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A NOTE ON THE BOUNDEDNESS OF AN INTEGRO-DIFFERENTIAL EQUATIONJuan Napoles Valdes aa Universidad de la Cuenca del Plata, Corrientes, Argentina,Universidad TecnologicaNacional, Chaco, ArgentinaPublished online: 18 Oct 2010.

To cite this article: Juan Napoles Valdes (2001) A NOTE ON THE BOUNDEDNESS OF AN INTEGRO-DIFFERENTIAL EQUATION,Quaestiones Mathematicae, 24:2, 213-216, DOI: 10.1080/16073606.2001.9639209

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Quaestiones Mathematicae 24(2001), 213–216.

c© 2001 NISC Pty Ltd, www.nisc.co.za

A NOTE ON THE BOUNDEDNESS OF AN

INTEGRO-DIFFERENTIAL EQUATION

Juan E. Napoles Valdes

Universidad de la Cuenca del Plata, Lavalle 50, (3400) Corrientes, Argentina andUniversidad Tecnologica Nacional, French 414, (3500) Resistencia, Chaco, Argentina.

E-Mail cuencadelplata@arnet.com.ar and napoles33@latinmail.com

Abstract. In this paper we study the prolongability and certain properties of theintegro-differential equation

(1) x′′ + a(t)f(t, x, x

′)x′ + g(t, x′) + h(x) =

∫ t

0

C(t, s)x′(s)ds.

Mathematics Subject Classification (2000): Primary 45J05; Secondary 45MXX.

Key words: Boundedness, Lyapunov functionals.

1. Introduction. We consider the integro-differential scalar equation:

(2) x′′ + a(t)f(t, x, x′)x′ + g(t, x′) + h(x) =

∫ t

0

C(t, s)x′(s)ds,

in which a is a positive continuous function on I, with I := [0,∞) , f is a continuousfunction satisfying f(t, x, x′)≥0 for (t, x, x′) ∈ I×R2, g and h are also continuousfunctions such that g(t, y)y > 0 for y 6=0 and xh(x) > 0 for x 6= 0 and C(s, t) iscontinuous for 0 ≤ s ≤ t < ∞. Here the functions indicated above are all scalarfunctions.To study the asymptotic behaviour of solutions of (1), the essential idea has

been to construct a nice Lyapunov functional or a Lyapunov function. In general,it is often difficult to construct nice Lyapunov functionals for a given equation,although recently many Lyapunov functionals are constructed experimentally.The purpose of this note is to give conditions to insure that all solutions of (1)

are bounded (global existence), via the construction of Lyapunov’s functionals forthis equation. We will also give a result for stability and L2[0,∞)-stability of thesolutions of equation (1). The approach utilized turns out to be very fruitful andhas led to several interesting results.Evidently, (1) can be regarded as a perturbed version of the equation:

(3) x′′ + a(t)f(t, x, x′)x′ + g(t, x′) + h(x) = 0,

and therefore, any qualitative property for (1) is interesting.

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214 J.E. Napoles Valdes

The definitions used here, are natural extensions of stability definitions forordinary differential equations, they have been used in integro-differential equationsas well as in delay-differential equations; see Driver [3], Miller [6] and Yoshizawa[8].

2. Results. As the problem of continuability of solutions is of paramount im-portance in the qualitative theory, it has received a considerable amount of at-tention in the last three decades. So, we now state our first result related tocontinuability (global existence) of the solutions of equation (2).Making y = x′ in (2) we obtain:

x′ = y,

y′ =

∫ t

0

C(t, s)y(s)ds− a(t)f(t, x, y)y − g(t, y)− h(x).(4)

The system (4) is equivalent to equation (2) and therefore any result for (4) is alsoa result for (2).Researchers from the early 1950’s to the present day have studied stability

properties of integro-differential equations by using Lyapunov’s functionals denotedby V (t, x(t), y(t)) (see [8]).Let

(5) V (t, x(t), y(t)) =y2(t)

2+H(x) +

∫ t

0

∫ ∞

t

|C(u, s)| duy2(s)ds,

where H(x) is defined as H(x) =∫ x

0h(s)ds. The functional V (t, x, y) plays a

central role in the derivation of the results in this section.

Theorem. Under the following assumptions:

i) there exists a constant R≥0 such that∫ t

0|C(t, s)| ds+

∫∞

t|C(u, t)| du ≤ R,

ii) [R− a(t)f(t, x, y)] ≤ 0 for all t ∈ [0,∞) and all x, y ∈ R,

iii) H(x)→∞ as x→∞,

all solutions of the system (4) are continuable to the future, that is, are defined forall t≥t0≥0.

Proof. The derivative V ′(2)(t, x(t), y(t)) of the function V defined by (5) along any

solution of system (4) satisfies:

(6) V ′(2)(t, x(t), y(t)) = y

∫ t

0

C(t, s)y(s)ds− a(t)f(t, x, y)y2 − g(t, y)y+

+

∫ ∞

t

|C(u, t)| duy2 −

∫ t

0

|C(t, s)| y2(s)ds.

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A note on the boundedness of an integro-differential equation 215

Thus,

V ′(2)(t, x(t), y(t)) ≤ y

∫ t

0

|C(t, s)| (y2(s) + y2)ds− a(t)f(t, x, y)y2 − g(t, y)y+

+

∫ ∞

t

|C(u, t)| duy2 −

∫ t

0

|C(t, s)| y2(s)ds.

Hence,

V ′(2)(t, x(t), y(t)) ≤ −a(t)f(t, x, y)y

2 − g(t, y)y+

+

[∫ t

0

|C(t, s)| ds+

∫ ∞

t

|C(u, t)| du

]

y2.

Using i), we may write

V ′(2)(t, x(t), y(t)) ≤ [R− a(t)f(t, x, y)] y2 − g(t, y)y,

and so we obtain from this and the assumptions that V ′(2)(t, x, y) ≤ 0.

It is known that the only way in which a solution (x(t), y(t)) of (4) fails to bedefined after some time T is if the condition

(7) Limt→T−

(

x2(t) + y2(t))

= +∞,

holds. Let (x(t), y(t)) be such a solution with initial condition (x0, y0). Since V ispositive definite and decreasing function on the trajectories of system (4), we have:

(8)y2(T )

2+H(x(T )) + k ≤ V0 ≡ V (0, x0, y0),

where k =∫ T

0

∫∞

T|C(u, s)| duy2(s)ds. From (8) we deduce:

y2(T )

2+H(x(T )) ≤ K = V0 − k.

From this inequality and by the condition v) we have that there exists a positiveconstant N such that |x(t)| ≤ N and |y(t)| ≤ N for all T > t0≥0.This shows that (7) is not possible. From this analysis we obtain the bound-

edness of all solutions of system (4) and therefore, of all solutions x(t) of equation(2). 2

Corollary. Under the assumptions of the theorem, suppose that

a(t)f(t, x, y)−R ≥ α > 0,

then we have for all solutions of (4) that y(t) ∈ L2[0,∞).

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216 J.E. Napoles Valdes

Proof. From the assumptions of the lemma we have that:

V ′(2)(t, x(t), y(t)) ≤ −αy

2(t).

If we integrate both sides from 0 to t we have:

0 ≤ V ′(2)(t, x(t), y(t)) ≤ V0 − α

∫ t

0

y2(s)ds,

implying∫∞

0y2(s)ds <∞. 2

Remark 1. As pointed out before, the equation (1) can be considered as a pertur-bation of equation (3). If in (1) we make C≡0, a(t)f(t, x, x′)≡0, g(t, x′) = g(x′)x′

with g(x′) > 0 for all x′ ∈ R, we have:

(9) x′′(t) + g(x′(t)) + h(x(t)) = 0.

In [7] Utz established sufficient conditions for the existence of periodic solutionsfor equations of type (7), which are similar to our theorem, see also [4].

Remark 2. Our results are also consistent with several earlier results on asymptoticbehaviour of solutions of simple cases of equation (1). We refer the reader to [1, 2,5, 9] for some interesting examples.

References

1. T.A. Burton, Liapunov functions and boundedness for differential and delay equa-tions, Hiroshima Math. J. 18 (1988), 341–350.

2. , A general stability result of Marachkov type, Ann. of Diff. Eqs.8(2) 1992, 148–1557.

3. R.D. Driver, Existence and stability of solutions of a delay-differential system,Archives of Rational Mechanics and Analysis 10 (1962), 401–426.

4. H.L. Guidorizzi, On the periodic solutions of systems of the type x′ = H(y), y′ =−

∑n

i=1 fi(x)Hi(y)− g(x), Universidade de Sao Paulo, RT-Mat 95–10, April 1995.

5. S. Murakami, Asymptotic behaviour of solutions of some differential equations, J.Math. Anal. Appl. 109 (1985), 534-545.

6. R.K. Miller, Asymptotic stability properties of linear Volterra integro-differntialequations, J. of Differential Equations 10 (1971), 485–506.

7. W.R. Utz, Periodic solution of second order differential equations with nonlinear,nondifferentiable damping, SIAM J. Appl. Math. 31 (1976), 504–510.

8. T. Yoshizawa, Stability by Lyapunov’s Second Method, Math. Soc. Japan, Tokyo,1966.

9. , Asymptotic behavior of solutions in nonautonomous systems, In:Trends in Theory and Practice of Nonlinear Equations, (V. Lakshmikantham, ed.),Dekker, 1984, 553–562.

Received 14 July, 1998 and in revised form 22 February, 2001.

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