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Explicit iterations for Lipschitzian semigroups with the Meir-Keeler type contraction in Banach spaces

Abstract

In this paper, we introduce and prove strong convergence theorems for a new viscosity iteration scheme for approximating common fixed points of a Lipschitzian semigroup on a compact and convex subset of a smooth Banach space. Our results extend and improve recent results.

MSC:47H09, 47H10.

1 Introduction

Let C be a nonempty, closed and convex subset of a Banach space E. Recall that a self mapping f:EE is an α-contraction on C if there exists a constant α(0,1) such that for any x,yE, we have

f ( x ) f ( y ) αxy.

A function ψ: R + R + is said to be an L-function if ψ(0)=0, ψ(t)>0 for any t>0, and for every t>0 and s>0, there exists u>s such that ψ(t)s for all t[s,u]. This implies that ψ(t)<t for all t>0.

A mapping f:EE is said to be a (ψ,L)-contraction if there exists an L-function ψ: R + R + such that

f ( x ) f ( y ) ψxy,x,yE with xy.

If ψ(t)=kt for all t>0, where k(0,1), then f is a contraction.

A mapping f is called a Meir-Keeler type mapping if for each ϵ>0, there exists δ(ϵ)>0 such that for all x,yE, if ϵ<xy<ϵ+δ, then f(x)f(y)<ϵ.

A mapping T:CC is said to be

  1. (i)

    nonexpansive if

    TxTyxy,x,yC;
  2. (ii)

    Lipschitzian with a Lipschitz constant l>0 if

    TxTylxy,x,yC;
  3. (iii)

    asymptotically nonexpansive if there exists a sequence { k n } of positive numbers satisfying the property lim n k n =1 and

    T n x T n y k n xy,x,yC.

Every nonexpansive mappings are asymptotically nonexpansive with respect to the sequence k n =1, nN. Also, every asymptotically nonexpansive mappings are uniformly l-Lipschitzian with l= sup n N k n .

Let S be a semigroup. Then a family S={ T s :sS} of mappings of C into itself is called Lipschitzian mappings on C if for each sS, the mapping T s is a Lipschitzian mapping on C with a Lipschitz constant k s , and T s t = T s T t for all s,tS. A family S is called a Lipschitzian semigroup on C if it satisfies the following:

  1. 1.

    T s t x= T s T t x for all s,tS and xC;

  2. 2.

    for each sS, T s is a Lipschitzian mapping of C into itself, i.e., there is k s 0 such that

    T s x T s y k s xyfor all x,yC.

A Lipschitzian semigroup S is called uniformly l-Lipschitzian if k s =l for all sS.

Let F(S) denote the common fixed point set {xC: T s x=x,sS} of the mappings T s .

For a semigroup S, we define a partial preordering on S by ab if and only if aSbS. If S is a left reversible semigroup (i.e., aSbS for a,bS), then it is a directed set. (Indeed, for every a,bS, applying aSbS, there exist a , b S with a a =b b ; by taking c=a a =b b , we have cSaSbS, and then ac and bc.)

Let S={ T s :sS} be a representation of a left reversible semigroup S as Lipschitzian mappings on C with Lipschitz constants { k s :sS}. We shall say that S is an asymptotically nonexpansive semigroup on C, if there holds the uniform Lipschitzian condition lim s k s 1 on the Lipschitz constants.

In 1953, Mann [1] introduced an iterative method as follows: a sequence { x n } defined by

x n + 1 = α n x n +(1 α n )T x n ,
(1.1)

where the initial guess element x 0 C is arbitrary and { α n } is a sequence of real numbers in [0,1]. The Mann iteration can guarantee in general only weak convergence. The Mann iteration has been extensively investigated for nonexpansive mappings and modified for strong convergence. Later, in 2000, Reich and Zaslavski [2] introduced the Krasnoselskii-Mann iterations of a generic nonexpansive operator on a closed and convex, but not necessarily bounded, subset of a hyperbolic space with a unique fixed point.

In 1967, Halpern [3] considered the following algorithm:

x n + 1 = α n x+(1 α n )T x n ,
(1.2)

where T is nonexpansive and the initial guess element xC is arbitrary.

In 2004, Xu [4] introduced and proved the following viscosity approximation methods for nonexpansive mappings in a uniformly smooth Banach space:

x n + 1 = α n f( x n )+(1 α n )T x n ,
(1.3)

where f:CC is a contraction mapping.

In 2007, Lau, Miyake and Takahashi [5] introduced the following Mann’s implicit iteration process:

x n + 1 = α n x+(1 α n )T( μ n ) x n ,
(1.4)

for a semigroup S={ T s :sS} of nonexpansive mappings on a compact and convex subset C of a smooth and strictly convex Banach space.

In the same year, Zhang et al. [6] introduced the following composite iteration scheme:

{ y n = β n x n + ( 1 β n ) T ( t n ) x n , x n + 1 = α n x + ( 1 α n ) y n ,
(1.5)

where {T(t):t0} is a nonexpansive semigroup, x is an arbitrary point in C. Under a suitable condition, they proved strong convergence theorems of an explicit composite iteration scheme for nonexpansive semigroups in a reflexive Banach space with a uniformly Gâteaux differentiable norm, uniformly smooth Banach space and uniformly convex Banach space with a weakly continuous normalized duality mapping.

In 2008, Shahram Saeidi [7] introduced the following viscosity iterative scheme:

x n + 1 = α n f( x n )+ β n x n + γ n T( μ n ) x n ,
(1.6)

for a representation of S as Lipschitzian mappings on a compact and convex subset C of a smooth Banach space E with respect to a left regular sequence { μ n } of means defined on an appropriate invariant subspace of l (S); for some related results, we refer the readers to [8, 9].

Motivated and inspired by the idea of Zhang et al. [6] and Saeidi [7], we introduce the explicit viscosity iterative process by Meir-Keeler type contraction in a smooth Banach space. Then we prove that the sequence { x n } converges strongly to a common fixed point of S={ T s :sS}, where S is a left reversible semigroup, which is the unique solution of the variational inequality

( f I ) q , J ( p q ) 0,pF(S).

2 Preliminaries

Let E be a Banach space and let E be the topological dual of E. The value of x E at xE will be denoted by x, x or x (x). With each xE, we associate the set

J(x)= { x E : x , x = x 2 = x 2 } .

Using the Hahn-Banach theorem, it immediately follows that J(x) for each xE. A Banach space E is said to be smooth if the duality mapping J of E is single-valued. We know that if E is smooth, then J is norm-to-weak continuous; see [8, 9]. Let E be a Banach space and let C be a closed and convex subset of E. Then

x y 2 = x 2 y 2 2xy,y,
(2.1)

and

λ x + ( 1 λ ) y 2 =λ x 2 +(1λ) y 2 λ(1λ) x y 2 ,
(2.2)

for all x,yE and λ[0,1].

Let S be a semigroup. We denote by l (S) the Banach space of all bounded real valued functions on S with a supremum norm. For each sS, we define l s and r s on l (S) by ( l s g)(t)=g(st) and ( r s g)(t)=g(ts) for each tS and g l (S). Let X be a subspace of l (S) containing 1 and let X be its topological dual. An element μ of X is said to be a mean on X if μ=μ(1)=1. We often write μ t (g(t)) instead of μ(g) for μ X and gX. Let X be left invariant (resp. right invariant), i.e., l s (X)X (resp. r s (X)X) for each sS. A mean μ on X is said to be left invariant (resp. right invariant) if μ( l s g)=μ(g) (resp. μ( r s g)=μ(g)) for each sS and gX. A subspace X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. A semigroup X is amenable if X is both left and right amenable. If a semigroup S is left amenable, then S is left reversible [10, 11].

A net { μ α } of means on X is said to be strongly left regular if

lim α l s μ α μ α =0

for each sS, where l s is the adjoint operator of l s . Let C be a nonempty, closed and convex subset of E. Throughout this paper, S will always denote a semigroup with an identity e. S is called left reversible if any two right ideals in S have nonvoid intersection, i.e., aSbS for any a,bS. In this case, we can define a partial ordering on S by ab if and only if aSbS. It is easy to see tts (for all t,sS). Further, if ts, then ptps for all pS. If a semigroup S is left amenable, then S is left reversible. But the converse is false. Denote by C a the set of almost periodic elements in C, i.e., all xC such that S={ T s x:sS} is relatively compact in the norm topology of E. We will call a subspace X of l (S), S-stable if the functions s T s x, x and s T s xy on S are in X for all x,yC and x E . We know that if μ is a mean on X and if for each x E , the function s T s x, x is contained in X and C is weakly compact, then there exists a unique point x 0 of E such that

μ s T s x , x = x 0 , x

for each x E . We denote such a point x 0 by T(μ)x. Note that T(μ)z=z for each zF(S); see [1214]. Let D be a subset of B where B is a subset of a Banach space E and let P be a retraction of B onto D. Then P is said to be sunny [15] if for each xB and t0 with Px+t(xPx)B,

P ( P x + t ( x P x ) ) =Px.

A subset D is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B onto D. If E is smooth and P is a retraction of B onto D, then P is sunny and nonexpansive if and only if for each xB and zD,

x P x , J ( z P x ) 0.
(2.3)

For more details, see [8, 9].

Lemma 2.1 ([16])

Let S be a left reversible semigroup and let S={ T s :sS} be a representation of S as Lipschitzian mappings from a nonempty, weakly compact and convex subset C of a Banach space E into C, with the uniform Lipschitzian condition lim s k s 1 on the Lipschitz constants of the mappings. Let X be a left invariant S-stable subspace of l (S) containing 1, and μ be a left invariant mean on X. Then F(S)=F(T(μ)) C a .

Corollary 2.2 ([7])

Let { μ n } be an asymptotically left invariant sequence of means on X. If z C a and lim inf n T( μ n )zz=0, then z is a common fixed point of S.

Lemma 2.3 ([7])

Let S be a left reversible semigroup and let S={ T s :sS} be a representation of S as Lipschitzian mappings from a nonempty weakly compact and convex subset C of a Banach space E into C, with the uniform Lipschitzian condition lim s k s 1 on the Lipschitz constants of the mappings. Let X be a left invariant subspace of l (S) containing 1 such that the mappings s T s x, x be in X for all xX and x E , and { μ n } be a strongly left regular sequence of means on X. Then

lim sup n sup x , y C ( T ( μ n ) x T ( μ n ) y x y ) 0.

Remark 2.4 From Lemma 2.3, taking

c n = sup x , y C ( T ( μ n ) x T ( μ n ) y x y ) ,n,
(2.4)

we obtain that lim sup n c n 0. Moreover,

T ( μ n ) x T ( μ n ) y xy+ c n ,x,yC.
(2.5)

Corollary 2.5 ([7])

Let S be a left reversible semigroup and let S={ T s :sS} be a representation of S as Lipschitzian mappings from a nonempty, compact and convex subset C of a Banach space E into C, with the uniform Lipschitzian condition lim s k s 1. Let X be a left invariant S-stable subspace of l (S) containing 1, and μ be a left invariant mean on X. Then T(μ) is nonexpansive and F(S). Moreover, if E is smooth, then F(S) is a sunny nonexpansive retract of C and the sunny nonexpansive retraction of C onto F(S) is unique.

Lemma 2.6 ([8, 9])

Let E be a real Banach space. Then, for any given x,yE and j(x+y)J(x+y), the following inequality holds:

x + y 2 x 2 +2 y , j ( x + y ) .

Lemma 2.7 ([17])

Let { x n } and { y n } be two bounded sequences in a Banach space E and let { β n } be a sequence in [0,1] with 0< lim inf n β n lim sup n β n <1. Suppose that x n + 1 =(1 β n ) y n + β n x n for all integers n0 and lim sup n ( y n + 1 y n x n + 1 x n )0. Then lim n y n x n =0.

Lemma 2.8 ([4])

Assume that { a n } is a sequence of nonnegative real numbers such that

a n + 1 (1 σ n ) a n + ρ n ,

where { α n } is a sequence in (0,1) and { δ n } is a sequence in such that

  1. (1)

    n = 1 α n =;

  2. (2)

    lim sup n δ n α n 0 or n = 1 | δ n |<.

Then lim n a n =0.

Lemma 2.9 ([18])

Let (Y,d) be a metric space and let f:YY be a mapping. The following assertions are equivalent:

  1. 1.

    f is a Meir-Keeler type mapping;

  2. 2.

    there exists a L-function ψ: R + R + such that f is a (ψ,L)-contraction.

Lemma 2.10 ([19])

Let E be a Banach space and let C be a convex subset of E. Let T:CC be a nonexpansive mapping and f be a (ψ,L)-contraction. Then the following assertions hold:

  1. 1.

    Tf is a (ψ,L)-contraction on C and has a unique fixed point in C;

  2. 2.

    for each α(0,1), the mapping xαf(x)+(1α)Tx is of Meir-Keeler-type and it has a unique fixed point in C.

Lemma 2.11 ([19])

Let E be a Banach space and let C be a convex subset of E. Let f:CC be a Meir-Keeler-type contraction. Then for each ϵ>0, there exists r(0,1) such that, for each x,yC with xyϵ, we have f(x)f(y)rxy.

3 Main results

In this paper, we suppose that ψ from the definition of (ψ,L)-contraction is continuous, strictly increasing. Let η(t)=tψ(t) for all t R + , we have that lim t η(t)= and η(t) is strictly increasing and onto. Consequently, we have that and η(t) is a bijection on  R + .

Theorem 3.1 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and S={ T s :sS} be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition lim s k s 1, and f be a Meir-Keeler contraction of C into itself. Let X be a left invariant S-stable subspace of l (S) containing 1, { μ n } be a strongly left regular sequence of means on X such that lim n μ n + 1 μ n =0 and { c n } be the sequence defined in (2.4) with lim sup n c n 0. Suppose the sequences { α n }, { β n }, { γ n } and { δ n } in (0,1) satisfy α n + β n + γ n =1, n1. The following conditions are satisfied:

  1. (i)

    lim n α n =0 and n = 0 α n =;

  2. (ii)

    lim n δ n =0;

  3. (iii)

    lim sup n c n α n 0;

  4. (iv)

    0< lim inf n β n lim sup n β n <1.

For arbitrary x 1 C, generate a sequence { x n } by

{ y n = δ n x n + ( 1 δ n ) T ( μ n ) x n , x n + 1 = α n f ( x n ) + β n x n + γ n y n .
(3.1)

Then { x n } converges strongly to qF(S), which is the unique solution of the variational inequality

( f I ) q , J ( p q ) 0,pF(S).

Equivalently, we have q=Pfq, where P is the unique sunny nonexpansive retraction of C onto F(S).

Proof First, we prove that { x n } is bounded. That is, if we take a point pF(S), we will show that x n pM for all nN. Let pF(S). It is obvious that M=max{ x 1 p, η 1 f(p)p}. By induction we suppose that M=max{ x k p, η 1 f(p)p}, where kN. Since T( μ k ) is nonexpansive, we have

y k p = δ n x k + ( 1 δ k ) T ( μ k ) x k p δ k x k p + ( 1 δ k ) T ( μ k ) x k p δ k x k p + ( 1 δ k ) ( x k p ) x k p .
(3.2)

Since f is a Meir-Keeler contraction, we have that

x k + 1 p = α k f ( x k ) + β k x k + γ k y k p = α k ( f ( x k ) p ) + β k ( x k p ) + γ n ( y k p ) α k f ( x k ) p + β k x k p + γ k y k p α k f ( x k ) f ( p ) + α k f ( p ) p + β k x k p + γ k y k p α k ψ x k p + α k f ( p ) p + β k x k p + γ k y k p = α k ψ x k p + ( β k + γ k ) x k p + α k η ( η 1 ) f ( p ) p = α k ψ x k p + ( 1 α k ) x k p + α k η ( η 1 ) f ( p ) p α k ψ ( M ) + ( 1 α k ) ( M ) + α k η ( M ) α k ψ ( M ) + ( 1 α k ) ( M ) + α k ( M ψ ( M ) ) = M .

That is, { x k + 1 } is bounded. By induction we have that { x n } is bounded, and so are the sequences {f( x n )}, {T( μ n ) x n } and { y n }. As T( μ n ) is bounded, we have { T s x n } is also bounded. Denote D= sup s S T s x n , then it follows that

T ( μ n + 1 ) x n T ( μ n ) x n = sup { | T ( μ n + 1 ) x n T ( μ n ) x n , z | : z E , z = 1 } = sup { | ( μ n + 1 ) s T s x n , z ( μ n ) s T s x n , z | : z E , z = 1 } μ n + 1 μ n sup s S T s x n z μ n + 1 μ n sup s S T s x n = μ n + 1 μ n D .

Since lim n μ n + 1 μ n =0, we obtain that

lim n T ( μ n + 1 ) x n T ( μ n ) x n =0.
(3.3)

Next, we will show that lim n x n + 1 x n =0 and by Lemma 2.3, we observe that

y n + 1 y n = δ n + 1 x n + 1 + ( 1 δ n + 1 ) T ( μ n + 1 ) x n + 1 ( δ n x n + ( 1 δ n ) T ( μ n ) x n ) = δ n + 1 x n + 1 δ n + 1 x n + δ n + 1 x n + ( 1 δ n + 1 ) T ( μ n + 1 ) x n + 1 ( 1 δ n + 1 ) T ( μ n ) x n + ( 1 δ n + 1 ) T ( μ n ) x n δ n x n ( 1 δ n ) T ( μ n ) x n = δ n + 1 ( x n + 1 x n ) + ( δ n + 1 δ n ) x n + ( 1 δ n + 1 ) ( T ( μ n + 1 ) x n + 1 T ( μ n ) x n ) + ( δ n δ n + 1 ) T ( μ n ) x n δ n + 1 x n + 1 x n + | δ n + 1 δ n | ( x n + T ( μ n ) x n ) + T ( μ n + 1 ) x n + 1 T ( μ n ) x n δ n + 1 x n + 1 x n + | δ n + 1 δ n | ( x n + T ( μ n ) x n ) + T ( μ n + 1 ) x n + 1 T ( μ n ) x n + 1 + T ( μ n ) x n + 1 T ( μ n ) x n δ n + 1 x n + 1 x n + | δ n + 1 δ n | ( x n + T ( μ n ) x n ) + T ( μ n + 1 ) x n + 1 T ( μ n ) x n + 1 + x n + 1 x n + c n .

Setting x n + 1 =(1 β n ) z n + β n x n , we see that

z n = x n + 1 β n x n 1 β n .

Then we compute

It follows that

z n + 1 z n x n + 1 x n ( α n + 1 1 β n + 1 + δ n + 1 ) x n + 1 x n + | α n + 1 1 β n + 1 α n 1 β n | ( f ( x n ) + y n ) + | δ n + 1 δ n | ( x n + T ( μ n ) x n ) + T ( μ n + 1 ) x n + 1 T ( μ n ) x n + 1 + c n .

From (i), (ii), (iv), (3.3) and Lemma 2.3, we have

lim sup n ( z n + 1 z n x n + 1 x n ) 0.

Applying Lemma 2.7, we obtain lim n z n x n =0 and also

x n + 1 x n =(1 β n ) z n x n 0,as n.

That is,

lim n x n + 1 x n =0.
(3.4)

Next, we will show that the set of all limit points of { x n } is a subset of F(S). Note that

x n + 1 x n = α n f ( x n ) + β n x n + γ n y n = α n f ( x n ) + β n x n + γ n ( δ n x n + ( 1 δ n ) T ( μ n ) x n ) x n = α n f ( x n ) ( 1 β n ) x n + γ n ( δ n x n + ( 1 δ n ) T ( μ n ) x n ) = α n f ( x n ) ( 1 β n ) x n + γ n δ n x n + γ n T ( μ n ) x n γ n δ n T ( μ n ) x n = α n f ( x n ) ( 1 β n ) x n + γ n δ n x n + ( 1 α n β n ) T ( μ n ) x n γ n δ n T ( μ n ) x n = α n ( f ( x n ) T ( μ n ) x n ) + ( 1 β n ) ( T ( μ n ) x n x n ) + γ n δ n ( x n T ( μ n ) x n ) = α n ( f ( x n ) T ( μ n ) x n ) + ( 1 + β n + γ n δ n ) ( x n T ( μ n ) x n ) α n f ( x n ) T ( μ n ) x n + ( 1 + β n + γ n δ n ) x n T ( μ n ) x n .

It follows that

x n T ( μ n ) x n 1 1 β n γ n δ n ( α n f ( x n ) T ( μ n ) x n x n + 1 x n ) .

From (i), (ii), (iv) and (3.4), we have

lim n x n T ( μ n ) x n =0.
(3.5)

Let p be a limit point of { x n } and { x n k } be a subsequence of { x n } converging strongly to p. From Lemma 2.3, we obtain that

lim sup k p T ( μ n k ) p lim sup k ( p x n k + x n k T ( μ n k ) x n k + T ( μ n k ) x n k T ( μ n k ) p ) lim sup k ( 2 p x n k + x n k T ( μ n k ) x n k + c n k ) 0 .

From (3.5), (2.4) and Corollary 2.2, we get pF(S).

We know that there exists a unique sunny nonexpansive retraction P of C onto F(S), and from the Banach contraction mapping principle, we known that Pf has a unique fixed point q which by (2.3) is the unique solution of

( f I ) q , J ( p q ) 0,pF(S).
(3.6)

Let { x n k } be a subsequence of { x n } converging to pC. From the smoothness of E and (3.6), we have that

lim sup n ( f I ) q , J ( x n q ) = lim sup k ( f I ) q , J ( x n k q ) = ( f I ) q , J ( p q ) 0 .
(3.7)

Finally, we show that the sequence { x n } converges strongly to q=Pfq. Now, we have

y n q = δ n x n + ( 1 δ n ) T ( μ n ) x n q = ( 1 δ n ) ( T ( μ n ) x n q ) + δ n ( x n q ) ( 1 δ n ) T ( μ n ) x n q + δ n x n q ( 1 δ n ) x n q + c n + δ n x n q = x n q + c n .
(3.8)

From Lemma 2.6, (3.8) and (2.2), we have

x n + 1 q 2 = α n f ( x n ) + β n x n + γ n y n q 2 = ( γ n ( y n q ) + β n ( x n q ) ) + α n ( f ( x n ) q ) 2 γ n ( y n q ) + β n ( x n q ) 2 + 2 α n f ( x n ) q , J ( x n + 1 q ) = ( 1 β n ) γ n 1 β n ( y n q ) + β n ( 1 β n 1 β n ) ( x n q ) 2 + 2 α n f ( x n ) f ( q ) , J ( x n + 1 q ) + 2 α n f ( q ) q , J ( x n + 1 q ) ( 1 β n ) γ n 1 β n ( y n q ) 2 + β n x n q 2 + 2 r α n x n q x n + 1 q + 2 α n f ( q ) q , J ( x n + 1 q ) γ n 2 1 β n y n q 2 + β n x n q 2 + r α n ( x n q 2 + x n + 1 q 2 ) + 2 α n f ( q ) q , J ( x n + 1 q ) γ n 2 1 β n x n q 2 + γ n 2 c n 1 β n + β n x n q 2 + r α n x n q 2 + r α n x n + 1 q 2 + 2 α n f ( q ) q , J ( x n + 1 q ) = ( γ n 2 1 β n + β n + r α n ) x n q 2 + γ n 2 c n 1 β n + r α n x n + 1 q 2 + 2 α n f ( q ) q , J ( x n + 1 q ) = ( ( ( 1 β n ) α n ) 2 1 β n + β n + r α n ) x n q 2 + γ n 2 c n 1 β n + r α n x n + 1 q 2 + 2 α n f ( q ) q , J ( x n + 1 q ) = ( ( 1 β n ) 2 2 ( 1 β n ) α n + α n 2 1 β n + β n + r α n ) x n q 2 + γ n 2 c n 1 β n + r α n x n + 1 q 2 + 2 α n f ( q ) q , J ( x n + 1 q ) = ( 1 β n 2 α n + α n 2 1 β n + β n + r α n ) x n q 2 + γ n 2 c n 1 β n + r α n x n + 1 q 2 + 2 α n f ( q ) q , J ( x n + 1 q ) = ( ( 1 r α n ) + ( 2 r α n 2 α n ) + α n 2 1 β n ) x n q 2 + γ n 2 c n 1 β n + r α n x n + 1 q 2 + 2 α n f ( q ) q , J ( x n + 1 q ) .

It follows that

x n + 1 q 2 ( 1 2 α n ( 1 r ) 1 r α n + α n 2 ( 1 r α n ) ( 1 β n ) ) x n q 2 + γ n 2 c n ( 1 r α n ) ( 1 β n ) + 2 α n 1 r α n f ( q ) q , J ( x n + 1 q ) ( 1 2 α n ( 1 r ) 1 r α n ) x n q 2 + α n 1 r α n ( α n 1 β n x n q 2 + γ n 2 c n α n ( 1 β n ) + 2 f ( q ) q , J ( x n + 1 q ) ) : = ( 1 σ n ) x n q 2 + ρ n ,

where σ n := 2 α n ( 1 r ) 1 r α n and ρ n := α n 1 r α n ( α n 1 β n x n q 2 + γ n 2 c n α n ( 1 β n ) +2f(q)q,J( x n + 1 q)). Now, from (i), (iii), (iv), (3.7) and Lemma 2.8, x n q0 as n. This proof is completed. □

Corollary 3.2 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and S={ T s :sS} be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition lim s k s 1, and f be an α-contraction of C into itself. Let X be a left invariant S-stable subspace of l (S) containing 1, { μ n } be a strongly left regular sequence of means on X such that lim n μ n + 1 μ n =0 and { c n } be the sequence defined by (2.4) with lim sup n c n 0. Suppose the sequences { α n }, { β n }, { γ n } and { δ n } in (0,1) satisfy α n + β n + γ n =1, n1. The following conditions are satisfied:

  1. (i)

    lim n α n =0 and n = 0 α n =;

  2. (ii)

    lim n δ n =0;

  3. (iii)

    lim sup n c n α n 0;

  4. (iv)

    0< lim inf n β n lim sup n β n <1.

For arbitrary given x 1 C, generate a sequence { x n } by

{ y n = δ n x n + ( 1 δ n ) T ( μ n ) x n , x n + 1 = α n f ( x n ) + β n x n + γ n y n .
(3.9)

Then { x n } converges strongly to qF(S), which is the unique solution of the variational inequality

( f I ) q , J ( p q ) 0,pF(S).

Equivalently, we have q=Pfq, where P is the unique sunny nonexpansive retraction of C onto F(S).

Corollary 3.3 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and S={ T s :sS} be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition lim s k s 1, and f be a Meir-Keeler contraction of C into itself. Let X be a left invariant S-stable subspace of l (S) containing 1, { μ n } be a strongly left regular sequence of means on X such that lim n μ n + 1 μ n =0 and { c n } be the sequence defined by (2.4) with lim sup n c n 0. Suppose the sequences { α n }, { β n } and { γ n } in (0,1) satisfy α n + β n + γ n =1, n1. The following conditions are satisfied:

  1. (i)

    lim n α n =0 and n = 0 α n =;

  2. (ii)

    lim sup n c n α n 0;

  3. (iii)

    0< lim inf n β n lim sup n β n <1.

For arbitrary x 1 C, generate a sequence { x n } by

x n + 1 = α n f( x n )+ β n x n + γ n T( μ n ) x n .
(3.10)

Then { x n } converges strongly to qF(S), which is the unique solution of the variational inequality

( f I ) q , J ( p q ) 0,pF(S).

Equivalently, we have q=Pfq, where P is the unique sunny nonexpansive retraction of C onto F(S).

Corollary 3.4 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and S={ T s :sS} be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition lim s k s 1, and f be an α-contraction of C into itself. Let X be a left invariant S-stable subspace of l (S) containing 1, { μ n } be a strongly left regular sequence of means on X such that lim n μ n + 1 μ n =0 and { c n } be the sequence defined by (2.4) with lim sup n c n 0. Suppose the sequences { α n }, { β n } and { γ n } in (0,1) satisfy α n + β n + γ n =1, n1. The following conditions are satisfied:

  1. (i)

    lim n α n =0 and n = 0 α n =;

  2. (ii)

    lim sup n c n α n 0;

  3. (iii)

    0< lim inf n β n lim sup n β n <1.

For arbitrary x 1 C, generate a sequence { x n } by

x n + 1 = α n f( x n )+ β n x n + γ n T( μ n ) x n .
(3.11)

Then { x n } converges strongly to qF(S), which is the unique solution of the variational inequality

( f I ) q , J ( p q ) 0,pF(S).

Equivalently, we have q=Pfq, where P is the unique sunny nonexpansive retraction of C onto F(S).

Corollary 3.5 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and S={ T s :sS} be a representation of S as nonexpansive mappings from C into itself and f be an α-contraction of C into itself. Let X be a left invariant S-stable subspace of l (S) containing 1, { μ n } be a strongly left regular sequence of means on X such that lim n μ n + 1 μ n =0 and { c n } be the sequence defined by (2.4) with lim sup n c n 0. Suppose the sequences { α n }, { β n }, { γ n } and { δ n } in (0,1) satisfy α n + β n + γ n =1, n1. The following conditions are satisfied:

  1. (i)

    lim n α n =0 and n = 0 α n =;

  2. (ii)

    lim n δ n =0;

  3. (iii)

    lim sup n c n α n 0;

  4. (iv)

    0< lim inf n β n lim sup n β n <1.

For arbitrary x 1 C, generate a sequences { x n } by

{ y n = δ n x n + ( 1 δ n ) T ( μ n ) x n , x n + 1 = α n f ( x n ) + β n x n + γ n y n .
(3.12)

Then { x n } converges strongly to qF(S), which is the unique solution of the variational inequality

( f I ) q , J ( p q ) 0,pF(S).

Equivalently, we have q=Pfq, where P is the unique sunny nonexpansive retraction of C onto F(S).

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Acknowledgements

The authors would like to express their thanks to referees for the careful reading of the paper and for the suggestions which improved the quality of this work.

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Correspondence to Siwaporn Saewan or Poom Kumam.

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Saewan, S., Kumam, P. Explicit iterations for Lipschitzian semigroups with the Meir-Keeler type contraction in Banach spaces. J Inequal Appl 2012, 279 (2012). https://doi.org/10.1186/1029-242X-2012-279

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Keywords

  • Meir-Keeler type mapping
  • Lipschitzian mapping
  • variational inequality
  • viscosity approximation method
  • fixed point