Error bounds for random generalized mixed variational inequality problems with random fuzzy mapping

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Adv. Fixed Point Theory, 7 (2017), No. 2, 183-206 ISSN: 1927-6303

ERROR BOUNDS FOR RANDOM GENERALIZED MIXED VARIATIONAL INEQUALITY PROBLEMS WITH RANDOM FUZZY MAPPING

SHAMSHAD HUSAIN∗, NISHA SINGH

Department of Applied Mathematics, Aligarh Muslim University , Aligarh 202001, India

Copyright c2017 Shamshad Husain and Nisha Singh. 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.

Abstract. In this paper, we define some new notions of gap functions of random generalized mixed variational inequality problems in a fuzzy environment. Further, we compute error bounds for random generalized mixed variational inequality problems in terms of the residual gap function, the regularized gap function and the D-gap function. The results obtained are new and generalize a number of known results for generalized mixed variational inequality problems with fuzzy mappings.

Keywords:random fuzzy mapping; sub-differential; strongly g-monotone; regularized gap function; error bound.

2010 AMS Subject Classification:49J40, 49J52, 90C30 .

1. Introduction

The theory of gap function was introduced for the study of a convex optimization problem and subsequently applied to variational inequality problems. One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of a gap function. Recently, some efforts has been made to develop gap

∗_{Corresponding author}

E-mail address: s [email protected] Received November 9, 2016

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functions for various classes of variational inequality problems; see for example [1, 2, 3, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23]. Besides these, gap functions also turned out to be very useful in designing new globally convergent algorithms and in analyzing the rate of convergence of some iterative methods and also in deriving the error bounds.

In 1965, Zadeh [24] introduced the fuzzy set theory. The applications of the fuzzy set theory can be found in control engineering and optimization problems of mathematical sciences. In the recent past, variational inequalities in the setting of fuzzy mappings have been introduced and studied which are closely related with fuzzy optimization problems. As a result, variational inequality problems have been generalized and extended in various directions using novel tech-niques of fuzzy theory.

In 1989, Chang and Zhu [5] introduced the concepts of variational inequality for fuzzy map-pings. The concept of a random fuzzy mapping was first introduced by Huang [11] while studying a new class of random multi-valued nonlinear generalized variational inclusions. For some related work, we refer to [6, 10, 11]. Recently, Dai [6] introduced a new class of general-ized mixed variational-like inequalities for random fuzzy mappings and established an existence theorem and an iterative algorithm for finding the solution of problems.

Throughout the paper, letH be a real Hilbert space, whose inner product and norm are de-noted byh·,·iandk · k, respectively. LetF be a collection of all fuzzy sets overH. A mapping T :H →F(H) is called a fuzzy mapping on H. If T is a fuzzy mapping on H, then T(x)

(denoted by T_x) is a fuzzy set on H and T_x(y) is the membership function of y in T_x. Let A∈F(H),q∈[0,1]. Then the set(A)q={x∈H :A(x)≥q}is called aq-cut set ofA.

In this paper, we denote by(Ω,Σ)a measurable space, whereΩis a set andΣis aσ-algebra

of subsets ofΩand also we denote byB(H),2H,CB(H)andH(·,·)the class of Borelσ-fields

in H, the family of all nonempty subsets of H, the family of all nonempty closed bounded subsets ofH, and the Hausd ¨orff metric onCB(H), respectively.

Let ˆT :Ω×H→F(H)be a random fuzzy mapping satisfying the following: Condition(I): There exists a mappingc:H→[0,1]such that

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By using the random fuzzy mapping ˆT, we can define a random multi-valued mapping T as follows:

T :Ω×H →CB(H), (t,x)→(Tˆt,x)c(x), ∀(t,x)∈Ω×H.

T is called the random multi-valued mapping induced by the random fuzzy mapping ˆT. Given mapping c:H→[0,1], the random fuzzy mapping ˆT :Ω×H →F(H)satisfies the condition(I)and random operatorg:Ω×H →H withIm g∩dom∂ φ 6=_∅, we consider the

following random generalized mixed variational inequality problem (for short, RGMVIP): Find measurable mappingsx,w:Ω→H, such that for allt∈Ω, y(t)∈H,

ˆ

T_t_,_x₍_t₎(w(t))≥c(x(t)),

w(t),y(t)−g(t,x(t))+φ g(t,x(t)),y(t)−φ g(t,x(t)),g(t,x(t))≥0, (1.1)

where∂ φ denotes the sub-differential of a proper, convex, and lower semi-continuous function φ(·,·):H×H→_R∪ {+∞}with its effective domain being closed.

The set of measurable mappings(x,w)is called a random solution of the RGMVIP (1.1). Special cases:

(i) Ifcis zero operator andT :H →H is a single valued mapping andg≡I, the identity operator then the RGMVIP (1.1) reduces to the generalized mixed variational inequality problem, denoted by GMVIP, which consists in findingx∈Hsuch that

hT x,y−xi+φ(x,y)−φ(x,x)≥0,∀y∈H. (1.2)

(ii) If φ(x,y) =φ(y),∀xthen problem (1.2) reduces to mixed variational inequality

prob-lem, denoted by MVIP, which consists in findingx∈H such that

hT x,y−xi+φ(y)−φ(x)≥0,∀y∈H, (1.3)

which was studied by Tang and Huang [20]. In this paper, he introduced two regularized gap functions for the MVIP (1.3) and studied their differentiable properties.

(iii) If the function φ(·) is an indicator function of a closed set K in H, then MVIP (1.3)

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findingx∈Ksuch that

hT x,y−xi ≥0,∀y∈K, (1.4)

which was studied by [1, 3, 7, 9, 15, 20]. They derived local and global error bounds for the VIP (1.4) in terms of the regularized gap functions and the D-gap functions.

Rest of the paper is organized as follows: In Section 2, we give some basic definitions and results which will be used in the paper. Furthermore, by using the residual vector we obtain the error bound for the solution of RGMVIP (1.1). In Section 3, we introduce a regularized gap function for RGMVIP (1.1) and derive the error bounds with and without Lipschitz continuity assumption. In Section 4, we introduce the D-gap function and derive global error bounds in terms of the D-gap function for the solution of RGMVIP (1.1).

2. Preliminaries

Definition 2.1. [11] A mapping x:Ω→H is said to be measurable if for anyB∈B(H),{t∈ Ω:x(t)∈B} ∈Σ.

Definition 2.2. [11] A mapping f :Ω×H →H is called a random operator if for any x∈ H, f(t,x) =x(t)is measurable. A random operator f is said to be continuous if for anyt∈Ω, the mapping f(t, .):H →His continuous.

Definition 2.3. [11] A multi-valued mapping T :Ω→2H is said to be measurable if for any B∈B(H), T−1(B) ={t∈Ω:T(t)∩B6=∅} ∈Σ.

Definition 2.4. [11] A mappingw:Ω→H is called a measurable selection of a multi-valued measurable mappingT :Ω→2H ifwis measurable and for anyt∈Ω, w(t)∈T(t).

Definition 2.5. [11] A mappingT :Ω×H →2H is called a random multi-valued mapping if for anyx∈H, T(.,x)is measurable. A random multi-valued mappingT :Ω×H →CB(H)is said to beH-continuous if for anyt∈Ω, T(t, .)is continuous in the Hausd ¨orff metric.

Definition 2.6. [11] A fuzzy mapping T : Ω→F(H) is called measurable, if for any ν ∈

(0,1], (T(.))_ν :Ω→2H is a measurable multi-valued mapping.

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Definition 2.8. A bi-functionφ :H×H→Ris said to be skew symmetric if,

φ(x(t),x(t))−φ(x(t),y(t))−φ(y(t),x(t)) +φ(y(t),y(t))≥0, ∀x,y∈H, t∈Ω.

Definition 2.9. A functionG:H→_Ris said to be a gap function for the RGMVIP (1.1), if it satisfies the following properties:

(i) G(x)≥0, ∀x∈H;

(ii) G(x_o) =0, if and only ifx_o∈H solves the RGMVIP (1.1). Now we first recall the following well-known results and concepts. For the VIP (1.4), it is well known thatx∈Kis a solution, if and only if

0=x−P_K[x−αT(x)],

whereP_Kis the orthogonal projector ontoKandα>0 is arbitrary. Hence, the norm of the right

hand side of the above equation can serve as a gap function for VIP (1.4), which is commonly known as the natural residual vector.

Then, motivated by the proximal map given in [17], we derive a similar characterization for the RGMVIP (1.1) in the random fuzzy environment by defining the mappingPφ

α(t):Ω×H→

domφ, as

Pφ,z

α(t)(t,z) =arg min_y₍_t₎_∈_H

φ(g(t,x(t)),y(t)) + 1

2α(t)k

y(t)−z(t)k2

, z(t)∈H, t∈Ω, α>0,

whereα :Ω→(0,+∞) a measurable function which is the so called proximal mapping inH for a random fuzzy mapping. Note that the objective function above is proper strongly convex. Sincedomφ is closed,Pφ,z

α(t)(t,z)is well defined and single-valued.

For any measurable functionα:Ω→(0,+∞)define the residual vector

Rφ_α(,x

t)(t,x(t)) =g(t,x(t))−P φ,x

α(t)[g(t,x(t))−α(t)w(t)], x(t)∈H. (2.1)

Next, we show thatRφ,x

α(t)(t,x(t))plays the role of the natural residual vector in random fuzzy

mapping for the RGMVIP (1.1).

Lemma 2.1.For any measurable functionα:Ω→(0,+∞)and for each t∈Ω, the measurable mapping x:Ω→H is a solution of the RGMVIP (1.1) if and only if Rφ,x

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Proof.LetRφ,x

α(t)(t,x(t)) =0, which implies thatg(t,x(t)) =P φ,x

α(t)[g(t,x(t))−α(t)w(t)].

It is equivalent to

g(t,x(t)) =arg min y(t)∈H

φ(g(t,x(t)),y(t)) + 1

2α(t)ky(t)−(g(t,x(t))−α(t)w(t))k

2

.

By the optimality conditions (which are necessary and sufficient, by convexity), the latter is equivalent to

0∈∂ φ(g(t,x(t)),y(t)) + 1 α(t)

g(t,x(t))−(g(t,x(t))−α(t)w(t))

=∂ φ(g(t,x(t)),y(t)) +w(t),

which implies

−w(t)∈∂ φ(g(t,x(t)),y(t)).

This in turn is equivalent, by the definition of the sub-gradient, to

φ g(t,x(t)),y(t)≥φ g(t,x(t)),g(t,x(t))−

D

w(t),y(t)−g(t,x(t))E, ∀y(t)∈H, t∈Ω,

which implies thatx(t)solves the RGMVIP (1.1).

Definition 2.10. A random multi-valued operator T :Ω×H →CB(H) is said to be strongly g-monotone, if there exists a measurable functionθ :Ω→(0,+∞)such that

D

w₁(t)−w₂(t),g(t,x₁(t))−g(t,x₂(t))E≥θ(t)kx1(t)−x2(t)k2,

∀w_i(t)∈T(t,x_i), ∀x_i(t)∈H, i=1,2, ∀t ∈Ω.

Definition 2.11. A random operatorg:Ω×H →H is said to be Lipschitz continuous, if there exists a measurable functionL:Ω→(0,+∞)such that

kg(t,x₁(t))−g(t,x₂(t))k ≤L(t)kx₁(t)−x₂(t)k, ∀x_i(t)∈H,i=1,2, ∀t ∈Ω.

Definition 2.12. A random multi-valued mappingT:Ω×H→CB(H)is said to be ˆH-Lipschitz continuous, if there exists a measurable functionλ :Ω→(0,+∞)such that

ˆ

H(T(t,x(t)),T(t,xo(t)))≤λ(t)kx(t)−xo(t)k, ∀x(t),xo(t)∈H.

Definition 2.13. Pφ,x_{is said to be non-expansive, if}

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Now, we give the following lemmas.

Lemma 2.2. [4] Let T :Ω×H →CB(H)be a H-Lipschitz continuous random multi-valuedˆ mapping, then for measurable mapping x:Ω→H, the multi-valued mapping T(·,x(·)):Ω→ CB(H)is measurable.

Lemma 2.3. [4]Let T1,T2:Ω→CB(H)be two measurable multi-valued mappings,ε >0be

a constant, and w₁:Ω→H be a measurable selection of T1, then there exists a measurable selection w2:Ω→H of T2such that for all t∈Ω,

kw1(t)−w2(t)k ≤(1+ε)Hˆ(T1(t),T2(t)).

Lemma 2.4. Let x:Ω→H be a measurable mapping and α :Ω→(0,+∞) be a measur-able function, then for all x(t)∈H and for each t ∈Ω, Rφ,x

α(t)(t,x(t)) is a gap function for the

RGMVIP (1.1).

Next we show that the RGMVIP (1.1) has a unique solution.

Theorem 2.1. Suppose that for each t ∈Ω, xo(t)∈H is a solution of the RGMVIP (1.1). Let

(Ω,Σ) be a measurable space, and H be a real Hilbert space. Let the random fuzzy mapping

ˆ

T :Ω×H →F(H) satisfy the condition (I) and T :Ω×H →CB(H) be the random multi-valued mapping induced by the random fuzzy mapping T . Let gˆ :Ω×H →H be a random mapping andφ :H×H→_R∪ {+∞}be a real valued function such that

(i) for each t ∈Ω, the measurable mapping T is strongly g-monotone and H-Lipschitzˆ continuous with the measurable functionsθ,λ :Ω→(0,+∞), respectively;

(ii) for each t∈Ω, the mapping g(t, .)is Lipschitz continuous with the measurable function

L:Ω→(0,+∞);

(iii) if there exists a measurable function K:Ω→(0,+∞)such that

Pφ,x

α(t)(z)−P φ,xo α(t)(z)

≤K(t)

x(t)−x_o(t), ∀x(t),x_o(t),z(t)∈H,

with

K(t)<1−

q

L2₍_t₎₋₂_α₍_t₎_θ₍_t_{) +}_α2₍_t₎₍₁₊_ε₎_λ₍_t₎

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Proof. (Uniqueness). Let the two measurable mappingsx₁,x₂:Ω→Hbe the two solutions of the RGMVIP (1.1) such thatx₁(t)6=x₂(t)∈H. Then, we have

w₁(t),y(t)−g(t,x₁(t))+φ g(t,x1(t)),y(t)

−φ g(t,x1(t)),g(t,x1(t))

≥0, (2.2)

w₂(t),y(t)−g(t,x₂(t))+φ g(t,x2(t)),y(t)

−φ g(t,x2(t)),g(t,x2(t))

≥0. (2.3)

Takingy(t) =g(t,x₂(t))in (2.2) andy(t) =g(t,x₁(t))in (2.3), adding the resultants, we have D

w1(t)−w2(t),g(t,x2(t))−g(t,x1(t)) E

≥0.

SinceT is stronglyg-monotone with measurable functionθ :Ω→(0,+∞), therefore 0≤Dw₁(t)−w₂(t),g(t,x₂(t))−g(t,x₁(t))E≤ −θ(t)x1(t)−x2(t)

2

,

which implies thatx1(t) =x2(t), ∀t∈Ω, the uniqueness of the solution of the RGMVIP (1.1). Now, by using normal residual vectorRφ,x

α(t)(t,x(t)), we derive the error bounds for the

solu-tion of RGMVIP (1.1).

Theorem 2.2. Suppose that for each t ∈Ω,xo(t)∈H is a solution of the RGMVIP (1.1). Let

(Ω,Σ)be a measurable space, and H be a real Hilbert space. Suppose that g:Ω×H→H be a random mapping andφ :H×H→_R∪ {+∞}be a real valued function. Let the random fuzzy mapping Tˆ :Ω×H →F(H) satisfy the condition (I), and a random multi-valued mapping T :Ω×H→CB(H)induced by the random fuzzy mapping such that

L:Ω→(0,+∞);

Pφ,x

α(t)(z)−P φ,xo α(t)(z)

≤K(t)

x(t)−x_o(t), ∀x(t),x_o(t),z(t)∈H,

then for any x(t)∈H, t∈Ωandα(t)>

h

L(t)K(t) θ(t)−K(t)λ(t)(1+ε)

i

, we have

x(t)−x_o(t)≤ "

α(t)λ(t)(1+ε) +L(t)

α(t)θ(t)−L(t)K(t)−K(t)α(t)λ(t)(1+ε)

# Rφ,x

α(t)(t,x(t))

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Proof.Let for eacht∈Ω, xo(t)∈H be a solution of the RGMVIP (1.1), then D

wo(t),y(t)−g(t,xo(t)) E

+φ g(t,xo(t)),y(t)

−φ g(t,xo(t)),g(t,xo(t)

≥0.

Substitutingy(t) =Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)]in the above inequality, we have

D

w_o(t),Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)]−g(t,xo(t))

E

+φ g(t,xo(t)),P_αφ₍,x_t₎o[g(t,x(t))−α(t)w(t)]

−φ g(t,xo(t)),g(t,xo(t)

≥0.

(2.4)

For any fixedx(t)∈H and measurable functionα :Ω→(0,+∞), we observe that g(t,x(t))−α(t)w(t)∈ I+α(t)∂ φ I+α(t)∂ φ−1(g(t,x(t))−α(t)w(t))

= I+α(t)∂ φP_α(φ,x_t₎[g(t,x(t))−α(t)w(t)],

which is equivalent to

−w(t) + 1

α(t)

g(t,x(t))−Pφ,x

α(t)[g(t,x(t))−α(t)w(t)]

∈∂ φ Pφ,x

.

By the definition of a sub-differential, we have D

w(t)− 1

α(t) g(t,x(t))−P

φ,xo

,

y(t)−Pφ,xo

E

+φ Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)],y(t)

−φ P_α(φ,x_t₎o[g(t,x(t))−α(t)w(t)],P_α(φ,x_t₎o[g(t,x(t))−α(t)w(t)]≥0.

Takingy(t) =g(t,x_o(t))in the above, we get D

w(t)− 1

α(t) g(t,x(t))−P

φ,xo

,

g(t,x_o(t))−Pφ,xo

E

+φ P_α(φ,_tx₎o[g(t,x(t))−α(t)w(t)],g(t,xo(t))

−φ P_α(φ,x_t₎o[g(t,x(t))−α(t)w(t)],Pφ,xo

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This implies that

D

−w(t) + 1

α(t)

g(t,x(t))−Pφ,xo

,

Pφ,xo

E

+φ Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)],g(t,xo(t))

−φ Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)],P φ,xo

≥0.

(2.5)

Adding (2.4) and (2.5), we get

D

w_o(t)−w(t) + 1

α(t)

g(t,x(t))−Pφ,xo

,

Pφ,xo

E

+φ g(t,xo(t)),P_αφ₍,_tx₎o[g(t,x(t))−α(t)w(t)]−φ g(t,xo(t)),g(t,xo(t))

+φ Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)],g(t,xo(t))

−φ Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)],P φ,xo

≥0.

Sinceφ is skew symmetric, we get

D

w_o(t)−w(t) + 1

α(t)

g(t,x(t))−Pφ,xo

,

Pφ,xo

E

≥0.

This can also be written as

α(t)

D

w_o(t)−w(t),Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)]−g(t,x(t))

E

+α(t)

D

w_o(t)−w(t),g(t,x(t))−g(t,x_o(t))

E

+Dg(t,x(t))−Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)],

Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)]−g(t,x(t))

E

α(t)[g(t,x(t))−α(t)w(t)],g(t,x(t))−g(t,xo(t))

E

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which implies that

α(t)

D

wo(t)−w(t),P_αφ₍,x_t₎o[g(t,x(t))−α(t)w(t)]−g(t,x(t))

E

≥α(t)

D

w_o(t)−w(t),g(t,x_o(t))−g(t,x(t))

E

g(t,x(t))−Pφ,xo

E

.

By using the strongg-monotonicity ofT, we get

α(t)

D

wo(t)−w(t),P_α(φ,x_t₎o[g(t,x(t))−α(t)w(t)]−g(t,x(t))

E

≥α(t)θ(t)x o(t)−x(t) 2

+Rφ,xo

α(t)(t,x(t))

2

.

Also the above inequality can be written as

α(t)Dwo(t)−w(t),P_αφ₍,x_t₎o[g(t,x(t))−α(t)w(t)]−g(t,x(t))

−Pφ,x

α(t)[g(t,x(t))−α(t)w(t)] +P φ,x

E

+

D

g(t,x(t))−Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)]−P φ,x

+Pφ,x

E

+Rφ,xo

α(t)(t,x(t))

2

.

By using the Cauchy-Schwarz inequality along with the triangular inequality, we have

α(t)wo(t)−w(t) Pφ,xo

α(t)[g(t,x(t))−α(t)w(t)]−P φ,x

+α(t)wo(t)−w(t)

P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)]−g(t,x(t))

+g(t,x(t))−P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)]

g(t,xo(t))−g(t,x(t))

+Pφ,x

α(t)[g(t,x(t))−α(t)w(t)]−P φ,xo

g(t,x_o(t))−g(t,x(t))

+Rφ,xo

α(t)(t,x(t))

2

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Now using the ˆH-Lipschitz continuity ofT, the Lipschitz continuity ofg, and assumption (iii) onPφ,x

α(t)(.), we have

α(t)λ(t)(1+ε)xo(t)−x(t)

K(t)x_o(t)−x(t)

+α(t)λ(t)(1+ε)xo(t)−x(t) Rφ,x

α(t)(t,x(t))

+Rφ,x

α(t)(t,x(t))

L(t)xo(t)−x(t)

+K(t)L(t)

x(t)−xo(t) 2

+Rφ,xo

α(t)(t,x(t))

2

.

The above can be again written as K(t)α(t)λ(t)(1+ε)xo(t)−x(t)

2

+α(t)λ(t)(1+ε)xo(t)−x(t) Rφ,x

α(t)(t,x(t))

+L(t)Rφ,x

α(t)(t,x(t))

x_o(t)−x(t)+K(t)L(t)

x(t)−x_o(t) 2

+Rφ,x

α(t)(t,x(t))

2

.

Therefore, we have

−K(t)α(t)λ(t)(1+ε)−L(t)K(t) +α(t)θ(t)xo(t)−x(t) 2

≤[α(t)λ(t)(1+ε) +L(t)]xo(t)−x(t) Rφ,x

α(t)(t,x(t))

Hence,

xo(t)−x(t) ≤

"

α(t)λ(t)(1+ε) +L(t)

α(t)θ(t)−L(t)K(t)−L(t)α(t)λ(t)(1+ε)

#

Rφ_α(,x_t₎(t,x(t)),

∀x(t)∈H,t∈Ω,

where

α(t)>

"

L(t)K(t)

θ(t)−K(t)λ(t)(1+ε)

#

.

3. Regularized gap functions

In this section our main motivation is to overcome the non differentiability of residual vector Rφ,x

α(t) i.e., the gap function defined by (2.1). Now by using an approach due to Fukushima

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viewed as a regularized gap function. Forα >0, the functionsGα is defined by

G_α(x(t)) = max y(t)∈H

nD

w(t),g(t,x(t))−y(t)E−φ g(t,x(t)),y(t)

+φ g(t,x(t)),g(t,x(t))− 1

2α

g(t,x(t))−y(t) 2o

(3.1)

which is finite valued everywhere and is differentiable whenever all operators involved in G_α(x(t)), are differentiable.

Lemma 3.1. For anyα>0,Gα(x(t))can be written as

G_α(x(t)) =Dw(t),g(t,x(t))−Pφ,x

E

−φ g(t,x(t)),

P_α(φ,x

t)[g(t,x(t))−α(t)w(t)]

+φ g(t,x(t)),g(t,x(t))

− 1

2α

g(t,x(t))−Pφ,x

2

, ∀x(t)∈H

(3.2)

Proof. Ifx(t)∈/domφ, then equation (3.2) holds, because φ ∼= +∞, while the other terms are

all finite (recall that Pφ,x

α(t)(t,z)∈domφ, for anyz(t)∈H). Consider now any x(t)∈domφ.

Denote byF(y)the function being maximized in (3.1). Letz(t)be the (unique, by concavity of F(y)) element at which the maximum is realized in (3.1). Then z(t)is uniquely characterized by the optimality condition

0∈∂(−F(z)) =w(t) +∂ φ(.,z) + 1

α(z(t)−g(t,x(t))

=∂ φ(.,z) + 1

α(t)[z(t)−(g(t,x(t))−α(t)w(t))].

But, this inclusion also uniquely characterizes the solution of the problem z=arg min

y(t)∈H n

φ(g(t,x(t)),y(t)) + 1

2α

y(t)−(g(t,x(t))−α(t)w(t)) 2o

=Pφ,x

where the above equation follows from the definition of the proximal mappingP_α(φ,x t)(.).

Next we show the functionG_α(x(t))forα >0 given by (3.1) is a gap function for RGMVIP

(1.1).

Theorem 3.1.Ifα >0, then we have

G_α(x(t))≥ 1

2α

Rφ,x

α(t)(t,x(t))

2

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In particular, G_α(x(t)) =0, if and only if x(t)is a solution of RGMVIP (1.1). Proof.For any fixedx(t)∈H,t∈Ωandα>0.

g(t,x(t))−α(t)w(t)∈(I+α(t)∂ φ)(I+α(t)∂ φ)−1[g(t,x(t))−α(t)w(t)]

=(I+α(t)∂ φ)Pφ,x

which is equivalent to

−w(t) + 1

α(t)

g(t,x(t))−Pφ,x

∈∂ φ Pφ,x

.

By the definition of a sub-differential, we have D

w(t)− 1

α(t)

g(t,x(t))−Pφ,x

,

y(t)−Pφ,x

E

+φ P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)],y(t)

−φ Pφ,x

α(t)[g(t,x(t))−α(t)w(t)],P φ,x

≥0.

Takingy(t) =g(t,x(t))in the above inequality, we get D

w(t)− 1

α(t)

g(t,x(t))−Pφ,x

,

g(t,x(t))−Pφ,x

E

+φ P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)],g(t,x(t))

−φ Pφ,x

≥0,

or

D

w(t),g(t,x(t))−Pφ,x

E

+φ Pφ,x

α(t)[g(t,x(t))−α(t)w(t)],g(t,x(t))

−φ Pφ,x

≥ 1

α(t)

D Rφ,x

α(t)(t,x(t)),R φ,x

α(t)(t,x(t))

E

.

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Combining (3.2) and (3.3) and by using the skew symmetry ofφ, we get

G_α(x(t))≥ 1

α(t)

D Rφ_α(,x

t)(t,x(t)),R φ,x

α(t)(t,x(t))

E

− 1

2α(t)

Rφ,x

α(t)(t,x(t))

2

= 1

2α(t)

Rφ_α(,x_t₎(t,x(t)) 2

.

Clearly, we haveGα(x(t))≥0, ∀x(t)∈H.

Now, from the above conclusion, ifG_α(x(t)) =0, thenRφ,x

α(t)(t,x(t)) =0. Hence by Lemma

2.1, we see thatx(t)∈His a solution of RGMVIP (1.1). Conversely, ifx(t)∈His a solution of RGMVIP (1.1), theng(t,x(t)) =Pφ,x

α(t)[g(t,x(t))−α(t)w(t)], consequently, from (3.2) we have

G_α(x(t)) =0.

As a consequence of Theorem 2.2 and Theorem 3.1, we have the following result on error bound in terms ofG_α₍_t₎(x(t))for RGMVIP (1.1).

Corollary 3.1. Suppose that for each t ∈Ω,xo(t)∈H is a solution of the RGMVIP (1.1). Let

ˆ

T :Ω×H →F(H) satisfy the condition (I) and T :Ω×H →CB(H) be the random multi-valued mapping induced by the random fuzzy mapping T . Let gˆ :Ω×H →H be a random mapping andφ :H×H→R∪ {+∞}be a real valued function such that

L:Ω→(0,+∞);

kPφ,x

α(t)(z)−P φ,xo

α(t)(z)k ≤K(t)kx(t)−xo(t)k,∀x(t),xo(t),z(t)∈H,

h _L₍_t₎_K₍_t₎

θ(t)−K(t)λ(t)(1+ε)

i

, we have

x(t)−x_o(t)≤ "

α(t)λ(t)(1+ε) +L(t)

#

√

2αpGα(x(t)).

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ˆ

T :Ω×H →F(H) satisfy the condition (I) and T :Ω×H →CB(H) be the random multi-valued mapping induced by the random fuzzy mapping T . Let gˆ :Ω×H →H be a random mapping and φ :H×H →R∪ {+∞}be a real valued function such that for each t ∈Ω, the measurable mapping T is strongly g-monotone andH-Lipschitz continuous with the measurableˆ functionsθ,λ :Ω→(0,+∞), respectively, then for any x(t)∈H,t∈Ω, we have

x(t)−x_o(t)≤ "

1 r

θ(t)− L

2₍_t₎

2α(t)

#

p

G_α(x(t)).

Proof.From (3.1), it can be written as

G_α(x(t))≥Dw(t),g(t,x(t))−g(t,x_o(t))E−φ g(t,x(t)),g(t,xo(t))

+φ g(t,x(t)),g(t,x(t))− 1

2α

g(t,x(t))−g(t,x_o(t)) 2

.

By using the strong g-monotonicity ofT, we have

G_α(x(t))≥Dw(t)−w_o(t),g(t,x(t))−g(t,x_o(t))E+Dw_o(t),g(t,x(t))−g(t,x_o(t))E

−φ g(t,x(t)),g(t,xo(t))

− 1

2α

g(t,x(t))−g(t,xo(t)) 2

or

≥θ(t)x(t)−xo(t) 2

+wo(t),g(t,x(t))−g(t,xo(t))

− 1

2α(t)L

2₍_t₎

x(t))−x_o(t)) 2

.

(3.4)

Sinceg(t,x_o(t))is a solution of RGMVIP (1.1). D

w_o(t),y(t)−g(t,x_o(t))

E

+φ g(t,xo(t)),y(t)

−φ g(t,xo(t)),g(t,xo(t))

≥0.

Takingy(t) =g(t,x(t))in the above inequality D

w_o(t),g(t,x(t))−g(t,x_o(t))E+φ g(t,xo(t)),g(t,x(t))

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Combining (3.4) and (3.5) and using the skew symmetry ofφ, we get

G_α(x(t))≥θ(t)x(t)−xo(t) 2

− 1

2α(t)L

2₍_t₎

x(t)−x_o(t) 2

G_α(x(t))≥ θ(t)− L

2₍_t₎

2α(t)

!

x(t)−x_o(t) 2

which implies

x(t)−x_o(t) 2

≤ 1

θ(t)−L

2₍_t₎

2α(t)

Gα(x(t))

x(t)−xo(t) ≤

1 r

θ(t)− L

2₍_t₎

2α(t)

p

G_α(x(t))

4. D-Gap Function

In this section, we consider another gap function associated with RGMVIP (1.1), which can be viewed as a difference of two regularized gap functions with distinct parameters, known as the D-gap function, which was introduced and studied by [18, 19, 23] for solving variational inequalities and complementarity problems.

For each x(t) ∈H, the difference of two regularized gap functions G_α(x(t))−G_β(x(t)), whereα>β>0 for RGMVIP (1.1) will not be well defined forx(t)∈/domφ, as both quantities

are not finite. Nevertheless, we shall define the D-gap function by taking a formal difference of equations (3.1) for the two parametersα >β >0.

The D-gap function associated with RGMVIP (1.1) is given by

D_α_,_β(x(t)) = max y(t)∈H

nD

w(t),g(t,x(t))−y(t)E−φ g(t,x(t)),y(t)

+φ g(t,x(t)),g(t,x(t))+ 1

2β

g(t,x(t))−y(t) 2

− 1

2α

g(t,x(t))−y(t) 2o

,x(t)∈H,α >β >0.

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The D-gap function defined by (4.1) can be written as

D_α_,_β(x(t)) =Dw(t),Pφ,x

β(t)[g(t,x(t))−β(t)w(t)]−P φ,x

E

−φ Pφ,x

β(t)[g(t,x(t))−β(t)w(t)],P φ,x

+φ Pφ,x

β(t)[g(t,x(t))−β(t)w(t)]

+ 1

2β

g(t,x(t))−Pφ,x

2 − 1 2α

g(t,x(t))−Pφ,x

2

.

Further, it can be written as

D_α_,_β(x(t)) =Dw(t),Rφ_α(,x

t)(t,x(t))−R φ,x

β(t)(t,x(t))

E

−φ Pφ,x

+φ Pφ,x

+ 1

2β

Rφ,x

β(t)(t,x(t))

2 − 1 2α Rφ_α(,x

t)(t,x(t))

2

.

(4.2)

Next, we derive global error bounds for RGMVIP (1.1). Theorem 4.1.For all x(t)∈H,t∈Ω, α >β >0, we have

1 2 1 β − 1 α Rφ,x

β(t)(t,x(t))

2

≤D_α_,_β(x(t))≤ 1 2 1 β − 1 α Rφ,x

α(t)(t,x(t)))

2

.

In particular D_α_,_β(x(t)) =0, if and only if, x(t)∈H solves RGMVIP (1.1). Proof.By the definition of a sub-differential, we have

D

w(t)− 1

α(t)

g(t,x(t))−Pφ,x

,

y(t)−Pφ,x

E

+φ P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)],y(t)

−φ Pφ,x

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Takingy(t) =Pφ,x

β(t)[g(t,x(t))−β(t)w(t)]in the above inequality, we get

D

w(t)− 1

α(t)

g(t,x(t))−Pφ,x

,

Pφ,x

β(t)[g(t,x(t))−β(t)w(t)]−P φ,x

E

+φ Pφ,x

−φ Pφ,x

≥0,

which implies that D

w(t),Rφ,x

α(t)(t,x(t))−R φ,x

β(t)(t,x(t))

E

≥ 1

α(t)

D Rφ,x

β(t)(t,x(t))

E

−φ Pφ,x

+φ P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)],P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)].

(4.3)

Combining (4.2) and (4.3) and using the skew symmetry ofφ, we get

D_α_,_β(x(t))≥ 1

α(t)

D Rφ,x

β(t)(t,x(t))

E

+ 1

2β

Rφ,x

β(t)(t,x(t))

2

− 1

2α(t)

Rφ,x

α(t)(t,x(t))

2 =1 2 1 β − 1 α Rφ,x

β(t)(t,x(t))

2 +1 α D Rφ,x

β(t)(t,x(t))

E

− 1

2α

Rφ_α(,x

t)(t,x(t))−R φ,x

β(t)(t,x(t))

2 −1 α D Rφ,x

β(t)(t,x(t)),R φ,x

β(t)(t,x(t))

E =1 2 1 β − 1 α Rφ,x

β(t)(t,x(t))

2 + 1 2α Rφ,x

β(t)(t,x(t))

2 ≥1 2 1 β − 1 α Rφ,x

β(t)(t,x(t))

2

,

(4.4)

which implies the left most inequality in the assertion. On the other hand,

−w(t) + 1

β

Rφ,x

β(t)(t,x(t))∈∂ φ P φ,x

(20)

which implies that D

w(t),Rφ,x

β(t)(t,x(t))

E

≤ 1

β

D Rφ,x

β(t)(t,x(t))

E

−φ P_α(φ,x_t₎[g(t,x(t))−α(t)w(t)],Pφ,x

+φ Pφ,x

.

(4.5)

Similarly, to the analysis above, we then obtain

D_α_,_β(x(t))≤ 1

β

D Rφ,x

β(t)(t,x(t))

E

+ 1

2β

Rφ,x

β(t)(t,x(t))

2 − 1 2α Rφ,x

α(t)(t,x(t))

2 =1 2 1 β − 1 α Rφ,x

α(t)(t,x(t))

2 − 1 2β Rφ,x

β(t)(t,x(t))

2 ≤1 2 1 β − 1 α Rφ,x

α(t)(t,x(t))

2

,

(4.6)

which implies the right most inequality in the assertion. Combining (4.4) and (4.6), we obtain the required result. The last assertion now follows from Lemma 2.1.

As a consequence of Theorem 2.2 and Theorem 4.1, we obtain the following result on the global error bound for RGMVIP (1.1).

Corollary 4.1. Suppose that for each t ∈Ω,xo(t)∈H is a solution of the RGMVIP (1.1). Let

ˆ

T :Ω×H →F(H) satisfy the condition (I) and T :Ω×H →CB(H) be the random multi-valued mapping induced by the random fuzzy mapping T . Let gˆ :Ω×H →H be a random mapping andφ :H×H→R∪ {+∞}be a real valued function such that

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kPφ,x

α(t)(z)−P φ,xo

α(t)(z)k ≤K(t)kx(t)−xo(t)k,∀x(t),xo(t),z(t)∈H,

h

L(t)K(t) θ(t)−K(t)λ(t)(1+ε)

i

, we have

x(t)−xo(t) ≤

"

α(t)λ(t)(1+ε) +L(t)

#s 2α β α−β

q

D_α_,_β(x(t)).

Now, we derive the global error bound for RGMVIP (1.1) without using the Lipschitz conti-nuity ofT.

ˆ

T :Ω×H →F(H) satisfy the condition (I) and T :Ω×H →CB(H) be the random multi-valued mapping induced by the random fuzzy mapping T . Let gˆ :Ω×H →H be a random mapping and φ :H×H →_R∪ {+∞}be a real valued function such that for each t ∈Ω, the measurable mapping T is strongly g-monotone andH-Lipschitz continuous with the measurableˆ functionsθ,λ :Ω→(0,+∞), respectively, then for any x(t)∈H, t∈Ω, we have

x(t)−xo(t) ≤

1 s

θ(t) +L

2₍_t₎

2

1

β −

1

α

q

D_α_,_β(x(t)).

Proof.From (4.1), it can be written as

D_α_,_β(x(t))≥Dw(t),g(t,x(t))−g(t,x_o(t))

E

+ 1

2β

g(t,x(t))−g(t,x_o(t)) 2

− 1

2α

g(t,x(t))−g(t,x_o(t)) 2

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By using the strongg-monotonicity ofT, we get

D_α_,_β(x(t))≥Dw(t)−w_o(t),g(t,x(t))−g(t,x_o(t))E+Dw_o(t),g(t,x(t))−g(t,x_o(t))E

+ 1

2β

g(t,x(t))−g(t,x_o(t)) 2

− 1

2α

g(t,x(t))−g(t,x_o(t) 2

.

≥Dw_o(t),g(t,x(t))−g(t,x_o(t))E+θ(t)x (t)−xo(t) 2

+ 1

2β

g(t,x(t))−g(t,x_o(t)) 2

− 1

2α

g(t,x(t))−g(t,x_o(t) 2

.

(4.7)

Sincex_o(t)∈H is a solution of RGMVIP (1.1) D

wo(t),y(t)−g(t,xo(t)) E

+φ g(t,xo(t)),y(t)−φ g(t,xo(t)),g(t,xo(t))≥0. Takingy(t) =g(t,x(t))in the above inequality

D

wo(t),g(t,x(t))−g(t,xo(t)) E

+φ g(t,xo(t)),g(t,x(t))

≥0.

(4.8)

Combining (4.7) with (4.8) and using the skew symmetry ofφ, we get

D_α_,_β(x(t))≥θ(t)x(t)−xo(t) 2 + 1 2β

g(t,x(t))−g(t,xo(t)) 2 − 1 2α

g(t,x(t))−g(t,x_o(t)) 2

.

≥θ(t)x(t)−xo(t) 2

+ 1

2βL

2₍_t₎

x(t)−xo(t) 2

− 1

2αL

2₍_t₎

x(t)−x_o(t) 2

D_α_,_β(x(t))≥

θ(t) +L

2₍_t₎

2β −

L2(t)

2α

x(t)−x_o(t) 2

which implies

x(t)−xo(t) ≤

1 q

θ(t) +L

2₍_t₎

2β −

L2₍_t₎

2α

q

D_α_,_β(t,x(t))

≤_q 1

θ(t) +L

2₍_t₎

2 1 β − 1 α q

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Conflict of Interests

The authors declare that there is no conflict of interests.

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