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Graphical Approximation of Common Solutions to Generalized Nonlinear Relaxed Cocoercive Operator Equation Systems with A,\eta-accretive Mappings Fixed Point Theory and Applications 2012,

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Graphical Approximation of Common Solutions to Generalized Nonlinear

Relaxed Cocoercive Operator Equation Systems with (A,\eta)-accretive

Mappings

Fixed Point Theory and Applications 2012, 2012:14 doi:10.1186/1687-1812-2012-14

Fang Li (lifang1687@163.com) Heng-you Lan (hengyoulan@163.com) Yeol JE Cho (yjcho@gsnu.ac.kr)

Article type Research

Submission date 24 April 2011

Acceptance date 15 February 2012

Publication date 15 February 2012

Article URL http://www.fixedpointtheoryandapplications.com/content/2012/1/14

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Fixed Point Theory and Applications manuscript No.

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Graphical approximation of common solutions to

generalized nonlinear relaxed cocoercive operator

equation systems with ( A, η ) -accretive mappings

Fang Li1, Heng-you Lan∗1 and Yeol Je

Cho2

∗Corresponding author:

hengyoulan@163.com

FL: lifang1687@163.com YJC:

yjcho@gsnu.ac.kr 1Department of

Mathematics, Sichuan University of

Science and Engineering, Zigong, 643000,

Sichuan, People’s Republic of China

2Department of Mathematics Education

and the RINS, College of Education,

Gyeongsang National University, Chinju

660-701, Korea

Abstract In this paper, we develop a new perturbed iterative algorithm framework with errors based on the variational graphical convergence of

op-erator sequences with (A, η)-accretive mappings in Banach space By using the generalized resolvent operator technique associated with (A, η)-accretive

mappings, we also prove the existence of solutions for a class of generalized nonlinear relaxed cocoercive operator equation systems and the variational convergence of the sequence generated by the perturbed iterative algorithm in

q-uniformly smooth Banach spaces The obtained results improve and

gener-alize some well-known results in recent literatures

Keywords

(A, η)-accretive mapping; Generalized resolvent operator technique;

Gener-alized nonlinear relaxed cocoercive operator equation systems; New perturbed iterative algorithm with errors; Variational graphical convergence

2000 Mathematics Subject Classification:

47H05; 49J40

1 Introduction

It is well known that standard Yosida regularizations/approximations have been tremendously effective to approximation solvability of general variational inclusion problems in the context of resolvent operators that turned out to be nonexpansive This class of nonlinear Yosida approximations have been applied

to approximation solvability of nonlinear inhomogeneous evolution inclusions

of the form

f (t) ∈ u 0 (t) + M u(t) − ωu(t), u(0) = u0

for almost all t ∈ [0, T ], where T ∈ (0, 1) is fixed, ω ∈ R (see [1]) For more

gen-eral details on approximation solvability of gengen-eral nonlinear inclusion prob-lems, we refer the reader to [2–18] and the references therein

On the other hand, it is well known that variational inequalities and vari-ational inclusions provide mathematical models to some problems arising in economics, mechanics, and engineering science and have been studied exten-sively There are many methods to find solutions of variational inequality and variational inclusion problems Among these methods, the resolvent opera-tor technique is very important For some literature, we recommend to the following example, and the reader [2–15, 17, 18] and the references therein

Example 1.1 ([19]) Let V : R n → R be a local Lipschitz continuous

function, and let K be a closed convex set in R n If x ∗ is a solution to the following problem:

min

x∈K V (x),

then

0 ∈ ∂V (x ∗ ) + N K (x ∗ ), where ∂V (x ∗ ) denotes the subdifferential of V at x ∗ and N K (x ∗) the normal

cone of K at x ∗

In 2006, Lan et al [7] introduced a new concept of (A, η)-accretive

map-pings, which provides a unifying framework for maximal monotone operators,

m-accretive operators, η-subdifferential operators, maximal η-monotone

oper-ators, H-monotone operoper-ators, generalized m-accretive mappings, H-accretive operators, (H, η)-monotone operators, and A-monotone mappings Recently,

by using the concept of (A, η)-accretive mappings and the resolvent opera-tor technique associated with (A, η)-accretive mappings, Jin [5] introduced and studied a new class of nonlinear variational inclusion systems with (A, η)-accretive mappings in q-uniformly smooth Banach spaces and developed some

new iterative algorithms to approximate the solutions of the mentioned nonlin-ear variational inclusion systems Furthermore, by using the resolvent operator technique, Petrot [14] studied the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point prob-lems for Lipschitz mappings in Hilbert spaces, and Agarwal and Verma [2] in-troduced and studied a new system of nonlinear (set-valued) variational

inclu-sions involving (A, η)-maximal relaxed monotone and relative (A, η)-maximal

monotone mappings in Hilbert spaces and proved its approximation solvabil-ity based on the variational graphical convergence of operator sequences For more literature, we recommend to the reader [9, 20] and the references therein Motivated and inspired by the above works, the purpose of this paper is

to consider and study the following generalized nonlinear operator equation

system with (A, η)-accretive mappings in real Banach space B1× B2:

Find (x, y) ∈ B1× B2 and u ∈ S(x), v ∈ T (y) such that

(

p(x) = R ρλ1,A1

η1,M1(·,x) [(1 − λ1)A1(p(x)) + λ1(A1(f (y)) − ρN1(u, y) + a)],

h(y) = R %λ2,A2

η2,M2(y,·) [(1 − λ2)A2(h(y)) + λ2(A2(g(x)) − %N2(x, v) + b)], (1.1) where for all (x, y) ∈ B1×B2, R ρλ1,A1

η1,M1(·,x) = (A1+ρλ1M1(·, x)) −1 and R %λ2,A2

η2,M2(y,·)=

(A2+ %λ2M2(y, ·)) −1 are two resolvent operators and two constants ρ, % > 0,

N1 : B1 × B2 → B1, N2 : B1 × B2 → B2, p : B1 → B1, h : B2 → B2,

f : B2 → B1, g : B1 → B2 are single-valued operators, λ1, λ2 > 0 are two

constants, (a, b) ∈ B1× B2 is an any given element, and S : B1 → 2 B1,

T : B2→ 2 B2, A i : B i → B i , η i : B i × B i → B i , M i : B i × B i → 2 B i (i = 1, 2) are any nonlinear operators such that for all x ∈ B1, M1(·, x) : B1 → 2 B1 is

an (A1, η1)-accretive mapping and M2(y, ·) : B2→ 2 B2 is an (A2, η2)-accretive

mapping for all y ∈ B2, respectively

Based on the definition of the resolvent operators associated with (A,

η)-accretive mappings, the Equation (1.1) can be written as

½

a ∈ A1(p(x)) − A1(f (y)) + ρN1(u, y) + ρM1(p(x), x),

b ∈ A2(h(y)) − A2(g(x)) + %N2(x, v) + %M2(y, h(y)) (1.2) Remark 1.1 For appropriate and suitable choices of B i , A i , η i , N i , M i (i =

1, 2), p, h, f, g, S, T , one can obtain a number (systems) of quasi-variational

inclusions, generalized (random) quasi- variational inclusions, quasi-variational inequalities, and implicit quasi-variational inequalities as special cases of the Equation (1.1) (or problem (1.2)) include Below are some special cases of problem

Example 1.2 If B i = B(i = 1, 2), p = f = h = g, N1(x, ·) = N2(·, y) =

N (·) and M1(·, x) = M1(·), M2(y, ·) = M2(·) for all (x, y) ∈ B1× B2 and a =

b = 0, then the problem (1.2) collapses to the following nonlinear variational

inclusion system with (A, η)-accretive mappings:

½

0 ∈ A1(g(x)) − A1(g(y)) + ρN (y) + ρM1(g(x)),

0 ∈ A2(g(y)) − A2(g(x)) + %N (x) + %M2(g(y)). (1.3) The system (1.3) was introduced and studied by Jin [5] Further, when A i=

A, M i = M (i = 1, 2) and y = x, the system (1.3) reduces to a nonlinear variational inclusion of find x ∈ B such that

0 ∈ N (x) + M (g(x)), which contains the variational inclusions with H-monotone operator, H-accretive mappings, or A-maximal (m)-relaxed monotone (AMRM) mappings in [2, 3]

as special cases

Example 1.3 If B i = H(i = 1, 2) is a Hilbert space, a = b = 0, S :

B1→ B1 and T : B2→ B2 are two single-valued mappings, p = f = h = g =

S = T = I is the identity operator and M1(·, x) = M2(y, ·) = M (·) for all (x, y) ∈ B1× B2, then the problem (1.2) is equivalent to solve the following

nonlinear variational inclusion system with (A, η)-monotone mappings:

½

0 ∈ A1(x) − A1(y) + ρN (y, x) + ρM (x),

0 ∈ A2(y) − A2(x) + %N (x, y) + %M (y), (1.4)

The system (1.4) was introduced and studied by Wang and Wu [18] and con-tains the generalized system for mixed variational inequalities with maximal

monotone operators in [14] as special cases Moreover, taking y = x, then the system (1.4) reduces to finding an element x ∈ H such that

0 ∈ N (x, x) + M (x),

which was considered by Verma [17]

Example 1.4 When B i = H, λ i = 1(i = 1, 2), p = h, A1 = A2 =

I, N1(x, ·) = N2(·, y) = N (·) and M1(·, x) = M1(·), N2(y, ·) = M2(·) for all (x, y) ∈ B1× B2, the system (1.1) becomes to the following nonlinear operator

equation systems: Finding (x, y) ∈ H × H such that

½

h(x) = J M ρ

1[f (y) − ρN (y)],

h(y) = J M %2[g(x) − %N (x)], (1.5)

where J M ρ1 = (I + ρM1)−1 and J M %2 = (I + %M2)−1 Based on the definition of the resolvent operators, we know that the system (1.5) is equivalent to solve the following system of general variational inclusions:

½

0 ∈ h(x) − f (y) + ρN (y) + ρM1(h(x)),

0 ∈ h(y) − g(x) + %N (x) + %M2(h(y)), (1.6)

which was studied by Noor et al [12] when M i = M is maximal monotone for

i = 1, 2 Moreover, some special cases of the problem (1.6) can be found in [4,

6] and the references therein

We also construct a new perturbed iterative algorithm framework with er-rors based on the variational graphical convergence of operator sequences with

(A, η)-accretive mappings in Banach space for approximating the solutions of

the nonlinear equation system (1.1) in smooth Banach spaces and prove the ex-istence of solutions and the variational convergence of the sequence generated

by the perturbed iterative algorithm in q-uniformly smooth Banach spaces.

The results present in this paper improve and generalize the corresponding results of [2, 3, 5, 12, 14, 17, 18] and many other recent works

2 Preliminaries

Let B be a real Banach space with dual space B ∗ , h·, ·i be the dual pair between

B and B ∗ , CB(B) denote the family of all nonempty closed bounded subsets of

B, and 2 B denote the family of all the nonempty subsets of B The generalized duality mapping J q : B → 2 B ∗

is defined by

J q (x) =©f ∗ ∈ B ∗ : hx, f ∗ i = kxk q , kf ∗ k = kxk q−1ª, ∀x ∈ B,

where q > 1 is a constant In particular, J2 is the usual normalized duality

mapping It is known that, in general, J q (x) = kxk q−2 J2(x) for all x 6= 0, and

J q is single-valued if B ∗ is strictly convex In the sequel, we always suppose

that B is a real Banach space such that J q is single-valued and H is a Hilbert space If B = H, then J2 becomes the identity mapping on H.

The modulus of smoothness of B is the function χ B : [0, ∞) → [0, ∞)

defined by

χ B (t) = sup

½ 1

2(kx + yk + kx − yk) − 1 : kxk ≤ 1, kyk ≤ t

¾

.

A Banach space B is called uniformly smooth if lim t→0 χ B t (t) = 0.

B is called q-uniformly smooth if there exists a constant c > 0 such that

χ B (t) ≤ ct q , q > 1 Remark that J q is single-valued if B is uniformly smooth In the study of characteristic inequalities in q-uniformly smooth

Ba-nach spaces, Xu [21] proved the following result:

Lemma 2.1 Let B be a real uniformly smooth Banach space Then, B is

q-uniformly smooth if and only if there exists a constant c q > 0 such that for

all x, y ∈ B,

kx + yk q ≤ kxk q + qhy, J q (x)i + c q kyk q

In the sequel, we give some concept and lemmas needed later

Definition 2.1 Let B be a q-uniformly smooth Banach space and T, A :

B → B be two single-valued mappings T is said to be

(i) accretive if

hT (x) − T (y), J q (x − y)i ≥ 0, ∀x, y ∈ B;

(ii) strictly accretive if T is accretive and

hT (x) − T (y), J q (x − y)i = 0

if and only if x = y;

(iii) r-strongly accretive if there exists a constant r > 0 such that

hT (x) − T (y), J q (x − y)i ≥ rkx − yk q , ∀x, y ∈ B;

(iv) γ-strongly accretive with respect to A if there exists a constant γ > 0

such that

hT (x) − T (y), J q (A(x) − A(y))i ≥ γkx − yk q , ∀x, y ∈ B;

(v) m-relaxed cocoercive with respect to A if, there exists a constant m > 0

such that

hT (x) − T (y), J q (A(x) − A(y))i ≥ −mkT (x) − T (y)k q , ∀x, y ∈ B;

(vi) (π, ι)-relaxed cocoercive with respect to A if, there exist constants

π, ι > 0 such that

hT (x) − T (y), J q (A(x) − A(y))i ≥ −πkx − yk q + ιkT (x) − T (y)k q , ∀x, y ∈ B;

(vii) s-Lipschitz continuous if there exists a constant s > 0 such that

kT (x) − T (y)k ≤ skx − yk, ∀x, y ∈ B.

In a similar way, we can define (relaxed) cocoercivity and Lipschitz

conti-nuity of the operator N (·, ·) : B × B → B in the first and second arguments.

Remark 2.1 (1) The notion of the cocoercivity is applied in several direc-tions, especially to solving variational inequality problems using the auxiliary problem principle and projection methods [16], while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoerciv-ity Several classes of relaxed cocoercive variational inequalities and variational inclusions have been studied in [2, 5, 7–10, 12, 16–18]

(2) When B = H, (i)–(iv) of Definition 2.1 reduce to the definitions of

monotonicity, strict monotonicity, strong monotonicity, and strong

monotonic-ity with respect to A, respectively (see [3, 18]).

Definition 2.2 A single-valued mapping η : B × B → B is said to be

τ -Lipschitz continuous if there exists a constant τ > 0 such that

kη(x, y)k ≤ τ kx − yk, ∀x, y ∈ B.

Definition 2.3 Let B be a q-uniformly smooth Banach space, η : B ×B →

B and A, H : B → B be single-valued mappings Then set-valued mapping

M : B → 2 Bis said to be

(i) η-accretive if

hu − v, J q (η(x, y))i ≥ 0, ∀x, y ∈ B, u ∈ M (x), v ∈ M (y);

(ii) r-strongly η-accretive if there exists a constant r > 0 such that

hu − v, J q (η(x, y))i ≥ rkx − yk q , ∀x, y ∈ B, u ∈ M (x), v ∈ M (y);

(iii) m-relaxed η-accretive if there exists a constant m > 0 such that

hu − v, J q (η(x, y))i ≥ −mkx − yk q , ∀x, y ∈ B, u ∈ M (x), v ∈ M (y);

(iv) ξ- ˆ H-Lipschitz continuous, if there exists a constant ξ > 0 such that

ˆ

H(M (x), M (y)) ≤ ξkx − yk, ∀x, y ∈ B,

where ˆH is the Hausdorff metric on CB(B);

(v) (A, η)-accretive if M is m-relaxed η-accretive and (A + ρM )(B) = B for every ρ > 0.

Remark 2.2 The (A, η)-accretivity generalizes the general (H, η)-accretivity, (I, η)-accretivity (so-called generalized m-accretivity), H-accretivity, classical

m-accretivity, (A, η)-monotonicity, A-monotonicity, (H, η)-monotonicity,

H-monotonicity, maximal η-H-monotonicity, and classical maximal monotonicity

as special cases (see, for example, [1, 7, 8, 13] and the references therein.)

Definition 2.4 Let A : B → B be a strictly η-accretive mapping and M :

B → 2 B be an (A, η)-accretive mapping The resolvent operator R ρ,A η,M : B → B

is defined by:

R ρ,A η,M (u) = (A + ρM ) −1 (u), ∀u ∈ B.

Remark 2.3 The resolvent operators associated with (A, η)-accretive

mappings include as special cases the corresponding resolvent operators

asso-ciated with (H, accretive mappings, (A, monotone operators [8], (H, η)-monotone operators, H-accretive operators, generalized m-accretive operators, maximal η-monotone operators, H-monotone operators, A-monotone opera-tors, η-subdifferential operaopera-tors, the classical m-accretive, and maximal

mono-tone operators See, for example, [1, 7, 8, 13] and the references therein

Lemma 2.2 ([7]) Let B be a q-uniformly smooth Banach space and η :

B × B → B be τ -Lipschitz continuous, A : B → B be a r-strongly η-accretive

mapping and M : B → 2 B be an (A, η)-accretive mapping Then, the resolvent operator R ρ,A η,M : B → B is τ q−1

r−ρm-Lipschitz continuous, i.e.,

°

°R ρ,A η,M (x) − R ρ,A η,M (y)

°

° ≤ τ

q−1

r − ρm kx − yk, ∀x, y ∈ B,

where ρ ∈ (0, m r) is a constant

Definition 2.5 Let M n , M : B → 2 B be (A, η)-accretive mappings on B for n = 0, 1, 2, Let A : B → B be r-strongly η-monotone and β-Lipschitz continuous The sequence M n is graph-convergent to M , denoted M n A−G −→ M ,

if for every (x, y) ∈ graph(M ), there exists a sequence (x n , y n ) ∈ graph(M n) such that

x n → x, y n → y as n → ∞.

Based on Definition 2.6 and Theorem 2.1 in [20], we have the following lemma

Lemma 2.3 Let M n , M : B → 2 B be (A, η)-accretive mappings on B for

n = 0, 1, 2, Then, the sequence M n A−G −→ M if and only if

R ρ,A η,M n (x) → R ρ,A η,M (x), ∀x ∈ B,

where R ρ,A η,M = (A + ρM n)−1 , R ρ,A η,M = (A + ρM ) −1 , ρ > 0 is a constant, and

A : B → B is r-strongly η-monotone and β-Lipschitz continuous.

3 Algorithms and graphical convergence

In this section, by using resolvent operator technique associated with (A,

η)-accretive mappings, we shall develop a new perturbed iterative algorithm framework with errors for solving the nonlinear operator equation system (1.1)

with (A, η)-accretive mappings and relaxed cocoercive operators and prove the

existence of solutions and the variational convergence of the sequence

gen-erated by the perturbed iterative algorithm in q-uniformly smooth Banach

spaces

Above all, we note that the equalities (1.1) can be written as











p(x) = R ρλ1,A1

η1,M1(·,x) (s),

s = (1 − λ1)A1(p(x)) + λ1(A1(f (y)) − ρN1(u, y) + a),

h(y) = R %λ2,A2

η2,M2(y,·) (t),

t = (1 − λ2)A2(h(y)) + λ2(A2(g(x)) − %N2(x, v) + b),

where ρ, λ > 0 are constants This formulation allows us to construct the

following perturbed iterative algorithm framework with errors

Algorithm 3.1 Step 1 For an arbitrary initial point (x0, y0) ∈ B1× B2,

take u0∈ S(x0) and v0∈ T (y0)

Step 2 Choose sequences {d n } ⊂ B1and {e n } ⊂ B2are two error sequences

to take into account a possible inexact computation of the operator points, which satisfy the following conditions:

lim

n→∞ d n= lim

n→∞ e n = 0,

∞

X

n=1

¡

kd n − d n−1 k + ke n − e n−1 k¢< ∞.

Step 3 Let the sequence {(s n , t n , x n , y n )} ⊂ B1× B2× B1× B2satisfy











s n = (1 − λ1)A1(p(x n )) + λ1(A1(f (y n )) − ρN1(u n , y n ) + a),

t n = (1 − λ2)A2(h(y n )) + λ2(A2(g(x n )) − %N2(x n , v n ) + b),

x n+1 = (1 − k)x n + k{x n − p(x n ) + R ρλ1,A1

η1,M n

1(·,x n)(s n )} + d n ,

y n+1 = (1 − κ)y n + κ{y n − h(y n ) + R %λ2,A2

η2,M n

2(y n ,·) (t n )} + e n ,

(3.1)

where R ρλ1,A1

η1,M n

1(·,x) = (A1+ρλ1M n

1(·, x)) −1 , R %λ2,A2

η2,M n

2(y,·) = (A2+%λ2M n

2(y, ·)) −1,

λ1, λ2, ρ, % are nonnegative constants and k, κ ∈ (0, 1] are size constants Step 4 Choose u n+1 ∈ S(x n+1 ) and v n+1 ∈ T (y n+1) such that (see [22])

ku n − u n+1 k ≤ (1 + 1

n+1) ˆH(S(x n ), S(x n+1 )),

kv n − v n+1 k ≤ (1 + 1

n+1) ˆH(T (y n ), T (y n+1 )). (3.2)

Step 5 If s n , t n , x n , y n , d n , and e n satisfy (3.1) and (3.2) to sufficient

accuracy, stop; otherwise, set n := n + 1 and return to Step 2.

Now, we prove the existence of a solution of problem (1.1) and the conver-gence of Algorithm 3.1

Theorem 3.1 For i = 1, 2, let B i be a q i-uniformly smooth Banach space

with q i > 1, η i , A i , M i , N i (i = 1, 2) and p, h, f, g be the same as in the Equation

(1.1) Also suppose that the following conditions hold:

(H1) η i is τ i -Lipschitz continuous, and A i is r i -strongly η i-accretive, and

σ i -Lipschitz continuous for i = 1, 2, respectively;

(H2) p is δ1-strongly accretive and l p -Lipschitz continuous, h is δ2-strongly

accretive and l h -Lipschitz continuous, f is l f -Lipschitz continuous and g is

l g -Lipschitz continuous, S : B1 → CB(B1) is ξ- ˆH-Lipschitz continuous and

T : B2→ CB(B2) is ζ- ˆH-Lipschitz continuous;

(H3) N1 is (π1, ι1)-relaxed cocoercive with respect to f1 and $2-Lipschitz

continuous in the second argument, and N2is (π2, ι2)-relaxed cocoercive with

respect to g2and $1-Lipschitz continuous in the first argument, and N1 is β1

-Lipschitz continuous in the first variable, and N2 is β2-Lipschitz continuous

in the second variable, where f1 : B2 → B1 is defined by f1(y) = A1◦ f (y) =

A1(f (y)) for all y ∈ B2 and g2 : B1 → B2 is defined by g2(x) = A2◦ g(x) =

A2(g(x)) for all x ∈ B1;

(H4) for n = 0, 1, 2, , M n

i : B i × B i → 2 B i (i = 1, 2) are any nonlinear operators such that for all x ∈ B1, M n

1(·, x) : B1→ 2 B1 is an (A1, η1)-accretive

mapping with M n

1(·, x) A1−G

−→ M1(·, x), and M n

2(y, ·) : B2→ 2 B2 is an (A2, η2

)-accretive mapping with M n

2(y, ·) A2−G

−→ M2(y, ·) for all y ∈ B2, respectively;

(H5) there exist constants ν i (i = 1, 2), ρ ∈ (0, r1/m1) and % ∈ (0, r2/m2) such that

°

°R ρλ1,A1

η1,M1(·,x) (z) − R ρλ1,A1

η1,M1(·,y) (z)

°

° ≤ ν2kx − yk, ∀x, y, z ∈ B1,

°

°R %λ2,A2

η2,M2(x,·) (z) − R %λ2,A2

η2,M2(y,·) (z)

°

° ≤ ν1kx − yk, ∀x, y, z ∈ B2, (3.3)

and















ν2+ q1p

1 − q1δ1+ c q1l q1

p +τ1q1−1 [(1−λ1)σ1l p +ρλ1β1ξ]

r1−ρλ1m1

+κλ2τ2q2−1 q2 √

σ q22 l q2 g −q2%ι2$ q21 +q2%π2+c q2 % q2 $ q21 k(r2−%λ2m2 ) < 1,

ν1+ q2p

1 − q2δ2+ c q2l q2

h +τ2q2−1 [(1−λ2)σ2l h +%λ2β2ζ]

r2−%λ2m2

+kλ1τ

q1−1

1 q1q

σ q11 l q1 f −q1ρι1$ q12 +q1ρπ1+c q1 ρ q1 $ q12 κ(r1−ρλ1m1 ) < 1

(3.4)

where c q1, c q2 are the constants as in Lemma 2.1 and k, κ ∈ (0, 1] are size

constants

Then, there exist (x ∗ , y ∗ ) ∈ B1 × B2 u ∗ ∈ S(x ∗ ), v ∗ ∈ T (y ∗) such that

(x ∗ , y ∗ , u ∗ , v ∗) is a solution of the Equation (1.1) and

x n → x ∗ , y n → y ∗ , u n → u ∗ , v n → v ∗ , as n → ∞,

where {x n }, {y n }, {u n } and {v n } are iterative sequences generated by

Algo-rithm 3.1

Proof Define k · k ∗ on B1× B2by

k(x, y)k ∗ = kxk + kyk, ∀(x, y) ∈ B1× B2.