0 Fixed Point Theory and Applications

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An approximate solution to the fixed point problems for an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, cocoercive quasivariational inclusions problems and mixed equilibrium problems in Hilbert spaces

Pattanapong Tianchai*

Correspondence:

pattana@mju.ac.th

Faculty of Science, Maejo University,

Chiangmai, 50290, Thailand

Abstract

We introduce a new iterative scheme by modifying Mann's iteration method to find a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu (Thai J. Math. 9(2):399-409,2011) and many others.

MSC: 46C05; 47H09; 47H10; 49J30; 49J40

Keywords: fixed point; asymptotically strictly pseudocontractive in the intermediate sense; variational inequalities; mixed equilibrium; strong convergence; Hilbert space

ringer

1 Introduction

Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by <•, •) and || • ||, respectively. For a sequence {xn} in H, we denote the strong convergence and the weak convergence of {xn} to x e H by xn ^ x and xn ^ x, respectively.

Recall that PC is the metric projection of H onto C; that is, for each x e H, there exists the unique point PCx e C such that ||x-PCx|| = minyeC ||x -y||. A mapping T: C ^ C is called nonexpansive if || Tx - Ty || < ||x - y || for all x,y e C, and uniformly L-Lipschitzian if there exists a constant L > 0 such that for each n e N, ||Tnx - Tny|| < L||x -y|| for all x,y e C, and a mapping f: C ^ C is called a contraction if there exists a constant a e (0,1) such that |f (x) -f (y) || < a ||x - y || for all x, y e C.A point x e C is a fixed point of T provided that Tx = x. We denote by F(T) the set of fixed points of T; that is, F(T) = {x e C: Tx = x}.If C

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is a nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [1]). Recall that a mapping T: C ^ C is said to be

(i) monotone if

(Tx - Ty,x -y)> 0, Vx,y e C, (1.1)

(ii) k-Lipschitz continuous if there exists a constant k >0 such that

\\Tx - Ty\\<k\\x -y||, Vx,y e C, (.2)

if k = 1, then A is nonexpansive,

(iii) a-strongly monotone if there exists a constant a >0 such that

(Tx - Ty,x -y)>a\\x -y\\2, Vx,y e C, (.3)

(iv) a-inverse-strongly monotone (or a-cocoercive) if there exists a constant a >0 such that

(Tx - Ty,x -y)>a\\Tx - Ty\\2, Vx,y e C, (.4)

if a = 1, then T is called firmly nonexpansive; it is obvious that any a-inverse-strongly monotone mapping T is monotone and (1/a)-Lipschitz continuous,

(v) k-strictly pseudocontractive [2] if there exists a constant k e [0,1) such that

\\Tx - Ty\\2 < \\x - y\\2 + k || (/ - T)x -(I - T )y||2, Vx, y e C. (1.5)

In brief, we use k-SPC to denote k-strictly pseudocontractive. It is obvious that T is nonexpansive if and only if T is 0-SPC,

(vi) asymptotically k-SPC [3] if there exists a constant k e [0,1) and a sequence |ynj of nonnegative real numbers with limn^TO yn = 0 such that

| Tnx - Tny|2 < (1 + Yn) \ \x -y\\2 + k || (I - Tn)x - (I - Tn)y|2, Vx,y e C, (1.6)

for all n e N.If k = 0, then T is asymptotically nonexpansive with kn = V1 + Yn for all n e N; that is, T is asymptotically nonexpansive [4] if there exists a sequence {kn} c [1, to) with limn^TO kn = 1 such that

|| Tnx - Tny|| < kn\\x - y\\<(1 + Yn)\x - y\\, Vx, y e C, (1.7)

for all n e N. It is known that the class of k-SPC mappings and the class of asymptotically k-SPC mappings are independent (see [5]). And the class of asymptotically nonexpansive mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate sense with en = YnK for all n e N and for some K > 0; that is, T is an asymptotically nonexpansive mapping in the intermediate sense if there exists a sequence {en} of nonnegative real numbers with limn^TO en = 0 such that

|| Tnx - Tny|| < \\x -y\\ + en, Vx,y e C, (.8)

for all n e N,

(vii) asymptotically k -SPC mapping in the intermediate sense [6] if there exists a constant k e [0,1) and a sequence {yn} of nonnegative real numbers with limn—TO yn =0 such that

limsup sup (|| Tnx - Tny|2 -(1 + Yn) ||x -y||2 - k || (I - Tn)x - (I - Tn)y ||2) < 0.

n—x,yeC

If we define

Tn = max J 0, sup (|| Tnx - Tny |22 -(1 + Yn) ||x - y ||:2 - k | (I - Tn)x - (I - Tn)y ||2)},

1 x,yeC '

then limn—TO Tn = 0, and the last inequality is reduced to

|| Tnx - Tny|2 < (1 + Yn) ||x -y ||2 + k || (I - Tn)x - (I - Tn)y|2 + Tn, Vx,y e C, (1.9)

for all n e N. It is obvious that if Tn =0 for all n e N, then the class of asymptotically k -SPC mappings in the intermediate sense is reduced to the class of asymptotically k-SPC mappings; and if Tn = k = 0 for all n e N, then the class of asymptotically k-SPC mappings in the intermediate sense is reduced to the class of asymptotically nonexpansive mappings; and if Yn = Tn = k = 0 for all n e N, then the class of asymptotically k-SPC mappings in the intermediate sense is reduced to the class of nonexpansive mappings; and the class of asymptotically nonexpansive mappings in the intermediate sense with {en} of nonnegative real numbers such that limn—TO en = 0 is reduced to the class of asymptotically k-SPC mappings in the intermediate sense with Tn = enK for all n e N and for some K > 0. Some methods have been proposed to solve the fixed point problem of an asymptotically k -SPC mapping in the intermediate sense (1.9); related work can also be found in [6-13] and the references therein.

Example 1.1 (Sahu etal. [6]) Let X = R and C = [0,1]. For each x e C,we define T: C — C by

Kx, ifx e [0,2], 0, if x e (¿,1],

where k e (0,1). Then

(1) T is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, T is an asymptotically k -SPC mapping in the intermediate sense.

(2) T is not continuous. Therefore, T is not an asymptotically k-SPC and asymptotically nonexpansive mapping.

Example 1.2 (Hu and Cai [7]) Let X = R, C = [0,1], and k e [0,1). For each x e C, we define T: C — C by

1Y i x + if x e [0,2],

yfx, if x e (¿,1].

(1) T is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, T is an asymptotically k -SPC mapping in the intermediate sense.

(2) T is continuous but not uniformly L-Lipschitzian. Therefore, T is not an asymptotically k -SPC mapping.

Example 1.3 Let X = R and C = [0,1]. For each x e C,we define T: C ^ C by

where k e [0,1). Then

(1) T is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, T is an asymptotically k -SPC mapping in the intermediate sense.

(2) T is not continuous. Therefore, T is not an asymptotically k-SPC and asymptotically nonexpansive mapping.

Iterative methods are often used to solve the fixed point equation Tx = x. The most well-known method is perhaps the Picard successive iteration method when T is a contraction. Picard's method generates a sequence {xn} successively as xn+1 = Txn for all n e N with x1 = x chosen arbitrarily, and this sequence converges in norm to the unique fixed point of T. However, if T is not a contraction (for instance, if T is nonexpansive), then Picard's successive iteration fails, in general, to converge. Instead, Mann's iteration method for a nonexpansive mapping T (see [14]) prevails, generates a sequence {xn} recursively by

xn+1 = anxn + (1-an)Txn, Vn e N, (1.10)

where xi = x e C chosen arbitrarily and the sequence {an} lies in the interval [0,1].

Mann's algorithm for nonexpansive mappings has been extensively investigated (see [2, 15, 16] and the references therein). One of the well-known results is proven by Reich [16] for a nonexpansive mapping T on C, which asserts the weak convergence of the sequence {xn} generated by (1.10) in a uniformly convex Banach space with a Frechet differentiate norm under the control condition ^an(1 - an) = to. Recently, Marino and Xu [17] developed and extended Reich's result to a SPC mapping in a Hilbert space setting. More precisely, they proved the weak convergence of Mann's iteration process (1.10) for a k-SPC mapping T on C, and subsequently, this result was improved and carried over the class of asymptotically k-SPC mappings by Kim and Xu [18].

It is known that Mann's iteration (1.10) is in general not strongly convergent (see [19]). The way to guarantee strong convergence has been proposed by Nakajo and Taka-hashi [20]. They modified Mann's iteration method (1.10), which is to find a fixed point of a nonexpansive mapping by the hybrid method, called the shrinking projection method (or the CQ method), as the following theorem.

2 k x 5 ,

ifx e (2,1],

Theorem NT Let Cbea nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping ofC into itself such thatF(T) = 0. Suppose thatx1 = x e C chosen

arbitrarily and {xn} is the sequence defined by

yn — &nxn + (1 &n) Txn, Cn = {z e C: llyn - z|| < ||xn - z\\}, Qn = {z e C: <xn - z, x1 - xn) > 0}, xn+1 = PCnHQn(x1), Vn e N,

where 0 < an < a <1. Then {xn} converges strongly to PF(T)(x1).

Subsequently, Marino and Xu [21] introduced an iterative scheme for finding a fixed point of a k-SPC mapping as the following theorem.

Theorem MX Let C be a nonempty closed convex subset of a real Hilbert space H and let T: C ^ C be a k-SPC mapping for some 0 < k <1. Assume that F (T) = 0. Suppose that x1 = x e C chosen arbitrarily and {xn} is the sequence defined by

yn — anxn + (1 an) Txn,

Cn = {z e C : llyn - z||2 < ||xn - z||2 + (1 - an)(K - an) x - Txn ||2}, Qn = {z e C: <xn - z, x1 - xn) > 0}, xn+1 = PCnnQn(x1), Vn e N,

where 0 < an <1. Then the sequence {xn} converges strongly to PF(T)(x1).

Quite recently, Kim and Xu [18] have improved and carried Theorem MX over a wider class of asymptotically k-SPC mappings as the following theorem.

Theorem KX Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C ^ C be an asymptotically k -SPC mapping for some 0 < k <1 with a bounded sequence {yn} c [0, to) such that limn^TO yn = 0. Assume thatF(T) is a nonempty bounded subset ofC. Suppose thatx1 = x e C chosen arbitrarily and {xn} is the sequence defined by

yn = an xn + (1 - an)Tn xn ,

Cn = {z e C : ||yn - z||2 < ||xn -z|2 + (k - an(1 - an))|xn - Tnxn||2 + On}, Qn = {z e C: <xn - z, x1 - xn) > 0}, xn+1 = PCnnQn(x1), Vn e N,

where On = An(1 - an)yn ^ 0 asn ^to, An = sup{||xn - z|: z e F(T)} < to and 0 < an < 1 such that limsupn_^TO an < 1 - k. Then the sequence {xn} converges strongly to PF(T)(x1).

The domain of the function y : C ^ R U {+to} is the set

domy = {x e C: y(x) < +to}.

Let y : C ^ R U{+to} be a proper extended real-valued function and let $ be a bifunction from C x C into R, where R is the set of real numbers. The so-called mixed equilibrium

problem is to find x e C such that

$(x,y) + y(y)-y(x) > 0, Vy e C. (1.11)

The set of solutions of problem (1.11) is denoted by MEP($, y), that is,

MEP($, y) = {x e C: $(x,y) + y(y) - y(x) > 0, Vy e C}.

It is obvious that if x is a solution of problem (1.11), then x e dom y. If y = 0, then problem (1.11) is reduced to finding x e C such that

$(x, y) > 0, Vy e C. (1.12)

We denote by EP($) the set of solutions of the equilibrium problem. The theory of equilibrium problems has played an important role in the study of a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity, and optimization and has numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis, and mechanics. The ideas and techniques of this theory are being used in a variety of diverse areas and have proved to be productive and innovative. Problem (1.12) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instance, [22, 23] and the references therein. Some methods have been proposed to solve equilibrium problem (1.12); related work can also be found in [7,12, 24-34].

For solving the mixed equilibrium problem, let us assume that the bifunction $ : C x C — R, the function y : C — R U {+kj}, and the set C satisfy the following conditions:

(A1) $(x, x) = 0 for all x e C;

(A2) $ is monotone; that is, $(x, y) + $(y, x) < 0 for all x, y e C;

(A3) for each x, y, z e C,

lim $tz +(1 - t)x, y) < $(x, y);

(A4) for each x e C, y — $(x,y) is convex and lower semicontinuous;

(A5) for each y e C, x — $(x,y) is weakly upper semicontinuous;

(B1) for each x e C and r >0, there exists a bounded subset Dx c C and yx e C such that for any z e C\Dx,

$(z,yx) + y(yx)- y(z) + - <yx - z,z - x) <0; r

(B2) C is a bounded set.

Variational inequality theory provides us with a simple, natural, general, and unified framework for studying a wide class of unrelated problems arising in elasticity, structural analysis, economics, optimization, oceanography, and regional, physical, and engineering sciences, etc. (see [35-41] and the references therein). In recent years, variational inequalities have been extended and generalized in different directions, using novel and innovative techniques, both for their own sake and for their applications. A useful and important generalization of variational inequalities is a variational inclusion.

Let B: H — H be a single-valued nonlinear mapping and let M: H — 2H be a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point x e H such that

where 0 is the zero vector in H. The set of solutions of problem (1.13) is denoted by VI (H, B, M).

A set-valued mapping T: H — 2H is called monotone if for all x,y e H,f e Tx and g e Ty imply <x - y,f -g) > 0. A monotone mapping T: H — 2H is maximal if the graph of G(T) of T is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping T is maximal if and only if for (x,f) e H x H, <x - y,f - g) > 0 for all (y,g) e G(T) impliesf e Tx.

Definition 1.4 (see [42]) Let M: H — 2H be a multi-valued maximal monotone mapping. Then the single-valued mapping JM,X : H — H defined by JM,X(u) = (I + kM)-1(u), for all u e H, is called the resolvent operator associated with M, where X is any positive number and I is the identity mapping.

Recently, Qin etal. [43] introduced the following algorithm for a finite family of asymptotically Ki-SPC mappings Ti on C .Letxo e C and {an}TO=0 be a sequence in [0,1]. Let N > 1 be an integer. The sequence {xn} generated in the following way:

xi = a0x0 + (1 - a0)Tix0, x2 = a1x1 + (1 - a1)T2x1,

xn = aN-1XN-1 + (1 - aN-1) TNXN-1,

xn+1 = aNxN + (1 - aN)T12XN, (1.14)

x2N = a2N-1x2N-1 + (1 - a2N-1)T'N[x2N-1, x2N+1 = a2Nx2N + (1 - a2N)T^x2N,

is called the explicit iterative scheme of a finite family of asymptotically Ki-SPC mappings Ti on C, where i = 1,2, ...,N. Since, for each n e N,it can be written as n = (h - 1)N + i, where i = i(n) e {1,2, ...,N}, h = h(n) > 1 is a positive integer and h(n) — to as n — to.

For each i = 1,2, ...,N,let {Ti: C — C} be a finite family of asymptotically Ki-SPC mappings with the sequence {y^TO C [0, to) such that limn—TO Yni = 0. One has

0 e Bx + Mx,

(1.13)

TS>x - T^MI2 ^ (1 + Yh(n),i(n))\\x -yll

Vx, y e C,

(1.15)

for all n e N, and we can rewrite (1.14) in the following compact form:

xn = an-1xn-1 + (1 - an^T^) xn-1, Vn e N.

To be more precise, they introduced an iterative scheme for finding a common fixed point of a finite family of asymptotically Ki-SPC mappings as the following theorem.

Theorem QCKS Let C be a nonempty closed convex subset of a real Hilbert space H. Let N > 1 bean integer. For each i = 1,2, ...,N, let {Ti: C ^ C} be a finite family of asymptotically Ki-SPC mappings defined as in (1.15), when Ki e [0,1) with the sequence {yn,i} c [0, to) such that limn^TO yn>i = 0. Let k = max{Ki: 1 < i < N} and yn = max{^/ 1 + yn,i: 1 < i < N}. Assume that Q := Pl^ F(Ti) is a nonempty bounded subset of C. For xQ = x e C chosen arbitrarily, suppose that {xn} is generated iteratively by

yn-1 = an-1xn-1 + (1 - an-1)T1hnn)xn-1,

Cn-1 = {z e C: ||yn-1 - z||2 < ||x„_1 - z||2 + 6^-1

-(1 - an_l)(an_l - k)HT'hln(nlxn-1 -Xn_l||2}, Qn-1 = {z e C: <xn-1 - z,x0 - xn-1) > 0}, xn = PCn^nQn-!(xq), Vn e N,

where &n-1 = (yh2n) - 1)(1 - an-1) • An-1 ^ 0 as n ^to such that An-1 = sup{yxn-1 - z\\ : z e Q} < to. If the control sequence {an}TO=Q is chosen such that 0 < an < a <1. Then the sequence {xn} converges strongly to PQ(xQ).

Recall that a discrete family S = {Tn : n > 0} of self-mappings of C is said to be an asymptotically k-SPC semigroup [44] if the following conditions are satisfied:

(1) TQ = I, where I denotes the identity operator on C;

(2) Tn+mx = Tn Tmx, Vn, m > 0, Vx e C;

(3) there exists a constant k e [0,1) and a sequence {yn} of nonnegative real numbers with limn^TO yn = 0 such that

||Tnx - Tny||2 < (1 + Yn)||x -y||2

+ k || (I - Tn)x -(I - TJyf, Vx, y e C, (1.16)

for all n > 0. Note that, for a single asymptotically k-SPC mapping T: C ^ C, (1.16) immediately reduces to (1.6) by taking Tn = Tn for all n > 0 such that TQ = I.

On the other hand, Tianchai [24] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of common fixed points for a discrete asymptotically k -SPC semigroup which is a subclass of the class of infinite families for the asymptotically k-SPC mapping as the following theorem.

Theorem T Let C be a nonempty closed convex subset of a real Hilbert space H, $ be a bifunctionfrom C x C into R satisfying the conditions (A1)-(A5), and let y : C ^ R U {+to} be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. Let S = {Tn : n > 0} be an asymptotically k-SPC semigroup on C for some

k e [0,1) with the sequence {Yn} C [0, to) suchthat limn—TO Yn = 0. Assume that Q := F(S) n MEP($, q>) = P|TO=0 F(Tn) n MEP($, q>) is a nonempty bounded subset of C. For x0 = x e C chosen arbitrarily, suppose that {xn}, {yn} and {un} are generated iteratively by

un e C such that

<b(un,y) +v(y)-v(un) + <y - Un, Un - xn) > 0, Vy e C,

yn = anun + (1 an)Tnun,

Cn+1 = {z e Cn n Qn : y - z||2 < ||xn - z||2

+ (1 -an)(0n + (k - an)||un - Tnun ||2)}, Qn+1 = {z e Cn n Qn : <xn - z,x0 - xn) > 0}, C0 = Q0 = C,

xn+1 = PCn+1nQn+1 (x0), Vn e N U {0},

where 0n = Yn • sup{||xn -z||2: z e Q} < to, {an} c [a, b] such thatk < a < b <1, {rn} c [r, to) for some r >0 and |rn+1 - rn| < to. Then the sequences {xn}, {yn}, and {un} converge strongly tow = Pq (x0).

Recently, Saha etal. [6] modified Mann's iteration method (1.10) for finding a fixed point of the asymptotically k-SPC mapping in the intermediate sense which is not necessarily uniformly Lipschitzian (see, e.g., [6, 7]) as the following theorem.

Theorem SXY Let C be a nonempty closed convex subset of a real Hilbert space H and let T: C — C be a uniformly continuous and asymptotically k-SPC mapping in the intermediate sense defined as in (1.9), when k e [0,1), with the sequences {Yn}, {Tn} C [0, to) such that limn—TO Yn = limn—TO xn = 0. Assume that F(T) is a nonempty bounded subset of C. Suppose thatx1 = x e C chosen arbitrarily and {xn} is the sequence defined by

yn — (1 an)xn + an T' xn, Cn = {z e C: ||yn - z||2 < ||xn - z||2 + 0n}, Qn = {z e C: <xn - z, x1 - xn) > 0}, xn+1 = PCnnQn(x1), "Vn e N,

where 0n = Yn • An + Tn — 0 asn — to, An = sup{||xn -z|: z e F(T)} < to and {an} c (0,1] such that 0<a < an < 1-k . Then the sequence {xn} converges strongly to PF(T)(x1).

Let N > 1 be an integer. For each i = 0,1,...,N - 1, let {Ti: C — C} be a finite family of asymptotically Ki-SPC mappings in the intermediate sense with the sequences {Ynj}^, {TnJTO^ C [0, to) such that limn—TO Yn,i = limn—TO V = 0. One has

llTffx- Thgy\\2 < (1 + Yh(n),i(n))||x-y|2

+ Ki(n)|(I - Tg)x - (I - rjg))y|2

+ Th(n),i(n), Vx,y e C, (1.17)

for all n e N U {0} such that n = (h - 1)N + i, where i = i(n) e{0,1, ...,N -1}, h = h(n) > 1 is a positive integer and h(n) ^to as n ^to.

Subsequently, Hu and Cai [7] modified Ishikawa's iteration method (see [45]) for finding a common element of the set of common fixed points for a finite family of asymptotically Ki-SPC mappings in the intermediate sense and the set of solutions of the equilibrium problems (see also Duan and Zhao [8]) as the following theorem.

Theorem HC Let C be a nonempty closed convex subset of a real Hilbert space H, $ : C x C ^ R be a bifunction satisfying the conditions (A1)-(A4), and let A : C ^ H be a ^-cocoercive mapping. LetN > 1 be an integer. For each i = 0,1,...,N-1, let {Ti: C ^ C} be a finite family of uniformly continuous and asymptotically Ki-SPC mappings in the intermediate sense defined as in (1.17) when Ki e [0,1) with the sequences {yn,i}, {rn,i} c [0, to) such that limn^TO yn>i = limn^TO rn>i = 0. Let k = max{Ki: 0 < i < N - 1}, yn = max{yn,i: 0 < i < N - 1} and Tn = max{rn,i: 0 < i < N - 1}. Assume that Q := f^Q1 F(Ti) n EP($) is a nonempty bounded subset of C. For xQ = x e C chosen arbitrarily, suppose that {xn} and {un} are generated iteratively by

un e C such that

$(un,y) + <Axn,y - un) + yn <y - un, un - xn) >0, Vy e C,

zn = (1 - Pn)u n + ^nTi(n) un,

yn = (1 - an)un + anzn, Cn = {z e H: ||yn - z||2 < ||xn - z||2 + Qn}, Qn = {z e C: <xn - z, xq - xn) > 0}, xn+1 = PCnnQn(xq), Vn e N U {0},

where Qn = yh(n) • A2n + xh(n) ^ 0 asn ^to such that An = sup{||xn - z\\: z e Q} < to. If the control sequences {an}, {pn} c (0,1], and {rn} c [a, b] are chosen such that 0 < a < an < 1, 0 < P < Pn < 1 — k, and 0 < a < rn < b < 2f, then the sequences {xn} and {un} converge strongly to Pq(xq).

In this paper, we study the sequences {xn,i}TO=1 generated by modifying Mann's iteration method (1.10) for an infinite family of asymptotically Ki-SPC mappings in the intermediate sense. For each i = 1,2,..., let {Ti: C ^ C} be an infinite family of asymptotically Ki -SPC mappings in the intermediate sense with the sequences {Yn,i}TO=1, {Tn,i}TO=1 c [0, to) such that limn^TO y„i = limn^TO rn,i = 0. One has

|| T"x - Tfy|2 < (1 + Yn,i)\x -y|2

+ Ki|(I - T,n)x - (I - Tf)y||2 + Tni, Vx,y e C, (1.18)

for all n e N. For each i = 1,2,..., let x1,i e C and {a^}^ be a sequence in [0,1], and let the sequences {x^}^ be generated in the following way:

xn+1,i = (1 - an,i)xn,i + an,iT"xn,i, Vn e N.

Quite recently, Ezeora and Shehu [9] introduced an iterative scheme for finding a common fixed point of an infinite family of asymptotically Ki-SPC mappings in the intermediate sense as the following theorem.

Theorem ES Let C be a nonempty closed convex subset of a real Hilbert space H. For each

1 = 1,2,..., let {Ti: C — C} be an infinite family of uniformly continuous and asymptotically Ki-SPC mappings in the intermediate sense defined as in (1.18) when Ki e [0,1) with the sequences {y^TO^, {^JTO C [0, to) such that limn—TO Yn,i = limn—TO rn,i = 0. Assume that Q := P|^ F(Ti) is a nonempty bounded subset ofC. For x1 = x e C chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

yni = (1 - an,i)xn + an,iT"xm n > 1, Cni = {z e C: ||yn,i - z||2 < ||xn - z||2 + 0n,i}, C = nTO C ■

Cn = i=1 Cn,i,

Qn = {z e Qn-1: <xn - z,x1 - xn) >0}, n > 2, Q1 = C,

xn+1 = PCnnQn(x1), Vn e N,

where 0n,i = Yn,i • A2n + Tn,i (i = 1,2,...), An = sup{||xn -z||: z e Q} < to and {a»^^ C (0,1] (i = 1,2,...) such that 0 < ai < an,i < 1 - Ki. Then the sequence {x^TO converges strongly to Pa fa).

Inspired and motivated by the works mentioned above, in this paper, we introduce a new iterative scheme (3.1) below by modifying Mann's iteration method (1.10) to find a common element for the set of common fixed points of an infinite family of asymptotically Ki -SPC mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu [9] and many others.

2 Preliminaries

We collect the following lemmas which be used in the proof of the main results in the next section.

Lemma 2.1 (see [1]) Let C be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds:

<x - PCx, PCx - y) > 0, Vx e H, y e C.

Lemma 2.2 (see [46]) Let H be a Hilbert space. Forallx, y, z e H and a, ¡3, y e [0,1] such that a + ¡3 + y = 1, one has

||ax + ¡y + yz||2 = a||x||2 + 3 ||y||2 + y ||z||2 - afi ||x -y||2 - aY ||x - z||2 - ¡y ||y - z||2.

Lemma 2.3 (see [25]) LetC be a nonempty closed convex subset of a real Hilbert space H, $ : C x C ^ R satisfying the conditions (A1)-(A5), and let y : C ^ R U {+to} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r >0, define a mapping Sr: C ^ C as follows:

Sr(x) = jz e C: $(z,y) + y(y) -y(z) + -<y-z,z-x) > 0,Vy e c|

for all x e C. Then the following statements hold:

(1) for each x e C, Sr(x) = 0;

(2) Sr is single-valued;

(3) Sr is firmly nonexpansive; that is, for any x, y e C,

||Srx - Sry||2 < <Srx - Sry,x -y);

(4) F(Sr) = MEP($, y);

(5) MEP($, y) is closed and convex.

Lemma2.4 (see [42]) LetM: H ^ 2H be a maximal monotone mapping and let B: H ^ H be an a-inverse-strongly monotone mapping. Then, the following statements hold:

(1) B is a (1/a)-Lipschitz continuous and monotone mapping.

(2) u e H is a solution of quasivariational inclusion (1.13) if and only if u = JM,x(u - XBu), for all X >0, that is,

VI(H,B,M) = F(Jm,x(I - XB)), VX > 0.

(3) If X e (0,2a], then VI(H, B, M) is a closed convex subset in H, and the mapping I - XB is nonexpansive, where I is the identity mapping on H.

(4) The resolvent operator JM,x associated with M is single-valued and nonexpansive for all X >0.

(5) The resolvent operator Jm,x is 1-inverse-strongly monotone; that is,

Jx (x) - Jm,x 0012 < (x - y, Jm,x (x) - Jm,x 00), Vx, y e H.

Lemma 2.5 (see [47]) LetM: H ^ 2H be a maximal monotone mapping and letB: H ^ H be a Lipschitz continuous mapping. Then the mapping S = M + B: H ^ 2H is a maximal monotone mapping.

Lemma 2.6 (see [6]) Let C be a nonempty closed convex subset of a real Hilbert space H and letT: C ^ C be a uniformly continuous and asymptotically k-strictlypseudocontrac-tive mapping in the intermediate sense defined as in (1.9) when k e [0,1) with the sequences {yn}, {Tn} c [0, to) such that limn^TO yn = limn^TO Tn = 0. Then the following statements hold:

(1) ||Tnx - Tny|| < ^(k ||x -y|| + V(1 + (1 - k)Yn) ||x -y||2 + (1 - k)tn),for all x,y e C and n e N.

(2) If {xn} is a sequence in C such that limn^TO ||xn - xn+11| = 0 and limn^TO ||xn - Tnxn\\ = 0. Then limn^TO ||xn - Txn|| = 0.

(3) I - T is demiclosed at zero in the sense that if {xn} is a sequence in C such that xn ^ x e C as n — to, andlimsupm—TOlimsupn—TO ||xn - TmxnH = 0, then

(I - T)x = 0.

(4) F(T) is closed and convex.

3 Main results

Theorem 3.1 LetH be a real Hilbert space, $ be a bifunction from H x H into R satisfying the conditions (A1)-(A5), and let y : H — R U {+to} be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. Let M: H — 2H be a maximal monotone mapping and let B : H — H be a %-cocoercive mapping. For each i = 1,2,..., let {Ti: H — H} be an infinite family of uniformly continuous and asymptotically Ki-SPC mappings in the intermediate sense defined as in (1.18) when Ki e [0,1) with the sequences {Yn,i}TO=1, {t^JTO^ C [0, to) such that limn—TO Yn,i = limn—TO rn,i = 0. Assume that Q := piJ^ F(Ti) n VI(H,B,M) n MEP($, y) is a nonempty bounded subset ofH. For xx = x e H chosen arbitrarily, suppose that {x^TO is generated iteratively by

un e H such that

<&(umy) +y(y)-y(un) + -n <y - Un, Un - Xn) > 0, Vy e H, yni = (1 - an,i - ßn,i)un + anTun + ßn,JM,x(un - XBun), Cn+u = {z e Cn n Qn : ||yn,i - z||2 < ||Xn - z||2 + 0n,il,

Cn+i = Hi =1 Cn+1,i,

Qn+i = {z e Cn n Qn : <Xn - z, xi - Xn) > 0}, C1,i = C1 = Q1 = H, Xn+1 = Pcn+1nQn+1 (xi), "in e N,

where dn>i = yn>i ■ A2n + rn,i (i = 1,2,...) and An = sup{||Xn - z|| : z e Q} < to satisfying the following conditions:

(CI) {an,ilTO=1 C [ai,i, bi,i] and {ßn,ilTOTO=1 C [a2,i, ¿2,il (i = 1,2,...) such that 0 < ajti < bji < 1

for each j = 1,2, and b1,i + b2,i < 1 - Ki (i = 1,2,...); (C2) X e (0,2%] and {rnl^ C [r, to) for some r >0. Then the sequence {Xn}TO=1 converges strongly tow = Pq(x1).

Proof Pickp e Q and fix i = 1,2,____From (3.1), by the definition of Srn in Lemma 2.3, we

un = SrnXn e domy, (.2)

and by Tip = p, Lemmas 2.3(4) and 2.4(2), we have

T?p = SmP = Jm,x(I - XB)p = p. (3.3)

From Lemma 2.3(3), we know that Srn is nonexpansive. Therefore, by (3.2) and (3.3), we have

||un -p|| = HSrnXn - SrnPH < ||Xn -p||. (.4)

Let tn = JM,\(un - kBun). From Lemmas 2.4(3) and 2.4(4), we know that JM,k and I - kB are nonexpansive. Therefore, by (3.3), we have

IItn -p\\ = \jM,k(Un - kBUn)-JM,k(P - kBp)\ < \\Un -p\\. (3.5)

By (3.3), (3.4), (3.5), Lemma 2.2, and the asymptotically Ki-SPC in the intermediate sense of Ti, we have

bni - p\\2 = II (1- an,i - ßn,i)(Un - p)+an^TilUn - p) + ßn,i(tn - p)\\

= (1 - an,i - ßn,i) \\Un -p\\2 + an,i|TnUn -p|2 + ßn,i\\tn -p\\

- an,i(1 - an,i - ßn,i) 1 Un - Ti Un 1

- ßn,i(1 - an,i - ßn,i) \\ Un - ^n \\ - an,ißn,i 1 Ti Un - tn 1 < (1- ani - ßn,i)\Un -p\\2

+ an,^ (1 + Yn,i) \ Un - p\2 + Ki\\Un - T^Un |2 + Tn,i) + ßn,i\tn -p\2 - an,i(1 - ani - ßn,i)1 Un - Tf Un |2

- ßn,i(1 - an,i - ßn,i) \\ Un - tn \\ - an,ißn,i 1 Ti Un - tn 1

< (1 + an,iYn,i)\\Un -p\2 + an,iTn,i

an,i(1 an,i Pn,i Ki)| un Ti un| - pn,i(1 - an,i - Pn,i) \ un - tn \ - an,i$n,i | Ti un - tn |

< (1 + Yn,i)\un -p||2 + Tni - an,i(1- ani - Pn,i - Ki)|un - Tfun|2

< ||xn -p||2 + Qni, (.6)

where Qn,i := yn,i • An + Tn,i and An := sup{||xn - z\\: z e Q}.

Firstly, we show that Cn n Qn is closed and convex for all n e N.It is obvious that C1 n Q1 is closed and, by mathematical induction, that Cn n Qn is closed for all n > 2, that is, Cn n Qn is closed for all n e N. Since, for any z e H, ||yn,i - z||2 < ||xn - z||2 + Qn,i is equivalent to

||yn,i - xn |2 + 2<yni - xn,xn - z) - Qn,i < 0 (.7)

for all n e N. Therefore, for any z1, z2 e Cn+1 n Qn+1 c Cn n Qn and e e (0,1), we have

||yn,i - xn\2 + 2yn,i - xn, xn - (ez1 + (1 - e)z^) - Qn,i

= e (||yn,i - xn \ + 2 <yn,i - xn, xn - z1) - Qn,i)

+ (1 -e)( ||yn,i - xn\2 + 2<yn,i - xn, xn - z2) - Qn,i) < 0, (3.8)

(xn - (ez1 + (1 - e)z^, x1 - xn)

= e<xn - z1,x1 -xn) + (1 - e)<xn - z2,x1 - xn)

> 0 (.9)

for all n e N. Since C1 n Q1 is convex, by putting n = 1 in (3.7), (3.8), and (3.9), we have that C2 n Q2 is convex. Suppose that xk is given and Ck n Qk is convex for some k > 2. It follows by putting n = k in (3.7), (3.8), and (3.9) that Ck+1 n Qk+1 is convex. Therefore, by mathematical induction, we have that Cn n Qn is convex for all n > 2, that is, Cn n Qn is convex for all n e N. Hence, we obtain that Cn n Qn is closed and convex for all n e N.

Next, we show that Q C Cn n Qn for all n e N.It is obvious that p e Q C H = C1 n Q1. Therefore, by (3.1) and (3.6), we have p e C2,i for all i, and so p e C2, and note that p e H = Q2, and so p e C2 n Q2. Hence, we have Q C C2 n Q2. Since C2 n Q2 is a nonempty closed convex subset of H, there exists a unique element x2 e C2 n Q2 such that x2 = PC2nQ2(x1). Suppose that xk e Ck n Qk is given such that xk = PCknQk(x1), and p e Q C Ck n Qk for some k > 2. Therefore, by (3.1) and (3.6), we have p e Ck+1,i for all i, and so p e Ck+1. Since xk = PCknQk (x1), therefore, by Lemma 2.1, we have

<xk - z, x1 - xk > > 0

for all z e Ck n Qk. Thus, by (3.1), we have p e Qk+1, and so p e Ck+1 n Qk+1. Hence, we have Q C Ck+1 n Qk+1. Since Ck+1 n Qk+1 is a nonempty closed convex subset of H, there exists a unique element xk+1 e Ck+1 n Qk+1 such that xk+1 = PCk+1 nQk+1(x1). Therefore, by mathematical induction, we obtain that Q C Cn n Qn for all n > 2, and so Q C Cn n Qn for all n e N, and we can define xn+1 = PCn+1 nQK+1(x1) for all n e N. Hence, we obtain that the iteration (3.1) is well defined, and by Lemmas 2.3(5), 2.4(3), and 2.6(4), we also obtain that PQ(x1) is well defined. Next, we show that {xn} is bounded. Since xn = PC„nQ„ (x1) for all n e N,we have

||xn - xj < ||z - xj, (.10)

for all z e Cn n Qn. It follows by p e Q C Cn n Qn that ||xn - x11| < ||p - x11| for all n e N.

This implies that {xn} is bounded, and so are {un}, {tn}, and {yn,i} for each i = 1,2,____

Next, we show that ||xn - xn+1| — 0 as n — to. Since xn+1 = PCn+1nQn+1 (x1) e Cn+1 n Qn+1 C Cn n Qn, therefore, by (3.10), we have ||xn - x11| < ||xn+1 - x11| for all n e N. This implies that {||xn - x1|} is a bounded nondecreasing sequence, there exists the limit of ||xn - x1|, that is,

lim ||xn - x1| = m1 (3.11)

n—TO

for some m1 > 0. Since xn+1 e Qn+1, therefore, by (3.1), we have

<xn - xn+1,x1 - xn> > 0. (3.12)

It follows that

11 xn xn+1|| — 11 xn x111 + 2 <xn x1, x1 xn > + 2 <xn x1, xn xn+1 > + | xn+1 x111

< ||xn+1 - x1 y - ||xn - x1 y .

Therefore, by (3.11), we obtain

||xn - xn+1| — 0 as n — to. (3.13)

Next, we show that {xn} is a Cauchy sequence. Observe that C1 n Q1 D C2 n Q2 D ••• D Cn n Qn D ••• D Q. It follows that

xn+m = PCn+mnQn+m (x1) e Cn+m n Qn+m c Cn+1 n Qn+1 c Qn+1

for each m > 1. Therefore, by (3.1), we have

<x n - xn + m , x 1 - xn ) > 0. (3.14)

Thus, we have

||xn+m - xn\\ = ||xn+m - x1W + ||xn - ^W -2<xn+m - x1, xn - x1)

— ||xn+m x1\\ 11 xn xj 2 <xn+m xn, xn x1)

< ||xn+m - x1W - ||xn - x1W . (3.15)

Hence, by (3.11), we obtain that ||xn+m - xn|| ^ 0 as n ^to, which implies that {xn} is a Cauchy sequence in H, and then there exists a point w e H such that xn ^ w as n ^to.

Next, we show that ||yn,i -xn || ^ 0 as n ^to. From (3.1), we have xn+1 = PCn+1nQn+1 (x1) e Cn+1 n Qn+1 c Cn+1 c Cn+1,i. Therefore, we have

||yn,i - xn+112 < ||xn - xn+11|2 + Qn,i. (.16)

It follows by (3.13) and limn^TO Qn,i = 0 that

||yn,i - xn+11| ^ 0 as n ^TO. (.7)

Since,

||yn,i - xn|| < ||yn,i - xn+1|| + ||xn+1 - xn ||. (.8)

Therefore, by (3.13) and (3.17), we obtain

||yn,i -xn||^0 as n ^ to. (.9)

Next, we show that ||un - xn || ^ 0 and ||un+1 - un|| ^ 0 as n ^to. By (3.2), (3.3), and the firmly nonexpansiveness of Srn in Lemma 2.3(3), we have

||un - p||2 < <Srnxn - Srnp, xn - p) = <un - p, xn - p) = 2 (||un -p\2 + ||xn -p|2 - ||un - xn\2), which implies that

||un -p||2 < ||xn -p||2 - ||un -xn\2. (.20)

By (3.4), (3.6), and (3.20), we have

Wyni - pf < (1 + Yni)\\Un - pf + Tni < \\Un - pf + Oni < \Xn - PW2- \\Un - Xn\\2+ Oni,

which implies that

Hun -xn|2 < ||xn -p|2- ||yn,i -p||2 + Oni

< ||xn -Уn,iH(HXn -p|| + ||yn,i -pH) + On,i. (3.21)

Therefore, by (3.19) and limn—TO On,i = 0, we obtain

||un - xn|| — 0 as n —to. (..2)

||un+1 - un|| < | un+1 - xn+1 | + |xn+1 - xn | + ||xn - un (3.23)

therefore, by (3.13) and (3.22), we obtain

||un+1 - un|| — 0 as n —to. (..4)

Next, we show that w e P|1F(Ti). From (3.4) and (3.6), we have

||yn,i -p|2 < (1 + Kn,i)|un -p|2 + Tni - an,i(1- ani - Pn,i - ^i)|un - Tfun|2

< ||un - p||2 + Oni - an,i(1- a„i - fin,i - K;)||un - Tfun|2,

which implies that

«1,i(1 - hi - b2,i - Ki) 1 un - Tfun |2

< an,i(1 - a„i - Pni - Ki)1 un - Tfun |2

< ||un -p||2- ||yn,i -p|2 + Oni

< ||un - yn,i||(||un - p|| + ||yn,i - p||) + On,i

< (||un - xj| + ||xn - yn,i||)(||un - pH + ||yn,i - p||) + On,i. (3.25) Therefore, by (3.19), (3.22), and limn—TO On,i = 0, we obtain

|un - Tnun| — 0 as n — to. (3.6)

From (3.24) and (3.26), by Lemma 2.6(2), we have

||un - Tiun H —0 as n — to. (3.7)

Therefore, by (3.27) and the uniform continuity of Ti, it is easy to see, by mathematical induction on m e N, that

\Tmun - Tm+1un\ ^ 0 as n ^TO (3.28)

for each m e N. Since, for any m e N,we have

\ un Ti un \ < || un Tiun || + \ Tiun Ti un \ + ••• + \ Ti un Ti un \

such that TQ = I, where I is the identity mapping on H. Therefore, by (3.27) and (3.28), we obtain

\un - Tm"un\ ^ 0 as n ^ to. (.9)

From (3.22) and xn ^ w, we have un ^ w as n ^ to. Therefore, from (3.29), by Lemma 2.6(3), we obtain that w e F(Ti) for all i = 1,2,...; that is, w e P|ITO1 FT). Next, we show that w e MEP($, y). From (3.1), we have

0 < $(un,y) + <p(y) - <p(un) + — <y - un, un - xn), Vy e H.

It follows by the condition (A2) that

$(y, un) < &(y, un) + $(un,y) + y(y)-y(un) + — <y - un, un - xn), Vy e H

< y(y)-y(un) + — <y - un, un - xn), Vy e H.

Hence,

v(y)-v(un) + (y - un, Un Xn ) > $(y, Un), Vy e W. (3.30)

From (3.22) and xn ^ w, we have «„ ^ w as n ^to. Therefore, we obtain

$(y, w) + y(w)-y(y) < 0, Vy e H. (.31)

For a constant t with 0 < t <1 and y e H, let yt = ty +(1 - t) w. Since y, w e H, thus, yt e H. So, from (3.31), we have

&(yt, w) + y(w) - y (y) < 0. (3.32)

By (3.32), the conditions (A1) and (A4), and the convexity of y, we have

0 = , yt ) + y(y )-y(yt )

< (t$(yt,y) + (1 - t)$(yt, w)) + (ty(y) + (1 - t)y(w)) - y(yt) = t($(yt,y) + y(y) - y(yt)) + (1 -t)($(yt, w) + y(w) - y(yt))

< t($(yt,y) + y(y)-y(yt)),

which implies that

®(yt, y) + y(y)-y(yt ) > 0.

Therefore, by the condition (A3) and the weakly lower semicontinuity of y, we have $(w,y) + y(y) - y(w) > 0 as t ^ 0 for all y e H, and hence we obtain that w e MEP($, y). Next, we show that w e VI(H, B, M). From (3.1), we have

yn,i = (1 - an,i - fin,i)Un + an,iTi Un + f3n,itn,

which implies that

a2,i II Un — tn y < fin,i II Un — tn y < Il Un — yn,i y + an,i | Un - Ti un | < IIun — xn y + IIxn - yn,i y + an,i 1 Un - Ti Un 1 .

Therefore, by (3.19), (3.22), and (3.26), we obtain

IIun - tny^0 as n ^to. (.33)

From Lemma 2.4(1), we have that B is (1/f )-Lipschitz continuous. Therefore, by Lemma 2.5, we have that M + B is maximal monotone. Let (y,g) e G(M + B), that is,

g - By e My. (3.4)

Since tn = JM,x(un - XBun), we have un - XBun e (I + XM)tn. Therefore,

— (un - tn - XBun) e Mtn. (3.5)

By (3.34), (3.35), and M is maximal monotone, we have

0 < (y - tn,My - Mtn) = [y - tn,g - By - X(Un - tn - XBUn) j.

It follows by the monotonicity of B that

(y - tn,g) > |y - tn,By + X(Un - tn - XBUn)

= |y - tn,(By - Btn) + (Btn - BUn) + X (Un - tn)

> (y - tn,Btn - BUn) + 1 (y - tn, Un - tn). (3.36)

From (3.33), we have yBtn - BUny ^ 0, and since xn ^ w, by (3.22) and (3.33), we have tn ^ w as n ^to. Therefore, by (3.33) and (3.36), we obtain that (y - w,g) > 0 as n ^to. It follows from the maximal monotonicity of M + B that 0 e (M + B)w; that is, w e VI(H,B,M), and so w e

Finally, we show that w = PQ(x1). Since Q c Cn+1 n Qn+1 c Qn+1. Therefore, by (3.1), we have

<xn - z,x1 - xn) > 0, Vz e Q. It follows by xn ^ w as n ^to that

<w - z,x1 - w) > 0, Vz e Q. Therefore, by Lemma 2.1, we obtain that w = PQ(x1). This completes the proof. □

Remark 3.2 The iteration (3.1) is different from the iterative scheme of Ezeora and Shehu [9] as follows:

1. The sequence {xn} is a projection sequence of x1 onto Cn n Qn for all n e N such that

C1 n Q1 D C2 n Q2 D • • O Cn n Qn D-Ofi.

2. The proof to the strong convergence of the sequence {xn} is simple by a Cauchy sequence.

3. An approximate solution to a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems by iteration is obtained.

We define the condition (B3) as the condition (B1) such that y = 0. If y = 0, then Theorem 3.1 is reduced immediately to the following result.

Corollary 3.3 Let H be a real Hilbert space and let $ be a bifunction from H x H into R satisfying the conditions (A1)-(A5) with the assumption that either (B2) or (B3) holds. Let M: H ^ 2H be a maximal monotone mapping and letB: H ^ H be a %-cocoercive mapping. For each i = 1,2,..., let {Ti: H ^ H} bean infinite family of uniformly continuous and asymptotically k;-SPC mappings in the intermediate sense defined as in (1.18) when k; e [0,1) with the sequences {yn,i}TO=1, {Tn,i}TO=1 c [0, to) such that limn^TO yn>i = limn^TO rn>i = 0. Assume that Q := f]F(Ti) n VI(H,B,M) n EP($) is a nonempty bounded subset ofH. For x1 = x e H chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

un e H such that $(un,y) + jr <y - un, un - xn) >0, Vy e H, yn,i = (1 - an,i - fin,i)un + an>T?un + Pn,JM,\(un - kBu^), Cn+1,i = {z e Cn n Qn : ||yn,i - z||2 < ||xn - z||2 + 6>n,;},

• Cn+1 = Hi =1 Cn+1,i,

Qn+1 = {z e Cn n Qn : <xn - z, x1 - xn) > 0},

C1,i = C1 = Q1 = H,

xn+1 = PCn+1nQn+1 (x1), Vn e N,

where dn>i = yn>i ■ A2n + rn>i (i = 1,2,...) and An = sup{\\xn - z\\ : z e < to satisfying the following conditions:

(CI) {«n,;}^ c [ai,i, bi,i] and {Pn,i}TO=1 c [a2,i, b2,i] (i = 1,2,...) sMch ihai 0 < ajti < b/,i < 1

for each j = 1,2, and b1,i + b2,i < 1 - (i = 1,2,...); (C2) X e (0,2%] and {rn^ C [r, to) for some r >0. Then the sequence {xn}TOi. converges strongly tow = P^(x1).

If $ = 0, then Corollary 3.3 is reduced immediately to the following result.

Corollary 3.4 Let H be a real Hilbert space, M: H ^ 2H be a maximal monotone mapping, and let B : H ^ Hbea%-cocoercive mapping. For each i = 1,2,..., let {T; : H ^ H} be an infinite family of uniformly continuous and asymptotically Ki-SPC mappings in the intermediate sense defined as in (1.18) when k; e [0,1) with the sequences {yn,i}TO=1, {Tn,i}TO=1 C [0, to) such that limn^TO yn>i = limn^TO Tn,i = 0. Assume that ^ := P|ITO=1 F(T;) n VI(H,B,M) is a nonempty bounded subset ofH. Forx1 = x e H chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

yn,i = (1 an,i fin,i)xn + an,iTi xn + fin,iJM,X(Xn XBxn),

Cn+1,i = {z e Cn n Qn : IIyn,i - zII2 < Ixn - zII2 + 6n,i},

Cn+1 = 0 TO=1 Cn+1,i,

Qn+1 = {z e Cn n Qn : (xn - z, X1 - Xn) > 0},

C1,i = C1 = Q1 = H,

xn+1 = Pcn+1nQn+1 (x1), Vn e N,

where 6n>i = yn>i ■ A2n + rn,i (i = 1,2,...) and An = sup{yxn - zy : z e < to satisfying the following conditions:

(Cl) {an,i}TO=1 C [a1,i, by] and {^n,i}TO=1 C [a2,i, b2,i\ (i = 1,2,...) such that 0 < a^ < j < 1

for each j = 1,2, and b1,i + b2,i < 1 - k; (i = 1,2,...); (C2) X e (0,2f \. Then the seqUence {xn}TO=i converges strongly tow = P^(xi).

If B = 0 and M = 0, then Theorem 3.1 is reduced immediately to the following result.

Corollary 3.5 Let C be a nonempty closed convex sUbset of a real Hilbert space H, $ be a bifUnctionfrom C x C into R satisfying the conditions (A1)-(A5), and let y : C ^ R U {+to} be a proper lower semicontinUoUs and convex fUnction with the assUmption that either (B1) or (B2) holds. For each i = 1,2,..., let {T; : C ^ C} be an infinite family of Uniformly continUoUs and asymptotically Ki-SPC mappings in the intermediate sense defined as in (1.18) when k; e [0,1) with the seqUences {yn>i}TO=1, {Tn,i}TO=1 C [0, to) sUch that limn^TO yn>i = limn^TO x„i = 0. AssUme that ^ := P|to=1 F(T;) n MEP($, y) is a nonempty boUnded sUbset

ofC. For x1 = x e C chosen arbitrarily, suppose that {xn}cTO=1 is generated iteratively by

un e C such that <&(un,y) + y(y)-y(un) + -n <y - Un, Un - Xn) > 0, Vy e C,

yn,i = (i - an,i)un + an,iTi un,

Cn+i,i = {z e Cn n Qn : ||yn,i - z||2 < ||Xn - z||2 + dn,i},

Cn+1 = ni =i Cn+1,i,

Qn+i = {z e Cn n Qn : <Xn - z, xi - Xn) > 0},

Ci,i = Ci = Qi = C,

Xn+i = PCn+inQn+i (xi), Vn e N,

where 6n>; = yn>; • A2n + rn,i (i = 1,2,...) and An = sup{||xn - z\\ : z e Q} < to satisfying the following conditions:

(C1) {an,i}TO—1 c [a, b] (i = 1,2,...) such that 0<a < b < 1 - k;;

(C2) {rn}TO=1 c [r, to) for some r >0. Then the sequence {xn}TO=1 converges strongly tow = PQ(x1).

If y = 0, then Corollary 3.5 is reduced immediately to the following result.

Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H and let $ be a bifunction from C x C into R satisfying the conditions (A1)-(A5) with the assumption that either (B2) or (B3) holds. For each i = 1,2,..., let {T; : C ^ C} be an infinite family of uniformly continuous and asymptotically k;-SPC mappings in the intermediate sense defined as in (1.18) when k; e [0,1) with the sequences ^^TOiv {t^ATO^ c [0, to) such that limn^TO y„i = limn^TO x„i = 0. Assume that Q := nTO1 F(T;) n EP($) is a nonempty bounded subset ofC. For x1 = x e C chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

un e C such that $(un,y) + <y - un, un - xn) >0, Vy e C,

yn,i = (1 - an,i)un + an,iT; un,

Cn+1,; = {z e Cn n Qn : ||yn,; - z|2 < ||xn - z||2 + 6n,;},

Cn+1 = OTOTO Cn+1,i,

Qn+1 = {z e Cn n Qn : <xn - z, x1 - xn) > 0}, C1,i = C1 = Q1 = C, xn+1 = PCn+1nQn+1 (x1), Vn e N,

where dn>i = yn>i ■ ¿S.« + rn,i (i = 1,2,...) and An = sup{||xn - z\\ : z e < to satisfying the following conditions:

(CI) {an,i}TO=1 C [a, b] (i = 1,2,...) such that 0<a < b < 1 - Ki; (C2) {rn}TO=1 C [r, to) for some r >0. Then the sequence {xn}TO=1 converges strongly tow = Pq (x1).

If $ = 0, then Corollary 3.6 is reduced immediately to the following result.

Corollary 3.7 Let Cbea nonempty closed convex subset of a real Hilbert space H. For each i = 1,2,..., let {Ti: C ^ C} be an infinite family of uniformly continuous and asymptotically Ki-SPC mappings in the intermediate sense defined as in (1.18) when k; e [0,1) with the sequences {yn,i}TO=1, {Tn,i}TO=1 C [0, to) such that limn^TO yn>i = limn^TO rn>i = 0. Assume that Q := P|TO=1 F(T;) is a nonempty bounded subset ofC. For x1 = x e C chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

yn,i = (1 an,i)xn + an,iTi xn,

Cn+v = {z e Cn n Qn : ||yn,i - z||2 < X - z||2 + dn,i},

Cn+i = 0 i=1 Cn+1,i,

Qn+i = {z e Cn n Qn : X - z,xi - Xn) > 0},

C1,i = C1 = Q1 = C,

Xn+i = PCn+inQn+i (xi), Vn e N,

where 0n,i = yn.i ■ A2n + xni (i = 1,2,...), An = sup{\\xn -z\\:z e < to and {an,i}TO=1 C [a,b] (i = 1,2,...) such that 0<a < b < 1 - k; . Then the sequence {xn}TO=1 converges strongly to w = Pq (x1).

Recall that for each i = 1,2,..., a mapping T; : C ^ C is said to be asymptotically nonexpansive if there exists a sequence {yn,i} C [0, to) with limn^TO yn>i = 0 such that

llTTx - Tfy|| < y/T+niUx-y\\, Vx,y e C (3.37)

for all n e N.If k; = 0 and rn,i = 0 for all i = 1,2,... and n e N, then Corollary 3.7 is reduced immediately to the following result.

Corollary 3.8 Let Cbea nonempty closed convex subset of a real Hilbert space H. For each i = 1,2,..., let {Ti: C ^ C} bean infinite family of asymptotically nonexpansive mappings defined as in (3.37) with the sequence {yn>i}TO=1 C [0, to) such that limn^TO yn>i = 0. Assume that Q := P| F (Ti) is a nonempty bounded subset ofC. For x1 = x e C chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

yn,i = (1 an,i)xn + an,iTi xn,

Cn+1,i = {z e Cn n Qn : \\yn,i - z\\2 < \\xn - z\2 + 0n,i},

Cn+1 = 0 TO=1 Cn+1,i,

Qn+1 = {z e Cn n Qn : {xn - z, x1 - xn) > 0},

C1,i = C1 = Q1 = C,

xn+1 = PCK+1nQK+1 (x1), "Vn e N,

where dn>i = y„ti ■ A2n (i = 1,2,...), An = sup{\xn - z\\: z e Q} < to, and {an,i}TO=1 C [a, b] (i = 1,2,...) such that 0 < a < b <1. Then the sequence {xn}TO=1 converges strongly tow = PQ(x1).

4 Applications

We introduce the equilibrium problem to the optimization problem:

min Z (x),

where C is a nonempty closed convex subset of a real Hilbert space H and Z : C ^ R U {+to} is proper convex and lower semicontinuous. We denote by Argmin(Z) the set of solutions of problem (4.1). We define the condition (B4) as the condition (B3) such that $ : C x C ^ R is a bifunction defined by $(x, y) = Z (y)-Z (x) for all x, y e C. Observe that EP($) = Argmin(Z). We obtain that Corollary 3.3 is reduced immediately to the following result.

Theorem 4.1 Let H be a real Hilbert space and let Z : H ^ R U {+to} be a proper lower semicontinuous and convex function with the assumption that either (B2) or (B4) holds. Let M: H ^ 2H be a maximal monotone mapping and letB: H ^ H be a %-cocoercive mapping. For each i = 1,2,..., let {T;: H ^ H} bean infinite family of uniformly continuous and asymptotically k;-SPC mappings in the intermediate sense defined as in (1.18) when k; e [0,1) with the sequences {y^TOv {Tn,;}TO=1 c [0, to) such that limn^TO yn>; = limn^TO rn>; = 0. Assume that Q := f^ F(T;) n VI(H, B, M) n Argmin(Z) is a nonempty bounded subset ofH. For x1 = x e H chosen arbitrarily, suppose that {xn}TO=1 is generated iteratively by

un e H such that Z (y) - Z(un) + <y - un, un - xn) >0, Vy e H, yn,i = (1 - an,; - Pn,i)un + anTun + Pn,;jM,\(un - kBun), Cn+1,i = {z e Cn n Qn : ||yn,; - z|2 < ||xn - z|2 + 0n,;},

Qn+i = {z e Cn n Qn : (x« - z,xi - x«) > 0},

Ci,i = Ci = Qi= H,

x«+i = PC„+inQ„+i (xi), Vn e N,

where dn>i = yn>i ■ a« + Tn,i (i = i, 2,...) and An = sup{|xn - z\\ : z e Q} < to satisfying the following conditions:

(Cl) {a«,i}TO=i C [ai,i, bi,i] and {^«,i}TO=i C [a2,i, b2,i] (i = i, 2,...) such that 0 < ajti < bji < i

for each j = i, 2, and bi,i + b2,i < i - Ki (i = i, 2,...); (C2) k e (0,2%] and {r«C [r, to) for some r >0. Then the sequence {xn}TO=i converges strongly tow = PQ (xi).

Competing interests

The author declares that he has no competing interests. Acknowledgements

The author would like to thank the Faculty of Science, Maejo University for its financialsupport. Received: 2 May 2012 Accepted: 7 November 2012 Published: 26 November 2012 References

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doi:10.1186/1687-1812-2012-214

Cite this article as: Tianchai: An approximate solution to the fixed point problems for an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, cocoercive quasivariational inclusions problems and mixed equilibrium problems in Hilbert spaces. Fixed Point Theory and Applications 2012 2012:214.

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