 (s) - q (s) ftx^-1 (s) |y(s)| As

+X y (t)| e\\ (t, 0), t > t0.

For t = t* we have

< (A - a) |yCT(t*)| eA (t*, 0)+ eA (t*, 0) / [am^r (s) - ^Mf-1q (s)] |y(s)| As

< (A - a) |yCT(t*)| eA (t*, 0).

If y(t) > 0, t > to, then from (4.1) it follows that, for t > to, the function y is decreasing. If y(t) < 0, t > t0, then y is increasing for t > t0. We conclude that |y(t)| , t > t0 has decreasing character. Then we obtain

which is a contradiction. Hence, we get

|y(t)| eA (t, 0) < KX1 for t > t0 and some A € (0, a).

The proof is complete. □

In [11], Cid et al. reformulated the problem of fluid motion in the pipe into the following periodic boundary value problem

u"(t) + au&(t) = —!— (e(i) - b(u&(t)&2) - c, t£ [0, u] ,

u(t) (5.1)

u(0) = u(w), u&(0) = u&(w),

where a > 0, b > 1, c > 0 and e is w-periodic continuous on R. By using the change of variables u = x1/(b+1), the singular problem (5.1) can be transformed to the following regular problem

f x&& (t) + ax& (t) + q (t) x^ (t) - r (t) xa (t) = 0, t € [0, w] , [ x(0) = x(w), x&(0) = x&(w),

r(t) = (b + l)e(t), q(t) = (6 + l)c, a = /3 =

with 0 < a < ^ < 1.

We will give new sufficient conditions ensuring the existence and the exponential stability of positive w-periodic solutions of the following dynamic equation

xAA (t)+ a (xCT)A (t) + (b + 1) [cx^ (t) - e (t) xa (t) ] = 0, t > t0. (5.2)

With respect to Theorems 4 and 5, we obtain the following theorem.

Theorem 6. Assume that a> 0, 0 < a < ,0 < 1 and there exist positive constants m and M, and a function k € Crd([t0, to) n T, R) such that (3.1) and (3.2) hold and

k(t)e0(o-fc) (ct (t) ,t0) = (b + 1) / e (s)expfa/ [e (a - k(v))] Av

, s 0 V fo 7 (5.3)

-cexp ^£ jf [e (a - k(v))] AvJ As, t > t0.

Then (5.2) has a positive w-periodic .solution.

Theorem 7. Assume that e € Crd([t0, to) n T, (t0, to)), a > 0, 0 < a < ft < 1, c > 0 and there exist positive constants m and M, and a function k € Crd([t0, to) n T, R) such that (3.1), (3.2) and (5.3) hold. Let, in addition, there exist constants m*,M* € (0, to) such that m* < m, M* > M and

ama-1e (t) - ftM*3-1c < 0 for t > t0.

Then (5.2) has a positive u-periodic solution which is exponentially stable.

In this paper, we provided the existence and exponential stability of positive periodic solutions with sufficient conditions for second-order dynamic equations on time scales. The main tools of this paper are the fixed point method and the Lyapunov method. However, by introducing new fixed mappings and suitable Lyapunov functionals, we get new existence and exponential stability conditions. An example illustrating our results is presented. The obtained results have a contribution to the related literature, and they improve and extend the results in [13] from the case of second-order differential equations to that case with second-order dynamic equations on time scales. It seems that the results of this paper can be extended to cover the case of delay second-order dynamic equations.

Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments and good advice.

REFERENCES