Some Existence and Convergence Theorems for Nonexpansive Type Mappings

Abstract

Some existence and convergence theorems for a class of nonexpansive type mappings are obtained in a normed space. The results obtained herein generalize certain known results.

1. Introduction

Let 𝐶 be a nonempty subset of a normed space 𝑋. A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦, (1) for all 𝑥,𝑦∈𝐶. Suppose 𝑆∶𝐶→𝐶 is another mapping on 𝐶. Then the mapping 𝑇 is said to be 𝑆-nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑆𝑥−𝑆𝑦, (2) for all 𝑥,𝑦∈𝐶. The class of 𝑆-nonexpansive mappings is more general than nonexpansive mappings [1–3].

We extend the above notion of 𝑆-nonexpansive mappings to a more general class of nonexpansive mappings.

Definition 1. Let 𝑋 be a normed space, 𝐶 a nonempty subset of 𝑋 and 𝑆,𝑇∶𝐶→𝐶. We say that 𝑇 is a generalized 𝑆-nonexpansive type mapping if 𝑇𝑥−𝑇𝑦≤𝑀𝑥,𝑦, (3) for all 𝑥,𝑦∈𝐶, where 𝑀,‖‖+𝑥,𝑦=max𝑆𝑥−𝑆𝑦𝑆𝑥−𝑇𝑥𝑆𝑦−𝑇𝑦2,𝑆𝑥−𝑇𝑦+𝑆𝑦−𝑇𝑥2. (4) Further 𝑇 will be called a generalized nonexpansive type mappings if 𝑇𝑥−𝑇𝑦≤𝑚𝑥,𝑦, (5) for all 𝑥,𝑦∈𝐶, where 𝑚,‖‖+𝑥,𝑦=max𝑥−𝑦𝑥−𝑇𝑥𝑦−𝑇𝑦2,𝑥−𝑇𝑦+𝑦−𝑇𝑥2. (6)

In this paper we obtain some common fixed point theorems for generalized nonexpansive type mappings in normed spaces. Specifically, in Section 3, we obtain some coincidence and common fixed point theorems while in Section 4, the weak convergence of Mann iterations (cf. Mann [4]) to a common fixed point for the above class of mappings is discussed. The results obtained herein generalize certain results of Kim et al. [1], Rhoades and Temir [3], and Shazad [5] among others.

2. Preliminaries

Let 𝐶 be a nonempty subset of a normed space 𝑋 and 𝑆,𝑇∶𝐶→𝑋. A point 𝑧∈𝐶 is called a coincidence point of 𝑆 and 𝑇 if 𝑆𝑧=𝑇𝑧 and a common fixed point if 𝑆𝑧=𝑇𝑧=𝑧.

Now onwards ℕ will denote the set of naturals while 𝐹(𝑆) and 𝐹(𝑇) the set of fixed points of 𝑆 and 𝑇, respectively.

Definition 2. Let 𝑋 be a normed space, 𝐶 a nonempty subset of 𝑋, and 𝑆,𝑇∶𝐶→𝐶. The pair of mappings (𝑆,𝑇) is called(i)commuting if 𝑇𝑆𝑥=𝑆𝑇𝑥 for all 𝑥∈𝐶.(ii)weakly commuting if for all 𝑥∈𝐶, ‖𝑆𝑇𝑥−𝑇𝑆𝑥‖≤‖𝑆𝑥−𝑇𝑥‖ (see [6]).(iii)𝑅-weakly commuting if for all 𝑥∈𝐶, there exists 𝑅>0 such that ‖𝑆𝑇𝑥−𝑇𝑆𝑥‖≤𝑅‖𝑆𝑥−𝑇𝑥‖ (see [7]).
The following example illustrates that weakly commuting mappings are 𝑅-weakly commuting but the converse is not true in general.

Example 3. Let 𝑋=𝐶=[1,∞) be endowed with usual norm ‖𝑥‖=|𝑥|. Let 𝑆,𝑇∶𝐶→𝐶 be mappings defined by 𝑆𝑥=𝑥²,𝑇𝑥=2𝑥²−1,∀𝑥∈𝑋. (7) Then ‖𝑆𝑇𝑥−𝑇𝑆𝑥‖=2𝑥²−4𝑥+2=2𝑥²,‖‖=𝑥−2𝑥+1𝑆𝑥−𝑇𝑥²−2𝑥+1. (8) Therefore ‖𝑆𝑇𝑥−𝑇𝑆𝑥‖=2|𝑥²−2𝑥+1|=2‖𝑆𝑥−𝑇𝑥‖ and the pair (𝑆,𝑇) is 𝑅-weakly commuting with 𝑅=2 but not weakly commuting.
In general, commuting ⇒ weakly commuting ⇒𝑅-weakly commuting.

Definition 4 (cf. [1]). Let 𝑋 be a normed space and 𝐶 a nonempty subset of 𝑋. The set 𝐶 is called 𝑞-starshaped with 𝑞∈𝐶 if for all 𝑥∈𝐶, the segment [𝑞,𝑥]={(1−𝑡)𝑞+𝑡𝑥} joining 𝑞 to 𝑥, is contained in 𝐶; where 0≤𝑡≤1.
Further if 𝐶 is a nonempty 𝑞-starshaped subset of a normed space 𝑋, then the mapping 𝑆∶𝐶→𝐶 is said to be 𝑞-affine if 𝑡𝑆𝑥+(1−𝑡)𝑆𝑞∈𝐶 (9) for all 𝑥∈𝐶 and 0≤𝑡≤1.

Definition 5 (cf. [1]). Let 𝑋 be a normed space, 𝐶 a nonempty subset of 𝑋, and 𝑆,𝑇∶𝐶→𝐶 such that 𝐹(𝑆)≠∅. Suppose 𝑞∈𝐹(𝑆) and 𝐶 is 𝑞-starshaped. Then the pair of mappings (𝑆,𝑇) is called 𝑅-subweakly commuting on 𝐶 if for all 𝑥∈𝐶, there exists a real number 𝑅>0 such that ‖𝑆𝑇𝑥−𝑇𝑆𝑥‖≤𝑅dist𝑆𝑥,𝑞,𝑇𝑥, (10) where dist(𝑆𝑥,[𝑞,𝑇𝑥])=inf{‖𝑆𝑥−𝑦‖∶𝑦∈[𝑞,𝑇𝑥]}.
We note that 𝑅-subweakly commuting mappings are 𝑅-weakly commuting but the converse is not true in general.

Example 6. Let 𝑋=ℝ (set of reals) with norm ‖𝑥‖=|𝑥| and 𝐶=[0,10]. Define 𝑆,𝑇∶𝐶→𝐶 by 𝑇𝑥=𝑥+12𝑥,𝑆𝑥=2. (11) Then ‖𝑇𝑆𝑥−𝑆𝑇𝑥‖=𝑥+24−𝑥+14=14,‖𝑇𝑥−𝑆𝑥‖=𝑥+12−𝑥2=12,∀𝑥∈𝐶. (12) Therefore ‖‖=1𝑇𝑆𝑥−𝑆𝑇𝑥4=1212=𝑅‖𝑇𝑥−𝑆𝑥‖ (13) holds for 𝑅=1/2 and 𝑇 and 𝑆 are 𝑅-weakly commuting on 𝐶.
On the other hand, 𝑞=0∈𝐹(𝑆), and for all 𝑥∈𝐶, =𝑥𝑆𝑥−𝑇𝑥,𝑞2−𝑥+12,0=0. (14) So, there does not exist any 𝑅>0 such that for all 𝑥∈𝐶, ‖‖=1𝑇𝑆𝑥−𝑆𝑇𝑥4≤𝑅𝑆𝑥−𝑇𝑥,𝑞=0 (15) holds. Thus 𝑇 and 𝑆 are not 𝑅-subweakly commuting on 𝐶.

Definition 7. Let 𝐶 be a nonempty subset of a normed space 𝑋 and 𝑇∶𝐶→𝐶. Let {𝑥_𝑛} be a sequence in 𝑋. We denote the weak and strong convergence of {𝑥_𝑛} to 𝑥 by 𝑥_𝑛⇀0𝑥00086𝑥 and 𝑥_𝑛→𝑥, respectively. The mapping 𝑇 is said to be demicontinuous if {𝑥_𝑛} is a sequence in 𝑋 such that 𝑥_𝑛→𝑥, then 𝑇𝑥_𝑛⇀0𝑥00086𝑇𝑥.

Definition 8. A Banach space 𝑋 is said to satisfy the Opial's condition (see [8]), if whenever a sequence {𝑥_𝑛} in 𝑋, converges weakly to 𝑥 ( 𝑥_𝑛⇀0𝑥00086𝑥), then lim_𝑛→∞inf𝑥_𝑛−𝑥<lim_𝑛→∞inf𝑥_𝑛−𝑦, (16) for all 𝑦∈𝑋, 𝑦≠𝑥.
We note that the 𝐿^𝑝 spaces, 𝑝≠2 do not satisfy Opial' condition while all 𝑙^𝑝 spaces (1<𝑝<∞) do (see for details Goebel and Kirk [9]).

3. Existence Results

The following common fixed point theorem is due to Shahzad [5, Theorem 2.1]. For related results we refer to [2, 10–12].

Theorem 9. Let (𝑋,𝑑) be a metric space and 𝐶 a nonempty subset of 𝑋. Let 𝑆,𝑇∶𝐶→𝐶 be a pair of mappings such that(i)𝑇(𝐶)⊆𝑆(𝐶); (ii)𝑑(𝑇𝑥,𝑇𝑦)≤𝑘max{𝑑(𝑆𝑥,𝑆𝑦),𝑑(𝑆𝑥,𝑇𝑥), 𝑑(𝑆𝑦,𝑇𝑦),(𝑑(𝑆𝑥,𝑇𝑦)+𝑑(𝑆𝑦,𝑇𝑥))/2}, 𝑘∈(0,1);(iii) the pair (𝑆,𝑇) is 𝑅-weakly commuting on 𝐶.
If cl(𝑇(𝐶)) (closure of 𝑇(𝐶)), is complete and 𝑇 is continuous, then 𝐹(𝑆)∩𝐹(𝑇)∩𝐶 is a singleton.

Now we obtain a more general version of the above theorem, where the continuity condition on 𝑇 has been dispensed with and the completeness of cl(𝑇(𝐶)) has been replaced by completeness of 𝑇(𝐶).

Theorem 10. Let (𝑋,𝑑) be a metric space and 𝐶 a nonempty subset of 𝑋. Let 𝑆,𝑇∶𝐶→𝐶 be a pair of mappings such that(i)𝑇(𝐶)⊆𝑆(𝐶);(ii)𝑑(𝑇𝑥,𝑇𝑦)≤𝑘max{𝑑(𝑆𝑥,𝑆𝑦),𝑑(𝑆𝑥,𝑇𝑥),𝑑(𝑆𝑦,𝑇𝑦),(𝑑(𝑆𝑥,𝑇𝑦)+𝑑(𝑆𝑦,𝑇𝑥))/2}, 𝑘∈(0,1);(iii) the pair (𝑆,𝑇) is 𝑅-weakly commuting on 𝐶.
Then we have the following:(a)𝐹(𝑆)∩𝐹(𝑇)∩𝑇(𝐶) is a singleton if 𝑇(𝐶) is complete,(b)𝐹(𝑆)∩𝐹(𝑇)∩𝑆(𝐶) is a singleton if 𝑆(𝐶) is complete.

Proof . Pick 𝑥₀∈𝐶. Since 𝑇(𝐶)⊆𝑆(𝐶), we can construct a sequence {𝑥_𝑛} in 𝐶 such that 𝑆𝑥_𝑛=𝑇𝑥_𝑛−1 for all 𝑛∈ℕ (see [12]). By (ii), we have 𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛=𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝑘max𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑇𝑥_𝑛,𝑑𝑆𝑥_𝑛−1𝑇𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑇𝑥_𝑛−1+𝑑𝑆𝑥_𝑛−1,𝑇𝑥_𝑛2=𝑘max𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1,𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛,𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛+12=𝑘max𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1,𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛,𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛+12≤𝑘max𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1,𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛,𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛+𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+12,=𝑘max𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1+𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛2. (17) Now if max𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1,𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1+𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛2=𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛+1+𝑑𝑆𝑥_𝑛−1,𝑆𝑥_𝑛2, (18) then 𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛≤𝑘2𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1+𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛. (19) Therefore 𝑑𝑆𝑥_𝑛+1,𝑆𝑥_𝑛≤𝑘𝑑𝑆𝑥_𝑛,𝑆𝑥_𝑛−1. (20) Since 𝑘<1, {𝑆𝑥_𝑛} is a Cauchy sequence in 𝐶 (see [1, 5]).
(a) Suppose that 𝑇(𝐶) is complete. Then there exists a point 𝑧∈𝑇(𝐶) such that 𝑇𝑥_𝑛→𝑧∈𝑇(𝐶). Thus, 𝑆𝑥_𝑛→𝑧. Since 𝑧∈𝑇(𝐶)⊆𝑆(𝐶), there exists 𝑢∈𝐶 such that 𝑧=𝑆𝑢. Again by (ii), we have 𝑑𝑇𝑢,𝑇𝑥_𝑛≤𝑘max𝑑𝑆𝑢,𝑆𝑥_𝑛(),𝑑,𝑑𝑆𝑢,𝑇𝑢𝑆𝑥_𝑛,𝑇𝑥_𝑛,𝑑𝑆𝑢,𝑇𝑥_𝑛+𝑑𝑆𝑥_𝑛,𝑇𝑢2. (21) Making 𝑛→∞, yields 𝑑()(),𝑑()𝑇𝑢,𝑆𝑢≤𝑘max0,𝑑𝑆𝑢,𝑇𝑢𝑆𝑢,𝑇𝑢2=𝑘𝑑(𝑆𝑢,𝑇𝑢)<𝑑(𝑆𝑢,𝑇𝑢), (22) a contradiction. Therefore 𝑑(𝑇𝑢,𝑆𝑢)=0 and 𝑆𝑢=𝑇𝑢=𝑧. Since the pair (𝑆,𝑇) is 𝑅-weakly commuting on 𝐶, it follows that 𝑑(𝑆𝑇𝑢,𝑇𝑆𝑢)≤𝑅𝑑(𝑆𝑢,𝑇𝑢)=𝑅𝑑(𝑧,𝑧)=0. (23) Therefore 𝑑(𝑆𝑇𝑢,𝑇𝑆𝑢)=0 and 𝑆𝑧=𝑇𝑧. Again by (ii), we have 𝑑𝑇𝑧,𝑇𝑥_𝑛≤𝑘max𝑑𝑆𝑧,𝑆𝑥_𝑛(),𝑑,𝑑𝑆𝑧,𝑇𝑧𝑆𝑥_𝑛,𝑇𝑥_𝑛,𝑑𝑆𝑧,𝑇𝑥_𝑛+𝑑𝑆𝑥_𝑛,𝑇𝑧2. (24) Making 𝑛→∞, yields 𝑑()𝑑()()(),𝑑()()𝑇𝑧,𝑧≤𝑘max𝑆𝑧,𝑧,𝑑𝑆𝑧,𝑇𝑧,𝑑𝑧,𝑧𝑆𝑧,𝑧+𝑑𝑧,𝑇𝑧2{𝑑()(()(),=𝑘max𝑆𝑧,𝑧,0,𝑑𝑆𝑧,𝑧)}=𝑘𝑑𝑆𝑧,𝑧=𝑘𝑑𝑇𝑧,𝑧 (25) which implies 𝑧=𝑇𝑧=𝑆𝑧. Using (ii), uniqueness of 𝑧 can be proved. Since 𝑧∈𝑇(𝐶), we conclude that 𝐹(𝑆)∩𝐹(𝑇)∩𝑇(𝐶)={𝑧}.
(b) Suppose 𝑆(𝐶) is complete. Then 𝑆𝑥_𝑛→𝑧 for some 𝑧∈𝑆(𝐶) and there exist 𝑢∈𝐶 such that 𝑧=𝑆𝑢. As in part (a), we can show that 𝑆𝑧=𝑇𝑧=𝑧. Thus 𝐹(𝑆)∩𝐹(𝑇)∩𝑆(𝐶)={𝑧}.

Recently Kim et al. [1] obtained the following result for 𝑆-nonexpansive type mappings in a normed space.

Theorem 11. Let 𝐶 be a nonempty 𝑞-star shaped subset of a normed space 𝑋 and 𝑆,𝑇∶𝐶→𝐶 two mappings satisfying the following conditions:(i)the mapping 𝑇 is 𝑆-nonexpansive and 𝑆 is 𝑞-affine with 𝑞∈𝐹(𝑆);(ii)𝑇(𝐶)⊆𝑆(𝐶);(iii) the pair (𝑆,𝑇) is 𝑅-subweakly commuting.

Suppose 𝑆(𝐶) is compact. Then we have the following:(a)𝑆 and 𝑇 have a coincidence point 𝑦∈𝑆(𝐶).(b)If 𝑆 or 𝑇 is demicontinuous, then 𝑦∈𝐹(𝑆)∩𝐹(𝑇).

We extend the above theorem to a more general class of nonexpansive type mappings. In the sequel we will need the following Lemma 12 and Proposition 13.

Lemma 12. Let (𝑋,𝑑) be a metric space and 𝐶 a nonempty subset of 𝑋. Let 𝑆,𝑇∶𝐶→𝐶 be a pair of mappings such that (i)𝑇(𝐶)⊆𝑆(𝐶);(ii)𝑑(𝑇𝑥,𝑇𝑦)≤𝑘max{𝑑(𝑆𝑥,𝑆𝑦),(𝑑(𝑆𝑥,𝑇𝑥)+𝑑(𝑆𝑦,𝑇𝑦))/2,(𝑑(𝑆𝑥,𝑇𝑦)+𝑑(𝑆𝑦,𝑇𝑥))/2}, 𝑘∈(0,1);(iii) the pair (𝑆,𝑇) is 𝑅-weakly commuting on 𝐶.
Then we have the following: (a)𝐹(𝑆)∩𝐹(𝑇)∩𝑇(𝐶) is a singleton if 𝑇(𝐶) is complete;(b)𝐹(𝑆)∩𝐹(𝑇)∩𝑆(𝐶) is a singleton if 𝑆(𝐶) is complete.

Proof. The proof can be completed on the lines of the proof of Theorem 10, where 𝑑(),𝑑,𝑑max𝑆𝑥,𝑆𝑦,𝑑𝑆𝑥,𝑇𝑥𝑆𝑦,𝑇𝑦𝑆𝑥,𝑇𝑦+𝑑𝑆𝑦,𝑇𝑥2 (26) is replaced by 𝑑,𝑑()max𝑆𝑥,𝑆𝑦𝑆𝑥,𝑇𝑥+𝑑𝑆𝑦,𝑇𝑦2,𝑑𝑆𝑥,𝑇𝑦+𝑑𝑆𝑦,𝑇𝑥2. (27)

Proposition 13. Let 𝐶 be a nonempty 𝑞-star shaped subset of a normed space 𝑋 and 𝑆,𝑇∶𝐶→𝐶 two mappings such that(i)𝑇 is a generalized 𝑆-nonexpansive type mapping and 𝑆 is 𝑞-affine with 𝑞∈𝐹(𝑆);(ii)𝑇(𝐶)⊆𝑆(𝐶); (iii)the pair (𝑆,𝑇) is 𝑅-subweakly commuting;(iv)𝑆(𝐶) is complete.
Then there exist exactly one point 𝑥_𝜆 such that 𝑥_𝜆=𝑆𝑥_𝜆=(1−𝜆)𝑞+𝜆𝑇𝑥_𝜆, (28) for all 𝜆∈(0,1).

Proof. Define 𝑇_𝜆∶𝐶→𝐶 by 𝑇_𝜆𝑥=(1−𝜆)𝑞+𝜆𝑇𝑥, for all 𝑥∈𝐶 and for each 𝜆∈(0,1).
Since (𝑆,𝑇) is 𝑅-subweakly commuting and 𝑆 is 𝑞-affine, we have 𝑆𝑇_𝜆𝑥−𝑇_𝜆=)−)‖‖𝑆𝑥(1−𝜆𝑞+𝜆𝑆𝑇𝑥(1−𝜆𝑞+𝜆𝑇𝑆𝑥=𝜆𝑇𝑆𝑥−𝑆𝑇𝑥≤𝜆𝑅𝑆𝑥−𝑇_𝜆𝑥, (29) for all 𝑥∈𝐶. Thus, the pair (𝑆,𝑇_𝜆) is 𝑅-weakly commuting on 𝐶.
Also 𝑇_𝜆𝑥−𝑇_𝜆𝑦,‖‖+=𝜆𝑇𝑥−𝑇𝑦≤𝜆max𝑆𝑥−𝑆𝑦𝑆𝑥−𝑇𝑥𝑆𝑦−𝑇𝑦2,𝑆𝑥−𝑇𝑦+𝑆𝑦−𝑇𝑥2, (30) for all 𝑥,𝑦∈𝐶. For 𝑥∈𝐶, we have 𝑇𝑥∈𝑇(𝐶)⊆𝑆(𝐶), that is, there exist a point 𝑦∈𝐶 such that 𝑇𝑥=𝑆𝑦∈𝑆(𝐶).
Observe that 𝑇_𝜆𝑥=(1−𝜆)𝑞+𝜆𝑇𝑥=(1−𝜆)𝑞+𝜆𝑆𝑦∈𝑆(𝐶). (31) It follows that 𝑇_𝜆(𝐶)⊆𝑆(𝐶) for all 𝜆∈(0,1). Now for each 𝜆∈(0,1), we conclude that  (i)*𝑇_𝜆(𝐶)⊆𝑆(𝐶), (ii)*‖𝑇_𝜆𝑥−𝑇_𝜆𝑦‖≤𝜆max{‖𝑆𝑥−𝑆𝑦‖,(‖𝑆𝑥−𝑇𝑥‖+‖𝑆𝑦−𝑇𝑦‖)/2,(‖𝑆𝑥−𝑇𝑦‖+‖𝑆𝑦−𝑇𝑥‖)/2}, (iii)*𝑆(𝐶) is complete,  (iv)*(𝑆,𝑇_𝜆) is 𝑅-weakly commuting on 𝐶.
Therefore by Lemma 12, there exist exactly one point 𝑥_𝜆∈𝑆(𝐶) such that 𝑥_𝜆=𝑆𝑥_𝜆=𝑇_𝜆𝑥_𝜆, (32) which implies that 𝑥_𝜆=𝑆𝑥_𝜆=(1−𝜆)𝑞+𝜆𝑇𝑥_𝜆.

Now we obtain a common fixed point theorem for generalized 𝑆-nonexpansive type mappings.

Theorem 14. Let 𝐶 be a nonempty subset of a normed space 𝑋. Let 𝑆,𝑇∶𝐶→𝐶 be two mappings satisfying conditions (i)–(iii) of Proposition 13. Suppose 𝑆(𝐶) is compact. Then we have the following:(a)𝑆 and 𝑇 have a coincidence point 𝑦∈𝑆(𝐶),(b)If 𝑆 or 𝑇 is demicontinuous, then 𝑦∈𝐹(𝑆)∩𝐹(𝑇).

Proof. Let {𝜆_𝑛} be a sequence in (0,1) such that 𝜆_𝑛→1. By Proposition 13, there exists exactly one point 𝑥_𝜆𝑛∈𝑆(𝐶) such that 𝑥_𝜆𝑛=𝑆𝑥_𝜆𝑛=1−𝜆_𝑛𝑞+𝜆_𝑛𝑇𝑥_𝜆𝑛, (33) for all 𝑛∈ℕ.
Set 𝑥_𝜆𝑛∶=𝑦_𝑛. Since 𝑆(𝐶) is compact, there exist a subsequence {𝑦_𝑛𝑗} of {𝑦_𝑛} such that lim_𝑗→∞𝑆𝑦_𝑛𝑗=𝑦∈𝑆(𝐶). (34) Thus 𝑦=𝑆𝑢 for some 𝑢∈𝐶.
The assumption (ii) implies that {𝑇𝑦_𝑛𝑗} is bounded. It follows that 𝑦_𝑛𝑗−𝑇𝑦_𝑛𝑗=1−𝜆_𝑚𝑞+𝜆_𝑚𝑇𝑦_𝑛𝑗−𝑇𝑦_𝑛𝑗=1−𝜆_𝑚𝑞−𝑇𝑦_𝑛𝑗→0,as𝑗→∞. (35) Thus lim_𝑗→∞𝑇𝑦_𝑛𝑗=lim_𝑗→∞𝑦_𝑛𝑗=𝑦. By the condition (3), we have 𝑇𝑦_𝑛𝑗−𝑇𝑢≤max𝑆𝑦_𝑛𝑗−𝑆𝑢,𝑆𝑦_𝑛𝑗−𝑇𝑦_𝑛𝑗+𝑆𝑢−𝑇𝑢2,𝑆𝑦_𝑛𝑗−𝑇𝑢+𝑆𝑢−𝑇𝑦_𝑛𝑗2=max𝑆𝑦_𝑛𝑗−𝑦,𝑆𝑦_𝑛𝑗−𝑇𝑦_𝑛𝑗+𝑦−𝑇𝑢2,𝑆𝑦_𝑛𝑗−𝑇𝑢+𝑦−𝑇𝑦_𝑛𝑗2. (36) Making 𝑗→∞, we get 1𝑦−𝑇𝑢≤max0,2,1𝑦−𝑇𝑢2=1𝑦−𝑇𝑢2𝑦−𝑇𝑢, (37) a contradiction. Therefore ‖𝑦−𝑇𝑢‖=0 and 𝑇𝑢=𝑦.(a)Since the pair (𝑆,𝑇) is 𝑅-subweakly commuting, we have ‖𝑆𝑇𝑢−𝑇𝑆𝑢‖≤𝑅dist𝑆𝑢,𝑇𝑢,𝑞=𝑅𝑆𝑢−1−𝜆_𝑛𝑞+𝜆_𝑛‖‖𝑇𝑢=𝑅𝑆𝑢−𝑆𝑢=0. (38) Thus 𝑆𝑦=𝑇𝑦.(b)Suppose 𝑆 is demicontinuous. Since lim_𝑚→∞𝑥_𝑚=lim_𝑚→∞𝑆𝑥_𝑚=𝑦, it follows from the demicontinuity of 𝑆 that 𝑆𝑦=𝑦. But 𝑆𝑦=𝑇𝑦. Thus we conclude that 𝑦∈𝐹(𝑆)∩𝐹(𝑇). Similarly we can prove that 𝑦∈𝐹(𝑆)∩𝐹(𝑇) when 𝑇 is demicontinuous.

The following example shows the generality of Theorem 14 over Theorem 11.

Example 15. Let 𝑋=ℝ (set of reals) with norm ‖𝑥‖=|𝑥| and 𝐶=[0,4]. Define 𝑆,𝑇∶𝐶→𝐶 by {},3𝑇𝑥=1,if𝑥∈1,3,42,if𝑥=2,2{},,otherwise,1𝑆𝑥=1,if𝑥=1,4,if𝑥∈2,32,if𝑥=4,2,otherwise. (39) For 𝑥=2 and 𝑦=3, we have 𝑇𝑥−𝑇𝑦=1>0=𝑆𝑥−𝑆𝑦, (40) and the condition (2) of Theorem 11 is not satisfied. Further, it can be easily verified that 𝑆 and 𝑇 satisfy all the hypotheses of Theorem 14 and 𝑆1=𝑇1=1, is a common fixed point of 𝑆 and 𝑇.

4. Convergence Results

Recently, Rhoades and Temir [3] obtained the following theorem.

Theorem 16. Let 𝑋 be a Banach space and 𝐶 a closed convex subset of 𝑋 which satisfies Opial's condition. Let 𝑆,𝑇∶𝐶→𝐶 be mappings such that(i)𝑇 is 𝑆-nonexpansive;(ii)𝑆 is nonexpansive. Suppose {𝑘_𝑛} is a real sequence in (0,1). Then the sequence of Mann iterates defined for an arbitrary 𝑥₀∈𝐶 by 𝑥_𝑛+1=1−𝑘_𝑛𝑥_𝑛+𝑘_𝑛𝑇𝑥_𝑛,𝑛∈{0}∪ℕ (41) converges weakly to a common fixed point of 𝑆 and 𝑇.

The following theorem extends Theorem 16 to generalized 𝑆-nonexpansive type mappings.

Theorem 17. Let 𝑋 be a Banach space and 𝐶 a closed convex subset of 𝑋 which satisfies Opial's condition. Let 𝑆,𝑇∶𝐶→𝐶 be such that(i)𝑇 is generalized 𝑆-nonexpansive type;(ii)𝑆 is nonexpansive.
Suppose {𝑘_𝑛} is a real sequence in (0,1). Then the sequence of Mann iterates defined for an arbitrary 𝑥₀∈𝐶 by 𝑥_𝑛+1=1−𝑘_𝑛𝑥_𝑛+𝑘_𝑛𝑇𝑥_𝑛,𝑛∈{0}∪ℕ (42) converges weakly to a common fixed point of 𝑆 and 𝑇.

Proof. If 𝐹(𝑇)∩𝐹(𝑆) is nonempty and singleton, then the proof is complete. Assume that 𝐹(𝑇)∩𝐹(𝑆) is nonempty and that 𝐹(𝑇)∩𝐹(𝑆) is not a singleton. Let 𝑧∈𝐹(𝑇)∩𝐹(𝑆). Then 𝑥_𝑛+1−𝑧=1−𝑘_𝑛𝑥_𝑛+𝑘_𝑛𝑇𝑥_𝑛=−𝑧1−𝑘_𝑛𝑥_𝑛−𝑧+𝑘_𝑛𝑇𝑥_𝑛−𝑧. (43) Since 𝑇 is generalized 𝑆-nonexpansive type, we have 𝑥_𝑛+1−𝑧≤1−𝑘_𝑛𝑥_𝑛−𝑧+𝑘_𝑛𝑀𝑥_𝑛,𝑧. (44) Now the following cases arise.
Case 1. (𝑀(𝑥_𝑛,𝑧)=‖𝑆𝑥_𝑛−𝑆𝑧‖). Then 𝑇𝑥_𝑛−𝑧≤𝑆𝑥_𝑛−𝑆𝑧. (45) Since 𝑆 is nonexpansive on 𝐶, the above inequality reduces to 𝑇𝑥_𝑛−𝑧≤𝑥_𝑛−𝑧. (46)
Case 2. (𝑀(𝑥_𝑛,𝑧)=(‖𝑆𝑥_𝑛−𝑇𝑥_𝑛‖+‖𝑆𝑧−𝑇𝑧‖)/2). Then 𝑇𝑥_𝑛−𝑧≤𝑆𝑥_𝑛−𝑇𝑥_𝑛+‖𝑆𝑧−𝑇𝑧‖2=𝑆𝑥_𝑛−𝑇𝑥_𝑛2≤𝑆𝑥_𝑛−𝑧+𝑧−𝑇𝑥_𝑛2=𝑆𝑥_𝑛−𝑆𝑧+𝑧−𝑇𝑥_𝑛2,𝑇𝑥_𝑛−𝑧≤𝑆𝑥_𝑛−𝑆𝑧. (47)
Nonexpansiveness of 𝑆 on 𝐶 implies 𝑇𝑥_𝑛−𝑧≤𝑥_𝑛−𝑧. (48)
Case 3. (𝑀(𝑥_𝑛,𝑧)=(‖𝑆𝑥_𝑛−𝑇𝑧‖+‖𝑆𝑧−𝑇𝑥_𝑛‖)/2). Then 𝑇𝑥_𝑛−𝑧≤𝑆𝑥_𝑛−𝑇𝑧+𝑆𝑧−𝑇𝑥_𝑛2=𝑆𝑥_𝑛−𝑇𝑧+𝑇𝑥_𝑛−𝑧2=𝑆𝑥_𝑛−𝑆𝑧+𝑇𝑥_𝑛−𝑧2,𝑇𝑥_𝑛−𝑧≤𝑆𝑥_𝑛−𝑆𝑧. (49)
Again, since 𝑆 is nonexpansive on 𝐶, it follows that 𝑇𝑥_𝑛−𝑧≤𝑥_𝑛−𝑧. (50) Therefore in all the cases, we get 𝑇𝑥_𝑛−𝑧≤𝑥_𝑛−𝑧. (51)
By (44) and (51), we get 𝑥_𝑛+1−𝑧≤1−𝑘_𝑛𝑥_𝑛−𝑧+𝑘_𝑛𝑥_𝑛=𝑥−𝑧_𝑛−𝑧. (52) Thus, for 𝑘_𝑛≠0, {‖𝑥_𝑛−𝑧‖} is a nonincreasing sequence. Hence, lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists. Now we show that {𝑥_𝑛} converges weakly to a common fixed point of 𝑆 and 𝑇. Let {𝑥_𝑛𝑘} and {𝑥_𝑚𝑘} be two subsequences of {𝑥_𝑛} which converge weakly to 𝑧 and 𝑧, respectively. We will show that 𝑧=𝑧. Suppose the contrary. Since 𝑋 satisfies Opial's condition and lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists for any 𝑧∈𝐹(𝑇)∩𝐹(𝑆), we get lim_𝑛→∞𝑥_𝑛−𝑧=lim_𝑘→∞𝑥_𝑛𝑘−𝑧<lim_𝑘→∞𝑥_𝑛𝑘−𝑧=lim_𝑛→∞𝑥_𝑛−𝑧=lim_𝑗→∞𝑥_𝑚𝑗−𝑧<lim_𝑗→∞𝑥_𝑚𝑗−𝑧=lim_𝑛→∞𝑥_𝑛−𝑧, (53) a contradiction. Hence 𝑧=𝑧.

Corollary 18. Theorem 16.

Proof. It comes from Theorem 17, when 𝑀(𝑥,𝑦)=‖𝑆𝑥−𝑆𝑦‖.

Corollary 19. Let 𝑋 be a Banach space and 𝐶 a closed convex subset of 𝑋 which satisfies Opial's condition. Let 𝑇∶𝐶→𝐶 be generalized nonexpansive type mapping.
Suppose {𝑘_𝑛} is a real sequence in (0,1). Then the sequence of Mann iterates defined for an arbitrary 𝑥₀∈𝐶 by 𝑥_𝑛+1=1−𝑘_𝑛𝑥_𝑛+𝑘_𝑛𝑇𝑥_𝑛,𝑛∈{0}∪ℕ (54) converges weakly to a common fixed point of 𝑆 and 𝑇.

Proof. It comes from Theorem 17, when 𝑆 is an identity mapping on 𝑋.

Corollary 20. Let 𝑋 be a Banach space and 𝐶 a closed convex subset of 𝑋 which satisfies Opial's condition. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping.
Suppose {𝑘_𝑛} is a real sequence in (0,1). Then the sequence of Mann iterates defined for arbitrary 𝑥₀∈𝐶 defined 𝑥_𝑛+1=1−𝑘_𝑛𝑥_𝑛+𝑘_𝑛𝑇𝑥_𝑛,𝑛∈{0}∪ℕ (55) converges weakly to a common fixed point of 𝑆 and 𝑇.

Proof. It comes from Corollary 19 when 𝑚(𝑥,𝑦)=‖𝑥−𝑦‖.

Acknowledgments

The authors would like to thank Professor M. Abbas for his useful comments. The authors would like to thank the referees for their constructive comments and useful suggestions.