##### Document Text Contents

Page 1

Bounded and Almost Periodic Solutions of

Nonlinear Operator Differential Equations

Page 2

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, U.S.S.R.

Yu. 1. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.

N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.

S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.

M. C. POL YVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Volume 55

Page 115

106 A.A. PANKOY

Bf,I'(R, E) CBf,'1(R, E).

However, regarding u(t) as a function with values in H, we have, by Proposition

3.12,

for all 1 ~s < 00. We note also that if q = 2, Y = 1 and f, I ES P' (R; E'), then u

belongs to S P (R; E) n CAP(R; H) and u' EBS P (R; E) n L 00 (R; H). Since u

satisfies (5.11), we additionally have u' EBP(R; E) (see the discussion at the end of

nO. 3.3).

6. Singular perturbation.

We consider (formally) the following inequality with a small parameter fL>O in

front of the derivative:

(p.u' (t) + A (t)u(t) - f(t), v - u(t» + q>(V)- q>(u(t» ;;;. 0, \tv E V,

for almost all t ER. If fL=O, then the problem turns into the stationary inequality

(A (t)u(t)-f(t), v-u(t»+q>(v)-q>(u(t»;;;. 0, \tVEV, (6.10>

where t ER plays the role of parameter. According to nO. 1.1, a weak setting of the

problem corresponding to (6.1 1') is stated in the following way. Let the conditions

of nO. l.1 (in particular, inequalities (1.3) and (1.5) and condition (c» be satisfied.

Given f EIJ{~ (R; V,), find a function u = u I' EIJ{oc (R; V) n C(R; H) such that

I,

![(uv'+Au-j,v-u)+</>(v)-</>(u)]dt ~ -II v-u 12 1:;,

I,

for any test function v. The corresponding limit inequality is

I,

j[(Au- j, v -u)+q>(v)-q>(u)]dt ;;;. 0, tl ~ t2,

I,

for any v EIJ{oc(R; V). This is clearly equivalent to (6.10)' It is easy to see that

under suitable conditions all previous results are applicable to (6.21') for any fixed

Page 116

CHAPTER 3 107

1-'->0 (it is sufficient to divide (6.2,,) by 1-'-.

The usual stationary theory (see, for example, [44, ll8]) is applicable to (6.20).

We present here the corresponding results in a suitable form.

PROPOSITION 6.1. Assume that condition (c) is satisfied and that inequalities (1.3)

and (1.5) hold. If f ELf;c (R; V'), then there is a solution U ELfoc (R; V) of inequality

(6.20) · Moreover, iff EBSI' (R; V'), then U EBSI(R; V).

Proof By general results [44, 118], for any fixed N>O problem (6.20) has a solu-

tion u=uNCU(-N, N; V) and

II U I I LP(-N,N;V) ",;;; CllfIILP'(-N,N;V), (6.3)

where C(x), x;;;'O, depends only on constants involved in (1.3) and (1.5). Moreover,

estimate (6.3) is valid for any solution of (6.20) on the interval (-N, N). It is easy

to see that UN I (a,b) is a solution of (6.20) on (a, b) C( - N, N) and is bounded in

LP(a, b; V). Hence passing a subsequence, if necessary, we may assume that

limuN=UO exists weakly in !,foc(R; V). Results in [142] imply that Uo is a solution

of (6.20) on R (but it is not difficult to see this immediately). The second assertion

easily follows from the estimate (6.3). 0

REMARK 6.2. If all operators A (t) are strictly monotone, then the solution con-

structed is unique. Assume now that condition (capv,p) is satisfied and that ine-

quality (3.1) holds with E= V. If fESP'(R; V), then we may consider a family of

inequalities,

J

f[(A (s + t)us(t) - f(s + t), u(t) - usCt» + q,(v(t» - tp(us(t»]dt ;;;. 0,

o

where sERB. Any such inequality has a unique solution US ELP(O, 1; V), and

us(t) = uo(s + t), s ER, t E(O, 1).

Results in [142) imply that UsOEU(O, 1; V) depends continuously on sERB and,

consequently, Uo ESP(R; V). Using results in [142) it can also be shown that

fECAP(R; V) implies Uo ECAP(R; V).

Now we consider the behaviour of bounded solutions, u,,' of (6.2 ,,) as JL~O.

Under the conditions of Theorem 2.7 such solutions are uniquely determined.

THEOREM 6.3. Assume that condition (c) is satisfied, inequalities (1.3) and (1.5)

hold, and one of inequalities (3.1) or (3.2) hold with q;;;'2 Then the family

Page 229

220 A.A.PANKOV

131. Grothendieck, A.: Produits tensoriells topo10giques et espaces nuclearres,

Mem. Amer. Math. Soc. 16, Amer. Math. Soc., 1955.

131a. Raraux, A.: Nonlinear Evolution Equations. Global Behavior of Solutions,

Lect. Notes Math. 841, Springer, 1981.

132. Ruet, D.: Perturbations singulieres d'inequations variationelles, Compt. Rend.

Acad Sci. Paris, 267 (1968),932-934.

133. Itoh, S. : Nonlinear random equations with monotone operators in Banach

spaces, Math. Ann., 236, no. 2 (1978), 133-146.

134. Itoh, S.: Random differential equations associated with accretive operators, J.

Diff. Eq., 31, no. 1 (1979), 139-154.

135. Ka-Sing, Lau.: On the Banach spaces of functions with bounded upper means,

Pacific J. Math., 91, no. 1 (1980), 153-172.

136. Kravvaritis, D.: Nonlinear random operators of monotone type in Banach

spaces, J. Math. Anal. Appl., 78, no. 2 (1980), 488-496.

137. Kravvaritis, D.: Existence theorems for nonlinear random equations and ine-

qualities, J. Math. Anal. Appl., 88, No.1 (1982),61-75.

138. Leray, 1., and Lions, 1.-L.: Quelques resultats de Visik sur les problemes ellip-

tiques non linearres par 1es methodes de Minty-Browder, Bull. Soc. Math.

France, 93 (1965),97-107.

139. Lions, 1.-L.: Sur certains systemes hyperboliques non linearres, Compt. Rend

Acad Sci. Paris, 257 (1963), 2057-2060.

140. Lions, 1.-L., and Stampacchia, G.: Variational inequalities, Comm. Pure Appl.

Math., 20 (1967), no. 3, 493-519.

141. Marcinkiewicz, 1.: Vne remarque sur 1es espaces de Besicovitch, Compt. Rend.

Acad Sci. Paris, 208 (1939), 152-159.

142. Mosco, U.: Convergence of convex sets and of solutions of variational ine-

qualities, Adv. Math., 3, No.4 (1969), 510-585.

143. Pozzi, G.: Sulle equazioni astratte lineare e nonlineare del tipo di Schrodinger.

Ann. Mat. Pura Appl., 78 (1968), 197-258.

144. Prouse, G.: Soluzioni quasi-periodiche dell'equazione di Navier-Stokes in due

dimensioni, Rend Sem. Mat. Univ. Padova, 33 (1963), 186-212.

145. Prouse, G.: Periodic or almost periodic solutions of a non-linear functional

equation, Rend. Accad. Naz. Lincei, Ser. 8,43 (1967), 161-167; 281-287; 448-

452; 44 (1968), 1-10.

146. Rabinowitz, P.R.: Free vibrations for a semilinear wave equation, Comm. Pure

Appl. Math, 31, no. 1 (1978), 31-68.

147. Tartar, L.: Interpolation non-linemre et regularite, J. Fun ct. Anal., 9 (1972),

469-489.

148. Zaidman, S.: Solutions presque periodiques des equations differentielles

abstraites, l'Ennseignem. Math., 24 (1978), no. 1-2, 87-110.

Page 230

Subject Index

almost periodic function in the sence of

Stepanov 25

almost periodic impulse measure 200

Besicovitch almost periodic function 11

Bohner transform 24

Bohner-Fejer operators 13

Bohr almost periodic function 7

Bohr compactification 5

character 6

coercive operator 52

compact hull 5

composition operator 49

coupled parabolic equations 147

evolution variational inequality 56

€-almost period 7

Fourier-Bohr transform 16

frequency group of a.p. function 16

group of characters 5

impulse measure 198

lower semicontinuous function 54

mean value 10

min-max procedure 92

monotone operator 50

nonlinear Schrodinger equation 144

positive boundary condition 135

proper function 54

p-coercive operator 159

p-monotone operator 159

relative Bohr compactification 21

relatively dense set 7

semi continuous operator 50

semi-permiable thin wall problem 119

singularly perturbed 106

variational inequality

spectrum of a.p. function 16

Stepavon almost periodic function 25

strictly hyperbolic operator 181

symmetric hyperbolic system 134

uniformly almost periodic function 7

variational inequality with shifted

variable 119

weak solution of evolution variational

inequality 58

variational inequality weakly almost

periodic function 28

Bounded and Almost Periodic Solutions of

Nonlinear Operator Differential Equations

Page 2

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, U.S.S.R.

Yu. 1. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.

N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.

S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.

M. C. POL YVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Volume 55

Page 115

106 A.A. PANKOY

Bf,I'(R, E) CBf,'1(R, E).

However, regarding u(t) as a function with values in H, we have, by Proposition

3.12,

for all 1 ~s < 00. We note also that if q = 2, Y = 1 and f, I ES P' (R; E'), then u

belongs to S P (R; E) n CAP(R; H) and u' EBS P (R; E) n L 00 (R; H). Since u

satisfies (5.11), we additionally have u' EBP(R; E) (see the discussion at the end of

nO. 3.3).

6. Singular perturbation.

We consider (formally) the following inequality with a small parameter fL>O in

front of the derivative:

(p.u' (t) + A (t)u(t) - f(t), v - u(t» + q>(V)- q>(u(t» ;;;. 0, \tv E V,

for almost all t ER. If fL=O, then the problem turns into the stationary inequality

(A (t)u(t)-f(t), v-u(t»+q>(v)-q>(u(t»;;;. 0, \tVEV, (6.10>

where t ER plays the role of parameter. According to nO. 1.1, a weak setting of the

problem corresponding to (6.1 1') is stated in the following way. Let the conditions

of nO. l.1 (in particular, inequalities (1.3) and (1.5) and condition (c» be satisfied.

Given f EIJ{~ (R; V,), find a function u = u I' EIJ{oc (R; V) n C(R; H) such that

I,

![(uv'+Au-j,v-u)+</>(v)-</>(u)]dt ~ -II v-u 12 1:;,

I,

for any test function v. The corresponding limit inequality is

I,

j[(Au- j, v -u)+q>(v)-q>(u)]dt ;;;. 0, tl ~ t2,

I,

for any v EIJ{oc(R; V). This is clearly equivalent to (6.10)' It is easy to see that

under suitable conditions all previous results are applicable to (6.21') for any fixed

Page 116

CHAPTER 3 107

1-'->0 (it is sufficient to divide (6.2,,) by 1-'-.

The usual stationary theory (see, for example, [44, ll8]) is applicable to (6.20).

We present here the corresponding results in a suitable form.

PROPOSITION 6.1. Assume that condition (c) is satisfied and that inequalities (1.3)

and (1.5) hold. If f ELf;c (R; V'), then there is a solution U ELfoc (R; V) of inequality

(6.20) · Moreover, iff EBSI' (R; V'), then U EBSI(R; V).

Proof By general results [44, 118], for any fixed N>O problem (6.20) has a solu-

tion u=uNCU(-N, N; V) and

II U I I LP(-N,N;V) ",;;; CllfIILP'(-N,N;V), (6.3)

where C(x), x;;;'O, depends only on constants involved in (1.3) and (1.5). Moreover,

estimate (6.3) is valid for any solution of (6.20) on the interval (-N, N). It is easy

to see that UN I (a,b) is a solution of (6.20) on (a, b) C( - N, N) and is bounded in

LP(a, b; V). Hence passing a subsequence, if necessary, we may assume that

limuN=UO exists weakly in !,foc(R; V). Results in [142] imply that Uo is a solution

of (6.20) on R (but it is not difficult to see this immediately). The second assertion

easily follows from the estimate (6.3). 0

REMARK 6.2. If all operators A (t) are strictly monotone, then the solution con-

structed is unique. Assume now that condition (capv,p) is satisfied and that ine-

quality (3.1) holds with E= V. If fESP'(R; V), then we may consider a family of

inequalities,

J

f[(A (s + t)us(t) - f(s + t), u(t) - usCt» + q,(v(t» - tp(us(t»]dt ;;;. 0,

o

where sERB. Any such inequality has a unique solution US ELP(O, 1; V), and

us(t) = uo(s + t), s ER, t E(O, 1).

Results in [142) imply that UsOEU(O, 1; V) depends continuously on sERB and,

consequently, Uo ESP(R; V). Using results in [142) it can also be shown that

fECAP(R; V) implies Uo ECAP(R; V).

Now we consider the behaviour of bounded solutions, u,,' of (6.2 ,,) as JL~O.

Under the conditions of Theorem 2.7 such solutions are uniquely determined.

THEOREM 6.3. Assume that condition (c) is satisfied, inequalities (1.3) and (1.5)

hold, and one of inequalities (3.1) or (3.2) hold with q;;;'2 Then the family

Page 229

220 A.A.PANKOV

131. Grothendieck, A.: Produits tensoriells topo10giques et espaces nuclearres,

Mem. Amer. Math. Soc. 16, Amer. Math. Soc., 1955.

131a. Raraux, A.: Nonlinear Evolution Equations. Global Behavior of Solutions,

Lect. Notes Math. 841, Springer, 1981.

132. Ruet, D.: Perturbations singulieres d'inequations variationelles, Compt. Rend.

Acad Sci. Paris, 267 (1968),932-934.

133. Itoh, S. : Nonlinear random equations with monotone operators in Banach

spaces, Math. Ann., 236, no. 2 (1978), 133-146.

134. Itoh, S.: Random differential equations associated with accretive operators, J.

Diff. Eq., 31, no. 1 (1979), 139-154.

135. Ka-Sing, Lau.: On the Banach spaces of functions with bounded upper means,

Pacific J. Math., 91, no. 1 (1980), 153-172.

136. Kravvaritis, D.: Nonlinear random operators of monotone type in Banach

spaces, J. Math. Anal. Appl., 78, no. 2 (1980), 488-496.

137. Kravvaritis, D.: Existence theorems for nonlinear random equations and ine-

qualities, J. Math. Anal. Appl., 88, No.1 (1982),61-75.

138. Leray, 1., and Lions, 1.-L.: Quelques resultats de Visik sur les problemes ellip-

tiques non linearres par 1es methodes de Minty-Browder, Bull. Soc. Math.

France, 93 (1965),97-107.

139. Lions, 1.-L.: Sur certains systemes hyperboliques non linearres, Compt. Rend

Acad Sci. Paris, 257 (1963), 2057-2060.

140. Lions, 1.-L., and Stampacchia, G.: Variational inequalities, Comm. Pure Appl.

Math., 20 (1967), no. 3, 493-519.

141. Marcinkiewicz, 1.: Vne remarque sur 1es espaces de Besicovitch, Compt. Rend.

Acad Sci. Paris, 208 (1939), 152-159.

142. Mosco, U.: Convergence of convex sets and of solutions of variational ine-

qualities, Adv. Math., 3, No.4 (1969), 510-585.

143. Pozzi, G.: Sulle equazioni astratte lineare e nonlineare del tipo di Schrodinger.

Ann. Mat. Pura Appl., 78 (1968), 197-258.

144. Prouse, G.: Soluzioni quasi-periodiche dell'equazione di Navier-Stokes in due

dimensioni, Rend Sem. Mat. Univ. Padova, 33 (1963), 186-212.

145. Prouse, G.: Periodic or almost periodic solutions of a non-linear functional

equation, Rend. Accad. Naz. Lincei, Ser. 8,43 (1967), 161-167; 281-287; 448-

452; 44 (1968), 1-10.

146. Rabinowitz, P.R.: Free vibrations for a semilinear wave equation, Comm. Pure

Appl. Math, 31, no. 1 (1978), 31-68.

147. Tartar, L.: Interpolation non-linemre et regularite, J. Fun ct. Anal., 9 (1972),

469-489.

148. Zaidman, S.: Solutions presque periodiques des equations differentielles

abstraites, l'Ennseignem. Math., 24 (1978), no. 1-2, 87-110.

Page 230

Subject Index

almost periodic function in the sence of

Stepanov 25

almost periodic impulse measure 200

Besicovitch almost periodic function 11

Bohner transform 24

Bohner-Fejer operators 13

Bohr almost periodic function 7

Bohr compactification 5

character 6

coercive operator 52

compact hull 5

composition operator 49

coupled parabolic equations 147

evolution variational inequality 56

€-almost period 7

Fourier-Bohr transform 16

frequency group of a.p. function 16

group of characters 5

impulse measure 198

lower semicontinuous function 54

mean value 10

min-max procedure 92

monotone operator 50

nonlinear Schrodinger equation 144

positive boundary condition 135

proper function 54

p-coercive operator 159

p-monotone operator 159

relative Bohr compactification 21

relatively dense set 7

semi continuous operator 50

semi-permiable thin wall problem 119

singularly perturbed 106

variational inequality

spectrum of a.p. function 16

Stepavon almost periodic function 25

strictly hyperbolic operator 181

symmetric hyperbolic system 134

uniformly almost periodic function 7

variational inequality with shifted

variable 119

weak solution of evolution variational

inequality 58

variational inequality weakly almost

periodic function 28