A shrinking projection approximant for the split equilibrium problems ...

arXiv:1803.02052v1 [math.FA] 6 Mar 2018

A shrinking projection approximant for the split equilibrium problems and fixed point problems in Hilbert spaces Abdul Ghaffar1 , Zafar Ullah2 , Muhammad Aqeel Ahmad Khan3,∗ and Faisal Mumtaz3 1 Department of Mathematical Sciences, Balochistan University of Information Technology, Engineering and Management Sciences, Quetta 87300, Pakistan 2 Department of Mathematics, University of Education Lahore, DG Khan Campus, DG Khan 32200, Pakistan 3 Department of Mathematics, COMSATS Institute of Information Technology Lahore, Lahore, 54000, Pakistan March 06, 2018 Abstract: This work is devoted to establish the strong convergence results of an iterative algorithm generated by the shrinking projection method in Hilbert spaces. The proposed approximation sequence is used to find a common element in the set of solutions of a finite family of split equilibrium problems and the set of common fixed points of a finite family of total asymptotically strict pseudo contractions in such setting. The results presented in this paper improve and extend some recent corresponding results in the literature. Keywords and Phrases: Split equilibrium problem; fixed point problem; total asymptotically strict pseudo contraction; inverse strongly monotone mapping; shrinking projection method; Hilbert space 2010 Mathematics Subject Classification: Primary: 47H05; 47H09; 47J05; Secondary: 49H05. 1. Introduction and Preliminaries Throughout this paper, we write xn → x (resp. xn ⇀ x) to indicate the strong convergence (resp. the weak convergence) of a sequence {xn }∞ n=1 . Let C be a nonempty subset of a real Hilbert space H and let T : C → C be a mapping. The set of fixed points of the mapping T is defined and denoted: F (T ) = {x ∈ C : T (x) = x}. A selfmapping T is said to be: (i) nonexpansive if kT x − T yk ≤ kx − yk ; (ii) asymptotically nonexpansive [15] if there exists a sequence {λn } ⊂ [0, ∞) with limn→∞ λn = 0 such that kT n x − T n yk ≤ (1 + λn ) kx − yk , n ≥ 1; (iii) Lipschitzian if kT n x − T n yk ≤ Θ kx − yk for some Θ > 0; (iv) firmly nonexpansive if kT x − T yk2 + k(I − T )x − (I − T )yk2 ≤ kx − yk2 ;

(1.1)

(v) pseudo-contraction [4], if kT x − T yk2 ≤ kx − yk2 + k(I − T )x − (I − T )yk2 ;

(1.2)

(vi) k-strict pseudo contraction [4], if there exists k ∈ [0, 1) such that kT x − T yk2 ≤ kx − yk2 + kk(I − T )x − (I − T )yk2 ;

(1.3)

*Corresponding author E-mail addresses: (A. Ghaffar) [email protected], (Z. Ullah) [email protected], (M.A.A. Khan) [email protected], [email protected] (F. Mumtaz) [email protected] 1

(vii). (k, {λn })-asymptotically strict pseudo contraction [22], if there exist a constant k ∈ [0, 1) and a sequence {λn } ⊂ [1, ∞) with limn→∞ λn = 1 such that kT n x − T n yk2 ≤ λn kx − yk2 + kk(I − T n )x − (I − T n )y)k2 ;

(1.4)

(viii). ({λn }, {µn }, ξ)-total asymptotically nonexpansive [1] if there exist nonnegative ∞ real sequences {λn }∞ n=1 , {µn }n=1 with limn→∞ λn = 0 = limn→∞ µn and a strictly increasing continuous function ξ : R+ → R+ with ξ(0) = 0 such that kT n x − T n yk ≤ kx − yk + λn ξ(kx − yk) + µn ;

(1.5)

(ix). (k, {λn }, {µn }, ξ)-total asymptotically strictly pseudo contraction [27], if there exist ∞ a constant k ∈ [0, 1) and nonnegative real sequences {λn }∞ n=1 , {µn }n=1 with limn→∞ λn = + 0 = limn→∞ µn and a strictly increasing continuous function ξ : R → R+ with ξ(0) = 0 such that kT n x − T n yk2 ≤ kx − yk2 + kk(I − T n )x − (I − T n )y)k2 + λn ξ(kx − yk) + µn , (1.6) holds for all x, y ∈ C. Remark 1.1. It is worth mentioning that the class of nonexpansive mappings have powerful applications to solve various problems arising in the field of applied mathematics, such as variational inequality problem, convex minimization, zeros of a monotone operator, initial value problems of differential equations, game-theoretic model and image recovery. It is therefore, natural to extend such powerful results of the class of nonexpansive mappings to the more general class of mappings. As a consequence, the notion of nonexpansive mapping has been generalized in several ways. In 1967, Browder and Petryshyn [4] introduced the concept of strict pseudo contraction as a generalization of nonexpansive mappings. Later on, Alber et al.[1] introduced the notion of total asymptotically nonexpansive mappings which is more general in nature and unifies various definitions of mappings associated with the class of asymptotically nonexpansive mappings. In 2011, Yang et al. [27] introduced the notion of total asymptotically strict pseudo contraction which contains properly the class of total asymptotically nonexpansive mappings and strict pseudo contractions. So, we study this general class of mappings to contribute in metric fixed point theory. Let C be a nonempty subset of a real Hilbert space H1 , Q be a nonempty subset of a real Hilbert space H2 and let A : H1 → H2 be a bounded linear operator. Let f : C × C → R and g : Q × Q → R be two bifunctions. The split equilibrium problem (SEP) is to find: x∗ ∈ C

such that f (x∗ , x) ≥ 0 for all x ∈ C,

(1.7)

and y ∗ = Ax∗ ∈ Q such that g (y ∗ , y) ≥ 0 for all y ∈ Q. (1.8) It is remarked that inequality (1.7) represents the classical equilibrium problem [12] and its solution set is denoted EP (f ). Moreover, inequalities (1.7) and (1.8) constitute a pair of equilibrium problems which aim to find a solution x∗ of an equilibrium problem (1.7) such that its image y ∗ = Ax∗ under a given bounded linear operator A also solves another equilibrium problem (1.8). The set of solutions of SEP (1.7) and (1.8) is denoted Ω = {z ∈ EP (f ) : Az ∈ EP (g)}. Equilibrium problem theory provides a unified approach to address a variety of mathematical problems arising in various disciplines. In 2012, Censor et al. [9] proposed the theory of split variational inequality problems (SVIP) whereas Moudafi [21] generalized the concept of SVIP to that of split monotone variational inclusions (SMVIP). The split equilibrium problems is a special case of SMVIP. The SMVIP have already been 2

studied and successfully employed as a model in intensity-modulated radiation therapy treatment planning, see [7, 8]. Moreover, this formalism is also at the core of modeling of many inverse problems arising for phase retrieval and other real-world problems; for instance, in sensor networks in computerized tomography and data compression; see, for example, [6, 11]. Some methods have been proposed and analyzed to solve SEP together with the fixed point problem in Hilbert spaces, see, for example [16, 17, 18, 24] and the references cited therein. In 2013, Chang et al. [10] studied the split feasibility problem for a total asymptotically strict pseudo contraction in infinitely dimensional Hilbert spaces. In 2015, Ma and Wang [19] established strong convergence results for the split common fixed point problem of total asymptotically strict pseudo contractions in Hilbert spaces. Quite recently, some methods have been proposed and analyzed in [17, 18] for the split equilibrium problem. Inspired and motivated by the above mentioned results and the ongoing research in this direction, we aim to employ a hybrid shrinking projection algorithm to find a common element in the set of solutions of a finite family of split equilibrium problems and the set of common fixed points of a finite family of total asymptotically strict pseudo contractions in Hilbert spaces. Our results can be viewed as a generalization and improvement of various existing results in the current literature.

2. Preliminaries This section is devoted to recall some definitions and results required in the sequel. Let C be a nonempty closed convex subset of a Hilbert space H1 . For each x ∈ H1 , there exists a unique nearest point of C, denoted by PC x, such that kx − PC xk ≤ kx − yk for all y ∈ C. Such a mapping PC : H1 → C is known as a metric projection or a nearest point projection of H1 onto C. Moreover, PC satisfies nonexpansiveness in a Hilbert space and hx − PC x, PC x − yi ≥ 0 for all x, y ∈ C. It is remarked that PC is firmly nonexpansive mapping from H1 onto C, that is, kPC x − PC yk2 ≤ hx − y, PC x − PC y i , for all x, y ∈ C. Recall that a nonlinear mapping A : C → H1 is λ-inverse strongly monotone if it satisfies hx − y , Ax − Ayi ≥ λ kAx − Ayk2 . Note that, if A := I − T is a λ-inverse strongly monotone mapping, then: 1 (i): A is a λ -Lipschitz continuous mapping; (ii): if T is a nonexpansive mapping, then A is a 21 -inverse strongly monotone mapping; (iii): if η ∈ (0, 2λ], then I − ηA is a nonexpansive mapping. The following lemma collects some well-known equations in the context of a real Hilbert space. Lemma 2.1. Let H1 be a real Hilbert space, then: (i): kx − yk2 = kxk2 − kyk2 − 2 hx − y, yi , for all x, y ∈ H1 ; (ii): kx + yk2 ≤ kxk2 + 2 hx − y, yi , for all x, y ∈ H1 ; (iii): kαx + (1 − α)yk2 = α kxk2 + (1 − α) kyk2 − α(1 − α) kx − yk2 for all x, y ∈ H1 and α ∈ [0, 1]. 3

Lemma 2.2 [20]. Let T : C → C be a (k, {λn }, {µn }, ξ)-total asymptotically strictly pseudo contraction. If F (T ) 6= ∅, then for each p ∈ F (T ) and for each x ∈ C, the following equivalent inequalities hold: kT n x − pk2 ≤ kx − pk2 + k kx − T n xk2 + λn ξ(kx − pk) + µn , 1−k λn µ kx − T n xk2 − ξ(kx − pk) − n , 2 2 2 λn µ 1+k kx − T n xk2 + ξ(kx − pk) + n . hx − T n x, p − T n xi ≤ 2 2 2 hx − T n x, x − pi ≥

(2.1) (2.2) (2.3)

Lemma 2.3 [20]. Let C be a nonempty subset of a real Hilbert space H1 and let S : C → C be a uniformly Θ-Lipschitzian and (k, {λn }, {µn }, ξ)-total asymptotically strictly pseudo contraction, then S is demiclosed at origin. That is, if for any sequence {xn } in C with xn ⇀ x and kxn − Sxn k → 0, we have x = Sx. Condition 2.4 [3, 12]. Let f : C × C → R be a bifunction satisfying the following conditions: 1. f (x, x) = 0 for all x ∈ C; 2. f is monotone, that is, f (x, y) + f (y, x) ≤ 0 for all x, y ∈ C; 3. f is upper hemicontinuous, that is, for each x, y, z ∈ C, lim f (tz + (1 − t)x, y) ≤ f (x, y);

t→0

4. for each x ∈ C, the function y 7→ f (x, y) is convex and lower semi-continuous. Lemma 2.5 [12]. Let C be a closed convex subset of a real Hilbert space H1 and let f : C × C → R be a bifunction satisfying Lemma 2.4. For r > 0 and x ∈ H1 , there exists z ∈ C such that 1 F (z, y) + hy − z, z − xi ≥ 0, for all y ∈ C. r F Moreover, define a mapping Tr : H1 → C by 1 f Tr (x) = z ∈ C : f (z, y) + hy − z, z − xi ≥ 0, for all y ∈ C , r for all x ∈ H1 . Then, the following hold: (i) Trf is single-valued; (ii) Trf is firmly nonexpansive, i.e., for every x, y ∈ H,

2 D E

f

Tr x − Trf y ≤ Trf x − Trf y, x − y

(ii) F (Trf ) = EP (f ); (iv) EP (f ) is closed and convex.

It is remarked that if g : Q × Q → R is a bifunction satisfying Lemma 2.4, then for s > 0 and w ∈ H2 we can define a mapping: 1 g Ts (w) = d ∈ C : g(d, e) + he − d, d − wi ≥ 0, for all e ∈ Q , s which is, nonempty, single-valued and firmly nonexpansive. Moreover, EP (g) is closed and convex, and F (Tsg ) = EP (g). 4

3. Main results We now prove our main result of this section. Theorem 3.1. Let H1 and H2 be two real Hilbert spaces and let C ⊆ H1 and Q ⊆ H2 be nonempty closed convex subsets of Hilbert spaces H1 and H2 , respectively. Let fi : C × C → R and gi : Q × Q → R be two finite families of bifunctions satisfying Condition 2.4 such that gi be upper semicontinuous for each i ∈ {1, 2, 3, · · · , N }. Let Si : C → C be a finite family of uniformly Θ-Lipschitzian and continuous total asymptotically strict pseudo contractions and let Ai : H1 → H2 be a finite family linear hT of bounded i N operators for each i ∈ {1, 2, 3, · · · , N }. Suppose that F := i=1 F (Si ) ∩ Ω 6= ∅, n o TN where Ω = z ∈ C : z ∈ i=1 EP (fi ) and Ai z ∈ EP (gi ) for 1 ≤ i ≤ N . Let {xn } be a sequence generated by: x1 ∈ C1 =C, un = Trfnn xn − γA∗n(mod N ) (I − Tsgnn ) An(mod N ) xn , n yn = αn un + (1 − αn ) Sn(mod N ) un , o n Cn+1 = z ∈ H 1 : kyn − zk 2 ≤ kxn − zk2 + θn , xn+1 = PCn+1 x1 , n ≥ 1,

(3.1)

where θn = (1 − αn ) {λn ξ n (Mn ) + λn Mn∗ Dn + µn } with Dn = sup {kxn − pk : p ∈ F}. Let {rn }, {sn } be two positive real sequences and let {αn } be in (0, 1). Assume that if the following set of conditions holds: (C1): 0 ≤ k < a ≤ αn ≤ b < 1 and γ ∈ 0, L1 where L = max {L1 , L2 , · · · , LN } and Li is the spectral radius of the operator A∗i Ai and A∗i is the adjoint of Ai for each i ∈ {1, 2, 3, · · · , N }; (C2): lim inf rn > 0 and lim inf sn > 0; n→∞ n→∞ ∞ ∞ P P (C3): λn < ∞ and µn < ∞; n=1

n=1

(C4): there exist constants Mi , Mi∗ > 0 such that ξ i (λi ) ≤ Mi∗ λi for all λi ≥ Mi , i = 1, 2, 3, · · · , N, then the sequence {xn } generated by (3.1) converges strongly to PF x1 . Proof. For the sake of simplicity, we define An = An(mod N ) and Sn = Sn(mod N ) for all n ≥ 1. We start our proof to establish that the sequence {xn } defined in (3.1) is well defined. In order to prove this assertion, we first show by mathematical induction that F ⊂ Cn for all n ≥ 1. Obviously, F ⊂ C1 = C. Now, assume that F ⊂ Ci for some i ≥ 1. Then it follows from (3.1) that

2

fi 2 fi ∗ gi kui − pk = Tri xi − γAi I − Tsi Ai xi − Tri p

2 ≤ xi − γA∗i I − Tsgii Ai xi − p

2

≤ kxi − pk2 + γ 2 A∗i I − Tsgii Ai xi + 2γ p − xi , A∗i I − Tsgii Ai xi

≤ kxi − pk2 + γ 2 Ai xi − Tsgii Ai xi , Ai A∗i I − Tsgii Ai xi

+2γ p − xi , A∗i I − Tsgii Ai xi

≤ kxi − pk2 + Lγ 2 Ai xi − Tsgii Ai xi , Ai xi − Tsgii Ai xi

+2γ p − xi , A∗i I − Tsgii Ai xi

2

= kxi − pk2 + Lγ 2 Ai xi − Tsgii Ai xi + 2γ p − xi , A∗i I − Tsgii Ai xi(3.2) . 5

Now letting Λ = 2γ hp − xi , A∗i (I − Tsgii ) Ai xn i , we have

Λ = 2γ p − xi , A∗i I − Tsgii Ai xi

= 2γ Ai (p − xi ) , Ai xi − Tsgii Ai xi

= 2γ Ai (p − xi ) + Ai xi − Tsgii Ai xi − Ai xi − Tsgii Ai xi , Ai xi − Tsgii Ai xi n

2 o = 2γ Ai p − Tsgii Ai xi , Ai xi − Tsgii Ai xi − Ai xi − Tsgii Ai xi

2

2 1 gi gi

≤ 2γ Ai xi − Tsi Ai xi − Ai xi − Tsi Ai xi 2

2 = −γ Ai xi − Tsgi Ai xi . i

Using the above simplification of Λ in (3.2), we get

2

i kui − pk2 ≤ kxi − pk2 + γ (Lγ − 1) Ai xi − Tsgn,i Ai xi .

Since γ ∈ 0, L1 by condition (C1), the above estimate then yields kui − pk2 ≤ kxi − pk2 .

(3.3)

(3.4)

Making use of (3.4), we have the following estimate:

2 kyi − pk2 = αi ui + (1 − αi ) Sii ui − p

2

2 = αi kui − pk2 + (1 − αi ) Sii ui − p − αi (1 − αi ) ui − Sii ui o n

2 ≤ αi kui − pk2 + (1 − αi ) kui − pk2 + k ui − Sii ui + λi ξ i (kui − pk) + µi

2 −αi (1 − αi ) ui − Sii ui o n

2 ≤ kui − pk2 + (1 − αi ) k ui − Sii ui + λi ξ i (Mi ) + λi Mi∗ kui − pk2 ) + µi

2 −αi (1 − αi ) ui − Sii ui

2 ≤ kxi − pk2 − (1 − αi ) (αi − k) ui − Sii ui o n (3.5) + (1 − αi ) λi ξ i (Mi ) + λi Mi∗ kxi − pk2 + µi . Since αi − k ≥ 0 by condition (C1), so (3.5) implies that kyi − pk2 ≤ kxi − pk2 + θi ,

(3.6)

n o where θi = (1 − αi ) {λi ξ i (Mi ) + λi Mi∗ Di + µi } and Di = sup kxi − pk2 : p ∈ F . It now follows from the estimate (3.6) that p ∈ Ci+1 . Hence, F ⊂ Cn for all n ≥ 1. Next, we show that the set Cn is closed and convex for all n ≥ 1. Since n o z ∈ H1 : kyn − zk 2 ≤ kxn − zk2 = z ∈ H1 : kyn k 2 − kxn k 2 ≤ 2 hyn − xn , zi +θn ,

it is closed and convex; hence the sequence {xn } defined in (3.1) is well-defined. Next, from xn = PCn x1 , we get 0 ≤ hxn − x1 , x∗ − xn i , 6

for all x∗ ∈ Cn . Since F 6= ∅, then for any p ∈ F , we get 0 ≤ hxn − x1 , p − xn i = hxn − x1 , p + x1 − x1 − xn i = hxn − x1 , x1 − xn i + hxn − x1 , p − x1 i = − kxn − x1 k2 + kxn − x1 k kp − x1 k . That is, kxn − x1 k ≤ kp − x1 k , for all p ∈ F and n ≥ 1. Hence, the sequence {xn } is bounded, so are {un } and {yn }. Moreover, from xn = PCn x1 and xn+1 = PCn+1 x1 ∈ Cn+1 ⊂ Cn , we have 0 ≤ hxn − x1 , xn+1 − xn i . Similarly, we get the following relation: kxn − x1 k ≤ kxn+1 − x1 k , for all n ≥ 1. From the above assertions, we conclude that the sequence {kxn − x1 k} is bounded and nondecreasing, therefore, we have lim kxn − x1 k exists.

n→∞

(3.7)

Further observe that kxn+1 − xn k2 = kxn+1 − x1 + x1 − xn k2 = kxn+1 − x1 k2 + kxn − x1 k2 − 2 hxn − x1 , xn+1 − x1 i = kxn+1 − x1 k2 + kxn − x1 k2 − 2 hxn − x1 , xn+1 − xn + xn − x1 i = kxn+1 − x1 k2 − kxn − x1 k2 − 2 hxn − x1 , xn+1 − xn i ≤ kxn+1 − x1 k2 − kxn − x1 k2 . From (3.7), we obtain that kxn+1 − x1 k2 −kxn − x1 k2 → 0 as n → ∞, hence the sequence {kxn − x1 k} is Cauchy. That is lim kxn+1 − xn k = 0.

n→∞

(3.8)

Since xn+1 ∈ Cn+1 , which implies that kyn − xn+1 k ≤ kxn − xn+1 k + θn,i . From (3.8), we conclude that lim kyn − xn+1 k = 0, for all n ≥ 1. (3.9) n→∞

Now using (3.8), (3.9) and the following triangular inequality, we get kyn − xn k ≤ kyn − xn+1 k + kxn+1 − xn k → 0

(3.10)

as n → ∞. Consider from (3.1), (3.3) and (3.6), we get

2 γ (1 − γL) An xn − Tsgnn An xn ≤ kxn − pk2 − kun − pk2

≤ kxn − pk2 − kyn − pk2 + θn ≤ (kxn − pk + kyn − pk) kxn − yn k + θn .

Utilizing the fact that γ (1 − γL) > 0 and the estimate (3.10), we have

2 lim An xn − Tsgnn An xn = 0 for all n ≥ 1. n→∞

7

(3.11)

For any p ∈ F and firm nonexpansiveness of Trfnn , we have

2

fn 2 fn ∗ gn kun − pk = Trn xn − γAn I − Tsn An xn − Trn p

≤ un − p, xn − γA∗n I − Tsgnn An xn − p

2

1 = {kun − pk2 + xn − γA∗n I − Tsgnn An xn − p 2

2 − un − xn − γA∗n I − Tsgnn An xn }

2 o 1n ≤ kun − pk2 + kxn − pk2 − un − xn − γA∗n I − Tsgnn An xn 2

2 1 {kun − pk2 + kxn − pk2 − (kun − xn k2 + γ 2 A∗n I − Tsgnn An xn = 2

−2γ un − xn , A∗n I − Tsgnn An xn )}

≤ kxn − pk2 − kun − xn k2 + 2γ kA (un − xn )k An xn − Tsgn An xn .(3.12) n

Using the following estimate:

kyn − pk2 ≤ αn kxn − pk2 + (1 − αn ) kun − pk2 + θn . in (3.12) and re-arranging the terms, we get (1 − αn ) kun − xn k2 ≤ (kxn − pk + kyn − pk) kxn − yn k

+2γ kA (un − xn )k An xn − Tsgnn An xn + θn .

Letting n → ∞ and utilizing (3.10) and (3.11), we have

lim kun − xn k = 0 for all n ≥ 1.

n→∞

(3.13)

Moreover, from (3.10) and (3.13), we obtain kyn − un k ≤ kyn − xn k + kxn − un k → 0

(3.14)

when n → ∞. Observe that kyn − un k = (1 − αn ) kSnn un − un k . Then it follows from condition (C1) and (3.14) that lim kSnn un − un k = 0 for all n ≥ 1.

n→∞

(3.15)

Since kSnn un − xn k ≤ kSnn un − un k + kxn − un k . Therefore from (3.13) and (3.15), we get lim kSnn un − xn k = 0 for all n ≥ 1.

n→∞

(3.16)

On a similar reasoning, we also obtain lim kSnn un − yn k = 0 for all n ≥ 1.

n→∞

Observe that each Sn is uniformly Θ-Lipschitzian, therefore, we have kSnn xn − xn k ≤ kSnn xn − Snn un k + kSnn un − xn k ≤ Θ kxn − un k + kSnn un − xn k . Now, using (3.13), (3.16) and the above estimate, we get lim kSnn xn − xn k = 0 for all n ≥ 1.

n→∞

8

(3.17)

Moreover, utilizing the uniform continuity of Sn and (3.17), the following estimate: kxn − Sn xn k ≤ kxn − Snn xn k + kSnn xn − Sn xn k implies that lim kxn − Sn xn k = 0 for all n ≥ 1.

n→∞

(3.18)

Similarly, we also have that lim kun − Sn un k = 0 for all n ≥ 1.

n→∞

(3.19)

Now, we show that ω(xn ) ⊂ F, where ω(xn ) is the set of all weak ω-limits of {xn }. Since {xn } is bounded, therefore ω(xn ) 6= ∅. Let q ∈ ω(xn ), then there exists a subsequence {xN n+i } of {xn } such that xN n+i ⇀ q. Using the fact that SN n+i = Si for all n ≥ 1 and the demiclosed principle (Lemma 2.3) for each Si , we have that x ∈ F (Si ) for T each 1 ≤ i ≤ N. Next, we show that q ∈ Ω, i.e., q ∈ N i=1 EP (fi ) and Ai q ∈ EP (gi ) TN for each 1 ≤ i ≤ N. In order to show that q ∈ i=1 EP (fi ), that is, q ∈ EP (fi ) for each 1 ≤ i ≤ N, we define subsequence {nj } of index {n} such that nj = N j + i for all a consequence, we can write fnj = fi for 1 ≤ i ≤ N. From unj = n ≥ 1. As gn

Trfnij I − γA∗nj I − Tsnjj Anj xnj for all n ≥ 1, we have E gn 1 D fi (unj , y) + y − unj , unj − xnj − γA∗nj I − Tsnjj Anj xnj ≥ 0, for all y ∈ C. rn j This implies that E gn 1 D 1

y − unj , unj − xnj − y − unj , γA∗nj I − Tsnjj Anj xnj ≥ 0 fi (unj , y) + rnj rnj From (A2), we have E gn 1 D 1

y − unj , unj − xnj − y − unj , γA∗nj I − Tsnjj Anj xnj ≥ fi (y, unj ), rnj rnj

for all y ∈ C. Since lim inf j→∞ rni > 0 (by (C2)), therefore it follows from (3.11) and (3.13) that fi (y, q) ≤ 0, for all y ∈ C and for 1 ≤ i ≤ N. Let yt = ty + (1 − t)q for some 0 < t < 1 and y ∈ C. Since q ∈ C, this implies that yt ∈ C. Using (A1) and (A4) from Condition 2.4, the following estimate: 0 = fi (yt , yt ) ≤ tfi (yt , y) + (1 − t)fi (yt , q) ≤ tfi (yt , y), implies that fi (yt , y) ≥ 0, for 1 ≤ i ≤ N. Letting t → 0, we have fi (q, y) ≥ 0 for all y ∈ C. Thus, q ∈ EP (fi ) for 1 ≤ i ≤ N. That T is, q ∈ N i=1 EP (Fi ). Reasoning as above, we show that Ai q ∈ EP (gi ) for each 1 ≤ i ≤ N. Since xnl −→ q and Anl is a bounded linear operator, therefore Anl xnl −→ Anl q. Hence, it follows from (3.11) that gn

Tsnll Anl xnl −→ Anl q

as

l → ∞.

Now, from Lemma 2.5, we have g E 1 D gn gn n gi Tsnll Anl xnl , z + z − Tsnll Anl xnl , Tsnll Anl xnl − Anl xnl ≥ 0, for all z ∈ Q. s nl 9

Since gi is upper hemicontinuous in the first argument for each 1 ≤ i ≤ N , therefore taking lim sup on both sides of the above estimate as l → ∞ and utilizing (C2) and (3.11), we get gi (Anl x, z) ≥ 0, for all z ∈ Q and for each 1 ≤ i ≤ N. Hence Ai q ∈ EP (gi ) for each 1 ≤ i ≤ N and consequently q ∈ F. It remains to show that xn → q = PF x1 . Let x = PF x1 , then from kxn − x1 k ≤ kx − x1 k , therefore, we have kx − x1 k ≤ kq − x1 k

≤ lim inf xnj − x1 j→∞

≤ lim sup xnj − x1 j→∞

≤ kx − x1 k .

This implies that

lim xnj − x1 = kq − x1 k .

j→∞

Hence xnj → q = PF x1 . From the arbitrariness of the subsequence xnj of {xn } , we conclude that xn → x as n → ∞. It is easy to see that yn,i → x and un,i → x. This completes the proof. Corollary 3.2. Let H1 and H2 be two real Hilbert spaces and let C ⊆ H1 and Q ⊆ H2 be nonempty closed convex subsets of Hilbert spaces H1 and H2 , respectively. Let fi : C × C → R and gi : Q × Q → R be two finite families of bifunctions satisfying Condition 2.4 such that gi be upper semicontinuous for each i ∈ {1, 2, 3, · · · , N }. Let Si : C → C be a finite family nonexpansive mappings and let Ai : H1 → H2 be a finite hT family of i bounded N linear operators for each i ∈ {1, 2, 3, · · · , N }. Suppose that F := i=1 F (Si ) ∩ Ω 6= ∅, n o TN where Ω = z ∈ C : z ∈ i=1 EP (fi ) and Ai z ∈ EP (gi ) for 1 ≤ i ≤ N . Let {xn } be a sequence generated by: x1 ∈ C1 =C, fn gn ∗ un = Trn xn − γAn(mod N ) (I − Tsn ) An(mod N ) xn , n yn = αn un + (1 − αn ) Sn(mod N ) un , o n 2 Cn+1 = z ∈ H 1 : kyn − zk ≤ kxn − zk2 + θn , xn+1 = PCn+1 x1 , n ≥ 1,

(3.20)


n=1

(C4): there exist constants Mi , Mi∗ > 0 such that ξ i (λi ) ≤ Mi∗ λi for all λi ≥ Mi , i = 1, 2, 3, · · · , N, then the sequence {xn } generated by (3.20) converges strongly to PF x1 . Theorem 3.3. Let H1 and H2 be two real Hilbert spaces and let C ⊆ H1 and Q ⊆ H2 10

be nonempty closed convex subsets of Hilbert spaces H1 and H2 , respectively. Let fi : C × C → R and gi : Q × Q → R be two finite families of bifunctions satisfying Condition 2.4 such that gi be upper semicontinuous for each i ∈ {1, 2, 3, · · · , N }. Let Si : C → C be a finite family of uniformly Θ-Lipschitzian and continuous total asymptotically strict pseudo contractions and let Ai : H1 → H2 be a finite family linear hT of bounded i N operators for each i ∈ {1, 2, 3, · · · , N }. Suppose that F := i=1 F (Si ) ∩ Ω 6= ∅, n o TN where Ω = z ∈ C : z ∈ i=1 EP (fi ) and Ai z ∈ EP (gi ) for 1 ≤ i ≤ N . Let {xn } be a sequence generated by: x1 ∈ C1 =C, un = Trfnn xn − γA∗n(mod N ) (I − Tsgnn ) An(mod N ) xn , n yn = αn un + (1 − αn ) Sn(mod N ) un , o n 2 Cn+1 = z ∈ H 1 : kyn − zk ≤ kxn − zk2 + θn , xn+1 = PCn+1 x1 , n ≥ 1,

(3.21)


n=1

(C4): there exist constants Mi , Mi∗ > 0 such that ξ i (λi ) ≤ Mi∗ λi for all λi ≥ Mi , i = 1, 2, 3, · · · , N, then the sequence {xn } generated by (3.21) converges strongly to PF x1 . Proof. Set H1 = H2 , C = Q and A = I(the identity mapping) then the desired result then follows from Theorem 3.1 immediately. Acknowledgment. The author M. A. A. Khan would like to acknowledge the support provided by the Higher Education Commission of Pakistan for funding this work through project No. NRPU 5332. References [1] Ya. I. Alber, C. E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., (2006), 2006:10673, 20 pp. [2] J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, 1998, Springer, New York, NY. [3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145. [4] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228. [5] R.E. Bruck, T. Kuczumow and S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Collo. Math., 65 (1993) 169-179. [6] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453. [7] Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071-2084. [8] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. 11

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