Three-Step Fixed Point Iteration for Generalized Multivalued Mapping in Banach Spaces

Abstract

The convergence of three-step fixed point iterative processes for generalized multivalued nonexpansive mapping was considered in this paper. Under some different conditions, the sequences of three-step fixed point iterates strongly or weakly converge to a fixed point of the generalized multivalued nonexpansive mapping. Our results extend and improve some recent results.

1. Introduction

Let 𝑋 be a Banach space and 𝐾 a nonempty subset of 𝑋. The set 𝐾 is called proximinal if for each 𝑥∈𝑋, there exists an element 𝑦∈𝐾 such that ‖𝑥−𝑦‖=𝑑(𝑥,𝐾), where 𝑑(𝑥,𝐾)=inf{‖𝑥−𝑧‖∶𝑧∈𝐾}. Let 𝐶𝐵(𝐾),𝐶(𝐾),𝑃(𝐾),𝐹(𝑇) denote the family of nonempty closed bounded subsets, nonempty compact subsets, nonempty proximinal bounded subsets of 𝐾, and the set of fixed points, respectively. A multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is said to be nonexpansive (quasi-nonexpansive) if𝐻≤𝐻≤(𝑇),𝑇𝑥,𝑇𝑦𝑥−𝑦,𝑥,𝑦∈𝐾,𝑇𝑥,𝑇𝑝𝑥−𝑝,𝑥∈𝐾,𝑝∈𝐹 (1.1) where 𝐻(⋅,⋅) denotes the Hausdorff metric on 𝐶𝐵(𝑋) defined by𝐻(𝐴,𝐵)∶=maxsup_𝑥∈𝐴inf_𝑦∈𝐵𝑥−𝑦,sup_𝑦∈𝐵inf_𝑥∈𝐴𝑥−𝑦,𝐴,𝐵∈𝐶𝐵(𝑋). (1.2) A point 𝑥 is called a fixed point of 𝑇 if 𝑥∈𝑇𝑥. Since Banach's Contraction Mapping Principle was extended nicely to multivalued mappings by Nadler in 1969 (see [1]), many authors have studied the fixed point theory for multivalued mappings (e.g., see [2]). For single-valued nonexpansive mappings, Mann [3] and Ishikawa [4], respectively, introduced a new iteration procedure for approximating its fixed point in a Banach space as follows:𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑦_𝑛,𝑦_𝑛=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑥_𝑛, (1.3) where {𝛼_𝑛} and {𝑏_𝑛} are sequences in [0,1]. Obviously, Mann iteration is a special case of Ishikawa iteration. Recently Song and Wang in [5, 6] introduce the following algorithms for multivalued nonexpansive mapping:𝑥_𝑛+1=1−𝛼_n𝑥_𝑛+𝛼_𝑛𝑠_𝑛, (1.4) where 𝑠_𝑛∈𝑇𝑥_𝑛,𝛾_𝑛∈(0,+∞) such that lim_𝑛→∞𝛾_𝑛=0 and ‖𝑠_𝑛+1−𝑠_𝑛‖≤𝐻(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛)+𝛾_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑟_𝑛,𝑦_𝑛=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑠_𝑛, (1.5) where ‖𝑠_𝑛−𝑟_𝑛‖≤𝐻(𝑇𝑥_𝑛,𝑇𝑦_𝑛)+𝛾_𝑛 and ‖𝑠_𝑛+1−𝑟_𝑛‖≤𝐻(𝑇𝑥_𝑛+1,𝑇𝑦_𝑛)+𝛾_𝑛 for 𝑠_𝑛∈𝑇𝑥_𝑛 and 𝑟_𝑛∈𝑇𝑦_𝑛. They show some strong convergence results of the above iterates for multivalued nonexpansive mapping 𝑇 under some appropriate conditions. However, the iteration scheme constructed by Song and Wang involves the following estimates, 𝑠_𝑛−𝑟_𝑛≤𝐻𝑇𝑥_𝑛,𝑇𝑦_𝑛+𝛾_𝑛,𝑠_𝑛+1−𝑟_𝑛≤𝐻𝑇𝑥_𝑛+1,𝑇𝑦_𝑛+𝛾_𝑛, (1.6) which are not easy to be computed and the scheme is more time consuming. It is observed that Song and Wang [6] did not use the above estimates in their proofs and the assumption on 𝑇, namely, 𝑇(𝑝)={𝑝} for any 𝑝∈𝐹(𝑇) is quite strong. It is noted that the domain of 𝑇 is compact, which is a strong condition. The aim of this paper is to construct an three iteration scheme for a generalized multivalued mappings, which removes the restriction of 𝑇, namely, 𝑇(𝑝)={𝑝} for any 𝑝∈𝐹(𝑇) and also relax compactness of the domain of 𝑇. The generalized multivalued mappings was introduced in [7], if12𝑑(𝑥,𝑇𝑥)≤𝑥−𝑦implies𝐻𝑇𝑥,𝑇𝑦≤𝑥−𝑦∀𝑥,𝑦∈𝐾, (1.7) where 𝑑 is induced by the norm. Obviously, the condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness, furthermore, there are some examples of a generalized nonexpansive multivalued mapping which is not a nonexpansive multivalued mapping (see [7, 8]).

Let 𝑇∶𝐾→𝑃(𝐾) be a generalized nonexpansive multivalued mapping and 𝑃_𝑇(𝑥)={𝑦∈𝑇(𝑥)∶‖𝑥−𝑦‖=𝑑(𝑥,𝑇(𝑥))}. The three-step mean multivalued iterative scheme is defined by 𝑥₀∈𝐾,𝑧_𝑛=1−𝑎_𝑛𝑥_𝑛+𝑎_𝑛𝑠_𝑛,𝑦_𝑛=1−𝑏_𝑛−𝑐_𝑛𝑥_𝑛+𝑏_𝑛𝑡_𝑛+𝑐_𝑛𝑠_𝑛,𝑥_𝑛+1=1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛+𝛼_𝑛𝑟_𝑛+𝛽_𝑛𝑡_𝑛+𝛾_𝑛𝑠_𝑛, (1.8) where {𝑎_𝑛},{𝑏_𝑛},{𝑐_𝑛},{𝑏_𝑛+𝑐_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝛼_𝑛+𝛽_𝑛+𝛾_𝑛} are appropriate sequence in [0,1], furthermore 𝑠_𝑛∈𝑃_𝑇(𝑥_𝑛),𝑡_𝑛∈𝑃_𝑇(𝑧_𝑛),𝑟_𝑛∈𝑃_𝑇(𝑦_𝑛). If 𝑎_𝑛=𝑐_𝑛=𝛽_𝑛=𝛾_𝑛≡0 or 𝑎_𝑛=𝑏_𝑛=𝑐_𝑛=𝛽_𝑛=𝛾_𝑛≡0, then iterative scheme (1.8) reduces to the Ishikawa and Mann multivalued iterative scheme. In fact let 𝛾_𝑛≡0 or 𝑐_𝑛=𝛽_𝑛=𝛾_𝑛≡0 or 𝑏_𝑛=𝑐_𝑛=𝛼_𝑛=𝛾_𝑛≡0, we also have the other three algorithms.

The mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is called hemicompact if, for any sequence 𝑥_𝑛 in 𝐾 such that 𝑑(𝑥_𝑛,𝑇(𝑥_𝑛))→0 as 𝑛→∞, there exists a subsequence 𝑥_𝑛𝑘 of 𝑥_𝑛 such that 𝑥_𝑛𝑘→𝑝∈𝐾. We note that if 𝐾 is compact, then every multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is hemicompact. The following definition was introduced in [9].

Definition 1.1. A multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is said to satisfy Condition (A) if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0,𝑓(𝑥)>0 for 𝑥∈(0,∞) such that 𝑑(𝑥,𝑇𝑥)≥𝑓(𝑑(𝑥,𝐹(𝑇)))∀𝑥∈𝐾. (1.9) where 𝐹(𝑇)≠∅  is the fixed point set of the multivalued mapping 𝑇. From now on, 𝐹(𝑇) stands for the fixed point set of the multivalued mapping  𝑇.

2. Preliminaries

A Banach space 𝑋 is said to be satisfy Opial's condition [10] if, for any sequence {𝑥_𝑛} in 𝑋, 𝑥_𝑛⇀𝑥(𝑛→∞) implies the following inequality:limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦, (2.1) for all 𝑦∈𝑋 with 𝑦≠𝑥. It is known that Hilbert spaces and 𝑙_𝑝(1<𝑝<∞) have the Opial's condition.

Lemma 2.1 (see [7, 11]). Let {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be sequence in uniformly convex Banach space 𝑋. Suppose that {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are sequence in [0,1] with 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,limsup_𝑛‖𝑥_𝑛‖≤𝑑,limsup_𝑛‖𝑦_𝑛‖≤𝑑,limsup_𝑛‖𝑧_𝑛‖≤𝑑, and lim_𝑛‖𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑦_𝑛+𝛾_𝑛𝑧_𝑛‖=𝑑. If liminf_𝑛𝛼_𝑛>0 and  liminf_𝑛𝛽_𝑛>0, then lim_𝑛‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.2 (see [7, 11]). Let 𝑋 be a uniformly convex Banach space and 𝐵_𝑟∶={𝑥∈𝑋∶‖𝑥‖≤𝑟},𝑟>0. Then there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝜆𝑥+𝜇𝑦+𝜉𝑧+𝜗𝜔²≤𝜆‖𝑥‖²+𝜇𝑦²+𝜉‖𝑧‖²+𝜗‖𝜔‖²−13𝜗𝜆𝑔(‖𝑥−𝜔‖)+𝜇𝑔𝑦−𝜔+𝜉𝑔(‖𝑧−𝜔‖), (2.2) for all 𝑥,𝑦,𝑧,𝜔∈𝐵_𝑟 and 𝜆,𝜇,𝜉,𝜗∈[0,1] with 𝜆+𝜇+𝜉+𝜗=1.

3. Main Results

Lemma 3.1. Let 𝑋 be a real Banach space and 𝐾 be a nonempty convex subset of 𝑋,𝑇∶𝐾→𝑃(𝐾) be a generalized multivalued nonexpansive mapping with 𝐹(𝑇)≠∅ such that 𝑃_𝑇 is nonexpansive. Let {𝑥_𝑛} be a sequence in 𝐾 defined by (1.8), then one has the following conclusion: lim_𝑛𝑥_𝑛−𝑝existsforany𝑝∈𝐹(𝑇). (3.1)

Proof. Let 𝑝∈𝐹(𝑇), then 𝑝∈𝑃_𝑇(𝑝)={𝑝}. Since 𝑇 is quasi-nonexpansive, thus we obtain 𝑧_𝑛−𝑝≤1−𝑎_𝑛𝑥_𝑛−𝑝+𝑎_𝑛𝑠_𝑛≤−𝑝1−𝑎_𝑛𝑥_𝑛−𝑝+𝑎_𝑛𝑑𝑠_𝑛,𝑃_𝑇𝑝≤1−𝑎_𝑛𝑥_𝑛−𝑝+𝑎_𝑛𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝≤1−𝑎_𝑛𝑥_𝑛−𝑝+𝑎_𝑛𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝, (3.2) similarly ‖𝑦_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, then we have 𝑥_𝑛+1−𝑝≤1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑟_𝑛−𝑝+𝛽_𝑛𝑡_𝑛−𝑝+𝛾_𝑛𝑠_𝑛≤−𝑝1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝐻𝑃_𝑇𝑦_𝑛,𝑃_𝑇𝑝+𝛽_𝑛𝐻𝑃_𝑇𝑧_𝑛,𝑃_𝑇𝑝+𝛾_𝑛𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝≤𝑥_𝑛−𝑝. (3.3) Then {‖𝑥_𝑛−𝑝‖} is a decreasing sequence and hence lim_𝑛‖𝑥_𝑛−𝑝‖ exists for any 𝑝∈𝐹(𝑇).

Lemma 3.2. Let 𝑋 be a uniformly convex Banach space and 𝐾 be a nonempty convex subset of 𝑋,𝑇∶𝐾→𝑃(𝐾) be a generalized multivalued nonexpansive mapping with 𝐹(𝑇)≠∅ such that 𝑃_𝑇 is nonexpansive. Let {𝑥_𝑛} be a sequence in 𝐾 defined by (1.8), if the coefficient satisfy one of the following control conditions:(i)liminf_𝑛𝛼_𝑛>0 and one of the following holds: (a)limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1 and limsup_𝑛(𝑏_𝑛+𝑐_𝑛)<1,(b)0<liminf_𝑛𝛽_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1 and limsup_𝑛𝑐_𝑛<1,(c)0<liminf_𝑛𝑏_𝑛≤limsup_𝑛(𝑏_𝑛+𝑐_𝑛)<1 and limsup_𝑛𝑎_𝑛<1,(d)0<liminf_𝑛𝑐_𝑛≤limsup_𝑛(𝑏_𝑛+𝑐_𝑛)<1; (ii)0<liminf_𝑛𝛽_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1 and limsup_𝑛𝑎_𝑛<1;(iii)0<liminf_𝑛𝛾_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1; (iv)0<liminf_𝑛(𝛼_𝑛𝑏_𝑛+𝛽_𝑛) and 0<liminf_𝑛𝑎_𝑛≤limsup_𝑛𝑎_𝑛<1;then we have lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0.

Proof. By Lemma 3.1, we know that lim_𝑛‖𝑥_𝑛−𝑝‖ exists for any 𝑝∈𝐹(𝑇), then it follows that {𝑠_𝑛−𝑝},{𝑡_𝑛−𝑝}, and {𝑟_𝑛−𝑝} are all bounded. We may assume that these sequences belong to 𝐵_𝑟 where 𝑟>0. Note that 𝑝∈𝑃_𝑇(𝑝)={𝑝} for any fixed point 𝑝∈𝐹(𝑇) and 𝑇 is quasi-nonexpansive. By Lemma 2.2, we get 𝑧_𝑛−𝑝²≤1−𝑎_𝑛𝑥_𝑛−𝑝²+𝑎_𝑛𝑠_𝑛−𝑝²≤1−𝑎_𝑛𝑥_𝑛−𝑝²+𝑎_𝑛𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝²≤𝑥_𝑛−𝑝²,𝑦_𝑛−𝑝²≤1−𝑏_𝑛−𝑐_𝑛𝑥_𝑛−𝑝²+𝑏_𝑛𝑡_𝑛−𝑝²+𝑐_𝑛𝑠_𝑛−𝑝²−131−𝑏_𝑛−𝑐_𝑛𝑏_𝑛𝑔𝑡_𝑛−𝑥_𝑛+𝑐_𝑛𝑔𝑠_𝑛−𝑥_𝑛≤1−𝑏_𝑛−𝑐_𝑛𝑥_𝑛−𝑝²+𝑏_𝑛𝐻𝑃_𝑇𝑧_𝑛,𝑃_𝑇𝑝²+𝑐_𝑛𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝²−131−𝑏_𝑛−c_𝑛𝑏_𝑛𝑔𝑡_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑝²−131−𝑏_𝑛−𝑐_𝑛𝑏_𝑛𝑔𝑡_𝑛−𝑥_𝑛, (3.4) and therefore we have 𝑥_𝑛+1−𝑝²≤1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝑟_𝑛−𝑝²+𝛽_𝑛𝑡_𝑛−𝑝²+𝛾_𝑛𝑠_𝑛−𝑝²−131−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝛼_𝑛𝑔𝑥_𝑛−𝑟_𝑛+𝛽_𝑛𝑔𝑥_𝑛−𝑡_𝑛+𝛾_𝑛𝑔𝑥_𝑛−𝑠_𝑛≤1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝐻𝑃_𝑇𝑦_𝑛,𝑃_𝑇𝑝²+𝛽_𝑛𝐻𝑃_𝑇𝑧_𝑛,𝑃_𝑇𝑝²+𝛾_𝑛𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝²−131−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝛼_𝑛𝑔𝑥_𝑛−𝑟_𝑛+𝛽_𝑛𝑔𝑥_𝑛−𝑡_𝑛+𝛾_𝑛𝑔𝑥_𝑛−𝑠_𝑛≤𝑥_𝑛−𝑝²−𝛼_𝑛31−𝑏_𝑛−𝑐_𝑛𝑏_𝑛𝑔𝑡_n−𝑥_𝑛−131−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛×𝛼_𝑛𝑔𝑥_𝑛−𝑟_𝑛+𝛽_𝑛𝑔𝑥_𝑛−𝑡_𝑛+𝛾_𝑛𝑔𝑥_𝑛−𝑠_𝑛. (3.5) Then 1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝛼_𝑛𝑔𝑥_𝑛−𝑟_𝑛≤3𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²,1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝛽_𝑛𝑔𝑥_𝑛−𝑡_𝑛≤3𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²,1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝛾_𝑛𝑔𝑥_𝑛−𝑠_𝑛≤3𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²𝛼_𝑛,1−𝑏_𝑛−𝑐_𝑛𝑏_𝑛𝑔𝑡_𝑛−𝑥_𝑛≤3𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (3.6)(3.7)(3.8)(3.9) Since lim_𝑛‖𝑥_𝑛−𝑝‖ exists for any 𝑝∈𝐹(𝑇), it follows from (3.6) that lim_𝑛(1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛)𝛼_𝑛𝑔(‖𝑥_𝑛−𝑟_𝑛‖)=0. From 𝑔 is continuous strictly increasing with 𝑔(0)=0 and 0<liminf_𝑛𝛼_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1, then lim_𝑛𝑥_𝑛−𝑟_𝑛=0. (3.10) Using a similarly method together with inequalities (3.7) and 0<liminf_𝑛𝛽_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1, then lim_𝑛𝑥_𝑛−𝑡_𝑛=0. (3.11) Similarly, from (3.8) and 0<liminf_𝑛𝛾_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1, we have lim_𝑛‖𝑥_𝑛−𝑠_𝑛‖=0, since 𝑠_𝑛∈𝑇𝑥_𝑛, then 0≤lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)≤lim_𝑛‖𝑥_𝑛−𝑠_𝑛‖=0, thus we get (iii). In the sequence we prove (i) (a). From iterative scheme (1.8), we have 𝑠_𝑛−𝑥_𝑛≤𝑠_𝑛−𝑟_𝑛+𝑟_𝑛−𝑥_𝑛≤𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑦_𝑛+𝑟_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑟_𝑛−𝑥_𝑛≤𝑏_𝑛𝑥_𝑛−𝑡_𝑛+𝑐_𝑛𝑥_𝑛−𝑠_𝑛+𝑟_𝑛−𝑥_𝑛. (3.12) To show that lim_𝑛‖𝑥_𝑛−𝑠_𝑛‖=0, it suffices to show that there exist a subsequence {𝑛_𝑗} of {𝑛} such that  lim_𝑛𝑗‖𝑥_𝑛𝑗−𝑠_𝑛𝑗‖=0. If  liminf_𝑗𝑏_𝑛𝑗>0, it follows from (3.9) that 𝛼_𝑛𝑗1−𝑏_𝑛𝑗−𝑐_𝑛𝑗𝑏_𝑛𝑗𝑔𝑡_𝑛𝑗−𝑥_𝑛𝑗≤3𝑥_𝑛𝑗−𝑝²−𝑥_𝑛𝑗+1−𝑝². (3.13) Since lim_𝑛‖𝑥_𝑛−𝑝‖ exists for any 𝑝∈𝐹(𝑇), we have lim_𝑛𝑗𝛼_𝑛𝑗1−𝑏_𝑛𝑗−𝑐_𝑛𝑗𝑏_𝑛𝑗𝑔𝑡_𝑛𝑗−𝑥_𝑛𝑗=0. (3.14) From 𝑔 is continuous strictly increasing with 𝑔(0)=0,liminf_𝑗𝛼_𝑛𝑗>0 and 0<liminf_𝑛𝑗𝑏_𝑛𝑗≤limsup_𝑛𝑗(𝑏_𝑛𝑗+𝑐_𝑛𝑗)<1, we have lim_𝑛𝑗𝑡_𝑛𝑗−𝑥_𝑛𝑗=0. (3.15) This together with (3.10), (3.12), (3.15) gives lim_𝑗1−𝑐_𝑛𝑗𝑠_𝑛𝑗−𝑥_𝑛𝑗=0. (3.16) Since liminf_𝑛𝑗(1−𝑐_𝑛𝑗)=1−limsup_𝑛𝑗𝑐_𝑛𝑗>0, we have lim_𝑗‖𝑠_𝑛𝑗−𝑥_𝑛𝑗‖=0. On the other hand, if liminf_𝑗𝑏_𝑛𝑗=0, then we may extract a subsequence {𝑏_𝑛𝑘} of {𝑏_𝑛𝑗} so that lim_𝑘𝑏_𝑛𝑘=0. This together with (i) (a) and (3.10), (3.12) gives lim_𝑘1−𝑐_𝑛𝑘𝑠_𝑛𝑘−𝑥_𝑛𝑘=0,andsolim_𝑘𝑠_𝑛𝑘−𝑥_𝑛𝑘=0. (3.17) By Double Extract Subsequence Principle, we obtain the result.
If 0<liminf_𝑛𝛽_𝑛≤limsup_𝑛(𝛼_𝑛+𝛽_𝑛+𝛾_𝑛)<1 and limsup_𝑛𝑎_𝑛<1, we will prove (ii), 𝑠_𝑛−𝑥_𝑛≤𝑠_𝑛−𝑡_𝑛+𝑡_𝑛−𝑥_𝑛≤𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑧_𝑛+𝑡_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑡_𝑛−𝑥_𝑛≤𝑎_𝑛𝑥_𝑛−𝑠_𝑛+𝑡_𝑛−𝑥_𝑛. (3.18) Since limsup_𝑛𝑎_𝑛<1, then liminf_𝑛1−𝑎_𝑛=1−limsup_𝑛𝑎_𝑛>0. (3.19) This together with (3.11), (3.18), we obtain the result.
We will prove (i) (b), let 𝑝∈𝐹(𝑇). By Lemma 3.1, we let lim_𝑛‖𝑥_𝑛−𝑝‖=𝑑 for some 𝑑≥0. From iterative scheme (1.8), we know 𝑑=lim_𝑛𝑥_𝑛+1−𝑝=lim_𝑛1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑟_𝑛−𝑝+𝛽_𝑛𝑡_𝑛−𝑝+𝛾_𝑛𝑠_𝑛−𝑝. (3.20) From Lemma 3.1, we have known that ‖𝑧_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖ and ‖𝑦_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, then limsup_𝑛𝑟_𝑛−𝑝≤limsup_𝑛𝐻𝑃_𝑇𝑦_𝑛,𝑃_𝑇𝑝≤limsup_𝑛𝑦_𝑛−𝑝≤𝑑,limsup_𝑛𝑡_𝑛−𝑝≤limsup_𝑛𝐻𝑃_𝑇𝑧_𝑛,𝑃_𝑇𝑝≤limsup_𝑛𝑧_𝑛−𝑝≤𝑑,limsup_𝑛𝑠_𝑛−𝑝≤limsup_𝑛𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝≤limsup_𝑛𝑥_𝑛−𝑝≤𝑑. (3.21) From (3.20) and Lemma 2.1, we have lim_𝑛𝑥_𝑛−𝑡_𝑛=lim_𝑛𝑟_𝑛−𝑥_𝑛=0. (3.22) Notice that 𝑥_𝑛−𝑠_𝑛≤𝑥_𝑛−𝑟_𝑛+𝑟_𝑛−𝑠_𝑛≤𝑥_𝑛−𝑟_𝑛+𝐻𝑃_𝑇𝑦_𝑛,𝑃_𝑇𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑥_𝑛−𝑟_𝑛≤𝑏_𝑛𝑥_𝑛−𝑡_𝑛+𝑐_𝑛𝑥_𝑛−𝑠_𝑛+𝑥_𝑛−𝑟_𝑛. (3.23) Since limsup_𝑛𝑐_𝑛<1, we have lim_𝑛‖𝑠_𝑛−𝑥_𝑛‖=0, therefore 0≤lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)≤lim_𝑛‖𝑥_𝑛−𝑠_𝑛‖=0.
We will prove (i) (c). From iterative scheme (1.8) and Lemma 3.1, we have 𝑥_𝑛+1−𝑝≤1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑦_𝑛−𝑝+𝛽_𝑛𝑧_𝑛−𝑝+𝛾_𝑛𝑥_𝑛≤−𝑝1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑦_𝑛−𝑝. (3.24) which implies 𝑥_𝑛+1−𝑝−𝑥_𝑛−𝑝+𝛼_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑦_𝑛−𝑝. (3.25) Notice that liminf_𝑛𝛼_𝑛>0 and lim_𝑛‖𝑥_𝑛−𝑝‖ exists. Hence from (3.25) we have 𝑑=lim_𝑛𝑥_𝑛−𝑝≤liminf_𝑛𝑦_𝑛−𝑝≤limsup𝑦_𝑛−𝑝≤𝑑. (3.26) Therefore, from iterative scheme (1.8) we have 𝑑=lim_𝑛𝑦_𝑛−𝑝=lim_𝑛1−𝑏_𝑛−𝑐_𝑛𝑥_𝑛−𝑝+𝑏_𝑛𝑡_𝑛−𝑝+𝑐_𝑛𝑠_𝑛−𝑝. (3.27) From Lemma 2.1, we have lim_𝑛𝑥_𝑛−𝑡_𝑛=0. (3.28) Notice that 𝑠_𝑛−𝑥_𝑛≤𝑠_𝑛−𝑡_𝑛+𝑡_𝑛−𝑥_𝑛≤𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑧_𝑛+𝑡_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑡_𝑛−𝑥_𝑛≤𝑎_𝑛𝑥_𝑛−𝑠_𝑛+𝑡_𝑛−𝑥_𝑛. (3.29) Since limsup_𝑛𝑎_𝑛<1, then 0≤lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)≤lim_𝑛‖𝑥_𝑛−𝑠_𝑛‖=0.
By (3.27) and Lemma 2.1, we can similarly prove (i) (d).
Finally, we will prove (iv). From iterative scheme (1.8) and Lemma 3.1, we have 𝑥_𝑛+1−𝑝≤1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑟_𝑛−𝑝+𝛽_𝑛𝑡_𝑛−𝑝+𝛾_𝑛𝑠_𝑛≤−𝑝1−𝛼_𝑛−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑦_𝑛−𝑝+𝛽_𝑛𝑧_𝑛−𝑝+𝛾_𝑛𝑥_𝑛≤−𝑝1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛1−𝑏_𝑛𝑥_𝑛−𝑝+𝑏_𝑛𝑧_𝑛−𝑝+𝛽_𝑛𝑧_𝑛−𝑝, (3.30) which implies 𝑥_𝑛+1−𝑝−𝑥_𝑛−𝑝+𝛼_𝑛𝑏_𝑛+𝛽_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑏_𝑛+𝛽_𝑛𝑧_𝑛−𝑝. (3.31) Notice that 0<liminf_𝑛𝛼_𝑛𝑏_𝑛+𝛽_𝑛,lim_𝑛𝑥_𝑛−𝑝exists. (3.32) Hence we have 𝑑=lim_𝑛𝑥_𝑛−𝑝≤liminf_𝑛𝑧_𝑛−𝑝≤limsup𝑧_𝑛−𝑝≤𝑑. (3.33) Thus, we have 𝑑=lim_𝑛𝑧_𝑛−𝑝=lim_𝑛1−𝑎_𝑛𝑥_𝑛−𝑝+𝑎_𝑛𝑠_𝑛−𝑝. (3.34) By Lemma 2.1  and 0<liminf_𝑛𝑎_𝑛≤limsup_𝑛𝑎_𝑛<1, we have  0≤lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)≤lim_𝑛‖𝑥_𝑛−𝑠_𝑛‖=0.

Theorem 3.3. Let 𝑋 be a uniformly convex Banach space and 𝐾 be a nonempty convex subset of 𝑋, 𝑇∶𝐾→𝑃(𝐾) be a generalized multivalued nonexpansive mapping with 𝐹(𝑇)≠∅ such that 𝑃_𝑇 is nonexpansive. Let {𝑥_𝑛} be a sequence in 𝐾 defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and 𝑇 satisfies Condition (A) with respect to the sequence {𝑥_𝑛}, then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. By Lemma 3.2, we have lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Since 𝑇 satisfies Condition (A) with respect to {𝑥_𝑛}. Then 𝑓𝑑𝑥_𝑛,𝐹(𝑇)≤𝑑𝑥_𝑛,𝑇𝑥_𝑛⟶0. (3.35) Thus, we get lim_𝑛𝑑(𝑥_𝑛,𝐹(𝑇))=0. The remainder of the proof is the same as in [6, Theorem 2.4], we omit it.

Theorem 3.4. Let 𝑋 be a uniformly convex Banach space and 𝐾 be a nonempty convex subset of 𝑋, 𝑇∶𝐾→𝑃(𝐾) be a generalized multivalued nonexpansive mapping with 𝐹(𝑇)≠∅ such that 𝑃_𝑇 is nonexpansive. Let {𝑥_𝑛} be a sequence in 𝐾 defined by (1.8), the coefficient satisfy the control conditions in Lemma 3.2 and 𝑇 is hemicompact, then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. By Lemma 3.2, we have lim_𝑛𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. Since 𝑇 is hemicompact, then there exist a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that lim_𝑘→∞‖𝑥_𝑛𝑘−𝑞‖=0 for some 𝑞∈𝐾. Thus, 𝑑𝑞,𝑇𝑞≤𝑞−𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘+𝐻𝑇𝑥_𝑛𝑘,𝑇𝑞≤2𝑞−𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘⟶0. (3.36) Hence, 𝑞 is a fixed point of 𝑇. Now on take on 𝑞 in place of 𝑝, we get that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. It follows that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Theorem 3.5. Let 𝑋,𝑇 and {𝑥_𝑛} be the same as in Lemma 3.2. If 𝐾 be a nonempty weakly compact convex subset of a Banach space 𝑋 and 𝑋 satisfies Opial's condition, then {𝑥_𝑛} converges weakly to a fixed point of 𝑇.

Proof. The proof of the Theorem is the same as in [6, Theorem 2.5], we omit it.

Remark 3.6. From the definition of iterative scheme (1.8), Theorems 3.3, 3.4, and 3.5 extend some results in [6, 12], and also give some new results are different from the [5]. In fact, we can present an example of a multivalued map 𝑇 for which 𝑃_𝑇 is nonexpansive. A multivalued map 𝑇∶𝐷→𝐶𝐵(𝑋) is ∗-nonexpansive [13] if for all 𝑥,𝑦∈𝐷 and 𝑢_𝑥∈𝑇(𝑥) with 𝑑(𝑥,𝑢_𝑥)=inf{𝑑(𝑥,𝑧)∶𝑧∈𝑇(𝑥)}, there exists 𝑢_𝑦∈𝑇(𝑦) with 𝑑(𝑦,𝑢_𝑦)=inf{𝑑(𝑦,𝑤)∶𝜔∈𝑇(𝑦)} such that 𝑑𝑢_𝑥,𝑢_𝑦≤𝑑𝑥,𝑦. (3.37) It is clear that if 𝑇 is ∗-nonexpansive, then 𝑃_𝑇 is nonexpansive. It is known that ∗-nonexpansiveness is different from nonexpansiveness for multivalued maps. Let 𝐷=[0,∞) and 𝑇 be defined by 𝑇𝑥=[𝑥,2𝑥] for 𝑥∈𝐷 [14]. Then 𝑃_𝑇(𝑥)=𝑥 for 𝑥∈𝐷 and thus it is nonexpansive. Note that 𝑇 is ∗-nonexpansive but not nonexpansive (see [14]).