On Convexity Properties of Set Valued Maps
by
Serkan Duzce
Anadolu University, Eskisehir
Coauthors: Kh.G. Guseinov, Orhan Ozer

This Research was supported by the Anadolu University Research Foundation

In this article the existence of the convex continuation of convex compact set valued maps is considered. The problem was studied in [1] for a special case. Conditions are obtained, based on the notion of the derivative of set valued maps, which guarantee the existence of convex continuation. Note that the results obtained can be applied in the investigation of some problems in differential inclusion theory (see, [2]).

Let t\leadsto W(t), t in [ t₀, t₁ ], be a set valued map, W(t₀) =/= \emptyset, W(t₁) =/= \emptyset, gr  W(·)={ (t, x) in [t₀, t₁] ×Rⁿ : x in W(t) }. For (t, x) in [t₀, t₁ ] ×Rⁿ define

D+W ( t, w) = { v in Rn : liminf \delta --> 0+     d( w+\deltav, W(t+\delta)) \delta =0 }

D-W ( t, w) = { v in Rn : liminf \delta --> 0+     d( w- \deltav, W(t- \delta)) \delta =0 }

The set D⁺W(t, w) (D^-W(t, w)) is said to be an upper right hand side (upper left hand side) derivative set of the set valued map t\leadsto W(t) calculated at the point (t, w). Note that the upper right hand side (left hand side) derivative set is closed and it has nearly connection with upper Bouligand contingent cone, used in many problems of the set valued and nonsmooth analysis (see, e.g. [ 3]).

Let us denote W(t₀)=W₀, W(t₁)=W₁. For given \alpha > 0, x₀,  x₁ in Rⁿ and W₀, W₁ subset Rⁿ, we define the set valued maps K_\alpha^L(x₀) | (·):[t₀-\alpha, t₁+\alpha] \leadsto Rⁿ and K_\alpha^R(x₀) | (·):[t₀-\alpha, t₁+\alpha] \leadsto Rⁿ, setting

K\alphaL(x0) | (t) = (1-  t-t0+\alpha \alpha )·x0+  t-t0+\alpha \alpha ·W0

K\alphaR(x1) | (t) = (1-  t1+\alpha-t \alpha ) ·x1+  t1+\alpha-t \alpha ·W1

Definition 1. Let \alpha > 0, t\leadsto W^*(t), t in [ t₀- \alpha, t₁+\alpha] be a set valued map, W^*(t₀-\alpha) =/= \emptyset, W^*(t₁+\alpha) =/= \emptyset, W^*(t)=W(t) for every t in [ t₀, t₁ ] and gr  W^*(·) = { (t, x) in [t₀-\alpha, t₁+\alpha] ×Rⁿ : x in W^*(t) } be convex set. Then the set valued map t\leadsto W^*(t), t in [ t₀-\alpha, t₁+\alpha] is said to be a convex continuation of the set valued map t\leadsto W(t), t in [ t₀, t₁ ].

Theorem 1. Let gr  W(·) subset [ t₀, t₁ ] ×Rⁿ be a convex, compact set, \alpha > 0, x₀ in Rⁿ, x₁ in Rⁿ. Assume that D⁺W(t₀, w₀) subset D⁺K_\alpha^L (x₀) | (t₀, w₀) for every w₀ in W₀, D^-W(t₁, w₁) subset D^-K_\alpha^R (x₁) | (t₁, w₁) for every w₁ in W₁ and

W*(t) = ì ï í ï î K\alphaL(x0) | (t), t in [ t0-\alpha, t0) W(t), t in [t0, t1] K\alphaR(x1) | (t), t in (t1, t1+\alpha]

for t in [ t₀-\alpha, t₁+\alpha]. Then the set valued map t\leadsto W^* (t), t in [ t₀-\alpha, t₁+\alpha], is a convex continuation of the set valued map t\leadsto W(t), t in [ t₀, t₁ ].

Theorem 2. Suppoze that for every fixed \alpha > 0 and x in Rⁿ there exists w in W₀ such that D⁺W(t₀, w) \not subset or equal D⁺K_\alpha^L (x) | (t₀, w) or there exists w in W₁ such that D^-W(t₁, w) \not subset or equal D^-K_\alpha^R (x) | (t₁, w). Then the set valued map t\leadsto W(t), t in [t₀, t₁], has no convex continuation.

References

[1] Guseinov, Kh., Ozer, O. and Duzce, S. A., On the Convex Continuation of the Convex Compact Multivalued Map, XII National Mathematics Symposium. 6-10 September, 1999. Malatya. Turkey. Abstracts, 27-30.

[2] Guseinov, Kh. G. and Ushakov, V. N. The Construction of Differential Inclusions with Prescribed Properties, Differ. Uravn. 2000. Vol.36. No.4, 438-445. Engl. transl. in: Diff. Equat., 2000. Vol.36, No.4, 488-496.

[3] Aubin, J-P. and Frankowska, H. Set Valued Analysis, Birkhauser, Boston, 1990.

Date received: June 8, 2001