UDC 519.4:517:513.88

MATHEMATICS

V. V. ZHIKOV

ALMOST-PERIODIC SOLUTIONS OF LINEAR AND NONLINEAR EQUATIONS IN A BANACH SPACE

(Presented by Academician A. N. Tikhonov on 6 VII 1970)

1. Introduction.

In a separable Banach space (B) we consider the equation

[
u' = Q(t)u,
\tag{1}
]

where (Q(t)) is, in general, an unbounded nonlinear operator depending on (t) almost periodically. Beginning with Favard’s work ((^1)), the conditions for the existence of almost-periodic (a.p.) solutions are formulated in terms of the so-called limiting equations.

Suppose that the domain of definition (D = D(Q(t))) does not depend on (t) and is a separable Banach space dense in (B). Let (Q(t)) be a continuous transformation (B \to B), depending on (t) almost periodically uniformly on each bounded subset (D). Then from every sequence ({t_m}) one can extract a subsequence ({t'_m}) such that (Q(t+t'_m)u \to Q(t)u). The set of elements (h) of the form (h=Q(s)) forms a compact metric group (G) with a system of shifts (L_t) and an invariant measure (\lambda). The limiting equations will be equations of the form

[
u'=\hat Q(t)u.
\tag{1′}
]

Below we shall use (in general, separately) the following assumptions concerning equations ((1′)):

a) the Cauchy problem is solvable for every initial value (u \in B), and the resolving operator (T_h(t)) is strongly continuous for (t \ge 0);

b) the resolving operator (T_h(t)) is weakly continuous for (t \ge 0);

c) a condition of continuous dependence on the “right-hand sides” is satisfied, namely: if (|u|\le c<\infty), (h_m\to h), then (|T_{h_m}(t)u - T_h(t)u|\to 0) for (t>0) uniformly in (u).

Further, if (1) is a linear equation of the form (u' = A(t)u + f(t)), then it is more convenient to use the resolving operator (V_h(t)) of the corresponding homogeneous problems. Here condition b) follows from a).

Put (X=B\times G) and (S_t x = S_t{u,h}={T_h(t)u,L_t h}). It is easy to verify (see ((^{3-5}))) that the transformations (S_t) form a dynamical system if (B) is endowed with the corresponding (strong or weak) topology.

2. A.p. in the sense of Besicovitch solutions of linear equations.

In the works of S. L. Sobolev ((^6)), L. Amerio ((^{10})), and others (see ((^9))), the question of a.p. solutions of hyperbolic equations was studied; the methods of investigation rely on the existence (for the homogeneous equation) of the energy integral. The results of Favard’s abstract theory ((^{7,8})) make it possible to study the case of energy dissipation, but no more than that. In the general case of variable energy, the problems of almost-periodicity can be solved only in the Besicovitch class. The corresponding theorem is formulated for a first-order equation in a Hilbert space.

Theorem 1. Let (A) be the generator of a strongly continuous semigroup, (B(t)) an a.p. in the sense of the operator norm function, and (f(t)) an a.p. function. Then, if the equation (u'=Au+B(t)u+f(t)) has a bounded solution, there exists a set (\Lambda \subset G) of full (\lambda)-measure such that, for (h\in\Lambda), the corresponding equation (u'=Au+\hat B(t)u+\hat f(t)) has

i.e., in the sense of Bohr, the solution has the same module of Fourier exponents as ({B(t), f(t)}).

Remark 1. Let (u(t)) be a bounded solution and let (X_0 \subset X) be the closure of the corresponding trajectory. Denote by (X_1) the closed invariant subset such that (X_0 \subset X_1) and the layer over some element (h \in G) is a convex set in (H). Then the solutions furnished by Theorem 1 may be chosen from (X_1).

Remark 2. For the hyperbolic equation
[
u''+A^2u+B(t)u=f(t)
]
the conditions of Theorem 1 are fulfilled if (A) is a positive definite operator and (A^{-1}B(t)) is a.p. in the sense of the operator norm of functions. Here boundedness of the solution means boundedness of the “energy” (|Au|^2+|u'|^2).

Remark 3. Theorem 1 generalizes to the case when (A) depends periodically on (t) and the usual solvability conditions for the Cauchy problem are satisfied. The general case, when the principal part (A) depends on (t) almost periodically, presents difficulties.

3. General theorems on a.p. solutions of nonlinear equations. Suppose that equations (2) possess the property of uniform positive stability, i.e., the family of transformations (T_h(t)) ((t \ge 0, h \in G)) is strongly uniformly continuous. This condition alone is apparently quite insufficient for selecting a.p. solutions. Below two special forms of stability are considered; with one of them the method of contracting shift operators is connected, with the other—Favard-type theorems of the first kind.

Theorem 2. Let conditions (a, c) be satisfied and
[
\varphi(T_h(t)u_1,\,T_h(t)u_2)\leqslant \varphi(u_1,u_2)\qquad (t\geqslant 0,\ u_1,u_2\in B),
\tag{2}
]
where (\varphi(u_1,u_2)) is strongly continuous on (B\times B) and satisfies the conditions:

1) (\varphi(u_1,u_2)=0), if and only if (u_1=u_2),

2) (\varphi(u_1,\alpha_1u_1+\alpha_2u_2)+\varphi(\alpha_1u_1+\alpha_2u_2,u_2)=\varphi(u_1,u_2)), where (\alpha_1,\alpha_2\geq 0), (\alpha_1+\alpha_2=1),

3) (\varphi(u_1,p)+\varphi(p,u_2)>\varphi(u_1,u_2)), if (p\not\equiv [u_1,u_2]).

Then, if equation (1) has a solution compact for (t\geq 0), it has at least one a.p. solution.

Let us note that, in applications, as the function (\varphi) one usually takes (|u_1-u_2|), provided only that the norm is strictly convex.

Theorem 3. Let inequality (2) be satisfied for a function (\varphi(u_1,u_2)) with properties 1), 2), 3), and let equation (1) have a weakly compact solution for (t\geq 0). Then, if the operators (T_h(t)) are weakly continuous for every (t\geq 0), equation (1) has at least one a.p. solution.

Theorem 3 reduces to Theorem 2 by proving the existence of a compact solution. This is achieved with the aid of the following lemma (see ((^8))).

Lemma. If the operators (T_h(t)) ((t\geq 0, h\in G)) are weakly continuous, condition (c_1) is satisfied, and
[
\sup_{t\geq 0}|T_h(t)u_1-T_h(t)u_2|\leq l|u_1-u_2|\quad (l<\infty,\ h\in G),
]
then every weakly recurrent solution is compact.

Considering, instead of equation (1), powers of a single transformation (T), we obtain new criteria for the existence of a fixed point of a contraction of a Banach space with strictly convex norm. It is assumed that (T) is defined on all of (B) and that the uniqueness condition is satisfied, i.e., (Tu_1\ne Tu_2) if (u_1\ne u_2).

Theorem 4. A contraction (T) has at least one fixed point if there exists an element (u_0\in B) for which the set ({T^m u_0}), where (m\geq 0), is compact. If, however, (T) is weakly continuous, then weak compactness of the same set is sufficient.

We now formulate the “strong” version of Favard’s first theorem.

Theorem 5. Suppose that conditions a, c, and d are satisfied for every (M>0):

[
\lim_{t\to+\infty}\ \sup_{h\in G,\ |u_1|,|u_2|\le M}
\left|T_h(t)u_1-T_h(t)u_2\right|=0.
\tag{3}
]

Then, if equation (1) has a solution bounded for (t\ge 0), it has a unique a.p. solution.

Let us note that if equation (1) is linear, then the condition of the existence of a bounded solution is superfluous—it follows from (3). There is also a more general assertion for linear equations.

Theorem 6. If (B) is reflexive, the transformations (V_h(t)) ((t\ge 0,\ h\in G)) are weakly uniformly continuous on the unit ball, and the homogeneous equation (1) has no bounded solutions, then equation (1) has a unique weak a.p. solution.

4. Applications. In a Banach space (B) with strictly convex dual (B^*), consider the equation

[
u' + Au = f(t),
\tag{4}
]

where (A) is monotone in the sense of Kato ((^{12})), (\sup_{t\in I}|f(t)|<\infty), and (f(t)) is an a.p. function in (L_2(0,1,B)). Put
(\Phi_1(u)=\operatorname{Re}\langle Au,F(u)\rangle),
(\Phi_2(u_1,u_2)=\operatorname{Re}\langle Au_1-Au_2,F(u_1-u_2)\rangle), where
(F:B\to B^*), (|F(u)|=|u|), (\langle u,F(u)\rangle=|u|^2).

Assume that the condition

[
\lim_{s\to\infty} 1/s \inf_{|u|\ge s}\Phi_1(u)=+\infty
]

is satisfied.

Theorem 7. Suppose there is a Banach space (B_1\subset B) with completely continuous embedding and the estimate

[
\Phi_1(u)\ge C_1|u|_1^2-C_2|u|^2\quad (C_1,C_2>0).
]

Then equation (4) has at least one a.p. solution.

Theorem 8. If the inequality

[
\Phi_2(u_1,u_2)\ge C(R)|u_1-u_2|^2
\quad (|u_1-u_2|\le R,\ R,C(R)>0),
]

holds, then equation (4) has a unique a.p. solution. This solution satisfies equation (4) in the ordinary sense if (f(t)) satisfies the Lipschitz condition
(|f(t_1)-f(t_2)|\le L|t_1-t_2|).

Theorems 7 and 8 are also valid in the case where (A) depends on (t), but the formulations are too cumbersome.

Let us now consider, in a Hilbert space, the equation

[
u''+B(u')+A^2u=f(t),
\tag{5}
]

where (A) is a positive definite operator, (\sup_{t\in I}|f(t)|<\infty), (f(t)) is an a.p. function in (L_2(0,1,H)), and the nonlinear operator (B) satisfies the following conditions:

1) (D(B)\supseteq D(A^m)) for some (m>0), ((Bu_1-Bu_2,u_1-u_2)\ge 0) for (u_1,u_2\in D(B));

2) the operator (A^{-m}BA^{-1}) extends to a continuous transformation (H\to H).

Denote by (H_A) the Hilbert space of elements (u\in D(A)) with norm (|u|_A=|Au|). In the energy space (E=H\times H_A), equation (5) reduces to the equation (z'+\widetilde A z=\widetilde f(t)), where the operator (\widetilde A), after a certain closure (or extension) procedure, is monotone in the sense of Kato. Starting from Kato’s solutions, generalized solutions are constructed for every (z(0)\in E).

Assume further that there exists a reflexive Banach space (D(A^n)\subseteq B_1\subseteq H) with norm (|u|_1), such that the operator (A^{-1}B) extends to ...

of a bounded and continuous transformation (K: B_1 \to H), and on bounded subsets of (B_1) the operator (K) acts (H_A)-continuously into (H).

Let
[
a_1(s)=\inf_{|u|1 \ge s}(Bu,u),\qquad
a_2(s)=\inf(Bu_1-Bu_2,u_1-u_2).
]
[
|Ku|\le \beta_1(|u|_1)\beta_2(|u|),\qquad
|Ku_1-Ku_2|\le \gamma_1(|u_1-u_2|_1)\gamma_2(|u_1-u_2|),
]
where the functions (\beta_i,\gamma_i) ((i=1,2)) are continuous on ([0,\infty)).

Theorem 9. Let
[
\lim_{s\to\infty} a_1/s=+\infty,
]
[
\lim_{s\to\infty}\beta_1/a_1<\infty,\qquad \lim_{s\to\infty}\beta_2/s=0.
]
Then for every solution (u(t)) the inequality
[
\sup_{t\in I}|u(t)|_E^2 \le |u(0)|_E^2+C
]
holds, where (C) does not depend on the solutions. If, moreover, the embedding (B_1\to H) is completely continuous, then equation (5) has at least one (E)-a.p. solution.

Theorem 10. Let
[
\lim_{s\to\infty} a_1/s=+\infty,
]
[
\lim_{s\to\infty}\beta_1/a_1<\infty,\qquad
\lim_{s\to\infty}\beta_2/s=0,
]
[
\lim_{s\to\infty}\gamma_1/a_2<\infty,\qquad
\lim_{s\to\infty}\gamma_2/s=0.
]
Then, if the operator (A^{-1}) is completely continuous, (a_2(s)\ne0) for (s\ne0), and the function (f(t)) is uniformly continuous on the entire axis, equation (5) has a unique (E)-a.p. solution. This solution satisfies equation (5) in the ordinary sense if (B_1\subset H_A), (a_2(s)\ge Cs^2) ((C>0,\ 0\le s\le1)), and (f(t)) satisfies the Lipschitz condition.

Let us note that the delicate results of Prouse follow easily from Theorem 10 ((^{11})). We also note that if (f(t)) is a periodic function, then the solutions given by Theorems 7–10 have the same period. There is no doubt that these results on periodic solutions (given, for example, by Theorem 9 for a Euclidean space) are difficult to obtain on the basis of the known fixed-point principles.

Note added in proof. We make two remarks concerning Favard’s theory.

1. Let (R) be a complete metric space with metric (\rho), and let (S_t) be a system on (R\times H) in the sense of item 1. The following “contraction mapping principle” holds (for comparison see Theorems 5 and 8): if for some (t_0>0) the inequality
   [
   \rho(S_h(t_0)u_1,S_h(t_0)u_2)\le q\rho(u_1,u_2)\qquad (q<1,\ h\in G)
   ]
   is satisfied, then equation (1) has a unique a.p. solution.

2. Let (B) be an arbitrary Banach space, and let equation (1) be linear and possess the property of uniform positive stability, i.e.
   [
   \sup_{t\ge0,\ h\in G}|V_h(t)|\le l<\infty.
   ]
   Then equation (1) has an a.p. solution if and only if it has a weakly compact solution (proof in ((^8))). This assertion is more general than that which follows from Theorem 3.

Vladimir Polytechnic Institute

Received
21 IV 1970

REFERENCES

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