CONVERGENCE OF PARALLEL ITERATIVE ALGORITHMS FOR A SYSTEM OF NONLINEAR VARIATIONAL INEQUALITIES IN BANACH SPACES

Abstract

In this paper, we consider the problems of convergence of parallel iterative algorithms for a system of nonlinear variational inequalities and nonexpansive mappings. Strong convergence theorems are established in the frame work of real Banach spaces.

1. Introduction

Let (E, ║ · ║) be a Banach space and C be a nonempty closed convex subset of E. This paper deals with the problems of convergence of iterative algorithms for a system of nonlinear variational inequalities: Find (x∗, y∗) ∈ C × C such that

where T1, T2 : C × C → E, g1, g2 : C → C are nonlinear mappings, J is the normalized duality mapping, j ∈ J and ρ1, ρ2 are two positive real numbers.

If T1, T2 : C → E are nonlinear mappings and g1 = g2 = I (I denotes the identity mapping), then (1.1) reduces to finding (x∗, y∗) ∈ C × C such that

which was considered by Yao et al. [13].

If E = H is a real Hilbert space and T1, T2 : C → H are nonlinear mappings and g1 = g2 = g, then (1.1) reduces to finding (x∗, y∗) ∈ C × C such that

which was studied by Yang et al. [12].

If g = I, then (1.3) reduces to finding (x∗, y∗) ∈ C × C such that

which was introduced by Ceng et al. [2].

In particular, if T1 = T2 = T, then (1.4) reduces to finding (x∗, y∗) ∈ C × C such that

which is defined by Verma [9].

Further, if x∗ = y∗, then (1.5) reduces to the following classical variational inequality (VI(T,C)) of finding x∗ ∈ C such that

We can see easily that the variational inequality (1.6) is equivalent to a fixed point problem. An element x∗ ∈ C is a solution of the variational inequality (1.6) if and only if x∗ ∈ C is a fixed point of the mapping PC(I − λT), where PC is the metric projection and λ is a positive real number. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Recent development of the variational inequality is to design efficient iterative algorithms to compute approximate solutions for variational inequalities and their generalization. Up to now, many authors have presented implementable and significant numerical methods such as projection method and it’s variant forms, linear approximation, descent method, Newton’s method and the method based on auxiliary principle technique.

However, these sequential iterative methods are only suitable for implementing on the traditional single-processor computer. To satisfy practical requirements of modern multiprocessor systems, efficient iterative methods having parallel characteristics need to be further developed for the system of variational inequalities (see [1,4,5,6,12,14]).

Motivated and inspired by the research work going on this field, in this paper, we construct an parallel iterative algorithm for approximating the solution of a new system of variational inequalities involving four different nonlinear mappings. Finally, we prove the strong convergence of the purposed iterative scheme in 2-uniformly smooth Banach spaces.

 

2. Preliminaries

Let C be a nonempty closed convex subset of a Banach space E with the dual space E∗. Let ⟨·, ·⟩ denote the dual pair between E and E∗. Let 2E denote the family of all the nonempty subsets of E. For q > 1, the generalized duality mapping Jq : E → 2E∗ is defined by

In particular, J = J2 is the normalized duality mapping. It is known that Jq(x) = ║x║q−2J(x) for all x ∈ E and Jq is single-valued if E∗ is strictly convex or E is uniformly smooth. If E = H is a Hilbert space, J = I, the identity mappings.

Let B = {x ∈ E : ║x║ = 1}. A Banach space E is said to be smooth if the limit

exists for all x, y ∈ B. The modulus of smoothness of E is the function ρE : [0,∞) → [0,∞) defined by

A Banach space E is called uniformly smooth if E is called q-uniformly smooth if there exists a constant c > 0 such that

If E is q-uniformly smooth, then q ≤ 2 and E is uniformly smooth.

Definition 2.1. Let T : C × C → E be a mapping. T is said to be

(i) (δ, ξ)-relaxed cocoercive with respect to the first argument if there exist j(x − y) ∈ J(x − y) and constants δ, ξ > 0 such that

for all x, y ∈ C;

(ii) μ-Lipschitz continuous with respect to the first argument if there exists a constant μ > 0 such that

for all x, y ∈ C;

(iii) γ-Lipschitz continuous with respect to the second argument if there exists a constant γ > 0 such that

for all x, y ∈ C.

Definition 2.2. Let g : C → C be a mapping. g is said to be

(i) ζ-strongly accretive if there exists a constant ζ > 0 such that

for all x, y ∈ C.

(ii) η-Lipschitz continuous if there exists a constant η > 0 such that

for all x, y ∈ C.

Let D be a subset of C and Q be a mapping of C into D. Then Q is said to be sunny if

whenever Q(x) + t(x − Q(x)) ∈ C for x ∈ C and t ≥ 0. A mapping Q of C into itself is called a retraction if Q2 = Q. If a mapping Q of C into itself is a retraction, then Q(z) = z for all z ∈ R(Q), where R(Q) is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.

In order to prove our main results, we also need the following lemmas.

Lemma 2.3 ([11]). Let E be a real 2-uniformly smooth Banach space. Then

where K is the 2-uniformly smooth constant of E.

Lemma 2.4 ([7]). Let C be a nonempty closed convex subset of a smooth Banach space E and let QC be a retraction from E onto C. Then the following are equivalent:

(i) QC is both sunny and nonexpansive;

(ii) ⟨x − QC(x), j(y − QC(x))⟩ ≤ 0 for all x ∈ E and y ∈ C.

Lemma 2.5 ([10]). Suppose {δn} is a nonnegative sequence satisfying the fol- lowing inequality:

with λn ∈ [0, 1], and σn = 0(λn). Then limn→∞ δn = 0.

Lemma 2.6 ([3]). Let {cn} and {kn} be two real sequences of nonnegative num-bers that satisfy the following conditions:

(i) 0 ≤ kn ≤ 1 for n = 1, 2, · · · and lim supn kn < 1;

(ii) cn+1 ≤ kncn for n = 1, 2, · · · .

Then cn converges to 0 as n → ∞.

 

3. Iterative algorithms

In this section, we suggest a parallel iterative algorithm for solving the system of nonlinear variational inequality (1.1). First of all, we establish the equivalence between the system of variational inequalities and fixed point problems.

Lemma 3.1. Let C be a nonempty closed convex subset of a smooth Banach space E. Let QC : E → C be a sunny nonexpansive retraction, Ti : C × C → E and gi : C → C be mappings for i = 1, 2. Then (x∗, y∗) with x∗, y∗ ∈ C is a solution of problem (1.1) if and only if

Proof. Applying Lemma 2.4, we have that

That is,

This completes the proof. □

This fixed point formulation allow us to suggest the following parallel iterative algorithms.

Algorithm 3.1. For any given x0, y0 ∈ C, computer the sequences {xn} and {yn} defined by

where ρ1, ρ2 are positive real numbers.

Also, we propose a relaxed parallel algorithm which can be applied to the approximation of solution of the problem (1.1) and common fixed point of two mappings.

Algorithm 3.2. For any given x0, y0 ∈ C, compute the sequences {xn} and {yn} defined by

where S1, S2 : C → C are nonexpansive mappings, {αn}, {βn} are sequences in [0,1], κ ∈ (0, 1) and ρ1, ρ2 are positive real numbers.

If T1, T2 : C → E are nonlinear mappings and g1 = g2 = I, then the algorithm 3.1 reduces to the following parallel iterative method for solving problem (1.2).

Algorithm 3.3. For any given x0, y0 ∈ C, compute the sequences {xn} and {yn} defined by

where ρ1, ρ2 are positive real numbers.

If E = H is a Hilbert space, T1, T2 : C → H are nonlinear mappings and g1 = g2 = g, Algorithm 3.1 reduces to the following parallel iterative method for solving problem (1.3).

Algorithm 3.4. For any given x0, y0 ∈ C, compute the sequences {xn} and {yn} defined by

where ρ1, ρ2 are positive real numbers.

 

4. Main results

We now state and prove the main results of this paper.

Theorem 4.1. Let E be a 2-uniformly smooth Banach space with the 2-uniformly smooth constant K, C be a nonempty closed convex subset of E and QC be a sunny nonexpansive retraction from E onto C. Let Ti : C × C → E be a non-linear mapping such that (δi, ξi)-relaxed cocoercive, μi-Lipschitz continuous with respect to the first argument and γi-Lipschitz continuous with respect to the sec-ond argument for i = 1, 2. Let gi : C → C be a ηi-Lipschitz continuous and ζi-strongly accretive mapping for i = 1, 2. Assume that the following assump-tions hold:

where τ1 = m1 +m2 + ρ2γ2, τ2 = m1 +m2 +ρ1γ1,

Then there exist x∗, y∗ ∈ E, which solves the problem (1.1). Moreover, the parallel iterative sequences {xn} and {yn} generated by the Algorithm 3.1 con-verge to x∗ and y∗, respectively.

Proof. To proof the result, we first need to evaluate ║xn+1 − xn║ for all n ≥ 0. From Algorithm 3.1 and the nonexpansive property of the sunny nonexpansive retraction QC, we can get

Using the strongly accretivity and Lipschitz continuity of g1 and Lemma 2.3, we find that

and

which imply that

and

where . Since T1 is (δ1, ξ1)-relaxed cocoercive and μ1-Lipschitz continuous with respect to the first argument, we have

Also,using the Lipschitz continuity of T1 with respect to second argument,

Combining (4.3)-(4.7), we have

where

Similarly, since g2 is η2-Lipschitz continuous and ζ2-strongly accretive, T2 is (δ2, ξ2)-relaxed cocoercive, μ2-Lipschitz continuous with respect to the first argument and γ2-Lipschitz continuous with respect to the second argument, we obtain

where . It follows from (4.8) and (4.9) that

where k = max{m1 + m2 + θ2 + ρ1γ1,m1 + m2 + θ1 + ρ2γ2}. From (4.1) and (4.2), we know that 0 ≤ k < 1. Let cn = ║xn − xn−1 ║ + ║yn − yn−1║. Then (4.10) can be rewritten as

It follows from Lemma 2.6 that {xn} and {yn} are both Cauchy sequences in E. There exist x∗, y∗ ∈ E such that xn → x∗ and yn → y∗ as n → ∞. By continuity, we know that x∗, y∗ satisfy

It follows from Lemma 3.1 that (x∗, y∗) is a solution of problem (1.1). This completes the proof. □

If T1, T2 : C → E are nonlinear mappings and g1 = g2 = I, the the following corollary follows immediately from Theorem 4.1.

Corollary 4.2. Let E be a 2-uniformly smooth Banach space with the 2-uniformly smooth constant K, C be a nonempty closed convex subset of E and QC be a sunny nonexpansive retraction from E onto C. Let Ti : C → E be a (δi, ξi)-relaxed cocoercive and μi-Lipschitz continuous mapping for i = 1, 2. Assume that the following assumptions hold:

Then there exist x∗, y∗ ∈ E, which solves the problem (1.2). Moreover, the par-allel iterative sequences{xn} and {yn} generated by the Algorithm 3.3 converge to x∗ and y∗, respectively.

Remark 4.1. (i) We note that Hilbert spaces and Lp(p ≥ 2) spaces are 2-uniformly smooth.

(ii) If E = H is a Hilbert space, then a sunny nonexpansive retraction QC is coincident with the metric projection PC from H onto C.

(iii) It is well known that the 2-uniformly smooth constant in Hilbert spaces.

We can obtain the following result immediately.

Corollary 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ti : C → H be a (δi, ξi)-relaxed cocoercive and μi-Lipschitz continuous mapping for i = 1, 2. Let g : C → C be a η-Lipschitz continuous and ζ-strongly monotone mapping. Assume that the following assumptions hold:

where

Then there exist x∗, y∗ ∈ H, which solve the problem (1.3). Moreover, the par-allel iterative sequences {xn} and {yn} generated by the Algorithm 3.4 converge to x∗ and y∗, respectively.

Let Fix(Si) denote the set of fixed points of the mapping Si, i.e., Fix(Si) = {x ∈ C : Six = x} and Ω the set of solutions of the problem (1.1).

Theorem 4.4. Let E be a 2-uniformly smooth Banach space with the 2-uniformly smooth constant K, C be a nonempty closed convex subset of E and QC be a sunny nonexpansive retraction from E onto C. Let Ti : C × C → E be a non-linear mapping such that (δi, ξi)-relaxed cocoercive, μi-Lipschitz continuous with respect to the first argument and γi-Lipschitz continuous with respect to the sec-ond argument for i = 1, 2. Let gi : C → C be a ηi-Lipschitz continuous and ζi-strongly accretive mapping for i = 1, 2. Let Si : C → C be a nonexpansive mapping with a fixed point for i = 1, 2. Let {αn}, {βn} be sequences in [0, 1]. Assume that the following assumptions hold:

(C1) 0 < Θ1,n = αn(1 − κ − (1 − κ)(m1 + ρ1γ1)) − βn(1 − κ)(m2 + θ2) < 1,

(C2) 0 < Θ2,n = βn(1 − κ − (1 − κ)(m2 + ρ2γ2)) − αn(1 − κ)(m1 + θ1) < 1,

(C3) , where

and

If Ω ∩ Fix(S1) ∩ Fix(S2) ≠ ϕ, then the sequences {xn} and {yn} generated by the Algorithm 3.2 converge to x∗ and y∗, respectively, where (x∗, y∗) ∈ Ω and x∗, y∗ ∈ Fix(S1) ∩ Fix(S2).

Proof. Letting (x∗, y∗) ∈ Ω, we obtain from Lemma 3.1 that

Since x∗, y∗ ∈ Fix(S1) ∩ Fix(S2), we have

Putting e1,n = κS1(xn) + (1 − κ)(xn − g1(xn) + QC[g1(yn) − ρ1T1(yn, xn)]) for each n = 0, 1, 2, · · · , we arrive at

Using the arguments as in the proof of Theorem 4.1, we obtain

and

where

From (4.11), we have

It follows that

Similarly, we obtain

where

Now (4.12) and (4.13) imply

where

Define the norm ║·║∗ on E × E by

Then (E × E, ║·║∗) is a Banach space. Hence, (4,14) implies that

From the conditions (C1)-(C3) and Lemma 2.5 to (4.15), we obtain that

Therefore, the sequences {xn} and {yn} converge to x∗ and y∗, respectively. This completes the proof. □

Remark 4.2. Theorem 4.1 and 4.4 extend the solvability of the systems of variational inequalities (1.2)-(1.6) to the more general system of variational inequalities (1.1). The underlying mapping Ti : C × C → E (i = 1, 2) in our paper needs to be relaxed (δi, ξi)-relaxed cocoercive while the underlying operators A,B in [13] needs to inverse strongly accretive. Hence, Theorem 4.1 and 4.4 extend and improve the main results of [9,12,13].