on the asymptotic behavior of the solution of a nonlinear integrodifferential equation

SIAM J. MATH. ANAL.Vol. 2, No. 3, August 1971

ON THE ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF ANONLINEAR INTEGRODIFFERENTIAL EQUATION*

STIG-OLOF LONDEN-

Abstract. A theorem concerning the asymptotic behavior of the solution of a nonlinear Volterraintegrodifferential equation is proved.

1. Introduction. We consider the equations

(1.1a) x’(t) b(t z)g(x(z))dr + [g(x(t))]-y(t) + f(t),

(1.1b) y’(t) g(x(t))- fly(t),

where 0 =< < , ]x(0)l < , y(0) > 0, > 0, fl > 0, and prove the followingtheorem.

THEOREM. Let

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

(.8)

where

g(x) c( , ), g(x) > O, g’(x) >= O,

lim g(x)= 0, lim g(x)= ,g’(x) =< (1/7)g(x), ]x] < oe, for some 7,

f(t) C[O, oo), sup If(t)l < oo,0<t<oo

lf(z) oo for some >_ O,F[ dr, F

b(t) C2[0, ) LI[O ),

b(t) <0, b’(t) < -pb"(t), 0 <= < ,

IX[

b(O) < O,

min (1//3, y/F) if F > O,P=

1/ ifF=O.Let x(t),y(t) be a solution of (1.1) on 0 <= < oe. Then limt_.oog(x(t)),

lim,_oo y(t) exist and satisfy

(1.9) lim g(x(t)) lim y(t) + ]b(z)[t- (Z t

While the existence of a solution x(t), y(t) on 0 =< < is assumed, we notethat the present hypothesis may also be used to obtain the a priori bounds necessaryfor an existence proof. Also observe that the assumptions above (in particularthe first part of (1.2)) are sufficient to guarantee uniqueness of the solution. Clearly,by (1.1b) and the second part of (1.2), y(t) > O, 0 <= < .

Received by the editors May 26, 1970, and in revised form November 27, 1970.q" Department of Mathematics, Technical University of Helsinki, Otaniemi, Finland.

356

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VOLTERRA INTEGRODIFFERENTIAL EQUATION 357

A considerable literature [2], [3], [6], [12] concerning the asymptotic behaviorof the solutions of

(1.10) x’(t) b(t- z)g(x(z))dr + f(t), 0 <= <

exists under the hypothesis xg(x) > 0, x 0. In [3], (- 1)kb(k)(t) <= 0 (k O, 1, 2, 30 < < m)isassumed, andin [6], b(t) is taken completely monotonic on 0 < < m(i.e., (- 1)kb(k)(t) N 0, k 0, 1, 2, 0 < < m). In [3] and [6], where f(t) O,the result (obtained with the aid of certain Lyapunov functions) is that if x(t) is asolution of (1.10) on 0 N < oe, then lim,_oo x()(t)= 0, k 0,1,2. In [2]a Popov-type condition is imposed on the kernel, and, in addition, b(t),f[ b(r)dr LI[0 0(3) L2[0 oo); f(t), f’(t)e LI[0, o) is assumed. The conclusionis that if x(t) is a solution of (1.10) on 0 __< < o, then lim_oo x(t) 0. In [4] anintegrated version of (1.9) is considered. If b(t) > O, g(x) x, then (1.10) becomesan equation of renewal type [1, Chap. 7].

Under the hypothesis g(x)>__ -2, Ixl < oe, 2 < oe, (1.10) has earlier beenstudied in [5], where results concerning the existence of bounded solutions on0 __< < oe under various assumptions on b(t) were obtained. Under the samehypothesis on g(x) (and making, in addition, certain growth assumptions), (1.1)has also been investigated [7]--[11], where different boundedness theorems wereproved. (Note that y(t) was taken more generally than here in [7]-[9] .)

In the present paper a result concerning the asymptotic behavior of the solu-tions of (1.1) is obtained. This result--stronger than mere boundedness--obviouslyrequires more restrictive hypotheses to be imposed on b(t), (1.8). Note that thesign hypothesis b(t) < 0, 0 < < oe, made in Theorems 3-8 in [7, Chap. 1], is ingeneral not alone sufficient to guarantee that lim,_o x(t) exists (not even b(t) < O,0 =< < m, suffices) as it is not difficult to construct numerical counterexamples.As to the nonlinear function g(x) we remark that (1.2) and (1.3) concern its mono-tonicity and (1.4) is a pointwise imposed growth condition.

We finally observe that (1.1)--with g(x) exp {x}--occurs in certain problemsin nonlinear nuclear reactor dynamics if the delayed neutrons are taken intoaccount; see [7, p. 11], for details.

2. Proof. Notice at first that we have the relation, 0 =< < ,

(2.1)

b’(r s)[g(x(s))- g(x(r))] 2 ds dr + 1/2 b(t- r)g2(x(r))dr

+ 1/2 b(r)g2(x(r)) dr b(O) g2(x(r)) dr

g(x(r)) b’(r s)g(x(s)) ds dr O,

which can be verified by expanding [g(x(s)) g(x(r))] 2 and performing an inter-change of the order of integration.

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358 STIG-OLOF LONDEN

We begin by examining the last term on the left side in (2.1),

(2.2) g(x(z)) b’(z s)g(x(s)) ds dz,

and consider the case F > 0. By (1.1a),

g(x(t)) F-l[x’(t)g(x(t))- y(t)- g(x(t)) b(t- r‘)g(x(r‘))dr‘

(2.3)g(x(t))[f(t)- F]], 0 < .

Substituting g(x(t)) from (2.3) into (2.2) one has

(2.2) -F-1 [x’(r‘)g(x(r‘))- y(r‘)- g(x(r‘)) b(r‘ s)g(x(s))ds

(2.4)

g(x(r‘))If(r‘) F ]] b’(r‘ s)g(x(s))ds, dr‘.

Define, for some Xo, IXol < ,(2.5) G(x) g(u) du, Ixl < oo.

We investigate the terms on the right side of (2.4) separately. Performing anintegration by parts in the first term yields, by (1.7),

(2.6)

F- x’(r‘)g(x(r‘)) b’(r‘ s)g(x(s)) ds

fiF- G(x(t)) b’(t r‘)g(x(r‘))

+ F- b"(r‘ s)g(x(s))G(x(r‘)) ds

+ b’(O)F- g(x(r‘))G(x(r‘)) dr,

+ F- b"(r‘ s)g(x(r‘))G(x(r‘)) ds dr‘

F- b"(r‘ s)g(x(r‘))G(x(r‘)) ds dr‘

F- G(x(t)) b’(t r‘)g(x(r‘)) dr‘

+ F- b"( s)[g(x(s))G(x(z)) g(x(r))G(x(r))] ds dz

+ F- b’(r‘)g(x(r‘))G(x(r‘)) dr‘.

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We need the following two lemmas.LEMMA 1. Let the hypothesis of the theorem hold. Then SUpo_<,< Ix(t)] < ,

SUPo_<,<oo y(t) <Proof Choose e > 0 sufficiently small. Solving (1.1b) for y(t) gives

(2.7) y(t) y(0)exp {-fit} + fO exp { flit r]}ag(x()) dr,

and, if is such that g(x(t)) <= [1 + e]g(x([)), 0 <= <= < o,

0=<t<,

(2.8) y(t) < Kxg(x(O + K2,

for some constants K1, K2 < O. Thus, after multiplying (1.1a) by g(x(t)), using(1.5) and (2.8), one obtains

(2.9) d-tG(x(t)) <= g(x(t)) b(t- z)g(x())d + K3g(x(0 + K2

(2.10) < K3g(x(0)+ K2,

where the final step follows from (1.2) and (1.8).Suppose SUpo=<,<oo x(t)= . Then, by (1.3), lim,_o supg(x(t))= , and,

from (1.2), lim,_ supG(x(t))= . Thus there exists {t,} such that G(x(t))<= G(x(t,)), 0 <= <= t,, (d/dt)G(x(t,)) >= O, lim,_ G(x(t,))= , lim,_o t, . By(2.9) we have that there exists a constant K4 < such that

(2.11) b(t, z)g(x(z)) dz => K,.

Let , max {tit < t,, g(x(t))= [1 + el-lg(x(t,))}.Then, after integrating (1.4),

g(x(t,)) f["Also, from (1.4),

1/2eg(x(t,)) <= g(x(t,)) g(x(,)) <= (1/7)g(x(t,))[x(t,) x(,)],

and x(t,) x(,) >= 1/2e. But, by (1.1a) and (2.8),

(2.12) x’(t) <= Ks, , <= <= t,.

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Thus t, t, is bounded away from zero and, as we may of course assume b(t) < 0,0 =< __< t, ,, and also, by (2.12), that x(t,)- x(r) is sufficiently small,=< t,, we have

lim b(t, r)g(x(r)) dr

which violates (2.11). Sup__<,<oo x(t) < oe follows. By (2.7), supo=<,< y(t) <Rather obvious modifications of the final part of Theorem 3 7, Chap. 1] then

show that the condition

(2.13) sup g(x [b(r)[ d: lim inf (t) + a d(r) dr < 0

---which there, together with supo_<,< x(t) < oe and the eventual monotonicity ofg(x) (decreasing) as x --, -oe, was shown to imply SUpo_<,< -x(t) < oe--may,under the assumption

[f(:)- F] dr < o,

be weakened to read

sup g(x) Ib(r)[ dr F a d(r) dr < 0.

As we now have F >= 0; ad(t) ae -t > 0; limx,_oo g(x) 0; g’(x) >= 0, Ixl < ,and b(t)e LI[0, oe); then supo_<t< Ix(t)l < follows. The lemma is proved.

LEMMA 2. Let hi(x) C(-oe, oe), Ixl < , and let hi(x) be monotone non-

decreasing for Ixl < 1, 2. Also let, for some rl > O,

(2.14) hi(x2)- h(x)<=1[h2(x2) h2(x)]

and let

(2.15) z(t)C[O, oe), sup [z(t)[ < ,0<t<oo

(2.16) a(t), ta(t) LO, oe) (’1 CO, ), a(t) >= O, 0 <= < oe.

Then, if hi(z(t)) >= O, 0 <= < , one has

f] ff a(r s)[h(z(r))h2(z(r)) h(z(s))hz(z(r))] ds

> rl a(r- s)[h2(z(r))- h(z(s))h(z(r))]dsdr K,

0=<t<oe,

for some constant K < oe, independent of t.

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Proof

flf a(r s) [h (z(r))he(z(r)) h (z(s))he(z(z)) ds

a(s) [hl(z)he(z hi(z, s)he(z)] dz ds

a(s) [hl(z)h2(z hl(Z s)h2(z)] dr ds,

where we write hi(z) hi(z(z)) and define

{hh(Z(-s)), s<<=t,(2.17) hl(z,s)

l(z(t- z)), 0 < z < s.

We assert that for any s, 0 __< s < t,

[hl(z)hz(z) hi(z, s)he(z)] dr _>_ q [h(z) l(z, s)hl(z)] dz.

To realize that the assertion holds we make the following observation. Suppose wehave 2n nonnegative real numbers, xi, yi, 1,..., n, such that x =< x2 <

Xn; Yl Y2 < Yn, and xi- xj __< (1/17)[y- yj], for any i,j such thatj. Then

(2.18) ExiYi- XqiYi] 17 IX XiXqi],i=1 i=1

where q l, q2, qn represents an arbitrary ordering of the integers 1, ..., n.

To see that (2.18) is true one only notices that any ordering q l, qe, "’", q, ofthe integers 1, ..., n may be obtained by a certain number of interchanges of thefollowing type:

where, for some io,

(rl, r2, "", r,)--+ (s1, $2, Sn)

ri=si, i-- 1,...,io- 1, io +2,...,n,

rio Sio + 1, rio+ Sio rio < rio+ 1.

At each such step needed to obtain q l, "’", q, from 1, ..., n we get the followingnonnegative contribution to the left side in (2.18)"

Yi[Xri Xsi] Yio[Xrio- Xsio] "+" Yio+ 1EXI’io+ Xsio+i=1

[Yio+l Yio] [Xr,o+, Xr,o] >- 17[Xio+1 Xio] [Xr,o+, Xrio]

17Xio[Xrio Xsio] "31- 17Xioq-1EXrio Xsio+ 1] q XiEXri Xsi]"i=1

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Invoking now the monotonicity of hi(x), (2.14), (2.15) and (2.17), one has (aftertaking the limit of a discrete version of the assertion) that the assertion holds. Thus

fl fJ a(’c s) (’C)h2(’c (S)h2(’c)]h ds dr,

>-_ ri a(s) [h2(z) x(-c, s)h (z)] d as

+ a(s) [h(x)- h(,s)h(x)] dr ds

a(s) [h(z)h2(z) h(z, s)h2(z)] dr ds

a(z s)[h(z) hl(S)hl(Z)J ds dr K,

where we have used (2.15) and (2.16). The lemma is proved.Using now Lemma 2, with g(x) hi(x), G(x) hz(x), 7 r/, x(t) z(t) and

-b"(t) a(t) gives, by (2.6), Lemma 1 and the hypothesis (note that (1.7) and(1.8) together imply tb"(t) Ll[0, ) and b"(t) =< 0, 0 =< < c and also that wecan of course, by Lemma 1, choose the Xo value defining G(x) such that G(x(t)) >= O,0__<t < oe)

(2.19)

F- x’(z)g(x(z)) b’(z s)g(x(s)) ds dr

>__ F -16(x(t)) b’(t z)g(x(z)) dr

7- b"(z s)[g(x(s))- g(x(’c))] 2 ds d’c

+ -ffY b’(t z)g2(x(z)) d -ff b’(z)g2(x(z)) dr

+ F- b’(z)g(x(z))a(x(z)) dr

for some constant k and where we have also used

(2.20)

fl f] b"(z s)[g(x(s))g(x(z)) g2(x(’c))] ds d

1/2 b"( s)[g(x(s)) g(x(z))] 2 ds d’c

+ 1/2 b’(t z)g2(x())d 1/2 b’()g2(x(z)) d.

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Substituting g(x(t)) as given by (1.1b) into the second term on the right side of(2.4) yields

F- y(z) b’(z s)g(x(s)) ds dr

-fl ;ifo 1 fl;aFb’(’c s)y(’c)y(s) ds dr, + b’(’c s)y(’c)y’(s) ds d’c

fl f s) +lb"(z s)]y(’c)y(s) dsdr

+ fib(O) y() daF

b’(O) fl+ - y() d

fib(O) y(z) draF

aft- b’()y(z) d

where we have integrated the term involving y’(s) by parts and applied the samereasoning that gives (2.20).

Analyzing next the third term on the right side of (2.4) one has, by (1.1a),

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g(u) du d + y() y’(r) + y() dF .,

x(O)

+ g2(x(z))[f(r) e dr + b(O) g2(x(z)) dr

g(x(t)) b(t r)g(x(r))dr2F

2Fg’(x(r))x’(z) b(r s)g(x(s)) ds d

b(O) xt)g2(u) aU + [2(t) 2(0) + FF axO)

+ g(x(rl[f(l- ; + b(O g(x(.Consider now the term

(. g’(x(rx’( b( sg(x(s s

above. Solving (1.1a) for ;b(r s)g(x(s))ds and substituting into (2.21) gives

(. ’(x(rx’(r[x’( f( ([g(x(-

’(x(x’(F(l[g(x()l]- g’(x(x’((rIf([g(x(rI]- rg’(x(x’( b(r sg(x(s s- f- (r[g(x(r -1 r

g’(x(rx’([ f(

g’(x(r))x’(r)y(r)[g(x(r)) - dr- g’(x(x’(r[f(r e([g(x(r?-i r

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Observe that the integrand in the first term after the last equality sign above is< 0, 0 < -c < oe. The term

(2.22)

[ Y(t)(2.22) +- g(x(t))

2Fg,(x(z))x,(z)yZ(z)[g(x(z))]- 2 dz

gives, after integrating by parts and using (1.1),

y2(0) ] 1-;-[y(t) y(0)] + y’2(qT)[g(x(z))]- dr.g-6)]

!

Integrating next

(2.23)

by parts we obtain

Let

g’(x(z))x’(z)y(z)[g(x(z))] - dr,

(2.23) -y(t)In {g(x(t))} + y(0)In {g(x(0))} + y’(-c) In {g(x(-c))} dz.

I lt-- {’clg(x(’c))>= (fl/oOy(’c), 0 <= "c <= t},

Izt-- {:lg(x()) < (/)Y(z), 0 = = t}.Then, as y’(z) 0 on I 1, and y’(z) < 0 on Izt

Collecting terms and substituting into (2.23) one has that there exists ka <such that (2.23) -k. Also, by Lemma 1,

(2.22) k + y’(r) [g(x(r));- d.

Thus, invoking (1.6), one has that there exists k5 < such that

Then, by Lemma 1 and (1.6),

k6 +flb(O) f ia y2(z) dz + b(O) g2(x(z)) dz

y,

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366 STIG--OLOF LONDEN

Collecting terms and substituting into (2.1) gives, as b(k)(t) LI[O 3), k O, 1,

and, remembering (1.8), and also that the hypothesis implies b"(t) <= O, 0 <= < ,we have

1 fly,2(2.24)flF

(z) [g(x(z))] dz

for some k7 < . By the hypothesis and Lemma 1, y"(t) exists on 0 __< < oand satisfies supo_<t<oo ly"(t)l < o. Thus we conclude from (2.24) that limt_.oo y’(t)exists and is equal to zero, or by (1.1b), that limt_oo [ag(x(t)) fly(t)] 0. Supposelimt_, g(x(t)) either does not exist or (if it exists) does not satisfy (1.9). Then thereclearly exist intervals It,, t, + T,] such that lim,_.oo t. lim,_oo T, o and suchthat, e.g.,

(2.25) x(t. + T.) x(t.) >= -6., lim 6. O,

ag(x(t)) fly(t)sufficiently small,

g(x(t)) >= + F z)l dz + 6,

for some > 0. Integrating (1.1a) over these intervals, remembering (1.6) and thatb(t)eLl[O, oe), readily gives a contradiction to (2.25). Thus lim_oog(x(t)),lim,_oo y(t) exist and satisfy (1.9).

Of course, if in addition to the hypothesis, g’(x)> 0 for x such thatg(x) g(x(oe)), then lim_, x(t) exists.

It is obvious that the arguments above remain much the same if F 0.One may then begin by considering (2.2) and replace g(x(t)) in this term by

x’(t)g(x(t))- y(t)- g(x(t)) b(t- z)g(x(’0)d’c -f(t)g(x(t))

(which is identically zero) and then proceed as above.This completes the proof.

3. Concluding remarks. In this paper we have investigated the solutions of acertain nonlinear Volterra integrodifferential equation and given a sufficienthypothesis under which the nonlinear function g(x(t)) of the solution x(t) tends to

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a limiting value as o. The equation considered may be viewed as a particularcase of the more general equation

fl fOfo d(z-s)g(x(s))dsdz+F(t)x(t) (t )g(x()) a + g(x())

on which extensions on the present work might be formulated.

REFERENCES

1] R. BELLMAN Aqr) K. L. CooE, Differential-difference Equations, Academic Press, New York, 1963.[2] C. CORDUYAYU, Sur une quation intOgrale de la thorie du rOglage automatique, C.R. Acad. Sci.

Paris S6r. A-B, 256 (1963), pp. 3564-3567.[3] J.J. LvIy, The asymptotic behavior ofthe solution ofa Volterra equation, Proc. Amer. Math. Soc.,

14 (1963), pp. 534-541; reprinted in V. LAKSHMIKANTHAM AND S. LEELA, Differential andIntegral Inequalities. I, Academic Press, New York, 1969, pp. 327-333.

[4], The qualitative behavior of a nonlinear Volterra equation, Proc. Amer. Math. Soc., 16(1965), pp. 711-718.

[5] ,Boundedness and oscillation ofsome Volterra anddelay equations, J. Differential Equations,5 (1969), pp. 369-398.

[6] J. J. LEVIN AND J. A. NOHEL, Note on a nonlinear Volterra equation, Proc. Amer. Math. Soc., 14(1963), pp. 924-929.

[7] S-O. LoyIN, On some nonlinear Volterra equations, Ann. Acad. Sci. Fenn. Ser. A, VI (1969), No.317.

[8] , On a Volterra integrodfferential equation, Comm. Phys. Math., 38 (1969), pp. 1-4.[9] , On a nonlinear Volterra integrodifferential equation, Ibid., 38 (1969), pp. 5-11.

[10], On a nonlinear retarded differential-difference equation, Ibid., 40 (1970), pp. 47-63.[11] , On a nonlinear retarded differential-difference equation. II, Ibid., 40 (1970), pp. 97-100.12] J. A. NoI-IL, Remarks on nonlinear Volterra equations, Proc. U.S.-Japan Seminar on Differential

and Functional Equations, W. A. Benjamin, New York, 1967, pp. 249-266.

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