Iterative Methods for Triple Hierarchical Variational Inequalities
with Mixed Equilibrium Problems, Variational Inclusions and
Variational Inequalities Constraints
Lu-Chuan Ceng
1, Yen-Cherng Lin
∗2, and Ching-Feng Wen
31Department of Mathematics, Shanghai Normal University, Shanghai 200234, China; and Scientific
Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China.
2
Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 40421, Taiwan. 3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
Email: L.-C. Ceng - [email protected]; Y.-C. Lin∗- [email protected]; C.-F. Wen - [email protected]; ∗Corresponding author
Abstract
In this paper, we introduce and analyze a step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converges strongly to a common element of the fixed point set of a strict pseudocontractive mapping, solution set of finitely many generalized mixed equilibrium problems, solution set of finitely many variational inclusions and solution set of general system of variational inequalities. Such element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions and general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many others.
Keywords: Triple hierarchical variational inequalities; Generalized mixed equilibrium problems; Variational
inclusions; General system of variational inequalities; Hybrid steepest-descent extragradient algorithm; Composite Mann-type viscosity algorithm; Fixed points.
AMS subject classifications. 49J30; 47H09; 47J20; 49M05
1
Introduction and FormulationsLet C be a nonempty closed convex subset of a real Hilbert space H and A : C → H be a nonlinear mapping on C. The variational inequality problem (VIP) defined by C and A is to find x ∈ C such that
The solution set of VIP (1.1) is denoted by VI(C,A). The theory of variational inequalities is well established area in nonlinear analysis and optimization. For further details on this topic, we refer to [25,38,34,31,47] and the references therein.
It is well known that if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (1.1) has a unique solution. Several iterative methods have been proposed in the literature to compute the approximate solutions of a VIP. Korpelevich’s extragradient method is one of them which was proposed by Korpelevich [30]. During the last two decades, this method received a great attention given by many authors, who improved and generalized it in different directions and ways, see, for example, [29,49,35,8,43, 7,33,22,19,18,11,16] and the references therein.
Let ϕ : C → R be a real-valued function, A : C → H be a nonlinear mapping and Θ : C × C → R be a bifunction.
The generalized mixed equilibrium problem (GMEP) is to find x ∈ C such that
Θ(x,y) + ϕ(y) − ϕ(x) + hAx,y − xi ≥ 0, ∀y ∈ C. (1.2)
We denote the set of solutions of GMEP (1.2) by GMEP( , ,AΘ ϕ ). The GMEP (1.2) is very general in the sense that it includes many problems as special cases, namely, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, etc. For different aspects and solution methods, we refer to [29,13,15,35,8,22,16] and the references therein.
If ϕ = 0, then GMEP (1.2) reduces to the generalized equilibrium problem (GEP) of finding x ∈ C such that
Θ(x,y) + hAx,y − xi ≥ 0, ∀y ∈ C.
It was considered and studied in [4,27] and further studied in [39]. The set of solutions of GEP is denoted by GEP( ,AΘ ).
If A ≡ 0, then GMEP (1.2) reduces to the mixed equilibrium problem (MEP) which is to find x ∈ C such that
Θ(x,y) + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C.
It was considered and studied in [21]. The set of solutions of MEP is denoted by MEP( ,Θ ϕ).
If ϕ ≡ 0, A ≡ 0, then GMEP reduces to the equilibrium problem (EP) which is to find x ∈ C such that
Θ(x,y) ≥ 0, ∀y ∈ C.
The set of solutions of EP is denoted by EP(Θ). It is worth to mention that the EP is an unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, etc.
The common assumptions on bifunction Θ : C × C → R are the following:
(A1) Θ(x,x) = 0 for all x ∈ C;
(A2) Θ is monotone, i.e., Θ(x,y) + Θ(y,x) ≤ 0 for any x,y ∈ C;
);
(A4) Θ(x,·) is convex and lower semicontinuous for each x ∈ C.
We also consider the following assumptions (B0) and (B1) or (B2) on the function ϕ : C → R:
(B0) ϕ is lower semicontinuous and convex;
(B1) For each x ∈ H and r > 0, there exists a bounded subset Dx ⊂ C and yx ∈ C such that for any
z ∈ C \ Dx,
0; or
(B2) C is a bounded set.
On the other hand, let B : C → H be a single-valued mapping and R be a multivalued mapping with D(R) = C. Consider the following variational inclusion: find x ∈ C such that
0 ∈ Bx + Rx. (1.3)
We denote by I(B,R) the solution set of the variational inclusion (1.3). In particular, if B ≡ R ≡ 0, then I(B,R) = C. If B ≡ 0, then problem (1.3) becomes the inclusion problem introduced by Rockafellar [36]. It is known that problem (1.3) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping R : D(R) ⊂ H → 2H be maximal monotone. We define the resolvent operator J
R, λ: H → D(R) associated
with R and λ by
JR, λ= (I + Rλ )−1, ∀x ∈ H,
where λ is a positive number.
Huang [28] studied problem (1.3) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with D(R) = C = H. Subsequently, Zeng et al. [48] further studied this problem in a more general setting than in [28]. Moreover, Ceng et al. [48] obtained the same strong convergence result as in [28]. In addition, Ceng et al. [48] also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, we refer to [23,7,50] and the references therein.
Let F1,F2 : C → H be two mappings. Consider the following general system of variational inequalities
(GSVI) of finding (x∗,y∗) ∈ C × C such that ½
hν1F1y∗+ x∗− y∗,x − x∗i ≥ 0, hν2F2x∗
+ y∗− x∗,x − y∗i ≥ 0,
∀x ∈ C,
where ν1 > 0 and ν2 > 0 are two constants. It was considered and studied in [14,43,19,18,11]. In particular, if F1
≡ F2 ≡ A, then the GSVI (1.4) reduces to the following problem of finding (x∗,y∗) ∈ C × C such that ½
hν1Ay∗∗++xy∗∗−− yx∗∗,x,x−−xy∗∗i ≥i ≥ 00,, ∀∀xx ∈∈ C,C, (1.5) hν2Ax
which is studied by Verma [40] and it is called a new system of variational inequalities (NSVI). Further, if x∗=
y∗additionally, then the NSVI reduces to the classical VIP (1.1). By considering G := P
C(I − ν1F1)PC(I − ν2F2) and
y∗= P
C(I − ν2F2)x∗, where PC denote the metric projection of H onto C, Ceng et al. [18] transformed GSVI (1.4)
into the following fixed point equation:
Gx∗= x∗. (1.6)
A variational inequality problem defined over the set of fixed points of a mapping is called hierarchical variational inequality problem.
Let S and T be two nonexpansive mappings. Yao et al. [44] considered the following hierarchical variational inequality problem (HVIP): find hierarchically a fixed point of T which is a solution to the VIP for monotone mapping I − S, namely, find ˜x Fix(∈ T) such that
h(I − S)x,p˜ − x˜i ≥ 0, ∀p Fix(∈ T). (1.7)
The solution set of HVIP (1.7) is denoted by Λ. It is easy to check that solving the HVIP (1.7) is equivalent to solving the fixed point problem of the composite mapping PFix(T)S, that is, find ˜x ∈ C such that x˜ = PFix(T)Sx˜.
Ceng et al. [44] introduced and analyzed an iterative algorithm for solving HVIP (1.7). They also studied the strong convergence of the sequences generated by their algorithm.
A variational inequality problem defined over the set of solutions of a hierarchical variational inequality problem is called a triple hierarchical variational inequality problem. For for further detail on triple hierarchical variational inequalities, we refer [2,9,10,12,8] and the references therein. Very recently, Kong et al. [29] introduced and studied the following triple hierarchical variational inequality problem (THVIP) (over the fixed point set of a strictly pseudocontractive mapping) with a variational inequality constraint.
Problem 1.1. [29, Problem II] Let F : C → H be κ-Lipschitzian and η-strongly monotone on the nonempty closed convex subset C of H, where κ and η are positive constants. Let A : C → H be a monotone and LLipschitzian mapping, V : C → H be a ρ-contraction with coefficient ρ [0∈ ,1), S : C → C be a nonexpansive mapping, and T : C → C be a ξ-strictly pseudocontractive mapping with Fix(T) ∩ VI(C,A) 6= . Let p∅
and 0 < γ ≤ τ, where τ = 1 − 1 − µ(2 η − µκ2). Then, the objective is to find x∗∈ Ξ such
that
h(µF − V γ )x∗,x − x∗i ≥ 0, ∀x ∈ ,Ξ (1.8)
where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding z∗∈ Fix(T) ∩ VI(C,A) such that
They proposed an algorithm for solving Problem 1.1 and studied the convergence analysis for the sequences generated by the proposed algorithm.
In this paper, we introduce and study the following triple hierarchical variational inequality problem (THVIP) (defined over the fixed point set of a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions and general system of variational inequalities:
Throughout the paper, M and N are assumed to be positive integers.
Problem 1.2. Assume that
(i) for each j = 1,2, Fj : C → H is ζj-inverse-strongly monotone and F : H → H is κ-Lipschitzian and
η-strongly monotone with positive constants , > κ η 0 such that 0 < γ ≤ τ and 0 where p
τ = 1 − 1 − µ(2 η − µκ2);
(ii) for each k {1∈ ,2,...,M}, Θk : C × C → R is a bifunction satisfying conditions (A1)–(A4) and ϕk : C → R ∪
{+∞} is a proper lower semicontinuous and convex function with restriction (B1) or (B2);
(iii) for k {1∈ ,2,...,M} and i {1∈ ,2,...,N}, Ri : C → 2H is a maximal monotone mapping, and Ak : H → H and Bi : C →
H are µk-inverse strongly monotone and ηi-inverse strongly monotone, respectively;
(iv) T : H → H is a ξ-strictly pseudocontractive mapping, S : H → H is a nonexpansive mapping and V : H → H is a ρ-contraction with coefficient ρ [0∈ ,1);
T
(v) VI(Ω,µF − Sγ ) =6 ∅ whereGMEP(GSVI(G)
Fix(T).
Then, the objective is to find x∗∈ Ξ such that
h(µF − V γ )x∗,x − x∗i ≥ 0, ∀x ∈ ,Ξ (1.10)
where Ξ := VI(Ω,µF − Sγ ), that is, the solution set of the following hierarchical variational inequality problem
(HVIP) of finding z∗∈ Ω such that
h(µF − Sγ )z∗,z − z∗i ≥ 0, ∀z ∈ Ω. (1.11)
Motivated and inspired by the above facts, we introduce and analyze two iterative methods for solving Problem 1.2. By combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method and Mann’s iteration method, we first propose a multi-step hybrid steepestdescent extragradient method. However, by combining Mann’s iteration method, Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method and projection method, we propose a multistep composite Mann-type viscosity iterative algorithm. We prove the strong convergence results for these methods. In particular, we prove that the proposed algorithms converge strongly to a common
el-T ementGMEP(GSVI(G) Fix(T) which is a unique solution of the THVIP 1.2. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions and GSVI (1.4). The results obtained in this paper improve and extend the corresponding results announced by many others.
2
PreliminariesThroughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by h·,·i and k·k, respectively. Let C be a nonempty closed convex subset of H. We write xn * x to indicate that the
sequence {xn} converges weakly to x and xn → x to indicate that the sequence {xn} converges strongly to x.
Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}, that is,
ωw(xn) := {x ∈ H : xni * x for some subsequence {xni} of {xn}}. Definition 2.1. A mapping A : C → H is called
(i) monotone if
hAx − Ay,x − yi ≥ 0, ∀x,y ∈ C;
(ii) η-strongly monotone if there exists a constant > η 0 such that
hAx − Ay,x − yi ≥ ηkx − yk2, ∀x,y ∈ C;
(iii) ζ-inverse-strongly monotone if there exists a constant > ζ 0 such that
It is easy to see that the projection PC is 1-inverse-strongly monotone. Inverse strongly monotone (also
referred to as co-coercive) operators have been applied widely in solving practical problems in various fields. It is obvious that if A is ζ-inverse-strongly monotone, then A is monotone and -Lipschitz continuous.
Moreover, we also have that, for all u,v ∈ C and > λ 0,
k(I − Aλ )u − (I − Aλ )vk2 = k(u − v) − λ(Au − Av)k2
= ku − vk2 − 2λhAu − Av,u − vi + λ2kAu − Avk2 (2.12) ≤ ku − vk2 +
λ( λ − 2ζ)kAu − Avk2.
So, if λ ≤ 2ζ, then I − A λ is a nonexpansive mapping from C to H.
The metric (or nearest point) projection from H onto C is the mapping PC : H → C which assigns to each
point x ∈ H the unique point PCx ∈ C satisfying the property
.
Some important properties of projections are gathered in the following proposition.
(ii) z = PCx k⇔ x − zk2 ≤ kx − yk2 − ky − zk2, ∀y ∈ C;
(iii) hPCx − PCy,x − yi ≥ kPCx − PCyk2, ∀y ∈ H. Consequently, PC is nonexpansive and monotone.
Definition 2.2. A mapping T : H → H is said to be:
(a) nonexpansive if
kTx − Tyk ≤ kx − yk, ∀x,y ∈ H;
(b) firmly nonexpansive if 2T − I is nonexpansive, or equivalently, if T is 1-inverse strongly monotone (1-ism),
hx − y,Tx − Tyi ≥ kTx − Tyk2, ∀x,y ∈ H;
alternatively, T is firmly nonexpansive if and only if T can be expressed as
,
where S : H → H is nonexpansive; projections are firmly nonexpansive. It can be easily seen that if T is nonexpansive, then I − T is monotone. Next we list some elementary conclusions for the MEP.
Proposition 2.2. [21] Assume that Θ : C × C → R satisfies (A1)-(A4) and let ϕ : C → R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a
mapping Tr( ,Θ ϕ) : H → C as follows:
for all x ∈ H. Then the following statements hold:
(i) For each x ∈ H, Tr( ,Θ ϕ)(x) is nonempty and single-valued;
(ii) Tr( ,Θ ϕ) is firmly nonexpansive, that is, for any x,y ∈ H,
;
(iii) Fix(Tr( ,Θ ϕ)) = MEP( ,Θ ϕ);
(iv) MEP( ,Θ ϕ) is closed and convex;
for all s,t > 0 and x ∈ H. Ceng et al. [18] transformed the GSVI (1.4) into a fixed point problem in the following way:
Proposition 2.3. [18] For given x,¯ y¯ ∈ C, (¯x,y¯) is a solution of the GSVI (1.4) if and only if x¯ is a fixed point of
Gx = PC(I − ν1F1)PC(I − ν2F2)x, ∀x ∈ C,
where y¯ = PC(I − ν2F2)¯x.
In particular, if the mapping Fj : C → H is ζj-inverse-strongly monotone for j = 1,2, then the mapping G is
nonexpansive provided νj ∈ (0,2ζj] for j = 1,2. We denote by GSVI(G) denote the fixed point set of the mapping
G.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below. Lemma 2.1. Let X be a real inner product space. Then there holds the following inequality
kx + yk2 ≤ kxk2 + 2hy,x + yi, ∀x,y ∈ X.
Lemma 2.2. Let H be a real Hilbert space. Then the following hold:
(a) kx − yk2 = kxk2 − kyk2 − 2hx − y,yi for all x,y ∈ H;
(b) k x λ + µyk2 = λkxk2 + µkyk2 − µλ kx − yk2 for all x,y ∈ H and ,µ λ ∈ ,1] with [0 λ + µ = 1;
(c) If {xn} is a sequence in H such that xn * x, it follows that
It is clear that, in a real Hilbert space H, T : C → C is ξ-strictly pseudocontractive if and only if the following inequality holds:
This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then -inverse strongly monotone; for further detail, we refer to [32] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.
Lemma 2.3. [32, Proposition 2.1] Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be a mapping.
(i) If T is a -strictly pseudocontractive mapping, then T satisfies the Lipschitzian conditionξ
(ii) If T is a -strictly pseudocontractive mapping, then the mapping I ξ − T is semiclosed at 0, that is, if {xn} is a
sequence in C such that xn * x˜ and (I − T)xn → 0, then (I − T)x˜ = 0.
(iii) If T is -(quasi-)strict pseudocontraction, then the fixed-point set ξ Fix(T) of T is closed and convex so
Lemma 2.4. [43] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a -strictlyξ pseudocontractive mapping. Let and be two nonnegative real numbers such that γ δ ( γ + δ) ξ ≤ . Thenγ
kγ(x − y) + δ(Tx − Ty)k ≤ ( γ + δ)kx − yk, ∀x,y ∈ C.
Lemma 2.5. [26, Demiclosedness principle]] Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C with Fix(S) 6= ∅. Then I − S is demiclosed. That is, whenever {xn} is a
sequence in C weakly converging to some x ∈ C and the sequence {(I − S)xn} strongly converges to some y, it
follows that (I − S)x = y. Here I is the identity operator of H.
Lemma 2.6. Let A : C → H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2.1 (i)) implies
u VI(∈ C,A) ⇔ u = PC(u − Auλ ), ∀ > λ 0.
Let C be a nonempty closed convex subset of a real Hilbert space H. Let λ be a number in (0,1] and let µ > 0. Associating with a nonexpansive mapping T : C → H, we define the mapping Tλ: C → H by
Tλx := Tx − µFλ (Tx), ∀x ∈ C,
where F : H → H is an operator such that, for some positive constants , > κ η 0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions:
kFx − Fyk ≤ κkx − yk and hFx − Fy,x − yi ≥ ηkx − yk2
for all x,y ∈ H.
Lemma 2.7. [41, Lemma 3.1] Tλis a contraction provided ; that is,
kTλx − Tλyk ≤ (1 − λτ)kx − yk, ∀x,y ∈ C,
p where τ = 1 − 1 −
µ(2 η − µκ2) (0∈ ,1].
Remark 2.1. (i) Since F is κ-Lipschitzian and η-strongly monotone on H, we get 0 < η ≤ κ. Hence, whenever 0
, we have which implies p 0 < 1 − 1 − 2µ η + µ2κ2 ≤ 1. p So, τ = 1 − 1 − µ(2 η − µκ2) (0∈ ,1].
.
Lemma 2.8. [41] Let {sn} be a sequence of nonnegative numbers satisfying the conditions
sn+1 ≤ (1 − αn)sn + αnβn, ∀n ≥ 1,
where {αn} and {βn} are sequences of real numbers such that
(i) {αn} [0⊂ ,1] and , or equivalently,
) = 0;
. Then, limn→∞ sn = 0.
Finally, recall that a set-valued mapping T : D(T) ⊂ H → 2H is called monotone if for all x,y ∈ D(T), f ∈ Tx and
g ∈ Ty imply
hf − g,x − yi ≥ 0.
A set-valued mapping T is called maximal monotone if T is monotone and (I+ Tλ )D(T) = H for each > λ 0, where I is the identity mapping of H. We denote by G(T) the graph of T. It is known that a monotone mapping
T is maximal if and only if, for (x,f) ∈ H × H, hf − g,x − yi ≥ 0 for every (y,g) ∈ G(T) implies f ∈ Tx. Next we provide an example to illustrate the concept of maximal monotone mapping.
Let A : C → H be a monotone, k-Lipschitz-continuous mapping and let NCv be the normal cone to C at v ∈ C,
that is,
NCv = {u ∈ H : hv − p,ui ≥ 0, ∀p ∈ C}.
Define
Then, Te is maximal monotone (see [36]) such that
0 ∈ Tve ⇐⇒ v VI(∈ C,A). (2.13)
Let R : D(R) ⊂ H → 2H be a maximal monotone mapping. Let ,µ > λ 0 be two positive numbers. Lemma 2.9. [6] There holds the resolvent identity
Remark 2.2. For ,µ > λ 0, there holds the following relation
(2.14) Indeed, whenever λ ≥ µ, utilizing Lemma 2.9 we deduce that
Similarly, whenever < µλ , we get
.
Combining the above two cases we conclude that (2.14) holds.
In terms of Huang [28] (see also [48]), there holds the following property for the resolvent operator
JR, λ: H → D(R).
Lemma 2.10. JR, λis single-valued and firmly nonexpansive, that is,
hJR,λx − JR,λy,x − yi ≥ kJR,λx − JR,λyk2, ∀x,y ∈ H.
Consequently, JR, λis nonexpansive and monotone.
Lemma 2.11. [7] Let R be a maximal monotone mapping with D(R) = C. Then for any given > λ 0, u ∈ C is a
solution of problem (1.5) if and only if u ∈ C satisfies
u = JR,λ(u − Buλ ).
Lemma 2.12. [48] Let R be a maximal monotone mapping with D(R) = C and let B : C → H be a strongly monotone, continuous and single-valued mapping. Then for each z ∈ H, the equation z (∈ B + Rλ )x has a unique solution xλfor > λ 0.
Lemma 2.13. [7] Let R be a maximal monotone mapping with D(R) = C and B : C → H be a monotone, continuous and single-valued mapping. Then (I + λ(R + B))C = H for each > λ 0. In this case, R + B is maximal monotone.
3
Hybrid Steepest-Descent Extragradient Method and Convergence ResultsIn this section, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm for finding a solution of THVIP 1.2. This algorithm is based on Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method and Mann’s iteration method. We prove the strong convergence of the proposed algorithm to a unique solution of THVIP 1.2 under suitable conditions. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem (HVIP).
Rest of the paper, unless otherwise specified, we assume that M and N are positive integers, and C is a nonempty closed convex subset of a real Hilbert space H.
Assumption 3.1. (a) For each k {1∈ ,2,...,M}, let Θk : C × C → R be a bifunction satisfying (A1)(A4) and ϕk : C → R
{+∞} be a proper lower semicontinuous and convex function with restriction (B1) or (B2). ∪
(b) For each k {1∈ ,2,...,M} and each i {1∈ ,2,...,N}, let Ri : C → 2H be a maximal monotone mapping, and Ak : H
→ H and Bi : C → H be µk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, (c) Let T : H → H be a ξ-strictly pseudocontractive mapping, S : H → H be a nonexpansive mapping and V : H
→ H be a ρ-contraction with coefficient ρ [0∈ ,1).
(d) For j = 1,2, let Fj : C → H be ζj-inverse-strongly monotone, and F : H → H be κ-Lipschitzian and
η-strongly monotone with positive constants , > κ η 0 such that 0 and 0 < γ ≤ τ, where p τ = 1 −
1 − µ(2 η − µκ2).
T
GMEP(GSVI(G) Fix(T).
(f) The solution set Ξ of HVIP (1.11) is nonempty.
(g) Let {αn}, {λn} (0⊂ ,1], {βn}, {γn}, {δn} [0⊂ ,1], {λi,n} [⊂ ai,bi] (0⊂ ,2ηi) and {rk,n} [⊂ ck,dk] (0⊂ ,2µk), where i
{1
∈ ,2,...,N} and k {1∈ ,2,...,M}.
We propose the following multi-step hybrid steepest-descent extragradient algorithm for finding a solution of THVIP 1.2.
Algorithm 3.1. For given arbitrarily x0 ∈ H, let {xn} be the sequence generated by
where G := PC(I − ν1F1)PC(I − ν2F2) with νj ∈ ,2ζ (0 j) for j = 1,2.
If V ≡ 0, then Algorithm 3.1 reduces to the following algorithm.
Algorithm 3.2. For given arbitrarily x0 ∈ H, let {xn} be the sequence generated by
u n = Tr(M,nΘM,ϕM)(I − rM,nAM)Tr(MΘ−M1−,n1,ϕM−1)(I − rM−1,nAM−1)···Tr(1Θ,n1,ϕ1)(I − r1,nA1)xn, vn = JRN,λN,n(I − λN,nBN)JRN−1,λN−1,n(I − λN−1,nBN−1)···JR1,λ1,n(I − λ1,nB1)un, yn = βnxn + γnGvn + δnTGvn, (3.16) xn+1 = λn(1 − αn) Sxγ n + (I − λnµF)yn, ∀n ≥ 0,
The following result provides the convergence of the sequences generated by the above algorithm.
Theorem 3.1. In addition to Assumption 3.1, suppose that
and ;
and ;
and
(v) βn + γn + δn = 1, (γn + δn) ξ ≤ γn (∀n ≥ 0), {βn} [⊂ a,b] (0⊂ ,1) and liminfn→∞ δn > 0;
(vi) kx − Txk ≥ k¯[d(x,Ω)]θ(∀x ∈ C) and lim
n→∞(λ1n/θ/αn) = 0 for some k, >θ ¯ 0.
Then, the following conclusions hold.
(a) If {Sxn} is bounded, then {xn} converges strongly to a point x∗∈ Ω which is a unique solution of Problem 1.2
provided that kyn − Txnk = o(αn).
(b) If {Sxn} is bounded, where {xn} is the sequence generated by (3.16), then {xn} converges strongly to a unique
solution x∗∈ of the following VIP provided that Ξ ky
n − Txnk = o(αn):
hFx∗,x − x∗i ≥ 0, ∀x ∈ .Ξ
Proof. Put
for all k {1∈ ,2,...,M} and n ≥ 1,
for all and , where I is the identity mapping on H. Then, we have
un = ∆Mn xn and vn = ΛNn un.
We divide rest of the proof into several steps.
Step 1. We prove that {xn} is bounded.
Indeed, take a fixed p ∈ Ω arbitrarily. Utilizing (2.12) and Proposition 2.2 (ii), we have kun − pk = kTr(M,nΘM,ϕM)(I − rM,nBM)∆Mn −1xn − Tr(M,nΘM,ϕM)(I − rM,nBM)∆Mn −1pk ≤ k(I − rM,nBM)∆Mn −1xn − (I − rM,nBM)∆Mn −1pk ≤ k∆Mn −1xn − ∆Mn −1pk ... ≤ k∆0nxn − ∆0npk = kxn − pk.
Utilizing (2.12) and Lemma 2.10, we have
kvn − pk = kJRN,λN,n(I − λN,nAN)ΛNn −1un − JRN,λN,n(I − λN,nAN)ΛNn −1pk ≤ k(I − λN,nAN)ΛNn −1un − (I − λN,nAN)ΛNn −1pk ≤ kΛNn −1un − ΛNn −1pk ... ≤ kΛ0nun − Λ0npk = kun − pk,
which together with the last inequality implies that
(3.17)
Since p = Gp = PC(I−ν1F1)PC(I−ν2F2)p, Fj is ζj-inverse-strongly monotone for j = 1,2, and 0 < νj ≤ 2ζj for j = 1,2, we
deduce that, for any n ≥ 0, kGvn − pk2 = kPC(I − ν1F1)PC(I − ν2F2)vn − PC(I − ν1F1)PC(I − ν2F2)pk2 ≤ k(I − ν1F1)PC(I − ν2F2)vn − (I − ν1F1)PC(I − ν2F2)pk2 = k[PC(I − ν2F2)vn − PC(I − ν2F2)p] − ν1[F1PC(I − ν2F2)vn − F1PC(I − ν2F2)p]k2 ≤ kPC(I − ν2F2)vn − PC(I − ν2F2)pk2 + ν1(ν1 − 2ζ1)kF1PC(I − ν2F2)vn − F1PC(I − ν2F2)pk2 (3.19) ≤ kPC(I − ν2F2)vn − PC(I − ν2F2)pk2 ≤ k(I − ν2F2)vn − (I − ν2F2)pk2 = k(vn − p) − ν2(F2vn − F2p)k2 ≤ kvn − pk2 + ν2(ν2 − 2ζ2)kF2vn − F2pk2 ≤ kvn − pk2.
(This shows that G : C → C is a nonexpansive mapping). Since (γn + δn) ξ ≤ γn for all n ≥ 0 and T is ξ-strictly
pseudocontractive, utilizing Lemma 2.4, we obtain from (3.15), (3.18) and (3.19) that kyn − pk = kβnxn + γnGvn + δnTGvn − pk = kβn(xn − p) + γn(Gvn − p) + δn(TGvn − p)k ≤ βnkxn − pk + kγn(Gvn − p) + δn(TGvn − p)k ≤ βnkxn − pk + (γn + δn)kGvn − pk (3.20) ≤ βnkxn − pk + (γn + δn)kvn − pk ≤ βnkxn − pk + (γn + δn)kxn − pk = kxn − pk,
Noticing the boundedness of {Sxn}, we get supn≥0 k Sxγ n −µFpk ≤ Mc for some M >c 0. Moreover, utilizing
Lemma 2.7, we deduce from (3.15), (3.20) and 0 < γ ≤ τ that for all n ≥ 0
By induction, we obtain
.
Thus, {xn} is bounded and so are the sequences {un}, {vn} and {yn}. Step 2. We prove that lim
(3.21)
where
for some M >f 0 and sup for some Mf0 > 0. Utilizing Proposition 2.2
(3.22)
where 0 is a constant such that for each n ≥ 0
.
Furthermore, define yn = βnxn + (1 − βn)wn for all n ≥ 0. It follows that
(3.23)
.
Since (γn + δn) ξ ≤ γn for all n ≥ 0, utilizing Lemma 2.4, we have
kγn+1(Gvn+1 − Gvn) + δn+1(TGvn+1 − TGvn)k ≤ (γn+1 + δn+1)kGvn+1 − Gvnk. (3.24)
Hence, it follows from (3.21) – (3.24) that
(3.25) In the meantime, simple calculation shows that
yn+1 − yn = βn(xn+1 − xn) + (1 − βn)(wn+1 − wn) + (βn+1 − βn)(xn+1 − wn+1). So, it follows
from (3.25) that
(3.26)
,
where sup for some
On the other hand, define zn := αnV xn + (1 − αn)Sxn for all n ≥ 0. Then, it is known that xn+1 =
λnγzn + (I − λnµF)yn for all n ≥ 0. Simple calculations show that
+ (I − λn+1µF)yn+1 − (I − λn+1µF)yn.
Since V is a ρ-contraction with coefficient ρ [0∈ ,1) and S is a nonexpansive mapping, we conclude that kzn+1 − znk ≤ |αn+1 − αn|kV xn − Sxnk + αn+1kV xn+1 − V xnk + (1 − αn+1)kSxn+1 − Sxnk
≤ |αn+1 − αn|kV xn − Sxnk + αn+1ρkxn+1 − xnk + (1 − αn+1)kxn+1 − xnk = (1 − αn+1(1
− ρ))kxn+1 − xnk + |αn+1 − αn|kV xn − Sxnk,
which together with (3.12) and 0 < γ ≤ τ, implies that
kxn+2 − xn+1k
≤ |λn+1 − λn|k zγ n − µFynk + λn+1γkzn+1 − znk + k(I − λn+1µF)yn+1 − (I − λn+1µF)ynk
≤ |λn+1 − λn|k zγ n − µFynk + λn+1γkzn+1 − znk + (1 − λn+1τ)kyn+1 − ynk ≤ |λn+1 − λn|k zγ n − µFynk + λn+1γ[(1 − αn+1(1 − ρ))kxn+1 − xnk XN + |αn+1 − αn|kV xn − Sxnk] + (1 − λn+1τ)[kxn+1 − xnk + Mf2( |λi,n+1 − λi,n| i=1 XM + |rk,n+1 − rk,n| + |γn+1 − γn| + |βn+1 − βn|)] k=1 zn+1 − zn = (αn+1 − αn)(V xn − Sxn) + αn+1(V xn+1 − V xn) +(1 − αn+1)(Sxn+1 − Sxn), xn+2 − xn+1 = (λn+1 − λn)( zγ n − µFyn) + λn+1γ(zn+1 − zn)
≤ (1 − λn+1( τ − γ) − λn+1αn+1γ(1 − ρ))kxn+1 − xnk + |λn+1 − λn|k zγ n − µFynk XN + |αn+1 − αn|kV xn − Sxnk + Mf2( |λi,n+1 − λi,n| i=1 XM + |rk,n+1 − rk,n| + |γn+1 − γn| + |βn+1 − βn|) k=1 XN XM ≤ (1 − λn+1αn+1γ(1 − ρ))kxn+1 − xnk + Mf3{ |λi,n+1 − λi,n| + |rk,n+1 − rk,n| i=1 k=1 + |λn+1 − λn| + |αn+1 − αn| + |βn+1 − βn| + |γn+1 − γn|},
where supn≥0{k zγ n − µFynk + kV xn − Sxnk + Mf2} ≤ Mf3 for some Mf3 > 0. Consequently,
where supn≥1{kxn − xn−1k + Mf3} ≤ Mf4 for some Mf4 > 0. From conditions (i)–(iv), it follows that
and
.
Thus, utilizing Lemma 2.8, we immediately conclude that
.
So, from αn → 0, it follows that
lim kxn+1 − xnk = 0. n→∞
Step 3. We prove that limn→∞ kxn − unk = 0, limn→∞ kxn − vnk = 0, limn→∞ kvn − Gvnk = 0 and limn→∞ kvn − Tvnk = 0.
(3.28) and hence,
(3.29) which together with {βn} [⊂ a,b] (0⊂ ,1), immediately yields
.
Since λn → 0, αn → 0, kxn+1 − xnk → 0 and {xn} is bounded, we have
lim kyn − xnk = 0.(3.30) n→∞
Observe that
≤ k(I − rk,nAk)∆kn−1xn − (I − rk,nAk)pk2
≤ k∆kn−1xn − pk2 + rk,n(rk,n − 2µk)kAk∆kn−1xn − Akpk2
≤ kxn − pk2 + rk,n(rk,n − 2µk)kAk∆kn−1xn − Akpk2, and
≤ kΛin−1un − pk2 + λi,n(λi,n − 2ηi)kBiΛin−1un − Bipk2 ≤ kun − pk2 + λi,n(λi,n − 2ηi)kBiΛin−1un − Bipk2 ≤ kxn − pk2 + λi,n(λi,n − 2ηi)kBiΛin−1un − Bipk2,
(3.32)
for i {1∈ ,2,...,N} and k {1∈ ,2,...,M}. Combining (3.28), (3.31) and (3.32), we get
which immediately leads to
(1 − βn)[rk,n(2µk − rk,n)kAk∆kn−1xn − Akpk2 + λi,n(2ηi − λi,n)kBiΛin−1un − Bipk2] ≤ kxn − pk2 − kyn − pk2
≤ kxn − ynk(kxn − pk + kyn − pk).
Since kxn − ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), {λi,n} [⊂ ai,bi] (0⊂ ,2ηi), {rk,n} [⊂ ck,dk] (0⊂ ,2µk), i {1∈ ,2,...,N},k ∈
{1,2,...,M} and {xn},{yn} are bounded sequences, we have
= 0 and lim ,
(3.33) for all k {1∈ ,2,...,M} and i {1∈ ,2,...,N}.
Furthermore, by Proposition 2.2 (ii) and Lemma 2.2 (a), we have
,
which implies that k∆knxn − pk2 ≤ k∆kn−1xn − pk2 − k∆nk−1xn − ∆knxn − rk,n(Ak∆kn−1xn − Akp)k2 = k∆kn−1xn − pk2 − k∆nk−1xn − ∆knxnk2 − rk,n2 kAk∆kn−1xn − Akpk2 + 2rk,nh∆kn−1xn − ∆knxn,Ak∆nk−1xn − Akpi ≤ k∆kn−1xn − pk2 − k∆nk−1xn − ∆knxnk2 + 2rk,nk∆kn−1xn − ∆knxnkkAk∆kn−1xn − Akpk ≤ kxn − pk2 − k∆kn−1xn − ∆knxnk2 + 2rk,nk∆nk−1xn − ∆knxnkkAk∆kn−1xn − Akpk. (3.34 )
which immediately leads to kΛinun − pk2
≤ kxn − pk2 − kΛin−1un − Λinun − λi,n(BiΛin−1un − Bip)k2
= kxn − pk2 − kΛin−1un − Λnkunk2 − λi,n2kBiΛin−1un − Bipk2 (3.35) + 2λi,nhΛni−1un − Λinun,BiΛin−1un − Bipi
≤ kxn − pk2 − kΛin−1un − Λinunk2 + 2λi,nkΛin−1un − ΛinunkkBiΛin−1un − Bipk.
Combining (3.28) and (3.35), we conclude that
which yields
(1 − βn)kΛin−1un − Λinunk2
≤ kxn − pk2 − kyn − pk2 + 2λi,nkΛin−1un − ΛniunkkBiΛin−1un − Bipk ≤ kxn − ynk(kxn −
pk + kyn − pk) + 2λi,nkΛin−1un − ΛinunkkBiΛin−1un − Bipk.
Since {βn} [⊂ a,b] (0⊂ ,1), {λi,n} [⊂ ai,bi] (0⊂ ,2ηi), i = 1,2,...,N, and {un},{xn} and {yn} are bounded sequences, we
deduce from (3.33) and kxn − ynk → 0 that
. (3.36)
Also, combining (3.17), (3.28) and (3.34), we deduce that
which yields
(1 − βn)k∆kn−1xn − ∆knxnk2
≤ kxn − pk2 − kyn − pk2 + 2rk,nk∆kn−1xn − ∆nkxnkkAk∆kn−1xn − Akpk ≤ kxn − ynk(kxn − pk + kyn − pk) +
2rk,nk∆kn−1xn − ∆knxnkkAk∆kn−1xn − Akpk. Since {βn} [⊂ a,b] (0⊂ ,1), {rk,n} [⊂ ck,dk] (0⊂ ,2µk) for k = 1,2,...,M, and
{xn},{yn} are bounded sequences, we deduce from (3.19) and kxn − ynk → 0 that
Hence, from (3.36) and (3.37), we get kxn − unk = k∆0nxn − ∆Mn xnk ≤ k∆0nxn − ∆1nxnk + k∆1nxn − ∆2nxnk + ··· + k∆Mn −1xn − ∆Mn xnk → 0 as n → ∞, and (3.38) ≤ kΛ0nun − Λ1nunk + kΛ1nun − Λ2nunk + ··· + kΛNn −1un − ΛNn unk → 0 as n → ∞,
respectively. Thus, from (3.38) and (3.39), we obtain
(3.39)
kxn − vnk ≤ kxn − unk + kun − vnk (3.40)
→ 0 as n → ∞.
On the other hand, for simplicity, we write ˜p = PC(I − ν2F2)p, ˜vn = PC(I − ν2F2)vn and kn = Gvn =
PC(I − ν1F1)v˜n for all n ≥ 1. Then,
p = Gp = PC(I − ν1F1)p˜ = PC(I − ν1F1)PC(I − ν2F2)p.
We now show that limn→∞ kGvn − vnk = 0, that is, limn→∞ kkn − vnk = 0. As a matter of fact, for p ∈ Ω, it follows
from (3.18), (3.19) and (3.28) that
(3.41) which immediately yields
(1 − βn)[ν2(2ζ2 − ν2)kF2vn − F2pk2 + ν1(2ζ1 − ν1)kF1v˜n − F1p˜k2]
≤ kxn − pk2 − kyn − pk2 ≤ kxn − ynk(kxn
− pk + kyn − pk).
Since kxn − ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), νj ∈ ,2ζ (0 j),j = 1,2, and {xn},{yn} are bounded sequences, we have
lim kF2vn − F2pk = 0 and lim kF1v˜n − F1p˜k = 0. (3.42) n→∞ n→∞
Also, in terms of the firm nonexpansivity of PC and the ζj-inverse strong monotonicity of Fj for j = 1,2, we obtain
from νj ∈ ,2ζ (0 j), j = 1,2 and (3.19) that
Thus, we have kv˜n−p˜k2 ≤ kvn−pk2−k(vn−v˜n)−(p−p˜)k2+2ν2h(vn−v˜n)−(p−p˜),F2vn−F2pi−ν22kF2vn−F2pk2, (3.43)
and kkn − pk2 ≤ kvn − pk2 − k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k. (3.44)
Consequently, from (3.18), (3.41) and (3.43), it follows that
kyn − pk2 ≤ βnkxn − pk2 + (1 − βn)[kv˜n − p˜k2 + ν1(ν1 − 2ζ1)kF1v˜n − F1p˜k2] ≤ βnkxn − pk2 + (1 − βn)kv˜n − p˜k2 ≤ βnkxn − pk2 + (1 − βn)[kvn − pk2 − k(vn − v˜n) − (p − p˜)k2 + 2ν2h(vn − v˜n) − (p − p˜),F2vn − F2pi − ν22kF2vn − F2pk2] ≤ βnkxn − pk2 + (1 − βn)[kxn − pk2 − k(vn − v˜n) − (p − p˜)k2 + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk] ≤ kxn − pk2 − (1 − βn)k(vn − v˜n) − (p − p˜)k2 + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk,
which hence leads to
(1 − βn)k(vn − v˜n) − (p − p˜)k2
≤ kxn − pk2 − kyn − pk2 + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk ≤ kxn − ynk(kxn − pk
+ kyn − pk) + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk.
Since kxn −ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), ν2 ∈ (0,2ζ2), and {xn}, {yn}, {vn}, {v˜n} are bounded sequences, we obtain
from (3.42) that
lim k(vn − v˜n) − (p − p˜)k = 0. (3.45) n→∞
Furthermore, from (3.18), (3.41) and (3.44), it follows that kyn − pk2 ≤ βnkxn − pk2 + (1 − βn)kkn − pk2 ≤ βnkxn − pk2 + (1 − βn)[kvn − pk2 − k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k] ≤ βnkxn − pk2 + (1 − βn)[kxn − pk2 − k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k] = kxn − pk2 − (1 − βn)k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k,
which hence yields
(1 − βn)k(v˜n − kn) + (p − p˜)k2
≤ kxn − pk2 − kyn − pk2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k ≤ kxn − ynk(kxn −
pk + kyn − pk) + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k.
Since kxn−ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), ν1 ∈ (0,2ζ1), and {xn}, {yn}, {kn}, {v˜n} are bounded sequences, we obtain
from (3.42) that
lim k(v˜n − kn) + (p − p˜)k = 0. (3.46) n→∞
kvn − knk ≤ k(vn − v˜n) − (p − p˜)k + k(v˜n − kn) + (p − p˜)k.
Hence, from (3.45) and (3.46), we get
lim kvn − Gvnk = lim kvn − knk = 0. (3.47) n→∞ n→∞
Also, observe that
yn − xn = γn(Gvn − xn) + δn(TGvn − xn), ∀n ≥ 0. Hence, we obtain δnkTGvn − vnk ≤ δnkTGvn − xnk + δnkxn − vnk = kyn − xn − γn(Gvn − xn)k + δnkxn − vnk ≤ kyn − xnk + γnkGvn − xnk + δnkxn − vnk ≤ kyn − xnk + γnkGvn − vnk + γnkvn − xnk + δnkxn − vnk = kyn − xnk + γnkGvn − vnk + (γn + δn)kxn − vnk ≤ kyn − xnk + kGvn − vnk + kxn − vnk.
So, from liminfn→∞ δn > 0, (3.30), (3.40) and (3.47), it follows that
lim kTGvn − vnk = 0. (3.48)
n→∞
In addition, noticing that
kTvn − vnk ≤ kTvn − TGvnk + kTGvn − vnk ≤ kvn − Gvnk
+ kTGvn − vnk,
we know from (3.47) and (3.48) that
lim kTvn − vnk = 0. (3.49
)
n→∞ Step 4. We prove that ωw(xn) ⊂ Ω.
Indeed, since H is reflexive and {xn} is bounded, there exists at least a weak convergence subsequence of
{xn}. Hence it is known that ωw(xn) 6= . Now, take an arbitrary ∅ w ∈ ωw(xn). Then there exists a subsequence
{xni} of {xn} such that xni * w. From (3.36) – (3.38) and (3.40) we have that uni * w,
and , where m {1∈ ,2,...,N} and k {1∈ ,2,...,M}. Utilizing Lemma 2.3 (ii), we deduce from vni * w and (3.49) that w Fix(∈ T). In the meantime, utilizing Lemma 2.5, we obtain from vni * w
and (3.47) that w GSVI(∈ G). Next, we prove that ). As a matter of fact,
since Bm is ηm-inverse strongly monotone, Bm is a monotone and Lipschitz continuous mapping. It follows
from Lemma 2.13 that Rm +Bm is maximal monotone. Let (v,g) ∈ G(Rm +Bm), that is, g −Bmv ∈ Rmv. Again, since
Λm
n un = JRm,λm,n(I − λm,nBm)Λmn −1un, n ≥ 1 , m {1∈ ,2,...,N}, we have
,
that is,
In terms of the monotonicity of Rm, we get , and hence, In particular, . Since kΛm
n un − Λmn −1unk → 0 (due to (3.36)) and kBmΛnmun − BmΛmn −1unk → 0 (due to the Lipschitz continuity of
Bm), we conclude from and {λi,n} [⊂ ai,bi] (0⊂ ,2ηi) that
.
It follows from the maximal monotonicity of Bm + Rm that 0 (∈ Rm + Bm)w, that is, w I(∈ Bm,Rm).
Therefore, ). Next we prove that GMEP(Θk,ϕk,Ak). Since
, we have
.
By (A2), we have
.
Let zt = ty + (1 − t)w for all t (0∈ ,1] and y ∈ C. This implies that zt ∈ C. Then, we have
(3.50)
.
By (3.37), we have . Furthermore, by the monotonicity of Ak, we
obtain 0. Then, by (A4), we obtain
hzt − w,Akzti ≥ ϕk(w) − ϕk(zt) + Θk(zt,w). (3.51)
Utilizing (A1), (A4) and (3.51), we have
0 = Θk(zt,zt) + ϕk(zt) − ϕk(zt)
≤ tΘk(zt,y) + (1 − t)Θk(zt,w) + tϕk(y) + (1 − t)ϕk(w) − ϕk(zt)
≤ t[Θk(zt,y) + ϕk(y) − ϕk(zt)] + (1 − t)hzt − w,Akzti
and hence,
0 ≤ Θk(zt,y) + ϕk(y) − ϕk(zt) + (1 − t)hy − w,Akzti.
Letting t → 0, we have, for each y ∈ C,
0 ≤ Θk(w,y) + ϕk(y) − ϕk(w) + hy − w,Akwi.
This implies that w ∈ GMEP(Θk,ϕk,Ak), and hence, GMEP(Θk,ϕk,Ak). Thus,
GMEP( ). Consequently, w ∈
T
GMEP(GSVI(G)Fix(T) =: Ω. This shows that ωw(xn) ⊂ Ω. Step 5. We prove that ωw(xn) ⊂ Ξ.
Indeed, take an arbitrary w ∈ ωw(xn). Then, there exists a subsequence {xni} of {xn} such that xni * w. Utilizing (3.29), we obtain that for all p ∈ Ω
,
which implies that
Since αn → 0, kxn − xn+1k → 0 and
,
from (3.52), we conclude that
h(µF − Sγ )p,w − pi = lim h(µF − Sγ )p,xni − pi i→∞ ≤ limsuph(µF − Sγ )p,xn − pi n→∞ ≤ 0, ∀p ∈ Ω, that is, h(µF − Sγ )p,w − pi ≤ 0, ∀p ∈ Ω. (3.53)
In addition, observe that
(3.54) and h(µF − Sγ )x − (µF − Sγ )y,x − yi = µhFx − Fy,x − yi − γhSx − Sy,x − yi
≥ µηkx − yk2 − γkx − yk2
= (µ η − γ)kx − yk2, ∀x,y ∈ H.
Since 0 < γ ≤ τ and κ ≥ η, we know that µ η ≥ τ ≥ γ and hence the mapping µF − S γ is monotone. Moreover, it is
h(µF − Sγ )w,p − wi ≥ 0, ∀p ∈ Ω. (3.55) This shows that w VI(∈ Ω,µF − Sγ ) =: Ξ. Thus, we derive ωw(xn) ⊂ Ξ according to the arbitrariness of w.
Step 6. We prove that limn→∞ kxn −x∗k = 0 provided kyn −Txnk = o(αn), where {x∗} = VI( ,µF Ξ − V γ ).
Indeed, it is clear that µF − V γ is (µ η − γρ)-strongly monotone and (µ κ + γρ)-Lipschitzian. Then it is
known that VI( ,µF Ξ − V γ ) is a singleton and hence we write VI( ,µF Ξ − V γ ) = {x∗}, that is, VI(VI(Ω,µF − Sγ ),µF
− V γ ) = {x∗}.
Utilizing (3.20) and (3.29) with p = x∗, we get
kxn+1 − x∗k2 ≤ [λn(αnγρkxn − x∗k + (1 − αn)γkxn − x∗k) + (1 − λnτ)kyn − x∗k]2 + 2λnαnh( V xγ ∗− µFx∗),xn+1 − x∗i + 2(1 − αn)λnh( Sxγ ∗− µFx∗),xn+1 − x∗i ≤ [λnγ(1 − αn(1 − ρ))kxn − x∗k + (1 − λnτ)kxn − x∗k]2 + 2λnαnh( V xγ ∗− µFx∗),xn+1 − x∗i + 2(1 − αn)λnh( Sxγ ∗− µFx∗),xn+1 − x∗i = [(1 − λn( τ − γ) − λnαnγ(1 − ρ))kxn − x∗k]2 + 2λnαnh( V γ − µF)x∗,xn+1 − x∗i (3.56) + 2(1 − αn)λnh( Sγ − µF)x∗,x n+1 − x∗i ≤ (1 − λn( τ − γ) − λnαnγ(1 − ρ))kxn − x∗k2 + 2λnαnh( V γ − µF)x∗,xn+1 − x∗i + 2(1 − αn)λnh( Sxγ ∗− µFx∗),xn+1 − x∗i ≤ (1 − λnαnγ(1 − ρ))kxn − x∗k2 + 2λnαnh( V γ − µF)x∗,xn+1 − x∗i + 2(1 − αn)λnh( S γ − µF)x∗,xn+1 − x∗i.
Now let us show that
. (3.57)
In fact, we may assume, without loss of generality, that there exists a subsequence {xnj} of {xn} such that xnj * xˆ and
. (3.58)
In terms of the fact that ωw(xn) ⊂ Ξ, we get ˆx ∈ Ξ. Since VI( ,µF Ξ − V γ ) = {x∗}, it is easy from (3.58) to see that
limsuph( V γ − µF)x∗,x
n − x∗i = h( V γ − µF)x∗,xˆ − x∗i ≤ 0, n→∞
that is, (3.57) holds.
In addition, from x∗∈ Ξ and condition (vi) we obtain that
(3.59) Utilizing Lemma 2.3 (i) we have
where sup for some 0. Hence, for a big enough constant
k¯1 > 0, from (3.59), we have
(3.61) Combining (3.56) and (3.61), we get
(3.62)
Since ) and 0, we conclude from (3.57)
and the assumption kyn − Txnk = o(αn) that ,
,
and
.
Therefore, applying Lemma 2.8 to (3.62) we infer that limn→∞ kxn − x∗k = 0. The proof of part (a) is complete.
It is easy to see that part (b) now becomes a straightforward consequence of part (a) since, if V ≡ 0, THVIP (1.10) reduces to the VIP in part (b). This completes the proof.
Remark 3.1. It is clear that the iterative scheme (3.15) is different from the one considered in
[44, 29]. We extend three-step iterative scheme in [29, Algorithm I] to four-step iterative scheme for THVIP (1.10) by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method [42] and Mann’s iteration method. It is worth pointing out that under the lack of the assumptions similar to those in [44, Theorem 3.2], for example, {xn} is bounded and Fix(T) ∩ intC 6= , the∅
sequence {xn} generated by (3.15) converges strongly to a point x∗∈
T ∗∈ Ξ of
GMEP(GSVI(G)Fix(T) =: Ω, which is a unique solution x
THVIP (1.10) (over the fixed point set of strictly pseudocontractive mapping T), that is, h(µF − V γ )x∗,p− x∗i ≥
0,∀p ∈ Ξ. We note that the nonexpansive mapping T in [44] is extended to strictly pseudocontractive mapping
Remark 3.2. Theorem 3.1 improves and extends Theorems 3.1 and 3.2 in [44] and Theorem 14 in [29] in the following aspects:
(a) THVIP (1.10) with the unique solution x∗∈ Ω satisfying
GMEP( GSVI(G)∩Fix(T)(I − (µF − Sγ ))x∗
is more general than the problem of finding a point ˜x ∈ C satisfying ˜x = PFix(T)Sx˜ in [44] and than the
problem of finding a point x∗∈ Fix(T)∩VI(C,A) satisfying x∗= P
Fix(T)∩VI(C,A)(I−(µF − Sγ ))x∗in [29, Theorem
14]. It is worth to point out that S is nonexpansive if and only if the complement -inverse
strongly monotone; see [19].
(b) Four-step iterative scheme (3.15) for THVIP (1.10) is more flexible and subtle than three-step iterative scheme considered in [29, Algorithm I] and than two-step iterative scheme studied in [44] because it can be used to solve several kinds of problems, for example, THVIP, HVIP and problem of finding a
common point of four sets: GMEP( ), GSVI(G) and Fix(T). In addition,
Theorem 3.1 drops the crucial requirements in [44, Theorem 3.2] that lim = 0,
= 0, Fix(T) ∩ intC 6= and {∅ xn} is bounded, and also removes the crucial ones in [29,
Theorem 14] that ) = 0 and kxn+1 − xnk + kxn − znk = o(λ2n). In the
meantime, Problem 1.1 (that is, Problem II in [29]) is extended and generalized to the setting of GSVI (1.4), finitely many GMEPs and finitely many variational inclusions in Problem 1.2.
(c) The technique used in Theorem 3.1 is different from the one used in [44, Theorems 3.1 and 3.2] and in [29, Theorem 14] because we used the properties of strictly pseudocontractive mappings (see Lemmas
2.3 and 2.4), the properties of resolvent operators and maximal monotone mappings (see Proposition
2.2, Remark 2.2 and Lemmas 2.9 – 2.13, the fixed point equation x∗= P
C(I − ν1F1)PC(I − ν2F2)x∗equivalent
to GSVI (1.4) (see Proposition 2.3 and contractive coefficient estimates for contractions associating with nonexpansive mappings (see Lemma 2.7).
(d) Compared with the restrictions on the parameter sequences in [44, Theorem 3.2] and [29, Theorem 14], respectively, the hypotheses (iii)–(iv) in Theorem 3.1 are additionally added because Theorem 3.1
involves the quite complex problem, that is, THVIP (1.10) (over the fixed point set Fix(T) of strictly pseudocontractive mapping T) with constraints of several problems: GSVI (1.4), finitely many GMEPs and finitely many variational inclusions.
We prove the strong convergence of the proposed algorithm to a unique solution of THVI Problem 1.2.
Theorem 3.2. In addition to Assumption 3.1, suppose that
(i) limn→∞ λn = 0, limn→∞ αn = 0 and ;
(iii) liminfn→∞ δn > 0 and {βn} [⊂ a,b] (0⊂ ,1);
(iv) limn→∞(λn/αn2) = 0 and kx − Txk ≥ k¯ · d(x,Ω),∀x ∈ C, for some k¯.
If {Sxn} is bounded, then
(a) ωw(xn) ⊂ Ω provided kxn − xn+1k → 0 (n → ∞);
(b) ωw(xn) ⊂ provided Ξ kxn − xn+1k = o(λn);
(c) {xn} converges strongly to a unique solution of Problem 1.2 provided kxn − xn+1k + kxn − Txnk =
.
Proof. Since the solution set Ξ of the HVIP (1.11) is nonempty, it is known that Ω 6= . As in the proof of∅ Theorem 3.1, put
for all k {1∈ ,2,...,M} and n ≥ 1,
for all i {1∈ ,2,...,N}, ∆0
n = I and Λ0n = I, where I is the identity mapping on H. Then, we have that un = ∆Mn xn and
vn = ΛNn un.
Rest of the proof is divided into several steps.
Step 1. As in the proof (Step 1) of Theorem 3.1, {xn} is bounded. Step 2. We prove that ωw(xn) ⊂ Ω provided kxn − xn+1k → 0 (n → ∞);.
Indeed, we first show that limn→∞ kxn − unk = 0, limn→∞ kxn − vnk = 0, limn→∞ kvn − Gvnk = 0 and limn→∞ kvn − Tvnk
= 0.
As a matter of fact, utilizing Lemmas 2.1 and 2.2 (b), from (3.15), (3.18), (3.19) and 0 < γ ≤ τ, we deduce
(3.63) and hence,
(3.64) which together with {βn} [⊂ a,b] (0⊂ ,1) yields
Since αn → 0, λn → 0, kxn+1 − xnk → 0 and {xn} is bounded, we have
Observe that lim kyn − xnk = 0. n→∞ (3.65) ≤ k(I − rk,nAk)∆kn−1xn − (I − rk,nAk)pk2 ≤ k∆kn−1xn − pk2 + rk,n(rk,n − 2µk)kAk∆kn−1xn − Akpk2 ≤ kxn − pk2 + rk,n(rk,n − 2µk)kAk∆kn−1xn − Akpk2, (3.66) and
= kJRi,λi,n(I − λi,nBi)Λin−1un − JRi,λi,n(I − λi,nBi)pk2 ≤ k(I − λi,nBi)Λin−1un − (I − λi,nBi)pk2
≤ kΛin−1un − pk2 + λi,n(λi,n − 2ηi)kBiΛin−1un − Bipk2 ≤ kun − pk2 + λi,n(λi,n − 2ηi)kBiΛin−1un − Bipk2 ≤ kxn − pk2 + λi,n(λi,n − 2ηi)kBiΛin−1un − Bipk2,
(3.67)
which immediately leads to
(1 − βn)[rk,n(2µk − rk,n)kAk∆kn−1xn − Akpk2 + λi,n(2ηi − λi,n)kBiΛin−1un − Bipk2] ≤ kxn − pk2 − kyn − pk2
≤ kxn − ynk(kxn − pk + kyn − pk).
Since kxn − ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), {λi,n} [⊂ ai,bi] (0⊂ ,2ηi), {rk,n} [⊂ ck,dk] (0⊂ ,2µk), i {1∈ ,2,...,N}, k ∈
{1,2,...,M} and {xn}, {yn} are bounded sequences, we have
= 0 and lim ,
(3.68) for all k {1∈ ,2,...,M} and i {1∈ ,2,...,N}.
Furthermore, by Proposition 2.2 (ii) and Lemma 2.2 (a), we have
,
which implies that k∆knxn − pk2 ≤ k∆kn−1xn − pk2 − k∆nk−1xn − ∆knxn − rk,n(Ak∆kn−1xn − Akp)k2 = k∆kn−1xn − pk2 − k∆nk−1xn − ∆knxnk2 − rk,n2 kAk∆kn−1xn − Akpk2 + 2rk,nh∆kn−1xn − ∆knxn,Ak∆nk−1xn − Akpi ≤ k∆kn−1xn − pk2 − k∆nk−1xn − ∆knxnk2 + 2rk,nk∆nk−1xn − ∆knxnkkAk∆kn−1xn − Akpk ≤ kxn − pk2 − k∆kn−1xn − ∆knxnk2 + 2rk,nk∆nk−1xn − ∆knxnkkAk∆kn−1xn − Akpk.
By Lemma 2.2 (a) and Lemma 2.10, we obtain
(3.69)
which immediately leads to kΛinun − pk2
≤ kxn − pk2 − kΛin−1un − Λinun − λi,n(BiΛin−1un − Bip)k2
≤ kxn − pk2 − kΛin−1un − Λinunk2 + 2λi,nkΛin−1un − ΛinunkkBiΛin−1un − Bipk.
Combining (3.63) and (3.70), we have
which yields
(1 − βn)kΛin−1un − Λinunk2
≤ kxn − pk2 − kyn − pk2 + 2λi,nkΛin−1un − ΛniunkkBiΛin−1un − Bipk ≤ kxn − ynk(kxn −
pk + kyn − pk) + 2λi,nkΛin−1un − ΛinunkkBiΛin−1un − Bipk.
Since {βn} [⊂ a,b] (0⊂ ,1), {λi,n} [⊂ ai,bi] (0⊂ ,2ηi),i = 1,2,...,N, and {un}, {xn} and {yn} are bounded sequences, we
deduce from (3.68) and kxn − ynk → 0 that
. (3.71)
Also, combining (3.17), (3.70) and (3.34)?, we deduce that
which yields
(1 − βn)k∆kn−1xn − ∆knxnk2
≤ kxn − pk2 − kyn − pk2 + 2rk,nk∆kn−1xn − ∆nkxnkkAk∆kn−1xn − Akpk ≤ kxn − ynk(kxn − pk +
kyn − pk) + 2rk,nk∆kn−1xn − ∆knxnkkAk∆kn−1xn − Akpk.
Since {βn} [⊂ a,b] (0⊂ ,1), {rk,n} [⊂ ck,dk] (0⊂ ,2µk) for k = 1,2,...,M, and {xn}, {yn} are bounded sequences, we
deduce from (3.68) and kxn − ynk → 0 that
. (3.72)
Hence, from (3.71) and (3.30), we get
≤ k∆0nxn − ∆1nxnk + k∆1nxn − ∆2nxnk + ··· + k∆Mn −1xn − ∆Mn xnk → 0 as n → ∞, and (3.73 ) ≤ kΛ0nun − Λ1nunk + kΛ1nun − Λ2nunk + ··· + kΛNn −1un − ΛNn unk → 0 as n → ∞,
respectively. Thus, from (3.73) and (3.32), we obtain
(3.74 )
kxn − vnk ≤ kxn − unk + kun − vnk (3.75
→ 0 as n → ∞.
On the other hand, for simplicity, we write ˜p = PC(I − ν2F2)p, ˜vn = PC(I − ν2F2)vn and kn = Gvn =
PC(I − ν1F1)v˜n for all n ≥ 1. Then,
p = Gp = PC(I − ν1F1)p˜ = PC(I − ν1F1)PC(I − ν2F2)p.
We now show that limn→∞ kGvn − vnk = 0, that is, limn→∞ kkn − vnk = 0.
As a matter of fact, for p ∈ Ω, it follows from (3.18), (3.19) and (3.63) that
(3.76) which immediately yields
(1 − βn)[ν2(2ζ2 − ν2)kF2vn − F2pk2 + ν1(2ζ1 − ν1)kF1v˜n − F1p˜k2]
≤ kxn − pk2 − kyn − pk2 ≤ kxn − ynk(kxn
− pk + kyn − pk).
Since kxn − ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), νj ∈ ,2ζ (0 j),j = 1,2, and {xn},{yn} are bounded sequences, we have
lim kF2vn − F2pk = 0 and lim kF1v˜n − F1p˜k = 0. (3.77) n→∞ n→∞
Also, in terms of the firm nonexpansivity of PC and the ζj-inverse strong monotonicity of Fj for j = 1,2, we obtain
from νj ∈ ,2ζ (0 j),j = 1,2 and (3.19) that
and
Thus, we have kv˜n−p˜k2 ≤ kvn−pk2−k(vn−v˜n)−(p−p˜)k2+2ν2h(vn−v˜n)−(p−p˜),F2vn−F2pi−ν22kF2vn−F2pk2, (3.78)
and kkn − pk2 ≤ kvn − pk2 − k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k. (3.79) Consequently,
kyn − pk2 ≤ βnkxn − pk2 + (1 − βn)[kv˜n − p˜k2 + ν1(ν1 − 2ζ1)kF1v˜n − F1p˜k2] ≤ βnkxn − pk2 + (1 − βn)kv˜n − p˜k2 ≤ βnkxn − pk2 + (1 − βn)[kvn − pk2 − k(vn − v˜n) − (p − p˜)k2 + 2ν2h(vn − v˜n) − (p − p˜),F2vn − F2pi − ν22kF2vn − F2pk2] ≤ βnkxn − pk2 + (1 − βn)[kxn − pk2 − k(vn − v˜n) − (p − p˜)k2 + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk] ≤ kxn − pk2 − (1 − βn)k(vn − v˜n) − (p − p˜)k2 + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk,
which hence leads to
(1 − βn)k(vn − v˜n) − (p − p˜)k2
≤ kxn − pk2 − kyn − pk2 + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk ≤ kxn − ynk(kxn − pk
+ kyn − pk) + 2ν2k(vn − v˜n) − (p − p˜)kkF2vn − F2pk.
Since kxn−ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), ν2 ∈ (0,2ζ2), and {xn}, {yn}, {vn}, {v˜n} are bounded sequences, we obtain
from (3.77) that
lim k(vn − v˜n) − (p − p˜)k = 0. (3.80) n→∞
Furthermore, from (3.18), (3.76) and (3.79), it follows that kyn − pk2 ≤ βnkxn − pk2 + (1 − βn)kkn − pk2 ≤ βnkxn − pk2 + (1 − βn)[kvn − pk2 − k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k] ≤ βnkxn − pk2 + (1 − βn)[kxn − pk2 − k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k] = kxn − pk2 − (1 − βn)k(v˜n − kn) + (p − p˜)k2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k,
which hence yields
(1 − βn)k(v˜n − kn) + (p − p˜)k2
≤ kxn − pk2 − kyn − pk2 + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k ≤ kxn − ynk(kxn −
pk + kyn − pk) + 2ν1kF1v˜n − F1p˜kk(v˜n − kn) + (p − p˜)k.
Since kxn−ynk → 0, {βn} [⊂ a,b] (0⊂ ,1), ν1 ∈ (0,2ζ1), and {xn}, {yn}, {kn}, {v˜n} are bounded sequences, we obtain
from (3.77) that
lim k(v˜n − kn) + (p − p˜)k = 0. (3.81) n→∞
Note that
kvn − knk ≤ k(vn − v˜n) − (p − p˜)k + k(v˜n − kn) + (p − p˜)k.
Hence, from (3.80) and (3.81), we get
lim kvn − Gvnk = lim kvn − knk = 0. (3.82) n→∞ n→∞
Also, observe that
yn − xn = γn(Gvn − xn) + δn(TGvn − xn), ∀n ≥ 0.
Hence, we find
δnkTGvn − vnk ≤ δnkTGvn − xnk + δnkxn − vnk
= kyn − xn − γn(Gvn − xn)k + δnkxn − vnk
≤ kyn − xnk + γnkGvn − vnk + γnkvn − xnk + δnkxn − vnk = kyn − xnk
+ γnkGvn − vnk + (γn + δn)kxn − vnk ≤ kyn − xnk + kGvn − vnk + kxn
− vnk.
So, from liminfn→∞ δn > 0, (3.65), (3.75) and (3.82), it follows that
lim kTGvn − vnk = 0. (3.83) n→∞
In addition, noticing that
kTvn − vnk ≤ kTvn − TGvnk + kTGvn − vnk ≤ kvn − Gvnk
+ kTGvn − vnk,
we know from (3.82) and (3.83) that
lim kTvn − vnk = 0. (3.84) n→∞ Secondly, we show that ωw(xn)
⊂ Ω.
In fact, since H is reflexive and {xn} is bounded, there exists at least a weak convergence subsequence of
{xn}. Hence, it is known that ωw(xn) =6 . Now, take an arbitrary ∅ w ∈ ωw(xn). Then there exists a subsequence
{xni} of {xn} such that xni * w. From (3.71) – (3.73) and (3.75), we have uni * w, vni * w,
and , where m {1∈ ,2,...,N} and k {1∈ ,2,...,M}. Utilizing Lemma 2.3 (ii), we deduce from vni * w and (3.84) that w Fix(∈ T). In the meantime, utilizing Lemma 2.5, we obtain from vni * w and (3.82) that w GSVI(∈ G).
Next, we prove that ). As a matter of fact, since Bm is ηm-inverse strongly monotone, Bm
is a monotone and Lipschitz continuous mapping. It follows from Lemma 2.13 that Rm +Bm is maximal
monotone. Let (v,g) ∈ G(Rm + Bm), that is, g − Bmv ∈ Rmv. Again, since
, we have
,
that is,
.
In terms of the monotonicity of Rm, we get
,
and hence,
In particular,
.
