Ceng et al. Journal of Inequalities and Applications (2015) 2015:150
DOI 10.1186/s13660-015-0646-z
R E S E A R C H Open Access
Hybrid extragradient viscosity method for general system of variational inequalities
Lu-Chuan Ceng1,2, Yeong-Cheng Liou3,4*, Ching-Feng Wen4 and Yuh-Jenn Wu5
*Correspondence: mailto:[email protected]
Web End [email protected]
3Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan
4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, TaiwanFull list of author information is available at the end of the article
Abstract
In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for nding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of a general system of variational inequalities, the set of solutions of a split feasibility problem (SFP), and the set of common xed points of nitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevichs extragradient method, viscosity approximation method, Manns iteration method, hybrid steepest-descent method and gradient-projection method (GPM) with regularization. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.
MSC: 49J30; 47H09; 47J20; 49M05
Keywords: Mann-type hybrid steepest-descent method; general mixed equilibrium; general system of variational inequalities; nonexpansive mapping; strict pseudocontraction; inverse-strongly monotone mapping
1 Introduction
Let H be a real Hilbert space with the inner product , and the norm , C be a nonempty closed convex subset of H and PC be the metric projection of H onto C. Let
S : C C be a self-mapping on C. We denote by Fix(S) the set of xed points of S and by R the set of all real numbers. A mapping A : C H is called L-Lipschitz continuous if there exists a constant L such that
Ax Ay L x y , x, y C.
In particular, if L = then A is called a nonexpansive mapping; if L [, ) then A is called a contraction. A mapping T : C C is called -strictly pseudocontractive if there exists a constant [, ) such that
Tx Ty x y +
(I T)x (I T)y , x, y C.
In particular, if = , then T is a nonexpansive mapping.
2015 Ceng et al.; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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Let A : C H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): nd a point x C such that
Ax, y x , y C. (.)
The solution set of VIP (.) is denoted by VI(C, A).
VIP (.) was rst discussed by Lions [] and now it is well known. Variational inequalities have extensively been investigated; see the monographs []. It is well known that if A is a strongly monotone and Lipschitz continuous mapping on C, then VIP (.) has a unique solution. In the literature, the recent research work shows that variational inequalities like VIP (.) cover several topics, for example, monotone inclusions, convex optimization and quadratic minimization over xed point sets; see [] for more details.
In , Korpelevich [] proposed an iterative algorithm for solving VIP (.) in the Euclidean space Rn:
yn = PC(xn Axn),xn+ = PC(xn Ayn), n ,
with > a given number, which is known as the extragradient method. The literature on the VIP is vast, and Korpelevichs extragradient method has received great attention given by many authors who improved it in various ways; see, e.g., [, ] and the references therein, to name but a few.
On the other hand, let C and Q be nonempty closed convex subsets of innite-dimensional real Hilbert spaces H and H, respectively. The split feasibility problem (SFP)
is to nd a point x with the property
x C and Ax Q, (.)
where A B(H, H) and B(H, H) denotes the family of all bounded linear operators from H to H. We denote by the solution set of the SFP.
In , the SFP was rst introduced by Censor and Elfving [], in nite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see, e.g., [] and the references therein. Recently, it has been found that the SFP can also be applied to study intensity-modulated radiation therapy (IMRT); see, e.g., [, ] and the references therein. In the recent past, a wide variety of iterative methods have been used in signal processing and image reconstruction and for solving the SFP; see, e.g., [, , , , ] and the references therein. A seemingly more popular algorithm that solves the SFP is the CQ algorithm of Byrne [, ] which is found to be a gradient-projection method (GPM) in convex minimization. However, it remains a challenge how to implement the CQ algorithm in the case where the projections PC and/or PQ fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm.
Very recently, Xu [] gave a continuation of the study on the CQ algorithm and its convergence. He applied Manns algorithm to the SFP and proposed an averaged CQ algo-
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 3 of 43
rithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained.
Throughout this paper, assume that the SFP is consistent, that is, the solution set of the SFP is nonempty. Let f : H R be a continuous dierentiable function. The minimization problem
min
xC f (x) :=
Ax PQAx +
where > is the regularization parameter.
Very recently, by combining the gradient-projection method with regularization and extragradient method due to Nadezhkina and Takahashi [], Ceng et al. [] proposed a Mann-type extragradient-like algorithm, and proved that the sequences generated by the proposed algorithm converge weakly to a common solution of SFP (.) and the xed point problem of a nonexpansive mapping.
Theorem CAY (see Theorem . in []) Let T : C C be a nonexpansive mapping such that Fix(T) = . Assume that < <
, and let {xn} and {yn} be the sequences in C generated by the following Mann-type extragradient-like algorithm:
Ax PQAx
is ill-posed. Therefore, Xu [] considered the following Tikhonov regularization problem:
min
xC f(x) :=
x ,
x = x C chosen arbitrarily,yn = ( n)xn + nPC(xn fn(xn)),xn+ = nxn + ( n)TPC(yn fn(yn)), n ,
where the sequences of parameters {n}, {n} and {n} satisfy the following conditions:(i)
n= n < ;(ii) {n} [, ] and < lim infn n lim supn n < ;(iii) {n} [, ] and < lim infn n lim supn n < .
Then both the sequences {xn} and {yn} converge weakly to an element z Fix(T) .
In this paper, we consider the following general mixed equilibrium problem (GMEP) (see also [, ]) of nding x C such that
(x, y) + h(x, y) , y C, (.)
where , h : C C R are two bi-functions. We denote the set of solutions of GMEP (.) by GMEP(, h). GMEP (.) is very general; for example, it includes the following equilibrium problems as special cases.
As an example, in [, , ] the authors considered and studied the generalized equilibrium problem (GEP) which is to nd x C such that
(x, y) + Ax, y x , y C.
The set of solutions of GEP is denoted by GEP(, A).
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In [, , ], the authors considered and studied the mixed equilibrium problem (MEP) which is to nd x C such that
(x, y) + (y) (x) , y C.
The set of solutions of MEP is denoted by MEP(, ).
In [], the authors considered and studied the equilibrium problem (EP) which is to nd x C such that
(x, y) , y C.
The set of solutions of EP is denoted by EP(). It is worth to mention that the EP is a unied model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, xed point problems, Nash equilibrium problems, etc.
Throughout this paper, it is assumed as in [] that : C C R is a bi-function satisfying conditions ()-() and h : C C R is a bi-function with restrictions (h)-(h), where
() (x, x) = for all x C;() is monotone (i.e., (x, y) + (y, x) , x, y C) and upper hemicontinuous in the rst variable, i.e., for each x, y, z C,
lim sup
t
+
z C : (z, y) + h(z, y) +
called the resolvent of and h.
Assume that C is the xed point set of a nonexpansive mapping T : H H, i.e., C =
Fix(T). Let F : H H be -strongly monotone and -Lipschitzian with positive constants
, > . Let u H be given arbitrarily and {n}n= be a sequence in [, ]. The hybrid steepest-descent method introduced by Yamada [] is the algorithm
un+ := Tn+un = (I n+F)Tun, n , (.)
where I is the identity mapping on H.
In , Xu and Kim [] proved the following strong convergence result.
Theorem XK (see Theorem . in []) Assume that < < /. Assume also that the control conditions hold for {n}n=: limn n = ,
n= n = and limn n/n+ =
tz + ( t)x, y (x, y);
() is lower semicontinuous and convex in the second variable; (h) h(x, x) = for all x C;
(h) h is monotone and weakly upper semicontinuous in the rst variable; (h) h is convex in the second variable.
For r > and x H, let Tr : H C be a mapping dened by
Trx =
r y z, z x , y C
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(or equivalently, limn(n n+)/n+ = ). Then the sequence {un} generated by algorithm (.) converges strongly to the unique solution u in Fix(T) to the hierarchical VIP:
Fu, v u , v Fix(T). (.)
Let F, F : C H be two mappings. Consider the following general system of variational inequalities (GSVI) of nding (x, y) C C such that
Fy + x y, x x , x C,
Fx + y x, x y , x C,
(.)
where > and > are two constants. The solution set of GSVI (.) is denoted by GSVI(C, F, F).
In particular, if F = F = A, then the GSVI (.) reduces to the following problem of nding (x, y) C C such that
Ay + x y, x x , x C,
Ax + y x, x y , x C,
which is dened by Verma [] and it is called a new system of variational inequalities (NSVI). Further, if x = y additionally, then the NSVI reduces to the classical VIP (.). In , Ceng et al. [] transformed GSVI (.) into the xed point problem of the mapping G = PC(I F)PC(I F), that is, Gx = x, where y = PC(I F)x. Throughout this paper, the xed point set of the mapping G is denoted by .
On the other hand, if C is the xed point set Fix(T) of a nonexpansive mapping T and S is another nonexpansive mapping (not necessarily with xed points), then VIP (.) becomes the variational inequality problem of nding x Fix(T) such that
(I S)x, x x , x Fix(T). (.)
This problem, introduced by Mainge and Mouda [, ], is called the hierarchical xed point problem. It is clear that if S has xed points, then they are solutions of VIP (.).
If S is a -contraction (i.e., Sx Sy x y for some < ), the solution set of VIP (.) is a singleton and it is well known as the viscosity problem. This was previously introduced by Mouda [] and also developed by Xu []. In this case, it is easy to see that solving VIP (.) is equivalent to nding a xed point of the nonexpansive mapping PFix(T)S, where PFix(T) is the metric projection on the closed and convex set Fix(T).In , Marino et al. [] introduced a multi-step iterative scheme
(un, y) + h(un, y) + rn y un, un xn , y C, yn, = n,Sun + ( n,)un,yn,i = n,iSiun + ( n,i)yn,i, i = , . . . , N,xn+ = nf (xn) + ( n)Tyn,N,
(.)
with f : C C a -contraction and {n}, {n,i} (, ), {rn} (, ), that generalizes the two-step iterative scheme in [] for two nonexpansive mappings to a nite family of non-expansive mappings T, Si : C C, i = , . . . , N, and proved that the proposed scheme (.)
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converges strongly to a common xed point of the mappings that is also an equilibrium point of GMEP (.).
More recently, Marino, Muglia and Yaos multi-step iterative scheme (.) was extended to develop the following relaxed viscosity iterative algorithm.
Algorithm CKW (see (.) in []) Let f : C C be a -contraction and T : C C be a -strict pseudocontraction. Let Si : C C be a nonexpansive mapping for each i =
, . . . , N. Let Fj : C H be j-inverse strongly monotone with < j < j for each j = , . Let : C C R be a bi-function satisfying conditions ()-() and h : C C R be a bi-function with restrictions (h)-(h). Let {xn} be the sequence generated by
(un, y) + h(un, y) + rn y un, un xn , y C, yn, = n,Sun + ( n,)un,yn,i = n,iSiun + ( n,i)yn,i, i = , . . . , N,yn = nf (yn,N) + ( n)Gyn,N,xn+ = nxn + nyn + nTyn, n ,
where G = PC(I F)PC(I F), {n}, {n} are sequences in (, ) with <
lim infn n lim supn n < , {n}, {n} are sequences in [, ] with lim infn n > and n +n +n = , n , {n,i} is a sequence in (, ) for each i = , . . . , N, (n +n) n, n , and {rn} is a sequence in (, ) with lim infn rn > .
The authors [] proved that the proposed scheme (.) converges strongly to a common xed point of the mappings T, Si : C C, i = , . . . , N, that is also an equilibrium point of
GMEP (.) and a solution of GSVI (.).
In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for nding a common element of the solution set GMEP(, h) of GMEP (.), the solution set
GSVI(C, F, F) (i.e., ) of GSVI (.), the solution set of SFP (.), and the common xed point set N
i= Fix(Si) Fix(T) of nitely many nonexpansive mappings Si : C C, i = , . . . , N, and a strictly pseudocontractive mapping T : C C, in the setting of the innite-dimensional Hilbert space. The iterative algorithm is based on Korpelevichs extragradient method, viscosity approximation method [] (see also []), Manns iteration method, hybrid steepest-descent method [] and gradient-projection method (GPM) with regularization. Our aim is to prove that the iterative algorithm converges strongly to a common element of these sets, which also solves some hierarchical variational inequality. We observe that related results have been derived say in [, , , , , , ].
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by , and , 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
w(xn) to denote the weak -limit set of the sequence {xn} and s(xn) to denote the strong
-limit set of the sequence {xn}, i.e.,
w(xn) := x H : xni x for some subsequence {xni} of {xn}
(.)
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and
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
x PCx = inf
yC x y =: d(x, C).
The following properties of projections are useful and pertinent to our purpose.
Proposition . Given any x H and z C, one has(i) z = PCx x z, y z , y C;(ii) z = PCx x z x y y z , y C;(iii) PCx PCy, x y PCx PCy , y H, which hence implies that PC is nonexpansive and monotone.
Denition . A mapping T : H H is said to be(a) nonexpansive if
Tx Ty x y , x, y H;
(b) rmly nonexpansive if T I is nonexpansive, or equivalently, if T is -inverse strongly monotone (-ism),
x y, Tx Ty Tx Ty , x, y H;
alternatively, T is rmly nonexpansive if and only if T can be expressed as
T =
(I + S),
where S : H H is nonexpansive; projections are rmly nonexpansive.
Denition . A mapping A : C H is said to be(i) monotone if
Ax Ay, x y , x, y C;
(ii) -strongly monotone if there exists a constant > such that
Ax Ay, x y x y , x, y C;
(iii) -inverse-strongly monotone if there exists a constant > such that
Ax Ay, x y Ax Ay , x, y C.
s(xn) := x H : xni x for some subsequence {xni} of {xn}
.
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It can be easily seen that if T is nonexpansive, then I T is monotone. It is also easy to see that the projection PC is -ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various elds.
On the other hand, it is obvious that if A : C H is -inverse-strongly monotone, then A is monotone and -Lipschitz continuous. Moreover, we also have that, for all u, v C and > ,
(I A)u (I A)v
u v + ( ) Au Av . (.)
So, if , then I A is a nonexpansive mapping from C to H.
In , Ceng et al. [] transformed problem (.) into a xed point problem in the following way.
Proposition . (see []) For given x, y C, (x, y) is a solution of GSVI (.) if and only if x is a xed point of the mapping G : C C dened by
Gx = PC(I F)PC(I F)x, x C,
where y = PC(I F)x.
In particular, if the mapping Fj : C H is j-inverse-strongly monotone for j = , , then the mapping G is nonexpansive provided j (, j] for j = , . We denote by the xed point set of the mapping G.
The following result is easy to prove.
Proposition . (see []) Given x H, the following statements are equivalent:(i) x solves the SFP;(ii) x solves the xed point equation
PC(I f )x = x,
where > , f = A(I PQ)A and A is the adjoint of A;(iii) x solves the variational inequality problem (VIP) of nding x C such that
f
x , x x , x C.
It is clear from Proposition . that
= Fix PC(I f )
= VI(C, f ), > .
Denition . A mapping T : H H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,
T ( )I + S,
where (, ) and S : H H is nonexpansive. More precisely, when the last equality holds, we say that T is -averaged. Thus rmly nonexpansive mappings (in particular, projections) are -averaged mappings.
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Proposition . (see []) Let T : H H be a given mapping.(i) T is nonexpansive if and only if the complement I T is -ism.(ii) If T is -ism, then for > , T is -ism.(iii) T is averaged if and only if the complement I T is -ism for some > /. Indeed, for (, ), T is -averaged if and only if I T is
-ism.
Proposition . (see [, ]) Let S, T, V : H H be given operators.(i) If T = ( )S + V for some (, ) and if S is averaged and V is nonexpansive, then T is averaged.
(ii) T is rmly nonexpansive if and only if the complement I T is rmly nonexpansive.(iii) If T = ( )S + V for some (, ) and if S is rmly nonexpansive and V is nonexpansive, then T is averaged.
(iv) The composite of nitely many averaged mappings is averaged. That is, if each of the mappings {Ti}Ni= is averaged, then so is the composite T TN. In particular, if T is
-averaged and T is -averaged, where , (, ), then the composite TT is
-averaged, where = + .
(v) If the mappings {Ti}Ni= are averaged and have a common xed point, then
N
i=
Fix(Ti) = Fix(T TN).
The notation Fix(T) denotes the set of all xed points of the mapping T, that is, Fix(T) = {x H : Tx = x}.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma . Let X be a real inner product space. Then there holds the following inequality:
x + y x + y, x + y , x, y X.
Lemma . Let H be a real Hilbert space. Then the following hold:(a) x y = x y x y, y for all x, y H;(b) x + y = x + y x y for all x, y H and , [, ] with
+ = ;(c) if {xn} is a sequence in H such that xn x, it follows that
lim sup
n
xn y = lim sup
n
xn x + x y , y H.
It is clear that, in a real Hilbert space H, T : C C is -strictly pseudocontractive if and only if the following inequality holds:
Tx Ty, x y x y
(I T)x (I T)y , x, y C.
This immediately implies that if T is a -strictly pseudocontractive mapping, then I T is -inverse strongly monotone; for further details, we refer to [] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 10 of 43
class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.
Lemma . (see Proposition . in []) 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 satises the Lipschitzian condition
Tx Ty
+ x y , x, y C.
(ii) If T is a -strictly pseudocontractive mapping, then the mapping I T is semiclosed at , that is, if {xn} is a sequence in C such that xn x and (I T)xn , then(I T)x = .
(iii) If T is -(quasi-)strict pseudocontraction, then the xed-point set Fix(T) of T is closed and convex so that the projection PFix(T) is well dened.
Lemma . (see []) Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C C be a -strictly pseudocontractive mapping. Let and be two nonnegative real numbers such that ( + ) . Then
(x y) + (Tx Ty) ( + ) x y , x, y C.
Lemma . (see Demiclosedness principle in []) 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) = .
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 . Let A : C H be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition .(i)) implies
u VI(C, A) u = PC(u Au), > .
Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let be a number in (, ] and let > . Associating with a nonexpansive mapping T : C C, we dene the mapping T : C H by
Tx := Tx F(Tx), x C,
where F : C H is an operator such that, for some positive constants , > , F is
-Lipschitzian and -strongly monotone on C; that is, F satises the conditions
Fx Fy x y and Fx Fy, x y x y
for all x, y C.
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Lemma . (see Lemma . in []) T is a contraction provided < < ; that is,
Tx Ty ( ) x y , x, y C,
where = ( ) (, ].
Lemma . (see Lemma . in []) Let {an} be a sequence of nonnegative real numbers satisfying
an+ ( n)an + nn + n, n ,
where {n}, {n} and {n} satisfy the following conditions:(i) {n} [, ] and
n= n = ;(ii) either lim supn n or
n= n < .
In the sequel, we will indicate with GMEP(, h) the solution set of GMEP (.).
Lemma . (see []) Let C be a nonempty closed convex subset of a real Hilbert space H. Let : C C R be a bi-function satisfying conditions ()-() and h : C C R be a bi-function with restrictions (h)-(h). Moreover, let us suppose that(H) for xed r > and x C, there exist bounded K C and x K such that for all z C \ K, (x, z) + h(z, x) + r x z, z x < .
For r > and x H, the mapping Tr : H C (i.e., the resolvent of and h) has the following properties:(i) Trx = ;(ii) Trx is a singleton;(iii) Tr is rmly nonexpansive;(iv) GMEP(, h) = Fix(Tr) and it is closed and convex.
Lemma . (see []) Let us suppose that ()-(), (h)-(h) and (H) hold. Let x, y H, r, r > . Then
Try Trx y x +
n= n|n| < ;
(iii) n for all n , and
Then limn an = .
Try y .
Lemma . (see []) Suppose that the hypotheses of Lemma . are satised. Let {rn} be a sequence in (, ) with lim infn rn > . Suppose that {xn} is a bounded sequence. Then the following statements are equivalent and true:(a) if xn Trnxn as n , each weak cluster point of {xn} satises the problem
(x, y) + h(x, y) , y C,
i.e., w(xn) GMEP(, h);(b) the demiclosedness principle holds in the sense that if xn x and xn Trnxn as n , then (I Trk)x = for all k .
r r
r
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Recall that a set-valued mapping T : D(T) H H is called monotone if for all x, y D(T), f Tx and g Ty imply
f g, x y .
A set-valued mapping T is called maximal monotone if T is monotone and (I +T)D(T) = H for each > , 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, f g, x y 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, i.e.,
NCv = u H : v p, u , p C
Dene
Tv v VI(C, A). (.)
3 Main results
We now propose the following hybrid extragradient viscosity iterative scheme:
(un, y) + h(un, y) + rn y un, un xn , y C, yn, = n,Sun + ( n,)un,yn,i = n,iSiun + ( n,i)yn,i, i = , . . . , N,
yn,N = PC(yn,N nfn(yn,N)),yn = PC[ n Vyn,N + (I nF)GPC(yn,N nfn(yn,N))],xn+ = nyn + nPC(yn,N nfn(yn,N)) + nTPC(yn,N nfn(yn,N))
for all n , where
F : C H is a -Lipschitzian and -strongly monotone operator with positive constants , > and V : C C is an l-Lipschitzian mapping with constant l ;
Fj : C H is j-inverse strongly monotone and G := PC(I F)PC(I F) with
j (, j) for j = , ;
T : C C is a -strict pseudocontraction and Si : C C is a nonexpansive mapping for each i = , . . . , N;
, h : C C R are two bi-functions satisfying the hypotheses of Lemma .; {n} is a sequence in (,
A
n= n < ; { n}, {n} are sequences in (, ) with < lim infn n lim supn n < ;
.
Av + NCv, if v C, , if v /
C.
Then it is known in [] that
T is maximal monotone and
Tv =
(.)
) with < lim infn n lim supn n <
( );
;
< < / and l < with :=
{n} is a sequence in (, ) with
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 13 of 43
{n}, {n} are sequences in [, ] with n + n + n = , n ; {n,i}Ni= are sequences in (, ) and (n + n) n, n ;{rn} is a sequence in (, ) with lim infn rn > and lim infn n > .
We start our main result from the following series of propositions.
Proposition . Let us suppose that = Fix(T)
N
i= Fix(Si)GMEP(, h) = .
Then the sequences {xn}, {yn}, {yn,i} for all i, {un} are bounded.
Proof Since < lim infn n lim supn n <
and < lim infn n
lim supn n < , we may assume, without loss of generality, that {n} [a, b] (,
A
)
and {n} [c, d] (, ). Now, let us show that PC(I f) is -averaged for each
(,
+ A
), where
= +
( + A ) (, ). (.)
Indeed, it is easy to see that f = A(I PQ)A is
-ism, that is,
f (x) f (y), x y
f (x) f (y)
. (.)
Observe that
+ A
f
(x) f(y), x y
= + A
x y +
f (x) f (y), x y
= x y +
f (x) f (y), x y
+ A x y
+ A
f (x) f (y), x y
x y +
f (x) f (y), x y
+ f (x) f (y)
= (x y) + f (x) f (y)
. (.)
Hence, it follows that f = I + A(I PQ)A is
+ A
-ism. Thus, f is
= f(x) f(y)
) -ism ac
cording to Proposition .(ii). By Proposition .(iii), the complement I f is (+ A
)
-
(+ A
averaged. Therefore, noting that PC is -averaged and utilizing Proposition .(iv), we know that for each (,
+ A
), PC(I f) is -averaged with
=
+
( + A )
+ ( + A )
(, ). (.)
This shows that PC(I f) is nonexpansive. Furthermore, for {n} [a, b] (,
A
),
( + A )
=
we have
a inf
n
n sup
n
n b <
= lim
n
n + A
. (.)
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 14 of 43
Without loss of generality, we may assume that
a inf
n
n sup
n
n b <
n + A
, n . (.)
Consequently, it follows that for each integer n , PC(I nfn) is n-averaged with
n = +
n(n + A )
n(n + A )
=
+ n(n + A )
(, ). (.)
This immediately implies that PC(I nfn) is nonexpansive for all n .
For simplicity, we write tn = PC(yn,N nfn(yn,N)) and
vn = n Vyn,N + (I nF)Gtn
for all n . Then yn = PCvn and xn+ = nyn + ntn + nTtn.
First of all, take a xed p arbitrarily. We observe that
yn, p un p xn p .
For all from i = to i = N, by induction, one proves that
yn,i p n,i un p + ( n,i) yn,i p un p xn p .
Thus we obtain that for every i = , . . . , N,
yn,i p un p xn p . (.)
For simplicity, we write p = PC(p Fp), tn = PC(tn Ftn) and zn = PC(tn Ftn) for each n . Then zn = Gtn and
p = PC(I F)p = PC(I F)PC(I F)p = Gp.
Since Fj : C H is j-inverse strongly monotone and < j < j for each j = , , we know that for all n ,
zn p = Gtn p
= PC(I F)PC(I F)tn PC(I F)PC(I F)p
(I F)PC(I F)tn (I F)PC(I F)p
= PC(I F)tn PC(I F)p
FPC(I F)tn FPC(I F)p
PC(I F)tn PC(I F)p
+ ( ) FPC(I F)tn FPC(I F)p
(I F)tn (I F)p + ( ) Ftn F p
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 15 of 43
= (tn p) (Ftn Fp) + ( ) Ftn F p
tn p + ( ) Ftn Fp + ( ) Ftn F p
tn p . (.)
From (.), (.) and the nonexpansivity of PC(I nfn), it follows that
yn,N p =
PC(I nfn)yn,N PC(I nf )p
PC(I nfn)yn,N PC(I nfn)p
+ PC(I nfn)p PC(I nf )p
yn,N p +
(I nfn)p (I nf )p
xn p + nn p . (.)
Utilizing Lemma ., we also have
yn,N p =
PC(I nfn)yn,N PC(I nf )p
= PC(I nfn)yn,N PC(I nfn)p
+ PC(I nfn)p PC(I nf )p
PC(I nfn)yn,N PC(I nfn)p
+ PC(I nfn)p PC(I nf )p, yn,N p
yn,N p +
PC(I nfn)p PC(I nf )p
yn,N p
xn p +
(I nfn)p (I nf )p
yn,N p
xn p + nn p yn,N p . (.)
Furthermore, utilizing Proposition .(ii), we have
tn p
yn,N nfn(yn,N) p
yn,N nfn(yn,N) tn
= yn,N p yn,N tn + n
f
n (yn,N), p tn
= yn,N p yn,N tn + n
f
n (yn,N) fn(p), p yn,N
+ f
n (p), p yn,N
+ f
n (yn,N), yn,N tn
yn,N p yn,N tn
+ n f
n (p), p yn,N
+ f
n (yn,N), yn,N tn
= yn,N p yn,N tn
+ n (nI + f )p, p yn,N
+ f
n (yn,N), yn,N tn
yn,N p yn,N tn + n
n p, p yn,N +
f
n (yn,N), yn,N tn
= yn,N p yn,N yn,N yn,N yn,N, yn,N tn yn,N tn
+ n n p, p yn,N +
fn(yn,N), yn,N tn
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 16 of 43
= yn,N p yn,N yn,N yn,N tn
+ yn,N nfn(yn,N) yn,N, tn yn,N
+ nn p, p yn,N . (.)
In the meantime, by Proposition .(i), we have
yn,N nfn(yn,N) yn,N, tn yn,N
= yn,N nfn(yn,N) yn,N, tn yn,N
+ nfn(yn,N) nfn(yn,N), tn yn,N
nfn(yn,N) nfn(yn,N), tn yn,N
n
f
n (yn,N) fn(yn,N)
tn yn,N
yn,N yn,N tn yn,N . (.)
So, from (.) and (.), we obtain
tn p yn,N p yn,N yn,N yn,N tn
+ yn,N nfn(yn,N) yn,N, tn yn,N
+ nn p, p yn,N
yn,N p yn,N yn,N yn,N tn
+ n n + A
yn,N yn,N tn yn,N + nn p, p yn,N
yn,N p yn,N yn,N yn,N tn
+ n n + A
yn,N yn,N + yn,N tn + nn p, p yn,N
= yn,N p + nn p p yn,N
+ n n + A
yn,N yn,N
yn,N p + nn p yn,N p
yn,N p + nn p
yn,N p + nn p
n
n + A
yn,N p + nn p yn,N p + nn p
= yn,N p + nn p
xn p + nn p
. (.)
Hence, utilizing Lemma . we deduce from (.) and (.) that
yn p = PCvn p
(I nF)Gtn (I nF)p + n ( V F)p
n l yn,N p + ( n) tn p + n
( V F)p
n (Vyn,N Vp) + (I nF)Gtn (I nF)p + n( V F)p
n Vyn,N Vp +
n l yn,N p + ( n)
yn,N p + nn p
+ n ( V F)p
n( l) yn,N p + n
( V F)p + nn p
n( l) xn p + n
( V F)p + nn p
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 17 of 43
= n( l) xn p + n( l)
( V F)p
l +
nn p
xn p ,
( V F)p
l
+ nn p . (.)
Since (n + n) n for all n , utilizing Lemma ., we obtain from (.) and (.) that
xn+ p =
n(yn p) + n(tn p) + n(Ttn p)
max
= n(yn p) + (n + n)
n + n
n(tn p) + n(Ttn p)
n yn p + (n + n)
n + n
n(tn p) + n(Ttn p)
n yn p + (n + n) tn p
n
max
+ nn p
xn p ,
( V F)p
l
+ ( n) xn p + nn p
= n max
xn p ,
( V F)p
l
+ ( n) xn p + nn p
max
xn p ,
( V F)p
l
+ nn p .
By induction, we can prove
xn+ p max
x p ,
( V F)p
l
n
k=
+ kk p , n .
) and
n= n < , we know that {xn} is bounded, and so are the sequences {un}, {vn}, {tn}, {tn}, {yn}, {yn,N}, {yn,i} for each i = , . . . , N. Since Ttn p
+ tn p , {Ttn} is also bounded.
Proposition . Let us suppose that = . Moreover, let us suppose that the following hold:
(H) limn n = and
Since {n} [a, b] (,
n= n = ;(H) limn |nn| n = and limn |nn| n = ; (H) limn |n,in,i| n = for each i = , . . . , N;
(H)
n= | n n| < or limn | n n| n = ; (H)
n= |rn rn| < or limn |rnrn| n = (H)
n= |n n| < or limn |nn| n = ; (H)
n= | nn
nn | = .
If un un = o( n), then limn xn+ xn = , i.e., {xn} is asymptotically regular.
Proof First, it is known that {n} [a, b] (,
nn | < or limn n | nn
) and {n} [c, d] (, ) as in the proof of Proposition .. Taking into account lim infn rn > , we may assume, without loss of generality, that {rn} [r, ) for some r > . First, we write xn = nyn + (
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 18 of 43
n)wn, n , where wn = xnnynn . It follows that for all n ,
wn wn = xn+
nyn
n
xn nyn
n
ntn + nTtn
= n
ntn + nTtn
n
=
n(tn tn) + n(Ttn Ttn) n +
n
n
n
n
tn
n
+ n
n
n
Ttn. (.)
Since (n + n) n for all n , utilizing Lemma . we have
n(tn tn) + n(Ttn Ttn) (n + n) tn tn . (.)
Next, we estimate yn yn . Indeed, according to n(n + A ) < ,
tn tn
yn,N nfn(yn,N)
yn,N nfn(yn,N)
yn,N yn,N +
nfn(yn,N) nfn(yn,N)
yn,N yn,N + |n n|
f
n (yn,N)
+ n f
n (yn,N) fn(yn,N)
yn,N yn,N + |n n|
f
n (yn,N)
+ n f
n (yn,N) fn(yn,N)
+ f
n (yn,N) fn(yn,N)
yn,N yn,N + |n n|
f
n (yn,N)
+ n |n n| yn,N +
n + A
yn,N yn,N
= yn,N yn,N + |n n|
f
n (yn,N)
+ n|n n| yn,N + n
n + A
yn,N yn,N
yn,N yn,N + |n n|
f
n (yn,N)
+ n|n n| yn,N + yn,N yn,N (.)
and
yn,N yn,N =
PC yn,N nfn(yn,N)
PC yn,N nfn(yn,N)
PC yn,N nfn(yn,N)
PC yn,N nfn(yn,N)
+ PC yn,N nfn(yn,N)
PC yn,N nfn(yn,N)
yn,N yn,N
+ yn,N nfn(yn,N)
yn,N nfn(yn,N)
= yn,N yn,N +
nfn(yn,N) nfn(yn,N)
yn,N yn,N + |nn nn| yn,N + |n n|
f (yn,N)
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 19 of 43
yn,N yn,N +
n|n n| + n|n n|
yn,N
. (.)
In the meantime, by the denition of yn,i one obtains that, for all i = N, . . . , ,
yn,i yn,i n,i un un + Siun yn,i |n,i n,i|
+ ( n,i) yn,i yn,i . (.)
In the case i = , we have
yn, yn, n, un un + Sun un |n, n,|
+ ( n,) un un
= un un + Sun un |n, n,|. (.)
Substituting (.) in all (.)-type one obtains, for i = , . . . , N,
yn,i yn,i un un +
i
k=
+ |n n|
f (yn,N)
Skun yn,k |n,k n,k|
+ Sun un |n, n,|, (.)
which together with (.) implies that
yn,N yn,N n
yn,N yn,N n +
n |
n n|
n +
n |
n n| n
yn,N
+ |
n n| n
f (yn,N)
un un
n +
N
k=
Skun yn,k |
n,k n,k|
n
+ Sun un |
n, n,|
n
+
n |
n n|
n +
n |
n n|
n
yn,N
. (.)
Since un un = o( n) and the sequences {un}, {yn,i}Ni= are bounded, we know that
lim
n
+ |
n n|
n
f (yn,N)
yn,N yn,N
n = .
On the other hand, we observe that
vn = n Vyn,N + (I nF)zn,vn = n Vyn,N + (I nF)zn, n .
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 20 of 43
Simple calculations show that
vn vn = (I nF)zn (I nF)zn
+ ( n n)( Vyn,N Fzn)
+ n (Vyn,N Vyn,N).
Then, passing to the norm and using the nonexpansivity of G, we get
yn yn vn vn
(I nF)zn (I nF)zn + | n n| Vyn,N Fzn
+ n Vyn,N Vyn,N
( n) zn zn +
M| n n| + n l yn,N yn,N
( n) tn tn +
M| n n| + n l yn,N yn,N , (.)
where supn Vyn,N Fzn
M for some
M > . Also, it is easy to see from (.) and
(.) that
wn wn
n(tn tn) + n(Ttn Ttn) n
+
n n
n
n
tn
+
n n
n
n
Ttn
(n + n) tn tn n +
n n
n
n
tn
+
n n
n
n
Ttn
= tn tn +
n n
n
n
tn + Ttn
. (.)
Moreover, by Lemma ., we know that
un un xn xn + L
rn
rn
,
where L = supn un xn .
Further, we observe that
xn+ = nyn + ( n)wn,xn = nyn + ( n)wn, n .
Simple calculations show that
xn+ xn = ( n)(wn wn) + (n n)(yn wn) + n(yn yn).
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 21 of 43
Consequently, passing to the norm we get from (.), (.) and (.)-(.)
xn+ xn
( n) wn wn + |n n| yn wn + n yn yn
( n)
tn tn +
n n
n
n
tn + Ttn
+ |n n| yn wn + n
( n) tn tn
+
M| n n| + n l yn,N yn,N
( n n) tn tn +
n n
n
n
tn + Ttn
+ |n n| yn wn +
M| n n| + n n l yn,N yn,N
( n n)
yn,N yn,N + |n n|
f
n (yn,N)
+ n|n n| yn,N + yn,N yn,N
+
n n
n
n
tn + Ttn
+ |n n| yn wn +
M| n n| + n n l yn,N yn,N
n n( l) yn,N yn,N + |n n|
f
n (yn,N)
+ n|n n| yn,N + yn,N yn,N
+
n n
n
n
tn + Ttn
+ |n n| yn wn +
M| n n|
un un +
n n( l)
N
k=
Skun yn,k |n,k n,k|
+ Sun un |n, n,|
+ |n n|
f
n (yn,N)
+ n|n n| yn,N + yn,N yn,N
+
n n
n
n
tn + Ttn
+ |n n| yn wn +
M| n n|
xn xn + L
n n( l)
rn
rn
N
k=
+ Skun yn,k |n,k n,k|
+ Sun un |n, n,|
+ |n n|
f
n (yn,N)
n n
n
n
+ n|n n| yn,N + yn,N yn,N +
tn + Ttn
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 22 of 43
+ |n n| yn wn +
M| n n|
c n( l) xn xn + L|
rn rn|
r
N
k=
+ Skun yn,k |n,k n,k|
+ Sun un |n, n,| + |n n|
f
n (yn,N)
n n
n
n
tn + Ttn
+ n|n n| yn,N + yn,N yn,N +
+ |n n| yn wn +
M| n n|
n( l)c xn xn +
M |rn rn|
r
+
M
N
k= |
n,k n,k| +
M|n, n,|
+
M|n n| +
M|n n| +
M yn,N yn,N +
M
n n
n
n
+
M|n n| +
M| n n|
= n( l)c xn xn +
M
|rn rn|
r
N
k= |
+ n,k n,k|
+ |n n| + |n n| + yn,N yn,N +
n n
n
n
+ |n n| + | n n|
= n( l)c xn xn
+ n( l)c
|rn rn|
nr
+
N
k=
|n,k n,k|
n
+ |
n n|
n +
|n n|
n +
|n n| n
+ | nn
n n |
n +
| n n|
n +
yn,N yn,N n
, (.)
where
sup
n
+ L +
M +
N
k=
Skun yn,k + Sun un
+ f
n (yn,N)
M
+ b yn,N + tn + Ttn + yn wn
for some
M > . Noticing limn yn,Nyn,N n = and using hypotheses (H)-(H) and Lemma ., we obtain the claim.
Proposition . Let us suppose that = . Let us suppose that {xn} is asymptotically regular. Then xn un = xn Trnxn and yn,N yn,N as n .
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 23 of 43
Proof Take xed p arbitrarily. We recall that, by the rm nonexpansivity of Trn, a standard calculation (see []) shows that for p GMEP(, h),
un p xn p xn un . (.)
Utilizing Lemmas . and ., we obtain from l < , (.) and (.) that
yn p
= n (Vyn,N Vp) + (I nF)zn (I nF)p + n( V F)p
n (Vyn,N Vp) + (I nF)zn (I nF)p + n ( V F)p, yn p
n Vyn,N Vp +
(I nF)zn (I nF)p + n ( V F)p, yn p
n l yn,N p + ( n) zn p
+ n ( V F)p, yn p
+ n ( V F)p, yn p
n
( l)
yn,N p + ( n) zn p + n
n
l
= yn,N p + ( n) zn p
( V F)p, yn p
( V F)p yn p
n yn,N p + tn p ( ) Ftn Fp
( ) Ftn F p + n
( V F)p yn p . (.)
Since (n + n) n for all n , utilizing Lemma . we have from (.), (.), (.), (.) and (.) that
xn+ p
= n(yn p) + n(tn p) + n(Ttn p)
= n(yn p) + (n + n)
n + n
n yn,N p + zn p + n
n(tn p) + n(Ttn p)
n yn p + (n + n)
n + n
n(tn p) + n(Ttn p)
n yn p + (n + n) tn p
= n yn p + ( n) tn p
n
n yn,N p + tn p ( ) Ftn Fp
( ) Ftn F p + n
( V F)p yn p
+ ( n) tn p
tn p n
( ) Ftn Fp + ( ) Ftn F p
( V F)p yn p
yn,N p + nn p p yn,N +
n n + A
+ n yn,N p + n
yn,N yn,N
n ( ) Ftn Fp + ( ) Ftn F p
+ n yn,N p + n
( V F)p yn p
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 24 of 43
un p + nn p p yn,N +
n n + A
yn,N yn,N
n ( ) Ftn Fp + ( ) Ftn F p
( V F)p yn p
xn p xn un + nn p p yn,N
+ n n + A
yn,N yn,N
n ( ) Ftn Fp + ( ) Ftn F p
+ n yn,N p + n
( V F)p yn p . (.)
So, we deduce from {n} [c, d] (, ) and {n} [a, b] (,
+ n yn,N p + n
) that
xn un +
b n + A
yn,N yn,N
+ c ( ) Ftn Fp + ( ) Ftn F p
xn un +
n n + A
yn,N yn,N
+ n ( ) Ftn Fp + ( ) Ftn F p
xn p xn+ p + nn p p yn,N
+ n yn,N p + n
( V F)p yn p
xn xn+
xn p + xn+ p
+ nb p p yn,N
+ n yn,N p + n
( V F)p yn p .
By Propositions . and . we know that the sequences {xn}, {yn}, {yn,N} and {yn,N} are bounded and that {xn} is asymptotically regular. Therefore, from n and n we obtain that
lim
n xn un =
lim
n Ftn Fp =
n yn,N yn,N = . (.)
Remark . By the last proposition we have w(xn) = w(un) and s(xn) = s(un), i.e., the sets of strong/weak cluster points of {xn} and {un} coincide.
Of course, if n,i i = as n , for all indices i, the assumptions of Proposition . are enough to assure that
lim
n
xn+ xn
n,i = , i {, . . . , N}.
In the next proposition, we estimate the case in which at least one sequence {n,k} is a
null sequence.
Proposition . Let us suppose that = . Let us suppose that (H) holds. Moreover, for an index k {, . . . , N}, limn n,k = and the following hold:
n Ftn F p =
lim
lim
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 25 of 43
(H) for each index i {, . . . , N},
lim
n
|n,i n,i| nn,k =
lim
n
|n n| nn,k =
lim
n
|n n| nn,k =
lim
n
|rn rn| nn,k
= lim
n
| n n| nn,k =
lim
n
nn,k
n n
n
n
|n n| nn,k = ;
(H) there exists a constant > such that
n |
n,k
= lim
n
n,k | < for all n .
If un un = o( nn,k), then
lim
n
xn+ xn n,k = .
Proof It is clear from (.) that
yn,N yn,N nn,k
un un nn,k +
N
k=
Skun yn,k |
n,k n,k| nn,k
+ Sun un |
n, n,| nn,k
+
n |
n n| nn,k +
n |
n n|
nn,k
yn,N
+ |
n n|
nn,k
f (yn,N)
.
According to (H) and un un = o( nn,k), we get
lim
n
yn,N yn,N
nn,k = . (.)
Consider (.). Dividing both the terms by n,k, we have
xn+ xn
n,k
n( l)c xn xn
n,k
+ n( l)c
|rn rn|
nn,k r
+
N
k=
|n,k n,k|
nn,k
+ |
n n|
nn,k +
|n n|
nn,k +
|n n| nn,k
+ | nn
n n |
nn,k +
| n n|
nn,k +
yn,N yn,N nn,k
.
So, by (H) we have
xn+ xn
n,k
n( l)c xn xn
n,k
+ n( l)c xn xn
n,k
n,k
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 26 of 43
+ n( l)c
|rn rn| nn,k r
+
N
k=
|n,k n,k| nn,k +
|n n| nn,k +
|n n| nn,k
+ |
n n|
nn,k +
| nn
n n |
nn,k +
| n n| nn,k +
yn,N yn,N nn,k
n( l)c xn xn
n,k + xn xn
n,k
n,k
+ n( l)c
|rn rn| nn,k r
+
N
k=
|n,k n,k| nn,k +
|n n| nn,k +
|n n| nn,k
+ |
n n|
nn,k +
| nn
n n |
nn,k +
| n n| nn,k +
yn,N yn,N nn,k
n( l)c xn xn
n,k +
n xn xn
+ n( l)c
|rn rn|
nn,k r
+
N
k=
|n,k n,k|
nn,k +
|n n|
nn,k +
|n n| nn,k
+ |
n n|
nn,k +
| nn
n n |
nn,k +
| n n|
nn,k +
yn,N yn,N nn,k
= n( l)c xn xn
n,k
+ n( l)c
( l)c
xn xn
+
M
|rn rn|
nn,k r
+
N
k=
|n,k n,k|
nn,k +
|n n|
nn,k +
|n n| nn,k
+ |
n n|
nn,k +
| nn
n n |
nn,k +
| n n|
nn,k +
yn,N yn,N nn,k
.
Therefore, utilizing Lemma ., from (.), (H), (H) and the asymptotical regularity of
{xn} (due to Proposition .), we deduce that
lim
n
xn+ xn
n,k = .
Proposition . Let us suppose that = . Let us suppose that (H)-(H) hold. If un
un = o( n), then zn tn as n .
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 27 of 43
Proof Let p . In terms of the rm nonexpansivity of PC and the j-inverse strong monotonicity of Fj for j = , , we obtain from j (, j), j = , and (.) that
tn p =
PC(I F)tn PC(I F)p
(I F)tn (I F)p, tn p
=
(I F)tn (I F)p + tn p
(I F)tn (I F)p (tn p)
tn p + tn p
(tn tn) (Ftn Fp) (p p)
=
tn p + tn p
(tn tn) (p p)
+ (tn tn) (p p), Ftn Fp
Ftn Fp
and
zn p =
PC(I F)tn PC(I F)p
(I F)tn (I F)p, zn p
=
(I F)tn (I F)p
+ zn p
(I F)tn (I F)p (zn p)
tn p + zn p
(tn zn) + (p p)
+ Ftn F p, (tn zn) + (p p)
Ftn F p
tn p + zn p
(tn zn) + (p p)
+ Ftn F p, (tn zn) + (p p)
.
Thus, we have
tn p tn p
(tn tn) (p p)
+ (tn tn) (p p), Ftn Fp
Ftn Fp (.)
and
zn p tn p
(tn zn) + (p p)
+ Ftn F p
(tn zn) + (p p)
. (.)
Consequently, from (.), (.), (.), (.) and (.), it follows that
xn+ p
n yn p + ( n) tn p
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 28 of 43
n
n yn,N p + zn p + n
( V F)p yn p
+ ( n) tn p
n
n yn,N p + tn p + n
( V F)p yn p
+ ( n) tn p
n
n yn,N p + tn p
(tn tn) (p p)
+ (tn tn) (p p)
Ftn Fp + n
( V F)p yn p
+ ( n) tn p
tn p + n yn,N p n
(tn tn) (p p)
+ (tn tn) (p p)
Ftn Fp + n
( V F)p yn p
xn p + nn p
+ n yn,N p n
(tn tn) (p p)
+ (tn tn) (p p)
Ftn Fp + n
( V F)p yn p ,
which yields
c (tn tn) (p p)
n
(tn tn) (p p)
xn p + nn p
xn+ p + n yn,N p
+ (tn tn) (p p)
Ftn Fp + n
( V F)p yn p
xn xn+ + nn p
xn p + xn+ p + nn p
+ n yn,N p +
(tn tn) (p p)
Ftn Fp
+ n ( V F)p yn p .
Since limn n = , limn n = , limn xn+ xn = , and {xn}, {yn}, {yn,N}, {tn} and {tn} are bounded, we deduce from (.) that
lim
n
(tn tn) (p p)
= . (.)
Furthermore, from (.), (.), (.) and (.), it follows that
xn+ p
n yn p + ( n) tn p
n
n yn,N p + zn p + n
( V F)p yn p
+ ( n) tn p
n
n yn,N p + tn p
(tn zn) + (p p)
+ Ftn F p
(tn zn) + (p p)
+ n ( V F)p yn p
+ ( n) tn p
tn p + n yn,N p n
(tn zn) + (p p)
+ Ftn F p
(tn zn) + (p p)
+ n ( V F)p yn p
xn p + nn p
+ n yn,N p n
(tn zn) + (p p)
+ Ftn F p
(tn zn) + (p p)
+ n ( V F)p yn p ,
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 29 of 43
which leads to
c (tn zn) + (p p)
n
(tn zn) + (p p)
xn p + nn p
xn+ p + n yn,N p
+ Ftn F p
(tn zn) + (p p)
+ n ( V F)p yn p
xn xn+ + nn p
xn p + xn+ p + nn p
+ n yn,N p + Ftn F p
(tn zn) + (p p)
+ n ( V F)p yn p .
Since limn n = , limn n = , limn xn+ xn = , and {xn}, {yn}, {yn,N}, {zn} and {tn} are bounded, we deduce from (.) that
lim
n
(tn zn) + (p p)
= . (.)
Note that
tn zn
(tn tn) (p p)
+ (tn zn) + (p p)
.
Hence from (.) and (.) we get
lim
n tn zn =
lim
n tn Gtn = . (.)
Proposition . Let us suppose that = . Let us suppose that < lim infn n,i lim supn n,i < for each i = , . . . , N. Moreover, suppose that un un = o( n) and (H)-(H) are satised. Then limn Siun un = for each i = , . . . , N provided Tyn
yn as n .
Proof First of all, it is clear that
tn yn,N =
PC yn,N nfn(yn,N)
PC yn,N nfn(yn,N)
yn,N nfn(yn,N)
yn,N nfn(yn,N)
= n f
n (yn,N) fn(yn,N)
n
n + A
yn,N yn,N
yn,N yn,N .
By Proposition ., we get
lim
n tn yn,N = ,
which together with (.) implies that
lim
n tn yn,N = . (.)
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 30 of 43
Note that
n Vyn,N + (I nF)zn tn
n Vyn,N Fzn + zn tn .
From Proposition . and n , we obtain
lim
n yn tn = . (.)
Also, observe that
xn+ xn + xn yn = xn+ yn
= n(tn yn) + n(Ttn yn)
= n(tn yn) + n(Ttn Tyn) + n(Tyn yn).
By Proposition . we know that {xn} is asymptotically regular. Utilizing Lemma . we have from (n + n) n that
yn xn =
( V F)p yn p
( V F)p yn p + nn p yn,N p + yn,N p
( V F)p yn p + nn p yn,N p
+ n,N SNun p + ( n,N) yn,N p n,N( n,N) SNun yn,N
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ n,N un p + ( n,N) un p n,N( n,N) SNun yn,N
= n yn,N p + n
( V F)p yn p + nn p yn,N p
+ un p n,N( n,N) SNun yn,N
yn tn
n(tn yn) n(Ttn Tyn) + n Tyn yn
xn+ xn + (n + n) tn yn + n Tyn yn
xn+ xn + tn yn + Tyn yn ,
which together with (.) and Tyn yn leads to
lim
n xn yn = . (.)
Let us show that for each i {, . . . , N}, one has Siun yn,i as n . Let p . When i = N, by Lemma .(b) we have from (.), (.), (.) and (.)
yn p
n yn,N p + zn p + n
( V F)p yn p
n yn,N p + tn p + n
xn+ xn n(tn yn) n(Ttn Tyn) n(Tyn yn)
xn+ xn +
n yn,N p + n
n yn,N p + n
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 31 of 43
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ xn p n,N( n,N) SNun yn,N .
So, we have
n,N( n,N) SNun yn,N
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ xn p yn p
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ xn yn
xn p + yn p
.
Since n , n , < lim infn n,N lim supn n,N < and limn xn yn = (due to (.)), it is known that { SNun yn,N } is a null sequence.
Let i {, . . . , N }. Then one has
yn p
n yn,N p + n
( V F)p yn p + nn p yn,N p + yn,N p
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ n,N SNun p + ( n,N) yn,N p
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ n,N xn p + ( n,N) yn,N p
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ n,N xn p + ( n,N)
n,N SNun p + ( n,N) yn,N p
( V F)p yn p + nn p yn,N p
+ n,N + ( n,N)n,N xn p +
N
k=N
( n,k) yn,N p ,
n yn,N p + n
and so, after (N i + )-iterations,
yn p
n yn,N p + n
( V F)p yn p + nn p yn,N p
+
!
n,N +
N
j=i+
! N
l=j
( n,l)
"
n,j
" xn p +
N
k=i+
( n,k) yn,i p
n yn,N p + n
( V F)p yn p + nn p yn,N p
+
!
n,N +
N
j=i+
! N
l=j
( n,l)
"
n,j
" xn p +
N
k=i+
( n,k) n,i Siun p
+ ( n,i) yn,i p n,i( n,i) Siun yn,i
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 32 of 43
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ xn p n,i
N
k=i
( n,k) Siun yn,i . (.)
Again we obtain that
n,i
N
k=i
( n,k) Siun yn,i
n yn,N p + n
( V F)p yn p
+ nn p yn,N p + xn p yn p
n yn,N p + n
( V F)p yn p
+ nn p yn,N p + xn yn
xn p + yn p
.
Since n , n , < lim infn n,i lim supn n,i < for each i = , . . . , N , and limn xn yn = (due to (.)), it is known that
lim
n Siun yn,i = .
Obviously for i = , we have Sun un .
To conclude, we have that
Sun un Sun yn, + yn, un = Sun yn, + n, Sun un
from which Sun un . Thus by induction Siun un for all i = , . . . , N since it is enough to observe that
Siun un Siun yn,i + yn,i Siun + Siun un
Siun yn,i + ( n,i) Siun yn,i + Siun un .
Remark . As an example, we consider N = and the sequences:(a) n = + n , n = n = n, n > ;(b) n =
A
n , n > A ;
(c) n =
n , n =
n , rn = n, n > ;(d) n, = n, n, =
n , n > .
Then they satisfy the hypotheses on the parameter sequences in Proposition ..
Proposition . Let us suppose that = and n,i i for all i as n . Suppose that there exists k {, . . . , N} such that n,k as n . Let k {, . . . , N} be the largest index such that n,k as n . Suppose that(i)
n+ n
n,k as n ;
(ii) if i k and n,i , then n,kn,i as n ;(iii) if n,i i = , then i lies in (, ).
Moreover, suppose that un un = o( nn,k) and (H), (H) and (H) hold. Then
limn Siun un = for each i = , . . . , N provided Tyn yn as n .
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 33 of 43
Proof First of all we note that if (H) holds then also (H)-(H) are satised. So {xn} is asymptotically regular.
Let k be as in the hypotheses. As in Proposition ., for every index i {, . . . , N} such that n,i i = (which leads to < lim infn n,i lim supn n,i < ), one has Siun
yn,i as n .
For all the other indices i k, we can prove that Siun yn,i as n in a similar manner. By the relation (due to (.), (.) and (.))
xn+ p
n yn p + ( n) tn p
n
n yn,N p + n
( V F)p yn p + nn p yn,N p
+ xn p n,i
N
k=i
( n,k) Siun yn,i
+ ( n) xn p + nn p
xn p + nn p
+ n yn,N p + n
( V F)p yn p
+ nn p yn,N p nn,i
N
k=i
( n,k) Siun yn,i ,
we immediately obtain that
c
N
k=i
( n,k) Siun yn,i
n
N
k=i
( n,k) Siun yn,i
xn p + nn p
xn+ p + n yn,N p
+ n ( V F)p yn p + nn p yn,N p
xn xn+ + nn p
n,i
xn p + xn+ p + nn p
( V F)p yn p + nn p yn,N p .
By Proposition . or by hypothesis (ii) on the sequences, we have
xn xn+
n,i =
xn xn+
n,k
+ n yn,N p + n
n,k
n,i .
So, the conclusion follows.
Remark . Let us consider N = and the following sequences:(a) n = + n , n = n =
n , n > ;
(b) n =
A
n , n > A ;
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 34 of 43
(c) n =
n , n =
n/ , rn =
n , n > ;
n/ , n > .
It is easy to see that all hypotheses (i)-(iii), (H), (H) and (H) of Proposition . are satised.
Remark . Under the hypotheses of Proposition ., analogously to Proposition ., one can see that
lim
n Siun yn,i = , i {, . . . , N}.
Corollary . Let us suppose that the hypotheses of either Proposition . or Proposition . are satised. Then w(xn) = w(un) = w(yn,), s(xn) = s(un) = s(yn,) and w(xn) .
Proof By Remark ., we have w(xn) = w(un) and s(xn) = s(un). Note that by Remark .,
lim
n SNun yn,N = .
In the meantime, it is known that
lim
n SNun un =
lim
n un xn =
(d) n, =
n/ , n, =
n , n, =
lim
n xn yn = .
Hence we have
lim
n SNun yn = . (.)
Furthermore, it follows from (.) that
lim
n yn,N yn,N =
lim
n
n,N SNun yn,N = ,
which together with limn SNun yn,N = yields
lim
n SNun yn,N = . (.)
Combining (.) and (.), we conclude that
lim
n yn yn,N = , (.)
which together with limn xn yn = leads to
lim
n xn yn,N = . (.)
Now we observe that
xn yn, xn un + yn, un = xn un + n, Sun un .
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 35 of 43
By Propositions . and ., xn un and Sun un as n , and hence
lim
n xn yn, = .
So we get w(xn) = w(yn,) and s(xn) = s(yn,).
Let p w(xn). Then there exists a subsequence {xni} of {xn} such that xni p. Since p w(un), by Proposition . and Lemma . (demiclosedness principle), we have p
Fix(Si) for each i = , . . . , N, i.e., p
N
i= Fix(Si). Combining (.) and (.), we obtain
xn tn as n . Taking into account p w(tn) and tn Gtn (due to (.)), by Lemma . (demiclosedness principle) we know that p Fix(G) =: . Also, since p w(yn) (due to (.)), in terms of Tyn yn and Lemma . (demiclosedness principle), we get p Fix(T). Moreover, by Lemma . and Proposition . we know that p GMEP(, h). Next we prove that p . As a matter of fact, from (.) and (.)
we know that yni p and yni,N p. Let
Tv =
f (v) + NCv, v C,
, v /
C,
where NCv = {u H : v p, u , p C}. Then
T is maximal monotone and
Tv if
and only if v VI(C, f ); see [] for more details. Let (v, u) G(
T). Since uf (v) NCv
and yn,N C, we have
v yn,N, u f (v)
.
On the other hand, from yn,N = PC(I nfn)yn,N and v C, we have
v yn,N, yn,N
yn,N nfn(yn,N)
,
and hence
# v yn,N, yn,N
yn,N
n + fn(yn,N)
$
.
Therefore we have
v yni,N, u
v yni,N, f (v)
v yni,N, f (v)
# v yni,N, yni,N
yni,Nni + fni (yni,N)
$
= v yni,N, f (v)
# v yni,N, yni,N
yni,Nni + f (yni,N)
$ ni v yni,N, yni,N
= v yni,N, f (v) f (yni,N)
+ v yni,N, f (yni,N) f (yni,N)
# v yni,N, yni,N
yni,N ni
$ ni v yni,N, yni,N
v yni,N, f (yni,N) f (yni,N)
# v yni,N, yni,N
yni,N ni
$ ni v yni,N, yni,N .
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 36 of 43
From (.) and since f is Lipschitz continuous, we obtain that limn f (yni,N) f (yni,N) = . From yni,N p, {n} [a, b] (,
) and (.), we have
T and hence p VI(C, f ), which implies p . Consequently, it is known that p Fix(T)
N
i= Fix(Si) GMEP(, h)
=: .
Theorem . Let us suppose that = . Let {n}, {n,i}, i = , . . . , N, be sequences in (, ) such that < lim infn n,i lim supn n,i < for each index i. Moreover, let us suppose that (H)-(H) hold. Then the sequences {xn}, {yn} and {un} dened by scheme (.) all converge strongly to x = P(I (F f ))x if and only if limn yn Tyn = , provided un un = o( n), where x = P(I (F f ))x is the unique solution of the hierarchical
VIP
( f F)x, x x , x . (.)
Proof First of all, we note that F : C H is -strongly monotone and -Lipschitzian on C and f : C C is an l-Lipschitz continuous mapping with l < . Observe that
%
+ +
.
It is clear that
(F f )x (F f )y, x y ( l) x y , x, y C.
Hence we deduce that F f is ( l)-strongly monotone. In the meantime, it is easy to see that F f is ( + l)-Lipschitz continuous with constant + l > . Thus, there exists a unique solution x in to VIP (.).
Now, observe that there exists a subsequence {xni} of {xn} such that
lim sup
n
( f F)x, xn x = lim
i
( f F)x, xn x = ( f F)x, p x . (.)
v p, u .
Since
T is maximal monotone, we have p
%
( f F)x, xni x . (.)
Since {xni} is bounded, there exists a subsequence {xnij } of {xni} which converges weakly to some p H. Without loss of generality, we may assume that xni p. Then, by Corollary ., we get p w(xn) . Hence, from (.) and (.), we have
lim sup
n
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 37 of 43
Since (H)-(H) hold, the sequence {xn} is asymptotically regular (according to Proposition .). In terms of (.) and Proposition ., xn yn and xn un as n .
Let us show that xn x as n . Indeed, putting p = x, we deduce from (.), (.), (.), (.) and (.) that
xn+ x
n
yn x + ( n) tn x
n
n (
l)
yn,N x + ( n) zn x + n ( V F)x, yn x
+ ( n) tn x
n
n (
l)
yn,N x + ( n) tn x + n ( V F)x, yn x
+ ( n) tn x
= ( n n) tn x + n n (
l)
yn,N x + n n ( V F)x, yn x
( n n)
xn x + nn x + n n (
l)
xn x
+ n n ( V F)x, yn x
= ( n n) xn x + nn x xn x + nn x
+ n n (
l)
xn x + n n ( V F)x, yn x
n n
xn x + n n ( V F)x, yn x
+ nn x xn x + nn x
=
n n
( l)
( l)
xn x
+ n n
( l)
( V F)x, yn x
+ nn x xn x + nn x . (.)
Since
n= n < ,
( l)
n= n = , {n} [a, b] (,
n= nn x ( xn x + nn x ) < ,
) and {n} [c, d] (, ), we
conclude from (.) that
n n
( l)
n=
c n
( l)
=
and
lim sup
n
( l)
( V F)x, yn x
= lim sup
n
( l)
( V F)x, xn x + ( V F)x, yn xn
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 38 of 43
( l)
Applying Lemma . to (.), we infer that the sequence {xn} converges strongly to x. This completes the proof.
In a similar way, we can conclude another theorem as follows.
Theorem . Let us suppose that = . Let {n}, {n,i}, i = , . . . , N, be sequences in (, ) such that n,i i for each index i as n . Suppose that there exists k {, . . . , N} for which n,k as n . Let k {, . . . , N} be the largest index for which n,k .
Moreover, let us suppose that (H), (H) and (H) hold and(i)
n+ n
n,k as n ;
(ii) if i k and n,i i, then n,kn,i as n ;(iii) if n,i i = , then i lies in (, ).
Then the sequences {xn}, {yn} and {un} dened by scheme (.) all converge strongly to x =
P (I (F f ))x if and only if limn yn Tyn = , provided un un = o( nn,k), where x = P(I (F f ))x is the unique solution of the hierarchical VIP
( f F)x, x x , x .
Remark . According to the above argument process for Theorems . and ., we can readily see that if in scheme (.) the iterative step yn = PC[ n Vyn,N +(I nF)GPC(yn,N
nfn(yn,N))] is replaced by the iterative one yn = PC[ n Vxn + (I nF)GPC(yn,N
nfn(yn,N))], then Theorems . and . remain valid.
Remark . Theorems . and . improve, extend, supplement and develop Theorems . and . in [] and Theorems . and . in [] in the following aspects.(i) The multi-step iterative scheme (.) of [] is extended to develop a hybrid extra-gradient viscosity iterative scheme (.) by virtue of Korpelevichs extragradient method, hybrid steepest-descent method [] and gradient-projection method (GPM) with regularization. The iterative scheme (.) is based on Korpelevichs extragradient method, viscosity approximation method [] (see also []), Manns iteration method, hybrid steepest-descent method [] and gradient-projection method (GPM) with regularization.(ii) The argument techniques in our Theorems . and . are very dierent from those techniques in Theorems . and . in [] and Theorems . and . in [] because we make use of the properties of strict pseudocontractions (see Lemmas . and .), the ones of the resolvent operator associated with and h (see Lemmas .-.), the xed point problem x = Gx ( GSVI (.)) (see Proposition .), the equivalence of inclusion problem
Tv to the VIP v VI(C, f ) for maximal monotone operator
T
(see (.)) and the contractive coecient estimates for the contractions T associating with nonexpansive mappings (see Lemma .).(iii) The problem of nding an element of Fix(T)
i= Fix(Si) GMEP(, h) in our Theorems . and . is more general and more subtle than the one of nding an element of Fix(T)
N
i= Fix(Si) GMEP(, h) in Theorems . and . in [] (where
= lim sup
n
.
( V F)x, xn x
N
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 39 of 43
N
i= Fix(Si)
GMEP(, h) in Theorems . and . in [] (where T is a strict pseudocontraction).(iv) Our Theorems . and . generalize Theorems . and . in [] from the non-expansive mapping T to the strict pseudocontraction T and extend them to the setting of GSVI (.), hierarchical VIP (.) and SFP (.). In the meantime, our Theorems . and . extend Theorems . and . in [] to the setting of hierarchical VIP (.) and SFP (.).
4 Applications
For a given nonlinear mapping A : C H, we consider the variational inequality problem (VIP) of nding x C such that
Ax, y x , y C. (.)
We will indicate with VI(C, A) the set of solutions of VIP (.).Recall that if u is a point in C, then the following relation holds:
u VI(C, A) u = PC(I A)u, > .
In the meantime, it is easy to see that the following relation holds:
GSVI (.) with F = VIP (.) with A = F. (.)
An operator A : C H is said to be an -inverse strongly monotone operator if there exists a constant > such that
Ax Ay, x y Ax Ay , x, y C.
As an example, we recall that the -inverse strongly monotone operators are rmly non-expansive mappings if and that every -inverse strongly monotone operator is also
-Lipschitz continuous (see []).
Let us observe also that if A is -inverse strongly monotone, the mappings PC(I A) are nonexpansive for all (, ] since they are compositions of nonexpansive mappings (see p. in []).
Let us consider S, . . . , SM be a nite number of nonexpansive self-mappings on C and A, . . . , AN be a nite number of -inverse strongly monotone operators. Let T : C C be a -strict pseudocontraction with xed points. Let us consider the following mixed problem of nding x Fix(T) GMEP(, h) such that
, y x , y Fix(T) GMEP(, h) , . . . ,
(I SM)x
, y x , y Fix(T) GMEP(, h) , Ax, y x , y C,
Ax, y x , y C, . . . ,
ANx, y x , y C.
T is a nonexpansive mapping) and the one of nding an element of Fix(T)
(I S)x
, y x , y Fix(T) GMEP(, h) , (I S)x
(.)
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 40 of 43
Let us call (SVI) the set of solutions of the (M + N)-system. This problem is equivalent to nding a common xed point of T, {PFix(T)GMEP(,h) Si}
Mi=, {PC(I Ai)}Ni=. So we
Theorem . Let us suppose that = Fix(T) (SVI) GMEP(, h) = . Fix
> . Let {n}, {n,i}, i = , . . . , (M +N), be sequences in (, ) such that < lim infn n,i
lim supn n,i < for all indices i. Moreover, let us suppose that (H)-(H) hold. Then the sequences {xn}, {yn} and {un} explicitly dened by scheme
(un, y) + h(un, y) + rn y un, un xn , y C,yn, = n,PFix(T)GMEP(,h) Sun + ( n,)un,yn,i = n,iPFix(T)GMEP(,h) Siun + ( n,i)yn,i, i = , . . . , M, yn,M+j = n,M+jPC(I Aj)un + ( n,M+j)yn,M+j, j = , . . . , N,
yn,M+N = PC(yn,M+N nfn(yn,M+N)),yn = PC[ n Vyn,M+N + (I nF)GPC(yn,M+N nfn(yn,M+N))],xn+ = nyn + nPC(yn,M+N nfn(yn,M+N)) + nTPC(yn,M+N nfn(yn,M+N)),
all converge strongly to x = P(I (F V))x if and only if limn yn Tyn = , provided un un = o( n), where x = P(I (F V))x is the unique solution of the hierarchical VIP
( V F)x, x x , x .
Theorem . Let us suppose that = . Fix > . Let {n}, {n,i}, i = , . . . , (M + N), be sequences in (, ) and n,i i for all i as n . Suppose that there exists k {, . . . , M +
N} such that n,k as n . Let k {, . . . , M + N} be the largest index for which
n,k . Moreover, let us suppose that (H), (H) and (H) hold and(i)
n+ n
n,k as n ;
(ii) if i k and n,i , then n,kn,i as n ;(iii) if n,i i = , then i lies in (, ).
Then the sequences {xn}, {yn} and {un} explicitly dened by scheme (.) all converge strongly to x = P(I (F V))x if and only if limn yn Tyn = , provided un un = o( nn,k), where x = P(I (F V))x is the unique solution of the VIP
( V F)x, x x , x .
Remark . If in system (.), F = F = A = = AN = and T is a nonexpansive mapping, we obtain a system of hierarchical xed point problems introduced by Mainge and
Mouda [, ].
On the other hand, recall that a mapping S : C C is called -strictly pseudocontractive if there exists a constant [, ) such that
Sx Sy x y +
(I S)x (I S)y , x, y C.
If = , then S is nonexpansive. Put A = I S, where S : C C is a -strictly pseudocon-tractive mapping. Then A is -inverse strongly monotone; see [].
claim that the following holds.
(.)
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 41 of 43
Utilizing Theorems . and ., we also give two strong convergence theorems for nding a common element of the solution set GMEP(, h) of GMEP (.), the solution set of SFP (.) and the common xed point set N
i= Fix(Si) Fix(S) of nitely many non-expansive mappings Si : C C, i = , . . . , N, and a -strictly pseudocontractive mapping
S : C C.
Theorem . Let (, ). Let us suppose that =
i= Fix(Si) Fix(S) GMEP(, h) = . Let {n}, {n,i}, i = , . . . , N, be sequences in (, ) such that < lim infn n,i
lim supn n,i < for all indices i. Moreover, let us suppose that there hold (H)-(H) with
n = , n . Then the sequences {xn}, {yn} and {un} generated explicitly by
(un, y) + h(un, y) + rn y un, un xn , y C, yn, = n,Sun + ( n,)un,yn,i = n,iSiun + ( n,i)yn,i, i = , . . . , N,
yn,N = PC(yn,N nfn(yn,N)),tn = PC(yn,N nfn(yn,N)),yn = PC[ n Vyn,N + (I nF)(( )tn + Stn)], xn+ = nyn + ( n)tn, n ,
all converge strongly to x = P(I (F V))x, provided un un = o( n), which is the unique solution of the VIP
( V F)x, x x , x .
Proof In Theorem ., put F = A = I S and F = . Then A is -inverse strongly monotone. Hence we deduce that Fix(S) = VI(C, A) = and
Gtn = PC(I F)PC(I F)tn
= PC(I F)tn
= ( )tn + Stn.
Thus, in terms of Theorem ., we obtain the desired result.
Theorem . Let (, ). Let us suppose that =
i= Fix(Si) Fix(S) GMEP(, h) = . Let {n}, {n,i}, i = , . . . , N, be sequences in (, ) such that n,i i for all i as n . Suppose that there exists k {, . . . , N} for which n,k as n . Let k {, . . . , N} be the largest index for which n,k . Moreover, let us suppose that there hold (H), (H) and (H) with n = , n and(i)
n+ n
n,k as n ;
(ii) if i k and n,i , then n,kn,i as n ;(iii) if n,i i = , then i lies in (, ).
Then the sequences {xn}, {yn} and {un} generated explicitly by (.) all converge strongly to x = P(I (F V))x, provided un un = o( nn,k), which is the unique solution of the hierarchical VIP
( V F)x, x x , x .
N
(.)
N
Ceng et al. Journal of Inequalities and Applications (2015) 2015:150 Page 42 of 43
Competing interests
The authors declare that they have no competing interests.
Authors contributions
All authors read and approved the nal manuscript.
Author details
1Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China. 2Scientic Computing Key Laboratory of Shanghai Universities, Shanghai, 200234, China. 3Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan. 4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan. 5Department of Mathematics, Chung Yuan Christian University, Taoyuan, 320, Taiwan.
Acknowledgements
LC Ceng was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and PhD Program Foundation of Ministry of Education of China (20123127110002). CF Wen was partially supported by grants from MOST. YC Liou was supported in part by grants from MOST NSC 101-2628-E-230-001-MY3 and NSC 103-2923-E-037-001-MY3. This research is supported partially by Kaohsiung Medical University Aim for the Top Universities Grant, grant No. KMU-TP103F00.
Received: 13 February 2015 Accepted: 30 March 2015
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The Author(s) 2015
Abstract
In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of a general system of variational inequalities, the set of solutions of a split feasibility problem (SFP), and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest-descent method and gradient-projection method (GPM) with regularization. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.
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