0 Fixed Point Theory and Applications

a SpringerOpen Journal

RESEARCH

Open Access

Fixed point problems and a system of generalized nonlinear mixed variational inequalities

Narin Petrot1,2 and Javad Balooee3

Correspondence: javad.balooee@gmail.com 3Department of Mathematics, Sari Branch, Islamic Azad University, Sari Iran

Fulllist of author information is available at the end of the article

Abstract

In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings Si (i = 1,2,3) to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (Si, S2, S3), which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed. MSC: Primary 47H05; secondary 47J20; 49J40; 90C33

Keywords: system of generalized mixed variational inequalities; fixed point problems; nearly uniformly Lipschitzian mapping; three-step resolvent iterative algorithm; convergence

ringer

1 Introduction

Variational inequality theory, which was initially introduced by Stampacchia [1] in 1964, is a branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using the novel and innovative techniques. An important and useful generalization is called the mixed variational inequality, or the vari-ational inequality of the second kind, involving the nonlinear term. For applications, numerical methods, and other aspects of variational inequalities, see, for example, [1-22] and the references therein. In recent years, much attention has been given to develop efficient and implementable numerical methods including projection method and its variant forms, Wiener-Hopf (normal) equations, linear approximation, auxiliary principle, and descent framework for solving variational inequalities and related optimization problems. It is well known that the projection method and its variant forms and Wiener-Hopf equations technique cannot be used to suggest and analyze iterative methods for solving mixed variational inequalities due to the presence of the nonlinear term. These facts motivated us to use the technique of resolvent operators, the origin of which can be traced back to

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Martinet [11] and Brezis [4]. In this technique, the given operator is decomposed into the sum of two (or more) maximal monotone operators, whose resolvents are easier to evaluate than the resolvent of the original operator. Such a method is known as the operator splitting method. This can lead to the development of very efficient methods, since one can treat each part of the original operator independently. The operator splitting methods and related techniques have been analyzed and studied by many authors including Peace-man and Rachford [15], Lions and Mercier [9], Glowinski and Tallec [7], and Tseng [18]. For an excellent account of the alternating direction implicit (splitting) methods, see [2]. A useful feature of the forward-backward splitting method for solving the mixed variational inequalities is that the resolvent step involves the subdifferential of the proper, convex and lower semicontinuous part only and the other part facilitates the problem decomposition.

Equally important is the area of mathematical sciences known as the resolvent equations, which was introduced by Noor [12]. Noor [12] established the equivalence between the mixed variational inequalities and the resolvent equations using essentially the resolvent operator technique. The resolvent equations are being used to develop powerful and efficient numerical methods for solving the mixed variational inequalities and related optimization problems. It is worth mentioning that if the nonlinear term involving the mixed variational inequalities is the indicator function of a closed convex set in a Hilbert space, then the resolvent operator is equal to the projection operator.

On the other hand, related to the variational inequalities, we have the problem of finding the fixed points of nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to consider a unified approach to these two different problems. Motivated and inspired by the research going in this direction, Noor and Huang [14] considered the problem of finding a common element of the set of solutions of variational inequalities and the set of fixed points of nonexpansive mappings. It is well known that every nonexpansive mapping is a Lipschitzian mapping. Lipschitzian mappings have been generalized by various authors. Sahu [23] introduced and investigated nearly uniformly Lipschitzian mappings as generalization of Lipschitzian mappings.

In the present paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID). We first verify the equivalence between the SGNMVID and the fixed point problems, and then by this equivalent formulation, we discuss the existence and uniqueness of the solution of the SGNMVID. Applying nearly uniformly Lipschitzian mappings Si (i = 1,2,3) and the aforesaid equivalent alternative formulation, we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding the element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (Si, S2, S3), which is the unique solution of the SGNMVID. Also, the convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, some comments on the results related to a class of strongly monotone mappings are discussed. The results presented in this paper extend and improve some known results in the literature.

2 Preliminaries and basic results

Throughout this article, we let H be a real Hilbert space which is equipped with an inner product <•, •) and the corresponding norm || • ||. Let Ti: H x H x H ^ H andgi: H ^ H (i = 1,2,3) be six nonlinear single-valued operators such that for each i = 1,2,3, gi is an

onto operator, and let d^i denote the subdifferential of the function q>i (i = 1,2,3), where for each i = 1,2,3, : H ^ R U is a proper convex lower semicontinuous function on H. For any given constants p, n, Y > 0, we consider the problem of finding x*, y*, z* e H such that

<p Ti(y*, z*, x*)+x* - gi(y*), gi(x)-x*> > p<Mx*)-p<Pi(gi(x)), Vx e H, <nT2(z*,x*,y*) + y* -g2(z*),g2(x) -y*> > n^2(y*) - W2(g2(x)), "Vx e H, (2.1) <Y T3(x*,y*,z*) + z* -g3(x*),g3(x)-z*> > y^3(z*) - m(g3(x)), Vx e H,

which is called the system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID).

If for each i = 1,2,3, gi = I, the identity operator, and q>i(x) = SK(x), for all x e K, where SK is the indicator function of a nonempty closed convex set K in H defined by

<YT3(x*,y*,z*) +z* -x*,x-z*> > 0, VxeK,

which was introduced and studied by Cho and Qin [6].

For different choices of operators and constants, we obtain different systems and problems considered and studied in [i, 5, 8, i3, i7, i9-2i] and the references therein.

Definition 2.1 A set-valued operator T: H ^ H is said to be monotone if, for any x,y e H,

(u - v,x -y> > 0, Vu e T(x), v e T(y).

A monotone set-valued operator T is called maximal if its graph, Gph(T) := {(x,y) e H x H: y e T(x)}, is not properly contained in the graph of any other monotone operator. It is well known that T is a maximal monotone operator if and only if (I + XT)(H) = H for all X >0, where I denotes the identity operator on H.

Definition 2.2 [4] For any maximal monotone operator T, the resolvent operator associated with T of parameter X is defined as

JT(u) = (I + XT)-i(u), Vu e H.

It is single-valued and nonexpansive, that is,

||jT (u) - JT (v) | < ||u - v||, Vu, v e H.

Sk (y)

0, y e K, то, yek,

then problem (2.i) reduces to the following system:

(pTi(y*,z*,x*)+x* -y*,x -x*> > 0, Vx e K, <nT2(z*,x*,y*) +y* - z*,x -y*> > 0, Vx e K,

If p is a proper, convex and lower-semicontinuous function, then its subdifferential dp is a maximal monotone operator, see Theorem 4 in [24]. In this case, we can define the resolvent operator associated with the subdifferential dp of parameter X as follows:

J\(u) = (I + X3<p)-1(u), Vu e H. The resolvent operator J^ has the following useful characterization. Lemma 2.1 [13] For a given z e H, x e H satisfies the inequality

(x - z, y - x) + Xp (y)- Xp (x) > 0, Vy e H,

if and only ifx = JX(z), where J'X is the resolvent operator associated with dp of parameter X >0.

It is well known that JX is nonexpansive, that is, ||/>)-/£(v)|| <||u - vy, Vu, v e H.

Let us recall that a mapping T: H ^ H is nonexpansive if ||Tx - Ty || < ||x - y|| for all x, y e H. In recent years, nonexpansive mappings have been generalized and investigated by various authors. In the next definitions, several generalizations of nonexpansive mappings are stated.

Definition 2.3 A nonlinear mapping T: H ^ H is called (a) L-Lipschitzian if there exists a constant L >0 such that

||Tx - Ty||<L||x -y||, Vx,y e H;

(b) generalized Lipschitzian [25] if there exists a constant L >0 such that

|| Tx - Ty||<L(||x -y|| + 1), Vx,y e H; (c) generalized (L, M)-Lipschitzian [23] if there exist two constants L, M >0 such that

||Tx - Ty ||<L( ||x - y|| + M, Vx, y e H;

(d) asymptotically nonexpansive [26] if there exists a sequence {kn} c [1, to) with kn = 1 such that for each n e N,

Tnx - Tny| < knUx-y|, Vx,y e H;

(e) pointwise asymptotically nonexpansive [27] if, for each integer n > 1, || Tnx - Tny| < an^x -y|, x,y e H,

where an ^ 1 pointwise on X;

(f) uniformly L-Lipschitzian if there exists a constant L >0 such that for each n e N,

Tnx - Tny|| < i||x -y||, Vx,y e H.

Definition 2.4 [23] A nonlinear mapping T: H ^ H is said to be (a) nearly Lipschitzian with respect to the sequence {an} if for each n e N, there exists a constant kn > 0 such that

|| Tnx - Tny|| < kn(||x -y|| + an), Vx,y e H, (2.3)

where {an} is a fix sequence in [0, to) with an ^ 0,as n ^ to. For an arbitrary, but fixed n e N, the infimum of constants kn in (2.3) is called nearly Lipschitz constant and is denoted by n(Tn). Notice that

{||Tnx - Tny|| 1

----: x, y e H, x = y [.

||x - y|| + an J

Definition 2.5 [23] A nearly Lipschitzian mapping T with the sequence {(an, n(Tn))} is said to be

(a) nearly nonexpansive if n(Tn) = 1 for all n e N, that is, || Tnx - Tny| < ||x -y || + an, Vx,y e H;

(b) nearly asymptotically nonexpansive if n(Tn) > 1 for all n e N and limn^TO n(Tn) = 1, in other words, kn > 1 for all n e N with limn^TO kn = 1;

(c) nearly uniformly L-Lipschitzian if n(Tn) < L for all n e N,in other words, kn = L for all n e N.

Remark 2.2 It should be pointed out that:

(a) Every nonexpansive mapping is an asymptotically nonexpansive mapping, and every asymptotically nonexpansive mapping is a pointwise asymptotically nonexpansive mapping. Also, the class of Lipschitzian mappings properly includes the class of pointwise asymptotically nonexpansive mappings.

(b) It is obvious that every Lipschitzian mapping is a generalized Lipschitzian mapping. Furthermore, every mapping with a bounded range is a generalized Lipschitzian mapping. It is easy to see that the class of generalized (L, M)-Lipschitzian mappings is more general than the class of generalized Lipschitzian mappings.

(c) Clearly, the class of nearly uniformly L-Lipschitzian mappings properly includes the class of generalized (L, M)-Lipschitzian mappings and that of uniformly L-Lipschitzian mappings. Note that every nearly asymptotically nonexpansive mapping is nearly uniformly L-Lipschitzian.

Some interesting examples to investigate relations between the mappings given in Definitions 2.3, 2.4 and 2.5 can be found in [3].

3 Existence of solution and uniqueness

In this section, we prove the existence and uniqueness theorem for a solution of the system of generalized nonlinear mixed variational inequalities (2.1). For this end, we need the following lemma, in which, by using the resolvent operator technique and Lemma 2.1, the equivalence between the system of generalized nonlinear mixed variational inequalities (2.1) and fixed point problems is stated.

Lemma 3.1 Let Ti, gi, pi (i = 1,2,3), p, p and y be the same as in SGNMVID (2.1). Then (x*, y*, z*) e H x H x H is a solution of SGNMVID (2.1) if and only if

'x* = JP1(g1(y* )-P T1(y*, z*, x*)),

• y* = JPp2 (g2(z*) - PT2(z*,x*,y*)), (3D

z = JY3(g3(x*)-y T3(x*, y*, z*)),

where Jpx is the resolvent operator associated with d p1 of parameter p, JP2 is the resolvent operator associated with d p2 of parameter p and J%3 is the resolvent operator associated with d p3 of parameter y.

Proof (x*,y*, z*) e H x H x H is a solution of SGNMVID (2.1) ifand onlyif

<x* - (g1(y*) - pT1(y*,z*,x*)),g1(x)-x*> + pP2(g2(x)) - pP1(x*)

> 0, Vx e H,

<y* - (g2(z*) - P T2 (z*, x*,y*)),g2(x) - y*> + pP2 (g2 (x)) - pP2(y*) ^

> 0, Vx e H, . <z* - (g3(x*) - Y T3(x*,y*,z*)),g3(x) -z*> + yP3(g3(x)) - YP3(z*)

> 0, Vx e H.

Since for each i = 1,2,3, gi is an onto operator, Lemma 2.1 implies that (x*,y*, z*) e H x H x H is a solution of (3.2) if and only if

x* = Jp1(g1(y* )-p T1(y*, z*, x*)),

• y* = Jp2 (g2(z*) - pT2(z*,x*,y*)),

.z* = J,Y2(g3(x*)-Y T3(x*, y*, z*)).

This completes the proof. □

Definition 3.1 Let T: H x H x H ^ H and g: H ^ H be two single-valued operators. Then the operator

(a) T is called monotone in the first variable if

T(x, •, •) - T(y, •, •),x -y) > 0, Vx,y e H;

(b) T is called r-strongly monotone in the first variable if there exists a constant r >0 such that

(T(x, •, •) - T(y, •, •),x -y) > r|x -y|2, Vx,y e H;

(c) T is called (k , 0 )-relaxed cocoercive in the first variable if there exist two constants k, 0 > 0 such that

(T (x, •, •) - T (y, •, •), x - y) > -k || T (x, •, •) - T (y, •, •) ||2 + 0 ||x - y y2, Vx, y e H;

(d) T is said to be ¡-Lipschitz continuous in the first variable if there exists a constant ¡x >0 such that

|| T(x, •, •)- T(y, •, 0|| < ¡||x -y||, Vx,y e H;

(e) g is called y -Lipschitz continuous if there exists a constant y >0 such that

|g(x)-g(y)| < y ||x -y||, Vx,y e H;

(f ) g is said to be v-strongly monotone if there exists a constant v >0 such that

(g(x)-g(y), x - y > v||x - y||2, Vx, y e H.

Theorem 3.2 Let Ti, gi, (i = 1,2,3), p, n and y be the same as in SGNMVID (2.1) such that for each i = 1,2,3, Ti is çi-strongly monotone and oi-Lipschitz continuous in the first variable andgi is ni-strongly monotone and Si-Lipschitz continuous. If the constants p, n and y satisfy the following conditions:

IP - f I <

In - Çf1 <

Vçf- -O12xi(2-xi)

Vçf- -o2>2(2-x2)

Vis2- -O3V3 (2-X3)

Çi > OiJ¡i(2- ¡i) (i = 1,2,3), ¡i = - (2ni - Sf ) <1 (i = 1,2,3), 2ni < 1 + Sf (i = 1,2,3),

then SGNMVID (2.1) admits a unique solution.

Proof Define the mappings ^, © : H x H x H ^ H by

^(x,y,z) J (gi(y) - pT,(y,z,x)),

®(x,y,z) J (g2(z) - nT2(z,x,y)), (3.4)

©(x,y,z) =JY3(g3(x) - y T3(x,y,z)), for all (x, y, z) e H x H x H. Define || • H on H x H x H by ||(x,y,z)||t = ||x|| + ||y|| + ||z||, V(x,y,z) e H x H x H.

It is obvious that (H x H x H, || • |U) is a Banach space. Moreover, define F : H x H x H ^ H x H x H as follows:

F (x, y, z) = (^ (x, y, z), $(x, y, z), ©(x, y, z)), V(x, y, z) e H x H x H. (3.5)

Now, we prove that F is a contraction mapping. For this end, let (x, y, z), (x, y, z) e H x H x H be given. By using the nonexpansivity property of the resolvent operator /p1, we get

11^ (x, y, z) - ^ (x, y, z)| = |/p1 (g100-pT1(y,z,x)) -/p1(g1(y)-pT1(y,z,x))1

< ||g1(y) -g1(y) - p{T(y,z,x) - T1(y,z,x))1

< |y-y- (g100-g1(y))1 + |y-y- p{T1(y,z,x)-TL(y,z,x))|. (3.6) Becauseg1 is ^-strongly monotone and 51-Lipschitz continuous, we have

|y -y - (g2(y)-g2(y)) |2

= ||y -y\\2 -2{g2(y)-g2(y),y -y) + |g1(y) g1 (y)|2

< (1 - 2^1) \y -yll2 + |g1(y)-g1(y) |2

< (1-2711 + 52) lly -y|2. (.7)

Since T1 is ^-strongly monotone and a1-Lipschitz continuous in the first variable, we conclude that

|y -y - p(T1(y,z,x)-T2(y,z,x))|2 = lly - yll22 - 2p{T (y, z,x)- T (y,z,x),y -y) + p21T (y, z,x)- T1 (y, z,x)12

< (1 -2psi)lly-yll2 + p2|T1(y,z,x)-T1(y,z,x)|2

< (1 - 2p^ + p2°i2) lly - yl2. (.8)

Substituting (3.7) and (3.8) in (3.6), we deduce that

(x,y,z)-*(x,y,z)| < (J 1-2711 + \2 ^1-2pq1 + p2^22)NУ-yll. (3.9)

Like in the proof of (3.9), we can establish that

|$(x, y, z)-$(x, y, z) | < (J 1 - 27T2 + 52 ^1-2p5-2 + p^llz - ¿ll (3.10)

|©(x,y,z)-&(x,y,z)| < (^ 1 - 2n3 + 52 ^1-2y^3 + Y^Dllx -x||. (3.11)

From (3.9)-(3.11), it follows that

(x, y, z)-^ (x, y, z)| + |$(x, y, z)-$(x, y, z)| + |©(x, y, z)-®(x, y, z)| < & 11 x - xll + e lly - yll + ellz - zll, (.12)

§ = - 2tT3 + Sj ^1-2x5-3 + y2a|,

0 = y 1 - 2tti + Sj + y 1 - 2/051 + p2a12, (3.13) g = yj 1-2n2 + + - 2)?52 + ^2a22. Applying (3.5) and (3.12), we conclude that

|F(x,y,z) -F(x,y,z)| < X|(x,y,z) - (x,y,z)|| , (314)

where X = max{§, 0, g}. Condition (3.3) implies that 0 < X <1 and so (3.14) guarantees that the mapping F is contraction. According to the Banach fixed point theorem, there exists a unique point (x*,y*,z*) e H x H x H such that F(x*,y*,z*) = (x*,y*,z*). It follows from (3.4) and (3.5) thatx* = Jpi(gi(y*)-pTtf,z*,x*)),y* = Jn2(g2(z*)-n^2(z*,x*,y*)) andz* = jY3(g3(x*) - yT3(x*,y*,z*)). Now, it follows from Lemma 3.1 that (x*,y*,z*) e H x H x H is a unique solution of SGNMVID (2.1). This completes the proof. □

4 Some new three-step resolvent iterative algorithms

In this section, applying nearly uniformly Lipschitzian mappings Si (i = 1,2,3) and by using the equivalent alternative formulation (3.1), we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of Q = (Si, S2, S3), which is the unique solution of SGNMVID (2.1).

Let S1 : H ^ H be a nearly uniformly L1 -Lipschitzian mapping with the sequence {^k}2=1, let S2: H ^ H be a nearly uniformly L2-Lipschitzian mapping with the sequence and let S3: H ^ H be a nearly uniformly L3-Lipschitzian mapping with the sequence {cnWe define the self-mapping Q of H x H x H as follows:

Q(x,y, z) = (S1x, S2y, S3z), Vx,y, z e H. (.1)

Then Q = (S1, S2, S3) :H x H x H ^ H x H x H is a nearly uniformly max{L1, L2, L3}-Lipschitzian mapping with the sequence {an + bn + cn with respect to the norm || • ||* in H x H x H.To see this fact, let (x, y, z), (x', y', z') e H x H x H be arbitrary. Then, for any n e N,we have

|Qn(x,y,z) - Qn(x',y',zO|* = | (Snx, S-y, S-z) - (%*!, Sn2y', Snz') |* = | (Six - Six', S-y - Sn/, Snz - Snz') | * — |S1x S-^x | + |Si—y S<2y | + |S3z Siz |

< L^ |x - x'| + an) + L2( |y - y| + bn) + L^ |z - z'| + c-)

< max{L1, L2, L3}( |x - x;| + |y - У| + |z - z;| + an + bn + cn) = max{L1,L2,L3}(|(x,y,z) -x',y',z!) | + an + bn + cn).

We denote the sets of all the fixed points of Si (i = 1,2,3) and Q by Fix(S;) and Fix(Q), respectively, and the set of all the solutions of system (2.1) by SGNMVID(H, Ti,gi, (pi, i = 1,2,3). It is clear that for any (x,y, z) e H x H x H, (x,y, z) e Fix(Q) if and only if x e Fix(S1), y e Fix(S2) and z e Fix(S3), that is, Fix(Q) = Fix(S1, S2, S3) = Fix(S1) x Fix(S2) x Fix(S3). We now characterize the problem. Let Ti, gi, ( (i = 1,2,3), p, p and y be the same as in SGNMVID (2.1). If (x*,y*, z*) e Fix(Q) n SGNMVID(H, Titgi, (, i = 1,2,3), then x* e Fix(S1), y* e Fix(S2), z* e Fix(S3) and (x*,y*, z*) e SGNMVID(H, Ti,gi, a, i = 1,2,3). Therefore, it follows from Lemma 3.1 that for each n e N,

x* = Snx* = /p1(g1(y*)-pT1(y*, z*, x*)) = Sn/p1(g1(y*)-pT1(y*, z*, x*)), • y* = Sny* = p (g2(z*) - pT2(z*,x*,y*)) = Sn/p2 (g2(z*) - pT2(z*,x*,y*)), (4.2)

z* = Snz* = /Y3(g3(x*) - YT3(x*,y*,z*)) = Sn/Y3(g3(x*) - yT3(x*,y*,z*)).

The fixed point formulation (4.2) is used to suggest the following three-step resolvent iterative algorithm with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S1, S2, S3), which is a unique solution of SGNMVID (2.1).

Algorithm 4.1 Let Ti, gi, ( (i = 1,2,3), p, p and y be the same as in SGNMVID (2.1). For an arbitrary chosen initial point (x1, y1, z1) e H x H x H, compute the iterative sequence {(xn,yn, zn)|J;=1 by the iterative processes

x«+i = (1 - a« - ßn)Xn + anS1Jpi(gi(yn+i) - pTi(y«+i,z«+i,x«))

+ anen + ßnjn + rn,

yn+i = (1 - an - ß'n )xn + an SJfefe+i) - vT2(Zn+i, Xn, yn))

+ a'nPn + ß'n qn + kn, Zn+i = (i - a'n - ß'n)Xn + a'nSrnJl3(g?,(Xn) - YT3(Xn,yn,Zn))

+ a'n&n + ß'n tn + In,

where Si : H ^ H (i = i, 2,3) are three nearly uniformly Lipschitzian mappings, {an}^=i, KI^p {an/}^£i, {ßn}n=v {ßn}~=i and {ßn'}~=i are sequences in the interval [0,i] such that

E~=i an = », E~=i ßn < E~=i ß'n < E~=i ß'n < an + ßn < i, an + ß'n < i, a'n + ß'n < i, an = i, an = i and {en}ii=i, {pn}ii=i, {sn}^i, {/n}SS=i, {qn}ii=i, {tn}i^=i, {rn}ii=i,

{kn}^=i, {/n}JÎ=i are nine sequences in H to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions:

en = en + C Pn = Pn + Ä sn = sn + sn,

limn^œ \\e'n\\ = 0, limn^œ iip/N = 0, limn^œ iis/H = 0,

Eco m //M v^œ M //M v^œ M //,,

n=i We«1 < œ, en=i \\Pn \\ < œ, en=i iis/,h < œ, EC Wr« W < œ, EC°=i \\kn W < œ, EC°=i Ill« W < œ.

If for each i = 1,2,3, Si = /, then Algorithm 4.1 reduces to the following algorithm.

Algorithm 4.2 Let Ti, gi, ( (i = 1,2,3), p, p and y be the same as in SGNMVID (2.1). For an arbitrary chosen initial point (x1, y1, z1) e H x H x H, compute the iterative sequence

{(xn,yn, Znin the following way:

Xn+1 = (1 - an - Pn)xn + aJh (gi(yn+i) - pTi(yn+i, Zn+i,Xn)) + anen + Pnjn + rn, ■ yn+i = (i - a'n - P'n)Xn + a'J^fefe+i) - ^fe+i,Xn,yn)) + a^Pn + P4n + K, Zn+i = (i - an' - ^n')Xn + aJ?, (g3 (Xn) - Y T3(xn, yn, Zn)) + a'^Sn + ^'¿n + ln,

where the sequences {an}~i, K^, K^i, {PnlTO=i, {enlTOSi, {PnlTOSi,

{Sn}~i, {jn}~=i, {qnl~=i, {inl~=i, {rnl~=i, {knl~=i and {«TO=i are the same as in Algorithm 4.i.

Remark 4.3 Equality (4.2) can be written as follows:

x* = S1Jpi (u), y* = SJ (v), z* = SnjY, (w),

u = gi(y*)-pTi(y*,z*,x*), v = g2(z*) - nT2(z*,x*,y*), (4.5)

w = g3(x*) - yT,(x*,y*,z*).

The fixed point formulation (4.5) enables us to suggest the following iterative algorithms.

Algorithm 4.4 Let T, gi, pi (i = i, 2,3), p, n and y be the same as in SGNMVID (2.i). For an arbitrary chosen initial point (ui, vi, wi) e H x H x H, compute the iterative sequence {(xn,yn, zn)}^ in the following way:

xn = S^h (un), yn = sy1^ (Vn), zn = Sj (Wn),

un+i = (i- an - Pn)Un + an(gi(yn)- pTi(y n, zn, xn)) + anen + ftnjn + rn, , ,

Vn+i = (i - an - Pn)Vn + an(g2(zn) - nT2(z n , xn , yn )) + anPn + Pnqn + kn,

Wn+i = (i - an - fin)Wn + an(g,(xn) -YT,(xmyn,zn)) + anSn + P n¿n + ln,

where Si: H ^ H (i = i, 2,3) are three nearly uniformly Lipschitzian mappings, {an}J;=i, {Pn lTOS are sequences in [0, i] such that ^^=i an = to, ^^=i pn < to, an + pn < i and the

sequences {enlTOi, {PnlTOi, {SnlTOSi, {/nlTO=i, {<=«lTO=i, {¿nlTO=i, {rnlTO=i, {knlTO=i, {lnlTO=i are the same as in Algorithm 4.i satisfying (4.4).

If pn = 0, for all n e N, then Algorithm 4.4 reduces to the following algorithm.

Algorithm 4.5 Let Ti, gi, pi (i = i, 2,3), p, n and y be the same as in SGNMVID (2.i). For an arbitrary chosen initial point (ui, vi, wi) e H x H x H, compute the iterative sequence {(xn,yn, zn)lTO=i in the following way:

xn = Snjpn(u n), yn = S2Jp>2 (vn), zn = s3JY3 (wn), un+i = (i - an)un + an(gi(yn) - pTi(y n, zn, xn)) + anen + rn, Vn+i = (i - an)Vn + an(g2(zn) - nT2(z n , xn , yn )) + anPn + kn, Wn+i = (i -an)Wn + an(g3(xn) -YT3(xn,yn,zn)) + anSn + ln,

where Si (i = i, 2,3), {anlTO=i, {enlTO=i, {PnlTOi, {SnlTOi, {rnlTO=i, {knlTOi and {lnlTO=i are the same as in Algorithm 4.i.

If Si = I (i = 1,2,3), then Algorithm 4.4 collapses to the following algorithm.

Algorithm 4.6 Let Ti, gi, ( (i = 1,2,3), p, p and y be the same as in SGNMVID (2.1). For an arbitrary chosen initial point (u1, v1, w1) e H x H x H, compute the iterative sequence {(xn,yn, zn)}Ss=1 in the following way:

xn = /p1 (Un), yn = /p2 (Vn), zn = /^ (Wn),

un+1 = (1 - an - fin)Un + an(g1(yn) - pT1(y n, zn, xn)) + anen + ftnjn + rn,

Vn+1 = (1 - an - Pn)Vn + an(g2(zn) - p T2 (z n , xn , yn )) + a„pn + Pnqn + kn, Wn+1 = (1 - an - fin)Wn + an(g3(xn) - YT3(xn,yn,zn)) + anSn + P ntn + In,

where the sequences {an}^, {Pn}~1, {en}^, {pn}~1, fe}^, {/n}~1, {qn}~1, {U~1,

1, {kn}^1 and {ln}%1 are the same as in Algorithm 4.1.

5 Main results

In this section, we discuss the convergence analysis of the suggested three-step resolvent iterative algorithms under suitable conditions. For this end, we need the following lemma.

Lemma 5.1 Let {an}, {bn} and {cn} be three nonnegative real sequences satisfying the following condition: There exists a natural number n0 such that

an+1 < (1 - tn)an + bntn + cn, Vn > n0,

where tn e [0,1], £tn = ro, limn^TO bn = 0, £cn < ro. Then limn^0 an = 0.

Proof The proof directly follows from Lemma 2 in Liu [10]. □

Theorem 5.2 Let Ti, gi, d (i = 1,2,3), p, p and y be the same as in Theorem 3.2 and let all the conditions of Theorem 3.2 hold. Suppose that S1 : H ^ H is a nearly uniformly L1-Lipschitzian mapping with the sequence {bn}^, that S2 : H ^ H is a nearly uniformly L2-Lipschitzian mapping with the sequence {cn}^, that S3 : H ^ H is a nearly uniformly L3-Lipschitzian mapping with the sequence {dn}J;=1, and that the self-mapping Q of H x H x H is defined by (4.1) such that Fix(Q) n SGNMVID(H, Titgi, (, i = 1,2,3) = 0. Further, let LiX < 1, where X is the same as in (3.14). Then the iterative sequence {(xn,yn, zn)}cHl1 generated by Algorithm 4.1, converges strongly to the only element of Fix(Q) n SGNMVID(H, Titgi, (, i = 1,2,3).

Proof According to Theorem 3.2, SGNMVID (2.1) has a unique solution (x*,y*, z*) e H x H x H. Accordingly, in view of Lemma 3.1, (x*, y*, z*) satisfies (3.1). Since SGNMVID(H, Ti, gi, (i, i = 1,2,3) is a singleton set, it follows from Fix(Q) n SGNMVID(H, Ti,gi, (ph i = 1,2,3) = 0 that (x*,y*, z*) e Fix(Q) and so x* e Fixfo), y* e Fix(S2) and z* e Fix(S3). Hence, for each n e N,we can write

x* = (1 - an - Pn)x* + anS1/p1l(g1(y*) - pTr(y*,z*,x*)) + pnx*,

y* = (1 - a'n - )y* + a'nS/2(g2(z*) - pT2(z*,x*,y*)) + py*, (5.1)

z = (1 - an - Oz* + a'nSn/l, (g3(x*) - y T3(x*,y*,z*)) +

where the sequences {a^p Kl^, {a^l^i, {Anl^i, {A^i and {A'l^i are the same as in Algorithm 4.i. Let r = sup„>i{y;„ - x*||, \\qn -y*||, ||i„ - z*|||. It follows from (4.3), (5.1) and the assumptions that

Xn+i X

< (1 - an - fin) ||x„ -x* || + an (gi(y„+i) - pTi(yn+i,Zn+i,xn)) - SJ (gi(y* ) - pTi(y*, z*, X*))|| + Pnjln - X*| + an\\en || + HuH

< (i- an - fin)|Xn - X*1

+ anLi(|fi(yn+i) -gi(/) - p(Ti(yn+i,Zn+i,Xn) - Ti(y*,z*,x*)) || + bn) + a^ |en| + |en ^ + ||rn|| + An r < (i - an - An) |Xn -x* 1 + anLi(|yn+i -y* - (gi(yn+i) -gi(y*)) 1 + |yn+i - y* - p{Ti(yn+i, Zn+i,Xn) - Ti(y*, z*,x*))| + bn)

+ an|en| + | e" | + ||r„ | + Anr. (5.2)

Sincegi is ^-strongly monotone and 5i-Lipschitz continuous, and Ti is ^i-strongly monotone and oi-Lipschitz continuous in the first variable, similar to the proofs of (3.7) and (3.8), one can prove that

|yn+i - y* - (gi(yn+i) -gi(y*)) | <yji-2ni + 52 |yn+i - y* | (5.3)

|yn+i -y* - p{Ti(yn+i,Zn+i,Xn) - Ti(y*,z*,x*))|

<yji - 2pgi + p2Oi2 |yn+i-y* |. (.4)

Substituting (5.3) and (5.4) in (5.2), we get

|xn+i- x^ < (i - an - An)|xn - x*| + anLid |yn+i - y*|

+ anLibn + an| e'n| + | e^| + ||rn || + Anr, (5.5)

where 0 is the same as in (3.i3). It follows from (4.3) and (5.i) that

yn+i- y

< (i - a'n - fin) |Xn -y*1 + a'n ||SJ (g2(Zn+i) - nT2(Zn+i,Xn,yn))

- SJ (g2 (z* ) - ^(z*, X*, y*))|| + fin hn - y*| + a!n\\Pn\\ + \\*n\\

< (i - a'n - fin) \ xn - y*|

+ a'nL2( ||g2 (Zn+i) -g2(z*) - n( T2(Zn+i, Xn, yn) - T2 (z*, X*, y*)) | + Cn + a'n dPnl + |K||) + \\*n\\ + fin r

< (i - a'n - fin) |Xn - y*| + a'n L^ |Zn+i - z* - (g2(Zn+i) -g2(z*))|

+ ||z„+1 - z* - p(T2(zn+1,xn,yn) - T^z*,x*,y*)) || + cn) + an \p'n\ + M + IlknN + P'n r.

Since g2 is n2-strongly monotone and 82-Lipschitz continuous, and T2 is ^2-strongly monotone and a2-Lipschitz continuous in the first variable, we can get

\zn+1 - z* - (g2(zn+1) -g2(z*))\ < y 1 -2TT2 + 8%\zn+1 - z*

\zn+1 - z* - p(T2(zn+1,xn,yn) - T2(z*,x*,y*)) \ < J1 - 2p^2 + p2022\zn+1 - z* \. Combining (5.6)-(5.8), we conclude that

\yn+1-y*\ < (1 - a'n - P'n)\xn - y*\ + a'n L2Q \zn+1 - z* \

+ a'nl2cn + a'n \p'n \ + \p'n \ + Nkn N + P'n r,

where q is the same as in (3.13). From (4.3) and (5.1), it follows that \zn+1 - z*\

< (1 - a'n - Pn')\xn - z*\ + a'n\Sn3/Y3 (g3(xn)-Y T3(xm yn, zn)) - Sn/3 (g3 (x*) - Y T3 (x*,y*, z*)) \ + P'n \ tn - z* \ + a'n ||s„ N + mm

< (1 - a'n - P'n)\xn - z* \ + a'nL3{\g3(xn) -g^x*)

- Y(T3(xn,yn,zn) - T3(x*,y*,z*))\ + dn) + an'(\sn\ + \si\) + ||lnll + P'nr

< (1 - a'n - P'n)\xn - z* \ + an'L3(\xn - x* - fa fa) -g3(x*)) \

+ \xn - x* - Y (T3(xn,yn, zn)-T^x*,y*, z*)) \ + dn)

+ a'n\s'n \ + \s'n \ + lllnll + Pn'r.

Because g3 is n3-strongly monotone and 83-Lipschitz continuous, and T3 is ^-strongly monotone and a3-Lipschitz continuous in the first variable, we can obtain

\xn - x* - (g3(xn) -g3(x*))\ ^ y 1 - 2TT3 + 83\xn - x*

\xn -x* - Y(T3(xn,yn,zn) - T3(x*,y*,z*))\ ^ y 1-2y?3 + Y2

Substituting (5.11) and (5.12) in (5.10), deduce that

\zn+1 - z* \ < (1 - a'n - Pn)\xn - z* \ + a'nL3$ \xn -x* \ + a'nL3dn + a'n \s'n \ + \s'n \ + llln N + P^r,

where is the same as in (3.13).

(5.11)

(5.12)

(5.13)

By using (5.i3) and the fact that L3û < i, we have

|Zn+i - z*| < (i - a! - fi!) ^n - z*| + a'Lû ^n -X*| + a'^dn + a'n ^| + ||s! | + \\/n \\ + fi'lF

< (i - a'n - p'D |Xn - X*| + a'L® |Xn -X*| + a'^dn

+ (i - a'n - p'D |x* - z* || + a'n ||s1 | + ||s! | + \\/n \\ + fi'^T

< (i - a'n - P'D ^n - X* | + a'n ^n - X* | + a'^dn

+ (i - a'n - p'D ||x* - z* || + a'n ||s1 | + ||s! || + \\/n \\ + fi'^T

< |Xn - x*| + a'^dn + (i - a'n - fi'n) |x* - z*|

+ a'Hs'nl + |s!| + HU + fiT. (.14)

It follows from (5.9), (5.i4) and the fact that L2Q < i that

|yn+i-y^ < (i-an - fin) ^n - y*| + a'n L2Q |Zn+i - z*|

+ anl2Cn + an |_p1 | + yn | + \\kn \\ + fin r

< (i-an - fi'n ) |xn - x* || + (i-an - fi'n ) ||x* - y* |

+ anL2Q|Zn+i - z*| + a'nL2Cn + a'n p| + + \\knH + fi'nT

< (i-an - fi'n ) |xn - x* || + (i-an - fin ) ||x* - y* |

+ a'n L2q{ |Xn - x*| + a'^dn + (i-an' - fi'D |x* - z*|

+ a'HSn\ + ||s" | + \\/n \ + fin'T) + a'nL2Cn + a'n \p'n\ + | + \\kn\\ + fin T

< |xn - x* | + (i-an - fin ) ||x* - y* || + a'n (i - a'n - fi'n)L2Q |x* - z* |

+ a'na'nL2L3Qdn + a'na'nL2Q |s' | + a'nL2Q ^| + a'nL2Q\\ln \\ + a'nL2Qfi'nr + \\kn \ + fin T. (5.i5)

Applying (5.5) and (5.i5), it follows that

|xn+i - x |

< (i - an - fin) |Xn - x*| + anlid |yn+i - y*| + anLi bn + an|én| + ^^ + \\rn\\ + finT

< (i - an - fin) |Xn - x*| + anlid (|Xn - x*| + (i - an - fiD |x* - y*|

+ a'n (i - a'n - fi'Dl2Q ||x* - z* | + a'na'nl2l3Qdn + a'na'nl2Q |s' | + a'nl2Q ^ + a'nl2Q\\ln\\ + a'nl2Qfi'nT + a'nl2cn + an |pn| + Xp'n | + \\kn\\ + fi'nT

+ anLi bn + an|én| + || e!| + i\rn \ + finT

< (i - an - fin ) |Xn - x*| + anlid ||xn - x*| + a^ i-an - fi'n) lid |x* - y*|

+ ana'n (i - a'n - fi^)LxL2Qq |x* - z*| + d^LL3^gdn + LxL20Cn

+ anLi bn + ana/na/1lLlL2dQ|s/n| + an^!nL1^2^Q ^ | + an^L^O Q\\ln\\

+ ana'nL1O \p'n \ + anL10 \p/'n \ + anL10 llkn N + an \en \ + \ e'n \ + lknll + (ana'n L^QP'n + anL^P'n + Pn) r < (1 - an (1 - L10)) \xn - x* \

f, , aJ (1 - an - Pn )L10 Nx* - y* N + an (1 - a'n - P^L^Q, Nx* - z* ll

+ an (1- L10 ) ---—-

1 - L10

a'n a'nL1L2L30Qdn + a'n L1L20 cn + Lb + a'n a'nL1L20 QNs'nN + a'nL10 Np'nN + Kl + 1 -L10

+ ana'nL1L20Q\sn\ + ana'nL1L20QNlnN + anL10 \p'n\ + anL10 NknN

+ \ e'n \ + NmN + {ana'n L^QP'n + anL^P'n + Pn) r. (5.16)

FromJ]^ Pn < ro, £^ P'n < ro an^En=1 P'n < ro,weinfer that limn^ro Pn = limn^ro Pn = limn^ro P'n = 0. Since L10 < 1, limn^ro a'n = limn^ro a'n = 1 and limn^ro bn = limn^ro cn = limn^ro dn = 0, in view of (4.4), it is evident that the conditions of Lemma 5.1 are satisfied and so Lemma 5.1 and (5.16) guarantee that xn ^ x*, as n ^ro. Because E^11ln N < ro, Ero=1 NknN < ro, £ro=1 Nsn'N < ro and NpnN < ro, we have NlnN ^ 0, NknN ^ 0, NsJiN ^ 0 and Np'nN ^ 0, as n ^ro. Now, it follows from (4.4), (5.14) and (5.15) that yn ^ y* and zn ^ z*, as n ^ro. Therefore, the sequence {(xn,yn, zn)}ro=1 generated by Algorithm 4.1 converges strongly to the unique solution (x*,y*, z*) of SGNMVID (2.1), that is, the only element of Fix(Q) n SGNMVID(W, Ti,gi, p, i = 1,2,3). This completes the proof. □

Theorem 5.3 Suppose that Ti, gi, pi (i = 1,2,3), p, p and y are the same as in Theorem 3.2 and let all the conditions of Theorem 3.2 hold. Then the iterative sequence {(xn,yn, zn))2=1 generated by Algorithm 4.2 converges strongly to the unique solution of SGNMVID (2.1).

Theorem 5.4 Let Ti, gi, pi, Si (i = 1,2,3), Q, p, p and y be the same as in Theorem 5.2 and let all the conditions of Theorem 5.2 hold. Then the iterative sequence {(xn, yn, zn)|J;=1 generated by Algorithm 4.4 converges strongly to the only element of Fix(Q) n SGNMVID (H, Ti,gi, pi, i = 1,2,3).

Proof Theorem 3.2 guarantees the existence of a unique solution (x*,y*, z*) e H x H x H for SGNMVID (2.1). Hence, Lemma 3.1 implies that x* = Jp1(g1(y*) - pT1(y*,z*,x*)), y* = Jp2(g2(z*) - nT2(z*,x*,y*)), z* = JY3(g3(x*) - YT3(x*,y*,z*)). Since SGNMVID(H, Ti,gi,p, i = 1,2,3) is a singleton set, by using Fix(Q) n SGNMVID(H, Ti,gi, p, i = 1,2,3) = 0, we conclude that (x*,y*, z*) e Fix(Q) andsox* e Fix(Si), y* e Fix(S2) andz* e Fix(S3). Hence, in view of Remark 4.3, for each n e N,we can write

x* = Sn1JPi (u), y* = SJ (v), z* = S"3JY3 (w),

u =(1 - an - Pn)u + an(g1(y*) - pT1(y*,z*,x*)) + finu,

v = (1 - an - fin)v + an(g2(z*) - pT2(z*,x*,y*)) + finv, w =(1 - an - fin)w + an(g3(x*) - yT3(x*,y*,z*)) + finW,

where the sequences {an|J;=i and {AnlS=i are the same as in Algorithm 4.4. Let r = supn>i{Hjn - u||, ||qn - v||, ||in - w|||. By using (4.6), (5.i7) and the assumptions, we have

|| Un+i - U||

< (i - an - An)||Un - U|| + an |gi(yn) -gi(y*) - p(Ti(yn, Zn,xn) - Ti(y*, z*,x*)) | + Mjn - u|| + an( |en| + | e'n|) + ||rn||

< (i - an - An)||Un - u|| + an |yn - y* - (giOn) -gi(y*))|

+ an |yn -y* - p{Ti(yn,zn,xn) - Ti(y*,z*,x*))|

+ an|en| + |en| + H^H + fir. (.18)

Sincegi is ni-strongly monotone and 5i-Lipschitz continuous, and Ti is ^i-strongly monotone and oi-Lipschitz continuous in the first variable, similar to the proofs of (3.7) and (3.8), one can prove that

|yn - y* - (gi(yn) -gi(y*)) | <7i-2ni + 52 |yn -y* | (5.i9)

|yn -y* - p(Ti(yn,zn,xn)-Ti(y*,z*,x*))| i-2psi + p2o2 |yn -y*|. (5.20) Combining (5.i8)-(5.20), we get

||Un+i - u|| < (i - an - An) ||Un - u|| + an0 |yn -y*|

+ an|en| + |e'"| + ||rn|| + An^, (5!i)

where 0 is the same as in (3.i3). It follows from (4.6) and (5.i7) that |yn - y*| = | J^n)-J(v)| < L2 ( |Jíp2 (Vn) - J<f2 (v) | + Cn)

< L2(|Vn - v|| + Cn). (..2)

Substituting (5.22) in (5.2i), conclude that

||Un+i - u|| < (i - an - An) ||Un - u|| + anL201|Vn - v|| + anL20Cn

+ an|en| + + Urn! + Anl". (5.3)

Like in the proofs of (5.i8)-(5.23), we can verify that

|| vn+i - v|| < (i - an - An) ||vn - v|| + anL3Q ||Wn - w|| + anL3gdn

+ an|pn|| + |p1| + \\kn\\ + finT (5.24)

\\Wn+i-w\\ < (i-an - fin ) \ \ Wn - w\\ + anLi»\\Kn - u\\ + anLi» bn

+ an|sn| + RI + \\ln\\ + fin T, (.5)

where ç and » are the same as in (3.i3).

Let L = max{Li: i = 1,2,3}. Then, applying (5.23)-(5.25), we obtain

|| (Un+1, Vn+1, Wn+l) - (u, V, W) || *

< (1 - an - ßn) I (Un, Vn, Wn) - (u, V, w) I *

+ anLX||(un, vn, wn) - (u, v, w)|| + anLX(bn + cn + dn)

+ an|(en,p,,OIL + I(«2,P-,si-') I* + II(rn,kn,ln)||* + 3ßnf

< (l-an(1- LX)) |(un, vn, wn) -(u, v, w)||

n ,, N II (en,K, s,) II * + LX(bn + Cn + dn)

+ an(1 - LX)-1-X-

+ II (C p"n< 4') II* + |(rn, kn, ln)|* + 3ßn f, (5.6)

where X is the same as in (3.14). Since ^an = to, ^ßn < to, LX <1 and limn^TO bn = limn^TO cn = limn^TO dn = 0, in view of (4.4), we note that all the conditions of Lemma 5.1 are satisfied. Hence, Lemma 5.1 and (5.26) guarantee that (un, vn, wn) ^ (u, v, w), as n ^ to. By using (4.6) and (5.17), we have

IK -x*II = Ijuj-ju)!

< L1(|jp1(un)-jp1(u)| + bn)

< L1 I un - uI + bn (5.27)

||*n - Z* | = I S,JY (Wn) - SJ3 (w) I

< L3 (IJY3 (w,) - JY3 (w) | + dn)

< L3(||wn - w|| + d^. (.28)

Since limn^TO un = u, limn^TO vn = v, limn^TO wn = w and limn^TO bn = limn^TO cn = limn^TO dn = 0, from inequalities (5.22), (5.27) and (5.28) it follows that yn ^ y*, xn ^ x* and zn ^ z*, as n ^to. Hence, the sequence {(xn,yn, zn)}cH=i generated by Algorithm 4.4 converges strongly to the unique solution (x*,y*, z*) of SGNMVID (2.i), that is, the only element of Fix(Q) n SGNMVID(W, Ti,gh pi, i = i,2,3). This completes the proof. □

Like in the proof of Theorem 5.4, one can prove the convergence of the iterative sequences generated by Algorithms 4.5 and 4.6, and we omit their proofs.

Theorem 5.5 Suppose that Ti, gi, pi, Si (i = i, 2,3), Q, p, n and y are the same as in Theorem 5.2 and let all the conditions of Theorem 5.2 hold. Then the iterative sequence {(xn,yn, zn)}TO=i generated by Algorithm 4.5 converges strongly to the only element of Fix(Q) n SGNMVID (W, Titgi, p, i = i, 2,3).

Theorem 5.6 Assume that Ti, gi, pi (i = i, 2,3), p, n and y are the same as in Theorem 3.2 and let all the conditions of Theorem 3.2 hold. Then the iterative sequence {(xn,yn, zn)}TO=i generated by Algorithm 4.6 converges strongly to the unique solution of SGNMVID (2.i).

6 An important remark on a relaxed cocoercive mapping

In view of Definition 3.1, we note that the relaxed cocoercivity condition is weaker than the strong monotonicity condition. In other words, the class of relaxed cocoercive mappings is more general than the class of strongly monotone mappings. However, it is worth to point out that if the considered mapping T is (k , 0 )-relaxed cocoercive and y -Lipschitz mapping such that 0 > ky2, then it must be a (0 - ky2)-strongly monotone mapping. Hence, the results that appeared in this paper can be also applied to a class of relaxed cocoercive mappings. In fact, one may rewrite the results considered under relaxed cocoercivity and Lipschitzian conditions of mappings and apply a known result on the strongly monotone condition to a new form. Below, we present an example of the mentioned situation.

For given three different nonlinear operators T1, T2,g: H x H ^ H and a continuous function p : H ^ R U {+ro}, Noor [13] introduced and considered the problem of finding (x*,y*) e H x H such that

(pT1(y*,x*)+x* -g(y*),g(x)-x*}>pp(x*) - pp(g(x)), Vx e H,

(pT2(x*,y*) + y* -g(x*),g(x) -y*} > pp(y*) - pp(g(x)), Vx e H,

which is called a system of general mixed variational inequalities involving three different nonlinear operators (SGMVID). He also considered some spacial cases of SGMVID (6.1).

He proposed the following two-step iterative algorithm and its special forms for solving SGMVID (6.1) and studied the convergence analysis of the proposed iterative algorithms under certain conditions.

Algorithm 6.1 ([13], Algorithm 3.1) For arbitrary chosen initial points xo,yo e H, compute the sequences {xn} and {yn} by

xn+1 = (1 - an)xn + ajp[g(yn) - pTx(yn,xn)], yn+1 = /p[g(xn+1) - p T2 (xn+1, yn)],

where an e [0,1] for all n > 0.

Theorem 6.2 ([13], Theorem 3.1) Letx*, y* be the solution of SGMVID (6.1). Suppose that T1: H x H ^ H is relaxed (y1, r1)-cocoercive and ¡x1 -Lipschitzian in the first variable, and T2: H x H ^ H is relaxed (y2, r 2)-cocoercive and ¡x2-Lipschitzian in the first variable. Let g be a relaxed (y3, r3)-cocoercive and ¡x3-Lipschitzian. If

r1 - Y1M2 < V(r1 - Y1 M2)2 - M2k(2 - k)

W (6.2)

> Y1M2 + mWk(2 - k), k < 1,

r2 - Y2M2

,V(r2- Y2 m2)2- ¿2k(2- k)

M2 (6.3)

r2 > Y2^2 + mWk(2 - k), k < 1,

k ^1-2(r3- Y3M3) + m2 (6.4)

and an e [0,1], E an = », then for arbitrarily chosen initial points xo, yo e H, xn andyn obtained from Algorithm 6.1 converge strongly to x* andy*, respectively.

Remark 6.3 In view of conditions (6.2) and (6.3) (conditions (4.1) and (4.2) in [13]), we note that k e (0,1). Now, condition (6.4) (condition (4.3) in [13]) and k >0 imply that 2(r3 - y3^J) <1 + ^3. Accordingly, the condition 2(r3 - y3^J) <1 + ^3 should be added to conditions (6.2)-(6.4). On the other hand, since k < 1, from condition (6.4) it follows that

r3 > Y3^i

Remark 6.4 The conditions ri > ym2 + №i>Jk(2 - k) (i = 1,2), and k < 1 in (6.2) and (6.3) imply that ri > ym2 for each i = 1,2. Since for each i = 1,2, Ti is Y ri)-relaxed cocoercive and Mi-Lipschitz continuous, the condition ri > ym2 (i = 1,2) guarantees that for each i = 1,2, the operator Ti is (ri - YiM2)-strongly monotone. Similarly, sinceg is (y3, r3)-relaxed cocoercive and M3-Lipschitz continuous, the condition r3 > y3M2 implies that the operator g is (r3 - Y3M3)-strongly monotone.

In view of the above remarks, one can rewrite Theorem 6.2 as follows.

Theorem 6.5 Let x*, y* be the solution of SGMVID (6.1). Let T : H x H ^ H be ^-strongly monotone and M1-Lipschitz continuous in the first variable, and let T2: H x H ^ H be %2-strongly monotone and m2-Lipschitz continuous in the first variable. Further, letg be %3-strongly monotone and M3-Lipschitz continuous. If the constants p and p satisfy the following conditions:

|p - ^21 < ^2 ,

p - . ^,J$-M22k(2-k) m\ 1 m2 ,

& > MiVk(2 -k) (i = 1,2),

k = - 2%3 + m2 <1, 2%3 < 1 + m2,

and ^2ro=0 an = ro, then the iterative sequences {xn} and {yn} generated by Algorithm 6.1 converge strongly to x* andy*, respectively.

7 Conclusion

In this paper, we have introduced and considered a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID). We have proved the equivalence between the SGNMVID and the fixed point problem, and then by this equivalent formulation, discussed the existence and uniqueness of solution of the SGNMVID. This equivalence and three nearly uniformly Lipschitzian mappings Si (i = 1,2,3) are used to suggest and analyze some new three-step resolvent iterative schemes with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S1, S2, S3), which is the unique solution of the SGNMVID. Several special cases are also considered. In Section 6, an important remark on a subclass of relaxed cocoercive mappings is discussed. It is expected that the results proved in this paper may stimulate further research regarding the numerical methods and their applications in various fields of pure and applied sciences.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors contributed equally and significantly in this paper. Allauthors read and approved the finalmanuscript. Author details

1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand. 2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand. 3Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.

Acknowledgements

The first author is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.

Received: 9 October 2012 Accepted: 19 June 2013 Published: 16 July 2013

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doi:10.1186/1687-1812-2013-186

Cite this article as: Petrot and Balooee: Fixed point problems and a system of generalized nonlinear mixed variational inequalities. Fixed Point Theory and Applications 2013 2013:186.