Semigroup Forum Vol. 71 (2005) 201–230

c 2005 Springer


DOI: 10.1007/s00233-005-0508-y

RESEARCH ARTICLE

Almost Automorphic Functions in Fr´ echet Spaces and Applications to Differential Equations Ciprian S. Gal, Sorin G. Gal, and Gaston M. N’Gu´ er´ ekata Communicated by Jerome A. Goldstein

Abstract In this paper we first develop a theory of almost automorphic functions with values in Fr´echet spaces. Then, we consider the semilinear differential equation x
(t) = Ax(t) + f (t, x(t)), t ∈ R in a Fr´ echet space X , where A is the infinitesimal generator of a C0 -semigroup satisfying some conditions of exponential stability. Under suitable conditions on f , we prove the existence and uniqueness of an almost automorphic mild solution to the equation. Keywords: Almost automorphic, asymptotically almost automorphic, mild solutions, semigroups of linear operators, semilinear differential equations, Fr´echet spaces. 1991 Mathematics Subject Classification: 43A60, 34G10.

1. Introduction Harald Bohr’s interest in
which functions could be represented by a Dirichlet ∞ −λn z series, i.e. of the form , where an , z ∈ C and (λn )n∈N is a n=1 an e monotone increasing sequence of real numbers (series which play an important role in complex analysis and analytic number theory), led him to devise a theory of almost periodic real (and complex) functions, founding this theory between the years 1923 and 1926. The theory of almost periodic functions was extended to abstract spaces, see for example the monographs [7], [8], [18], [19] (for Banach space valued functions), [6], [18], [31] (for Fr´echet space valued functions). Also, in the recent paper [1] (see also Chapter 3 of the book [19]), the theory of real-valued almost periodic functions has been extended to the case of fuzzy-number-valued functions. The concept of almost automorphy is a generalization of almost periodicity. It has been introduced in the literature by S. Bochner in relation to some aspects of differential geometry [2-5]. Important contributions to the theory of almost automorphic functions have been made, for example, with the papers [28], [23]–[26], [15] and the books [27], [18], [19] (concerning almost automorphic functions with values in Banach spaces), and the paper [22] (concerning almost automorphy on groups). Also, the theory of almost automorphic functions with

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values in fuzzy-number-type spaces was developed in the papers [12], [11] (see also Chapter 4 in [19]). However the theory of almost automorphic functions with values in a locally convex space (Fr´echet space) was not yet developed. It is the main goal of the present paper to develop this theory in Section 2. Section 3 contains elements of semigroups of operators on locally convex spaces while Section 4 deals with the existence and uniqueness of almost automorphic mild solutions with values in Fr´echet spaces, for the differential equations x (t) = Ax(t) + f (t), t ∈ R, and

x (t) = Ax(t) + f (t, x(t)),

t ∈ R,

where A is the infinitesimal generator of a C0 -semigroup of linear operators on a Fr´echet space, (T (t))t≥0 , satisfying some exponential-type conditions of stability. 2. Almost automorphic functions in Fr´ echet spaces In this section we develop a theory of almost automorphic functions with values in Fr´echet spaces. First we recall the following: Definition 2.1. A linear space (X, +, ·) over R is called Fr´echet space if X is a metrizable, complete, locally convex space. Remark. It is a classical fact that Fr´echet spaces are characterized by the existence of a countable, sufficient and increasing family of seminorms (pi )i∈N (that is pi (x) = 0 , for all i ∈ N implies x = 0, and pi (x) ≤ pi+1 (x) , for all x ∈ X , i ∈ N ), which define the pseudonorm: |x|X =

∞  1 pi (x) , 2i 1 + pi (x) i=0

x ∈ X,

and the metric d(x, y) = |x − y|X invariant with respect to translations, such that d generates a complete metric topology equivalent to that of locally convex space. That is, d has the properties: d(x, y) = 0 iff x = y , d(x, y) = d(y, x) , d(x, y) ≤ d(x, z) + d(z, y) , d(x + u, y + u) = d(x, y) for all x, y, z, u ∈ X . Also,
∞ pi (x) notice that since 1+p ≤ 1 and i=0 21i = 1 , it follows that |x|X ≤ 1 , for all i (x) x ∈ X. Moreover, d has the following properties: Theorem 2.2. (i) d(cx, cy) ≤ d(x, y) for |c| ≤ 1 ; (ii) d(x + u, y + v) ≤ d(x, y) + d(u, v); (iii) d(kx, ky) ≤ d(rx, ry) if k, r ∈ R, 0 < k ≤ r ; (iv) d(kx, ky) ≤ kd(x, y), for all k ∈ N, k ≥ 2 ; (v) d(cx, cy) ≤ (|c| + 1)d(x, y), for all c ∈ R .

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Proof. Property (i) is well known. (ii) We have d(x + u, y + v) = d(x + (u − v) + v, y + v) = d(x + u − v, y) = d(y, x + u − v) ≤ d(y, x) + d(x, x + u − v) = d(x, y) + d(x + v, x + u) = d(x, y) + d(v, u). (iii) Denote t = kr . Since 0 < t ≤ 1 , we get d(kx, ky) = d(t(rx), t(ry)) ≤ d(rx, ry) . (iv) Since d(2x, 2y) = d(x + x, y + y) ≤ d(x, y) + d(x, y) = 2d(x, y) , by mathematical induction we easily obtain d(kx, ky) ≤ kd(x, y) , for all k ∈ N, k > 2. (v) If |c| ≤ 1 then d(cx, cy) ≤ d(x, y) ≤ (|c| + 1)d(x, y) . Let |c| > 1 . If c > 1 then we get d(cx, cy) ≤ d(([c] + 1)x, ([c] + 1)y) ≤ ([c] + 1)d(x, y) ≤ (c + 1)d(x, y) ≤ (|c| + 1)d(x, y). For c < 0, |c| > 1 , we obtain d(cx, cy) = d(−c(−x), −c(−y)) ≤ (−c + 1)d(x, y) = (|c| + 1)d(−x, −y) ≤ (|c| + 1)d(x, y), since d(−x, −y) ≤ d(x, y) ; which proves the theorem. Remark. If (X, ⊕, , ρ) is a fuzzy-number-type space (see e.g. [11]), it is known that ρ is a complete metric (on X ), which is invariant with respect to translations, satisfies the property (ii) in Theorem 2.2 as well as the stronger property ρ(cx, cy) = |c|ρ(x, y), for all c ∈ R. Everywhere in the rest of the paper, (X, (pi )i∈N , d) will be a Fr´echet space with (pi )i∈N and d as in the Remark following Definition 2.1. We start with the following Bochner-type definition: Definition 2.3. We say that a continuous function f : R → X is almost automorphic, if every sequence of real numbers (rn )n contains a subsequence (sn )n , such that for each t ∈ R , there exists g(t) ∈ X with the property lim d(g(t), f (t + sn )) = lim d(g(t − sn ), f (t)) = 0.

n→+∞

n→+∞

(The above convergence on R is pointwise).

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Remark. Almost automorphy, as defined above, is a more general concept than almost periodicity in Fr´echet spaces, as it was defined in [18, p. 51]. Indeed, by the Bochner’s criterion (see e.g. [18, p. 55, Theorem 3.1.8]), a function with values in a Fr´echet space is almost periodic if and only if for every sequence of real numbers (rn )n , there exists a subsequence (sn )n , such that the sequence (f (t + sn ))n converges uniformly in to t ∈ R , with respect to the metric d . Obviously this is a stronger condition than the pointwise convergence in Definition 2.3. Also, note that the limits with respect to d in Definition 2.3, are equivalent to the corresponding limits with respect to each seminorm pj , that is to lim pj (g(t) − f (t + sn )) = lim pj (g(t − sn ) − f (t)) = 0,

n→+∞

n→+∞

for all j ∈ N.

The following elementary properties hold. Theorem 2.4. Let (X, (pi )i∈N , d) be a Fr´echet space. If f, f1 , f2 : R → X are almost automorphic functions, then we have: (i) f1 + f2 is almost automorphic; (ii) cf is almost automorphic for every scalar c ∈ R ; (iii) fa (t) := f (t + a), for all t ∈ R , is almost automorphic for each fixed a ∈ R; (iv) For all i ∈ N , we have sup{pi [f (t)]; t ∈ R} < +∞ and sup{pi [g(t)]; t ∈ R} < +∞ , where g is the function attached to f in Definition 2.3; (v) The range Rf = {f (t); t ∈ R} of f is relatively compact in the complete metric space (X, d); (vi) The function h defined by h(t) := f (−t), t ∈ R is almost automorphic; (vii) If f (t) = 0 for all t > a for some real number a, then f (t) = 0 for all t ∈ R ; (viii) If A: X → Y is continuous, where Y is another Fr´echet space, then A(f ): R → Y is almost automorphic too; (ix) Let hn : R → X, n ∈ N be a sequence of almost automorphic functions such that hn (t) → h(t) when n → +∞, uniformly in t ∈ R with respect to the metric d . Then h is almost automorphic. Proof.

(i) It is immediate from the property d (u + v, w + e) ≤ d (u, w) + d (v, e) ,

for all u, v, w, e ∈ X

and from Definition 2.3. (ii) It follows from the property d(c · u, c · v) ≤ (|c| + 1)d(u, v), ∀u, v ∈ X, (see Theorem 2.2,(v) ) and Definition 2.3.

for all c ∈ R

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(iii) The proof is immediate by Definition 2.3. (iv) Let us suppose that it exists i ∈ N such that sup{pi [f (t)]; t ∈ R} = +∞ . Then there exists a sequence of real numbers (rn )n such that pi [f (rn )] → +∞ , when n → +∞. Since f is almost automorphic, by Definition 2.3 for t = 0 , we can extract a subsequence (sn )n of (rn )n such that limn→+∞ d(g(0), f (sn )) = 0, where g(0) ∈ X . By the expression of d in the Remark after Definition 2.1, we immediately get pi [f (sn ) − g(0)] → 0 when n → ∞. 1 + pi [f (sn ) − g(0)] This obviously implies pi [f (sn ) − g(0)] → 0,

when n → ∞.

It follows that pi [f (sn )] ≤ pi [f (sn ) − g(0)] + pi [g(0)], from which, by passing to the limit with n → ∞ , we obtain the contradiction +∞ ≤ pi [g(0)]. The proof for g is similar, considering the relation limn→+∞ d(g(−sn ) , f (0)) = 0 , in Definition 2.3 for t = 0 . (v) Let (f (rn ))n be an arbitrary sequence in X . From Definition 2.3, there exists a subsequence (sn )n of (rn )n such that limn→+∞ d(g(0), f (sn )) = 0 , i.e. (f (sn ))n is a convergent subsequence of (f (rn ))n in the complete metric space (X, d) , which proves that Rf is relatively compact in (X, d). (vi) The proof is similar to the proof of Theorem 2.1.4 in [18, p. 13]. (vii) The proof is identical to the proof of Theorem 2.1.8 in [18, p. 17]. (viii) It is an immediate consequence of Definition 2.3 and the continuity of A . (ix) The proof is identical to the proof of Theorem 2.1.10 in [18, p. 18– 19], by using the fact that (X, d) is a complete metric space and the triangle inequality. Let us recall now that for f : R → X , the derivative of f at x ∈ R, denoted by f  (x) ∈ X , is defined by the relation lim d(f  (x),

h→0

f (x + h) − f (x) ) = 0. h

Regarding the derivative of almost automorphic functions, we present: Theorem 2.5. Let (X, (pi )i∈N , d) be a Fr´echet space. If f : R → X is almost automorphic and the derivative f  : R → X exists and is uniformly continuous on R (as a function between two metric spaces), then f  is almost automorphic.

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Proof. By [17, vol. II, p. 47, Theorem 1], for any a, b ∈ R , with a < b, the b Leibniz-Newton formula holds, i.e. f (b) − f (a) = a f  (t) dt . (Here the integral of f on [a, b] is defined as the (unique) limit in the metric d of the Riemann integral sums, when the norm of partitions of [a, b] tends to zero.) This immediately implies 

1/n

n

[f  (t + s) − f  (t)] ds = n[f (t + 1/n) − f (t)] − f  (t).

0

Let i be fixed and pi the seminorm on X . Let 0 < ε < From the above equality we easily obtain 



1/n

pi (n[f (t + 1/n) − f (t)] − f (t)) ≤ n

1 2

be given.

pi ([f  (t + s) − f  (t)]) ds.

0

Since f  is uniformly continuous on R , there exists δ > 0 such that for any |t1 − t2 | < δ , we have d[f  (t1 ), f  (t2 )] < 2εi . By the definition of d , it easily follows pi (f  (t1 ) − f  (t2 )) < ε, 1 + pi (f  (t1 ) − f  (t2 )) which implies pi (f  (t1 ) − f  (t2 )) < for all |t1 − t2 | < δ . But there exists n0 such that t ∈ R and n ≥ n0 , we obtain

1 n

ε < 2ε, 1−ε

< δ , for all n ≥ n0 , that is for arbitrary

pi (n[f (t + 1/n) − f (t)] − f  (t)) ≤ n



1/n

pi ([f  (t + s) − f  (t)]) ds < 2ε.

0

Consequently,

lim pi (n[f (t + 1/n) − f (t)] − f  (t)) = 0,

n→∞

uniformly with respect to t , for all i ∈ N . Now, we will prove that the sequence Fn (t) = n[f (t + 1/n) − f (t)], n ∈ N, converges uniformly to f  (t) in the topology of d . Since the family of seminorms is increasing, we get d[Fn (t), f  (t)] =

m ∞   1 pi (Fn (t) − f  (t)) 1 pi (Fn (t) − f  (t)) + 2i 1 + pi (Fn (t) − f  (t)) i=m+1 2i 1 + pi (Fn (t) − f  (t)) i=1

≤

m ∞  pm (Fn (t) − f  (t))  1 1 +  i 1 + pm (Fn (t) − f (t)) i=1 2 2i i=m+1

≤

1 pm (Fn (t) − f  (t)) 1 + m+1 ≤ pm (Fn (t) − f  (t)) + m+1 .  1 + pm (Fn (t) − f (t)) 2 2

207

´re ´kata Gal, Gal, and N’Gue Let ε > 0 be fixed and choose an m0 ∈ N such that same ε , there exists n0 ∈ N , such that pm0 (Fn (t) − f  (t)) ≤ ε,

1 2m+1

< ε . For the

for all n > n0 , t ∈ R.

Then, from the previous inequality we obtain d[Fn (t), f  (t)] < 2ε,

for all n > n0 , t ∈ R,

that is the sequence Fn (t), n ∈ N , converges uniformly on R to f  , in the metric d . By Theorem 2.4, (i), (ii), (iii) it follows that each Fn (t) is almost automorphic and by Theorem 2.4, (ix), we get the required conclusion. Regarding the integral of almost automorphic functions, we present: Theorem 2.6. Let f : R → X bealmost automorphic and consider the funct tion F : R → X defined by F (t) = 0 f (s) ds . Then F is almost automorphic if and only if its range RF = {F (t); t ∈ R} is relatively compact in the Fr´echet space (X, (pi )i∈N , d). Proof. We use the ideas in the proof of Theorem 2.4.4 in [18, pp. 27–29] adapted to Fr´echet spaces. According to Theorem 2.4,(v), it suffices to prove that if RF is relatively compact, then F is almost automorphic. Since f is almost automorphic and RF is relatively compact in X , given (sn )n a sequence of real numbers, there exist a subsequence (sn )n and α1 ∈ X such that lim d(f (t + sn ), g(t)) = lim d(f (t), g(t − sn )) = lim d(F (sn ), α1 )) = 0.

n→+∞

n→+∞

n→+∞

But by [17, vol. II, p. 47, Theorem 1], we get  t   F (t + sn ) = F (sn ) + f (r + sn ) dr. 0

We will prove that    t lim d F (t + sn ), α1 + g(r) dr = 0.

n→+∞

0

f (r+sn ),

Indeed, denoting gn (r) := it is obvious that limn→+∞ d(gn (r), g(r)) = 0 , pointwise in r ; which immediately implies that limn→+∞ pi [gn (r)−g(r)] = 0 , pointwise in r , for all i ∈ N . But      t  t  t   d F (t + sn ), α1 + g(r) dr = d F (sn ) + gn (r) dr, α1 + g(r) dr 0

0

≤

d(F (sn ), α1 )+d



0



t

gn (r) dr, 0

t

 g(r) dr .

0

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t Let pi be an arbitrary fixed seminorm. We easily get pi [ 0 gn (r) dr − t t g(r) dr] ≤ p [g (r) − g(r)] dr , where from above we have 0 0 i n limn→+∞ pi (gn (r), g(r)) = 0 . Also, we have pi [gn (r) − g(r)] ≤ pi [gn (r)] + pi [g(r)] = pi [f (r + sn )] + pi [g(r)], and passing to supremum with r on the right-hand side, by Theorem 2.4, (iv), it follows that pi [gn (r) − g(r)] ≤ Ci , for all r ∈ R , where Ci > 0 is independent of n and r . By the well-known Lebesgue’s dominated convergence theorem and by  t lim pi [gn (r) − g(r)] dr = 0, n→+∞

we get

0

 n→+∞



t

t

gn (r) dr −

lim pi 0

 g(r) dr = 0.

0

Since this happens for all i ∈ N , reasoning at the end of the proof  t exactlyas t of Theorem 2.5, we obtain limn→+∞ d( 0 gn (r) dr, 0 g(r) dr) = 0 , and consequently we get the required relation  t  lim d(F (t + sn ), α1 + g(r) dr) = 0. n→+∞

Now, denoting G(t) := α1 +

0

t 0

g(r) dr , from the relation

lim d(F (t + sn ), G(t)) = 0,

n→+∞

for all t ∈ R,

it follows that the range RG of G satisfies RG ⊂ RF , which implies RG ⊂ RF . Since RF is compact by hypothesis, it follows that RG also is compact (as a closed subset of compact set), that is RG is relatively compact. Let i ∈ N be fixed. Since RF and RG are relatively compact in X and the seminorm pi is continuous (by sequences) on X (viewed as a metric space), it easily follows that pi [RF ] and pi [RG ] are relatively compact subsets of the real axis R endowed with the usual metric topology. That means that both pi [RF ] and pi [RG ] are bounded, that is there exists Mi > 0 such that sup{pi (F (t)); t ∈ R} < Mi and sup{pi (G(t)); t ∈ R} < Mi . Also, the inclusion RG ⊂ RF immediately implies that sup{pi (G(t)); t ∈ R} ≤ sup{pi (F (t)); t ∈ R}. Since RG is relatively compact, there is a subsequence (sn )n of (sn )n and α2 ∈ X , such that limn→+∞ d(G(−sn ), α2 ) = 0 . Then reasoning exactly as in [18, p. 28] we get lim d(G(t − sn ), α2 + F (t)) = 0.

n→+∞

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209

It remains to prove that α2 = 0 . As in [18, p. 29], we get As (F )(t) = α2 + F (t), for all t ∈ R , where we use the notations s = (sn ) , As (F ) ≡ Ts [T−s (F )] and Ts is defined by Ts (F ) = H , with H given by the relation limn→+∞ d(F (t + sn ), H(t)) , for all t ∈ R . Denoting Ans = As [An−1 ] we get Ans (F )(t) = n · α2 + F (t) . Firstly, let s us prove sup{pi (Ans (F )(t)); t ∈ R} ≤ sup{pi (F (t)); t ∈ R}. For that, it suffices to prove the inequality sup{pi (As (F )(t)); t ∈ R} ≤ sup{pi (F (t)); t ∈ R}. In this sense we need the following result in (X, (pi )i∈N , d) : if limn→+∞ pi (xn − l) = 0 , then pi (l) ≤ pi (l − xn ) + pi (xn ) , wherefrom passing to limit, we get pi (l) ≤ limn→+∞ pi (xn ) . Now, applying this result for xn = G(t − sn ) and l = α2 + F (t) = As (t) , we obtain pi (As (F )(t)) ≤

lim pi (G(t − sn )) ≤ sup{pi (G(t)); t ∈ R}

n→+∞

≤ sup{pi (F (t)); t ∈ R},

for all t ∈ R.

Passing to supremum, with t ∈ R , we then obtain the desired inequality. Finally, from n · α2 = [Ans (F )(t) − F (t)] , we have n · pi (α2 ) = pi (n · α2 ) = pi (Ans (F )(t) − F (t)) ≤ pi (Ans (F )(t)) + pi (F (t)) ≤ 2 sup{pi (F (t)); t ∈ R}. Passing to limit with n → +∞ we obtain a contradiction if α2 = 0 , since we have sup{pi (F (t)); t ∈ R} < +∞ . The theorem is thus completely proved. In the study of almost automorphic solutions of nonlinear differential equations in Fr´echet spaces, the following concepts and results can be useful. We follow here the ideas in [18, Section 2.2]. In general, the results in the case of Banach spaces in [18] remain the same for the case of Fr´echet spaces (X, (pi )i∈N , d) . Definition 2.7. A continuous function f : R × X → X is said to be almost automorphic in t ∈ R for each x ∈ X , if for every sequence of real numbers (rn )n , there exists a subsequence (sn )n such that for all t ∈ R and x ∈ X , there exists g(t, x) with the property lim d(f (t + sn , x), g(t, x)) = lim d(g(t − sn , x), f (t, x)) = 0.

n→+∞

n→+∞

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´re ´kata Gal, Gal, and N’Gue The following simple properties hold.

Theorem 2.8. (i) If f1 , f2 : R × X → X are almost automorphic in t for each x ∈ X , then f1 + f2 and c · f1 , where c ∈ R are also almost automorphic in t for each x ∈ X . (ii) If f (t, x) is almost automorphic in t for each x ∈ X then for all i ∈ N and x ∈ X , we have sup{pi (f (t, x)); t ∈ R} < +∞ . Also, for the corresponding function g in Definition 2.3 we have sup{pi (g(t, x)); t ∈ R} < +∞. (iii) If f (t, x) is almost automorphic in t for each x ∈ X and if d(f (t, x), f (t, y)) ≤ Ld(x, y), for all x, y ∈ X and t ∈ R , where L is independent of x, y and t , then for the corresponding g in Definition 2.3 we have d(g(t, x), g(t, y)) ≤ Ld(x, y), for all x, y ∈ X and t ∈ R . (iv) Let f (t, x) be almost automorphic in t for each x ∈ X and ϕ: R → X be almost automorphic. (a) If d(f (t, x), f (t, y)) ≤ Ld(x, y), for all x, y ∈ X and t ∈ R , where L is independent of x, y and t then the function F : R → X defined by F (t) = f (t, ϕ(t)) is almost automorphic. (b) If for any j ∈ N there exist i ∈ N and Lj > 0 independent of t, x, y such that pj [f (t, x) − f (t, y)] ≤ Lj pi (x − y),

for all t ∈ R, x, y ∈ X,

then the function F : R → X defined by F (t) = f (t, ϕ(t)) is almost automorphic. Proof. The proof of (i) is similar to that of Theorem 2.4, (i), (ii). The proof of (ii) is similar to the proof of Theorem 2.4, (iv). The proofs of (iii) and (iv),(a), are similar to the proofs of Theorem 2.2.5 and 2.2.6 in [18, pp. 22–23], respectively, by using d(x, y) instead of x − y (here  ·  denotes the norm in Banach space). The proof of (iv),(b) is identical with the proof of Theorem 2.2.6 in [18, pp. 22–23], by replacing the norm in reasonings with each seminorm pj and taking into account the Remark after Definition 2.3. Similar to the case of Banach spaces (see e.g. [18, p. 37]), the concept in Definition 2.3 can be generalized as follows. Definition 2.9. Let (X, (pi )i∈N , d) be a Fr´echet space. A continuous function f : R+ → X is said to be asymptotically almost automorphic if it admits the decomposition f (t) = g(t) + h(t), t ∈ R+ , where g: R → X is almost automorphic and h: R+ → X is a continuous function with limt→+∞ |h(t)|X = 0 . Here g and h are called the principal and the corrective terms of f , respectively. Remark. Every almost automorphic function restricted to R+ is asymptotically almost automorphic, by taking h(t) = 0 , for all t ∈ R+ . Regarding this new concept, the following results similar to those in the case of Banach spaces hold.

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211

Theorem 2.10. Let f, f1 , f2 be asymptotically almost automorphic. Then we have: (i) f1 + f2 and c · f, c ∈ R are asymptotically almost automorphic; (ii) For any fixed a ∈ R+ , the function fa (t) := f (t + a) is asymptotically almost automorphic; (iii) For any i ∈ N, we have sup{pi (f (t)); t ∈ R+ } < +∞; (iv) Let (X, (pi )i∈N , d), (Y, (qj )j∈N , ρ) be two Fr´echet spaces and f : R+ → X be an almost automorphic function, f = g + h . Let φ: X → Y be continuous and assume there is a compact set B in (X, d) which contains the closures of {f (t); t ∈ R+ } and {g(t); t ∈ R+ } . Then φ ◦ f : R+ → Y is asymptotically almost automorphic; (v) The decomposition of an asymptotically almost automorphic function is unique. Proof. (i) Let c ∈ R , f1 = g1 + h1 , f2 = g2 + h2 , f = g + h , where the decompositions are those in Definition 2.9. We have f1 +f2 = [g1 +g2 ]+[h1 +h2 ] and c · f = c · g + c · h . By Theorem 2.4, (i), (ii), it follows that g1 + g2 , c · g are almost automorphic. Also, from the properties of pseudonorm |x|X = d(x, 0) derived from the properties of the metric d (see Theorem 2.2, (v), too), we get lim |h1 (t) + h2 (t)|X ≤ lim |h1 (t)|X + lim |h2 (t)|X = 0,

t→+∞

t→+∞

t→+∞

and limt→+∞ |c · h(t)|X ≤ [|c| + 1] limt→+∞ |h(t)|X = 0 . (ii) Let f = g + h be the decomposition in Definition 2.9. Then fa (t) = g(t + a) + h(t + a), where by Theorem 2.4, (iii), g(t + a) is almost automorphic and by Definition 2.9, we get limt→+∞ |h(t + a)|X = 0 . (iii) Let F = G + H . We have sup{pi (F (t)); t ∈ R+ } ≤ sup{pi (G(t)); t ∈ R+ } + sup{pi (H(t)); t ∈ R+ }. By Theorem 2.4, (iv), we have sup{pi (G(t)); t ∈ R+ } < +∞ . Also, denoting Qi (t) = pi (H(t)) , obviously Qi is continuous on [0, +∞) . By hypothesis, limt→+∞ d(H(t), 0) = 0 , which immediately implies that for any i ∈ N , we have lim pi ((H(t)) = lim Qi (t) = 0. t→+∞

t→+∞

Let ε > 0 be fixed. There exists δ > 0 , such that pi (H(t)) < ε , for all t > δ . From the continuity of Qi on [0, δ] , there exists M > 0 such that Qi (t) ≤ M , for all t ∈ [0, δ]. Thus 0 ≤ Qi (t) ≤ M + ε , for all t ∈ R+ , which implies the desired conclusion. (iv) Let f = g + h be the decomposition in Definition 2.9. By Theorem 2.4, (viii), φ ◦ g: R+ → Y is almost automorphic and also by assumption, φ ◦ f , φ ◦ g , are continuous on R+ . Denote Γ(t) = φ(f (t)) − φ(g(t)) . Let ε > 0 . By the uniform continuity of φ on the compact set B , there exists δ > 0 , such that ρ(φ(x), φ(y)) < ε , for all d(x, y) < δ , x, y ∈ B .

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On the other hand, since by assumption we have limt→+∞ d(h(t), 0) = 0 , there exists t0 (depending on δ ), such that |h(t)|X = d(f (t), g(t)) < δ , for all t > t0 . Then, for t > t0 we obtain |Γ(t)|Y = ρ[φ(f (t)), φ(g(t))] < ε, for all t > t0 ; which means limt→+∞ |Γ(t)|Y = 0 . (v) Let us suppose that f has two decompositions f = g1 + h1 = g2 + h2 . For all t ≥ 0, we get g1 (t) − g2 (t) = h2 (t) − h1 (t) , which implies lim |g1 (t) − g2 (t)|X ≤ lim |h2 (t)|X + lim |h1 (t)|X = 0.

t→+∞

t→+∞

t→+∞

Consider the sequence (n) . Since g1 − g2 is almost automorphic, there exists a subsequence (nk ) such that lim d[g1 (t + nk ) − g2 (t + nk ), F (t)] = 0

k→+∞

and lim d[F (t − nk ), g1 (t) − g2 (t)] = 0,

k→+∞

pointwise on R . But |F (t)|X = d[F (t), 0] ≤ d[F (t), g1 (t + nk ) − g2 (t + nk )] + |g1 (t + nk ) − g2 (t + nk )|X . Passing to limit with k → +∞ and taking into account the above relations, it follows that |F (t)|X = 0 , for all t ∈ R+ , which implies g1 (t) − g2 (t) = 0 , for all t . Therefore, h2 (t) − h1 (t) = 0 , for all t ∈ R+ , which proves the theorem. Concerning the differentiation and integration of asymptotically almost automorphic functions, we present two results. First we need the following concept suggested by Definition 7.1.7 in [18, pp. 103–104]. Definition 2.11. A Fr´echet space (X, (pi )i∈N , d) is said to be automorphicperfect, if the conditions sup{pi (f (t)); t ∈ R} < +∞ , for all i ∈ N and f  almost automorphic, implies f is almost automorphic (here f : R → X ). Theorem 2.12. Let (X, (pi )i∈N , d) be an automorphic-perfect Fr´echet space and f : R+ → X be an asymptotically almost automorphic functions, f = g +h , where g and h are those in Definition 2.9. Assume that g  (t) exists for every t ∈ R and f  (t) exists for every t ∈ R+ . If moreover f  is asymptotically almost automorphic, then g  and h will be its principal and corrective terms, respectively. Proof. We follow the ideas in the case of Banach spaces in [18, pp. 40–41], adapted to Fr´echet spaces. First note that h (t) exists for all t ∈ R+ . Since f  is asymptotically almost automorphic, let us write f  (t) = G(t) + H(t), t ∈ R+ , where G and H are the principal and corrective terms, respectively.

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213

We will show that G(t) = g  (t), t ∈ R and H(t) = h (t), t ∈ R+ . For a fixed η > 0 , let us consider the functions  t+η  t+η α(t) = G(s) ds, t ∈ R, and β(t) = H(s) ds, t ∈ R+ . t

t

Since β is continuous on R+ , for any i we have  t+η pi (β(t)) ≤ pi (H(t)) dt ≤ |η| sup{pi (H(t)); t ∈ Iη }, t

where Iη = [t + η, t] or Iη = [t, t + η] according to the sign of η . But limt→+∞ d[H(t), 0] = 0 , which implies that for all i we have limt→+∞ pi (H(t)) = 0 . From the previous inequality, we obtain lim pi (β(t)) ≤ lim pi (h(t)) = 0,

t→+∞

t→+∞

for all i ∈ N . Reasoning exactly as in the proof of Theorem 2.5, it follows that limt→+∞ d(β(t), 0) = limt→+∞ |β(t)|X = 0 . Since G is almost automorphic, the function α is continuous on R and by Theorem 2.4, (iv), for all i ∈ N , we have sup{pi [G(t)]; t ∈ R} < +∞ . Also, by [17, p. 47, Theorem 1] (which can also be seen as the Leibniz-Newton formula in a Fr´echet space), we get α (t) = G(t + η) − G(t) , which is almost automorphic; and since X is an automorphic-perfect Fr´echet space, it follows that α is almost automorphic. But obviously f (t + η) − f (t) = [g(t + η) − g(t)] + [h(t + η) − h(t)], and by the same result in [17, p. 47, Theorem 1], we obtain the equality f (t + η) − f (t) = α(t) + β(t), where t ∈ R+ and η is chosen so that t + η ≥ 0 . By the uniqueness of decomposition in Theorem 2.10, (v) of the asymptotically almost automorphic function f (t + η) − f (t) , we get α(t) = g(t + η) − g(t), t ∈ R, and β(t) = h(t + η) − h(t), t ∈ R+ . t Denoting A(t) = 0 G(s) ds , we have A (t) = G(t) and α(t) = A(t + η) − A(t) , which implies     A(t + η) − A(t) α(t) lim d G(t), = lim d G(t), η→0 η→0 η η   g(t + η) − g(t) = lim d G(t), = 0, η→0 η i.e. G(t) = g  (t), t ∈ R .

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´re ´kata Gal, Gal, and N’Gue t

Similarly, denoting B(t) := B(t + η) − B(t) , which implies

0

G(s) ds , we have B  (t) = H(t) and β(t) =

    B(t + η) − B(t) β(t) lim d H(t), = lim d H(t), η→0 η→0 η η   h(t + η) − h(t) = lim d H(t), = 0, η→0 η i.e. H(t) = h (t), t ∈ R+ . The theorem is proved. Theorem 2.13. Let (X, (pi )i∈N , d) be a Fr´echet space and f : R+ → X an asymptotically almost automorphic function, f = g+h , where thedecomposition t is that in Definition 2.9. Define F : R+ → X by F (t) := 0 f (s) ds and t G: R → X by G(t) := 0 g(s) ds . Assume that RG is relatively compact in  +∞ X and for all i ∈ N , we have 0 pi (h(s)) ds < +∞ . Then F is asymptotically almost automorphic, with principal term G(t)+  +∞  +∞ h(s) ds and corrective term H(t) = − t h(s) ds . 0  +∞ Proof. First let us prove that the integral 0 h(s) ds exists in X . Define n the sequence an := 0 h(s) ds ∈ X , n ∈ N . We will show that (an )n is a Cauchy sequence in X with respect to d , which by the completeness of X will show that there exists I ∈ X with limn→+∞ d[an , I] = 0 , where I will be  +∞ denoted by 0 h(s) ds . By a reasoning similar to that in the proof of Theorem 2.5, it can be proved that (an )n is a Cauchy sequence with respect to d , if and only if, for each i ∈ N , (an )n is a Cauchy sequence with respect to all the seminorms pi , i ∈ N. Let i ∈ N be fixed. We have 



n+p

pi (an+p − an ) = pi

h(s) ds n



n+p

≤

pi (h(s) ds. n

n Since by assumption, the sequence 0 pi (h(s)) ds , n ∈ N is convergent, it follows that it is a Cauchy sequence, which by the previous inequality implies that (an )n is a Cauchy sequence with respect to pi . Thus (an )n is Cauchy with respect to  +∞ d , which proves the existence of 0 h(s) ds in X .  +∞ Since by Theorem 2.6, G is almost automorphic, so is G + 0 h(s) ds .  +∞ h(s) ds , Let us now prove that the continuous function H(t) = − t t ∈ R+ has the property limt→∞ |H(t)|X = 0 . Indeed, by  +∞ pi (H(t)) ≤ pi (h(s)) ds, t

215

´re ´kata Gal, Gal, and N’Gue we get 

+∞

lim pi (H(t)) ≤ lim

t→+∞

for all i ∈ N,

pi (h(s)) ds = 0,

t→+∞

t

which by a reasoning similar to that in the proof of Theorem 2.5 implies lim |H(t)|X = lim d[H(t), 0] = 0.

t→∞

t→∞

The proof follows now from the relation 

+∞

F (t) = G(t) +

h(s) ds + H(t),

t ∈ R+ .

0

Theorem 2.14. If (X, (pi )i∈N , d) is a Fr´echet space, then the space of almost automorphic X -valued functions AA(X) is a Fr´echet space with respect to the countable family of seminorms given by qi (f ) = sup{pi (f (t)); t ∈ R}, i ∈ N , which generates the metric D on AA(X) defined by

D(f, g) :=

+∞  1 qi (f − g) . 2i 1 + qi (f − g) i=0

Proof. First note that the convergence of a sequence (fn )n ∈ AA(X) to f ∈ AA(X) with respect to D , is equivalent to the uniform convergence with respect to t ∈ R , in each seminorm pi , i ∈ N , which is also equivalent to the uniform convergence with respect to t ∈ R , in the metric d . Now, by Theorem 2.4, (i), (ii), (iv), AA(X) is a linear subspace of the space of all f : R → X , continuous, bounded (i.e. sup{pi (f (t)); t ∈ R} < +∞ , for all i ∈ N ) functions, denoted by Cb (R; X) . Since Cb (R; X) is complete and AA(X) is closed by Theorem 2.4, (ix), it follows that AA(X) is complete. 3. Semigroups of operators on Fr´ echet spaces First let us recall some known concepts and results in locally convex (Fr´echet) spaces. Theorem 3.1. (see e.g. [13, p. 128]) Let (X, (pi )i∈J1 ), (Y, (qj )j∈J2 ) be two locally convex spaces, where (pi )i and (qj )j are the corresponding families of seminorms. A linear operator A: X → Y is continuous on X if and only if for any j ∈ J2 , there exists i ∈ J1 and a constant Mj > 0 , such that qj (A(x)) ≤ Mj pi (x),

for all x ∈ X.

The space of all linear and continuous operators from X to Y is denoted by B(X, Y ) . If X = Y , then B(X, Y ) will be denoted by B(X) .

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Remark. For A ∈ B(X) , let us denote Ai,j = sup{pj (A(x)); x ∈ X, pi (x) ≤ 1} . Then it is well-known that A ∈ B(X) if and only if for every j there exists i (depending on j ) such that Ai,j < +∞ . Definition 3.2. (see e.g. [16], [21]) Let (X, (pj )j∈J ) be a locally convex space. A family T = (T (t))t≥0 with T (t) ∈ B(X) , for all t ≥ 0 is called C0 -semigroup on X if: (i) T (0) = I (the identity operator on X ); (ii) T (t + s) = T (t)T (s) ; for all t, s ≥ 0 (here the product means composition); (iii) For all j ∈ J , x ∈ X and t0 ∈ R+ we have limt→t0 pj [T (t)(x) − T (t0 )(x)] = 0 . An operator A is called the (infinitesimal) generator of a C0 -semigroup T = (T (t))t≥0 on X , if for every j ∈ J we have   T (t)(x) − x lim pj A(x) − = 0, t t→0+ for all x ∈ X , and the domain D(A) of A is the set of all x ∈ X such that the above limit exists. Remark. In a similar manner we can define a C0 -group on X by replacing R+ with R . Definition 3.3. (see e.g. [18, p. 99, Definition 7.1.1]) Let (X, (pj )j∈J ) be a complete, Hausdorff locally convex space. A family F = (Ai )i∈Γ , Ai ∈ B(X) , for all i, is said to be equicontinuous, if for any j1 ∈ J there exists j2 ∈ J such that pj1 [Ai x] ≤ pj2 (x), for all x ∈ X, i ∈ Γ. According to e.g. [18, pp. 100–103, Theorems 7.1.2, 7.1.3, 7.1.5, 7.1.6], we can state the following. Theorem 3.4. Let (X, (pj )j∈J ) be a complete, Hausdorff locally convex space and A ∈ B(X) such that the countable family {Ak ; k = 1, 2, . . . , } is
m k equicontinuous. For x ∈ X and t ≥ 0 , let us define Sm (t, x) = k=0 tk! Ak x. It follows: (i) For each x ∈ X and t ≥ 0 , the sequence Sm (t, x), m = 1, 2, . . . , is convergent in X , that is there exists an element in X denoted by etA x, such that lim pj (etA x − Sm (t, x)) = 0, for all j ∈ J m→+∞


+∞ k and we write etA x = k=0 tk! Ak x; (ii) For any fixed t ≥ 0 , we have etA ∈ B(X); (iii) e(t+s)A = etA esA , for all t, s ≥ 0 ;

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217

(iv) For every j ∈ J we have   etA x − x lim pj Ax − = 0, t t→0+ for all x ∈ X ; d tA (iv) dt [e x] = AetA x and the function etA x0 : R → X is the unique solution of the Cauchy Problem x (t) = Ax(t), t ∈ R , x(0) = x0 . Remark. Theorem 3.4, (i)–(iii), show that T (t) = etA , t ≥ 0 is a C0 semigroup of linear operators as in Definition 3.2. Also, let us prove the following. Theorem 3.5. Let (X, (pi )i∈N , d) be a Fr´echet space. (i) Let (T (t))t∈R be a C0 -group of bounded linear operators on X . Assume that the function x(t) = T (t)x0 : R → X is almost automorphic for some x0 ∈ X . Then there exists i ∈ N such that inf t∈R pi [x(t)] > 0 , or x(t) = 0, for all t ∈ R . (ii) Let x: R+ → X , f : R → X be two continuous functions, and T = (T (t))t∈R+ be a C0 -semigroup of bounded linear operators on X . Suppose that  t x(t) = T (t)x(0) + T (t − s)f (s) ds, t ∈ R+ . 0

Then for t given in R and b > a > 0 , a + t > 0 , we have  t x(t + b) = T (t + a)x(b − a) + T (t − s)f (s + b) ds. −a

Proof. (i) Let us suppose that for all k ∈ N we have inf t∈R pk [x(t)] = 0 . In this case, because T (t) ∈ B(X) , for arbitrary fixed t and i, let us denote by j the smallest index which corresponds to the continuity of T (t) in Definition 3.1, i.e. such that pi [T (t)u] ≤ Mi,t pj (u),

for all u ∈ X.

We have two subcases: a) j ≤ i; b) j > i. Subcase a). Since the family of seminorms (pi )i is increasing, from the above inequality we get pi [T (t)u] ≤ Mi,t pi (u),

for all u ∈ X.

Let (sn )n be a sequence of real numbers such that limn→+∞ pi (x(sn )) = 0 . Since by assumption x(t) is almost automorphic, we can extract a subsequence (sn )n of (sn )n such that for all t ∈ R , there exists y(t) ∈ X with the property lim d(y(t), x(t + sn )) = lim d(y(t − sn ), x(t)) = 0,

n→+∞

n→+∞

the above convergence on R being pointwise.

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But the convergence in d implies the convergence with respect to all seminorms on X , i.e. lim pk (y(t) − x(t + sn )) = lim pk (y(t − sn ) − x(t)) = 0,

n→+∞

n→+∞

for all k ∈ N,

i.e. for i too. Also, we can easily derive x(t + sn ) = T (t + sn )x0 = T (t)T (sn )x0 = T (t)x(sn ). From the above limits, we obtain pi (y(t)) ≤ pi (y(t) − x(t + sn )) + pi (x(t + sn )) ≤ pi (y(t) − x(t + sn )) + Mi,t pi [x(sn )], wherefrom passing to limit with n → +∞ it follows that pi (y(t)) = 0 . Subcase b). Let (sn )n be a sequence of real numbers such that limn→+∞ pj (x(sn )) = 0 . Since by hypothesis x(t) is almost automorphic, by Definition 2.3 we can extract a subsequence (sn )n of (sn )n such that for all t ∈ R , there exists y(t) ∈ X with the property lim d(y(t), x(t + sn )) = lim d(y(t − sn ), x(t)) = 0,

n→+∞

n→+∞

the above convergence on R being pointwise. It follows that lim pj (y(t) − x(t + sn )) = lim pj (y(t − sn ) − x(t)) = 0.

n→+∞

n→+∞

As in the above subcase a), we get pi (y(t)) ≤ pi (y(t) − x(t + sn )) + pi (x(t + sn )) ≤ pj (y(t) − x(t + sn )) + Mi,t pj (x(sn )), wherefrom passing to limit with n → +∞ it follows pi (y(t)) = 0 . Thus in both subcases we get pi (y(t)) = 0 , for all i ∈ N, t ∈ R ; which implies y(t) = 0 , for all t ∈ R , since the family of seminorms is sufficient. By lim d(y(t), x(t + sn )) = lim d(y(t − sn ), x(t)) = 0,

n→+∞

n→+∞

we immediately obtain x(t) = 0 , for all t ∈ R , which proves (i).

219

´re ´kata Gal, Gal, and N’Gue (ii) As in the proof of Theorem 2.4.7 in [18, pp. 32–33], we get





b−a

x(t + b) = T (t + a) x(b − a) −

T (b − a − s)f (s) ds 0



t+b

T (t + b − s)f (s) ds.

+ 0

Then from the above relation we get 



b−a

T (b − a − s)f (s) ds

x(t + b) + T (t + a) 0

 = T (t + a)x(b − a) +

t+b

T (t + b − s)f (s) ds. 0

Taking into account that T commutes with the integral (since it is linear and continuous operator), by the semigroup property T (u + v) = T (u)T (v) , for all u, v ∈ R+ and making under the integrals the substitution u = s − b , we obtain 

−a

x(t + b) + −b

 T (t − u)f (u + b) du = T (t + a)x(b − a) +

t

−b

T (t − u)f (u + b) du.

But because t > −a, we can write 



t

−b

T (t − u)f (u + b) du =

−a

−b

 T (t − u)f (u + b) du +

t

−a

T (t − u)f (u + b) du,

which yields to the required relation. The theorem is proved. In what follows, we are concerned with the behavior of asymptotically almost automorphic semigroups of linear operators T = T (t), t ∈ R+ on Fr´echet spaces. We will investigate some of their topological and asymptotic properties. Definition 3.6. Let (X, (pi )i∈N , d) be a Fr´echet space. A mapping u: R+ × X → X is called a dynamical system if: (i) u(0, x) = x, for all x ∈ X ; (ii) u(·, x): R+ → X is continuous for any t > 0 and right-continuous at t = 0 , for each x ∈ X . (The mapping u(·, x) is called the motion originating at x ∈ X ). (iii) u(t, ·): X → X is continuous for each t ≥ 0 ; (iv) u(t + s, x) = u(t, u(s, x)) , for all x ∈ X , t, s ∈ R+ .

220

´re ´kata Gal, Gal, and N’Gue We begin with

Theorem 3.7. Every C0 -semigroup (T (t))t≥0 on (X, (pi )i∈N , d) determines a dynamical system. Conversely, by defining u(t, x) = T (t)x, t ∈ R+ , x ∈ X , we obtain a semigroup (T (t))t≥0 of (possibly nonlinear) operators on X . It is a C0 -semigroup if each u(t, ·): X → X is linear. Proof.

It is similar to the proof of [18, Theorem 2.7.2, p. 43].

In the rest of this section, T = (T (t))t∈R+ will be a C0 -semigroup of linear operators on the Fr´echet space (X, (pi )i∈N , d) such that for fixed x0 ∈ X , the motion T (t)(x0 ): R+ → X is an asymptotically almost automorphic function with principal term f and corrective term g . Definition 3.8. A function ϕ: R → X is said to be a complete trajectory of T if it satisfies the functional equation ϕ(t) = T (t − a)(ϕ(a)) , for all a ∈ R , t ≥ a. We have the following. Theorem 3.9. for T .

The principal term f of T (t)x0 is a complete trajectory

Proof. It is similar to the proof of [18, Theorem 2.7.4, p. 44], by considering the limits there with respect to the metric d . Definition 3.10.

Let (X, (pi )i∈N , d) be a Fr´echet space.

The set ω + (x0 ) = {y ∈ X ; there exists 0 ≤ tn → +∞ , limn→+∞ d(T (t)x0 , y) = 0} is called the ω -limit set of T (t)x0 . ωf+ (x0 ) = {y ∈ X ; there exists 0 ≤ tn → +∞ , limn→+∞ d(f (tn ), y) = 0} is called the ω -limit set of f , the principal term of T (t)x0 . γ + (x0 ) = {T (t)x0 ; t ∈ R+} is called the trajectory of T (t)x0 . A set B ⊆ X is said to be invariant under the semigroup T = (T (t))t∈R+ , if T (t)y ∈ B , for all y ∈ B , t ∈ R+ . e ∈ X is called a rest-point for the semigroup T if T (t)e = e, for all t ≥ 0. Also, the following properties hold. Theorem 3.11. (i) ω + (x0 ) = ωf+ (x0 ) is not empty; ω + (x0 ) is invariant under T and is closed in X (with respect to d ); ω + (x0 ) is compact if γ + (x0 ) is relatively compact. Also, if x0 is a rest-point of the semigroup T , then ω + (x0 ) = {x0 } . (ii) If we denote γf (x0 ) = {f (t); t ∈ R} then γf (x0 ) is relatively compact (by Theorem 2.4,(v)); moreover it is invariant under the semigroup T . (iii) If we denote ν(t) = inf{d(T (t)x0 , y); y ∈ ω + (x0 )} , then limt→+∞ ν(t) = 0.

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221

Proof. (i) As in the proof of Theorem 2.7.6 in [18, p. 46], from the almost automorphy of f , let (tnk )k∈N be the sequence satisfying limk→+∞ d[f (tnk ), g(0)] = 0. We have d[T (tnk )(x0 ), f (tnk )] = d[f (tnk ) + g(tnk ), f (tnk )] = |g(tnk )|X , which implies limk→+∞ d[T (tnk )(x0 ), f (tnk )] = 0 . We then immediately obtain limk→+∞ d[T (tnk )(x0 ), g(0)] , which means that g(0) ∈ ω + (x0 ) , i.e. ω + (x0 ) is nonempty. The equality ω + (x0 ) = ωf+ (x0 ) is immediate from the relation lim d[T (t)x0 , f (t)] = 0,

t→+∞

which can be proved as above. To prove that ω + (x0 ) is invariant under T , we reason exactly as in the proof of Theorem 2.7.9 in [18, p. 47], taking into account that the continuity of a linear operator on a Fr´echet space X with respect to the family of seminorms (pi )i∈N , is equivalent to the continuity with respect to the metric d . Reasoning as in the proof of Theorem 2.7.10 in [18, pp. 47–48], we immediately obtain that ω + (x0 ) is closed in X . Applying reasonings similar to those in the proof of Theorem 2.7.11 in [18, p. 48], we get that ω + (x0 ) is compact if γ + (x0 ) is relatively compact. Also, reasoning as in the proof of Theorem 2.7.16 in [18, p. 49], (considering the limits there in the metric d ), we immediately obtain that ω + (x0 ) = {x0 } if x0 is a rest-point of the semigroup T . (ii) It is similar to the proof of [18, Theorem 2.7.12, p. 48]. (iii) It is similar to the proof of [18, Theorem 2.7.13, p. 48], by replacing a − b with d(a, b) and considering the limits with respect to the metric d . 4. Applications to differential equations in Fr´ echet spaces If (X, (pi )i∈N , d) is a Fr´echet space, then let us recall that for f : R → X , the derivative of f at x ∈ R , denoted by f  (x) ∈ X , is defined by the relation 

f (x + h) − f (x) lim d f (x), h→0 h 

 = 0.

It easily follows that this is equivalent to   f (x + h) − f (x) lim pi f  (x) − = 0, ∀i ∈ N. h→0 h For A ∈ B(X) , denote by (T (t))t≥0 the C0 -semigroup of operators on X generated by A (according to Definition 3.2).

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´re ´kata Gal, Gal, and N’Gue

First let us consider the following Abstract Cauchy Problem in the Fr´echet space (X, (pi )i∈N , d) : x (t) = Ax(t) + f (t),

t ∈ R+ , x(0) = x0 .

By similar reasonings with those in Banach spaces, see e.g. [14, p. 84, proof of Theorem], (i.e. replacing the norm with an arbitrary seminorm from (pi )i∈N in the limit process of derivative), we note that if x(t) is solution of the above problem, then d [T (t − s)x(s)] = T (t − s)f (s), ds which by integration from 0 to t (see the Leibniz-Newton formula in Fr´echet spaces in e.g. [17, p. 47, Theorem 1]) implies 

t

T (t − s)f (s) ds.

x(t) = T (t)x0 + 0

Such a continuous function will be called a mild solution of the above Cauchy Problem. The above considerations show that every classical solution of the Cauchy Problem is a mild solution too, but the converse is not in general true, since a mild solution is not necessarily differentiable (see for instance [14]). Now if we consider the following Abstract Cauchy Problem x (t) = Ax(t) + f (t, x(t)), t ∈ R+ , x(0) = x0 , by similar considerations, it can be proved that any solution of this Problem can be represented in as 

t

T (t − s)f (s, x(s)) ds;

x(t) = T (t)x0 + 0

and we call a mild solution of the Cauchy Problem, any continuous x(t) with such an integral representation. Obviously, because of the absence, in general, of its differentiability, a mild solution is not a classical solution of the Cauchy Problem. Remark.

Let us note that if x(t) is a mild solution of x (t) = Ax(t) + f (t),

t ∈ R+ ,

x(0) = x0 ,

then it has also the representation  x(t) = T (t − a)x(a) +

t

T (t − s)f (s) ds,

for all t ≥ a, a ∈ R.

a

Indeed, the proof is similar to that in Banach space, by replacing the norm in all the reasonings by an arbitrary seminorm in the family (pi )i∈N (see also

´re ´kata Gal, Gal, and N’Gue

223

the proofs (in the case of fuzzy-type spaces) of Theorem 3.9 in [10] and of Theorem 2.4 in [11], by replacing the relations with respect to the metric there, with the corresponding relations with respect to each seminorm in the family (pi )i∈N ). Similarly, if x(t) is a mild solution of x (t) = Ax(t) + f (t, x(t)),

t ∈ R+ ,

x(0) = x0 ,

then it has also the representation  x(t) = T (t − a)x(a) +

t

T (t − s)f (s, x(s)) ds,

for all t ≥ a,

a ∈ R.

a

This section is concerned with almost automorphic mild solutions of the above two differential equations. The first result is the following. Theorem 4.1. Let (X, (pi )i∈N , d) be a Fr´echet space and let us assume that A generates a C0 -semigroup (T (t))t≥0 on X which satisfies the condition: for any j ∈ N there exists i ∈ N, Kj > 0, ωj < 0 , such that T (t)i,j ≤ Kj eωj t ,

for all t ≥ 0.

(Here  · i,j is defined as in the Remark after Theorem 3.1.) If f ∈ AA(X) then the equation x (t) = Ax(t) + f (t),

t ∈ R+ ,

x(0) = x0 ,

has a unique almost automorphic mild solution. t Proof. Let x(t) = T (t−a)x(a)+ a T (t−s)f (s) ds, for all a ∈ R, t ≥ a, be a mild solution. Let us prove that it is almost automorphic. t t Consider u(t) := −∞ T (t − s)f (s) ds , defined as limr↓−∞ d[u(t), r T (t − s)f (s) ds] . First we will prove that this improper integral really exists in X . t Indeed, clearly for each r < t , the integral r T (t − s)f (s) ds exists, because f is continuous on R and because of Definition 3.2, (iii). From the Cauchy-Bolzano criterion, we need to prove that for any ε > 0 , there exists M ∈ R such that

    t

t

T (t − s)f (s) ds,

d a

b

T (t − s)f (s) ds = d b

for all

T (t − s)f (s) ds, 0X < ε, a

− ∞ < a < b < M < 0 < t.

224

´re ´kata Gal, Gal, and N’Gue

Reasoning as in the proof of Theorem 2.5, from the definition of d , we get the general inequality d(α, β) ≤ pm (α − β) +

1 , 2m+1

for all α, β ∈ X,

for all m ∈ N.

Then we obtain  d[

b

 T (t − s)f (s) ds, 0] ≤ pm [

a

b

T (t − s)f (s) ds] +

a

1 , 2m+1

for all m ∈ N.

1 Let j ∈ N be fixed such that 2j+1 < ε/2 . The condition on the C0 -semigroup in the assumption and the definition of  · i,j , immediately implies the condition

pj [T (t)x] ≤ T (t)i,j pi (x),

for all x ∈ X.

It follows that  pj [



b

T (t − s)f (s) ds] ≤

a



b

a

=

b

pj (T (t − s)f (s)) ds ≤

Kj eωj (t−s) pi [f (s)] ds a

Kj Mi ωj (t−b) Kj Mi −ωj b − eωj (t−a) ] ≤ < ε/2, [e e |ωj | |ωj |

for b < 0 sufficiently small. Here Mi = sup{pi (f (s)); s ∈ R} < +∞ according to Theorem 2.4, (iv), since f ∈ AA(X) . The Cauchy-Bolzano  t criterion is then satisfied; which means that the improper integral u(t) = −∞ T (t − s)f (s) ds exists in X , for any t ≥ 0 . Also, by the above reasonings, for any j ∈ N , there exist i ∈ N, Kj > 0, ωj < 0 , such that 



t

T (t − s)f (s) ds ≤

pj r

K j Mi , |ωj |

for all r < t

where Mi = sup{pi (f (s)); s ∈ R} < +∞ . Passing to limit with r → −∞, it follows pj (u(t)) ≤

K j Mi , |ωj |

for all t ≥ 0.

Now let (sn ) be an arbitrary sequence of real numbers. Since f ∈ AA(X) , there exists a subsequence (sn ) of (sn ) and g: R → X such that limn→+∞ d[f (t + sn ), g(t)] = 0 and limn→+∞ d[g(t − sn ), f (t)] = 0 , for each t ∈ R.

´re ´kata Gal, Gal, and N’Gue Let us consider  t+sn  u(t + sn ) = T (t + sn − s)f (s) ds = −∞



225

t

−∞

T (t − σ)f (σ + sn ) dσ

t

= −∞

T (t − σ)fn (σ)dσ,

where fn (σ) = f (σ + sn ) , n = 1, 2, . . .. Then clearly for any j ∈ N , there exist i ∈ N , Kj > 0 , ωj < 0 , such that pj (u(t + sn )) ≤

K j Mi , |ωj |

for all t ≥ 0, n ∈ N.

By the continuity of the semigroup, T (t − σ)fn (σ) → T (t − σ)g(σ) , as n → ∞ for each σ ∈ R fixed and any t ≥ σ , where the convergence is considered with respect to each seminorm pj (see Definition 3.2, (iii)). t If we let v(t) = −∞ T (t − s)g(s) ds , by Theorem 2.4, (iv), we obtain Ci = sup{pi (g(s)); s ∈ R} < +∞, and by similar reasonings with those for u(t) , it follows that v(t) exists in X as an improper integral, for all t ≥ 0 . For any j ∈ N , we get  pj [u(t + sn ) − v(t)] ≤

t

−∞

pj (T (t − s)fn (s) − T (t − s)g(s)) ds,

where there exist i ∈ N , Kj > 0 , ωj < 0 , such that pj (T (t − s)fn (s) − T (t − s)g(s)) ≤ Kj eωj (t−s) ,

for all − ∞ < s ≤ t.

t 0 t Then from the relation −∞ = −∞ + 0 and the previous inequality, it follows that the sequence of functions of variable s, given by pj (T (t − s)fn (s) − T (t − s)g(s)), is bounded on (−∞, t] . We have proved earlier that this sequence converges to 0 ; so by the Lebesgue’s dominated convergent theorem, it follows that lim pj [u(t + sn ) − v(t)] = 0,

n→+∞

for all t ∈ R.

Since j ∈ N is arbitrary, the above convergence holds with respect to the metric d too.

226

´re ´kata Gal, Gal, and N’Gue We can show in a similar way that for any j ∈ N , lim pj [v(t − sn ) − u(t)] = 0,

n→+∞

for all t ∈ R,

which implies the convergence in the metric d as well. In conclusion, u ∈ AA(X) . a a Now let u(a) = −∞ T (a − s)f (s) ds . So T (t − a)u(a) = −∞ T (t − s)f (s) ds . If t ≥ a, then 



t

T (t − s)f (s) ds = a



t

−∞

T (t − s)f (s) ds −

a

−∞

T (t − s)f (s) ds

= u(t) − T (t − a)u(a), t so that, u(t) = T (t − a)u(a) + a T (t − s)f (s) ds . If we fix x(a) = u(a), then x(t) = u(t) , that is x ∈ AA(X) . We finally prove the uniqueness of the almost aumorphic solution. Assume x and y are two such solutions and we let z(t) = x(t) − y(t) . Note that z(t) satisfies also the equation z(t) = T (t − s)z(s),

for all s ∈ R, t ≥ s.

For any j ∈ N , there exists i ∈ N, Kj > 0, ωj < 0 , such that pj (z(t)) = pj (T (t − s)z(s) ≤ Kj eωj (t−s) Mi ,

for all s ∈ R, t ≥ s,

where Mi = sup{pi (z(s)); s ∈ R} ≤ sup{pi (x(s)); s ∈ R}+sup{pi (y(s)); s ∈ R} < +∞, from Theorem 2.4, (iv) (since x(t) and y(t) are almost automorphic). Take a sequence of real numbers (sn ) such that sn → −∞. For any fixed t ∈ R , we then can find a subsequence (snk ) of (sn ) with snk < t for all k = 1, 2, 3 . . .. Using the fact that ωj < 0 , the inequality pj (z(t)) ≤ Kj Mi eωj (t−s) ,

for all s ∈ R, t ≥ s,

implies pj (z(t)) = 0,

for all t ∈ R, j ∈ N.

Since (pj )j∈N is a sufficient family of seminorms, it follows z(t) = 0 , for all t ∈ R , which shows the uniqueness of the solution and ends the proof. The second result is the following.

´re ´kata Gal, Gal, and N’Gue

227

Theorem 4.2. Let (X, (pi )i∈N , d) be a Fr´echet space and let us assume that A generates a C0 -semigroup (T (t))t≥0 on X which satisfies the condition: for any j ∈ N there exist Kj > 0, ωj < 0 , such that T (t)α(j),j ≤ Kj eωj t ,

for all t ≥ 0,

where α: N → N is a function satisfying the condition α(α(j)) = α(j), for all j . Also, assume that f (t, x) is almost automorphic in t for each x ∈ X , and that f : R × X → X satisfies the Lipschitz-type condition in x uniformly in t : pj [f (t, x) − f (t, y)] ≤ Lj pj (x − y),

for all x, y ∈ X, t ∈ R, j ∈ N

|ω |

where Lα(j) < Kjj , for all j ∈ N. Then the equation x (t) = Ax(t) + f (t, x(t)),

t ∈ R+ ,

has a unique almost automorphic mild solution. Proof. equation

Let x(t) be a mild solution. It is continuous and satisfies the integral 

t

x(t) = T (t − a)x(a) +

T (t − s)f (s, x(s)) ds,

for all a ∈ R, and t ≥ a.

a

t Consider a T (t − s)f (s, x(s) ds and the nonlinear operator G: AA(X) → AA(X) given by 

t

(Gφ)(t) := −∞

T (t − s)f (s, φ(s)) ds.

In view of Theorem 2.8,(iv),(b), and the proof of Theorem 4.1, if φ ∈ AA(X) , we deduce that Gφ ∈ AA(X) , so G is well defined. By Theorem 2.14, AA(X) is a Fr´echet space with respect to the countable family of seminorms given by qj (f ) = sup{pj (f (t)); t ∈ R} , j ∈ N . Now for φ1 , φ2 ∈ AA(X) , we have:  qj [Gφ1 − Gφ2 ] = sup pj [

−∞

t∈R

 ≤ sup t∈R

≤ sup t∈R

T (t − s)[f (s, φ1 (s)) − f (s, φ2 (s))] ds]

t

−∞



t

pj (T (t − s)[f (s, φ1 (s)) − f (s, φ2 (s))]) ds

t

−∞

T (t − s)α(j),j pα(j) [f (s, φ1 (s)) − f (s, φ2 (s))] ds

228

´re ´kata Gal, Gal, and N’Gue  ≤ Lα(j) qα(j) (φ1 − φ2 ) sup

−∞

t∈R

=

t

Kj eωj (t−s) ds

Lα(j) Kj qα(j) (φ1 − φ2 ). |ωj |

So qj [Gφ1 − Gφ2 ] ≤

Lα(j) Kj qα(j) (φ1 − φ2 ), |ωj |

which by e.g. [9, p. 92, Theorem 1] implies that there exists a unique u ∈ AA(X) , t such that Gu = u , that is u(t) = −∞ T (t − s)f (s, u(s)) ds. a If we let u(a) = −∞ T (a − s)f (s, u(s)) ds, then  T (t − a)u(a) =

a

−∞

T (t − s)f (s, u(s)) ds.

But for t ≥ a, 



t

T (t − s)f (s, u(s)) ds = a

t

−∞



−

T (t − s)f (s, u(s)) ds

a

−∞

T (t − s)f (s, u(s)) ds

= u(t) − T (t − a)u(a). t So u(t) = T (t − a)u(a) + −∞ T (t − s)f (s, u(s)) ds is a mild solution of the equation and u ∈ AA(X) . The proof is now complete. Remark. Theorems 4.1 and 4.2 represent extensions to Fr´echet spaces of those for Banach spaces in [20]. References [1] Bede, B. and S. G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems, 147(3) (2004), 385–403. [2] Bochner, S., Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. USA 52 (1964), 907–910. [3] Bochner, S., Uniform convergence of monotone sequences of functions, Proc. Nat. Acad. Sci. USA 47 (1961), 582–585. [4] Bochner, S., A new approach in almost-periodicity, Proc. Nat. Acad. Sci. USA 48 (1962), 2039–2043. [5] Bochner, S. and J. von Neumann, On compact solutions of operationaldifferential equations, I, Ann. Math. 36 (1935), 255–290.

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The University of Memphis Department of Mathematical Sciences Memphis, TN, 38152, USA [email protected] Department of Mathematics Morgan State University 1700 E. Cold Spring Lane Baltimore, MD 21251, USA [email protected]

Received November 22, 2004 Online publication October 20, 2005

Department of Mathematics University of Oradea Romania, 3700 Oradea, Romania [email protected]