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Metric Projection Operator and Continuity of the Set-Valued Metric Generalized Inverse in Banach Spaces.

1. Introduction and Preliminaries

Let (X, [parallel]x[parallel]) be a real Banach space. Let S(X) and B(X) denote the unit sphere and the unit ball of X, respectively. By X* we denote the dual space of X. Let T be a linear bounded operator from X into Y. Let D(T), R(T), and N(T) denote the domain, range, and null space of T, respectively. The Chebyshev radius and Chebyshev center of set A are defined, respectively, by

[mathematical expression not reproducible]. (1)

Moreover, it is easy to see that if A is a convex set, then c(A) is a convex set. The set-valued mapping [P.sub.c] : X [right arrow] C,

[mathematical expression not reproducible]. (2)

is said to be the metric projection from X onto C.

Continuity of metric projection is an important content in theory of geometry of Banach spaces. Metric projection has important applications in the optimization, computational mathematics, theory of equation, and control theory. It is well known that if closed convex set C is approximatively compact, then [P.sub.c] is upper semicontinuous. It is very natural to ask in which Banach spaces metric projection [P.sub.c] is upper semicontinuous and C is not approximative compact.

The theory of generalized inverses has its genetic in the context of the so-called "ill-posed" linear problems. Such problems cannot be solved in the sense of a solution of a nonsingular problem. In order to solve the best approximation problems for an ill-posed linear operator equation in Banach spaces, Nashed and Votruba introduced the concept of the set-valued metric generalized inverse of linear operator in [1]. Moreover, Nashed and Votruba (see [1]) raised the following study suggestion: "the problem of obtaining selections with nice properties for the metric generalized inverse merits is worth studying." In [2] upper semicontinuity of the setvalued metric generalized inverse [T.sup.[partial derivative]] in approximatively compact Banach spaces is investigated by means of the methods of geometry of Banach spaces. It is very natural to ask whether the approximative compactness of Banach space is necessary for upper semicontinuity of the set-valued metric generalized inverse [T.sup.[partial derivative]]. In this paper, authors, by putting different equivalent norms on [I.sup.2], show that there exists a proximinal hyperplane H such that [P.sub.H] is continuous and H is not approximative compact. Moreover, the authors give some examples of continuous metric projection and lower semicontinuous metric projection. Finally, continuous homogeneous selection and continuity for the set-valued metric generalized inverses [T.sup.[partial derivative]] in 3-strictly convex spaces are investigated by continuity of metric projection. Hence approximative compactness of Banach space is not necessary for upper semicontinuity of the set-valued metric generalized inverse [T.sup.[partial derivative]]. The results are an answer to the problem posed by Nashed and Votruba. Other researches on generalized inverses of linear operators are visible in [1-8]. First let us recall some definitions that will be used in the further part of the paper.

Definition 1 (see [9]). A nonempty set C is said to be Chebyshev set if [P.sub.c](x) is a singleton for all x [member of] X. A nonempty set C is said to be proximinal if [P.sub.c](x) [not equal to] 0 for all x [member of] X.

Definition 2 (see [9]). A Banach space X is said to be k-strictly convex if for any k + 1 elements [x.sub.1], [x.sub.2], ..., [x.sub.k+1] [member of] S(X), if [parallel] [x.sub.1] + [x.sub.2] + ... + [x.sub.+11][parallel] = k + 1, then [x.sub.1], [x.sub.2], ..., [x.sub.k+1] are linearly dependent.

It is well known that X is a 1-strictly convex space if and only if X is a strictly convex space.

Definition 3 (see [10]). A nonempty subset C of X is said to be approximatively compact if for any [{[y.sub.n]}.sup.[infinity].sub.n=1] [subset] C and any x [member of] X satisfying [parallel]x - [y.sub.n][parallel] [right arrow] [inf.sub.y[member of]C] [parallel]x - y[parallel] as n [right arrow] [infinity], the sequence [{[y.sub.n]}.sup.[infinity].sub.n=1] has a subsequence converging to an element in C.

Definition 4 (see [11]). Set-valued mapping F : X [right arrow] Y is said to be upper semicontinuous at [x.sub.0], if for each norm open set W with F([x.sub.0]) [subset] W, there exists a norm neighborhood U of [x.sub.0] such that F(x) [subset] W for all x in U. F is called lower semicontinuous at [x.sub.0], if for any y [member of] F([x.sub.0]) and any [{[x.sub.n]}.sup.[infinity].sub.n=1] in X with [x.sub.n] [right arrow] [x.sub.0], there exists [y.sub.n] [member of] F([x.sub.n]) such that [y.sub.n] [right arrow] y as n [right arrow] [infinity]. F is called continuous at [x.sub.0], if F is upper semicontinuous and is lower semicontinuous at [x.sub.0].

Definition 5 (see [12]). A point x [member of] S(X) is said to be H-point if [{[y.sub.n]}.sup.[infinity].sub.n=1] [subset] S(X) and [mathematical expression not reproducible]; one has [x.sub.n] [right arrow] x as n [right arrow] [infinity]. Moreover, if the set of all H-points is equal to S(X), then X is said to have the H-property.

Definition 6 (see [1]). A point [x.sub.0] [member of] D(T) is said to be the best approximative solution to the operator equation Tx = y, if

[mathematical expression not reproducible]. (3)

Definition 7 (see [1]). Let X, Y be Banach spaces and T be a linear operator from X to Y. The set-valued mapping [T.sup.[partial derivative]] : Y [right arrow] X defined by

[T.sup.[partial derivative]] (y) = {[x.sub.0] [member of] D (T) :

[x.sub.0] is a best approximative solution to T(x) = y}

for any y [member of] D([T.sup.[partial derivative]]) is said to be the (set- valued) metric generalized inverse of T, where

D([T.sup.[partial derivative]]) = {y [member of] Y : T(x)

= y has a best approximative solution in X}. (5)

2. Continuity of Metric Projection Operator and Approximative Compactness

Theorem 8. Let f [member of] S([X.sup.*]), H ={x [member of] X : f(x) = 0}, and the set [A.sub.f] = {x [member of] X : f(x) = 1} is a nonempty compact set. Then

(1) [P.sub.H](x) = x - f(x)[A.sub.f] for any x [member of] X

(2) The metric projector [P.sub.H] is continuous.

Proof. (1) Let x [member of] X. Pick z [member of] H and y [member of] S(X). Then there exists [alpha] [member of] R such that x - z = [alpha]y. It is easy to see that [alpha] = f(x)/f(y). Then x - z = (f(x)/f(y))y. Hence [parallel]x - z[parallel] = [absolute value of f(x)]/[absolute value of f(y)] [greater than or equal to] [absolute value of f(x)]. Then it is easy to see that z [member of] [P.sub.H](x) if and only if y [member of] [A.sub.f]. Hence [P.sub.H](x) = x - f(x)[A.sub.f] for any x [member of] X.

(2) Suppose that PH is not upper semicontinuous at x0. Then there exist a sequence [{[y.sub.n]}.sup.[infinity].sub.n=1] [subset] X and an open set W [contain] [P.sub.H]([x.sub.0]) such that [P.sub.N(T)]([x.sub.n]) [not subset] W and [x.sub.n] [right arrow] [x.sub.0] as n [right arrow] [infinity]. Then there exists [z.sub.n] [member of] [P.sub.N(T)]([x.sub.n]) such that [z.sub.n] [not member of] W. By (1), we have [z.sub.n] = [x.sub.n] - f([x.sub.n])[y.sub.n], where [y.sub.n] [member of] [A.sub.f]. Since [A.sub.f] is compact, there exists a subsequence [mathematical expression not reproducible] and

[mathematical expression not reproducible], (6)

a contradiction. This implies that [P.sub.H] is upper semicontinuous.

Let [x.sub.n] [right arrow] [x.sub.0] as n [right arrow] [infinity]. Pick [z.sub.0] [member of] [P.sub.H]([x.sub.0]). Then, by (1), there exists [y.sub.0] [member of] [A.sub.f] such that [z.sub.0] = [x.sub.0] - f([x.sub.0])[y.sub.0]. By (1), we have [z.sub.n] = [x.sub.n] - f([x.sub.n])[y.sub.0] [member of] [P.sub.N(T)] ([x.sub.n]) and

[mathematical expression not reproducible]. (7)

This implies that [P.sub.H] is lower semicontinuous at [x.sub.0]. Hence we obtain that [P.sub.H] is continuous. This completes the proof.

Theorem 9. Suppose that every proximinal hyperplane of X is approximatively compact. Then X has the H-property.

Proof. Let [mathematical expression not reproducible]. Then there exists [x.sup.*] [member of] S([X.sup.*]) such that [x.sup.*](x) = 1. Hence the hyperplane [mathematical expression not reproducible] is proximinal. Suppose that the sequence [{[x.sub.n]}.sup.[infinity].sub.n=1] does not converge to x. Then we may assume without loss of generality that [parallel][x.sub.n] - x[parallel] > [epsilon] for every n [member of] N. Since [mathematical expression not reproducible], is a proximinal set, there exists [mathematical expression not reproducible]. Since

[mathematical expression not reproducible], (8)

we obtain that

[mathematical expression not reproducible]. (9)

This implies that the sequence [{[y.sub.n]}.sup.[infinity].sub.n=1] is relatively compact. Hence the sequence [{[x.sub.n]}.sup.[infinity].sub.n=1] is relatively compact. Then there exists a subsequence [mathematical expression not reproducible] is a Cauchy sequence. Since [mathematical expression not reproducible], a contradiction. Hence [x.sub.0] [right arrow] x as n [right arrow] [infinity]. This implies that X has the H-property.

Example 10. There exists a proximinal hyperplane H of X such that [P.sub.H] is continuous and H is not approximative compact. Let ([l.sup.2], [[parallel] x [parallel].sub.1]) and ([l.sup.2], [[parallel] x [parallel].sub.1]) be two Banach spaces, where

[mathematical expression not reproducible]. (10)

Then

[mathematical expression not reproducible]. (11)

This implies that [[parallel] x [parallel].sub.1] [less than or equal to] [[parallel] x [parallel].sub.2]. Hence [[parallel] x [parallel].sub.1] and [[parallel] x [parallel].sub.2] are equivalent. This implies that ([l.sup.2], [[parallel] x [parallel].sub.2]) is reflexive and if [x.sup.*] [member of] ([l.sup.2], [[parallel] x [parallel].sub.2]),then [x.sup.*] [member of] ([l.sup.2], [[parallel] x [parallel].sub.1]). Let[([e.sub.n]).sup.[infinity].sub.n=1] be the orthonormal basis of ([l.sup.2], [[parallel] x [parallel].sub.1] and [x.sub.n] = [t.sub.n][e.sub.1] + [e.sub.n+1], where 0 < [t.sub.n] [less than or equal to] 1 and [t.sub.n] [right arrow] 1 as n [right arrow] [infinity]. Then it is easy to see that [mathematical expression not reproducible] is not a Cauchy sequence in ([l.sup.2], [[parallel] x [parallel].sub.2]). This implies that ([l.sup.2], [[parallel] x [parallel].sub.2]) does not have the H-property.

We claim that there exists a proximinal hyperplane H of ([l.sup.2], [[parallel] x [parallel].sub.1]) such that [P.sub.H] is continuous and H is not approximative compact. Since ([l.sup.2], [[parallel] x [parallel].sub.1]) is reflexive, we obtain that [A.sub.f] [not equal to] 0 for any f [member of] S(([l.sup.2], [[parallel] x [parallel].sub.2])). Let f = ([[eta].sub.1], [[eta].sub.2], ...), x = ([[xi].sub.1], [[xi].sub.2], ...), and f(x) = [parallel] f [parallel] = [parallel] x [parallel] = 1. Then, by Cauchy inequality and holder inequality, we have

[mathematical expression not reproducible]. (12)

This implies that [mathematical expression not reproducible] whenever i [greater than or equal to] 3. It is easy to see that [A.sub.f] is a compact set. Therefore, by Theorem 8, we obtain that [P.sub.H] is continuous, where H =[x [member of] [l.sup.2], [[parallel] x [parallel].sub.2]) : f(x) = 0}. Hence, for any [lambda] [member of] R, we obtain that [P.sub.H([lambda])] is continuous, where H([lambda]) = [x [member of] ([l.sup.2], [[parallel] x [parallel].sub.2]) : f(x) = [lambda]}. Suppose that every proximinal hyperplane is approximative compact. Then, by Theorem 9, we obtain that ([l.sup.2], [[parallel] x [parallel].sub.1]) has the H-property, a contradiction.

Theorem 11. Let [H.sub.1] be a closed subspace of [X.sub.1] and [H.sub.2] be a closed subspace of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is lower semicontinuous on [mathematical expression not reproducible] is lower semicontinuous on [X.sub.2]. Then the metric projection operator[mathematical expression not reproducible] is lower semicontinuous on ([X.sub.1] x [X.sub.2], [[parallel] x [parallel]), where [parallel]([x.sub.1] x [x.sub.2])[parallel] = [parallel] [x.sub.1] [parallel] + [parallel] [x.sub.2] [parallel].

Proof. Let ([x.sub.1n], [x.sub.2n]) [right arrow] ([x.sub.1], [x.sub.2]) as n [right arrow] [infinity]. Then [x.sub.1n] [right arrow] [x.sub.1] and [x.sub.2n] [right arrow] [x.sub.2] as n [right arrow] [infinity]. Moreover, it is easy to see that [mathematical expression not reproducible] is lower semicontinuous.

Let X be k-strictly convex and H = [x [member of] X : f(x) = 0, f [member of] S([X.sup.*])}. Then [A.sub.f] = [x [member of] X : f(x) = 1} is a nonempty compact set. Then, by Theorem 8, the metric projector operator [P.sub.H] is lower semicontinuous. Let Y be strictly convex and M is an approximatively compact closed subspace of Y. Then the metric projector operator [P.sub.M] is continuous. Therefore, by Theorem 11, we obtain that [P.sub.HxM] is lower semicontinuous on (X x Y, [parallel] x [parallel]), where H([x.sub.1], [x.sub.2])H = [parallel] [x.sub.1] [parallel] + [parallel] [x.sub.2] [parallel].

3. Continuous Selections and Continuity of the Set-Valued Metric Generalized Inverse

Theorem 12. Let X be a 3-strictly convex space, Y be a Banach space, D(T) be a closed subspace of X, and R(T) be an approximatively compact Chebyshev subspace of Y. Then

(1) [P.sub.N(T)] is upper semicontinuous if and only if [T.sup.[partial derivative]] is upper semicontinuous

(2) [P.sub.N(T)] is continuous if and only if [T.sup.[partial derivative]] is continuous

(3) If [P.sub.N(T)] is continuous, then there exists a homogeneous selection [T.sup.[sigma]] of [T.sup.[partial derivative]] such that [T.sup.[sigma]] is continuous on [z [member of] Y : lim [inf.sub.h[right arrow]z] diam(c([T.sup.[partial derivative]](h))) [greater than or equal to] diam(c([T.sup.[partial derivative]] (z)))}.

Proof. (1) Let "[??]" [y.sub.0] [member of] Y. We first will prove that [T.sup.[partial derivative]] is upper semicontinuous at [y.sub.0], that is, for any [{[y.sub.n]}.sup.[infinity].sub.n=1] [subset] Y, [y.sub.n] [right arrow] [y.sub.0] [member of] Y, and any norm open set W with [T.sup.[partial derivative]]([y.sub.0]) [subset] W, there exists a natural number [N.sub.0] such that [T.sup.[partial derivative]]([y.sub.n]) [subset] W whenever n > [N.sub.0]. Pick [x.sub.0] [member of] [T.sup.-1] ([P.sub.R(T)]([y.sub.0])). Then, by the definition of set-valued metric generalized inverse, we obtain that Td(y0) = [x.sub.0] - [P.sub.N(T)]([x.sub.0]). Since T is a bounded linear operator, we obtain that N(T) is a closed subspace of D(T). Let

[bar.T] : D(T)/N(T) [right arrow] R(T),

[bar.T][x] = Tx, (13)

where [x] [member of] D(T)/N(T) and x [member of] D(T). Then it is easy to see that R([bar.T]) = R(T). Moreover, [bar.R(T)] = R(T). In fact, suppose that [bar.R(T)] [not equal to] R(T). Then there exists y' [member of] [bar.R(T)] such that y' [member of] R(T). It is easy to see that {y [member of] R(T) : [parallel]y' - y[parallel] = dist(y', R(T))} = 0. This implies that R(T) is not a Chebyshev subspace of Y, a contradiction. Since [bar.R(T)] = R(T), we obtain that R(T) is a Banach space. Moreover, it is easy to see that T is a bounded linear operator and N([bar.T]) = {0}. This implies that the bounded linear operator [bar.T] is both injective and surjective. Therefore, by the inverse operator theorem, we obtain that [[bar.T].sup.1] is a bounded linear operator. Pick [x.sub.n] [member of] [[bar.T].sup.1] ([P.sub.R(T)]([y.sub.n])). Since Y is approximatively compact and R(T) is a Chebyshev subspace of Y, we obtain that the metric projector operator [P.sub.R(T)] is continuous. Hence [P.sub.R(T)]([y.sub.n]) [right arrow] [P.sub.R(T)]([y.sub.0]) as n [right arrow] [infinity].

Since [[bar.T].sup.1] is a bounded linear operator, we obtain that

[mathematical expression not reproducible] (14)

Hence we may assume without loss of generality that [x.sub.n] [right arrow] [x.sub.0] as n [right arrow] [infinity]. Since [T.sup.[partial derivative]]([y.sub.0]) = [x.sub.0] - [P.sub.N(t)]([x.sub.0]), we obtain that [x.sub.0] - [P.sub.N(T)] ([x.sub.0]) [subset] W. Hence, for any z [member of] [P.sub.N(T)] ([x.sub.0]), we obtain that [x.sub.0] - z [member of] W. Hence there exist [[delta].sub.z] > 0 and [r.sub.z] >0 such that B([x.sub.0], [[delta].sub.z]) - B(z, [r.sub.z]) [member of] W. Since X is a 3- strictly convex space, we obtain that [P.sub.N(T)]([x.sub.0]) is compact. Therefore, by

[mathematical expression not reproducible], (15)

there exist [z.sub.1] [member of] [P.sub.N(T)] ([x.sub.0]), [z.sub.2] [member of] [P.sub.N(T)] ([x.sub.0]), ..., [z.sub.k] [member of] [P.sub.N(T)] ([x.sub.0]) such that

[mathematical expression not reproducible]. (16)

Let [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (17)

This implies that

[mathematical expression not reproducible]. (18)

Since [P.sub.N(T)] is upper semicontinuous, there exists [n.sub.0] [member of] N such that [parallel][x.sub.n] - [x.sub.0][parallel] < [delta] and

[mathematical expression not reproducible] (19)

whenever n > [n.sub.0]. Since [mathematical expression not reproducible]. Hence

[mathematical expression not reproducible]. (20)

This implies that [T.sup.[partial derivative]] is upper semicontinuous at [y.sub.0]. Hence [T.sup.[partial derivative]] is upper semicontinuous.

"[??]" Suppose that [P.sub.N(T)] is not continuous. Then there exist [{[x.sub.n]}.sup.[infinity].sub.n=1] [subset] X, [x.sub.0] [member of] X, and an open set W such that [x.sub.n] [right arrow] [x.sub.0], [P.sub.N(T)]([x.sub.0]) [subset] W, and [P.sub.N(T)]([x.sub.n]) [subset] W. Hence there exists [[pi].sub.N(T)] ([x.sub.n]) [member of] [P.sub.N(T)] ([x.sub.n]) such that [[pi].sub.N(T)] ([x.sub.n]) [not member of] W. We claim that there exists [delta] > 0 such that

[mathematical expression not reproducible]. (21)

Otherwise, there exists [z.sub.n] [member of] [P.sub.N(T)]([x.sub.0]) such that B([z.sub.n], 1/n) [not member of] W. Since [P.sub.N(T)]([x.sub.0]) is compact, we may assume that [z.sub.n] [right arrow] [z.sub.0] [member of] [P.sub.N(T)]([x.sub.0]) as n [right arrow] [infinity]. Hence there exists [eta] > 0 such that B([z.sub.0], 4[eta]) [subset] W. Moreover, there exists n0 e N such that [mathematical expression not reproducible]. Hence, for any [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (22)

This implies that [mathematical expression not reproducible], a contradiction. Let [y.sub.n] = [Tx.sub.n] and [y.sub.0] = [Tx.sub.0]. Then

[mathematical expression not reproducible]. (23)

Since [P.sub.N(T)]([x.sub.0]) [subset] W, we obtain that [T.sup.[partial derivative]]([y.sub.0]) = [x.sub.0] - [P.sub.N(T)]([x.sub.0]) [subset] [x.sub.0] - W. We claim that

[mathematical expression not reproducible] (24)

whenever [parallel][x.sub.n] - [x.sub.0][parallel] < S. Infact, suppose that [x.sub.n] - [[pi].sub.N(T)]([x.sub.n]) [member of] [mathematical expression not reproducible]. Then

[mathematical expression not reproducible], (25)

a contradiction. Since [mathematical expression not reproducible] we obtain that [T.sup.[partial derivative]] is not upper semicontinuous at [y.sub.0], a contradiction.

(2) Let [y.sub.0] [member of] Y and [y.sub.n] [right arrow] [y.sub.0] as n [right arrow] [infinity]. Then, by the previous proof, there exist [x.sub.0] [member of] X and [{[x.sub.n]}.sup.[infinity].sub.n=1] [subset] X such that [P.sub.R(T)]([y.sub.0]) = [Tx.sub.0], [P.sub.R(T)]([y.sub.n]) = [Tx.sub.n], and [x.sub.n] [right arrow] [x.sub.0] as n [right arrow] [infinity]. Then [T.sup.[partial derivative]]([y.sub.0]) = [x.sub.0] - [T.sup.[partial derivative]]([x.sub.0]) and [T.sup.[partial derivative]]([y.sub.n]) = [x.sub.n] - [P.sub.R(T)]([x.sub.n]). Since [P.sub.N(T)] is continuous, we obtain that, for any z [member of] [P.sub.N(T)]([x.sub.0]), there exists [z.sub.n] [member of] [P.sub.N(T)]([x.sub.n]) such that [z.sub.n] [right arrow] [z.sub.0] as n [right arrow] [infinity]. Hence, for any [x.sub.0] - z [member of] [x.sub.0] - [P.sub.N(T)]([x.sub.0]), there exists [x.sub.n] - [z.sub.n] [member of] [x.sub.n] - [P.sub.N(T)]([x.sub.n]) such that [x.sub.n] - [z.sub.n] [right arrow] [x.sub.0] - z as n [right arrow] [infinity]. This implies that [T.sup.[partial derivative]] is lower semicontinuous at [y.sub.0]. Therefore, by (1), we obtain that [T.sup.[partial derivative]] is continuous at [y.sub.0].

"[??]" Let [y.sub.0] [member of] Y and [y.sub.n] [right arrow] [y.sub.0] as n [right arrow] [infinity]. Then, by the previous proof, there exist [x.sub.0] [member of] X and [{[y.sub.n]}.sup.[infinity].sub.n=1] [subset] X such that [P.sub.R(T)]([y.sub.0]) = [Tx.sub.0] [P.sub.R(T)]([y.sub.n]) = [Tx.sub.n] and [x.sub.n] [right arrow] [x.sub.0] as n ^ [infinity]. Then [T.sup.[partial derivative]]([y.sub.0]) = [x.sub.0] - [P.sub.N(T)]([x.sub.0]) and [T.sup.[partial derivative]]([y.sub.n]) = [x.sub.n] - [P.sub.N(T)]([x.sub.n]). Since [T.sup.[partial derivative]] is continuous, we obtain that for any [x.sub.0] - z [member of] [x.sub.0] - [P.sub.N(T)]([x.sub.0]), there exists [x.sub.n] - [z.sub.n] [member of] [x.sub.n] - [P.sub.N(T)]([x.sub.n]) such that [x.sub.n] - [z.sub.n] [right arrow] [x.sub.0] - z as n [right arrow] [infinity]. Hence, for any z [member of] [P.sub.N(T)]([x.sub.n]), there exists [z.sub.n] [member of] [P.sub.N(T)]([x.sub.0]) such that [z.sub.n] [rho] [z.sub.0] as n [right arrow] [infinity]. This implies that [P.sub.N(T)] is lower semicontinuous at [y.sub.0]. Therefore, by (1), we obtain that [P.sub.N(T)] is continuous at [y.sub.0].

We next will prove that condition (3) is true. For clarity, we will divide the proof into some parts.

(3a) Define a set-valued mapping F : Y [right arrow] X such that F(y) = c([T.sup.[partial derivative]](y)). We claim that if [y.sub.n] [right arrow] y as n [right arrow] [infinity], then

[mathematical expression not reproducible], (26)

where [y.sub.n] [member of] Y and y [member of] Y. Otherwise, we may assume without loss of generality that [mathematical expression not reproducible]. Then there exists z(n) [member of] [T.sup.[partial derivative]](y) such that [mathematical expression not reproducible]. Since X is a 3-strictly convex space, we obtain that [P.sub.N(T)](x) is compact. From the previous proof, there exists x [member of] X such that [T.sup.[partial derivative]](y) = x - [P.sub.N(T)](x). This implies that [T.sup.[partial derivative]](y) is compact. Hence we may assume without loss of generality that z(n) [right arrow] [z.sub.0] as n [right arrow] [infinity]. This implies that [z.sub.0] [member of] [T.sup.[partial derivative]](y). Hence we may assume without loss of generality that

[mathematical expression not reproducible] (27)

for all n [member of] N. Since [P.sub.N(T)] is continuous, by [z.sub.0] [member of] [T.sup.[partial derivative]](y), there exist [h.sub.n] [member of] [T.sup.[partial derivative]]([y.sub.n]) such that [h.sub.n] [right arrow] [z.sub.0] as n [right arrow] [infinity], which contradicts formula (27).

We next will prove that F is upper semicontinuous. Suppose that F is not upper semicontinuous. Then there exist [T.sup.[partial derivative]] [subset] X, [y.sub.0] [member of] Y, and a norm open set W such that c([T.sup.[partial derivative]]([y.sub.0])) [subset] W, c([T.sup.[partial derivative]]([y.sub.n])) [not member of] W, and [y.sub.n] [right arrow] [y.sub.0] as n [right arrow] [infinity]. Hence there exists [x.sub.n] [member of] c([T.sup.[partial derivative]]([y.sub.n])) such that [x.sub.n] [not member of] W. Since is continuous, we obtain that Td is upper semicontinuous. Hence, for any [epsilon] > 0, there exists [n.sub.0] [member of] N such that

[mathematical expression not reproducible] (28)

whenever n > [n.sub.0]. This implies that dist[{[x.sub.n]}.sup.[infinity].sub.n=1], [T.sup.[partial derivative]]([y.sub.0])) = 0. Hence there exists x(n) [member of] c([T.sup.[partial derivative]]([y.sub.0])) such that dist([{[x.sub.n]}.sup.[infinity].sub.n=1], x(n)) < 1/n. Since [T.sup.[partial derivative]]([y.sub.0]) is compact, we may assume without loss of generality that x(n) [right arrow] [x.sub.0] as n [right arrow] [infinity]. This implies that dist([{[x.sub.n]}.sup.[infinity].sub.n=1], [x.sub.0]) = 0. Hence we may assume without loss of generality that [x.sub.n] [right arrow] [x.sub.0] as n [right arrow] [infinity]. We claim that [x.sub.0] [member of] c([T.sup.[partial derivative]]([y.sub.0])). In fact, suppose that [x.sub.0] [not member of] c([T.sup.[partial derivative]]([y.sub.0])). Let [r.sub.0] = r([T.sup.[partial derivative]]([y.sub.0])). Then there exist [h.sub.0] [member of] [T.sup.[partial derivative]]([y.sub.0]) and [delta] > 0 such that [parallel][h.sub.0] - [x.sub.0][parallel] [greater than or equal to] [r.sub.0] + [delta]. Moreover, we claim that

[mathematical expression not reproducible]. (29)

Otherwise, we may assume that there exist [h.sub.n] [member of] [T.sup.[partial derivative]]([y.sub.n]) and [eta] > 0 such that [mathematical expression not reproducible]. Since [P.sub.N(T)] is continuous, we obtain that [T.sup.[partial derivative]] is continuous. From the previous proof, we may assume without loss of generality that [h.sub.n] [member of] [h.sub.n] [member of] [T.sup.[partial derivative]](y), a contradiction. Therefore, by formulas (26) and (29), we may assume that

[mathematical expression not reproducible] (30)

for every n [member of] N. Therefore, by formula (30) and [x.sub.n] [right arrow] [x.sub.0], we may assume that there exists [h.sub.n] [member of] [T.sup.[partial derivative]]([y.sub.n]) such that

[mathematical expression not reproducible], (31)

for every n e N. Then

[mathematical expression not reproducible] (32)

for every n [member of] N. Therefore, by [x.sub.n] [member of] c([T.sup.[partial derivative]]([y.sub.n])), we obtain that

r([T.sup.[partial derivative]]([y.sub.n])) > r([T.sup.[partial derivative]]([y.sub.0])) + 3/4 [delta] (33)

for every n [member of] N. Pick z [member of] c([T.sup.[partial derivative]]([y.sub.0])). Therefore, by formula (30), there exists [z.sub.n] [member of] c([T.sup.[partial derivative]]([y.sub.n])) such that [parallel]z - [z.sub.n][parallel] < [delta]/64. Since the set [T.sup.[partial derivative]]([y.sub.n])) is compact, there exists [w.sub.n] [member of] [T.sup.[partial derivative]]([y.sub.n]) such that [parallel][w.sub.n] - [z.sub.n][parallel] [greater than or equal to] r([T.sup.[partial derivative]]([y.sub.n])). Moreover, by formula (30), there exists w(n) [member of] c([T.sup.[partial derivative]]([y.sub.0])) such that [parallel]w(n) - [w.sub.n][parallel] < [delta]/64. Since the set [T.sup.[partial derivative]]([y.sub.0]) is compact, we may assume without loss of generality that w(n) [right arrow] w as n [right arrow] [infinity]. Hence we may assume without loss of generality that [parallel]w - [w.sub.n][parallel] < [delta]/60. Therefore, by z [member of] c([T.sup.[partial derivative]]([y.sub.0])), we obtain that

[mathematical expression not reproducible], (34)

which contradicts formula (33). This implies that F is upper semicontinuous.

(3b) We will prove that if X is a 3-strictly convex space and z [member of] X, then there exists [x.sub.z] [member of] X and a 2-dimensional space [X.sub.z] such that [P.sub.N(T)](z) [subset] [x.sub.z] + [X.sub.z]. We may assume that z = 0. Pick [x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4] [member of] [P.sub.N(T)](0) such that [x.sub.1], [x.sub.2], [x.sub.3] are linearly independent. Then ([x.sub.1] + [x.sub.2] + [x.sub.3] + [x.sub.4])/4 [member of] N(T). Therefore, by the Hahn-Banach theorem, there exists [x.sup.*] [member of] S([X.sup.*]) such that [x.sup.*] ([x.sub.1] + [x.sub.2] + [x.sub.3] + [x.sub.4]) = 4. Then

[mathematical expression not reproducible], (35)

We may assume without loss of generality that [x.sub.4] = [t.sub.1][x.sub.1] + [t.sub.2][x.sub.2] + [t.sub.3][x.sub.3]. Then [x.sup.*]([x.sup.4]) = [x.sup.*]([t.sub.1][x.sub.1] + [t.sub.2][x.sub.2] + [t.sub.3][x.sub.3]) = 1. Hence [t.sub.1] + [t.sub.2] + [t.sub.3] = 1. Since [x.sub.1] + [x.sub.2] + [x.sub.3] are linearly independent, we obtain that for any x [member of] [P.sub.N(T)](0), if [x.sub.3] = [t.sub.1][x.sub.1] + [t.sub.2][x.sub.2] + [t.sub.3]x, then [t.sub.3] [not equal to] 0. Hence

[mathematical expression not reproducible]. (36)

This implies that, for any x [member of] [P.sub.N(T)](0), we have x = [[lambda].sub.1][x.sub.1] + [[lambda].sub.2][x.sub.2] + [[lambda].sub.3][x.sub.3], where [[lambda].sub.1] + [[lambda].sub.2] + [[lambda].sub.3] = 1. Then

x = [[lambda].sub.1] ([x.sub.1] - [x.sub.3]) + [[lambda].sub.2] ([x.sub.2] - [x.sub.3]) + [x.sub.3]. (37)

This implies that [P.sub.N(T)](0) [subset] span{[x.sub.1] - [x.sub.3], [x.sub.2] - [x.sub.3]} + [x.sub.3]. Hence, if X is a 3-strictly convex space and z [member of] X, then there exists [x.sub.z] [member of] X and a two-dimensional space [X.sub.z] such that [P.sub.N(T)](z) [subset] [x.sub.z] + [X.sub.z]. Moreover, we know that, for any y [member of] Y, there exists x [member of] X such that [T.sup.[partial derivative]](y) = x - [P.sub.N(T)](x). Hence, for any y [member of] Y, there exists [x.sub.z] [member of] X and a two-dimensional space [X.sub.z] such that [T.sup.[partial derivative]](y) [subset] [x.sub.z] + [X.sub.z].

(3c) We next will prove that, for any y [member of] Y, the set c([T.sup.[partial derivative]](y)) is a line segment. In fact, suppose that {[z.sub.1], [z.sub.2], [z.sub.3]} [subset] [T.sup.[partial derivative]](y) - [x.sub.z] and [z.sub.1] [not member of] [[z.sub.2], [z.sub.3]]. Since c([T.sup.[partial derivative]](y)) - [x.sub.z] is a convex set, we have co{[z.sub.1], [z.sub.2], [z.sub.3]} [subset] c([T.sup.[partial derivative]](y)) - [x.subz] [subset] [X.sub.z]. Then there exists [eta] > 0 such that

[mathematical expression not reproducible]. (38)

Since ([z.sub.1] + [z.sub.2] + [z.sub.3])/3 [member of] c([T.sup.[partial derivative]](y)) - [x.sub.z], there exists z [member of] c([T.sup.[partial derivative]](y)) - [x.sub.z] such that

[mathematical expression not reproducible]. (39)

Moreover, by formula (38), there exists t [member of] (1, +[infinity]) such that

[mathematical expression not reproducible], (40)

a contradiction. This implies that the set [T.sup.[partial derivative]](y) - [x.sub.z] is a line segment. Hence the set c([T.sup.[partial derivative]](y)) is a line segment.

(3d) From the proof of (3c), we obtain that the set c([T.sup.[partial derivative]](z)) is a line segment for all z [member of] Y. Let c([T.sup.[partial derivative]](z)) = [x(1, z), x(2, z)]. Define

[T.sup.[sigma]](z) = 1/2(x(1, z) + x(2, z)) (41)

for any z [member of] Y. We next will prove that [T.sup.[sigma]] is continuous at y, where

[mathematical expression not reproducible]. (42)

Let [y.sub.n] [right arrow] y as n [right arrow] [infinity]. Then

[mathematical expression not reproducible]. (43)

Since the set c([T.sup.[partial derivative]](y)) is a line segment for any y [member of] Y, there exist two sequences [{x(1, [y.sub.n])}.sup.[infinity].sub.n=1] and [{x(2, [y.sub.n])}.sup.[infinity].sub.n=1] such that

[mathematical expression not reproducible]. (44)

Since y [member of] {z [member of] Y : lim [inf.sub.h[right arrow]z] diam(c([T.sup.[partial derivative]](h))) [greater than or equal to] diam(c([T.sup.[partial derivative]](z)))}, we obtain that

[mathematical expression not reproducible]. (45)

We claim that lim [sup.sub.n[right arrow][infinity]] [parallel]x(1, [y.sub.n]) - x(2, [y.sub.n])[parallel] [less than or equal to] [parallel]x(1, y) - x(2, y)[parallel]. Otherwise, there exists a subsequence {[n.sub.k]} of {n} such that

[mathematical expression not reproducible]. (46)

Since F is upper semicontinuous, by the proof of (3a), we may assume without loss of generality that

[mathematical expression not reproducible]. (47)

This implies that

[mathematical expression not reproducible], (48)

which contradicts [mathematical expression not reproducible] and formula (45), we obtain that

[mathematical expression not reproducible] (49)

and [[x.sub.1], [x.sub.2]] = [x(1, y), x(2, y)]. Suppose that [T.sup.[sigma]] is not continuous at y. Then we may assume that there exists [delta] > 0 such that [parallel][T.sup.[sigma]]([y.sub.n]) - [T.sup.[sigma]](y)[parallel] [greater than or equal to] [delta] for all n [member of] N. Moreover, since [mathematical expression not reproducible], we may assume that [x.sub.1] = x(1, y) and [x.sub.2] = x(2, y). This implies that

[mathematical expression not reproducible] (50)

which contradicts [parallel][T.sup.[sigma]]([y.sub.n]) - [T.sup.[sigma]](y)[parallel] [greater than or equal to] [delta] for all n [member of] N. Hence we obtain that [T.sup.[sigma]] is continuous on {z [member of] Y : lim [inf.sub.h[right arrow]z]diam(c([T.sup.[partial derivative]](h))) [greater than or equal to] diam(c([T.sup.[partial derivative]](z)))}.

(3e) We next will prove that [T.sup.[sigma]] is a homogeneous selection of [T.sup.[partial derivative]]. Pick y [member of] Y. Then, by the previous proof, there exists x [member of] X such that Tx = [[P.sub.R(T)](y) and [T.sup.[partial derivative]](y) = x - [P.sub.N(T)](x). Since

[mathematical expression not reproducible], (51)

we have T([lambda]x) = [lambda]Tx = [lambda][P.sub.R(T)](y) = [P.sub.R(T)]([lambda]y). Therefore, by the definition of the set-valued metric generalized inverse, we have [T.sup.[partial derivative]]([lambda]y) = [lambda]x - [P.sub.N(T)]([lambda]x). Let c(x - [P.sub.N(T)](x)) = [[x.sub.1], [x.sub.2]]. Then c([P.sub.N(T)](x)) = [x - [x.sub.i], x - [x.sub.2]]. Let

[X.sub.0] = [[alpha]x + z : z [member of] N (T), [alpha] [member of] R}. (52)

Then [X.sub.0] is a closed subspace of X. Since X is a 3-strictly convex space, we obtain that [X.sub.0] is a 3-strictly convex space. Moreover, by the Hahn-Banach theorem, there exists [f.sub.x] [member of] S([X.sup.*.sub.0]) such that

N(T) = {z [member of] [X.sub.0] : [f.sub.x](z) = 1}. (53)

Since [X.sub.0] is a 3-strictly convex space, we obtain that [mathematical expression not reproducible] is compact. Therefore, by Theorem 8, we have [mathematical expression not reproducible]. Since [mathematical expression not reproducible], we obtain that

[mathematical expression not reproducible]. (54)

This implies that

c([T.sup.[partial derivative]]([lambda]y)) = c([lambda]x - [P.sub.N(T)]([lambda]x)) = [[lambda][x.sub.1], [lambda][x.sub.2]]. (55)

Therefore, by c(x - [P.sub.N(T)](x)) = [[x.sub.1], [x.sub.2]] and formula (55), we have [T.sup.[sigma]]([lambda]y) = ([lambda][x.sub.1] + [lambda][x.sub.2])/2 and [T.sup.[sigma]](y) = ([x.sub.1] + [x.sub.2])/2. It is easy to see that [T.sup.[sigma]]([lambda]y) = [lambda][T.sup.[sigma]](y). Hence there exists a homogeneous selection [T.sup.[sigma]] of [T.sup.[partial derivative]] such that [T.sup.[sigma]] is continuous on [z [member of] Y : lim [inf.sub.h[right arrow]z]diam(c([T.sup.[partial derivative]](h))) [greater than or equal to] diam(c([T.sup.[partial derivative]](z)))}, which completes the proof.

Corollary 13. Let X be a 2-strictly convex space, Y be a Banach space, D(T) be a closed subspace of X, and R(T) be an approximatively compact Chebyshev subspace of Y. Then

(1) [P.sub.N(T)] is upper semicontinuous if and only if [T.sup.[partial derivative]] is upper semicontinuous

(2) [P.sub.N(T)] is continuous if and only if [T.sup.[partial derivative]] is continuous

(3) If [P.sub.N(T)] is continuous, then there exists a homogeneous selection [T.sup.[sigma]] of [T.sup.[partial derivative]] such that [T.sup.[sigma]] is continuous on Y.

Proof. By Theorem 8, it is easy to see that (1) and (2) are true. Since X is a 2-strictly convex space, we obtain that [P.sub.N(T)](x) is a line segment for all x [member of] X (see [8]). Then c([T.sup.[partial derivative]](y)) is a singleton for all y [member of] Y. Therefore, by Theorem 12, we obtain that Corollary 13 is true.

Corollary 14. Let X be a strictly convex space, Y be a Banach space, D(T) be a closed subspace of X, and R(T) be an approximatively compact Chebyshev subspace of Y. Then the following statements are equivalent:

(1) [P.sub.N(T)] is upper semicontinuous.

(2) [P.sub.N(T)] is continuous.

(3) [T.sup.[partial derivative]] is a continuous homogeneous single-valued mapping.

Proof. By Corollary 13, it is easy to see that Corollary 14 is true.

Example 15. There exist a 3-strictly convex space X and a closed subspace H of X such that [P.sub.H] is continuous, where H is not a hyperplane of X. Let ([l.sup.2], [parallel] x [parallel]), where

[mathematical expression not reproducible]. (56)

Let

[mathematical expression not reproducible]. (57)

Let [{[x.sub.n]}.sup.4.sub.n=1] [subset] S(X) and [parallel][x.sub.1] + [x.sub.2] + [x.sub.3] + [x.sub.4][parallel] = 4. Then, by the Hahn-Banach theorem, there exists f = ([[eta].sub.1], [[eta].sub.1], ...) [member of] S([X.sup.*]) such that f([x.sub.1] + [x.sub.2] + [x.sub.3] + [x.sub.4]) = 4. Then f([x.sub.1]) = f([x.sub.2]) = f([x.sub.3]) = f([x.sub.4]) = 1. Let f(x) = [parallel]f[parallel] = [parallel]x[parallel] = 1, where x = ([[xi].sub.1], [[xi].sub.2], ...). Then, by Example 10, we obtain that [mathematical expression not reproducible] whenever i [greater than or equal to] 4. It is easy to see that dim span [A.sub.f] [less than or equal to] 3. Hence [x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4] are linearly dependent. This implies that X is 3-strictly convex. Hence [P.sub.H] is upper semicontinuous and X is 3-strictly convex. Pick x = ([[xi].sub.1], [[xi].sub.2], ...) [member of] X. Then

[mathematical expression not reproducible]. (58)

It is easy to see that [P.sub.H] is lower semicontinuous. Hence [P.sub.H] is continuous.

https://doi.org/10.1155/2017/7151430

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by "Foundation of Heilongjiang Educational Committee under Grant 12541187" and "China Natural Science Fund under Grant 11401084."

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Shaoqiang Shang (1) and Jingxin Zhang (2)

(1) Department of Mathematics, Northeast Forestry University, Harbin 150040, China

(2) Department of Mathematics and Applied Mathematics, Harbin University of Commerce, Harbin 150028, China

Correspondence should be addressed to Shaoqiang Shang; sqshang@163.com

Received 17 April 2017; Accepted 28 June 2017; Published 31 July 2017

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