Journal of Applied Analysis Vol. 15, No. 1 (2009), pp. 129–138
DIRECTIONAL CONVEX EXTENSIONS OF THE CONVEX VALUED MAPS ¨ S. A. DUZCE and KH. G. GUSEINOV Received November 28, 2007 and, in revised form, September 19, 2008 Abstract. In this paper, the concept of directional convex extension of the convex valued maps is introduced. Necessary and sufficient conditions for the existence of the directional convex extension of the convex valued map are obtained.
1. Introduction The existence of the convex extension of convex compact set valued map has been studied in [2], [3]. Some maximality properties of the convex extension are considered in [5]. The existence of the convex extension of the convex set valued map arises in the solutions of some inverse problems of the differential inclusion theory, where a differential inclusion with prescribed attainable sets or integral funnel is constructed (see [2], [4], [6]). The family of all non-empty compact (non-empty compact, convex) subsets of Rn is denoted by comp(Rn ) (conv(Rn )). The Hausdorff distance 2000 Mathematics Subject Classification. Primary: 26E25, 47L07. Key words and phrases. Set valued map, directional convex extension, derivative set.
ISSN 1425-6908
c Heldermann Verlag.
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between the sets A ⊂ by
Rn and C ⊂ Rn is denoted by h (A, C) and defined (
)
h (A, C) = max sup d (x, C) , sup d (y, A) x∈A
y∈C
where d (x, C) = inf {kx − ck : c ∈ C}, k·k means the Euclidean norm. It is known that, (comp (Rn ) , h (·, ·)) is a metric space (see, e.g., [1], [7]). For A ⊂ Rn , the convex hull of A is denoted by co(A), the interior of A by int(A), the closure of A by cl(A), the boundary of A by ∂A. Let K ⊂ Rn and F (·) : K → comp(Rm ) be a set valued map. The graph of the set valued map F (·) is denoted by gr F (·) and defined as gr F (·) = {(x, y) ∈ K × Rm : y ∈ F (x)} .
The set valued map F (·) : K → comp(Rm ) (K ⊂ Rn ) is said to be compact (convex) iff its graph is compact (convex). Let K ∈ conv (Rn ) and x0 ∈ K. The Bouligand contingent cone TK (x0 ) (simply called tangent cone) to K at x0 is defined by ! [ 1 TK (x0 ) = cl (K − x0 ) . (1) δ δ>0
According to [1] TK (x0 ) is a closed convex cone. For (t, x) ∈ [t0 , θ] × Rn and the set valued map V (·) : [t0 , θ] → comp(Rn ), we denote 1 − n D V (t, x) = v ∈ R : lim inf d(x − δv, V (t − δ)) = 0 . δ→0+ δ The set D− V (t, x) is said to be an upper left hand side derivative set of the set valued map V (·) calculated at the point (t, x). Note that upper 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., [1], [2], [6], [7]). In this article, the concept of directional convex extension of convex valued maps is introduced. This concept is a theoretical generalization of the convex extension notion given in [2], [3]. The necessary and sufficient conditions for the existence of the directional convex extension of convex valued set valued map are obtained (Theorem 3). To prove our existence results for directional convex extension for convex valued maps we need the following definition and propositions given in [3]. For given α > 0, x∗ ∈ Rn and the set valued map V (·) : [t0 , θ] → comp(Rn ), we define the set valued map VαR (x∗ )|(·) : [t0 , θ + α] → comp(Rn )
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setting VαR (x∗ )|(t)
θ+α−t θ+α−t = 1− x∗ + V (θ). α α
It is obvious that VαR (x∗ )|(θ) = V (θ). If V (θ) ⊂ VαR (x∗ )|(·) is a convex set valued map.
(2)
Rn is a convex set, then
Definition 1 ([3]). Let V (·) : [t0 , θ] → comp(Rn ) be a convex compact set valued map and α > 0 be a fixed number. If there is a convex compact set valued map W (·) : [t0 , θ + α] → comp(Rn ) such that W (t) = V (t) for all t ∈ [t0 , θ] then the set valued map W (·) is said to be a right-hand convex α-extension of the set valued map V (·). Proposition 1 ([3]). Let α > 0, Vα (·) : [t0 , θ + α] → comp(Rn ) and V (·) : [t0 , θ] → comp(Rn ) be convex set valued maps. Suppose that Vα (θ) = V (θ) and V (t) ⊂ Vα (t) for all t ∈ [t0 , θ). Then the set valued map W (·) : [t0 , θ + α] → comp(Rn ) defined as ( V (t), t ∈ [t0 , θ] W (t) = Vα (t), t ∈ (θ, θ + α] is a right-hand convex α-extension of the set valued map V (·). Proposition 2 ([3]). Let α > 0, x1 ∈ a convex set valued map. Assume that
Rn and V (·) : [t0, θ] → comp(Rn) be
D− V (θ, v) ⊂ D− VαR (x1 )|(θ, v) for every v ∈ V (θ). Then V (t) ⊂ VαR (x1 )|(t) for every t ∈ [t0 , θ] where the set valued map VαR (x1 )|(·) defined in (2). Proposition 3 ([3]). Let the set valued map W : [t0 , θ + α] → comp(Rn ) be a right hand convex α-extension of the convex set valued map V (·) : [t0 , θ] → comp(Rn ). Then for every fixed α∗ ∈ (0, α] and x∗ ∈ W (θ+α∗ ) the inclusion V (t) ⊂ VαR∗ (x∗ ) | (t) holds for any t ∈ [t0 , θ].
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2. Preliminary studies Let
S = {x ∈ Rn : kxk = 1} .
(3)
δ∗ (x0 , v) = min {δ ∈ (−∞, 0] : x0 + δv ∈ K} .
(4)
For given K ∈ conv (R
n ),
x0 ∈ K and v ∈ S we set
The next proposition characterizes the case when δ∗ (x0 , v) < 0.
Proposition 4. Let K ∈ conv (Rn ), x0 ∈ K, v ∈ S and δ∗ = δ∗ (x0 , v) be defined by (4). Then δ∗ < 0 if and only if [ 1 (−v) ∈ (K − x0 ) . (5) δ δ>0
Proof. Let δ∗ = δ∗ (x0 , v) < 0 and let us prove that the inclusion (5) is satisfied. Assume the contrary. Let [ 1 (K − x0 ) . (−v) 6∈ δ δ>0
Then (−v) 6∈ (1/δ) (K − x0 ) for every δ > 0 and consequently x0 +τ v 6∈ K for every τ < 0. Thus, we get from (4) that δ∗ = δ∗ (x0 , v) = 0 which contradicts to inequality δ∗ = δ∗ (x0 , v) < 0. So, we have that if δ∗ = δ∗ (x0 , v) < 0 then the inclusion (5) is valid. Now, let [ 1 (−v) ∈ (K − x0 ) . δ δ>0
Then there exists δ0 > 0 such that (−v) ∈ (1/δ0 ) (K − x0 ). Since K is convex and compact, then we get that x0 + δv ∈ K for every δ ∈ [−δ0 , 0]. So we obtain from (4) that δ∗ = δ∗ (x0 , v) ≤ δ0 < 0. From Proposition 4 we obtain the validity of following corollary. Corollary 1. Let K ∈ conv (Rn ), x0 ∈ K and v ∈ S. int (TK (x0 )) then δ∗ = δ∗ (x0 , v) < 0.
If (−v) ∈
Proof. It is known that if E ⊂ Rn is convex (see, e.g. [8]) then int (cl (E)) = int (E) .
Then we have from (1) and (6) int (TK (x0 )) = int cl
!! [ 1 (K − x0 ) = int δ
δ>0
! [ 1 (K − x0 ) . δ
δ>0
(6)
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The last equality implies that int (TK (x0 )) ⊂
[ 1 (K − x0 ). δ
(7)
δ>0
The validity of the corollary follows from (7) and Proposition 4. For given K ∈ conv (Rn ) and S, where S is defined by (3), we set ne (K, S) = {(x0 , v) ∈ ∂K × S : x0 + δv ∈ / K for every δ > 0} .
(8)
The following proposition characterizes the set ne (K, S). Proposition 5. Let K ∈ conv (Rn ), x0 ∈ ∂K and v ∈ S. Then (x0 , v) ∈ ne (K, S) iff [ 1 v∈ / (K − x0 ) . δ δ>0
Proof. Let (x0 , v) ∈ ne (K, S) where S x0 ∈ δK, v ∈ S. Then x0 + δv ∈ /K for every δ > 0 and consequently v ∈ / (1/δ) (K − x ). 0 δ>0 S Now, let v ∈ / δ>0 (1/δ) (K − x0 ) where x0 ∈ δK, v ∈ S. Then x0 + δv ∈ / K for every δ > 0 and consequently (x0 , v) ∈ ne (K, S). Now let us give the definition of directional convex extension. Note that the directional convex extension of the set valued map F (·) : K → conv (Rm ) will be given at the point x0 in the direction v such that (x0 , v) ∈ ne (K, S) and δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4). Let K ∈ conv (Rn ), F (·) : K → conv (Rm ) be a set valued map, x0 ∈ K, v ∈ S and δ∗ = δ∗ (x0 , v) be defined by (4). We define the set valued map Φ(x0 , v) | (·) : [δ∗ , 0] → conv (Rm ), setting Φ (x0 , v) | (t) = F (x0 + tv) .
(9)
Definition 2. Suppose α∗ > 0, K ∈ conv (Rn ), F (·) : K → conv (Rm ) is a set valued map, (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4) and Φ(x0 , v) | (·) : [δ∗ , 0] → conv (Rm ) defined by (9) is a convex set valued map. If there exists a convex set valued map Ψα∗ (·) : [δ∗ , α∗ ] → conv(Rm )
(10)
such that Ψα∗ (t) = Φ(x0 , v) | (t) for all t ∈ [δ∗ , 0], then the set valued map Ψα∗ (·) is said to be a convex α∗ -extension of the set valued map F (·) at x0 in the direction v.
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So, the set valued map F (·) has a convex α∗ -extension at x0 in the direction v ∈ S iff the set valued map Φ(x0 , v) | (·) has a right hand convex α∗ -extension where the set Φ(x0 , v) | (t) is defined by (9). 3. Existence of the directional convex extension In this section, we discuss the existence of the convex extension of the set valued map F (·) : K → conv(Rm ) at the point x0 in the direction v where K ∈ conv (Rn ), (x0 , v) ∈ ne (K, S).
Proposition 6. Let α∗ > 0, K ∈ conv (Rn ), F (·) : K → conv(Rm ), (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4). Suppose that the set valued map Φ(x0 , v) | (·) : [δ∗ , 0] → conv (Rm ) defined by (9) is compact convex one and there exists a convex set valued map Φα∗ (·) : [δ∗ , α∗ ] → conv(Rm ) such that Φα∗ (0) = Φ(x0 , v) | (0) and Φ(x0 , v) | (t) ⊂ Φα∗ (t) for all t ∈ [δ∗ , 0]. Then the set valued map Ψα∗ (x0 , v) | (·) : [δ∗ , α∗ ] → conv(Rm ) defined by ( Φ(x0 , v) | (t), t ∈ [δ∗ , 0] Ψα∗ (x0 , v) | (t) = Φα∗ (t), t ∈ (0, α∗ ] is a convex α∗ -extension of the set valued map F (·) at x0 in the direction v ∈ S. The proof of Proposition 6 follows from Proposition 1. Let α∗ > 0, y∗ ∈ Rm , K ∈ conv (Rn ), F (·) : K → conv (Rm ) be a set valued map, x0 ∈ K, v ∈ S, δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4). Now we define the set valued map setting
Γα∗ (x0 , v, y∗ ) | (·) : [δ∗ , α∗ ] → conv(Rm )
t t Γα∗ (x0 , v, y∗ ) | (t) = y∗ + 1 − F (x0 ). α∗ α∗
(11)
It is obvious that Γα∗ (x0 , v, y∗ ) | (0) = F (x0 ), Γα∗ (x0 , v, y∗ ) | (α∗ ) = y∗ and Γα∗ (x0 , v, y∗ ) | (·) : [δ∗ , α∗ ] → conv(Rm ) is convex compact set valued map. Proposition 7. Let α∗ > 0, y∗ ∈ Rm , K ∈ conv(Rn ), F (·) : K → conv(Rm ) be a set valued map, (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4), the set valued map Φ(x0 , v) | (·) : [δ∗ , 0] → conv (Rm ) defined by (9) be a compact convex one. Assume that D− Φ(x0 , v) | (0, w) ⊂ D− Γα∗ (x0 , v, y∗ ) | (0, w) for every w ∈ F (x0 ).
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Then Φ(x0 , v) | (t) ⊂ Γα∗ (x0 , v, y∗ ) | (t) for every t ∈ [δ∗ , 0]
(12)
where the set valued map Γα∗ (x0 , v, y∗ ) | (·) : [δ∗ , α∗ ] → conv(Rm ) is defined by (11). The proof of Proposition 7 follows from Proposition 2. From Propositions 6 and 7 we obtain following theorem which gives us a sufficient condition for the existence of directional convex extension at the given point. Theorem 1. Let α∗ > 0, y∗ ∈ Rm , K ∈ conv(Rn ), F (·) : K → conv(Rm ) be a set valued map, (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4), the set valued map Φ(x0 , v) | (·) : [δ∗ , 0] → conv (Rm ) defined by (9) be a compact convex one. Assume that D− Φ(x0 , v) | (0, w) ⊂ D− Γα∗ (x0 , v, y∗ ) | (0, w) for every w ∈ F (x0 ).
Then the set valued map Ψα∗ (x0 , v, y∗ ) | (·) : [δ∗ , α∗ ] → conv(Rm ) defined by ( Φ(x0 , v) | (t), t ∈ [δ∗ , 0] Ψα∗ (x0 , v, y∗ ) | (t) = Γα∗ (x0 , v, y∗ ) | (t), t ∈ (0, α∗ ] is a convex α∗ -extension of F (·) at x0 in the direction v. Proposition 8. Let α∗ > 0, K ∈ conv(Rn ), F (·) : K → conv(Rm ) be a set valued map, (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4), Ψα∗ (·) : [δ∗ , α∗ ] → conv(Rm ) be a convex α∗ -extension of F (·) at x0 in the direction v. Then for every fixed α ∈ (0, α∗ ] and y∗ ∈ Ψα∗ (α) the inclusion Φ(x0 , v) | (t) ⊂ Γα (x0 , v, y∗ ) | (t)
(13)
holds for every t ∈ [δ∗ , 0], where the set Φ(x0 , v) | (t), t ∈ [δ∗ , 0] is defined by (9), the set Γα (x0 , v, y∗ ) | (t), t ∈ [δ∗ , α] is defined by (11). Proof. The proof of this proposition follows from (9), (11), that is from definitions of the set valued maps Φ(x0 , v) | (·), t ∈ [δ∗ , 0], Γα (x0 , v, y∗ ) | (·), t ∈ [δ∗ , α∗ ] and from Proposition 3. From Proposition 6 we get the following theorem.
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Theorem 2. Let α∗ > 0, K ∈ conv(Rn ), F (·) : K → conv(Rm ) be a set valued map, (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4). Assume that Ψα∗ (·) : [δ∗ , α∗ ] → conv(Rm ) is a convex α∗ -extension of F (·) at x0 in the direction v, α ∈ (0, α∗ ] and y∗ ∈ Ψα∗ (α).Then D− Φ(x0 , v) | (0, w) ⊂ D− Γα∗ (x0 , v, y∗ ) | (0, w) for every w ∈ F (x0 ). (14) Here the set Φ(x0 , v) | (t), t ∈ [δ∗ , 0], is defined by (9), the set Γα (x0 , v, y∗ ) | (t), t ∈ [δ∗ , α], is defined by (11). We obtain from Theorem 1 and Theorem 2, the validity of the following theorem which gives a necessary and sufficient condition for the existence of directional convex extension of the given convex valued set valued map. Theorem 3. Let α∗ > 0, K ∈ conv(Rn ), F (·) : K → conv(Rm ) be a set valued map, (x0 , v) ∈ ne (K, S), δ∗ = δ∗ (x0 , v) < 0 where δ∗ (x0 , v) is defined by (4), Φ(x0 , v) | (·) : [δ∗ , 0] → conv (Rm ) defined by (9) be a convex set valued map. The set valued map F (·) has a convex α∗ -extension at x0 in the direction v if and only if there exists y∗ ∈ Rm such that D− Φ(x0 , v) | (0, w) ⊂ D− Γα∗ (x0 , v, y∗ ) | (0, w) for every w ∈ F (x0 ).
Example. Let K = (x1 , x2 ) ∈ R2 : x21 + x22 ≤ 1 , F (·) : K → conv(R) q 2 2 F (x1 , x2 ) = y : 0 ≤ y ≤ 1 − x1 − x2 (15)
and x∗ = (1, 0). It is obvious that x∗ ∈ ∂K. Now let us choose an arbi trary v = (v1 , v2 ) ∈ S = (y1 , y2 ) ∈ R2 : y12 + y22 = 1 and calculate δ∗ (x∗ , v) where δ∗ (x∗ , v) = min {δ ∈ (−∞, 0]|(1 + δv1 , δv2 ) ∈ K} . According to the definition of K it is possible to show that (1+δv1 , δv2 ) ∈ K iff δ 2 + 2v1 δ ≤ 0. The last inequality implies that ( −2v1 δ∗ (x∗ , v) = 0
if v1 > 0 if v1 ≤ 0.
Denote S∗ = {(v1 , v2 ) ∈ S : v1 > 0} . It follows from (16) that δ∗ (x∗ , v) = −2v1 < 0
(16)
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for every v = (v1 , v2 ) ∈ S∗ . Thus, we get that (x∗ , v) ∈ ne(K, S) and δ∗ (x∗ , v) < 0 for every v = (v1 , v2 ) ∈ S∗ . For given x∗ = (1, 0) and v = (v1 , v2 ) ∈ S∗ and we define a set valued map Φ(x∗ , v) | (·) : [−2v1 , 0] → conv(R) setting It is obvious that
Φ(x∗ , v) | (t) = F (x∗ + tv).
Φ(x∗ , v) | (t) = F (x∗ + tv) = F (1 + tv1 , tv2 ) n o p = y : 0 ≤ y ≤ −2tv1 − t2 .
So, the graph of the set valued map t → Φ(x∗ , v) | (t), t ∈ [δ∗ (x∗ , v) , 0] = [−2v1 , 0], is the upper half of the closed ball centered at (−v1 , 0) with radius v1 . Now, we prove that for any α∗ > 0 the set valued map F (·) has no convex α∗ -extension at x∗ in direction v ∈ S∗ , that is the set valued map Φ(x∗ , v) | (·) : [−2v1 , 0] → conv(R) has no right hand convex α∗ -extension. Assume that there is a convex set valued map Ψα∗ (·) : [−2v1 , α∗ ] → conv(R)
such that Ψα∗ (t) = Φ(x∗ , v∗ ) | (t) for every t ∈ [−2v1 , 0]. It means that the set valued map Ψα∗ (·) : [−2v1 , α∗ ] → conv(R) is a right hand convex α∗ -extension of the set valued map Φ(x∗ , v) | (·) : [−2v1 , 0] → conv(R). Since Ψα∗ (t) = Φ(x∗ , v∗ ) | (t) for every t ∈ [−2v1 , 0] then, (−2v1 , 0) ∈ gr Ψα∗ (·) and (−v1 , v1 ) ∈ gr Ψα∗ (·). Let us take an arbitrary y∗ ∈ Ψα∗ (α∗ ). Since Ψα∗ (·) is convex set valued map, then we have that (1 − λ)(α∗ , y∗ ) + λ(−2v1 , 0) ∈ gr Ψα∗ (·), (1 − λ)(α∗ , y∗ ) + λ(−v1 , v1 ) ∈ gr Ψα∗ (·)
for every λ ∈ [0, 1]. Let y∗ < 0 and λ∗ = Then
Since
α∗ ∈ (0, 1). 2v1 + α∗
(1 − λ∗ )(α∗ , y∗ ) + λ∗ (−2v1 , 0) = 0, Ψα∗ (0) = Φ(x∗ , v∗ ) | (0) = {0},
we obtain that
2v1 y∗ . 2v1 + α∗
2v1 y∗ 6= 0, 2v1 + α∗
(1 − λ∗ )(α∗ , y∗ ) + λ∗ (−2v1 , 0) 6∈ gr Ψα∗ (·)
(17) (18)
¨ S. A. DUZCE AND KH. G. GUSEINOV
138
which contradicts to (17). Now let y∗ ≥ 0 and λ∗ = Then
α∗ ∈ (0, 1). v1 + α∗
(1 − λ )(α∗ , y∗ ) + λ (−v1 , v1 ) = 0,
v1 (α∗ + y∗ ) . v 1 + α∗
Ψα∗ (0) = Φ(x∗ , v∗ ) | (0) = {0},
v1 (α∗ + y∗ ) 6= 0, v 1 + α∗
∗
∗
Since
we get
(1 − λ∗ )(α∗ , y∗ ) + λ∗ (−v1 , v1 ) 6∈ gr Ψα∗ (·)
which contradicts to (18). So we conclude that for any α∗ > 0 the set valued map Φ(x∗ , v) | (·) : [−2v1 , 0] → conv(R)
has no right hand convex α∗ -extension that is the set valued map F (·) has no convex α∗ -extension at x∗ in direction v ∈ S∗ . References [1] Aubin, J.-P., Frankowska, H., Set Valued Analysis, Birkhauser, Boston, 1990. [2] Duzce, S. A., Convex continuation of the convex set valued maps and applications to the differential inclusion theory (in Turkish), PhD thesis, Anadolu University, Eskisehir, 2003. [3] Guseinov, Kh. G., Duzce, S. A., Ozer, O., Convex extensions of the convex set valued maps, J. Math. Anal. Appl. 314(2) (2006), 672–688. [4] Guseinov, Kh. G., Duzce, S. A., Ozer, O., The construction of differential inclusions with prescribed attainable sets, J. Dyn. Control Systems 14(4) (2008), 441–452. [5] Guseinov, Kh. G., Ozer, O., Duzce, S. A., Maximal convex continuation for convex compact set valued map (in Turkish), Anadolu Univ. J. Sci. Tech. 3 (2002), 383–388. [6] Guseinov, Kh. G., Ushakov, V. N., The construction of differential inclusions with prescribed properties, Differ. Equ. 36 (2000), 488–496. [7] Hu, Sh., Papageorgiou, N. S., Handbook of Multivalued Analysis. Vol. I. Theory, Kluwer Academic, Dordrecht, 1997. [8] Rockafellar, R. T., Convex Analysis, Princeton Math. Ser. 28, Princeton Univ. Press, N. J., 1970.
¨ zce Serkan Ali Du Kh. G. Guseinov Anadolu University Anadolu University Department of Mathematics Department of Mathematics 26470 Eskisehir, Turkey 26470 Eskisehir, Turkey e-mail:
[email protected] e-mail:
[email protected]
