NONLINEAR VISCOSITY ALGORITHM WITH PERTURBATION FOR NONEXPANSIVE MULTI-VALUED MAPPINGS Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2021, Volume 23, Issue 1, P. 60-76

УДК 519.65

DOI 10.46698/e7204-1864-5097-s

NONLINEAR VISCOSITY ALGORITHM WITH PERTURBATION FOR NONEXPANSIVE MULTI-VALUED MAPPINGS

H. R. Sahebi1

1 Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, P. O. Box 39618-13347, Iran E-mail: sahebi@aiau.ac.ir

Abstract. The viscosity iterative algorithms for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors. The viscosity technique allow us to apply this method to convex optimization, linear programming and monoton inclusions. In this paper, based on viscosity technique with perturbation, we introduce a new nonlinear viscosity algorithm for finding an element of the set of fixed points of nonexpansive multi-valued mappings in a Hilbert spaces. Furthermore, strong convergence theorems of this algorithm were established under suitable assumptions imposed on parameters. Our results can be viewed as a generalization and improvement of various existing results in the current literature. Moreover, some numerical examples that show the efficiency and implementation of our algorithm are presented.

Key words: fixed point problem, generalized equilibrium problem, nonexpansive multi-valued mapping, Hilbert space.

Mathematical Subject Classification (2010): 47H09, 47H10, 47J20.

For citation: Sahebi, H. R. Nonlinear Viscosity Algorithm with Perturbation for Nonexpansive MultiValued Mappings, Vladikavkaz Math. J., 2021, vol. 23, no. 1, pp. 60-76. DOI: 10.46698/e7204-1864-5097-s.

1. Introduction

Throughout the paper unless otherwise stated, H denotes a real Hilbert space, we denote the norm and inner product of H by (.,.) and norm ||.||, respectively. The set C (C being a nonempty closed convex subset of H) is called proximinal if for each x € H, there exists an element y € C such that ||x — y|| = d(x,C), where d(x,C) = inf{||x — z|| : z € C}. Let CB(D), K(C) and P(C) be the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of C, respectively. The Hausdorff metric on CB(C) is defined by

H (A, B) = max < sup d(x,B), sup d(y,A)l, A, B € CB(C).

I xeA y€B )

A multi-valued mapping T : C — 2C is said to be nonexpansive if H(Tx,Ty) ^ ||x — y|| for all x, y € C. An element p € C is called a fixed point of T : C — 2C if p € Tp. The fixed points set of T is denoted by Fix(T).

The problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points for single-valued mappings in the framework of Hilbert spaces

© 2021 Sahebi, H. R.

has been intensively studied by many authors, for instance, see [1-5] and the references cited therein.

Ceng et al. [6], introduced the following generalized equilibrium problem with perturbation: Find x* € C such that

f (x*,y) + ((A + B)x*,y - x*> ^ 0 (Vy € C), (1.1)

where A,B : C — H are nonlinear mappings and f : C x C — r is a bifunc-tion. The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others [1, 4, 5, 7, 8].

In 2016, Azhini and Taherian [9], motivated by [6, 10], proposed the following iteration process for finding a common element of the set of solutions of variational inequality (1.1) and the set of common fixed points of infinitely many nonexpansive mappings {Sn} of C into itself and proved the strong convergence of the sequence generated by this iteration process to an element of F(PcS) = f|n=i F(PcS.n)-

F(un,y) + ((M + N)xn,y - un> + —{y - un,un — xn) ^ 0 (Vy € C),

rn

xn+1 = PnPc f (xn) + Ynxn + KPcSn[anZ + (1 - an)un] (Vn € n),

where Pn + Yn + K = 1.

In 2019, Sahebi et al. [11] by intuition from [12-15] considered a general viscosity iterative algorithm for finding a common element of the set general equilibrium problem system and the set of fixed points of a nonexpansive semigroup in a Hilbert space as follows:

un,i — i (xn rni^iXn), k

Wn — fa Xy (12)

i=1 V • /

sn

Xn+1 = anYf(xn) + /3nBxn + ((1 - en)I - (3nB - anA) ^ / T(s)wn ds.

n 0

They proved that, the sequence generated by this algorithm under the certain conditions imposed on parameters strongly convergence to a common solution of general equilibrium problem system.

Many authors have shown the existence of fixed points of multi-valued mappings in Hilbert spaces (see [16-19]). The study of multi-valued mappings is much more complicated and difficult than that of single-valued mappings.

In this paper, motivated by the research going on in this direction, we introduce the iterative algorithm for finding a common element of the set of fixed point of a nonexpansive multi-valued mapping in a real Hilbert space. Some strong convergence theorems and lemmas of the proposed algorithm are proven under new techniques and some mild assumption on the control conditions. Finally, some numerical examples that show the efficiency and implementation of our algorithm are presented.

The paper is structured as follows. In Section 2, we collect some lemmas, which are essential to prove our main results. In Section 3, we introduce a new algorithm for finding a common element of the set of fixed point of a nonexpansive set-valued mapping in a real Hilbert space. Then, we establish and prove the strong convergence theorem under some proper conditions. In Section 4, we also give some numerical examples to support our main theorem.

2. Preliminaries

Let H be a Hilbert space and C be a nonempty closed and convex subset of H. For each point x € H, there exists a unique nearest point of C, denote by Pcx, such that ||x — Pcx|| ^ ||x — y|| for all y € C. Pc is called the metric projection of H onto C. It is well known that Pc is nonexpansive mapping. Also, a mapping M : C — H is said to be monotone, if

(Mx — My, x — y) ^ 0 (Vx, y € C).

M is called a-inverse-strongly-monotone if there exist a positive real number a such that

(Mx — My,x — y) ^ a||Mx — My||2 (Vx,y € C).

It is obvious that any a-inverse-strongly-monotone mapping M is monotone and Lipschitz continuous.

Recall that a mapping T : H — H is said to be firmly nonexpansive if

(Tx — Ty,x — y) ^ ||Tx — Ty||2 (Vx,y € H).

It is also known that H satisfies Opial's condition [20], i.e., for any sequence {xn} with xn ^ x, the inequality

lim inf ||xn — x|| < lim inf ||xn — y|| (2.1)

holds for every y € H with y = x. The following lemmas will be used for proving the convergence result of this paper in the sequel.

Lemma 2.1 [21]. Let C be a nonempty and weakly compact subset of a Banach space E with the Opial condition and T : C — K(E) a nonexpansive mapping. Then I — T is demiclosed.

Lemma 2.2 [22]. The following inequality holds in real space H: ||x + y||2 < ||x||2 + 2(y,x + y) (Vx,y € H).

Lemma 2.3 [23]. Let C be a closed and convex subset of a real Hilbert space H. Let T : C — CB(C) be a nonexpansive multi-valued map with Fix(T) = 0, and Tp = {p} for each p € Fix(T). Then Fix(T) is a closed and convex subset of C.

Lemma 2.4 [24]. Let F : C x C — R be a bifunction satisfying Assumption 2.1 and let TF be defined as in Lemma 2.5, for r > 0. Let x, y € H and t, s > 0. Then,

IITFy - TFx|| < ||x - y|| +

|TsF y -■

Lemma 2.5 [25]. Let C be a nonempty, closed convex subset of H and let F : C x C — R be a bifunction satisfying Assumption 2.1. Then for r > 0 and x € H, there exists z € C such that F(z,y) + ^(y — z,z — x) ^ 0 for all y € C. Further define

Tfx = jz€C: + ^ oj (Vy € C)

for all r > 0 and x € H. Then, the following hold: (i) TF is single-valued.

(ii) TF is firmly nonexpansive, i. e.,

\T?(x) - TrF(y)||2 ^TrF(x) - TrF(y),x - y) (Vx,y € H).

(iii) Fix(TrF) = EP(F).

(iv) EP(F) is compact and convex.

Lemma 2.6 [26]. Assume that B is a strong positive linear bounded self adjoint operator on a Hilbert space H with coefficient 7 > 0 and 0 < p ^ ||B||-1. Then ||1 - pB|| ^ 1 - pq.

Lemma 2.7 [27, 28]. Let C be a closed and convex subset of a real Hilbert space H and let Pc be the metric projection from H onto C. Given x € H and z € C. Then z = Pcx if and only if

(x - z,y - z> < 0 (Vy € C).

Lemma 2.8 [29]. Let {xn} and {yn} be bounded sequences in a Banach space X and {Pn} be a sequence in [0,1] with 0 < liminfn^^ pn ^ limsupn^rc Pn < 1. Suppose xn+1 = (1 - Pn)yn + Pnxn for all integers n ^ 0 and lim supn^(||yn+1 - yn|| - ||xn+1 - xn||) ^ 0. Then limn^rc, ||yn - xn|| = 0.

Lemma 2.9 [10]. Let F : C x C — r be a bifunction satisfying Assumption 2.1 and let T.'f be defined as in Lemma 2.5, for r > 0. Let x € H and s,t > 0. Then,

TFx - TFx

2 s -1 <-

(tsf(x) - TtF(x),TF(x) - x

Lemma 2.10 [30]. Let {an} be a sequence of nonnegative real numbers such that an+1 ^ (1 — an)an + ön, n ^ 0, where an is a sequence in (0,1) and 5n is a sequence in r such that

(i) X£°=1an =

(ii) lim sup^oo or E~=15ra < 00. Then limn^^> an = 0.

Assumption 2.1. Let F : C xC ^ r be a bifunction satisfying the following assumptions:

1. F(x,x) ^ 0 (Vx € C);

2. F is monotone, i. e., F(x,y) + F(y,x) ^ 0 (Vx € C);

3. F is upper hemicontinuouse, i. e., for each x,y,z € C,

lim sup F(tz + (1 — t)x, y) ^ F(x, y). t^o

For each x € C fixed, the function x F(x, y) is convex and lower semicontinuous.

3. A Nonlinear Iterative Algorithm

Let C be a nonempty closed convex subset of real Hilbert space H. Let F : C x C — r be a bifunction satisfying Assumption 2.1. Let M, N be two a-inverse strongly monotone and ß-inverse strongly monotone mappings from C into H, respectively. Recall that the set of all solutions of problem (1.1) is denoted by GEPP, i.e.

GEPP = {x € C : F(x, y) + ((M + N)x,y — x) ^ 0 (V y € C)}.

Let T be a nonexpansive multi-valued mapping on C into K(H) such that 0 = Fix(T) n GEPP = 0. Also f : C — H be a a-contraction mapping and A, B be a strongly positive

bounded linear self adjoint operators on H with coefficient 71 > 0 and 72 > 0 respectively such that 0<7<^<7 + ^,7i^ ||A|| ^ 1 and ||B|| = 72.

Algorithm 3.1. For given xo € C arbitrary, let the sequence {xn} be generated by:

fun = TF (xn — rn(M + N )xn); (31)

|xn+1 = an7/(xn) + /n Bxn + ((1 — en) I — ^nB — anA)Zn,

where zn € Tun such that ||zn+1 — zn|| ^ H(Tun+1,Tun).

Let {an}, {/n}, {en} are sequences in (0,1), {rn} C [r, 00) with r > 0 satisfied the following

n} { n} { n

conditions:

(C1) limn^tt an = 0, XttUan = (C2) limsupn^tt /n = 1;

(C3) limn^tt |rn+1 — rn| = 0, liminfn^tt rn > 0, 0 < b < rn <a< 2min{a,/7}.

Lemma 3.1. Let p € 0. Then the sequence {xn} generated by Algorithm 3.1 is bounded.

< We may assume without loss of generality that an ^ (1 — en — /n||B||)||A||-1. Since A and B are linear bounded self adjoint operators, we have

= sup{|(Ax,x)| : x € H, ||x|| = 1}, B|| = sup{|(Bx,x)| : x € H, ||x|| = 1}

observe that

(((1 — en) I — /nB — anA)x,x) = (1 — en)(x,x) — /n(Bx,x) — an(Ax,x) ^ 1 — en — /n||B|| — an||A|| ^ 0.

Therefore, (1 — en)I — / nB — anA is positive. Then, by strong positivity of A and B, we get ||(1 — en)I — /nB — anA|| = sup {(((1 — en)I — /nB — x € H, ||x|| = 1}

= sup j(1 — en)(x,x) — /n(Bx,x) — an(Ax,x) : x € H, ||x|| = 1| ^ 1 — en — /n72 — an 71 ^ 1 — /n72 — an71.

(3.2)

Let p € 0 := Fix(T) n GEPP. Since p € GEPP, from [4, Theorem 3.1] we have

Un — p||2 < ||xn — p||2 + rn(rn — 2a)||Mxn — Mp|| +rn(rn — 2/)||Nxn — Np||2 < ||xn — p||2.

(3.3)

Then ||un — p|| ^ ||x„ — p||. We obtain

||x„+i — p|| = ||an7/(xn) + ^„Bx„ + ((1 — e„)/ — ^„B — araA)zra — < a„||7/(x„) — Ap|| + ^n|Bx„ — Bp|| + ejp|| + ||((1 — e„)I — ^„B — anA)||||zn — p|| < an(|7/(x„) — 7/(p)| + ||Y/(p) — Ap||) + £n||Bx„ — Bp|| + ejp||

+(1 — ^„72 — a„7i)d(z„, Tp) ^ a„7a|xn — p|| + a„||7/(p) — Ap|| + ^n72|x„ — p|| + a„||p||

+(1 — & 72 — a„7i)H (Tu„, Tp) ^ a„7a|xn — p|| + an||7/(p) — Ap|| + ^n72||x„ — p|| + a„||p|| (3-4)

+(1 — ^„72 — a„7i)|un — p||

< (1 — (7i — 7CK)an)|xn — p|| + a„(||p|| + ||y/(p) — Ap||)

... ^ ma^ ||x0 — p||,

7i — 7a ||7/(p)-Ap|| + ||

7i — 7a

Hence {xn} is bounded. This implies that the sequences {un}, {zn} and {/(xn)} are bounded. >

Lemma 3.2. The following properties are satisfying for the Algorithm 3.1. P1. lim„^ ||x„+i — x„|| = 0. P2. ||xn inH = 0-

P3. lim„^ ||Mx„ — Mp|| = 0 and lim„^ ||Nx„ — Np|| = 0. P4. ||x„ — = 0.

< P1: We have

||u„+i — u„|| = ¡Trn+1 (x„+i — r„+i(M + N )x„+i) — Trn (x„ — r„(M + N )x„)^ ^ ¡(x„+i — r„+i(M + N)x„+i) — (x„ — r„(M + N)x„)!

+ |r"+1~rral||Trn+1(xra+i - rn+l{M + N)xn+l) - (xra+i - rn+l{M + A0xra+i)||

rn+i

^ I|x„+1 - xn\\ + \rn+i - rnI II(M + N)(xn+1 - xn)\\ + ——an+i, (3.5)

r„+i

where = sup„eN ||Trn+i(x„+i — r„+i(M + N)x„+i) — (x„+i — r„+i(M + N)x„+i)||. Setting xn+i = enxn + (1 — en)en, then we have

a„+iY/(x„+i)+ ^„+iBx„+i + ((1 — e„+i)1—^n+iB — a„+iA)z„+i e„+ixn+i

Cri+1 Cri —

1 — e„+i

a„Y/(x„) + ^„Bx„ + ((1 — e„)1 — ^„B anA)zn

1 — e«

Cln+l -(7/(xra+i) - Azn+1) + (Xn (Azn - 7f(xn))

-I V / tf — «+1/ I -,

1 — en+1 1 — en

+ f_j3n±1--@n )B(xra+1 - xra) + (zra+i - zra)

V 1 — e«+1 1 — en /

+ ( T^--)B(zn+1 - zra) + f —^—--£ra+1 )(xra -xn+i).

VI — en 1 — era+1/ V1 — en 1 —era+1/

Using (3.5), we have

||e„+i - e„|| ^

®n+1

1 — en+i _l_ Pn+l

\\lf (xn+i) — Azn+i\\ +

A '

an

+ ||zn+i zn || + <

An

1 — £n+i 1 — e An+i

1 — en 1 — en+1

an+i

1 - en ||B|| ||Xn+i xn||

en+i

\\Yf (xn) — Azn

||£||||Zn+i — Zn || + an

hf{xn+i) -Azn+i\\ +

1 — en+1 1 — en

1 en+i 1 en \\lf (Xn) — Azn\\

| xn+i n

+

An+i

An

+ H (Tun+i,Tun) +

+

1 — en+i 1 — en

' An

||B|| ||Xn+i — X,

An+i

1 — en 1 — en+i e n+i e n

||B ||H (Tun+i,Tun)

<

1 — en+ i 1 — e

Cln+l \\lf{xn+i) - Azn+i || +

1 — en+ i

+

| xn+i Xn||

(Xn \\lf{xn) - Az„

An+ i

An

+ ||un+i un

|| +

+

1 — en+ i 1 — en

An

1 en ||B|| ||Xn+i — X,

An+ i

1 — en 1 — en+ i

e n+i e n

1 — en+ i 1 — en

<

«ra+1 1 — en+ i

\\lf (Xn+i) — AZn+i\\ +

HBHHun+i un H

||Xn+i — n|

||7¡{Xn) - Az,n

1 en

+

An+ i

An

1 en+i 1 e

||Xn+i — Xn| + ||Xn+i — Xn|| + |rn+i — rn| \\(M + N)(Xn+i — Xn)\

, \rn+1 - rn I

H--CTra+1 +

rn+i

An

An+i

1 en

1 — en+i

x||(M + AT)(a;ra+1-a;ra)||+|rra+1 r'

rn+i

-Cra+1 +

72 ( ||Xn+i — Xn| + |rn+i — rn|

en+i en

1 — en+i 1 — en

||Xn+i " Xn || ,

which implies

||en+i — en|| — ||Xn+i — Xn| ^

an

+

1e

'Hlf (Xn) —Azn || +

Cln+l \\lf(xn+i) - Azn+i\\ ||Xn+i — n|

1 — en+i

An+ i A

+

1 — en+i 1 — e

+ \rn+l - rn\\\(M + N)(xn+l - xn)\\ + |r"+1~r"1

An An+i

rn+i

0"n+i

1 en

1 — en+ i

, K+l - rn I

H--&n+l +

rn+i

72 ( ||Xn+i — Xn|| + |rn+i — rn| ||(M + N)(Xn+i — Xn)

en+ i ei

1 — en+i 1 — en

| Xn+i Xn

e

n

Hence, it follows by conditions (C1)-(C4) that

lim sup (||era+i - e„|| - ||xn+i - £n||) ^ 0. (3.6)

From (3.6) and Lemma 2.8, we get limn^œ ||en — xn|| = 0, and then

lim ||xn+i - = lim (1 - e„)||era - = 0. (3.7)

n^-œ n^-œ

P2: We can write

||xn - ¿J ^ ||xn+1 - xJ + ||an7/ (xn) + ßnBxn + ((1 - £n)1 - ßnB anA)zn zn < ||xn+1 - Xn|| + an|7/(Xn) - Azn|| + ßn|BXn - BZn|| + Én 11 zn

^ ||xn+i - Xn| + an||Y/(xn) - Azn|| + ßn7211x n zn y +

Then

(1 - ßn72)||Xn - Zn|| ^ ||Xn+1 - Xn|| + an|Y/(Xn) - Azn| + Én Therefore

1 a

IIXn - znII ^ ---— \\xn+i -xn\\ + --= ||7/(xn) - AtnII +

1 - 1 - ßn7 ' 1 - ßn72

1 a

< 1-7r—\\Xn+l - Xn\\ + --^—(\\qf(xn) - Azn\\ + ||zra||).

1 - Pn72 1 - Pn72

Since an — 0, ||xn+1 - xn|| — 0 and (C2) we obtain

lim ||xn - Zn|| = 0. (3.8)

P3: From (3.3), we have

||xn+1 - p||2 = ||an7/(xn) + ßnBxn + ((1 - en)/ - ßnB - anA)zn - p|2 = ¡an (7/(xn) - Ap) + ßn(Bxn - Bp) + ((1 - en)/ - ßnB - an A) (zn - p) - ÉnP|2 ^ ¡((1 - en)/- ßnB - anA)(zn - p) + ßn(Bxn - Bp) - enP|2

+ 2(an(Y/(xn) - Ap),xn+1 - p> < ((1 - ßn72 - an71)d(Zn, Tp) + ßn|B||xn - Zn| + ejp||) 2 + 2a^7/(xn) - Ap,xn+1 - p) ^ ((1 - ßn72 - an70H(Tun,Tp) +ßn||B||||xn - Zn| + ejp||) + 2an(7/(xn) - Ap,xn+1 - p)

^ ((1 - ßn72 - an71) ||Un - p| + ßn||B||xn - Zn| + Én|p^ + 2a^7/(xn) - Ap, xn+1 - p> = (1 - ßn72 - an71)2||un -p||2 + (ßn)2||B||2||xn - Zn|2 + (en)2||p||2

+ 2(1 - ßn72 - an70ßn|B|||un - p||xn - Zn||

+ - ßn72 - an70en|p||un - py + 2ßnen||B

n - Zn

+2a^7/(xn) - Ap, xn+1 - p>

(3.9)

< (1 — 72 — a„7i) (j|x„ — p||2 + r„(r„ — 2a)||Mx„ — Mp||2 + r„(r„ — 2^)|Nx„ — Np||2) + (^n)2yвy2yxn — z„||2 + M^pll2

+ 2(1 — /„72 — a„7i) 1^1111^ — p|| ||x„ — z„|| + 2(1 — 72 — a„7i) e„||p| p|| + 2^nenУB ||||p||||x + 2a„(7/(x„) — Ap,x„+i — p) ^ ||x„ — p||2 + (/„72 + an7l)2||xn — p||2 + (1 — /„72 — a„ 7i)2( r„(r„ — 2a)||Mx„ — Mp||2 + r„(r„ — 2£)||Nx„ — Np||2)

+C0n)2||B||2||Xn — z„||2 + (e„)2 ||p||2 + 2(1 — /„72 — ^^IIB |||K — pHK — z„ + 2(1 — 72 — a„7i)

e„||p| p|| + 2^nenУB UNpNNx „ H + 2a„(7/(x„) — Ap,x„+i — p).

By (C3), we can write

% /i)2( r„(2a — r„)HMx„ — Mp||2 ^ /„v

(1 — /„72 — a„7i)2( r„(2a — r„)HMx„ — Mp||2 + r„(2/ — r„)HNx„ — Np||2)

< l|x„ —pH2 — ||x„+i — p||2 + (/„72 + an7l)2||xn — pH2 + (/„)2HBH2Hx„ — z„||2 + (a„)2||p||2 + 2(1 — /„72 — a„7i) HBHHu„ — p| ||x„ — z„|| + 2(1 — /„72 — a„7i) an||p||||«n — pH + 2/„ enЦBЦЦpЦЦx H + 2a„( Y/(x„) — Ap, x„+i — p> ^ (l|x„ — p| + ||x„+i — p^|x„ — x„+iH + (/„72 + an702Цxn — pH2 + (/„)2HB||2||x„ — z„||2 +(a„)2HpH2 + 2(1 - /„72 — a„7i)/„||BHHu„ — pHHx„ — z„|| + 2(1 — /„72 — a„7i)

a„HpHp|| + 2& B HHpHHx „ z„ H + 2a„( y/(x„) — Ap, x„+i — p).

By a„ — 0, ||x„+i — x„|| — 0 and ||x„ — z„|| — 0, then we obtain ||Mx„ — Mp|| — 0 and ||Nx„ — Np|| — 0 as n —y to.

P4: Since p € 0, we can obtain

IK — pll2 < ||x„ — pll2 — IK — x„||2 + 2r„Hu„ — x„H(H— MpH + ||Nx„ — Np||). It follows from (3.9) that

||x„+i — pll2 < (1 — //„72 — аn702Цun—pH2 + (^n)2ЦBЦ2Цxn — z„||2 + (£„) + 2(1 —/„72 — a„7i) HBHHu„ — pHHx„ — z„||

+ — ^ra 72 — a„70 e^N1^ — p|H + 2/nínЦB ||HpHHx „ z„ H + 2a„( Y/(x„) — Ap, x„+i — p>

22

^ (1 — 72 — a„7i) [ ||x„ — pH2 — ||u„ — x„|

2

+2r„Hu„ — x„H (H— MpH + ||Nx„ — Np^ + (/„)2HBH2Hx„ — z +(e„)2||p||2 + — 72 — a„70 /„HB HHu„ — pHHx„ — z„H

+ — 72 — a„70 e^b1111^ — py + 2/„emB H||p||||x „ z„H +2a^ y/(x„) — Ap, x„+i — p).

Therefore

(1 - 72 - awYi)2||ura - x„||2 ^ ||x„ - p||2 - ||x„+i - p||2 + (^n72 + a„7i)2||x„ - p||2 + 2r„( 1 - ^„72 - a„7i)2|K - x„|| (||Mx„ - Mp|| + ||Nx„ - Np||) + )2||B||2||x„ - z„||2 +(e„)2||p||2 + 2(1 - ^„72 - a„7i)^„||£||||u„ - p||||x„ - z„|| + 2(1 - ^„72 - a„7i)e„||p||||u„ - p|| + 2^„e„||£||||p||||x„ - z„|| + 2a„(7/(x„) - Ap, x„+i - p> ^ (||x„ - p|| + ||x„+i - p||)||x„ - x„+i|| + (^„72 + a„7i)2|x„ - p||2

+ 2rn(1 - ßn72 - an702|un - xn||(||Mxn - Mp|| + ||Nxn - Np||) + (ßn)2||B||2||xn - Zn|2

+(e„)2|p|2 +2(1 - ^„72 - a„7i)^„||£||||u„ -p||||x„ - z„|| + 2(1 - ^„72 - a:„7i)e„||p||||u„ - p|| + 2^„e„||£||||p||||x„ - z„|| + 2a„(7/(x„) - Ap, x„+i - p).

Since a„ — 0, ||x„+i - x„|| — 0, ||Mx„ - Mp|| — 0, ||Nx„ - Np|| — 0 and ||x„ - z„|| — 0 as n — to and we obtain

lim ||x„ - u„|| = 0. (3.10)

Using (3.8) and (3.10), we obtain ||z„ - u„|| ^ ||z„ - x„|| + ||x„ - u„|| — 0, as n — to. Then lim„^ ||z„ - u„|| =0. >

4. Strong Convergence Algorithm

Theorem 4.1. The Algorithm defined by (3.1) convergence strongly to z € Fix(T) n GEPP, which is a unique solution in of the variational inequality ((7/ - A)z,y - z) ^ 0 for all y € 0.

< Let s = P©. We get

||s(1 - A + 7/)(x) - s(1 - A + 7/)(y)|| < ||(1 - A + 7/)(x) - (I - A + 7/)(y)|| < ||1 - A|| ||x - y|| + 7||/(x) - /(y) || < (1 - 7i) ||x - y|| + 7«||x - y|| = (1 - (7i - 7a))||x - yy.

Then s(1 - A + 7/) is a contraction mapping from H into itself. Therefore by Banach contraction principle, there exists z € H such that z = s(1 - A + 7/)z = PFix(T)nEPP(1 - A + 7/)z.

We show that ((7/ - A)z, x„ - z) ^ 0. To show this inequality, we choose a subsequence {x„} of {x„} such that

lim sup ((7/ - A)z, x„ - z) = lim ((7/ - A)z, x„ - z). (4.1)

Since {x„} is bounded, there exists a subsequence {xrai. } of {x„} which converges weakly to some w € C. Without loss of generality, we can assume that xni — w. Now, we prove that w € Fix(S) n GEPP. Let us first show that w € Fix(S). From ||x„ - u„|| — 0, we obtain urai — w. On the other hand ||z„-u„|| = 0 and by Lemma 2.1,1-T is demiclosed at 0.

Thus, we obtain w € Fix(T). We show that w € GEPP. Since u„ = Trn(x„ - r„(M + N)x„). we have

F(un,y) + ((M + N)xn, y — un) + — (y - ura,ura -xra) ^ 0 (Vy € C).

It follows from the monotonicity of F that

((M + N)xn, y — un) + —(y-un,un -xn) ^ F(y,un) (Vy € C)

rn

which implies that

((M + N)xni,y - uni) + — (y -uni,uni -xni) ^ F(y,uni) (Vy € C). Let ut = ty + (1 — t)w for all t € (0,1]. Since y € C and w € C, we get ut € C. It follows that

(ut — urai, (M + N)u> ^ (ut — urai, (M + N)ut) — (ut — urai, (M + N)xrai>

/ u _ x \

-{Ut-uni,^-^ ) + = («t — uni, (M + — (M + N)uni)

+(ut ~ uni, (M + - (M + A0xrai) - ( % - ^-^ ) + F(ut,urai)

ni /

uni xn; 'n.

= (ut — uni, Mut — Muni) + (ut — uni, Nut — Nun J + (ut — uni, Muni — MxnJ + (ut — uni, Nuni — Nxni)

uni xni \ . T-,/ \

-< Ut -Uni,- )

rni /

Since ||uni — XniII — 0, we have ((Mu^ — Mxni|| — 0 and ||Nuni — Nxni|| — 0. Further from monotonically of M and N, we obtain

(ut — uni, Mut — Muni) ^ 0, (ut — uni, Nut — Nun J ^ 0,

so as i —y to from Assumption 2.1, we have (ut — w, (M + N)ut) ^ F(ut, w). Therefore

0 = F(ut, ut) < tF(ut, y) + (1 — t)F(ut, w) < tF (ut, y) + (1 — t) (ut — w, (M + N )ut) < tF (ut, y) + (1 — t)t(y — w, (M + N)ut),

then 0 ^ F(ut, y) + (1 — t)(y — w, (M + N)ut).

Letting t — 0, we obtain 0 ^ F(w,y) + (y — w, (M + N)w). This implies that w € GEPP. Now from Lemma 2.7, we have

lim sup ((7/ — A)z, xn — z) ^ lim sup ((7/ — A)z, xni — z) = ((7/ — A)z, w — z) ^ 0. (4.2)

n—^^o i—^^o

Now we prove that xn is strongly convergence to z.

It follows from (3.3) that

||x«+i - z||2 = a«(7/(x«} - Az, x«+i - z) + /«(Bx« - Bz, x«+i - z>

-e«(z, Xn+i - z) + ^((1 - e«}1 - ^«B - A(z« - z),x«+i - z^

^ a« (7(/(x«} - /(z},x«+i - z) + (7/(z) - Az,x«+i - z)) + /«||B||||x« - z||||x«+i - z||

-e«||z||||x«+i - z|| + ||(1 - e«}1 - /«B - a«A||||z« - z||||x«+i - z|| ^ a« (7(/(x«} - /(z},x«+i - z) + (7/(z) - Az,x«+i - z)) + /«||B||||x« - z||||x«+i - z||

-e«||z|| ||x«+i - z|| + ||(1 - e«}1 - /«B - a«A||d(z«, Tz)||x«+i - z|| ^ a«(7(/(x«} - /(z},x«+i - z) + (7/(z} - Az,x«+i - z)) + /«||B||||x« - zЦЦxn+l - z|| -e«||z||||x«+i - z|| + ||(1 - e«}1 - /«B - a«A||H(Tu«, Tz)||x«+i - z|| ^ a«aY||x« - z||||x«+i - z|| + a«(7/(z} - Az,x«+i - z) + /«/||x« - zЦЦxn+l - z|| -en||zЦЦxn+l - z|| + (1 - /«72 - a«7i}||x« - zЦЦxn+l - z|| = (1 - a«(7i - aY}) ||x« - z^xb+i - z|| - enЦzЦЦxn+l - z|| + a«(7/(z) - Az,x«+i - z)

^ 1—ajq_1—_ zf + \\Xn+l _ z||2) 6n||z||||Xfi-\-1 - z||+ara(7/(z)-Az,xra+i - z) ^ 1——07) _ + 1 _ z||2_era||z||||a;ra+1 _ z||+ara(7/(z) - Az,xra+i - z>.

This implies that

2||x«+i - z||2 ^ (1 - a«(7i - a7)) ||x« - z||2 + ||x«+i - z||2 -2a«||z|| ||x«+i - z|| + 2a«(7/(z) - Az,x«+i - z).

Then

I|xn+1 - z||2 ^ (1 - an (71 - 0:7)) ||xn - z||2 - 2an||z||||xra+i - z| + 2an(7/(z) - Az,xra+i - z) = (1 - kn)||xn - z||2 + 2a„1„,

(4.3)

where k« = a«(7i - a7} and 1« = (7/(z) - Az,x«+i - z) - ||zЦЦxn+l - z||.

Since lim«^^, a« = 0 and oa« — to, it is easy to see that k« — 0, — 00

and limsup«^^ 1« ^ 0. Hence, from (4.2) and (4.3) and Lemma 2.10, we deduce that xn — z, where z = P©(/ - A + 7/)z. >

Remark 4.1. Putting A = B = M = N = 0, 7 =1, we obtain methods introduced in [31, Theorem 3.1].

5. Numerical Examples

In this section, we give some examples and numerical results for supporting our main theorem.

All the numerical results have been produced in Matlab 2017 on a Linux workstation with a 3.8 GHZ Intel annex processor and 8 Gb of memory.

Example 5.1. Let H = r, the set of all real numbers, with the inner product defined by (x,y) = xy for all x,y € r, and induced usual norm | . |. Let C = [0,2]; let F : C x C — r be defined by F(x,y) = (x - 4)(y - x) for all x,y € C; let M, N : C — H be defined by M(x) = x and N(x) = 2x for all x € C, such that a = \ and (3 = | respectively, and let for each x € r, we define /(x) = |x, A(x) = 2x, B(x) = |x and

Tx i {xh 0 < x < 1;

!{*}, 1<*<2.

Then there exist unique sequences {xn} C r and {un} C C generated by the iterative schemes

xn+1 =

un = TF (xn — rn(M + N)x^;

1

11

1

3n2 /

1

2n2 — 3

I —\B - — A )zn

n2

n

(4.4)

(4.5)

where an = /?„ = e„ = ^4z3 and rn = 1. Then {x„} converges to {1} e Fix(T)nGEPP.

It is easy to prove that the bifunction F satisfy the Assumption 2.1. Further, / is contraction mapping with constant a = -g and A is a strongly positive bounded linear operator with constant 71 = 1 on r. Therefore, we can choose 7 = 1 which satisfies 0<7<-^-<7 + ^- Furthermore, it is easy to observe that Fix(T) = [0,1] and GEPP = {1}. Hence Fix(T) n GEPP = {1} = 0. After simplification, schemes (4.6) and (4.7) reduce to un — 2 x<n.

Tun =

{2 — xn}, 0 < un < 1 or (1 < xn < 2);

1 < un ^ 2 or (0 ^ xn < 1).

If zn = 2 — xn for xn € [1, 2], we have

1

- f _i 11 JL

Xn+1 V 8n + 3772 + 277,2 - 3

If zn = | for xn € [0,1), we have

xn+1

1 1 8n 3n2

xn +

xn + 2 1 —

1

1

2n2 3

1

3n2

1

2n2 3

1

3n2

Following the proof of Theorem 4.1, we obtain that {xn}, {un} converges strongly to w {1} € Fix(T) n GEPP as n — to.

10 1 5 20 25 30 35 Iteration steps=3Q

Fig. 1. The graph of {xn} with initial value xi = 1.

Example 5.2. Let H = r, the set of all real numbers, with the inner product defined by (x,y) = xy for all x,y € r, and induced usual norm | . |. Let C = [—1,3]; let F : C x C — r be defined by F(x, y) = x(y — x) for all x, y € C; let M, N : C — H be defined by M(x) = 2x

and N(x) = 3x for all x € C, such that a = | and ¡3 = | respectively, and let for each x € R, we define /(x) = ^x, A(x) = f, £>(x) = ^ x and

Tx =

0 < x < 3; -1 < x < 0.

1

2

Then there exist unique sequences {xn} C r and {un} C C generated by the iterative schemes

(4.6)

Un = TrFn(xra - r„(M + N)x„);

Xn+1

+

3 y/n 10(77 + l)2

xn +

1 -

n2

I-

(n + 1):

:B -

n

Az

(4.7)

where an

= = (ra+i)2' = ^ and = 1 + Then i®»} converges to {0} €

Fix(T) n GEPP.

It is easy to prove that the bifunction F satisfy the Assumption 2.1. Further, / is contraction mapping with constant a = and A is a strongly positive bounded linear operator with constant 71 = 1 on r. Therefore, we can choose 7 = 2 which satisfies 0<7<^-<7 + ^. Furthermore, it is easy to observe that Fix(T) = {0} and GEPP = {0}. Hence Fix(T) n GEPP = {0} = 0. After simplification, schemes (4.6) and (4.7) reduce to

_ f-An Un~ V 277 + 1 r"'

Tun =

If Zn = ^xn for xn € [-1,0], we have

-15 ^ un < 0 or (0 < xn ^ 3); 0 < u„ < 2 or (-1 < xn < 0).

Xn+1

+

Sy/ri 10(77 + l)2 If zn = 0 for xn € (0, 3], we have

xra + I 1 o

772 1 0 ( 77 + l)2 2y/n J V 477 + 2

— 4n — 5

11

X n+1 = I ~ F= +

xn.

Jy/n 10(77 + 1)2^

Following the proof of Theorem 4.1, we obtain that {xn}, {un} converges strongly to w {0} € Fix(T) n GEPP as n — to.

Fig. 2. The graph of {xn} with initial value x1 = 1.

Acknowledgments. The author would like to thanks the referees for their remarks that helped us very much in revising the paper.

1

2

1

x

n

1

1

x

n

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Received July 7, 2019

Hamid Reza Sahebi

Department of Mathematics,

Ashtian Branch, Islamic Azad University,

Ashtian, P. O. Box 39618-13347, Iran,

Assistant Professor

E-mail: sahebi@aiau.ac.ir

https://orcid.org/0000-0002-1944-5670

Владикавказский математический журнал 2021, Том 23, Выпуск 1, С. 60-76

АЛГОРИТМ НЕЛИНЕЙНОЙ ВЯЗКОСТИ С ВОЗМУЩЕНИЕМ ДЛЯ НЕРАСШИРЯЮЩИХ МНОГОЗНАЧНЫХ ОТОБРАЖЕНИЙ

Сахеби Х. Р.1

1 Исламский университет Азад, Аштиан, Иран E-mail: sahebi@aiau.ac.ir

Аннотация. Итерационные алгоритмы вязкости для поиска общего элемента множества неподвижных точек нелинейных операторов и множества решений вариационных неравенств исследовались многими авторами. Соответствующая техника позволяет применить этот метод к выпуклой оптимизации, линейному программированию и монотонным включениям. В этой статье на основе метода вязкости с возмущением, мы вводим новый алгоритм нелинейной вязкости для нахождения элемента множества неподвижных точек нерасширющих многозначных отображений в гильбертовом

пространстве. Установлены теоремы о сильной сходимости этого алгоритма при подходящих предположениях относительно параметров. Наши результаты можно рассматривать как обобщение и усиление имеющихся в текущей литературе результатов. Представлены также некоторые числовые примеры, показывающие эффективность и применитость предложенного алгоритма.

Ключевые слова: проблема неподвижной точки, обобщенного проблема равновесия, нерасширя-ющее многозначное отображение, гильбертово пространство.

Mathematical Subject Classification (2010): 47H09, 47H10, 47J20.

Образец цитирования: Sahebi, H. R. Nonlinear Viscosity Algorithm with Perturbation for Nonexpansive Multi-Valued Mappings // Владикавк. мат. журн.—2020.—Т. 23, № 1.—C. 60-76 (in English). DOI: 10.46698/e7204-1864-5097-s.