# On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

Authors: F. O. Isiogugu; P. Pillay; P. U. Nwokoro
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed pointsF(T) of a multivalued (or single-valued) k-strictly pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence {Kn}n=1∞ of closed convex subsets of H from an arbitrary x0∈H and a sequence {xn}n=1∞ of the metric projections of x0 into Kn. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space with an inner product 〈.,.〉 and a norm ., respectively and let K be a nonempty closed convex subset of H. Let A:H→H be an operator on H and F:K×K→R be a bifunction on K, where R is the set of real numbers. The variational inequality problem of A in K denoted by VIP(A,K) is to find an x∗∈K such that (1)x-x∗,Ax∗≥0,∀x∈K,while the equilibrium problem for F is to find x∗∈K such that (2)Fx∗,x≥0,∀x∈K.The set of solutions of (2) is denoted by EP(F). Suppose F(x,y)=〈y-x,Ax〉 for all x,y∈K, then w∈EP(F) if and only if w is a solution of (1). Many problems in optimization, economics, and physics reduce to finding a solution of (1) (see, for example, [1–3]) and the references therein. The following conditions are assumed for solving the equilibrium problems for a bifunction F:K×K→R,(A1)
F(x,x)=0 for all x∈K.(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0, for all x,y∈K.(A3)
For eachx,y,z∈K, limt↓0F(tz+(1-t)x,y)≤F(x,y).(A4)
For eachx∈K, y↦F(x,y) is convex and lower semicontinuous.Several algorithms have been introduced by authors for approximating the solutions of an equilibrium problem for a bifunction (or the common elements of the sets of solutions of equilibrium problems for a finite family of bifunctions). Many authors have also approximated the common elements of the set of fixed pointsF(T) of a multivalued (or single-valued) mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F (or the common elements of the sets of fixed points of a finite family of multivalued (or single-valued) mappings and the sets of solutions of equilibrium problems for a finite family of bifunctions) (see, for example, [4–10] and references therein). In a real Hilbert space, many authors have studied the algorithms involving the construction of the sequences of sets {Kn}n=1∞ and the metric projections {xn}n=1∞, from an arbitrary x0∈H, where Kn+1={z∈Kn:z-un2≤z-xn2}, xn+1=PKn+1x0, while PKn is the projection map and {un}n=1∞ is the sequence of the resolvent of the bifunctions, (see, for example, [4, 9] and references therein).Among the iteration schemes studied are the modified Reich-Sabach-type Algorithm 1 and Mann-Reich-Sabach-type Algorithm 2 below defined for the approximation of (i) the solutions of an equilibrium problem for a bifunction; (ii) the common elements of the set of fixed pointsF(T) of a multivalued (or single-valued) mapping T, and the set of solutions EP(F) of an equilibrium problem for a bifunction F respectively.(i) LetH be a real Hilbert space and K a closed and convex subset of H. Let F:K×K→R be a bifunction and r∈[a,∞) for some a>0. Then from an arbitrary x0∈H the algorithm is generated as follows.Algorithm 1.
(3) y n = x n , u n ∈ K such  that F u n , y + 1 r y - u n , u n - y n ≥ 0 ,  ∀ y ∈ K , K n + 1 = z ∈ K n : z - u n 2 ≤ z - x n 2 x n + 1 = P K n + 1 x 0 .(ii) Let H be a real Hilbert space, K a closed and convex subset of H, F:K×K→R a bifunction, and T:K→P(K) multivalued k-strictly pseudocontractive-type mapping. Let {αn}n=1∞⊂[0,1] and r∈[a,∞) for some a>0. Then from an arbitrary x0∈H the algorithm is generated as follows.Algorithm 2.
(4) y n = α n x n + 1 - α n v n , u n ∈ K such  that F u n , y + 1 r y - u n , u n - y n ≥ 0 ,  ∀ y ∈ K , K n + 1 = z ∈ K n : z - u n 2 ≤ z - x n 2 x n + 1 = P K n + 1 x 0 ,where vn∈Txn for multivalued mapping T.However, despite the fact that most of these algorithms yield strong convergence theoretically, the difficulty encountered by computers with the construction of the sequence of the metric projection{xn}n=1∞ and the sequence of sets {Kn}n=1∞ has made such algorithms almost impossible for real life applications. This noncomputability and nonapplicability of such algorithms has led to the introduction of other algorithms which do not involve the construction of these two sequences but require stronger conditions and many parameters in the hypothesis of their convergence theorems.One of these important algorithms is the algorithm of Zhaoli Ma et al. [10].The purpose of this research is to develop a computable version of Algorithms 1 and 2. In particular, it is established that given the modified Reich-Sabach-types Algorithm1 for approximating the solutions of an equilibrium problem EP(F) for a bifunction F:K×K→R in a real Hilbert space H involving the construction of the sequences {Kn}n=1∞ and {xn}n=1∞ from an arbitrary x0∈H, where K=K0 is a closed and convex subset of H, Kn+1={z∈Kn:z-un2≤z-xn2}, and xn+1=PKn+1x0, while PKn is the metric projection of H into Kn and {un}n=1∞ is the sequence of the resolvents of the bifunction; there exists a selection {kn}n=1∞ of {Kn}n=1∞ which converges strongly to a solution of the equilibrium problem. Furthermore, if the norm on H is order inclusion transitive on the closed convex subsets CC(H) of H, then xn+1=PKn+1x0 and the selection converges strongly to PEP(F)x0. Where a norm . on a Hilbert space H is order inclusion transitive on CC(H) if given any A,B∈CC(H) with A⊆B and arbitrary x∈H, then d(x,B)=infb¯∈Bb¯-x=b-x and d(b,A)=infa¯∈Aa¯-b=a-b imply that d(x,A)=infa¯∈Aa¯-x=a-x and EP(F) is the set of the solutions of the equilibrium problem for the bifunction. Also if we set yn=αnxn+(1-αn)vn in Algorithm 1, where {αn}n=1∞⊂[0,1] satisfying some conditions and T:K→P(K) is a multivalued k-strictly pseudocontractive-type mapping a similar selection existing as well which is a selection of Algorithm 2, the numerical example of the computation is presented for the selection of Algorithm 2 which is the generalization of the selections of Algorithm 1. The results of this research are great contributions towards the resolution of the controversy over the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of the sequences {Kn}n=1∞ and {xn}n=1∞ above.


## 2. Preliminaries

LetX be a nonempty set and let T:X→X be a map. A point x∈X is called a fixed point of T if x=Tx. If T:X→2X is a multivalued map from X into the family of nonempty subsets of X, then x is a fixed point of T if x∈Tx. If Tx={x}, x is called a strict fixed point of T. The set F(T)={x∈D(T):x∈Tx} (respectively, F(T)={x∈D(T):x=Tx}) is called the fixed point set of multivalued (respectively, single-valued) map T while the set Fs(T)={x∈D(T):Tx={x}} is called the strict fixed point set of T.LetX be a normed space. A subset K of X is called proximinal if for each x∈X there exists k∈K such that (5)x-k=inf⁡x-y:y∈K=dx,K.It is known that every closed convex subset of a uniformly convex Banach space is proximinal. We shall denote the family of all nonempty closed and bounded subsets of X by CB(X), the family of all nonempty subsets of X by 2X, the family of all nonempty closed and convex subsets of X by CC(X), and the family of all proximinal subsets of X by P(X), for a nonempty set X.LetH denote the Hausdorff metric induced by the metric d on X; that is, for every A,B∈CB(X), (6)HA,B=max⁡supa∈A⁡da,B,supb∈B⁡db,A.Let X be a normed space. Let T:D(T)⊆X→2X be a multivalued mapping on X. A multivalued mapping T:D(T)⊆X→2X is called L-Lipschitzian if there exists L≥0 such that for all x,y∈D(T)(7)HTx,Ty≤Lx-y.In (7), if L∈[0,1)T is said to be a contraction while T is nonexpansive if L=1.Definitions 3 (see [11]).
T is said to be k-strictly pseudocontractive-type of Isiogugu [11] if there exists k∈(0,1) such that given any pair x,y∈D(T) and u∈Tx, there exists v∈Ty satisfying u-v≤H(Tx,Ty) and (8)H2Tx,Ty≤x-y2+kx-u-y-v2.Definitions 4.
A multivalued mapT:D(T)⊆X→2X is said to be of type-one (see for example [12, 13]) if given any pair x,y∈D(T), then (9)u-v≤HTx,Ty,for  allu∈PTx,v∈PTy.Lemma 5 (see [14]).
LetH be a real Hilbert space and let T:D(T)⊆H→2H be a k-strictly pseudocontractive-type mapping. Then T is an L-Lipschitzian.Lemma 6 (see [7]).
LetK be a nonempty subset of a real Hilbert space H and let T:K→P(K) be a k-strictly pseudocontractive-type mapping such that Fs(T) is nonempty. Then Fs(T) is closed and convex.Lemma 7.
LetH be a real Hilbert space and let K be a nonempty closed convex subset of H. Let PK be the convex projection onto K. Then, convex projection is characterized by the following relations:(i)
x∗=PK(x)⇔〈x-x∗,y-x∗〉≤0, for all y∈K.(ii)
x-PKx2≤x-y2-y-PKx2.(iii)
x-PKy2≤x-y2-PKy-y2.Lemma 8 (see [1]).
LetK be a nonempty closed convex subset of a real Hilbert space H and F:K×K→R a bifunction satisfying (A1)-(A4). Let r>0 and x∈H. Then, there exists z∈K such that (10)Fz,y+1ry-z,z-x≥0,∀y∈K.Lemma 9 (see [2]).
LetK be a nonempty closed convex subset of a real Hilbert space H. Assume that F:K×K→R satisfies (A1)-(A4). Let r>0 and x∈H; define Tr:H→2K by (11)Trx=z∈K:Fz,y+1ry-z,z-x≥0,∀y∈K. Then the following conditions hold:(1)
Tr is single-valued.(2)
Tr is firmly nonexpansive, that is for any x,y∈H, Trx-Try2≤〈Trx-Try,x-y〉.(3)
F(Tr)=EP(F).(4)
EP(F) is closed and convex.Lemma 10 (see [15]).
LetK be a nonempty closed convex subset of a real Hilbert space H and F:K×K→R a bifunction satisfying (A1)-(A4). Let r>0 and x∈H. Then for all x∈H and p∈F(Tr)(12)p-Trx2+Trx-x2≤p-x2.Definition 11 (see [16, 17]).
LetE be a Banach space. Let T:D(T)⊆E→2E be a multivalued mapping. I-T is said to be weaklydemiclosed at zero if for any sequence {xn}n=1∞⊆D(T) such that {xn} converges weakly to p and a sequence {yn} with yn∈Txn for all n∈N such that {xn-yn} converges strongly to zero. Then p∈Tp (i.e., 0∈(I-T)p).


## 3. Main Results

Definition 12.
Let{Kn}n=1∞ be sequence of sets. Then a sequence {zn}n=1∞ is called a selection of {Kn}n=1∞ if zn∈Kn for each n.Definition 13.
A norm. on a Hilbert space H is order inclusion transitive on CC(H) if given any A,B∈CC(H) with A⊆B and arbitrary x∈H, then d(x,B)=infb¯∈Bb¯-x=b-x and d(b,A)=infa¯∈Aa¯-b=a-b imply that d(x,A)=infa¯∈Aa¯-x=a-x (i.e., if b is the point in B closet to x and a is the point in A closest to b then a is the point in A closest to x).Definition 14.
A HilbertH is said to have order inclusion transitive property on CC(H) if its norm is order inclusion transitive on CC(H).
It is easy to see that the set of real numbers with the usual norm has order inclusion transitive property.Proposition 15.
In the definition of the setKn+1≔{z∈Kn:z-un2≤z-xn2}, if we define xn+1=1/2(un+xn), then the following conditions are true:(C1)
{xn}n=1∞ is a selection of {Kn}n=1∞.(C2)
xn+1=PKn+1xn.(C3)
IfH has order inclusion transitive property on CC(H) then, xn+1=PKn+1x0Proof.
( C 1 ) (13) x n + 1 - u n 2 = 1 2 u n + x n - u n 2 = 1 4 u n - x n 2 and (14)xn+1-xn2=12un+xn-xn2=14un-xn2. Therefore, xn+1∈Kn+1; thus, {xn}n=1∞ is a selection of {Kn}n=1∞.
( C 2 ) Let y∈Kn+1 arbitrary. Then (15)xn-xn+1,y-xn+1=xn-12un+xn,y-12un+xn=12xn-un,-12un-y+xn-y=14xn-un,-un-y-xn-y=14xn-y+y-un,y-un+y-xn=14xn-y,y-un+xn-y,y-xn+y-un,y-un+y-un,y-xn=14xn-y,y-un-y-xn,y-xn+y-un,y-un+y-un,y-xn=14xn-y,y-un-y-xn2+y-un2+y-un,y-xn=14-y-xn,y-un-y-xn2+y-un2+y-xn,y-un≤14-y-xn,y-un-y-xn2+y-xn2+y-xn,y-un=0. Consequently, xn+1=PKn+1xn.
( C 3 ) Since Kn+1⊆Kn and Kn is closed and convex for each n, condition C2 and order inclusion transitive property of H on CC(H) guarantee that xn+1=PKn+1x0.We now consider the following algorithm which we shall refer to as a selection of Algorithm1.LetH be a real Hilbert space, K be a nonempty closed convex subset of H, and F:K×K→R be a bifunction. Let r∈[a,∞) for some a>0. Then from an arbitrary x0∈H we generate the sequence {xn}n=1∞ as follows.Algorithm 16.
(16) y n = x n , u n ∈ K such  that F u n , y + 1 r y - u n , u n - y n ≥ 0 ,  ∀ y ∈ K , x n + 1 = 1 2 u n + x n .Theorem 17.
LetH, {xn}n=1∞ and F and r be as in Algorithm 16. Suppose F satisfying (A1)-(A4), EP(F)≠∅, then (i) {xn} converges strongly to p∈EP(F); (ii) if H has order inclusion transitive property, then {xn}n=1∞ converges strongly to PEP(F)x0.Proof.
Sinceun=Tryn=Trxn, given arbitrary q∈EP(F), we have that (17)xn+1-q2=12un+xn-q2=12un-q+12xn-q2=12un-q2+12xn-q2-12xn-un2≤12xn-q2+12xn-q2-14xn-un2=xn-q2-14xn-un2≤xn-q2. Therefore, {xn-p}n=0∞ is monotone, nonincreasing, and bounded; hence, limn→∞xn-q exists. Also, (18)14xn-un2≤xn-q2-xn+1-q2. Hence ∑n=0∞‍xn-un2<∞ which implies that limn→∞xn-un=xn-Trxn=0. Also, (19)xn+1-xn=12xn-un≤xn-un, so that limn→∞xn+1-xn=0. Consequently {xn}n=0∞, {yn}n=0∞={xn}n=0∞, and {un}n=0∞={Trxn}n=0∞ are bounded. From limn→∞xn+1-xn=0 we have that {xn}n=0∞ is a Cauchy in K and hence converges strongly to p∈K. From the Opial condition of H, the firmly nonexpansive and demiclosedness property of (I-Tr) established that p∈F(Tr)=EP(F). (ii) If H has order inclusion transitive property then xn=PKnx0, consequently, from Lemma 7(i) (20)xn-y,x0-xn≥0,∀y∈Kn.Since EP(F)⊆Kn for all n≥1, we have that (21)xn-q,x0-xn≥0,∀q∈EPF.Taking the limit as n→∞ in (21) we have (22)p-q,x0-p≥0,∀q∈EPF.Thus, from Lemma 7(i) p=PEP(F)x0. This completes the proof.Remark 18.
It is important to note that the strong convergence of Algorithm16 to a p∈EP(F) does not depend on the order inclusion transitive property condition on H.Motivated by Algorithm16 we now obtain the following algorithm which is a selection of Algorithm 2.LetH be a real Hilbert space, K be a closed and convex subset of H, F:K×K→R be a bifunction, and T:K→P(K) be a multivalued k-strictly pseudocontractive-type mapping. Let {αn}n=1∞⊂[0,1] and rn∈[a,∞) for some a>0. Then from an arbitrary x0∈H the algorithm is generated as follows.Algorithm 19.
(23) y n = α n x n + 1 - α n v n , u n ∈ K such  that F u n , y + 1 r n y - u n , u n - y n ≥ 0 ,  ∀ y ∈ K , x n + 1 = 1 2 u n + x n ,where vn∈Txn.Theorem 20.
LetH, K, F, T, {αn}, and {rn} be as in Algorithm 19. Suppose that (I-T) is weakly demiclosed at zero, F satisfies (A1)-(A4), F=FsT∩EP(F)≠∅, and {αn} satisfies (i) αn→α<1-k; (ii) α>0; (iii) ∑n=1∞‍αn(1-αn)=∞. Then {xn} converges strongly to q∈F. Also, if H has order inclusion transitive property, then q=PFx0.Proof.
Letp∈F be arbitrary. Then (24)
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. It then follows that limn→∞xn-p exists; hence, {xn} is bounded. Also, (25)∑n=1∞12αn1-αn+kxn-vn2≤x0-p2<∞. Since α>0 from (ii), we have that limn→∞xn-vn=0. Furthermore, limn→∞xn-un=0. Consequently, limn→∞xn+1-xn2=limn→∞1/2(xn-un)2=0 which implies that {xn} is a Cauchy sequence in K. Also, since K is closed and convex, {xn} converges strongly to some q∈K. From the Opial condition of H, weakly demiclosed of (I-T) at zero, we have that q∈Tq.
It remains to show thatq is in EP(F) since (26)limn→∞⁡xn-un=0.It follows from limn→∞xn-q=0 and (26) that (27)limn→∞⁡un-q=0.Also, (28)yn-q2≤xn-q2-αn1-αn+kxn-vn2. Observe that (29)q-xn2-q-un2=xn2-un2-2q,xn-un≤xn-unxn+un+2qxn-un. It follows from (26) that (30)limn→∞⁡q-xn-q-un=0.Now from (28), (31)q-yn≤q-xn.Also, using un=Trnyn, Lemma 10, and (31) we have (32)un-yn2=Trnyn-yn2≤q-yn2-q-Trnyn2≤q-xn2-q-Trnyn2=q-xn2-q-un2. Therefore, from (30) and (32), (33)limn→∞⁡un-yn=0.Consequently, from (27) and (33), (34)limn→∞⁡yn-q=0.From the assumption that rn≥a>0, (35)limn→∞⁡un-ynrn=0.Since un=Trnyn, we have (36)Fun,y+1rny-un,un-yn≥0. We then deduce from (A2) that (37)un-yn2rn≥1rny-un,un-yn≥-Fun,y≥Fy,un,∀y∈K. By taking limit as n→∞ of the above inequality and from (A4), (27), and (34), F(y,q)≤0, for all y∈K. Let t∈(0,1) and for all y∈K, since q∈K, yt=ty+(1-t)q∈K. Hence F(yt,q)≤0. Therefore, from (A1), (38)0=Fyt,yt≤tFyt,y+1-tFyt,q≤tFyt,y; that is, F(yt,y)≥0. Letting t↓0, from (A3) we obtain F(q,y)≥0 for all y∈K so that q∈EP(F) for all i=1,2,…,N. Hence q∈F.
Finally, ifH has order inclusion transitive property, xn=PKnx0 consequently, from Lemma 7(i) (39)xn-y,x0-xn≥0,∀y∈Kn.Since EP(F)⊆Kn for all n≥1, we have that (40)xn-p,x0-xn≥0,∀p∈EPF.Taking the limit as n→∞ in (40) we have (41)q-p,x0-q≥0,∀p∈EPF. Thus, from Lemma 7(i) q=PEP(F)x0. This completes the proof.Remark 21.
The above proof shows that the strong convergence of Algorithm19 to a common solution q∈F does not depend on order inclusion transitive property condition on H. However, order inclusion transitive property is only required on H if we want to have that p=PEP(F)x0.Remark 22.
It is also of a great interest to us to get the same results in normed spaces which enjoy the order inclusion transitive property.


## 4. Numerical Examples of the Computations

We shall use Algorithm19 to recompute the example presented by Isiogugu et al. [13], when i=1 is defined as follows.LetH=R (the reals with the usual metric and inner product) and K=[-4,10]; we define(i)T:[-4,10]→P([-4,10]) by (42)Tx=-2x,-9x4,x∈-4,0-3x9,x∈0,10.We have that, for any x,y∈[-4,0], (43)H2Tx,Ty=94x-y2=942x-y2=x-y2+942-1x-y2.Also, given any u∈Tx, u=-αx, 2≤α≤9/4, and we can choose v=-αy∈Ty so that (44)x-u-y-v2=1+α2x-y2.It then follows that (45)H2Tx,Ty=x-y2+92-16161+α2x-u-y-v2≤x-y2+92-16161+22x-u-y-v2.Similarly, for any x∈[-4,0], y∈(0,10], (46)H2Tx,Ty=94x-3y92≤94x-94y2≤x-y2+92-16161+22x-u-y-v2.Furthermore, for any x,y∈(0,10], (47)H2Tx,Ty=39x-y2≤x-y2+92-16161+22x-u-y-v2.Thus, T is (9)2-16/16(1+2)2-strictly pseudocontractive-type. It is easy to see that, given any pair x,y∈[-4,0], we have that H(Tx,Ty)=9/4x-y. Therefore T is not nonexpansive and Fs(T)={0}≠∅. We then set(ii)(48)vn=-9xn4,xn∈-4,0-3xn9,xn∈0,10.(iii)(49)αnn=1∞=92-16+161+22n+321+222+321+22321+22n+321+222.We also let F:R×R→R, {rn}n=1∞, and {un}n=1∞ to be as defined in [9] for i=1. That is,(iv)F(x,y)=-x2+y2.Observe thatEP(F)={0}, FsT∩EP(F)={0}, and (50)Fz,y+1ry-z,z-x≥0⇒y2-z2+1ry-zz-x≥0,⇒y2-z2+1ryz-xy-z2+xz≥0,⇒ry2-rz2+yz-xy-z2+xz≥0,⇒ry2+z-xy-rz2-z2+xz≥0.Now F(y)=ry2+(z-x)y-rz2-z2+xz is a quadratic function of y with coefficients a=r, b=z-x, and c=-rz2-z2+xz. Therefore, we can compute the discriminant Δ of F as follows:(51)Δ=z-x2+4rrz2+z2-xz=z2+x2-2xz+4r2z2+4rz2-4rxz=1+4r2+4rz2-22r+1xz+x2=1+2r2z2-21+2rxz+x2=1+2rz-x2.Obviously, F(y)≥0 for all y∈R if it has at most one solution in R. Thus Δ≤0 and hence z=Trn(x)=x/1+2r. Consequently,(v){un}n=1∞={Trn(yn)}n=1∞={yn/2rn+1}n=1∞,(vi){rn}n=1∞={(n+1)/n}n=1∞.The algorithm is computed with Microsoft word Excel 97-2003 Workbook. Table1 shows the sequences {xn} and {Kn} generated from our computation using two different values of x0=2,-3.Table 1
Sequences generated by Algorithms2 and 19.
Algorithm 2 Algorithm 2 Algorithm 4 Algorithm 2 Algorithm 2 Algorithm 4 n K_n x_n x_n K_n x_n x_n  0 [-4, 10] 2 2 [-4, 10] -3 -3  1 [-4, 1.126855044] 1.126855044 1.126855044 [-1.532563756, 10] -1.532563756 -1.532563756  2 [-4, 0.652769535] 0.652769535 0.652769535 [-0.787076007, 10] -0.787076007 -0.787076007  3 [-4, 0.38284414] 0.38284414 0.38284414 [-0.405188015, 10] -0.405188015 -0.405188015  4 [-4, 0.226111845] 0.226111845 0.226111845 [-0.208877029, 10] -0.208877029 -0.208877029  5 [-4, 0.134146663] 0.134146663 0.134146663 [-0.107772707, 10] -0.107772707 -0.107772707  6 [-4, 0.079836206] 0.079836206 0.079836206 [-0.055641074, 10] -0.055641074 -0.055641074  7 [-4, 0.047623897] 0.047623897 0.047623897 [-0.028739594, 10] -0.028739594 -0.028739594  8 [-4, 0.028459121] 0.028459121 0.028459121 [-0.014849721, 10] -0.014849721 -0.014849721  9 [-4, 0.01703056] 0.01703056 0.01703056 [-0.007674974, 10] -0.007674974 -0.007674974  10 [-4, 0.010203099] 0.010203099 0.010203099 [-0.003967653, 10] -0.003967653 -0.003967653  11 [-4, 0.006118509] 0.006118509 0.006118509 [-0.002051501, 10] -0.002051501 -0.002051501  12 [-4, 0.003672014] 0.003672014 0.003672014 [-0.001060909, 10] -0.001060909 -0.001060909  13 [-4, 0.002205248] 0.002205248 0.002205248 [-0.00054871, 10] -0.00054871 -0.00054871  14 [-4, 0.001325149] 0.001325149 0.001325149 [-0.00028383, 10] -0.00028383 -0.00028383  15 [-4, 0.000796697] 0.000796697 0.000796697 [-0.000146831, 10] -0.000146831 -0.000146831  16 [-4, 0.0004792] 0.0004792 0.0004792 [-0.000075965, 10] -0.000075965 -0.000075965  17 [-4, 0.000288345] 0.000288345 0.000288345 [-0.000039304, 10] -0.000039304 -0.000039304  18 [-4, 0.000173565] 0.000173565 0.000173565 [-0.000020337, 10] -0.000020337 -0.000020337  19 [-4, 0.000104508] 0.000104508 0.000104508 [-0.000010523, 10] -0.000010523 -0.000010523  20 [-4, 0.000062945] 0.000062945 0.000062945 [-0.000005445, 10] -0.000005445 -0.000005445  21 [-4, 0.000037921] 0.000037921 0.000037921 [-0.000002817, 10] -0.000002817 -0.000002817  22 [-4, 0.000022851] 0.000022851 0.000022851 [-0.000001457, 10] -0.000001457 -0.000001457  23 [-4, 0.000013772] 0.000013772 0.000013772 [-0.000000754, 10] -0.000000754 -0.000000754  24 [-4, 0.000008302] 0.000008302 0.000008302 [-0.00000039, 10] -0.00000039 -0.00000039  25 [-4, 0.000005005] 0.000005005 0.000005005 [-0.000000201, 10] -0.000000201 -0.000000201  26 [-4, 0.000003018] 0.000003018 0.000003018 [-0.000000103, 10] -0.000000103 -0.000000103  27 [-4, 0.00000182] 0.00000182 0.00000182 [-0.000000053, 10] -0.000000053 -0.000000053  28 [-4, 0.000001097] 0.000001097 0.000001097 [-0.000000027, 10] -0.000000027 -0.000000027  29 [-4, 0.000000661] 0.000000661 0.000000661 [-0.000000013, 10] -0.000000013 -0.000000013  30 [-4, 0.000000398] 0.000000398 0.000000398 [-0.000000006, 10] -0.000000006 -0.000000006  31 [-4, 0.00000024] 0.00000024 0.00000024 [-0.000000003, 10] -0.000000003 -0.000000003  32 [-4, 0.000000144] 0.000000144 0.000000144 [-0.000000001, 10] -0.000000001 -0.000000001  33 [-4, 0.000000086] 0.000000086 0.000000086 { 0 } 0 0  34 [-4, 0.000000051] 0.000000051 0.000000051  35 [-4, 0.00000003] 0.00000003 0.00000003  36 [-4, 0.000000018] 0.000000018 0.000000018  37 [-4, 0.00000001] 0.00000001 0.00000001  38 [-4, 0.000000005] 0.000000005 0.000000005  39 [-4, 0.000000002] 0.000000002 0.000000002  40 [-4, 0.000000001] 0.000000001 0.000000001  41 { 0 } 0 0



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*Source: 7218487-2018-10-01.xml*