Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 16, 1-8; http://www.math.u-szeged.hu/ejqtde/

ALMOST AUTOMORPHIC MILD SOLUTIONS TO SOME SEMILINEAR ABSTRACT DIFFERENTIAL EQUATIONS WITH

DEVIATED ARGUMENT IN FRÉCHET SPACES

CIPRIAN G. GAL

Abstract. In this paper we consider the semilinear differential equation with deviated argument in a Fréchet space x′(t) = Ax(t) + f(t, x(t), x[α(x(t), t)]), t ∈ ℝ, where A is the infinitesimal (bounded) generator of a C_{0}-semigroup satisfying some conditions of exponential stability. Under suitable conditions on the functions f and α we prove the existence and uniqueness of an almost automorphic mild solution to the equation.

1991 Mathematics Subject Classification: 43A60, 34G10.

Key words and phrases: almost automorphic, mild solutions, semigroups of linear operators, semilinear differential equations with deviated arguments, Fréchet spaces.

1. Introduction

In a very recent paper [2], the existence and uniqueness of almost automorphic mild solutions with values in Banach spaces, for the differential equation

(1.1)  x′(t) = Ax(t) + f(t, x(t), x[α(x(t), t)]), t ∈ ℝ,

is proved, where A is the infinitesimal (bounded) generator of a C_{0}-semigroup of operators (T (t))_{t≥0} on a Banach space, satisfying some exponential-type conditions of stability and f and α satisfy suitable conditions.

The goal of the present note is to prove the existence and uniqueness of almost automorphic mild solutions for the differential equation (1.1), but in the more general setting of Fréchet spaces. We now give the framework which is necessary to study (1.1) in locally convex spaces. We recall the following:

Definition 1.1 A linear space (X, +, ·) over ℝ is called Fréchet space if X is a metrizable, complete, locally convex space.

Remark 1.1. It is a known fact that the Fréchet spaces are characterized by the existence of a countable, sufficient and increasing family of seminorms (p_{i})_{i∈ℕ} (that is p_{i}(x) = 0, ∀i ∈ ℕ implies x = 0_{X} and p_{i}(x) ≤ p_{i+1}(x), ∀x ∈ X, i ∈ ℕ), which define the pseudonorm

(1.2)  |x|_{X} = ∑_{i=0}^{∞} \frac{1}{2^{i}} \frac{p_{i}(x)}{1 + p_{i}(x)}, x ∈ X.

EJQTDE, 2006 No. 16, p. 1 The metric d(x, y) = |x − y|_{X} is invariant with respect to translations and generates a complete (by sequences) topology equivalent to that of locally convex space. Moreover, 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 ∈ X. Also, we note that since \frac{p_{i}(x)}{1+p_{i}(x)} ≤ 1 and ∑_{i=0}^{∞} \frac{1}{2^{i}} = 1, it follows that |x|_{X} ≤ 1, ∀x ∈ X.

Furthermore, d has the following properties:

Theorem 1.2(see e.g. [3])

(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), ∀k ∈ ℕ, k ≥ 2;

(v) d(cx, cy) ≤ (|c| + 1)d(x, y), ∀c ∈ ℝ.

Remark 1.2. Everywhere in the rest of this paper, (X, (p_{i})_{i∈ℕ}, d) will be a Fréchet space with (p_{i})_{i∈ℕ} and d as in the Remark 1.1 following Definition 1.1.

The concept of almost automorphy is a generalization of periodicity. It has been introduced in the literature by S. Bochner in relation to some aspect of differential geometry. There are many important contributions that have been made to the theory of almost automorphic functions with values in Banach spaces. We refer the reader to the book [8] and the references therein. Moreover, in [3] , the authors develop the theory of almost automorphic functions with values in Fréchet spaces and apply it to abstract differential equations of the form (1.1) in the special case when the semilinear term f (in (1.1)) depends only on the first two arguments t and x (t). Our goal is to generalize (at least, partially) such results when f is as in (1.1) and α is an almost automorphic function that satisfies suitable conditions.

We start with the following Bochner-kind definition.

Definition 1.3 (see e.g. [3]) We say that a continuous function f : ℝ → X is almost automorphic, if every sequence of real numbers (r_{n})_{n} contains a subsequence (s_{n})_{n} such that for each t ∈ ℝ, there exists g(t) ∈ X with the property

(1.3)  lim_{n→+∞} d(g(t), f(t + s_{n})) = lim_{n→+∞} d(g(t − s_{n}), f(t)) = 0.

(The above convergence on ℝ is pointwise).

The set of all almost automorphic functions with values in X is denoted by AA(X).

2. Basic Result

First let us recall some known concepts and results in locally convex (Fréchet) spaces. Theorem 2.1 (see e.g. [ 4, p. 128]) Let (X, (p_{i})_{i∈J_{1}}), (Y, (q_{j})_{j∈J_{2}}) be two locally

convex spaces, where (p_{i})_{i} and (q_{j})_{j} are the corresponding families of semi-norms. A EJQTDE, 2006 No. 16, p. 2 linear operator A : X → Y is continuous on X if and only if for any j ∈ J_{2}, there exists i ∈ J_{1} and a constant M_{j} > 0, such that

(2.1)  q_{j}(A(x)) ≤ M_{j}p_{i}(x), ∀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).

Remark 2.1. For A ∈ B(X), let us denote

||A||_{i,j} = sup{p_{j}(A(x)); x ∈ X, p_{i}(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 ||A||_{i,j} < +∞.

Definition 2.2 (see e.g. [6],[9]) Let (X, (p_{j})_{j∈J}) be a locally convex space. A family T = (T (t))_{t≥0} with T (t) ∈ B(X), ∀t ≥ 0 is called C_{0}-semigroup on X if :

(i) T (0) = I (the identity operator on X) ;

(ii) T (t + s) = T (t)T (s), ∀t, s ≥ 0 (here the product means composition) ; (iii) For all j ∈ J, x ∈ X and t_{0} ∈ ℝ_{+}, we have lim_{t→t_{0}} p_{j}[T (t)(x) − T (t_{0})(x)] = 0. (iv) The operator A is called the (infinitesimal) (possibly unbounded) generator of

the C_{0}-semigroup T on X, if for every j ∈ J we have

(2.2)  lim_{t→0^{+}} p_{j}[A(x) − \frac{T (t)(x) − x}{t}] = 0,

for all x ∈ X.

Remark 2.2. In a similar manner, we can define a C_{0}-group on X by replacing ℝ_{+} with ℝ.

Definition 2.3 (see e.g. [8, p. 99, Definition 7.1.1]) Let (X, (p_{j})_{j∈J}) be a complete, Hausdorff locally convex space. A family F = (A_{i})_{i∈Γ}, A_{i} ∈ B(X), ∀i, is called equicontinuous, if for any j_{1} ∈ J there exists j_{2} ∈ J such that

(2.3)  p_{j_{1}}[A_{i}(x)] ≤ p_{j_{2}}(x), ∀x ∈ X, i ∈ Γ.

According to e.g. [8, p. 100-103, Theorems 7.1.2, 7.1.3, 7.1.5, 7.1.6], we can state the following:

Theorem 2.4 Let (X, (p_{j})_{j∈J}) be a complete, Hausdorff locally convex space and A ∈ B(X) such that the countable family {A^{k}; k = 1, 2, ..., } is equicontinuous. For x ∈ X and t ≥ 0, let us define S_{m}(t, x) = ∑_{k=0}^{m} \frac{t^{k}}{k!}A^{k}(x). It follows :

(i) For each x ∈ X and t ≥ 0, the sequence S_{m}(t, x), m ∈ℕ is convergent in X, that is, there exists an element in X denoted by e^{tA}(x), such that

(2.4)  lim_{m→+∞} p_{j}(e^{tA}(x) − S_{m}(t, x)) = 0, ∀j ∈ J

and we write e^{tA}(x) = ∑_{k=0}^{+∞} \frac{t^{k}}{k!}A^{k}(x); EJQTDE, 2006 No. 16, p. 3 (ii) For any fixed t ≥ 0, we have e^{tA} ∈ B(X); (iii) e^{(t+s)A} = e^{tA}e^{sA}, ∀t, s ≥ 0; (iv) For every j ∈ J, we have

(2.5)  lim_{t→0^{+}} p_{j}[A(x) − \frac{e^{tA}(x) − x}{t}] = 0,

for all x ∈ X;

(iv) \frac{d}{dt}[e^{(t−a)A}(x)] = A[e^{(t−a)A}(x)], for every t ≥ a, a ∈ ℝ and the function e^{(t−a)A}(x (a)) : ℝ → X is the unique solution of the problem x′(t) = A[x(t)], for every t ≥ a, a ∈ ℝ.

If (X, (p_{i})_{i∈ℕ}, d) is a Fréchet 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

(2.6)  lim_{h→0}d(f′(x), \frac{f(x + h) − f(x)}{h}) = 0.

It easily follows that this is equivalent to

lim_{h→0}p_{i}[f′(x) − \frac{f(x + h) − f(x)}{h}] = 0, ∀i ∈ ℕ.

For A ∈ B(X), denote by (T (t))_{t≥0} a C_{0}-semigroup of operators on X generated by A (according to Definition 2.2).

Now, let us consider the following abstract differential equation with deviated argument in the Fréchet space (X, (p_{i})_{i∈ℕ}, d),

(2.7)  x′(t) = Ax(t) + f(t, x(t), x[α(x(t), t)]), t ∈ ℝ.

It is easy to prove (see [3, proof of Theorem 3.5]) that if x(t) is a mild solution of the differential equation (2.7), then it has the form

x(t) = T (t − a)[x (a)] + ∫_{a}^{t} T (t − s)[f(s, x(s), x(α(x(s), s)))]ds,

for every a ∈ ℝ, every t ≥ a and we refer to any continuous x ∈ C (ℝ, X) satisfying the above relation as a mild solution of the above problem . Obviously, because of the absence, in general, of its differentiability, a mild solution is not a strong solution of the problem.

This section is concerned with existence and uniqueness of almost automorphic mild solutions of the differential equation (2.7) with deviated argument. The almost automorphic property of the deviation function α(s, t) with respect to t and a Lipschitz condition in s, uniformly with respect to t, permits us to generalize some of the results found in the literature for the semilinear ordinary differential equations with deviated arguments in Fréchet spaces.

The main result is the following:

EJQTDE, 2006 No. 16, p. 4

Theorem 2.5 Let (X, (p_{i})_{i∈ℕ}, d) be a Fréchet space and let us assume that A ∈ B (X) generates a C_{0}-semigroup (T (t))_{t≥0} on X which satisfies the condition : for any j ∈ ℕ there exist K_{j} > 0, ω_{j} < 0, such that

(2.8)  ||T (t)||_{β(j),j} ≤ K_{j}e^{ω_{j}t}, ∀t ≥ 0,

where β : ℕ → ℕ is an application satisfying the condition β[β(j)] = β(j), ∀j ∈ ℕ. Also, assume that f(t, x, y) is almost automorphic in t for each x, y ∈ X, and that f : ℝ × X × X → X satisfies the Lipschitz-type conditions uniformly in t of the form

(2.9) p_{j}[f(t, x, u)−f(t, y, v)] ≤ C_{j}[p_{j}(x−y)+p_{j}(u−v)], ∀x, y, u, v ∈ X, t ∈ ℝ, j ∈ ℕ,

α : X × ℝ → ℝ is almost automorphic in t ∈ ℝ for each x ∈ X and satisfies the conditions

(2.10)  p_{j}[α(u, t) − α(v, t)] ≤ S_{j}p_{j}(u − v), ∀u, v ∈ X, t ∈ ℝ,

and that ||A||_{k,k} < +∞, ∀k ∈ ℕ.

Denoting M_{j} = sup{p_{j}[f(t, x, y)]; t ∈ ℝ, x, y ∈ X} < +∞, for all j ∈ ℕ and

(2.11)  L_{j} = M_{j} + M_{β(j)} \frac{K_{j}}{|ω_{j}|}||A||_{β(j),β(j)},

for all j ∈ ℕ, under the conditions

(2.12)  C_{β_{j}} (S_{β(j)}L_{β(j)} + 2) \frac{K_{j}}{|ω_{j}|} < 1, ∀j ∈ ℕ,

then the equation

x′(t) = A[x(t)] + f[t, x(t), x(α(x(t), t))], t ∈ ℝ,

has a unique almost automorphic mild solution in the Fréchet space

AA_{(L_{j} )_{j}} (X ) = {Φ ∈ AA(X ); p_{j} [Φ(u) − Φ(v)] ≤ L_{j} |u − v|, ∀u, v ∈ ℝ, j ∈ ℕ}.

Proof. Let x(t) be a mild solution of (2.7). It is continuous and satisfies the integral equation

x(t) = T (t − a)[x(a)] + ∫_{a}^{t} T (t − s)[f(s, x(s), x(α(x(s), s)))]ds, ∀a ∈ ℝ, ∀t ≥ a.

Since by [3, Theorem 2.14], AA(X) is a Fréchet space with respect to the countable family of seminorms q_{j}(f) = sup{p_{j}(f(t)); t ∈ ℝ}, j ∈ ℕ, it is easy to show that AA_{(L_{j})_{j}}(X) is closed under the convergence with respect to the family of seminorms (q_{j})_{j}. It follows that AA_{(L_{j})_{j}}(X) is also a Fréchet space with respect to the same family of seminorms.

EJQTDE, 2006 No. 16, p. 5

Consider now ∫_{a}^{t} T (t − s)[f(s, x(s), x(α(x(s), s)))]ds and the nonlinear operator G : AA_{(L_{j})_{j}}(X) → AA(X) given by

(GΦ)(t) := ∫_{−∞}^{t} T (t − s)[f(s, Φ(s), Φ(α(Φ(s), s)))]ds.

First we show that GΦ ∈ AA_{(L_{j})_{j}}(X) for Φ ∈ AA_{(L_{j})_{j}}(X). Denote

F (s) = f(s, Φ(s), Φ[α(Φ(s), s)]), s ∈ ℝ,

with Φ ∈ AA_{(L_{j})j}(X).

Since Φ ∈ AA(X), by the hypothesis on α and by Theorem 2.8, (iv) (see also [3]), it follows that α(Φ(s), s) : ℝ → ℝ is in AA(ℝ). Since Φ also is continuous, by Theorem 2.4, (viii), we get that Φ[α(Φ(·), ·)] ∈ AA(X). Denoting γ(s) = Φ[α(Φ(s), s)], we have γ ∈ AA(X) and since f(t, u, v) is almost automorphic in t for each u and v, and Lipschitz in u and v, by similar reasonings with those in the proof of Theorem 2.8, (iv) in [3], we immediately get γ ∈ AA(X), F (s) = f (s, Φ(s), β(s)) ∈ AA(X), s ∈ ℝ. Because F (s) ∈ AA (X) , then it is bounded in norm so that M_{j} = sup_{s∈ℝ} p_{j}[F (s)] exist and are finite for all j ∈ ℕ. Moreover, as in the proof of Theorem 4.1 in [3] , we immediately obtain that GΦ ∈ AA(X). Thus, the map G is well defined.

As in e.g. [5] , we obtain

(GΦ)′(t) = F (t) + ∫_{−∞}^{t} T (t − s)A[F (s)]ds,

which implies

p_{j}[(GΦ)′(t)] ≤ p_{j}[F (t)] + ∫_{−∞}^{t} p_{j}(T (t − s)A[F (s)])ds ≤
M_{j} + ∫_{−∞}^{t} ||T (t − s)||_{β(j),j} · p_{β(j)}(A[F (s)])ds ≤
M_{j} + ∫_{−∞}^{t} K_{j}e^{ω_{j}(t−s)}||A||_{β(j),β(j)}p_{β(j)}(F (s))ds ≤

M_{j} + K_{j}M_{β(f)} \frac{||A||_{β(j),β(j)}}{|ω_{j}|} = L_{j}.

Then by the mean value theorem in locally convex spaces (see e.g [7, p. 15, Proposition 3]), we obtain

p_{j}[(GΦ)(u) − (GΦ)(v)] ≤ L_{j}|u − v|, ∀u, v ∈ ℝ,

that is GΦ ∈ AA_{(L_{j})_{j}}(X) for Φ ∈ AA_{(L_{j})_{j}}(X).

Finally, it remains to check that G is a contraction. Let Φ_{1}, Φ_{2} ∈ AA_{(L_{j})_{j}}(X). We calculate

EJQTDE, 2006 No. 16, p. 6

q_{j}[GΦ_{1} − GΦ_{2}]

= sup_{t∈ℝ}p_{j}[∫_{−∞}^{t} T (t − s)[f(s, Φ_{1}(s), Φ_{1}(α(Φ_{1}(s), s))) − f(s, Φ_{2}(s), Φ_{2}(α(Φ_{2}(s), s)))]ds] ≤

sup_{t∈ℝ}∫_{−∞}^{t} p_{j}(T (t − s)[f(s, Φ_{1}(s), Φ_{1}(α(Φ_{1}(s), s))) − f(s, Φ_{2}(s), Φ_{2}(α(Φ_{2}(s), s)))])ds ≤

sup_{t∈ℝ}∫_{−∞}^{t} ||T (t − s)||_{β(j),j}·

p_{β(j)}[f(s, Φ_{1}(s), Φ_{1}(α(Φ_{1}(s), s))) − f(s, Φ_{2}(s), Φ_{2}(α(Φ_{2}(s), s)))]ds ≤

sup_{t∈ℝ}∫_{−∞}^{t} ||T (t − s)||_{β(j),j}C_{β(j)}[q_{β(j)}(Φ_{1} − Φ_{2}) + p_{β(j)}(Φ_{1}(α(Φ_{1}(s), s)) − Φ_{1}(α(Φ_{2}(s), s)))+

p_{β(j)}(Φ_{1}(α(Φ_{2}(s), s)) − Φ_{2}(α(Φ_{2}(s), s)))ds ≤

sup_{t∈ℝ}∫_{−∞}^{t} ||T (t−s)||_{β(j),j}C_{β(j)}[2q_{β(j)}(Φ_{1} −Φ_{2})+L_{β(j)}p_{β(j)}(α(Φ_{1}(s), s)−α(Φ_{2}(s), s))]ds ≤

C_{β(j)} sup_{t∈ℝ}∫_{−∞}^{t} K_{j}e^{ω_{j}(t−s)}C_{β(j)}[2q_{β(j)}(Φ_{1} − Φ_{2}) + L_{β(j)}S_{β(j)}q_{β(j)}(Φ_{1} − Φ_{2})]ds ≤

C_{β(j)}[S_{β(j)}L_{β(j)} + 2]\frac{K_{j}}{|ω_{j}|}q_{β(j)}(Φ_{1} − Φ_{2}) < q_{β(j)}(Φ_{1} − Φ_{2}),

because of the assumption (2.12). It follows from [1, p. 92, Theorem 1] that there exists a unique u ∈ AA_{(L_{j})_{j}}(X) such that Gu = u, that is,

u(t) = ∫_{−∞}^{t} T (t − s)[f(s, u(s), u(α(u(s), s)))]ds.

Reasoning now exactly as in the proof of Theorem 4.2 in [3], the proof is complete. We omit the details.

In conclusion, in the present note, we have obtained an existence and uniqueness result concerning almost automorphic mild solutions for differential equations of the form (1.1) in locally convex (Fréchet) spaces and in Banach spaces [2], but only when the generator A is a bounded operator. The next natural step is to extend such results to the case when A is a (possibly) unbounded operator in Banach and Fréchet spaces. Such problems will be investigated in future articles.

EJQTDE, 2006 No. 16, p. 7

References

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[3] C. G. Gal, S. G. Gal and G. M. N’Guerekata, Almost automorphic functions in Fréchet spaces and applications to differential equations, Semigroup Forum, 75 (2005), No. 2, 23-48.

[4] D. Gaspar, Functional Analysis (in Romanian), Facla Press, Timisoara, 1981.

[5] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985.

[6] T. Komura, Semigroups of operators in locally convex spaces, J. Funct. Analysis, 2 (1968), 258-296.

[7] G. Marinescu, Treatise of Functional Analysis (in Romanian), vol. 2, Academic Press, Bucharest, 1972.

[8] G. M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001.

[9] S. Ouchi, Semigroups of operators in locally convex spaces, J. Math. Soc. Japan, 25 (1973), 265-276.

(Received January 26, 2006)

Morgan State University, Department of Mathematical Sciences, Baltimore, MD, 21212, U.S.A.

  E-mail address: cgal@morgan.edu

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