CONTINUOUS SELECTION THEOREMS MichaÃl Kisielewicz

CONTINUOUS SELECTION THEOREMS MichaÃl Kisielewicz

Faculty of Mathematics Computer Science and Econometrics

University of Zielona G´ora

Prof. Z. Szafrana 4a, 65–516 Zielona G´ora, Poland e-mail: M.Kisielewicz@wmie.uz.zgora.pl

Abstract

Continuous approximation selection theorems are given. Hence, in some special cases continuous versions of Fillipov’s selection theorem follow.

Keywords: set-valued mappings, continuous selection, Fillipov’s selection theorem,

2000 Mathematics Subject Classification: 34A60, 34C11.

The existence of continuous selections of multifunctions has been investi- gated by many authors in connection with a lot of problems of the theory of differential inclusions and their applications in the optimal control the- ory. Usually such type theorems are obtained as some generalizations of the famous Michael continuous selection theorem ([1, 2]). The present paper deals with some continuous version of the general implicit function theorem.

In the measurable case it is known as the Fillipov selection theorem. Let (X, ρ), (Y, | · |) and (Z, k · k) be Polish and Banach spaces, respectively and denote by Cl(Y ) a family of all nonempty closed subsets of Y . Let P(Y ) denote a family of all nonempty subsets of Y . Recall that a set-valued map- ping F : X → P(Y ) is said to be lower semicontinuous (l.s.c.) at x ∈ X if for every open set U in Y with F (x) ∩ U 6= ∅ there is a neighbourhood V _x of x such that F (x) ∩ U 6= ∅ for every x ∈ V _x .

Lemma 1. Let v : [0, T ] × X × Y → Z be continuous. Then a family

{v(t, ·, ·)} _{t∈[0,T ]} is equicontinuous on X × Y .

P roof. Suppose {v(t, ·, ·)} _{t∈[0,T ]} is not equicontinuous on X × Y . Then there are (x, u) ∈ X × Y and ε ₀ > 0 such that for every σ = _n ¹ there are t _n ∈ [0, T ] and (x _n , u _n ) ∈ X × Y such that |x _n − x| + |u _n − u| < ¹ _n and kv(t _n , x _n , u _n ) − v(t _n , x, u)k ≥ ε ₀ for n = 1, 2, . . . . Let (t _n

) ^∞ _k=1 be a subsequence of (t _n ) ^∞ _n=1 converging to t ∈ [0, T ]. For every k = 1, 2, . . . we also have |x _n

− x| + |u _n

− u| < _n ¹

k

. Therefore we have (t _n

, x _n

, u _n

) → (t, x, u) as k → ∞ and kv(t _n

, x _n

, u _n

) − v(t _n

, x, u)k ≥ ε ₀ for k = 1, 2, . . . . A contradiction, because v is continuous on [0, T ] × X × Y .

Corollary 1. Let λ : [0, T ] × X × Y → Z and u : [0, T ] × X → Z be continuous. Then for every (x, u) ∈ X × Y one has

(i) lim

(x,u)→(x,u) sup

t∈[0,T ]

kλ(t, x, u) − λ(t, x, u)k = 0

and (ii) lim

x→x sup

t∈[0,T ]

|u(t, x) − u(t, x)| = 0.

Lemma 2. Let λ : [0, T ] × X × Y → Z and u : [0, T ] × X → Z be continuous and let F : X → P(Y ) be lower semicontinuous and such that u(t, x) ∈ λ(t, x, F (x)) for (t, x) ∈ [0, T ] × X. Then for every lower semicontinuous function ε : X → (0, ∞) a set-valued mapping Φ : X → P(Y ) defined by

Φ(x) = F (x) ∩

½

u ∈ Y : sup

0≤t≤T kλ(t, x, u) − u(t, x)k < ε(x)

¾ (1)

is lower semicontinuous on X.

P roof. Let η > 0 be given and let (x, u) ∈ Graph(Φ). There is σ > 0 such

that sup _0≤t≤T kλ(t, x, u) − u(t, xk = ε(x) − σ. By (i) of Corollary 1, there

is δ > 0 such that sup _0≤t≤T kλ(t, x, u) − λ(t, x, uk < 1/3σ for every (x, u) ∈

X × Y satisfying max(ρ(x, x), |u − u|) < δ. By the lower semicontinuity of

F there is σ ₁ > 0 such that for every x ∈ X satisfying ρ(x, x) < σ ₁ there is

y _x ∈ F (x) such that |y _x − u| < min(η, 1/3σ, δ). By (ii) of Corollary 1 there

is σ ₂ > 0 such that sup _0≤t≤T ku(t, x) − u(t, xk < 1/3σ. Furthermore, by the

lower semicontinuity of ε, there is σ ₃ > 0 such that ε(x) > ε(x) − 1/3σ for

every x ∈ X such that ρ(x, x) < σ ₃ .

Now, for every x ∈ X satisfying ρ(x, x) < min(δ, σ ₁ , σ ₂ , σ ₃ ) one gets sup

0≤t≤T kλ(t, x, y _x ) − u(t, x)k ≤ sup

0≤t≤T kλ(t, x, y _x ) − λ(t, x, uk + sup

0≤t≤T kλ(t, x, u) − u(t, x)k + sup

0≤t≤T ku(t, x) − u(t, x)k

< 1/3σ + ε(x) − σ + 1/3σ < ε(x).

Then y _x ∈ Φ(x) and ky _x − uk < η. Therefore, (y _x + ηB ₀ ) ∩ Φ(x) 6= ∅ and for every x ∈ X satisfying ρ(x, x) < min(δ, σ ₁ , σ ₂ , σ ₃ ) one has (y _x +ηB ⁰ )

∩ Φ(x) 6= ∅.

Taking into account properties of measurable set-valued mappings ([1], Proposition II.3.3) in a similar way we can get the following result.

Lemma 3. Let λ : [0, T ] × X × Y → Z and u : [0, T ] × x → Z be measurable in t ∈ [0, T ] and continuous with respect to (x, y) ∈ X × Y and x ∈ X, respectively. Assume Y is a separable Banach space and F : [0, T ] × X → P(Y ) is measurable and such that F (t, ·) is l.s.c. for fixed t ∈ [0, T ]. If u(t, x) ∈ λ(t, x, F (t, x)) for (t, x) ∈ [0, T ] × X, then for every ε > 0 a set- valued mapping Φ : [0, T ] × X → P(Y ) defined by

Φ(t, x) = F (t, x) ∩ {u ∈ Y : kλ(t, x, u) − u(t, x)k < ε}

(2)

is measurable and such that Φ(t, ·) is l.s.c. for fixed t ∈ [0, T ].

Immediately from Michael’s continuous selection theorem ([1], Theorem II.4.1) and its generalization given in [3] we obtain the following continuous approximation selection theorems.

Theorem 4. Let λ : [0, T ]×X×Y → Z and u : [0, T ]×X → Z be continuous and let F : X → Cl(Y ) be l.s.c. and such that u(t, x) ∈ λ(t, x, F (x)) for (t, x) ∈ [0, T ] × X. Assume λ(t, x, ·) is affine for fixed (t, x) ∈ [0, T ] × X and F (x) are convex subsets of Y . Then for every ε > 0 there is a continuous function f _ε : X → Y such that f _ε (x) ∈ F (x) and sup _0≤t≤T kλ(t, x, f _ε (x)) − u(t, x)k ≤ ε for x ∈ X.

P roof. By virtue of Lemma 2 for every ε > 0 a set valued mapping Φ _ε :

X → P(Y ) defined by (1) with ε(x) ≡ ε is l.s.c. . Hence also its closure cl(Φ _ε )

is l.s.c. on X. Furthermore, by the convexity of F (x) and the properties of λ(t, x, ·), it follows that Φ _ε (x) and cl(Φ _ε )(x) are convex subsets of Y for every x ∈ X. Hence, by Michael’s continuous selection theorem there is a continuous selector f _ε for cl(Φ _ε ). It is clear that f _ε satisfies the conditions of our theorem.

Theorem 5. Let λ : [0, T ]×X×Y → Z and u : [0, T ]×X → Z be continuous and let F : [0, T ]×X → Cl(Y ) be l.s.c. and such that u(t, x) ∈ λ(t, x, F (t, x)) for (t, x) ∈ [0, T ] × X. Assume λ(t, x, ·) is affine and F (t, x) are convex subsets of Y for fixed (t, x) ∈ [0, T ] × X. Then for every ε > 0 there is a continuous function f _ε : [0, T ] × X → Y such that f _ε (t, x) ∈ F (t, x) and kλ(t, x, f _ε (t, x)) − u(t, x)k ≤ ε for (t, x) ∈ [0, T ] × X.

Theorem 6. Let λ : [0, T ]×X×Y → Z and u : [0, T ]×X → Z be measurable in t ∈ [0, T ] and continuous with respect to (x, y) ∈ X × Y and x ∈ X, respectively. Assume Y is a separable Banach space, λ(t, x, ·) is affine for fixed (t, x) ∈ [0, T ] × X and F : [0, T ] × X → Cl(Y ) is measurable and such that F (t, ·) is l.s.c. for fixed t ∈ [0, T ]. If u(t, x) ∈ λ(t, x, F (t, x)) and F (t, x) are convex subsets of Y for every (t, x) ∈ [0, T ]×X then for every ε > 0 there is a measurable function f _ε : [0, T ] × X → Y such that f _ε (t, ·) is continuous for fixed t ∈ [0, T ], f _ε (t, x) ∈ F (t, x) and kλ(t, x, f _ε (t, x)) − u(t, x)k < ε for every (t, x) ∈ [0, T ] × X.

Immediately from Theorem 4 and Theorem 6 we obtain the following Fillipov’s type continuous selection theorems.

Theorem 7. Assume λ, u and F satisfy conditions of Theorem 4 and let F be bounded. If furthermore λ(t, x, ·) is linear for every (t, x) ∈ [0, T ] × X and there is a number L > 0 such that kλ(t, x, x, u)k ≥ L|u| for every (t, x) ∈ [0, T ] × X and u ∈ Y , then there is a continuous selector f for F such that u(t, x) = λ(t, x, f (x)) for (t, x) ∈ [0, T ] × X.

Theorem 8. Assume λ, u, F and Y satisfy conditions of Theorem 6 and

let F be bounded. If furthermore λ(t, x, ·) is linear for every (t, x) ∈ [0, T ] ×

X and there is a number L > 0 such that kλ(t, x, u)k ≥ Lkuk for every

(t, x) ∈ [0, T ] × X and u ∈ Y , then there is a measurable selector for F such

that f (t, ·) is continuous for fixed t ∈ [0, T ] and u(t, x) = λ(t, x, f (t, x)) for

(t, x) ∈ [0, T ] × X.

References

[1] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad.

Publ., New York 1991.

[2] E. Michael, Continuous selections, Annales Math. 63 (1956), 361–382.

[3] L. Rybi´ nski, On Carath´eodory type selections, Fund. Math. 125 (1985), 187–193.

Received 8 May 2005