Lobachevskii Journal of Mathematics Vol. 5, 1999, 3–11 \begin{matrix}c \\ ◯ \end{matrix}B. Marzouki

B. Marzouki

ALMOST PERIODIC SOLUTIONS FOR SOME MULTIVALUED DIFFERENTIAL EQUATIONS IN BANACH SPACES

(submitted by A.M.Elizarov)

Abstract. It is known that in the frame of ordinary differential equations x′ = f(t, x) in Banach space with f almost-periodic in t uniformly for x and strongly dissipative with respect to x, the existence of a bounded solution is equivalent to the existence of an almost-periodic one ( see [5],[6]). In this note we want to generalize the result in the frame of a multivalued differential equation x′ ∈ F (t, x).

Preliminaries

Let (E, ∥.∥) be a real Banach space and d a metric from ∥.∥ defined by d(x, y) = ∥x − y∥, we denote by C(E) the set of nonempty compact subsets of E provided with the Hausdorff metric defined by

d_{H}(A, B) = Max(e(A, B), e(B, A)) where e(A, B) = sup_{x∈A} d(x, B).

It is known that (C(E), d_{H}) is a complete metric space. A multivalued map F : ℝ × E ⟶ C(E) is said to be :

1) Upper semicontinuous ( u.s.c for short ) at x_{0} ∈ ℝ × E if ∀ε > 0 ∃δ > 0 such that x ∈ B(x_{0}, δ) ⟹ F x ⊂ F x_{0} + B(0, ε), where B(a, r) means a ball of center a and radius r.

2) Bounding if it makes bounded subsets of ℝ × E into bounded subsets of E. iii) Continuous, if it is with respect to d_{H} . iv) strongly dissipative if, ∃c > 0, r ≥ 2 such that

∀(t, x, y) ∈ ℝ × E × E < F (t, x) − F (t, y), x − y >_{−}≤ −c∥x − y∥^{r} which means that ∀x_{1} ∈ F (t, x), ∀y_{1} ∈ F (t, y) < x_{1} − y_{1}, x − y >_{−}≤ −c∥x − y∥^{r} where

1991 Mathematical Subject Classification. 34A 34C.

Key words and phrases. Banach space, multivalued differential equation, boundness, strongly monotone, almost periodicity.

< x, y >_{−}= ∥y∥ lim_{h→0^{−}} \frac{∥y + hx∥ − ∥y∥}{h}

is the lower semi-inner product on E. We denote by ⇀ the convergence for the weak topology σ(E, E^{∗}) We recall that x : J ⊂ ℝ ⟶ E is said to be absolutely continuous ( a.c for short ) if given ε > 0 there is δ > 0 such that ∑ ∥x(t_{i})−x(s_{i})∥ ≤ ε whenever the finitely many intervals [s_{i}, t_{i}] ⊂ J do not overlap and ∑ | t_{i} − s_{i} |≤ δ. In particular every Lipschitzean map is a.c. When E is of finite dimension it is known that x is a.c if and only if x is differentiable almost everywhere ( a.e for short ) and x′ ∈ L^{1}(J, E), but if E is of infinite dimension and is not reflexive, then an a.c function need not be differentiable at any point ( see e.g Deimling [3] p138 ). So a solution of the Cauchy problem

x′ ∈ F (t, x) ; x(t_{0}) = x_{0} (∗)

defined in some interval I ( with t_{0} ∈ I ) is understood to be a continuous function on I a.c in every compact subset of I which is differentiable a.e and (*) is satisfied a.e on I.

Almost periodic multivalued map

Definition 1. A continuous multivalued map F : ℝ ⟶ C(E) is said to be almost periodic if ∀ε > 0, ∃l_{ε} such that ∀a ∈ ℝ ∃τ ∈ [a, a + l_{ε}] / d_{H}(F (t + τ), F (t)) ≤ ε ∀t ∈ ℝ. the real number τ is called an almost period of F associated to ε, in particular every periodic multivalued map is almost periodic.

For the property of almost periodic functions, we can see [4, 10, 11]. Characterization of almost periodic multivalued maps. In the practice we use the following Bochner’s characterization of almost periodicity.

Definition 2. A continuous multivalued map F : ℝ ⟶ C(E) is said to be almost periodic if and only if it’ s set of translates is relatively compact for the topology of uniform convergence in the space of continuous and bounded multivalued maps on ℝ, which is a complete metric space for the metric defined

by : D_{H}(F, G) = sup_{t∈ℝ} d_{H}(F (t), G(t)).

On the other word F is almost periodic if and only if from every real sequence t′_{n} one can extract a subsequence t_{n} such that F (t + t_{n}) converges uniformly on ℝ to a multivalued G(t) which is also continuous and almost periodic.

Almost periodic multivalued map with parameter. For the study of differential inclusions, the multivalued map is of the form F (t, x), so the definition of almost periodicity become as follow Definition 3. A continuous multivalued map F : ℝ × E ⟶ C(E) is said to be almost periodic in t uniformly for x if ∀ε > 0 ∀K compact of E ∃l_{ε,K} / ∀a ∈ R ∃τ ∈ [a, a + l_{ε,K}] / d_{H}(F (t + τ, x), F (t, x)) ≤ ε ∀t ∈ ℝ ∀x ∈ K. On the other words F is almost periodic in t for x if from every sequence t′_{n} one can extract a subsequence t_{n} such that F (t + t_{n}, x) converges uniformly on ℝ × K where K is any compact of E, the limit is also continuous and almost periodic in t for x.

Strongly monotone differential inclusions

Let (E, ∥.∥) be a real and reflexive Banach space, F : ℝ × E ⟶ CV (E) u.s.c and bounding, where CV(E) is the set of nonempty compact and convex subsets of E. We consider the differential inclusion

x′ ∈ F (t, x) (∗)

And we make the following hypothesis : H1) ∃c > 0, ∃r ≥ 2 such that

∀(t, x, y) ∈ ℝ × E × E < F (t, x) − F (t, y), x − y >_{−}≤ −c∥x − y∥^{r}

H2) ∃M > 0 / F (t, 0) ⊂ B(0, M) ∀t ∈ ℝ . Under the hypothesis H1) F is in particular dissipative, so ∀(a, x_{0}) ∈ ℝ × E the Cauchy problem:

\{ \begin{matrix}x′(t) & ∈ & F (t, x(t)) & (1) \\ x(a) & = & x_{0} (a ∈ ℝ) & (2) \end{matrix}

has a unique solution defined on [a, +∞[ ( see [8] ). Let x(t, a, x_{0}) and y(t, a, y_{0}) (x_{0} ≠ y_{0}) are two solutions of the inclusion (*) which satisfy x(a) = x_{0}, y(a) = y_{0} then we have the following lemma using hypothesis H1)

Lemma 1.

∥x(t, a, x_{0}) − y(t, a, y_{0})∥ ≤ e^{−c(t−a)}∥x_{0} − y_{0}∥ where t ≥ a and r = 2.

∥x(t, a, x_{0}) − y(t, a, y_{0})∥ ≤ \frac{∥x_{0} − y_{0}∥}{\{c(r − 2)∥x_{0} − y_{0}∥^{r−2}(t − a) + 1\}^{\frac{1}{r−2}}} where

t ≥ a and r > 2.

Now using hypothesis H2) we have

Lemma 2. Every solution x(.) of the Cauchy problem(1),(2) satisfies :

∥x(t)∥ ≤ e^{−c(t−a)}∥x(a)∥ + \frac{M}{c} (1 − e^{−c(t−a)}) where t ≥ a and r = 2.

∥x(t)∥ ≤ \frac{∥x(a)∥}{\{c(r − 2)∥x(a)∥^{r−2}(t − a) + 1\}^{\frac{1}{r−2}}} + (\frac{M}{c} )^{\frac{1}{r−1}} where

t ≥ a and r > 2.

Using these two lemma we have

Theorem 1. Under hypothesis H1) and H2) the differential inclusion (*) has a unique solution x defined and bounded on ℝ, furthermore x satisfies

sup_{t∈ℝ} ∥x(t)∥ ≤ (\frac{M}{c})^{\frac{1}{r−1}}

Proof : We consider a sequence of initial conditions

a_{n} = −n and x_{n} = \frac{nx_{0}}{1 + n} with ∥x_{0}∥ ≤ (\frac{M}{c})^{\frac{1}{r−1}}

then by lemma 1 we have : If r = 2 and n > m then

∥x(t, a_{n}, x_{n}) − x(t, a_{m}, x_{m})∥ ≤ e^{−c(t+m)}∥x(−m, a_{n}, x_{n}) − x_{m}∥ (t ≥ −m)

Using lemma 2 we deduce that

∥x(−m, a_{n}, x_{n})∥ ≤ e^{−c(−m+n)}∥x_{n}∥ + \frac{M}{c}

So

∥x(−m, a_{n}, x_{n}) − x_{m}∥ ≤ ∥x_{n}∥ + \frac{M}{c} + ∥x_{m}∥
≤ \frac{3M}{c}

Consequently

∥x(t, a_{n}, x_{n}) − x(t, a_{m}, x_{m})∥ ≤ \frac{3M}{c} e^{−c(t+m)}

Now let α > 0 such that n > m > α then

∥x(t, a_{n}, x_{n}) − x(t, a_{m}, x_{m})∥ ≤ \frac{3M}{c} e^{−c(m−α)}

So

sup_{t∈[−α,α]} ∥x(t, a_{n}, x_{n}) − x(t, a_{m}, x_{m})∥ →_{m} 0

If r > 2 then

∥x(t, a_{n}, x_{n}) − x(t, a_{m}, x_{m})∥ ≤ \frac{∥x(−m, a_{n}, x_{n}) − x_{m}∥}{\{c(r − 2)∥x(−m, a_{n}, x_{n}) − x_{m}∥^{r−2}(t + m) + 1\}^{\frac{1}{r−2}}}

≤ \frac{1}{\{c(r − 2)(t + m)\}^{\frac{1}{r−2}}}

So

sup_{t∈[−α,α]} ∥x(t, a_{n}, x_{n}) − x(t, a_{m}, x_{m})∥ ≤ \frac{1}{\{c(r − 2)(m − α)\}^{\frac{1}{r−2}}} →_{m} 0

Hence ∀r ≥ 2 the sequence x_{n}(t) = x(t, a_{n}, x_{n}) converges uniformly in every compact subset of ℝ to a continuous map which we denote by x(t, −∞, x_{0}) furthermore using lemma 2 we have

sup_{t∈ℝ} ∥x(t, −∞, x_{0})∥ ≤ (\frac{M}{c})^{\frac{1}{r−1}}

Now we have to show that the limit x(t, −∞, x_{0}) is a solution of the differential inclusion (*), put J = [−α, α] with n > α, we denote x(t, −∞, x_{0}) by x(t), then we have x′_{n}(t) ∈ F (t, x_{n}(t)) ∀t ∈ J, but F is bounding, so x′_{n} ∈ L^{∞}(J, E) ⊂ L^{2}(J, E), since the Banach space L^{2}(J, E) is reflexive, x′_{n} has a subsequence ( which we denote by the same ) such that x′_{n} ⇀ y ∈ L^{2}(J, E) for the weak topology σ(L^{2}(J, E), L^{2}(J, E^{∗})), hence ∀z ∈ L^{2}(J, E^{∗}) < z, x′_{n} >→_{n}< z, y > consequently

∫_{J} < z(t), x′_{n}(t) > dt →_{n} ∫_{J} < z(t), y(t) > dt

So

x_{n}(t) = x_{n}(0) + ∫_{0}^{t} x′_{n}(s)ds
= x_{n}(0) + ∫_{J} χ_{[0,t]}(s)x′_{n}(s)ds ⇀ x(0) + ∫_{J} χ_{[0,t]}(s)y(s)ds
= x(0) + ∫_{0}^{t} y(s)ds

But x_{n}(t) →_{n} x(t) hence x_{n}(t) ⇀ x(t) and consequently x(t) = x(0) + ∫_{0}^{t} y(s)ds which shows that x′(t) = y(t) a.e in J ( since L^{2}(J, E) ⊂ L^{1}(J, E)) so we deduce that x′_{n} ⇀ x′ in L^{2}(J, E) for the weak topology σ(L^{2}(J, E), L^{2}(J, E^{∗})). Now let ε > 0 arbitrary and put

A_{ε} = \{z ∈ L^{2}(J, E)/z(t) ∈ F (t, x(t)) + \overline{B}(0, ε) p.p\}

Then A_{ε} nonempty ( because x_{n}(t) →_{n} x(t) and F is u.s.c, so x′_{n} ∈ A_{ε} for n large ), A_{ε} is closed and convex, so A_{ε} is weakly closed, since x′_{n} ∈ A_{ε} and x′_{n} ⇀ x′ we deduce that x′(t) ∈ \overline{F (t, x(t))} = F (t, x(t)) a.e Finally the inclusion (*) has a solution defined and bounded on ℝ . Uniqueness of the bounded solution : Let x and y are two solutions defined and bounded on ℝ for the inclusion (*), and let a ∈ ℝ , then using lemma 1 we have for r = 2

∥x(t, a, x(a)) − y(t, a, y(a))∥ ≤ e^{−c(t−a)}∥x(a) − y(a)∥

So since ∥x(a) − y(a)∥ is bounded, and when a → −∞ we deduce that x(t) = y(t) ∀t ∈ ℝ. we have the same for r > 2 since

∥x(t, a, x(a)) − y(t, a, y(a))∥ ≤ \frac{∥x(a) − y(a)∥}{\{c(r − 2)∥x(a) − y(a)∥^{r−2}(t − a) + 1\}^{\frac{1}{r−2}}}

Almost periodic solution

Theorem 2. Let E be a real and reflexive Banach space, and F : ℝ × E ⟶ CV (E) bounding and strongly dissipative, we suppose furthermore that F is continuous and almost periodic in t uniformly for x, then the inclusion (*) has a unique almost periodic solution.

Proof : We know that the inclusion (*) has a unique bounded solution on ℝ, because F is almost periodic, hence ∃M > 0 such that F (t, 0) ⊂ B(0, M) and the conclusion becomes from the previous result. We note u(.) this solution and we show that it is almost periodic, for to do this, we use the Bochner’s characterization of almost periodicity, given any real sequence t′_{n} we have to extract a subsequence t_{n} such that u(t + t_{n}) converges uniformly on ℝ. F is almost periodic in t uniformly for x, so from t′_{n} we can extract a subsequence t_{n} such that F (t + t_{n}, x) converges uniformly on ℝ × K ( where K is any compact subset of E ) to some multivalued map G(t, x) which is continuous and almost periodic in t uniformly for x, put u_{n}(t) = u(t + t_{n}) and F_{n}(t, x) = F (t + t_{n}, x), now let us show that G is strongly dissipative and for every (t, x), G(t, x) is convex, for this we use the following lemma which becomes from the definition of the Hausdorff metric d_{H}.

Lemma 3. Let A, B ∈ C(E), then ∀x ∈ A ∃y ∈ B / ∥x − y∥ ≤ 2d_{H}(A, B).

G(t,x) is convex for every (t,x) : Indeed, let x_{0}, x_{1} ∈ G(t, x) and λ ∈ [0, 1], from lemma 3 ∃y_{0}^{n} ∈ F_{n}(t, x) and y_{1}^{n} ∈ F_{n}(t, x) such that

∥x_{0} − y_{0}^{n}∥ ≤ 2d_{H}(F_{n}(t, x), G(t, x)) et ∥x_{1} − y_{1}^{n}∥ ≤ 2d_{H}(F_{n}(t, x), G(t, x))

So

d(λx_{0} + (1 − λ)x_{1}, G(t, x)) ≤ ∥λx_{0} + (1 − λ)x_{1} − λy_{0}^{n} − (1 − λ)y_{1}^{n}∥ +
+ d(λy_{0}^{n} + (1 − λ)y_{1}^{n}, G(t, x))
≤ 2d_{H}(F_{n}(t, x), G(t, x)) + d_{H}(F_{n}(t, x), G(t, x))
= 3d_{H}(F_{n}(t, x), G(t, x))

Consequently when n → +∞ we deduce that λx_{0} + (1 − λ)x_{1} ∈ \overline{G(t, x)} = G(t, x). G is strongly dissipative : Indeed let x_{1} ∈ G(t, x) and y_{1} ∈ G(t, y) then from lemma 3 ∃α_{n} ∈ F_{n}(t, x), ∃β_{n} ∈ F_{n}(t, y) such that

∥x_{1} − α_{n}∥ ≤ 2d_{H}(F_{n}(t, x), G(t, x)) and ∥y_{1} − β_{n}∥ ≤ 2d_{H}(F_{n}(t, y), G(t, y))

So

< x_{1} − y_{1}, x − y >_{−} = < x_{1} − α_{n} + α_{n} − β_{n} + β_{n} − y_{1}, x − y >_{−}
≤ < α_{n} − β_{n}, x − y >_{−} + < x_{1} − α_{n}, x − y >_{+} +
+ < β_{n} − y_{1}, x − y >_{+}
≤ −c∥x − y∥^{r} + 2∥x − y∥[d_{H}(F_{n}(t, x), G(t, x)) +
+ d_{H}(F_{n}(t, y), G(t, y))]

Consequently when n → +∞ we deduce that

< x_{1} − y_{1}, x − y >_{−}≤ −c∥x − y∥^{r}.

On the other hand G is bounding, so the limit equation

x′ ∈ G(t, x)

has a unique bounded solution defined on ℝ which we denote by u_{0}(.), now we show that u(t + t_{n}) →_{n} u_{0}(t) uniformly on ℝ, we begin by showing that the convergence is pointwise. Put Φ_{n}(t) = ∥u_{n}(t) − u_{0}(t)∥ where u_{n}(t) = u(t + t_{n}), then

Φ_{n}(t)D^{−}Φ_{n}(t) =< u′_{n}(t) − u′_{0}(t), u_{n}(t) − u_{0}(t) >_{−}

On the other hand u′_{0}(t) ∈ G(t, u_{0}(t)) so from lemma 3 ∃v_{n} ∈ F_{n}(t, u_{0}(t)) such that

∥v_{n} − u′_{0}(t)∥ ≤ 2d_{H}(F_{n}(t, u_{0}(t)), G(t, u_{0}(t)))

So

Φ_{n}(t)D^{−}Φ_{n}(t) = < u′_{n}(t) − v_{n} + v_{n} − u′_{0}(t), u_{n}(t) − u_{0}(t) >_{−}
≤ < u′_{n}(t) − v_{n}, u_{n}(t) − u_{0}(t) >_{−} + < v_{n} − u′_{0}(t), u_{n}(t) − u_{0}(t) >_{+}
≤ −∥u_{n}(t) − u_{0}(t)∥^{r} + 2∥u_{n}(t) − u_{0}(t)∥d_{H}(F_{n}(t, u_{0}(t)), G(t, u_{0}(t)))

Consequently

D^{−}Φ_{n}(t) ≤ −c(Φ_{n}(t))^{r−1} + θ_{n}(t) (∗∗)

Where

θ_{n}(t) = 2d_{H}(F_{n}(t, u_{0}(t)), G(t, u_{0}(t)))

Hence if r=2 then by integrating (**) we deduce that

∥u_{n}(t) − u_{0}(t)∥ ≤ e^{−c(t−t_{0})}∥u_{n}(t_{0}) − u_{0}(t_{0})∥ + ∫_{t_{0}}^{t} e^{−c(t−s)}θ_{n}(s)ds
≤ ke^{−c(t−t_{0})} + ∫_{t_{0}}^{t} e^{−c(t−s)}θ_{n}(s)ds (∗ ∗ ∗)

But θ_{n}(s) →_{n} 0 pointwise on ℝ and θ_{n} is bounded on [t_{0}, t] so by the Lebesgue’s convergence theorem

∫_{t_{0}}^{t} e^{−c(t−s)}θ_{n}(s)ds →_{n} 0

So

lim sup_{n→+∞} ∥u_{n}(t) − u_{0}(t)∥ ≤ ke^{−c(t−t_{0})}

Now when t_{0} → −∞ we deduce that u_{n}(t) →_{n} u_{0}(t) pointwise on E. If r > 2, then we put ψ_{n}(t) = (Φ_{n}(t))^{r−1} so Φ_{n} is positive and bounded on ℝ, and there exists two constants strictly positive l and L such that l ≤ Φ_{n}(t) ≤ L so α := l^{r−2} ≤ (Φ_{n}(t))^{r−2} ≤ L^{r−2} := β hence

D^{−}ψ_{n}(t) = (r − 1)(Φ_{n}(t))^{r−2}D^{−}Φ_{n}(t) ≤ −c(r − 1)αψ_{n}(t) + β(r − 1)θ_{n}(t)

the constant γ = c(r−1)α is strictly positive, so we deduce that ψ_{n}(t) →_{n} 0 and consequently Φ_{n}(t) →_{n} 0 pointwise on ℝ, which shows that u_{n}(t) →_{n} u_{0}(t) pointwise on E. Finally we have showed that from every sequence t′_{n} we can extract a subsequence t_{n} such that u_{n}(t) = u(t + t_{n}) converges pointwise to u_{0}(t) on E,( notice that the solution u_{0}(.) depends on the sequence t′_{n} ), so K = \overline{u(ℝ)} is compact, and hence \overline{u_{0}(ℝ)} is compact, consequently F_{n}(t, u_{0}(t)) converges uniformly on ℝ × K to G(t, u_{0}(t)) and so θ_{n} converges to 0 uniformly on ℝ, Now let t_{0} → −∞ in the inequality (***) then

∥u_{n}(t) − u_{0}(t)∥ ≤ ∫_{−∞}^{t} e^{−c(t−s)}θ_{n}(s)ds

And consequently u(t + t_{n}) converges to u_{0}(t) uniformly on ℝ, which shows that u is almost periodic.

Remark : Using lemma 3 we can show that the almost periods of the solution u are the same as for the multivalued map F.

1. Consequence of theorem 1

Theorem 3. If F is periodic of period T > 0, and if we assume the hypothesis H1 then the bounded solution from theorem 1 is periodic of period T.

Proof : F is periodic, so the hypothesis H2 is satisfied, on the other hand let x(t) be the unique bounded solution of the differential inclusion (*), then y(t) = x(t + T) is also a bounded solution of the same inclusion, so by uniqueness we deduce that x(t) = x(t + T).

ALMOST PERIODIC SOLUTIONS 11

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Université Mohammed I Faculté des Sciences Département de Mathématiques, Oujda. Maroc. E-mail address: marzoukisciences.univ-oujda.ac.ma