APPROXIMATING A COMMON SOLUTION OF A FINITE FAMILY OF GENERALIZED EQUILIBRIUM AND FIXED POINT PROBLEMS

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1 SINET: Ethiop. J. Sci., 38():7 28, 205 ISSN: (Print) College of Natural Sciences, Addis Ababa University, (Online) APPROXIMATING A COMMON SOLUTION OF A FINITE FAMILY OF GENERALIZED EQUILIBRIUM AND FIXED POINT PROBLEMS Tesfalem Hadush Meche, Mengistu Goa Sangago 2 and Habtu Zegeye 3 Department of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, PO Box 76, Addis Ababa, Ethiopia. tesfalemh78@gmail.com 2 Department of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, PO Box 367, Addis Ababa, Ethiopia. mengistu.goa@aau.edu.et 3 Department of Mathematics, College of Sciences, Botswana International University of Science and Technology, Private Mail Bag 6, Palapye, Botswana. habtuzh@yahoo.com ABSTRACT: In this paper, we introduce and investigate an iterative scheme for finding a common element of the set of common solutions of a finite family of generalized equilibrium problems and the set of fixed points of a Lipschitz and hemicontractive-type multi-valued mapping. We obtain strong convergence theorems of the proposed iterative process in real Hilbert space settings. Our results improve, generalize and extend most of the recent results that have been proved by many authors in this research area. Key words/phrases: Continuous monotone mapping, demiclosedness principle, fixed point problem, generalized equilibrium problems, hemicontractive-type multi-valued mapping INTRODUCTION Let CC be a nonempty subset of a real Hilbert space HH with inner product.,. and norm.. A single-valued mapping TT: CC HH is said to be k-strictly pseudocontractive in the sense of Browder and Petryshtn (967) if there exists kk [0,) such that TTTT TTTT 2 xx yy 2 + kk xx TTTT (yy TTTT) 2, xx, yy CC... (.) If kk = in (.), then TT is called pseudocontractive mapping. A mapping TT: CC HH is called Lipschitzian if there exists LL 0 such that TTTT TTTT LL xx yy, xx, yy CC. If LL =, then TT is called nonexpansive and if LL [0,), then TT is called contraction. Observe that the class of pseudocontractive mappings contains the class of kk-strictly pseudocontractive mappings and nonexpansive mappings (see Browder and Petryshyn, 967; Chidume et al., 203). A mapping TT: CC HH is said to be firmly nonexpansive if TTTT TTTT 2 TTTT TTTT, xx yy for all xx, yy CC. It is known that every firmly nonexpansive mapping is nonexpansive mapping, but the inclusion is proper (see Mongkolkeha et al., 203). A mapping TT: CC HH with FF(TT) = {xx CC: xx = TTTT} nonempty is said to be quasi-nonexpansive if TTTT pp xx pp holds for all pp FF(TT), xx CC and TT is called hemicontractive if TTTT pp 2 xx pp 2 + xx TTTT 2 holds for all pp FF(TT), xx CC. We remark that the class of hemicontractive mappings contains the class of pseudocontractive mappings with FF(TT) and the class of quasi-nonexpansive mappings. The following examples show that the inclusion is proper. Example.. Let HH = R and CC = [0,]. Let TT: CC HH be defined by TTTT = xx 2 sin if xx 0 x and TT0 = 0. Then, zero is the only fixed point of TT and for all xx CC, we have TTTT 0 2 = xx 2 sin xx 2 xx 4 xx 2 xx xx TTTT 2. Hence, TT is hemicontractive mapping. However, if we take xx = 2 and yy =, then we get that ππ ππ TTTT TTTT 2 = 4 ssssss ππ ππ 2 2 ππ 2 ssssssss 2 = 6 ππ 4. But,

2 8 Tesfalem Hadush et al. xx yy 2 + xx yy (TTTT TTTT) 2 = ππ 2 + ππ 4 ππ 2 2 = 2ππ2 8ππ + 6 ππ 4 < 6 ππ 4 = TTTT TTTT 2, which shows that TT is not pseudocontractive. Example.2. Let HH = R and CC = [0,]. Let TT: CC HH be defined by TTTT = 2, iiii xx 0, 2, 0, iiii xx 2,. Then, is the only fixed point of TT. If xx 0,, 2 2 we have TTTT TT( 2 ) 2 = 0 xx xx TTTT 2. And if xx,, we get 2 TTTT TT 2 2 = < 4 xx2 xx xx TTTT 2. Thus, TT is hemicontractive mapping. However, TT is not quasi-nonexpansive mapping. In fact, for xx = 3, we have 5 TTTT TT = > = 3 = xx Let CCCC(CC) denote the family of nonempty, closed and bounded subsets of CC. The Pompeiu- Hausdorff metric (see Berinde and Paa curar, 203) on CCCC(CC) is defined by DD(AA, BB) = max ssssss xx AA dd(xx, BB), ssssss yy BB dd(yy, AA), for all AA, BB CCCC(CC), where dd(xx, BB) = inf{ xx bb bb BB}. A multi-valued mapping TT: CC CCCC(CC) is called Lipschitzian if there exists LL 0 such that DD(TTTT, TTTT) LL xx yy, xx, yy CC.... (.2) If LL = in (.2), then TT is called nonexpansive and if LL [0,), then TT is called contraction mapping. A multi-valued mapping TT: CC CCCC(CC) is said to be kk-strictly pseudocontractive in the sense of Chidume et al. (203) if there exists a constant kk [0,) such that DD 2 (TTTT, TTTT) xx yy 2 + kk (xx uu) (yy vv) 2,. (.3) for all xx, yy CC and uu TTTT, vv TTTT, where DD 2 (TTTT, TTTT) = (DD(TTTT, TTTT)) 2. If kk = in (.3), then TT is said to be pseudocontractive mapping. Let TT: CC CCCC(CC) be a multi-valued mapping, then an element xx CC is called fixed point of TT if xx TTTT. We denote the set of fixed points of a mapping TT by FF(TT). We also write weak convergence and strong convergence of a sequence {xx nn } to xx in HH as xx nn xx and xx nn xx, respectively. A multi-valued mapping TT: CC CCCC(CC) with nonempty set of fixed points is called: i) Quasi-nonexpansive if for all pp FF(TT), xx CC, we have DD(TTTT, TTTT) xx pp. ii) Hemicontractive-type in the sense of Sebsibe Teferi et al. (205) if for all pp FF(TT), xx CC DD 2 (TTTT, TTTT) xx pp 2 + xx uu 2 holds for all uu TTTT.... (.4) We observe that every nonexpansive mapping TT with FF(TT) is quasi-nonexpansive mapping, and every pseudocontractive mapping TT with FF(TT) and TT(pp) = {pp}, pp FF(TT) is hemicontractive-type mapping (Habtu Zegeye et al., 207). In recent years, the existence and approximation of fixed points for multi-valued (including hemicontractive-type) mappings in various spaces under different assumptions has been studied by several authors; (see, for example, Nadler, 969; Panyanak, 2007; Shahzad and Habtu Zegeye, 2008; Yu et al., 202; Chidume et al., 203; Isiogugu and Osilike, 204; and references therein). Sebsibe Teferi et al. (205) proved the following result: Theorem WSZ. Let CC be a nonempty, closed and TT ii : CC CCCC(CC), ii =,2,,, be a finite family of Lipschitz hemicontractive-type mappings with Lipschitz constants LL ii, ii =, 2,,, respectively. Assume that (II TT ii ), ii =, 2,,, are demiclosed at zero and Ƒ = ii= FF(TT ii ) is nonempty, closed and convex with TT ii (pp) = {pp}, pp Ƒ. Let {xx nn } be the sequence generated from an arbitrary xx, ww CC by yy nn = ( ββ nn )xx nn + ββ nn uu nn, uu nn TT nn xx nn, zz nn = ( γγ nn )xx nn + γγ nn ww nn, ww nn TT nn yy nn,... (.5) xx nn+ = αα nn ww + ( αα nn )zz nn, nn,

3 SINET: Ethiop. J. Sci., 38(), where TT nn TT nnnnnnnn ()+ and {αα nn }, {ββ nn }, {γγ nn } (0,) satisfy the following conditions: i) 0 < αα nn cc <, nn such that lim nn αα nn 0 and αα nn = ii) 0 < αα γγ nn ββ nn ββ < nn= ; 4LL 2 ++, nn, for LL max{ll ii : ii =, 2,, }. Then, {xx nn } converges strongly to some point pp Ƒ nearest to ww. Recall that a mapping AA: CC HH is called monotone if AAAA AAAA, xx yy 0, xx, yy CC AA is called αα-inverse strongly monotone if there exists a positive real number αα such that AAAA AAAA, xx yy αα AAAA AAAA 2, xx, yy CC. We note that the class of αα-inverse strongly monotone mappings is properly contained in the class of monotone mappings (see Habtu Zegeye et al., 207). Let CC be a nonempty, closed and convex subset of a real Hilbert space HH. Let FF: CC CC R be a bifunction and AA: CC HH be a nonlinear mapping. Takahashi and Takahashi (2008) considered the following generalized equilibrium problem: Finding a point zz CC such that FF(zz, yy) + AAAA, yy zz 0, yy CC.... (.6) In this paper, we denote the set of solutions of problem (.6) by EEEE(FF, AA), i.e., EEEE(FF, AA) = {zz CC FF(zz, yy) + AAAA, yy zz 0, yy CC}. If in (.6) we have AA 0, then problem (.6) reduces to the equilibrium problem of finding an element zz CC such that FF(zz, yy) 0, yy CC,... (.7) which was studied by Blum and Oettli (994) and many others (Combettes and Hirstoaga, 2005; Takahashi and Takahashi, 2007; Wang et al., 2007; Ali, 2009; Cholamjiak et al., 205). The set of solutions of problem (.7) is denoted by EEEE(FF). If in (.6) we have FF 0, then the generalized equilibrium problem (.6) reduced to finding a point zz CC such that AAAA, yy zz 0, yy CC,... (.8) which is called the classical variational inequality problem. The set of solutions of problem (.8) is denoted by VVVV(CC, AA). Problem (.8) has been considered by many authors (see, for instance, Ali, 2009; Habtu Zegeye and Shahzad, 20a; 202; Tesfalem Hadush et al., 206) and references therein. We note that if a point zz VVVV(CC, AA) EEEE(FF), then zz EEEE(FF, AA), however, the converse is not true (see Habtu Zegeye et al., 207). Assumption.. Let CC be a nonempty, closed and convex subset of a real Hilbert space HH. In the sequel, let FF be a bifunction of CC CC into R satisfying the following assumptions: (A) FF(xx, xx) = 0, xx CC; (A2) FF is monotone, i.e., FF(xx, yy) + FF(yy, xx) 0, xx, yy CC; (A3) lim tt 0 FF(tttt + ( tt)xx, yy) FF(xx, yy), xx, yy, zz CC; (A4) For each xx CC, yy FF(xx, yy) is convex and lower semicontinuous. For instance, the bifunction FF: [0, ) [0, ) R given by FF(xx, yy) = yy xx satisfies Assumption.. Remark.2. Let CC be a nonempty, closed and FF be a bifunction from CC CC into R satisfying Assumption. and let AA: CC HH be a continuous monotone mapping. Define GG: CC CC R by GG(xx, yy) = FF(xx, yy) + AAAA, yy xx, then it is easy to see that the bifunction GG satisfies Assumption.. Thus, the generalized equilibrium problem (.6) is equivalent to the equilibrium problem of finding a point zz CC such that GG(zz, yy) 0, for all yy CC. Generalized equilibrium problem is more general in the sense that it includes, as special case, equilibrium problems and hence variational inequality, optimization problems, Nash equilibrium problems, fixed point problems, etc. Consequently, many authors have shown their interest in constructing an iterative algorithms for approximating common solution of generalized equilibrium and fixed point problems (see, for example, Hao, 20; Kamraksa and Wangkeeree, 20; Razani and Yazdi, 202; Zhang and Hao, 206 and references cited therein). Takahashi and Takahashi (2008) introduced and considered the following iterative algorithm for finding a common point of the set of solu-

4 20 Tesfalem Hadush et al. tions of problem (.6) and the set of fixed points of nonexpansive single-valued mapping TT and then they obtained a strong convergence theorem in Hilbert space settings. and Hao (206) and some other recent results that have been obtained previously in this research area. FF(zz nn, yy) + AAxx nn, yy zz nn + λλ nn yy zz nn, zz nn xx nn 0, yy CC, yy nn = αα nn uu + ( αα nn )zz nn, xx nn+ = ββ nn xx nn + ( ββ nn )TTyy nn, nn, where uu, xx CC are arbitrary, FF: CC CC R is a bifunction satisfying Assumption. and AA is an αα-inverse strongly monotone mapping from CC into HH, and {αα nn }, {ββ nn } [0,] and {λλ nn } [0,2αα] satisfy some appropriate control conditions. Recently, Huang and Ma (204) extended the results of Takahashi and Takahashi (2008) from nonexpansive mapping to kk-strictly pseudocontractive mapping. In fact, they proved the following weak convergence theorem. Theorem.3. Let CC be a nonempty, closed and AA: CC HH be an αα-inverse strongly monotone mapping and FF be a bifunction from CC CC into R which satisfies Assumption.. Let TT: CC CC be a kk-strictly pseudocontractive mapping such that Ƒ EEEE(FF, AA) FF(TT) is nonempty and let {ee nn } be a bounded sequence in CC. Let {xx nn } be a sequence generated by FF(uu nn, uu) + AAxx nn, yy uu nn + rr nn uu uu nn, uu nn xx nn 0, uu CC, xx nn+ = αα nn xx nn + ββ nn (δδ nn uu nn + ( δδ nn )TTuu nn )+γγ nn ee nn, nn N,... (.9) where the sequences {αα nn }, {ββ nn }, {γγ nn }, {δδ nn } (0,) and {rr nn } [0,2αα] satisfy some mild restrictions. Then the sequence {xx nn }, generated by (.9), converges weakly to a point pp Ƒ, where pp = lim nn PP Ƒ xx nn. In this paper, motivated and inspired by the results surveyed above, we introduce an iterative algorithm for finding a common element of the common solution set of a finite family of generalized equilibrium problems (.6) and the fixed point set of a multi-valued Lipschitz hemicontractive-type mapping. The results presented in this paper generalize, improve and extend the corresponding results of Huang and Ma (204), Tesfalem Hadush et al. (206), Habtu Zegeye et al. (207), Takahashi and Takahashi (2008), Ceng et al. (200) and Zhang PRELIMINARIES Throughout this section unless otherwise stated, CC denotes a nonempty, closed and convex subset of a real Hilbert space HH. For every point xx HH, there exists a unique nearest point in CC, denoted by PP CC xx, such that xx PP CC xx = inf{ xx yy yy CC}. PP CC is called the metric projection of HH onto CC. The following characterizes the metric projection PP CC : for given xx HH and zz CC, zz = PP CC xx xx zz, zz yy 0, yy CC.... (2.) Definition 2.. Let {xx nn } be a sequence in CC such that xx nn xx and let TT: CC CCCC(CC) be a multivalued mapping. Then, (II TT) is said to be demiclosed at zero if lim nn dd(xx nn, TTxx nn ) = 0 implies xx TTTT, where II is the identity mapping on CC. We note that if the mapping TT: CC CC in Definition 2. is a single-valued nonexpansive mapping, then (II TT) is demiclosed at zero (see Agrawal et al., 2009). In the proof of our main result, we also need the following lemmas. Lemma 2.2. (Habtu Zegeye and Shahzad, 20b). nn Let HH be a real Hilbert space and {xx ii } ii= HH. Then, for αα ii [0,], ii =,2,, nn, such that αα + αα αα nn =, we have the following identity: αα xx + αα 2 xx αα nn xx nn 2 = ii= αα ii xx ii 2 αα ii αα jj xx ii xx jj 2 ii,jj nn. Lemma 2.3. (Agrawal et al., 2009). Let HH be a real Hilbert space. Then, for every xx, yy HH, we have the following: i) xx yy 2 = xx 2 + yy 2 2 xx, yy ; ii) xx + yy 2 = xx yy, xx + yy. Lemma 2.4. (Blum and Oettli, 994; Combettes and Hirstoaga, 2005). Let FF be a bifunction from CC CC into R which satisfies Assumption.. For rr > 0, define TT rr : HH CC as follows:

5 SINET: Ethiop. J. Sci., 38(), TT rr xx = zz CC FF(zz, yy) + yy zz, zz xx, yy CC. rr Then, the following hold: () TT rr is nonempty and single-valued; (2) TT rr is firmly nonexpansive, i.e., TT rr xx TT rr yy 2 TT rr xx TT rr yy, xx yy, xx, yy HH; (3) FF(TT rr ) = EEEE(FF); (4) EEEE(FF) is closed and convex. Lemma 2.5. (Nadler, 969). Let (XX, dd) be a metric space and let AA, BB CCCC(XX). Then, for any uu AA and εε > 0, there exists a point vv BB such that dd(uu, vv) DD(AA, BB) + εε. This implies that for every element uu AA, there exists an element vv BB such that dd(uu, vv) 2DD(AA, BB). Lemma 2.6. (Xu, 2002). Let {aa nn } be a sequence of nonnegative real numbers such that aa nn + ( αα nn )aa nn + αα nn δδ nn, for nn nn 0, where {αα nn } (0,) and {δδ nn } R satisfying the following conditions: lim nn αα nn = 0, nn= αα nn =, and limsup δδ nn 0. nn Then, lim nn αα nn = 0. Lemma 2.7. (Mainge, 2008). Let {bb nn } be a sequence of real numbers such that there exists a subsequence nn jj of {nn} such that bb nnjj < bb nnjj +, for all jj N. Then, there exists a nondecreasing sequence {nn kk } N such that nn kk and the following properties are satisfied by all (sufficiently large) numbers kk N: b nk b nk + and bb kk bb nnkk + In fact, nn kk = max{ii kk bb ii bb ii+ }. MAIN RESULT In this section, we define an iterative algorithm and prove its strong convergence to a common solution of a finite family of generalized equilibrium problems and a fixed point problem for a multi-valued Lipschitz hemicontractive-type mapping. Theorem 3.. Let CC be a nonempty, closed and TT: CC CCCC(CC) be a Lipschitz hemicontractivetype multi-valued mapping with Lipschitz constant LL. Let AA mm : CC HH be a continuous monotone mapping and let FF mm : CC CC R be a bifunction satisfying Assumption., for each m {,2,.., N}. N Assume that Θ = m= EP(F m,a m ) F(T) is nonempty and TTTT = {qq} for all qq ΘΘ. Let rr mm,nn (0, ) and let {aa nn }, {bb nn }, {cc nn }, {ee nn } and dd mm,nn be sequences in (0,) such that i) bb nn + cc nn + ee nn = ; ii) mm = dd mm,nn = ; iii) bb nn + cc nn aa nn dd < +4LL 2 +. arbitrary xx, vv CC by FF mm yy mm,nn, zz + AA mm yy mm,nn, zz yy mm,nn + zz yy rr mm,nn, yy mm,nn xx nn 0, zz CC, mm =,2,,, mm,nn ww nn = dd mm,nn yy mm,nn, mm = zz nn = aa nn vv nn + ( aa nn )ww nn, xx nn+ = bb nn vv + cc nn uu nn + ee nn ww nn,...(3.) for all nn, where vv nn TTww nn, uu nn TTzz nn such that vv nn uu nn 2DD(TTww nn, TTzz nn ). Then, the sequence {xx nn } is bounded. Proof. Let qq ΘΘ. Then, we have TTTT = {qq} and FF mm (qq, zz) + AA mm qq, zz qq 0, for all mm =, 2,, and zz CC. Define GG mm : CC CC R by GG mm (xx, zz) FF mm (xx, zz) + AA mm xx, zz xx for all xx, zz CC and mm {, 2,, }. Then, in view of Remark.2, GG mm is a bifunction satisfying Assumption., for each mm {, 2,, } and qq EEEE(FF mm, AA mm ) is equivalent to GG mm (qq, zz) 0 for all zz CC. Hence, using Lemma 2.4, yy nn,mm can be rewritten as yy mm,nn = TT rrmm,nn xx nn and hence we obtain qq = TT rrmm,nn qq. In view of the fact that TT rrmm,nn is nonexpansive, by Lemma 2.4, we have that yy mm,nn qq = TT rrmm,nn xx nn TT rrmm,nn qq xx nn qq...(3.2) Then, from (3.2) and condition (ii), we have the following: ww nn qq = dd mm,nn yy mm,nn qq mm = = dd mm,nn yy mm,nn dd mm,nn qq mm = mm = dd mm,nn yy mm,nn qq mm =

6 22 Tesfalem Hadush et al. dd mm,nn xx nn qq mm= = xx nn qq.... (3.3) Using Lemma 2.2, the fact that TT is hemicontractive-type mapping and vv nn TTww nn, we find that zz nn qq 2 = aa nn (vv nn qq) + ( aa nn )(ww nn qq) 2 = aa nn vv nn qq 2 + ( aa nn ) ww nn qq 2 aa nn ( aa nn ) ww nn vv nn 2 aa nn DD 2 (TTww nn, TTTT) + ( aa nn ) ww nn qq 2 aa nn ( aa nn ) ww nn vv nn 2 aa nn ( ww nn qq 2 + ww nn vv nn 2 ) +( aa nn ) ww nn qq 2 aa nn ( aa nn ) ww nn vv nn 2 = ww nn qq 2 + aa nn 2 ww nn vv nn 2. From (3.3), it follows that zz nn qq 2 xx nn qq 2 + aa nn 2 ww nn vv nn (3.4) In addition, since TT is hemicontractive-type mapping and uu nn TTzz nn, from (3.) and (3.4), we get that uu nn qq 2 DD 2 (TTzz nn, TTTT) zz nn qq 2 + zz nn uu nn 2 xx nn qq 2 + aa nn2 ww nn vv nn 2 + zz nn uu nn 2...(3.5) It follows from (3.) that ww nn zz nn 2 = ww nn (aa nn vv nn + ( aa nn )ww nn ) 2 = aa 2 nn ww nn vv nn 2...(3.6) Thus, since vv nn uu nn 2DD(TTww nn, TTzz nn ) and TT is a LL Lipschitzian mapping, from (3.6) and Lemma 2.2, we get that zz nn uu nn 2 = aa nn vv nn + ( aa nn )ww nn uu nn 2 = aa nn vv nn uu nn 2 + ( aa nn ) ww nn uu nn 2 aa nn ( aa nn ) ww nn vv nn 2 ( aa nn ) ww nn uu nn 2 + 4aa nn DD 2 (TTww nn, TTzz nn ) aa nn ( aa nn ) ww nn vv nn 2 ( aa nn ) ww nn uu nn 2 + 4aa nn LL 2 ww nn zz nn 2 aa nn ( aa nn ) ww nn vv nn 2 = ( aa nn ) ww nn uu nn 2 + 4aa 3 nn LL 2 ww nn vv nn 2 aa nn ( aa nn ) ww nn vv nn 2 = ( aa nn ) ww nn uu nn 2 +aa nn (4LL 2 aa 2 nn + aa nn ) ww nn vv nn (3.7) Hence, substituting (3.7) into (3.5), we have that uu nn qq 2 xx nn qq 2 + aa nn 2 ww nn vv nn 2 +( aa nn ) ww nn uu nn 2 +aa nn (4LL 2 aa nn 2 + aa nn ) ww nn vv nn 2 = xx nn qq 2 + ( aa nn ) ww nn uu nn 2 +aa nn (4LL 2 aa nn 2 + 2aa nn ) ww nn vv nn 2... (3.8) Thus, from (3.3), (3.8), Lemma 2.2 and condition (i), we obtain that xx nn+ qq 2 = bb nn vv + cc nn uu nn + ee nn ww nn qq 2 bb nn vv qq 2 + cc nn uu nn qq 2 +ee nn ww nn qq 2 cc nn ee nn ww nn uu nn 2 bb nn vv qq 2 + cc nn ( xx nn qq 2 +( aa nn ) ww nn uu nn 2 +aa nn (4LL 2 aa nn 2 + 2aa nn ) ww nn vv nn 2 ) +ee nn ww nn qq 2 cc nn ee nn ww nn uu nn 2 bb nn vv qq 2 + ( bb nn ) xx nn qq 2 +cc nn ( ee nn aa nn ) ww nn uu nn 2 cc nn aa nn ( 4LL 2 aa nn 2 2aa nn ) ww nn vv nn 2 = bb nn vv qq 2 + ( bb nn ) xx nn qq 2 cc nn aa nn ( 4LL 2 aa nn 2 2aa nn ) ww nn vv nn 2 +cc nn (bb nn + cc nn aa nn ) ww nn uu nn 2,... (3.9) and from condition (iii), we have that 4LL 2 aa nn 2 2aa nn 4LL 2 dd 2 2dd > 0 and bb nn + cc nn aa nn 0, for all nn.... (3.0) Thus, from (3.9) and (3.0), we find that xx nn+ qq 2 bb nn vv qq 2 + ( bb nn ) xx nn qq 2 max{ vv qq 2, xx nn qq 2 }. Hence, by induction, the sequence {xx nn } is bounded. This completes the proof. Theorem 3.2. Let CC be a nonempty, closed and TT: CC CCCC(CC) be a Lipschitz hemicontractive-type multi-valued mapping with Lipschitz constant LL. Let AA mm : CC HH be a continuous monotone mapping and let FF mm : CC CC R be a bifunction satisfying Assumption., for each mm {, 2,, }. Assume that ΘΘ = mm = EEEE(FF mm, AA mm ) FF(TT) is nonempty, closed and convex, (II TT) is demiclosed at zero and TTTT = {qq} for all qq ΘΘ. Let rr mm,nn (0, ) such that lim nn rr mm,nn = rr mm for some 0 < rr mm < and for each mm =, 2,,, and let {aa nn }, {bb nn }, {cc nn }, {ee nn } and dd mm,nn be sequences in (0,) such that i) bb nn + cc nn + ee nn = and 0 < aa cc nn, ee nn bb < ; ii) lim nn bb nn = 0, nn= bb nn = ; iii) mm = dd mm,nn = and 0 < cc dd mm,nn ; iv) bb nn + cc nn aa nn dd < +4LL 2 +. arbitrary xx, vv CC by

7 SINET: Ethiop. J. Sci., 38(), FF mm yy mm,nn, zz + AA mm yy mm,nn, zz yy mm,nn + zz yy rr mm,nn, yy mm,nn xx nn 0, zz CC, mm =,2,,, mm,nn ww nn = dd mm,nn yy mm,nn, mm = zz nn = aa nn vv nn + ( aa nn )ww nn, xx nn + = bb nn vv + cc nn uu nn + ee nn ww nn,... (3.) for all nn, where vv nn TTww nn, uu nn TTzz nn such that vv nn uu nn 2DD(TTww nn, TTzz nn ). Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). Proof. Since ΘΘ is nonempty, closed and convex subset of HH, then we see that PP ΘΘ is well defined. Obviously, from Theorem 3. the sequence {xx nn } and hence yy nn,mm, {ww nn } and {zz nn } are bounded. Now, let qq ΘΘ. Then, using the fact that TT rrnn,mm is firmly nonexpansive and TT rrmm,nn qq = qq, for all mm =, 2,,, and Lemma 2.3 (i), we find that yy mm,nn qq 2 = TT rrmm,nn xx nn TT rrmm,nn qq 2 yy mm,nn qq, xx nn qq = yy 2 mm,nn qq 2 + xx nn qq 2 yy mm,nn xx nn 2, which implies that yy mm,nn qq 2 xx nn qq 2 yy mm,nn xx nn 2. This gives that ww nn qq 2 dd mm,nn yy mm,nn qq 2 mm = xx nn qq 2 dd mm,nn yy mm,nn xx nn 2. mm =... (3.2) On the other hand, from (3.), Lemma 2.2 and Lemma 2.3 (ii), we have xx nn + qq 2 = bb nn vv + cc nn uu nn + ee nn ww nn qq 2 cc nn (uu nn qq) + ee nn (ww nn qq) 2 +2bb nn vv qq, xx nn + qq cc nn uu nn qq 2 + ee nn ww nn qq 2 cc nn ee nn ww nn uu nn 2 +2bb nn vv qq, xx nn + qq....(3.3) Thus, substituting (3.8) and (3.2) into (3.3), we obtain that xx nn+ qq 2 cc nn ( xx nn qq 2 + aa nn (4LL 2 aa 2 nn + 2aa nn ) ww nn vv nn 2 ) +cc nn ( aa nn ) ww nn uu nn 2 +ee nn xx nn qq 2 dd mm,nn yy mm,nn xx nn 2 mm = cc nn ee nn ww nn uu nn 2 + 2bb nn vv qq, xx nn + qq = ( bb nn ) xx nn qq 2 cc nn aa nn ( 4LL 2 aa nn 2 2aa nn ) ww nn vv nn 2 +cc nn (bb nn + cc nn aa nn ) ww nn uu nn 2 ee nn dd mm,nn yy mm,nn xx nn 2 mm= +2bb nn vv qq, xx nn+ qq... (3.4) Now, we consider the following two cases: Case. Suppose that there exists nn 0 N such that { xx nn qq } nn nn0 is nonincreasing sequence. Then, the boundedness of { xx nn qq } implies that { xx nn qq } is convergent. From (3.0) and (3.4), it follows that cc nn aa nn ( 4LL 2 aa 2 nn 2aa nn ) ww nn vv nn 2 ( bb nn ) xx nn qq 2 xx nn+ qq 2 +2bb nn vv qq, xx nn + qq Thus, from (3.0), the assumptions of {cc nn } and {aa nn }, and the fact that bb nn 0 as nn and {xx nn } is bounded, we find that ww nn vv nn 0 as nn.... (3.5) This implies that lim nn dd(ww nn, TTww nn ) = (3.6) From (3.0) and (3.4), we see that ee nn dd mm,nn yy mm,nn xx nn 2 ( bb nn ) xx nn qq 2 xx nn+ qq 2 +2bb nn vv qq, xx nn + qq. This together with conditions (i), (ii) and (iii) imply that lim nn yy mm,nn xx nn = 0....(3.7) In addition, from the Lipschitz condition of TT, (3.6) and (3.5), we get that ww nn uu nn ww nn vv nn + vv nn uu nn ww nn vv nn + 2LL ww nn zz nn = ww nn vv nn + 2LLaa nn ww nn vv nn. Hence lim nn ww nn uu nn = (3.8) Again from (3.) and triangle inequality, we get that xx nn+ xx nn xx nn+ ww nn + ww nn xx nn = bb nn (vv ww nn ) + cc nn (uu nn ww nn ) + mm = dd mm,nn yy mm,nn xx nn bb nn vv ww nn + cc nn uu nn ww nn + mm = dd mm,nn yy mm,nn xx nn. Therefore, from (3.7), (3.8) and the fact that bb nn 0 as nn, we obtain lim nn xx nn+ xx nn = 0....(3.9)

8 24 Tesfalem Hadush et al. It follows from (3.0) and (3.4) that xx nn + qq 2 ( bb nn ) xx nn qq 2 +2bb nn vv qq, xx nn+ qq.... (3.20) Now, let pp = PP ΘΘ (vv). Then, we show that limsup nn vv pp, xx nn+ pp 0. Since {xx nn+ } is a bounded sequence in a real Hilbert space HH, which is a reflexive Banach space, then there exists a subsequence xx nnii + of {xx nn + } such that xx nnii + yy as ii and limsup vv pp, xx nnii + pp = lim vv pp, xx nnii + pp. nn ii Since, CC is weakly closed, we have yy CC and from (3.9), it follows that xx nnii yy as ii. Using (3.7), we see that ww nnii xx nnii 0 as ii and so ww nnii yy as ii. Thus, demiclosedness of (II TT) at zero and (3.6) imply that yy FF(TT). On the other hand, from the fact that (II TT rrmm,nn ) is demiclosed at zero and (3.7), we obtain ii that yy = TT rrmm,nn yy and thus yy EEEE(FF ii mm, AA mm ), for all mm {, 2,, }. That is, yy EEEE(FF mm, AA mm ). mm = Therefore, yy ΘΘ. Hence, since pp = PP ΘΘ (vv) and xx nnii yy as ii, from the property of a metric projection, equation (2.), we have that limsup vv pp, xx nnii + pp = lim vv pp, xx nnii + pp nn ii = v p, y p 0....(3.2) Thus, since pp ΘΘ, from (3.20), (3.2), condition (ii) and Lemma 2.6, we conclude that xx nn pp 0 as nn. That is, xx nn pp = PP ΘΘ (vv). Case 2. Suppose that there exists a subsequence nn jj of {nn} such that xx nnjj qq < xx nnjj qq, for all jj N. Then, by Lemma 2.7, there exists a nondecreasing sequence {nn kk } N such that nn kk, and xx nnkk qq xx nnkk + qq and xx kk qq xx nnkk + qq,...(3.22) for all kk N. Thus, replacing nn by nn kk and using (3.22), (3.4), (3.0) and the fact that bb nn 0 as nn, we find that ww nnkk vv nnkk 0 and yy mm,nnkk xx nnkk 0 as kk. Hence, following an argument similar to that in Case, we obtain that limsup ii vv pp, xx nnkk + pp 0... (3.23) Now, since pp ΘΘ, from (3.20), we get that xx nnkk + pp 2 bb nnkk xx nnkk pp 2 +2bb nnkk vv pp, xx nnkk + pp... (3.24) and hence, since pp ΘΘ, (3.22) and (3.24) imply that bb nnkk xx nnkk pp 2 xx nnkk pp 2 xx nnkk + pp 2 +2bb nnkk vv pp, xx nnkk + pp 2bb nnkk vv pp, xx nnkk + pp. Then, from the fact that bb nnkk > 0, we have that xx nnkk pp 2 2 vv pp, xx nnkk + pp. It follows from (3.23) that xx nnkk pp 0 as kk. This together with (3.24) and the fact that bb nnkk 0 imply that xx nnkk + pp 0 as kk. Since pp ΘΘ, we get that xx kk pp xx nnkk + pp for all kk N. Thus, we obtain that xx kk pp as kk. Therefore, we conclude that the sequence {xx nn } generated by (3.) converges strongly to the point pp = PP ΘΘ (vv). This completes the proof. As a direct consequence of our main result, we obtain the following results. Corollary 3.3. Let CC be a nonempty, closed and TT: CC CCCC(CC) be a Lipschitz pseudocontractive multi-valued mapping with Lipschitz constant LL. Let AA mm : CC HH be a continuous monotone mapping and let FF mm : CC CC R be a bifunction satisfying Assumption., for each mm {, 2,, }. Assume that ΘΘ = mm = EEEE(FF mm, AA mm ) FF(TT) is nonempty, closed and convex, (II TT) is demiclosed at zero and TTTT = {qq} for all qq FF(TT). Let rr mm,nn (0, ) such that lim nn rr mm,nn = rr mm for some 0 < rr mm < and for each mm =, 2,,, and let {aa nn }, {bb nn }, {cc nn }, {ee nn } and dd nn,mm be sequences in (0,) such that i) bb nn + cc nn + ee nn = and 0 < aa cc nn, ee nn bb < ; ii) lim nn bb nn = 0, nn= bb nn = ; iii) mm = dd mm,nn = and 0 < cc dd mm,nn ; iv) bb nn + cc nn aa nn dd < +4LL 2 +.

9 SINET: Ethiop. J. Sci., 38(), arbitrary xx, vv CC by FF mm yy mm,nn, zz + AA mm yy mm,nn, zz yy mm,nn + zz yy rr mm,nn, yy mm,nn xx nn 0, zz CC, mm =,2,,, mm,nn ww nn = dd mm,nn yy mm,nn, mm= zz nn = aa nn vv nn + ( aa nn )ww nn, xx nn+ = bb nn vv + cc nn uu nn + ee nn ww nn, for all nn, where vv nn TTww nn, uu nn TTzz nn such that vv nn uu nn 2DD(TTww nn, TTzz nn ). Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). Proof. Since a Lipschitz pseudocontractive multivalued mapping TT with FF(TT) and TTTT = {qq}, qq FF(TT) is Lipschitz hemicontractive-type mapping, we obtain the desired result from Theorem 3.2. If, in Theorem 3.2, we assume that AA mm 0, for all mm {,2,, }, then we obtain the following corollary on a finite family of equilibrium problems and fixed point problem of multi-valued Lipschitz hemicontractive-type mapping. Corollary 3.4. Let CC be a nonempty, closed and TT: CC CCCC(CC) be a Lipschitz hemicontractive-type multi-valued mapping with Lipschitz constant LL. Let FF mm : CC CC R be a bifunction satisfying Assumption., for each mm {, 2,, }. Assume that ΘΘ = mm = EEEE(FF mm ) FF(TT) is nonempty, closed and convex, (II TT) is demiclosed at zero and TTTT = {qq} for all qq ΘΘ. Let rr mm,nn (0, ) such that lim nn rr mm,nn = rr mm for some 0 < rr mm < and for each mm =, 2,,, and let {aa nn }, {bb nn }, {cc nn }, {ee nn } and dd mm,nn be sequences in (0,) such that i) bb nn + cc nn + ee nn = and 0 < aa cc nn, ee nn bb < ; ii) lim nn bb nn = 0, nn= bb nn = ; iii) mm= dd mm,nn = and 0 < cc dd mm,nn ; iv) bb nn + cc nn aa nn dd < +4LL 2 +. arbitrary xx, vv CC by FF mm yy mm,nn, zz + zz yy rr mm,nn, yy mm,nn xx nn 0, mm,nn zz CC, mm =,2,,, ww nn = mm = dd mm,nn yy mm,nn, zz nn = aa nn vv nn + ( aa nn )ww nn, xx nn + = bb nn vv + cc nn uu nn + ee nn ww nn, for all nn, where vv nn TTww nn, uu nn TTzz nn such that vv nn uu nn 2DD(TTww nn, TTzz nn ). Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). If, in Theorem 3.2, we assume that FF mm 0, for all mm {, 2,, }, then we have the following result on the problem of finding a common point of the common solution set of a finite family of variational inequality problems and fixed point set of Lipschitz hemicontractive-type mapping. Corollary 3.5. Let CC be a nonempty, closed and TT: CC CCCC(CC) be a Lipschitz hemicontractive-type multi-valued mapping with Lipschitz constant LL. Let AA mm : CC HH be a continuous monotone mapping, for each mm {, 2,, }. Assume that ΘΘ = mm = VVVV(CC, AA mm ) FF(TT) is nonempty, closed and convex, (II TT) is demiclosed at zero and TTTT = {qq} for all qq ΘΘ. Let rr mm,nn (0, ) such that lim nn rr mm,nn = rr mm for some 0 < rr mm < and for each mm =, 2,,, and let {aa nn }, {bb nn }, {cc nn }, {ee nn } and dd mm,nn be sequences in (0,) such that bb nn + cc nn + ee nn = and 0 < aa cc nn, ee nn bb < ; i) lim nn bb nn = 0, nn= bb nn = ; ii) mm = dd mm,nn = and 0 < cc dd mm,nn ; iii) bb nn + cc nn aa nn dd < +4LL 2 +. arbitrary xx, vv CC by AA mm yy mm,nn, zz yy mm,nn + zz yy rr mm,nn, yy mm,nn xx nn 0, nn,mm zz CC, mm =,2,,, ww nn = mm= dd mm,nn yy mm,nn, zz nn = aa nn vv nn + ( aa nn )ww nn, xx nn + = bb nn vv + cc nn uu nn + ee nn ww nn, for all nn, where vv nn TTww nn, uu nn TTzz nn such that vv nn uu nn 2DD(TTww nn, TTzz nn ). Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). If, in Theorem 3.2, we assume that TT is a singlevalued hemicontractive mapping from CC into itself, then we obtain the following result. Corollary 3.6. Let CC be a nonempty, closed and TT: CC CC be a Lipschitz hemicontractive mapping with Lipschitz constant LL. Let AA mm : CC HH be a continuous monotone mapping and let FF mm : CC CC R be a bifunction satisfying Assumption. for each, mm {, 2,, }. Assume that ΘΘ = mm = EEEE(FF mm, AA mm ) FF(TT) is nonempty, closed and convex, (II TT) is demiclosed at zero. Let rr mm,nn (0, ) such that lim nn rr mm,nn = rr mm for some 0 < rr mm < and for each mm =, 2,,, and let {aa nn }, {bb nn }, {cc nn }, {ee nn } and dd mm,nn be sequences in (0,) such that

10 26 Tesfalem Hadush et al. i) bb nn + cc nn + ee nn = and 0 < aa cc nn, ee nn bb < ; ii) lim nn bb nn = 0, nn= bb nn = ; iii) mm= dd mm,nn = and 0 < cc dd mm,nn ; iv) bb nn + cc nn aa nn dd < +4LL 2 +. arbitrary xx, vv CC by FF mm yy nn,mm, zz + AA mm yy nn,mm, zz yy nn,mm + zz yy rr nn,mm, yy nn,mm xx nn 0, zz CC, mm =,2,,, mm,nn ww nn = mm = dd mm,nn yy nn,mm, zz nn = aa nn TTTT nn + ( aa nn )ww nn, xx nn+ = bb nn vv + cc nn TTTT nn + ee nn ww nn, for all nn. Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). If, in Theorem 3.2, we assume that =, then we get the following corollary. Corollary 3.7. (Habtu Zegeye et al., 207). Let CC be a nonempty, closed and convex subset of a real Hilbert space HH. Let TT: CC CCCC(CC) be a Lipschitz hemicontractive-type multi-valued mapping with Lipschitz constant LL. Let AA: CC HH be a continuous monotone mapping and let FF: CC CC R be a bifunction satisfying Assumption.. Assume that ΘΘ = EEEE(FF, AA) FF(TT) is nonempty, closed and convex, (II TT) is demiclosed at zero and TTTT = {qq} for all qq ΘΘ. Let {rr nn } (0, ) such that lim nn rr nn = rr for some 0 < rr <, and let {aa nn }, {bb nn }, {cc nn }, and {ee nn } be sequences in (0,) such that i) bb nn + cc nn + ee nn = and 0 < aa cc nn, ee nn bb < ; ii) lim nn bb nn = 0, bb nn = iii) bb nn + cc nn aa nn dd < nn= ;. +4LL 2 + arbitrary xx, vv CC by FF(yy nn, zz) + AAyy nn, zz yy nn + zz yy rr nn, yy nn xx nn 0, zz CC, nn zz nn = aa nn vv nn + ( aa nn )yy nn, xx nn+ = bb nn vv + cc nn uu nn + ee nn yy nn, for all nn, where vv nn TTyy nn, uu nn TTzz nn such that vv nn uu nn 2DD(TTyy nn, TTzz nn ). Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). If, in Theorem 3.2, we assume that TT = II, where II is the identity mapping on CC, then we obtain the following corollary on a finite family of generalized equilibrium problems. Corollary 3.8. Let CC be a nonempty, closed and AA mm : CC HH be a continuous monotone mapping and let FF mm : CC CC R be a bifunction satisfying Assumption., for each mm {, 2,, }. Assume that ΘΘ = mm = EEEE(FF mm, AA mm ) FF(TT) is non- empty. Let rr mm,nn (0, ) such that lim nn rr mm,nn = rr mm for some 0 < rr mm < and for each mm =, 2,,, and let {bb nn } and dd mm,nn be sequences in (0,) such that i) lim nn bb nn = 0, nn= bb nn = ; ii) mm = dd mm,nn = and 0 < cc dd mm,nn arbitrary xx, vv CC by FF mm yy mm,nn, zz + AA mm yy mm,nn, zz yy mm,nn + zz yy rr mm,nn, yy mm,nn xx nn 0, zz CC, mm =,2,,, mm,nn ww nn = mm = dd mm,nn yy mm,nn, xx nn + = bb nn vv + ( bb nn )ww nn, for all nn. Then, the sequence {xx nn } converges strongly to pp = PP ΘΘ (vv). Remark 3.9. Theorem 3.2 extends the results of Tesfalem Hadush et al. (206), Habtu Zegeye et al. (207), Razani and Yazdi (202), Hao (20), Wang et al. (2007), Ceng et al. (200), Huang and Ma (204) in the sense that our iterative algorithm provides strong convergence to a common element of the set of common solutions of a finite family of generalized equilibrium problems and the set of fixed points of a Lipschitz hemicontractive-type multi-valued mapping. We have used the demiclosedness principle in the proof of our Theorem 3.2 which makes a little simpler than using Assumption.. ACKNOWLEDGEMENTS This work was completed with the financial support of Simons Foundation based at Botswana International University of Science and Technology. The authors are thankful to the anonymous reviewers for their valuable suggestions to enhance the quality of our research paper. REFERENCES. Agrawal, R.P., O Regan, D and Sahu, D.R. (2009). Fixed Point Theory for Lipschitzian-type Mappings with Application. Springer, New York. 2. Ali, B. (2009). Fixed points approximation and solutions of some equilibrium and variational inequalities problems. International Journal of Mathematics and Mathematical Sciences, 7 pages, doi: 0.52/2009/

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