Iterative Algorithms for a System of Random Nonlinear Equations in Hilbert Spaces with Fuzzy Mappings (Nonlinear Analysis and Convex Analysis)

Iterative

_Algorithms

for

a

System

of Random

Nonlinear

_Equations

in Hilbert

_Spaces

with

Fuzzy Mappings

Jong Kyu

Kiml

and

Salahuddin2

1Department

of Mathematics_{Education, Kyungnam University}

Changwon, Gyeongnam 51767, Korea

‐mail: _{[email protected]}

2Department

ofMathematics,Jazan_University,

Jazan, Kingdomof Saudi Arabia

‐mail: _{[email protected]}

Abstract

The purpose of this paper, by usingthe resolvent operatortechnique associated with

randomly

(A, $\eta$, m)

‐monotone_operator, istoestablishanexistence and convergence the‐ oremforaclass of_systemof random nonlinearequationswith

fuzzy

_mappingsin Hilbert

spaces. Our worksare_improvementsandgeneralizationsof the

corresponding

well‐known

results.

Keywords:

system of random nonlinear_equations, relaxed cocoerciveoperators, ran‐

domly

(A, $\eta$, m)

‐proximal

operatorequations,

fuzzy mappings,

Hilbertspaces.

AMS Mathematics _Subject Classification: _{49\mathrm{J}40,} 47\mathrm{H}06.

1

Introduction

The

_fuzzy

sets

_theory

is anextension of a

_crisp

set

_by

_enlarging

the truth valued set

_\{0

,1

\}

tothe real unit interval

_[0

,1

] ([27]).

A

fuzzy

setis characterized and identified

by

a

mapping

calleda

membership grade

function from the wholesetinto

_[0

,1

]

.

Heilpern

[13]

introduced the

concept of

_{fuzzy mappings}

and

_proved

a fixed

_point

theorem for

fuzzy

contraction

_mapping

Jong Kyu

Kimand Salahuddin

whichisa

fuzzy analogue

of Nadler’s fixed

_point

theoremfor multi‐valued

_mappings.

In

_1989,

_Chang

and Zhu

_[6]

first introduced and studiedaclass of variational

inequalities

for

_{fuzzy mappings.}

Since then several classes of variational

_{inequalities, quasi}

variational

inequalities

and

_{complementarity problems}

with

_{fuzzy mappings}

wereconsidered

_by

Agarwal

etal.

_[1],

_Chang

and

_Huang

_[8],

_Ding

et al.

_[9],

_Huang

_[10],

Lee etal.

_[21],

Salahuddin

_[25]

in

the

_setting

of Hilbertspacesand Banachspaces.

Lan

_[20]

introduced a new _concepts of

(A, $\eta$)

‐monotone _operator which

_generalizes

the

(H, $\eta$)

‐monotonicity

and A

_{‐monotonicity}

in Hilbert _spaces and studied some

properties

of

(A, $\eta$)

‐monotone _operators and

_applied

resolvent _operators associated with

_{(A, $\eta$)}

‐monotone

operators to

_approximate

the solution ofa newclass of nonlinear

(A, $\eta$)

‐monotone _operator

inclusion

_problems

with relaxed cocoercive _operators in Hilbert spaces.

Recently

Kim et al.

[16]

introduced the

_{(A, $\eta$, m)}

_‐proximal

_operator to

_study

the_system of

_equations

in Hilbert

spaces.

Recently

some_systemsof variational

inequalities,

variational

inclusions, complementarity

problems

and

_{equilibrium problems}

have been studied

_by

some authors _{in recent years} be‐

cause of their close relationto Nash

_{equilibrium problems. Huang}

and

_Fang

_[11]

introduced

a

_system

of order

complementarity problems

and establishedsomeexistence results for these

using

fixed

_point

_theory.

Kim and Kim

_[18]

introduced and studiedsome_systemof variational

inequalities

and

_developed

someiterative

algorithms

for

approximately

the solutionsof_system

of variational

_{inequalities.}

On the other

_hand,

random variational

_{inequality problems,}

random

_quasi

variational

inequality problems

and random variational inclusions and

_{complementarity problems}

have

been studied

_{by Chang}

_[5],

_Chang

and

_Huang

_[7],

_Huang

_[10],

Khan andSalahuddin

_[15]

and

Bharucha\ulcornerRed

[3],

etc.

The_conceptsof random

_{fuzzy mapping}

wasfirst introduced

by Huang

[10].

Subsequently

the random variational inclusion

_problems

for random

_fuzzy

mappings

isstudied

_by

Anastas‐

siou et al.

_[2],

Salahuddin

_[25],

_Zhang

and Bi

_[28].

Inspired

and motivated

_by

the works

_[2,

_{12, 14, 17, 23,}

_26],

weestablish the existence and

convergencetheorem forsystemof random nonlinear

_equations

with

_fuzzy

_mapping

inHilbert

spaces

by using

random

(A, $\eta$, m)

‐proximal

operator

equations

2

Preliminaries

Throughout

this paper,

( $\Omega$, $\Sigma$)

is a measurable _space with aset $\Omega$ and a

a‐algebra

$\Sigma$ of a

A_systemof random nonlinear

_equations

inHilbertspaces

Notations

_{\mathcal{B}(H)}

, 2^{H} and

CB(H)

denote the class of Borel a‐fields in H, the

family

of all _nonempty subsets of H and the

_family

of all _nonempty closed bounded subset of

_H,

respectively.

Definition 2.1. A

_mapping

u: $\Omega$\rightarrow Hissaidtobe measurable if for_any

_{B\in B(H)}

,u^{-1}=

\{t\in $\Omega$, u(t)\in B\in $\Sigma$\}.

Definition 2.2. A

_mapping

_f

: $\Omega$\times H\rightarrow H is called a random

_mapping

if for each fixed

u\in H, a

mapping

f(\cdot, u)

: $\Omega$\rightarrow H is measurable. A random

mapping

f

is said to be

continuous if for each fixed t\in $\Omega$, a

mapping

f(t, \cdot)

: H\rightarrow H iscontinuous.

Definition 2.3. A multi‐valued

_mapping

T : $\Omega$\rightarrow 2^{H} issaid to be measurable if for_any

B\in B(H)

,

T^{-1}(B)=\{t\in $\Omega$ : T(t)\cap B\neq\emptyset\}\in $\Sigma$.

Definition 2.4. A

_mapping

u : $\Omega$\rightarrow H is called a measurable selection of a measurable

multi‐valued

_mapping

T: $\Omega$\rightarrow 2^{H}_, ifuis measurable and for_any

t\in $\Omega$,

u(t)\in T_{t}(u(t))

. Definition 2.5. A

_mapping

T: $\Omega$\times H\rightarrow 2^{H} is called arandom multi‐valued

mapping

if

for each fixed

_{x\in H,}

T

_x)

:

$\Omega$\rightarrow 2^{H}

is a measurable multi‐valued

mapping.

A random

multi‐valued

_mapping

_{T: $\Omega$\times H\rightarrow CB(H)}

issaidtobe \mathfrak{D}‐continuousif for each fixed

_{t\in $\Omega$,}

T(t, \cdot)

:

$\Omega$\times H\rightarrow 2^{H}

is

randomly

continuouswith_respect totheHausdorffmetric\mathfrak{D}.

Definition 2.6. A multi‐valued

_mapping

T :

$\Omega$\times H\rightarrow 2^{H}

is called random if for _any

x\in H,

T

_x)

ismeasurable

_(denoted

by

T_{t,x}

or

_{T_{t}}

).

Let $\Omega$ be asetand

_ff(H)

beacollection of

fuzzy

setsoverH. A

mapping

\tilde{F}

: $\Omega$\times H\rightarrow

\mathrm{f}\mathrm{f}(H)

is calleda

fuzzy

_mapping

on H. If

\tilde{F}

isa

fuzzy

mapping

on H thenforany

t\in $\Omega$,

\tilde{F}(t)

(denote

it

_by

F inthe

_sequel)

isa

fuzzy

mapping

onHand

\tilde{F}_{t}(x)

isthe

membership

grade

of

x in

\tilde{F}_{t}

. Let

A\in ff(H)

_,

$\alpha$\in(0,1].

Then theset

A_{ $\alpha$}=\{x\in H:A(x)\geq $\alpha$\}

iscalledan $\alpha$‐cutof A.

Definition 2.7. A

_fuzzy

_{mapping \tilde{F}}

:

$\Omega$\times H\rightarrow \mathfrak{F}(H)

_is said to be

_measurable,

if for _any

$\alpha$\in(0,1], (\tilde{F}(\cdot))_{ $\alpha$}

: $\Omega$\rightarrow 2^{H} isameasurable multi‐valued

_mapping.

Definition 2.8. A

_fuzzy

_{mapping \tilde{F}}

:

$\Omega$\times H\rightarrow \mathrm{f}\mathrm{f}(H)

_isarandom

fuzzy mapping

if for_any

x\in H,

F

_x)

:

$\Omega$\times H\rightarrow \mathfrak{F}(H)

isameasurable

_{fuzzy mapping}

(denoted

_by

\tilde{F}_{t,x}

short down

Jong Kyu

Kimand Salahuddin

Let

\tilde{T}

:

$\Omega$\times H\rightarrow \mathfrak{F}(H)

bearandom

_{fuzzy mapping satisfying}

the

_following

condition:

(*)

: there exists a function $\alpha$ :

H\rightarrow(0,1]

such that for all

(t, x)\in $\Omega$\times H

, we have

(\tilde{T}_{t.x(t)})_{ $\alpha$(x(t))}\in CB(H)

, where

T_{t,x}

denotes the value of T at

(t, x)

. Induced multi‐valued random

_mapping

\tilde{T}_{t}

from Tasfollows:

T: $\Omega$\times H\rightarrow CB(H) , T_{t}=\tilde{T}(t, x(t))_{ $\alpha$(x(t))} , (t, x)\rightarrow T_{t,x_{ $\alpha$(x)}},\forall(t, x)\in $\Omega$\times H.

In this paper we consider the

following

random

(A_{t}, $\eta$_{t}, m_{t})

‐proximal

_operator

_equation

systemwith

_{fuzzy mappings,}

weconsider for each fixed t\in $\Omega$

finding

(x(t), y(t))

_,

(z(t), w(t))\in

H_{1}\times H_{2},

u(t)\in T_{t}(x(t))

and

E_{\mathrm{t}}(x(t), y(t))+$\rho$^{-1}R_{ $\rho$,A_{1,\ell}}^{M_{\mathrm{t}}(\cdot,x(t))}(z(t))=0,

G_{t}(u(t), y(t))+$\rho$^{-1}R_{ $\rho$,A_{2,t}}^{N_{t}(\cdot,y(t))}(w(t))=0

(2.1)

where T :

H_{1}\times $\Omega$\rightarrow \mathfrak{F}(H_{1})

is a

fuzzy mapping,

E :

H_{1}\times H_{2}\times $\Omega$\rightarrow H_{1},

G :

H_{1}\times

H_{2}\times $\Omega$\rightarrow H_{2}, g:H_{1}\times $\Omega$\rightarrow H_{1}, h:H_{2}\times $\Omega$\rightarrow H_{2}, $\eta$_{1}:H_{1}\times H_{2}\times $\Omega$\rightarrow H_{1}

and

$\eta$_{2} :

H_{2}\times H_{2}\times $\Omega$\rightarrow H_{2}

are nonlinear random

_{single‐valued mappings, A_{1}}

:

H_{1}\times $\Omega$\rightarrow

H_{1},

A_{2}:H_{2}\times $\Omega$\rightarrow H_{2},

M:H_{1}\times H_{1}\times $\Omega$\rightarrow 2^{H_{1}}

and

_{N:H_{2}\times H_{2}\times $\Omega$\rightarrow 2^{H_{2}}}

are

any nonlinear operators such that for all

_{(z(t), t)\in H_{1}\times $\Omega$,}

_{M_{t}}

z_{t}

)

:

H_{1}\rightarrow 2^{H_{1}}

isa ran‐

domly

_{(A_{1,t}, $\eta$_{1,t}, m_{1,t})}

‐monotone_operatorwith

_{f_{t}(H_{1})\cap d $\sigma$ m(M_{t}(\cdot, z(t)))\neq\emptyset}

and for all

_{(w, t)\in}

H_{2}\times $\Omega$, N_{t}

w(t))

:

H_{2}\rightarrow 2^{H_{2}}

isa

randomly

(A_{2,t}, $\eta$_{2,t}, m_{2,t})

‐monotone_operatorwith

_{g_{t}(H_{2})\cap}

dom(N_{t}(\cdot, w(t))\neq\emptyset,

R_{$\rho$_{\mathrm{t}},A_{1.t}}^{M_{t}(\cdot,x(t))}=I-A_{1,t}(J_{ $\rho$ \mathrm{r},A_{1,\mathrm{t}}}^{M_{\mathrm{t}}(\cdot,x(\mathrm{t}))}), R_{ $\rho$ \mathrm{t}}^{N_{t}(,y(t))}=A_{2,\mathrm{t}}I-A_{2,t}(J_{$\rho$_{\mathrm{t}},A_{2,\mathrm{t}}}^{N_{\mathrm{t}}(\cdot,y(t))}),

I is the

_{identity mapping,}

A_{1,t}(J_{$\rho$_{t},A_{1,\mathrm{t}}}^{M_{\mathrm{t}}(\cdot,x(t))}(z(t)))=A_{1,t}(J_{$\rho$_{\mathrm{t}},A_{1,\mathrm{t}}}^{M_{\mathrm{t}}(\cdot,x(t))})(z(t))

,

A_{2,t}(J_{$\rho$_{t},A_{2,\mathrm{t}}}^{N_{t}(,y(t))}(w(t)))=

A_{2,\mathrm{t}}(J_{ $\rho \iota$,A_{2,\mathrm{t}}}^{N_{t}(\cdot,y(t))})(w(t))

,

J_{$\rho$_{\mathrm{t}},A_{1,t}}^{M_{\ell}(\cdot,x(t))}=(A_{1,t}+$\rho$_{t}M_{t}(\cdot, x(t)))^{-1}

and

J_{ $\rho$ t}^{N_{t}(\cdot,y(t))}A_{2.\mathrm{t}}=(A_{2,t}+$\rho$_{t}N_{t}(\cdot, y(t)))^{-1},

for all

_{(x(t), z(t))\in H_{1}, (y(t), w(t))\in H_{2}}

and _{$\rho$, $\rho$}:

$\Omega$\rightarrow(0,1)

aremeasurable

mappings.

For

_appropriate

and suitable choice of

_{T, E,}

_{G, M, N, f,}

g,

A_{i},

$\eta$_{i}and

H_{i}

fori=1,2we see

that

_(2.1)

is

_generalized

version ofsome

problems

which include the _system

(random)

vari‐

ational

_inclusions,

_(random)

_{generalized quasi}

variational

_inequalities

and

_(random)

_implicit

quasi

variational

_inequalities

for

_fuzzy

_mappings,

see

[17, 18].

Lemma2.9.

_[4]

Let M:

$\Omega$\times H\rightarrow CB(H)

bea\mathfrak{D}‐continuous random multi‐valued

_mapping.

Then for a measurable

_mapping

x : $\Omega$\rightarrow H, the

mapping

M x :

$\Omega$\rightarrow CB(H)

is measurable.

Lemma 2.10.

_[4]

Let

_M,

V :

$\Omega$\rightarrow CB(H)

be twomeasurable multi‐valued

_mappings

and

A_systemof random nonlinear

_equations

inHilbert spaces

measurable selectiony: $\Omega$\rightarrow H of V such that for all t\in $\Omega$

\Vert x(t)-y(t)\Vert\leq(1+ $\epsilon$)\mathfrak{D}(M(t), V(t))

.

Lemma 2.11.

_[22]

Let

_{(H, d)}

bea

complete

metric space.

Suppose

that

G:H\rightarrow CB(H)

satisfies

\mathfrak{D}(G(x), G(y))\leq $\omega$ d(x, y) , \forall x, y\in H,

where

_{$\omega$\in(0,1)}

isaconstant. Then the

_mapping

G has afixed

point

in H.

Definition 2.12. Letx, y,w: $\Omega$\rightarrow H be random

mappings

and t\in $\Omega$. A random

mapping

T: $\Omega$\times H\times H\rightarrow Hissaidtobe:

(i)

randomly

monotone inthe first _argument if

\{T_{t}(x(t), w(t))-T_{t}(y(t), w(t)) , x(t)-y(t))\geq 0,

for all

_x(t)

,

y(t)\in H.

(ii)

randomly strictly

monotone in thefirst_argument if

_{T_{t}}

is monotoneand

\{T_{t}(x(t), w(t))-T_{t}(y(t), w(t)) , x(t)-y(t)\rangle=0

if and

_only

if

_x(t)=y(t)

_;

(iii)

randomly

r_{t}

‐strongly

monotone inthefirst_argumentif thereexistsameasurable function

r_{t} :

$\Omega$\rightarrow(0, \infty)

such that

\{T_{\mathrm{t}}(x(t), w(t))-T_{t}(y(t), w(t)) , x(t)-y(t))\geq r_{t}\Vert x(t)-y(t)\Vert^{2},

for all

_x(t)

,

y(t)\in H.

(iv)

randomly

m_{t}‐relaxedmonotone inthe first_argumentif thereexistsameasurable function

m_{t}:

$\Omega$\rightarrow(0, \infty)

such that

\{T_{t}(x(t), w(t))-T_{t}(y(t), w(t)), x(t)-y(t)\}\geq-m_{t}\Vert x(t)-y(t)\Vert^{2},

for all

_x(t)

,

y(t)\in H.

(v)

randomly

s_{t}‐cocoercive in the first _argument if there exists a measurable function s_{t} :

$\Omega$\rightarrow(0, \infty)

such that

\langle T_{t}(x(t),w(t))-T_{t}(y(t), w(t)) , x(t)-y(t))\geq s_{t}\Vert T_{t}(x(t), w(t))-T_{t}(y(t), w(t))\Vert^{2},

Jong Kyu

Kimand Salahuddin

(vi)

randomly

$\gamma$_{t}‐relaxed cocoercive withrespectto

A_{t}

inthefirst argumentif thereexists a

measurable function

_{$\gamma$_{t}\rightarrow(0, \infty)}

such that

\{T_{t}(x(t), w(t))-T_{t}(y(t), w(t))

,

A_{t}(x(t))-A_{t}(y(t))\rangle\geq-$\gamma$_{t}\Vert T_{t}(x(t), w(t))-T_{t}(y(t), w(t))\Vert^{2},

for all

_x(t)

,

y(t)

,

w(t)\in H\times H\times H.

(vii)

randomly

($\gamma$_{t}, $\alpha$_{t})

‐relaxedcocoercivewith_respectto

_{A_{t}}

inthe first_argumentif there exist

measurable functions_{$\gamma$_{\mathrm{t}},}$\alpha$_{t}:

$\Omega$\rightarrow(0, \infty)

such that

\langle T_{t}(x(t), w(t))-T_{t}\cdot(y(t), w(t)) , A_{t}(x(t))-A_{t}(y(t))\rangle

\geq-$\gamma$_{\mathrm{t}}\Vert T_{t}(x(t), w(t))-T_{t}(y(t), w(t))\Vert^{2}+$\alpha$_{\mathrm{t}}\Vert x(t)-y(t)\Vert^{2},

for all

_x(t)

,

y(t)

,

w(t)\in H\times H\times H.

(viii)

randomly

$\mu$_{t}

‐Lipschitz

continuous inthefirst_argumentif thereexists ameasurablefunc‐

tion_{$\mu$_{t}}:

$\Omega$\rightarrow(0, \infty)

such that

\Vert T_{t}(x(t), w(t))-T_{t}(y(t), w(t))\Vert\leq$\mu$_{t}\Vert x(t)-y(t)\Vert,

for all

_x(t)

,

y(t)

,

w(t)\in H\times H\times H.

Inasimilar_way,we candefine a

randomly Lipschitz continuity

of the_operatorT .,

)

in

the second

_argument.

Definition 2.13. Let T:H\times $\Omega$\rightarrow 2^{H} bearandom multi‐valued

_mapping.

Then T issaid

to be

_randomly

$\tau$_{t^{-}}\tilde{D}

_‐Lipschitz

continuous in the first _argument if there exists a measurable

mapping

$\tau$ :

$\Omega$\rightarrow(0,1)

such that

\overline{D}(T_{t}(x(t)), T_{t}(y(t)))\leq$\tau$_{t}\Vert x(t)-y(t)\Vert,

for all

_x(t)

,

y(t)\in H,

t\in $\Omega$,where

\overline{D}:2^{H}\times 2^{H}\rightarrow(-\infty, +\infty)\cup\{+\infty\}

isthe Hausdorffmetric

i. e.,

\displaystyle \overline{D}(A, B)=\max\{\sup_{x(t)\in A}\inf_{y(t)\in B}\Vert x(t)-y(t)\Vert

,

\displaystyle \sup_{x(t)\in B}\inf_{y(t)\in A}\Vert x(t)-y(t)\Vert\}

,

\forall A,

B\in 2^{H}.

Inasimilar_waywe candefine

randomly

\tilde{D}

‐Lipschitz continuity

of the T

)

inthe second

A_systemof random nonlinear

_equations

inHilbert_spaces

Lemma 2.14. Let

_{(H, d)}

be a

complete

metric _space and

T_{1}, T_{2}

:

H\rightarrow CB(H)

be two

set‐valued contractive

_mappings

withsamecontractive constant

t\in(0,1)

_{i. e.,}

\tilde{\mathcal{D}}(T_{i}(x), T_{i}(y))\leq td(x, y) , \forall x, y\in H, i=1, 2

.

Then

\displaystyle \overline{\mathcal{D}}(F(T_{i}), F(T_{i}))\leq\frac{1}{1-t}\sup_{x\in H}\overline{\mathcal{D}}(T_{1}(x), T_{2}(x))

, where

_{F(T_{1})}

and

_{F(T_{2})}

arethesetsof fixed

_points

of_{T_{1}} and_{T_{2}},

respectively.

Definition 2.15. Let

_{A:H\times $\Omega$\rightarrow H,}

_$\eta$ : H\times H\times $\Omega$\rightarrow H be two random

_single

valued

_mappings.

The set‐valued

_mapping

M : H\times H\times $\Omega$\rightarrow 2^{H} is said to be

randomly

(A_{t}, $\eta$_{t}, m_{t})

‐monotoneif

(1)

M is

_randomly

m_{t}‐relaxed$\eta$_{t}‐monotone

mapping;

(2) (A_{t}+$\rho$_{t}M_{t})(H)=H

,where

$\rho$: $\Omega$\rightarrow(0,1)

isa measurable

mapping.

Definition 2.16. Let A: $\Omega$\times H\rightarrow H be a

randomly

_{r_{t}}

‐strongly

_{$\eta$_{t}‐monotone}

mapping

and M : $\Omega$\times H\rightarrow 2^{H} be a

randomly

(A_{t}, $\eta$_{t})

‐monotone_operator. Then random _operator

(A_{t}+$\rho$_{t}M_{t})^{-1}

isa

single‐valued

random

_mapping

for_anymeasurable function

$\rho$:H\rightarrow(0, \infty)

and t\in $\Omega$.

Definition2.17. LetA: $\Omega$\times H\rightarrow H bea

randomly strictly

_{$\eta$_{t}‐monotone}

mapping

and M:

$\Omega$\times H\rightarrow 2^{H} bea

randomly

(A_{t}, $\eta$_{t}, m_{t})

‐monotone

_mapping.

Then forany

given

measurable

mapping

$\rho$:

$\Omega$\rightarrow(0,1)

_,the random resolvent _operator

J_{ $\beta$ t}^{$\eta$_{\mathrm{t}},M_{\mathrm{t}}}A_{t}

:H\rightarrow H isdefined

by

\sqrt{}^{\mathrm{t}}$\rho$_{t},A_{t}M{}^{\mathrm{t}}(x(t))=(A_{t}+$\rho$_{t}M_{t})^{-1}(x(t)) , \forall t\in $\Omega$, x(t)\in H.

Proposition

2.18.

_[19]

Let H be a Hilbert_space and _$\eta$: $\Omega$\times H\times H\rightarrow H be a

randomly

$\tau$_{t}

‐Lipschitz

continuous

mapping,

A : $\Omega$\times H\rightarrow H be a

randomly

r_{\mathrm{t}}

‐strongly

$\eta$_{t}‐monotone

mapping

and M : $\Omega$\times H\rightarrow 2^{H} be a

randomly

(A_{t}, $\eta$_{t}, m_{t})

‐monotone

_mapping.

Then the

random resolvent _operator

_{\sqrt{}^{0,M_{\mathrm{t}}} $\beta$ \mathrm{t}^{A $\iota$}}

: H\rightarrow H is a

_randomly

(\displaystyle \frac{$\tau$_{\mathrm{t}}}{r_{\mathrm{t}}-$\rho$_{t}m_{t}})

_‐Lipschitz

continuous

mapping

i. e.,

\displaystyle \Vert J_{$\rho$_{\mathrm{t}},A_{t}}^{$\eta$_{l},M_{l}}x(t)-J_{$\eta$_{\mathrm{t}},M_{l}}^{$\rho$_{t},A_{t}}y(t)\Vert\leq\frac{$\tau$_{t}}{r_{t}-$\rho$_{\mathrm{t}}m_{t}}\Vert x(t)-y(t)\Vert,

where

_{$\rho$_{t}\in(0,r\lrcorner_{-)}m_{k}}

isareal‐valued random variable for all t\in $\Omega$.

In connection with a

randomly

(A_{t}, $\eta$_{t}, m_{t})

‐proximal

_operator

_equation

_system

(2.1),

we

Jong Kyu

Kim and Salahuddin

mappings

x,

u: $\Omega$:\rightarrow H_{1}, y: $\Omega$\rightarrow H_{2}

such that for all t\in $\Omega$ and each fixed

\tilde{T}_{t,x(t)}(u(t))\geq

$\alpha$(x(t))

and

0\in E_{t}(x(t), y(t))+M_{t}(x(t),x(t))

,

0\in G_{t}(u(t), y(t))+N_{t}(y(t), y(t))

.

(2.2)

Lemma 2.19. For

_{t\in $\Omega$,}

x,

u:Q\rightarrow H_{1}

and

y: $\Omega$\rightarrow H_{2},

(x(t), y(t), u(t))

isasolution of

problem

(2.2)

if and

only

if

_{(x(t), u(t))\in H_{1}, y(t)\in H_{2}}

such that

x(t)=J_{A_{1,\mathrm{t}}, $\mu$}^{M_{\mathrm{t}}(\cdot,x(t))}[A_{1,t}(x(t))-$\rho$_{t}E_{t}(x(t),y(t))]

y(t)=J_{A_{2,t},$\rho$_{\mathrm{t}}}^{N_{l}(\cdot,y(t))}[A_{2,t}(y(t))-$\rho$_{t}G_{t}(u(t), y(t))]

(2.3)

where

J_{A_{1,t},p $\iota$}^{M_{t}(\cdot,x(t))}=(A_{1,t}+$\rho$_{t}M_{t}(\cdot, x(t)))^{-1}

and

J_{A_{2,\mathrm{t}}, $\rho$ t}^{N_{\mathrm{t}}(\cdot,y(t))}=(A_{2,t}+$\rho$_{t}N_{t} y(t)))^{-1}

are corre‐

sponding

random resolvent _operator in the first _argument ofa random

(A_{1,t}, $\eta$_{1,t})

‐monotone

operator

M_{t}

random

_{(A_{2,t}, $\eta$_{2,t})}

‐monotone_operator.

_{N_{t}(\cdot}

,

respectively,

A_{i,t}

isa

randomly

r_{i,t}

‐strongly

monotone_operatorfor i=1,2and $\rho$, $\rho$:

$\Omega$\rightarrow(0,1)

aremeasurable

mappings.

Nowwe _provethat

problem

(2.1)

is

equivalent

to

_problem

_(2.3).

Lemma 2.20. For t\in $\Omega$ the

_problem

_(2.1)

hasasolution

(x(t), y(t), u(t))

with

u(t)\in\tilde{T}_{\mathrm{t}}(x(t))

if and

_only

if the

_problem

_(2.3)

has asolution

(x(t), y(t), u(t))

with

u(t)\in\tilde{T}_{t}(x(t))

_, where

x(t)=J_{A_{1,t, $\beta$ \mathrm{f}}}^{M_{t}(\cdot,x(t))}(z(t)) , y(t)=J_{A_{2,t},$\rho$_{\mathrm{t}}}^{N_{\mathrm{t}}(\cdot,y(t))}(w(t))

(2.4)

and

z(t)=A_{1,t}(x(t))-$\rho$_{t}E_{t}(x(t), y(t))

,

w(t)=A_{2,t}(y(t))-$\rho$_{t}G(u(t), y(t))

,

where _{$\rho$, $\rho$}:

$\Omega$\rightarrow(0,1)

aremeasurable

_mappings.

3

Main Results

Inthis

_section,

wefirst discuss theexistencethorem. And thenwe

developed

an

algorithm

for

the

_problem

and

_proved

the_convergenceof the randomsequence

generated by given algorithm.

Theorem 3.1. Let

_{( $\Omega$, $\sigma$)}

be a measurable _space. Let

$\Lambda$_{i}

:

H_{i}\times $\Omega$\rightarrow H_{i}

be a

randomly

r_{i,t}

‐strongly

monotone and

randomly

s_{i,t}

‐Lipschitz

continuous

mapping

for each i=1,

2,

T :

A_systemof random nonlinear

_equations

inHilbertspaces

$\alpha$ :

H_{1}\rightarrow(0,1)

and

\tilde{T}_{t,x(t)}(x(t))\geq $\alpha$(x(t))

satisfying

the condition

(*)

. Let

\tilde{T}

:

H_{1}\times $\Omega$\rightarrow H_{1}

be the

_randomly

$\kappa$_{t}-\overline{D}

_‐Lipschitz

continuous

_mapping

induced

_by

T,where

\overline{\mathcal{D}}

isthe Hausdorff

pseudo

metric on 2^{H_{i}}, for each i=1,2. Let M :

H_{1}\times H_{1}\times $\Omega$\rightarrow 2^{H_{1}}

be a

randomly

(A_{1,t}, $\eta$_{1,t})

‐monotone

_mapping

with measurable

_mapping

m_{1} :

$\Omega$\rightarrow(0,1)

inthefirst variable

andN:

H_{2}\times H_{2}\times $\Omega$\rightarrow 2^{H_{2}}

bea

_randomly

(A_{2,\mathrm{t}}, $\eta$_{2,\mathrm{t}})

‐monotone

_mapping

with measurable

mapping

m_{2}:

$\Omega$\rightarrow(0,1)

inthefirst variable. Let _{$\eta$_{1}} :

H_{1}\times H_{1} $\Omega$\rightarrow H_{1}

bea

randomly

_{$\tau$_{2,t^{-}}}

Lipschitz

continuous

_mapping,

_{$\eta$_{2}} :

H_{2}\times H_{2}\times $\Omega$\rightarrow H_{2}

be

randomly

_{$\tau$_{2,t}}

‐Lipschitz

continuous

mapping,

E :

H_{1}\times H_{2}\times $\Omega$\rightarrow H_{1}

be the

randomly

Lipschitz

continuous

_mapping

with

respecttofirst variable with measurable

_{mapping $\beta$}

:

$\Omega$\rightarrow(0,1)

_, and second

_argument

with

respecttothe measurable

mapping $\xi$

:

$\Omega$\rightarrow(0,1)

and

randomly

($\gamma$_{1,t}, $\alpha$_{1,t})

‐relaxed_cocoercive

with _respect to

_{A_{1,t}}

and first variable of

_{E_{t}}

with measurable

_mappings

_$\gamma$, $\alpha$ :

$\Omega$\rightarrow(0,1)

. Let G :

H_{1}\times H_{2}\times $\Omega$\rightarrow H_{2}

be the

randomly Lipschitz

continuous with _respect to first and second variables with measurable

_mappings

$\mu$,

$\zeta$

:

$\Omega$\rightarrow(0,1)

,

respectively.

Let G be a

randomly

_{($\gamma$_{2,t}, $\alpha$_{2,t})}

‐relaxed cocoercive

_mapping

with_respectto

_{A_{2,t}}

withmeasurable

_mappings

$\gamma$_{2},$\alpha$_{2} :

$\Omega$\rightarrow(0,1)

_,

respectively.

Ifinaddition _$\rho$ :

$\Omega$\rightarrow(0, \rightarrow mr1,t)

and _$\rho$ :

$\Omega$\rightarrow(0,mr\text{∽^{}2\mathrm{t}}2,\mathrm{t})

are

measurable

mappings

and

\Vert J_{$\rho$_{t},A_{1,t}}^{M_{t}(\cdot,x(t))}(z(t))-J_{ $\rho$ t,A_{1,\mathrm{t}}}^{M_{\mathrm{t}}(\cdot,y(t))}(z(t))\Vert\leq v_{1,t}\Vert x(t)-y(t)\Vert

,

(3.1)

for all

_{(x(t), y(t), z(t), t)\in H_{1}\times H_{1}\times H_{1}\times $\Omega$,}

\Vert J_{$\rho$_{t},A_{2.t}}^{N_{t}(\cdot,x(t))}(z(t))-J_{$\rho$_{t},A_{2,\mathrm{t}}}^{N_{t}(\cdot,y(t))}(z(t))\Vert\leq v_{2,t}\Vert x(t)-y(t)\Vert

,

(3.2)

for all

_{(x(t), y(t), z(t), t)\in H_{2}\times H_{2}\times H_{2}\times $\Omega$}

, where x,u :

$\Omega$\rightarrow H_{1}

and y :

$\Omega$\rightarrow H_{2}

are

measurable

_mappings,

then

_problem

_(2.1)

has arandom solution

(x^{*}(t), y^{*}(l), u^{*}(t))

.

4

Iterative

_algorithms

and

_convergence

_analysis

In this

_section,

based onLemma 2.20 and Nadler results

_[23],

we shallconstruct a newclass

of iterative

_algorithms

for

_{solving problems}

_(2.1)

and discuss the_convergence

_analysis

of the

algorithms.

Algorithm

4.1. Assume that

_{H_{i},}

_{A_{ $\eta$}\cdot,}

$\eta$_{i},

M, N, E, G, T,

\tilde{T}

aresameasinthe

problem

(2.1)

for

eachi=1,2andx_{0} :

$\Omega$\rightarrow H_{1},

y_{0} :

$\Omega$\rightarrow H_{2}

aremeasurable

mappings.

Fora :

H_{2}\rightarrow(0,1)

,

n\geq 0 and the random element

_{(x(t), y(t), u(t))\in H_{1}\times H_{2}\times H_{1}}

, we define the iterative

sequences

\{x_{n}(t)\}, \{y_{n}(t)\}, \{u_{n}(t)\}

by

Jong Kyu

Kim and Salahuddin

y_{n+1}(t)=(1-$\lambda$_{n}(t))y_{n}(t)+$\lambda$_{n}(t)[J_{$\rho$_{\mathrm{t}},A_{2,t}}^{N_{t}(\cdot,y_{n}(t))}(A_{2,t}(y_{n}(t))-$\rho$_{t}G_{t}(u_{n}(t), y_{n}(t)))]+q_{n}(t)

,

(4.2)

\tilde{T}_{t,x(t)}(u_{n}(t))\geq a(x_{n}(t)) , \Vert u_{n}(t)-u(t)\Vert\leq(1+ $\iota$)\overline{\mathcal{D}}(\tilde{T}_{t}(x_{n}(t)),\tilde{T}_{t}(x(t)))

,

(4.3)

where _$\rho$, $\rho$ :

$\Omega$\rightarrow(0,1)

are

measurable,

\{$\lambda$_{n}(t)\}

is a measurable _{sequence in}

(0,1],

and

p_{n}(t)

,

q_{n}(t)

are two random error sequences

satisfying

the same conditions in

H_{1}

and

H_{2},

respectively.

Lemma 4.2.

_[24]

Let

_{{an}, \{b_{n}\}}

and

_{\{c_{n}\}}

be three _sequences of

_nonnegative

real numbers

satisfying

the

_following

conditions:

(i)

0\leq b_{n}<1,

n=0,

1, 2,

\cdots and

\displaystyle \lim\sup_{n}b_{n}<1

;

(ii)

$\Sigma$_{n=0^{C_{n}}}^{\infty}<+\infty

;

(ii)

a_{n+1}\leq b_{n}a_{n}+c_{ $\eta$}

,n=0,

1, 2,

\cdots

Then

_{\displaystyle \lim_{n\rightarrow\infty}a_{n}=0.}

Theorem4.3. Let

_{H_{1}, H_{2}, T_{t},}

\tilde{T}_{t},

_{$\eta$_{1,t}, $\eta$_{2,t},}

_{A_{1,t}, A_{2,t},}

_{M_{t}, N_{t}, E_{t}, G_{t}}

be thesame as inTheorem

3.1. Assumethat all the conditions of Theorem3.1 hold and

\displaystyle \lim\sup_{n}$\lambda$_{n}(t)<1, $\Sigma$_{n=0}^{\infty}(\Vert p_{n}(t)\Vert+\Vert q_{n}(t)\Vert)<+\infty

.

(4.4)

Then the random iterative_sequences

_{(x_{n}(t), y_{n}(t))}

with

u_{n}(t)\in\tilde{T}_{t}(x(t))

defined

_by

_Algorithm

4.1,

converges

strongly

tothe random solution

(x^{*}(t), y^{*}(t), u^{*}(t))

of

(2.1).

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