Báo cáo hóa học: " Research Article Approximation of Fixed Points of Nonexpansive Mappings and Solutions of Variational " ppt

If E H, a real Hilbert space, the variational inequality problem reduces to the following.. 1.7 Starting with an arbitrary initial guess x0∈ H, let a sequence {x n} be generated by the

Volume 2008, Article ID 284345, 12 pages

doi:10.1155/2008/284345

Research Article

Approximation of Fixed Points of Nonexpansive

Mappings and Solutions of Variational Inequalities

C E Chidume, 1 C O Chidume, 2 and Bashir Ali 3

1 The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy

2 Department of Mathematics and Statistics, College of Sciences and Mathematics, Auburn University, Auburn, AL 36849, USA

3 Department of Mathematical Sciences, Bayero University, 3011 Kano, Nigeria

Correspondence should be addressed to C E Chidume, chidume@ictp.it

Received 3 July 2007; Accepted 17 October 2007

Recommended by Siegfried Carl

Let E be a real q-uniformly smooth Banach space with constant d q , q ≥ 2 Let T : E → E and

G : E → E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian,

respectively Let{λ n } be a real sequence in 0, 1 that satisfies the following condition: C1: lim λ n 0 and 

λ n  ∞ For δ ∈ 0, qη/d q k q1/q−1  and σ ∈ 0, 1, define a sequence {x n } iteratively in E

by x0∈ E, x n1  T λ n1 x n  1 − σx n  σTx n − δλ n1 GTx n , n ≥ 0 Then, {x n} converges strongly

to the unique solution x∗of the variational inequality problem VIG, K search for x ∗ ∈ K such

thatGx∗, j q y − x∗ ≥ 0 for all y ∈ K, where K : FixT  {x ∈ E : Tx  x} / ∅ A convergence

theorem related to finite family of nonexpansive maps is also proved.

Copyright q 2008 C E Chidume et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Let E be a real-normed space and let E∗be its dual space For some real number q 1 < q < ∞, the generalized duality mapping J q : E → 2 E∗is defined by

J q x f∗∈ E∗:x, f∗  x q ,f∗  x q−1

where·, · denotes the pairing between elements of E and elements of E∗.

Let K be a nonempty closed convex subset of E, and let S : E → E be a nonlinear operator The variational inequality problem is formulated as follows Find a point x∗ ∈ K

such that

VIS, K :Sx∗, j q



y − x∗

≥ 0 ∀y ∈ K. 1.2

If E  H, a real Hilbert space, the variational inequality problem reduces to the following Find

a point x∗∈ K such that

VIS, K :Sx∗, y − x∗

A mapping G : DG ⊂ E → E is said to be accretive if for all x, y ∈ DG, there exists j q x −y ∈

J q x − y such that



where DG denotes the domain of G For some real number η > 0, G is called η-strongly

accre-tive if for all x, y ∈ DG, there exists j q x − y ∈ J q x − y such that



Gx − Gy, j q x − y ≥ η x − y q 1.5

G is κ-Lipschitzian if for some κ > 0, Gx − Gy ≤ κ x − y for all x, y ∈ DG and G is called nonexpansive if k  1.

In Hilbert spaces, accretive operators are called monotone where inequalities 1.4 and

1.5 hold with j q replaced by the identity map of H.

It is known that if S is Lipschitz and strongly accretive, then VIS, K has a unique

solu-tion An important problem is how to find a solution of VIS, K whenever it exists Consid-erable efforts have been devoted to this problem see, e.g., 1,2 and the references contained therein

It is known that in a real Hilbert space, the VIS, K is equivalent to the following fixed-point equation:

x∗ P Kx∗− δSx∗

where δ > 0 is an arbitrary fixed constant and P K is the nearest point projection map from H onto K, that is, P K x  y, where x − y  inf u∈K x − u for x ∈ H Consequently, un-der appropriate conditions on S and δ, fixed-point methods can be used to find or

approx-imate a solution of VIS, K For instance, if S is strongly monotone and Lipschitz, then a

mapping G : H → H , defined by Gx  P K x − δSx, x ∈ H with δ > 0 sufficiently small,

is a strict contraction Hence, the Picard iteration, x0 ∈ H, x n1  Gx n , n ≥ 0 of the classical

Banach contraction mapping principle, converges to the unique solution of the VIK, S

It has been observed that the projection operator P Kin the fixed-point formulation1.6 may make the computation of the iterates difficult due to possible complexity of the convex set

K In order to reduce the possible difficulty with the use of P K , Yamada 2 recently introduced

a hybrid descent method for solving the VIK, S Let T : H → H be a map and let K : {x ∈

H : Tx  x} /  ∅ Let S be η-strongly monotone and κ-Lipschitz on H Let δ ∈ 0, 2η/κ2 be arbitrary but fixed real number and let a sequence{λ n } in 0, 1 satisfy the following conditions:

C1: lim λ n  0; C2: λ n  ∞; C3: lim λ n − λ n1

λ2

n

 0. 1.7

Starting with an arbitrary initial guess x0∈ H, let a sequence {x n} be generated by the follow-ing algorithm:

x n1  Tx n − λ n1 δS

Tx n



Then, Yamada2 proved that {x n } converges strongly to the unique solution of VIK, S.

In the case that K  r i1 FT i  / ∅, where {T i}r

i1 is a finite family of nonexpansive mappings, Yamada2 studied the following algorithm:

x n1  T n1 x n1 − λ n1 δS

T n1 x n

where T k  T k mod r for k ≥ 1, with the mod function taking values in the set {1, 2, , r}, where

the sequence {λ n } satisfies the conditions C1, C2, and C4:|λ n − λ nN | < ∞ Under these

conditions, he proved the strong convergence of{x n } to the unique solution of the VIK, S.

Recently, Xu and Kim1 studied the convergence of the algorithms 1.8 and 1.9, still

in the framework of Hilbert spaces, and proved strong convergence with condition C3 replaced

by C5: limλ n − λ n1 /λ n1   0 and with condition C4 replaced by C6: limλ n − λ nr /λ nr 

0 These are improvements on the results of Yamada In particular, the canonical choice λ n :

1/n  1 is applicable in the results of Xu and Kim but is not in the result of Yamada 2 For further recent results on the schemes1.8 and 1.9, still in the framework of Hilbert spaces, the reader my consult Wang3, Zeng and Yao 4, and the references contained in them Recently, the present authors5 extended the results of Xu and Kim 1 to q-uniformly smooth Banach spaces, q ≥ 2 In particular, they proved theorems which are applicable in L p

spaces, 2≤ p < ∞ under conditions C1, C2, and C5 or C6 as in the result of Xu and Kim.

It is our purpose in this paper to modify the schemes1.8 and 1.9 and prove strong convergence theorems for the unique solution of the variational inequality VIK, S

Further-more, in the case T i : E → E, i  1, 2, , r, is a family of nonexpansive mappings with

K  r i1 FT i  / ∅, we prove a convergence theorem where condition C6 is replaced by

limn→∞ T n1 x n − T n x n  0 An example satisfying this condition is given see, for example,

6 All our theorems are proved in q-uniformly smooth spaces, q ≥ 2 In particular, our theo-rems are applicable in L pspaces, 2≤ p < ∞.

2 Preliminaries

Let E be a real Banach space and let K be a nonempty, closed, and convex subset of E Let P

be a mapping of E onto K Then, P is said to be sunny if P P x  tx − P x  P x for all x ∈ E and t ≥ 0 A mapping P of E into E is said to be a retraction if P2  P A subset K is said to be

sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto K A

retraction P is said to be orthogonal if for each x, x − P x is normal to K in the sense of James

7

It is well knownsee 8 that if E is uniformly smooth and there exists a nonexpansive retraction of E onto K, then there exists a nonexpansive projection of E onto K If E is a real smooth Banach space, then P is an orthogonal retraction of E onto K if and only if P x ∈ K and

Px − x, j q Px − y ≤ 0 for all y ∈ K It is also known see, e.g., 9 that if K is a convex

subset of a uniformly convex Banach space whose norm is uniformly Gˆateaux differentiable

and T : K → K is nonexpansive with FT /  ∅, then FT is a nonexpansive retract of K Let K be a nonempty closed convex and bounded subset of a Banach space E and let the diameter of K be defined by dK : sup{ x − y : x, y ∈ K} For each x ∈ K, let rx, K :

sup{ x − y : y ∈ K} and let rK : inf{rx, K : x ∈ K} denote the Chebyshev radius

of K relative to itself The normal structure coefficient NE of E see, e.g., 10 is defined by

NE : inf{d K/rK : K is a closed convex and bounded subset of E with dK > 0}.

A space E such that NE > 1 is said to have uniform normal structure It is known that all

uniformly convex and uniformly smooth Banach spaces have uniform normal structuresee, e.g.,11,12

We will denote a Banach limit by μ Recall that μ is an element of l∞∗such that μ 

1, lim inf n→∞ a n ≤ μ n a n ≤ lim supn→∞ a n and μ n a n  μ n1 a n for all {a n}n≥0 ∈ l∞ see, e.g.,

11,13

Let E be a normed space with dim E ≥ 2 The modulus of smoothness of E is the function

ρ E:0, ∞ → 0, ∞ defined by

ρ E τ : sup x  y  x − y

2 − 1 : x  1; y  τ 2.1

The space E is called uniformly smooth if and only if lim t→0ρ E t/t  0 For some positive constant q, E is called q-uniformly smooth if there exists a constant c > 0 such that ρ E t ≤ ct q,

t > 0 It is known that

L porl p spaces are

 2-uniformly smooth if 2≤ p < ∞,

p-uniformly smooth if 1 < p ≤ 2 2.2

see, e.g., 13 It is well known that if E is smooth, then the duality mapping is singled-valued, and if E is uniformly smooth, then the duality mapping is norm-to-norm uniformly continuous

on bounded subset of E.

We will make use of the following well-known results

Lemma 2.1 Let E be a real-normed linear space Then, the following inequality holds:

x  y 2 ≤ x 2 2y, jx  y

∀x, y ∈ E, ∀jx  y ∈ Jx  y. 2.3

In the sequel, we will also make use of the following lemmas

Lemma 2.2 see 14 Let a0, a1,  ∈ l∞ such that μ n a n  ≤ 0 for all Banach limit μ and

lim supn→∞ a n1 − a n  ≤ 0 Then, lim sup n→∞ a n ≤ 0.

Lemma 2.3 see 15 Let {x n } and {y n } be bounded sequences in a Banach space E and let {β n } be

a sequence in 0, 1 with 0 < lim inf β n ≤ lim sup β n < 1 Suppose x n1  β n y n  1 − β n x n for all integers n ≥ 0 and lim sup y n1 − y n − x n1 − x n  ≤ 0 Then, lim y n − x n  0.

Lemma 2.4 see 16 Let {a n } be a sequence of nonnegative real numbers satisfying the following

relation:

a n1≤1− α na n  α n σ n  γ n , n ≥ 0, 2.4

where i {α n } ⊂ 0, 1,α n  ∞; ii lim sup σ n ≤ 0; iii γ n ≥ 0; n ≥ 0,γ n < ∞ Then, a n→ 0

as n → ∞.

Lemma 2.5 see 17 Let E be a real q-uniformly smooth Banach space for some q > 1, then there

exists some positive constant d q such that

x  y q ≤ x q  qy, j q x  d q y q ∀x, y ∈ E, j q x ∈ J q x. 2.5

Lemma 2.6 see 12, Theorem 1 Suppose E is a Banach space with uniformly normal structure,

K is a nonempty bounded subset of E, and T : K → K is uniformly k-Lipschitzian mapping with

k < NE 1/2 Suppose also that there exists a nonempty bounded closed convex subset of C of K with the following propertyP:

where ω w x is the ω-limi set of T at x, that is, the set



y ∈ E : y  weak-lim

j T n j x for some n j −→ ∞. 2.6

Then, T has a fixed point in C.

3 Main results

We first prove the following lemma which will be central in the sequel

Lemma 3.1 Let E be a real q-uniformly smooth Banach space with constant d q , q ≥ 2 Let T : E → E and G : E → E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian, respectively For δ ∈ 0, qη/d q κ q1/q−1 , σ ∈ 0, 1, and λ ∈ 0, 2/pp − 1, define a map T λ : E →

E by T λ x  1 − σx  σTx − λδGTx, x ∈ E Then, T λ is a strict contraction Furthermore,

T λ x − T λ y  ≤ 1 − λα x − y , x,y ∈ E, 3.1

where α  q/2 −

q2/4 − σδqη − δ q−1 d q κ q  ∈ 0, 1.

Proof For x, y ∈ E,

T λ x − T λ yq1 − σx − y  σTx − Ty − λδGTx − GTy q

≤ 1 − σ x − y q  σ Tx − Ty q − qλδGTx − GTy, j q Tx − Ty

 d q λ q δ qGTx − GTyq

≤ 1 − σ x − y q  σ Tx − Ty q − qλδη Tx − Ty q  d q λ q δ q κ q Tx − Ty q

≤1− σλδqη − d q λ q−1 δ q−1 κ q

x − y q

≤1− σλδqη − d q δ q−1 κ q

x − y q

3.2 Define

fλ : 1 − σλδ

qη − d q δ q−1 κ q

 1 − λτ q for some τ ∈ 0, 1 say. 3.3

Then, there exists ξ ∈ 0, λ such that

1− σλδqη − d q δ q−1 κ q

 1 − qτλ 1

2qq − 11 − ξτ q−2 λ2τ2. 3.4

This implies that

1− σλδqη − d q δ q−1 κ q

≤ 1 − qτλ 1

2qq − 1λ2τ2. 3.5

Then, we have τ ≤ q/2 −



q2/4 − σδqη − d q δ q−1 κ q .

Set

α : q

2 −



q2

4 − σδqη − d q δ q−1 κ q

and the proof is complete

We note that in L pspaces, 2≤ p < ∞, the following inequality holds see, e.g., 13 For

each x, y ∈ L p, 2≤ p < ∞,

x  y 2≤ x 2 2y, jx

 p − 1 y 2. 3.7 Using this inequality and following the method of proof ofLemma 3.1, the following corollary

is easily proved

Corollary 3.2 Let E  L p , 2 ≤ p < ∞ Let T : E → E, G : E → E be a nonexpansive map, an

η-strongly monotone, and κ-Lipschitzian map, respectively For λ, σ ∈ 0, 1 and δ ∈ 0, 2η/p − 1κ2,

define a map T λ : E → E by T λ x  1 − σx  σTx − λδGTx, x ∈ E Then, T λ is a contraction In particular,

T λ x − T λ y  ≤ 1 − λα x − y , x,y ∈ H, 3.8

where α  1 −

1− σδ2η − p − 1δκ2 ∈ 0, 1.

Corollary 3.3 Let H be a real Hilbert space, T : H → H, G : H → H a nonexpansive map and an

η-strongly monotone map which is also κ-Lipschitzian, respectively For λ, σ ∈ 0, 1 and δ ∈ 0, 2η/κ2,

define a map T λ : H → H by T λ x  1 − σx  σTx − λδGTx, x ∈ H Then, T λ is a contraction.

In particular,

T λ x − T λ y  ≤ 1 − λα x − y , x,y ∈ H, 3.9

where α  1 −

1− σδ2η − δκ2 ∈ 0, 1.

Proof Set p  2 inCorollary 3.2and the result follows

Corollary 3.3is a result of Yamada2 and is the main tool used in 1 4

We now prove our main theorems

Theorem 3.4 Let E be a real q-uniformly smooth Banach space with constant d q , q ≥ 2 Let T : E → E and G : E → E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian, respectively Let {λ n } be a real sequence in 0, 1 satisfying

C1: lim λ n 0; C2: 

For δ ∈ 0, qη/d q κ q1/q−1  and σ ∈ 0, 1, define a sequence {x n } iteratively in E by x0∈ E,

x n1  T λ n1 x n  1 − σx n  σTx n − δλ n1 G

Tx n



Then, {x n } converges strongly to the unique solution x∗of the variational inequality VI G, K.

Proof Let x∗∈ K : Fix T, then the sequence {x n} satisfies

x n − x∗ ≤ maxx

0− x∗, δ

αG

It is obvious that this is true for n  0 Assume that it is true for n  k for some k ∈ N.

From the recursion formula3.11, we have

x k1 − x∗  T λ k1 x k − x∗

≤T λ k1 x k − T λ k1 x∗  T λ k1 x∗− x∗

≤1− λ k1 αx k − x∗  λ k1 δG

x∗

≤ maxx

0− x∗, δ

αG

x∗ ,

3.13

and the claim follows by induction Thus, the sequence{x n } is bounded and so are {Tx n} and

{GTx n }.

Define two sequences{β n } and {y n } by β n: 1−σλn1 σ and y n: xn1 −x n β n x n /β n

Then,

y n 1 − σλ n1 x n  σ



Tx n − λ n1 δG

Tx n



Observe that{y n} is bounded and that

y n1 − y n  − x n1 − x n

≤

β σ n1 − 1

x n1 − x n  σ

β n1− σ

β n



Tx n   λ n2 1 − σ

β n1

x n1 − x n

 1 − σ

λ n2

β n1 −λ n1

β n



x n   λ n1 σδ

β n

G

Tx n



− GTx n1   σδλ n1

β n − λ n2

β n1



G

Tx n1.

3.15 This implies that lim supn→∞ ||y n1 − y n || − ||x n1 − x n || ≤ 0, and byLemma 2.3,

lim

n→∞y n − x n   0. 3.16 Hence,

x n1 − x n   β ny n − x n  −→ 0 as n −→ ∞. 3.17 From the recursion formula3.11, we have that

σx n1 − Tx n  ≤ 1 − σx n1 − x n   λ n1 σδG

Tx n  −→ 0 as n −→ ∞, 3.18 which implies that

x n1 − Tx n  −→ 0 as n −→ ∞. 3.19

From3.17 and 3.19, we have

x n − Tx n  ≤ x n − x n1   x n1 − Tx n  −→ 0 as n −→ ∞. 3.20

We now prove that lim supn→∞ −Gx∗, jx n1 − x∗ ≤ 0.

Define a map φ : E → R by

Then, φx → ∞ as x → ∞, φ is continuous and convex, so as E is reflexive, there exists

y∗∈ E such that φy∗  minu∈E φu Hence, the set

K∗:x ∈ E : φx  min

u∈E φu

/

By Lemma 2.6, K∗∩ K / ∅ Without loss of generality, assume that y∗  x∗ ∈ K∗∩ K Let

t ∈ 0, 1 Then, it follows that φx∗ ≤ φx∗− tGx∗ and usingLemma 2.1, we obtain that

x n − x∗ tGx∗2≤x n − x∗2 2tG

x∗

, j

x n − x∗ tGx∗

3.23 which implies that

μ n



− Gx∗

, j

x n − x∗ tGx∗

Moreover,

μ n



− Gx∗

, j

x n − x∗

 μ n− Gx∗

, j

x n − x∗

− jx n − x∗ tGx∗

 μ n



− Gx∗

, j

x n − x∗ tGx∗

≤ μ n− Gx∗

, j

x n − x∗

− jx n − x∗ tGx∗

.

3.25

Since j is norm-to-norm uniformly continuous on bounded subsets of E, we have that

μ n



− Gx∗

, j

x n − x∗

Furthermore, since x n1 − x n → 0, as n → ∞, we also have

lim sup

n→∞



− Gx∗

, j

x n − x∗ −− Gx∗

, j

x n1 − x∗ ≤ 0, 3.27

and so we obtain byLemma 2.2that lim supn→∞ −Gx∗, jx n − x∗ ≤ 0.

From the recursion formula3.11 andLemma 2.1, we have

x n1 − x∗2T λ n1 x n − T λ n1 x∗ T λ n1 x∗− x∗2

≤T λ n1 x n − T λ n1 x∗2 2λ n1 δ

− Gx∗

, j

x n1 − x∗

≤1− λ n1 αx n − x∗2 2λ n1 δ

− Gx∗

, j

x n1 − x∗ ,

3.28

and byLemma 2.4, we have that x n → x∗as n → ∞ This completes the proof.

The following corollaries follow fromTheorem 3.4.

Corollary 3.5 Let E  L p , 2 ≤ p < ∞ Let T : E → E and G : E → E be a nonexpansive map and an

η-strongly accretive map which is also κ-Lipschitzian, respectively Let {λ n } be a real sequence in 0, 1

that satisfies conditions C1 and C2 as in Theorem 3.4 For δ ∈ 0, 2η/p − 1κ2 and σ ∈ 0, 1, define

a sequence {x n } iteratively in E by 3.11 Then, {x n } converges strongly to the unique solution x∗of the variational inequality VI G, K.

Corollary 3.6 Let E  H be a real Hilbert space Let T : H → H and G : H → H be a nonexpansive

map and an η-strongly monotone map which is also κ-Lipschitzian, respectively Let {λ n } be a real

sequence in 0, 1 that satisfies conditions C1 and C2 as in Theorem 3.4 For δ ∈ 0, 2η/κ2 and

σ ∈ 0, 1, define a sequence {x n } iteratively in H by 3.11 Then, {x n } converges strongly to the

unique solution x∗of the variational inequality VI G, K.

Finally, we prove the following more general theorem

Theorem 3.7 Let E be a real q-uniformly smooth Banach space with constant d q , q ≥ 2 Let T i :

E → E, i  1, 2, , r, be a finite family of nonexpansive mappings with K : r i1FixTi  / ∅ Let

G : E → E be an η-strongly accretive map which is also κ-Lipschitzian Let {λ n } be a real sequence in

0, 1 satisfying

C1: lim λ n 0; C2: 

For a fixed real number δ ∈ 0, qη/d q κ q1/q−1 , define a sequence {x n } iteratively in E by x0∈ E :

x n1  T λ n1

n1 x n  1 − σx n  σT n1 x n − δλ n G

T n1 x n

, n ≥ 0, 3.30

where T n  T n mod r Assume also that

K  Fix

T r T r−1 · · · T1



 FixT1T r · · · T2



 · · ·  FixT r−1 T r−2 · · · T r



3.31

and lim n→∞ T n1 x n − T n x n  0 Then, {x n } converges strongly to the unique solution x∗ of the variational inequality VI G, K.

Proof Let x∗∈ K, then the sequence {x n} satisfies that

x n − x∗ ≤ maxx0− x∗, δ

αG

It is obvious that this is true for n  0 Assume it is true for n  k for some k ∈ N.

From the recursion formula3.30, we have

x k1 − x∗  T λ k1

k1 x k − x∗

≤T λ k1

k1 x k − T λ k1

k1 x∗  T λ k1

k1 x∗− x∗

≤1− λ k1 αx k − x∗  λ k1 δG

x∗

≤ maxx

0− x∗, δ

αG

x∗ ,

3.33

and the claim follows by induction Thus, the sequence {x n } is bounded and so are {T n x n} and{GT n x n }.

Define two sequences{β n } and {y n } by β n: 1−σλn1 σ and y n: xn1 −x n β n x n /β n

Then,

y n 1 − σλ n1 x n  σ



T n1 x n − λ n1 δG

T n1 x n



Observe that{y n} is bounded and that

y n1 − y n  − x n1 − x n ≤ σ

β n1 − 1

x n1 − x n

 σ

β n1

T n2 x n − T n1 x n  σ

β n1− σ

β n



T n1 x n

 λ n2 1 − σ

β n1

x n1 − x n   1 − σλ n2

β n1 − λ n1

β n



x n

 λ n1 σδ

β n

G

T n1 x n



− GT n2 x n1

 σδ

λ n1

β n −λ n2

β n1



G

T n2 x n1.

3.35

This implies that lim supn→∞  y n1 − y n − x n1 − x n  ≤ 0, and byLemma 2.3,

lim

n→∞y n − x n   0. 3.36 Hence,

x n1 − x n   β ny n − x n  −→ 0 as n −→ ∞. 3.37 From the recursion formula3.30, we have that

σx n1 − T n1 x n  ≤ 1 − σx n1 − x n   λ n1 σδG

T n1 x n  −→ 0 as n −→ ∞ 3.38

which implies that

From3.37 and 3.39, we have

x n − T n1 x n  ≤ x n − x n1   x n1 − T n1 x n  −→ 0 as n −→ ∞. 3.40 Also,

x nr − x n  ≤ x nr − x nr−1   x nr−1 − x nr−2   ···  x n1 − x n, 3.41 and so

x nr − x n  −→ 0 as n −→ ∞. 3.42

Proof Let x∗∈...