An Iterative Shrinking Metric 𝑓-Projection Method for Finding a Common Fixed Point of a Closed and Quasi-Strict𝑓-Pseudocontraction and a Countable Family of Firmly Nonexpansive Mappings and Applications in Hilbert Spaces

Abstract

We create some new ideas of mappings called quasi-strict 𝑓-pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric 𝑓-projection, we propose an iterative shrinking metric 𝑓-projection method for finding a common fixed point of a quasi-strict 𝑓-pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to find a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results.

1. Introduction

It is well known that the metric projection operators in Hilbert spaces and Banach spaces play an important role in various fields of mathematics such as functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problem (see, e.g., [1, 2]). In 1994, Alber [3] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized 𝑓-projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by [3] and proved some properties of the generalized 𝑓-projection operator. Fan et al. [5] presented some basic results for the generalized 𝑓-projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.

Let 𝐻 be a real Hilbert space; a mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐻 is called firmly nonexpansive if 𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑇𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐷(𝑇), (1) nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐷(𝑇). (2) Throughout this paper, 𝐼 stands for an identity mapping. The mapping 𝑇 is said to be a strict pseudocontraction if there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐷(𝑇). (3) In this case, 𝑇 may be called a 𝑘-strict pseudocontraction. We use 𝐹(𝑇) to denote the set of fixed points of 𝑇 (i.e. 𝐹(𝑇)={𝑥∈𝐷(𝑇)∶𝑇𝑥=𝑥}). 𝑇 is said to be a quasi-strict pseudocontraction if the set of fixed point 𝐹(𝑇) is nonempty and if there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑝²≤𝑥−𝑝²+𝑘‖𝑥−𝑇𝑥‖²,∀𝑥∈𝐷(𝑇),𝑝∈𝐹(𝑇). (4)

Construction of fixed points of nonexpansive mappings via Mann’s algorithm [6] has extensively been investigated in the literature; see, for example, [6–10] and references therein. However, we note that Mann’s iterations have only weak convergence even in a Hilbert space (see, e.g., [11]). Nakajo and Takahashi [12] modified the Mann iteration method so that strong convergence is guaranteed, later well known as a hybrid projection iteration method. Since then, the hybrid method has received rapid developments. For the details, the readers are referred to papers [13–26] and the references therein.

On the other hand, for a real Banach space 𝐸 and the dual 𝐸^∗, let 𝐶 be a nonempty closed convex subset of 𝐸. Let Θ∶𝐶×𝐶→ℝ be a bifunction, let 𝜑∶𝐶→ℝ be a real-valued function, and let 𝐴∶𝐶→𝐸^∗ be a nonlinear mapping. The generalized mixed equilibrium problem is to find 𝑥∈𝐶 such that Θ𝑥,𝑦+𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (5) The solution set of (5) is denoted by 𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑); that is, =+𝑦(𝑥).𝐺𝑀𝐸𝑃Θ,𝐴,𝜑𝑥∈𝐶∶Θ𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜑−𝜑≥0,∀𝑦∈𝐶 (6) If 𝐴=0, the problem (5) reduces to the mixed equilibrium problem for Θ, denoted by MEP(Θ,𝜑), which is to find that 𝑥∈𝐶 such that Θ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (7) If Θ=0, the problem (5) reduces to the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐴,𝜑), which is to find that 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (8) If 𝐴=0 and 𝜑=0, the problem (5) reduces to the equilibrium problem for Θ (for short, EP), denoted by EP(Θ), which is to find that 𝑥∈𝐶 such that Θ𝑥,𝑦≥0,∀𝑦∈𝐶. (9) If Θ=0 and 𝐴=0, the problem (5) reduces to the minimization problem for 𝜑, denoted by Argmin(𝜑), which is to find that 𝑥∈𝐶 such that 𝜑(𝑥)≤𝜑𝑦,∀𝑦∈𝐶. (10)

The previous formulation, (8), was shown in [27] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem, and the optimization problem, which can also be written in the form of (9). However, (5) is very general; it covers the problems mentioned above as special cases.

In 2007, S. T. Takahashi and W. T. Takahashi [28] and Tada and Takahashi [29, 30] proved weak and strong convergence theorems for finding a common element of the set of solutions of the equilibrium problem (9) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Takahashi et al. [22] studied a strong convergence theorem by the hybrid method for a family of nonexpansive mappings in Hilbert spaces as follows: 𝑥₀∈𝐻, 𝐶₁=𝐶, and 𝑥₁=𝑃_𝐶1𝑥₀, and let 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛∈ℕ, (11) where 0≤𝛼_𝑛≤𝑎<1, for all 𝑛∈ℕ, and {𝑇_𝑛} is a sequence of nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠𝜙. They proved that if {𝑇_𝑛} satisfies some appropriate conditions, then {𝑥_𝑛} converges strongly to 𝑃_⋂^∞_{𝑛=1𝐹(𝑇𝑛)}𝑥₀.

Motivated by Takahashi et al. [22], Takahashi and Zembayashi [31] (see also [32]) introduced and proved a hybrid projection algorithm for solving equilibrium problems and fixed point problems of a relatively nonexpansive mapping within the framework of a uniformly smooth and uniformly convex Banach space.

In 2011, Saewan and Kumam [33] introduced a new hybrid projection method based on the modified Mann iterative scheme by the generalized 𝑓-projection operator for a countable family of relatively quasi-nonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Later, they [34] also studied the new hybrid Ishikawa iteration process by the generalized 𝑓-projection operator for finding a common element of the fixed point set for two countable families of weak relatively nonexpansive mappings and the set of solutions of the system of generalized Ky Fan inequalities in a uniformly convex and uniformly smooth Banach space.

Recently, Li et al. [35] have studied the following hybrid iterative scheme for a relatively nonexpansive mapping by using the generalized 𝑓-projection operator in Banach spaces: 𝑥₀∈𝐶,𝐶₀𝑦=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1. (12) Under some appropriate assumptions, they obtained strong convergence theorems in Banach spaces.

Motivated and inspired by the work mentioned above, in this paper, we are interested to study our theorems within the framework of a real Hibert space, and we create some new ideas of mappings called quasi-strict 𝑓-pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric 𝑓-projection, we propose an iterative shrinking metric 𝑓-projection method for find a common fixed point of an quasi-strict 𝑓-pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to finding a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results.

2. Preliminaries

In this section, some definitions are provided, and some relevant lemmas which are useful to prove in the following section are collected. Most of them are known, and others are not hard to find or understand their proofs. Throughout this paper, we will use the notation ⇀ for weak convergence and → for strong convergence.

Lemma 1 (see Takahashi [36]). Let {𝑎_𝑛} be a sequence of real numbers. Then, lim_𝑛→∞𝑎_𝑛=0 if and only if, for any subsequence {𝑎_𝑛𝑖} of {𝑎_𝑛}, there exists a subsequence {𝑎_𝑛𝑖𝑗} of {𝑎_𝑛𝑖} such that lim_𝑗→∞𝑎_𝑛𝑖𝑗=0.

Definition 2 (see [37–40]). Let 𝐶 be a nonempty, closed, and convex subset of a Banach space 𝐸, and let {𝑇_𝑛} be a sequence of mappings of 𝐶 into itself such that ⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. Then, {𝑇_𝑛} is said to satisfy the NST-condition if, for each bounded sequence {𝑧_𝑛}⊂𝐶, lim_𝑛→∞𝑧_𝑛−𝑇_𝑛𝑧_𝑛=0 (13) implies that 𝜔_𝑤(𝑧_𝑛)⊂⋂^∞_𝑛=1𝐹(𝑇_𝑛), where 𝜔_𝑤(𝑧_𝑛) is the set of all weak cluster points of {𝑧_𝑛} (i.e., 𝜔_𝑤(𝑧_𝑛)={𝑧∣∃{𝑧_𝑛𝑖}⊂{𝑧_𝑛}suchthat𝑧_𝑛𝑖⇀𝑧}).

Let 𝐻 be a real Hilbert space, and let 𝐶 be nonempty, closed, and convex subset of 𝐻. Let (⋅,⋅)_𝑓∶𝐶×𝐻→(−∞,+∞] be a functional defined as follows (see Li et al. [35] (see also [4])): 𝑦,𝑥_𝑓∶=𝑦²−2𝑦,𝑥+‖𝑥‖²+2𝜌𝑓𝑦=𝑦−𝑥²+2𝜌𝑓𝑦, (14) where 𝑦∈𝐶, 𝑥∈𝐻, 𝜌 is positive number, and 𝑓∶𝐶→(−∞,+∞] is proper, convex, and lower semicontinuous. From the definitions of (⋅,⋅)_𝑓 and 𝑓, it is easy to see the following properties: (i)(𝑦,𝑥)_𝑓 is convex and continuous with respect to 𝑥 when 𝑦 is fixed; (ii)(𝑦,𝑥)_𝑓 is convex and lower semicontinuous with respect to 𝑦 when 𝑥 is fixed.

Definition 3 (see Li et al. [35] (see also [4])). Let 𝐻 be a real Hilbert space, and let 𝐶 be nonempty, closed, and convex subset of 𝐻. We say that 𝑃^𝑓_𝐶∶𝐻→2^𝐶 is a metric 𝑓-projection operator if 𝑃^𝑓_𝐶𝑥=𝑢∈𝐶∣(𝑢,𝑥)_𝑓=inf_𝜉∈𝐶(𝜉,𝑥)_𝑓,∀𝑥∈𝐻. (15)

Lemma 4 (see Li et al. [35, Lemma 3.1(ii)]). Let 𝐻 be a real Hilbert space, and let 𝜙≠𝐶⊂𝐻. Then, for every 𝑥∈𝐻, 𝑥=𝑃^𝑓_𝐶𝑥 if and only if ⟨𝑥−𝑦,𝑥−𝑥⟩+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶. (16)

Lemma 5 (see Li et al. [35, Lemma 3.2]). Let 𝐻 be a real Hilbert space, let 𝐶 be a nonempty, closed, and convex subset of 𝐻, and let 𝑥∈𝐻, 𝑥=𝑃^𝑓_𝐶𝑥. Then, 𝑦−𝑥²+(𝑥,𝑥)_𝑓⩽𝑦,𝑥_𝑓,∀𝑦∈𝐶. (17)

Lemma 6 (see Deimling [41]). Let 𝐻 be a real Hilbert space, and 𝑓∶𝐻→ℝ∪{+∞} be a lower semicontinuous convex functional. Then, there exist 𝑧∈𝐻 and 𝛼∈ℝ such that 𝑓(𝑥)≥⟨𝑥,𝑧⟩+𝛼,∀𝑥∈𝐻. (18)

Due to the properties of 𝑓, we have the motivation and ideas to create a new type of mappings which is general and covers a quasi-strict pseud-contraction as follows.

Definition 7. Let 𝐻 be a real Hilbert space, and a mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐻 is called quasi-strict 𝑓-pseudocontraction if the set of fixed points 𝐹(𝑇) is nonempty and if there exists a constant 0≤𝑘<1 such that, for each 𝑝∈𝐹(𝑇), 𝑝,𝑇𝑥_𝑓⩽𝑝,𝑥_𝑓+𝑘(𝑥,𝑇𝑥)_𝑓−2𝜌𝑓𝑝,∀𝑥∈𝐷(𝑇). (19)

It is obvious from the previous definition that (19) is equivalent to 𝑝−𝑇𝑥²≤𝑝−𝑥²+𝑘‖𝑥−𝑇𝑥‖²+2𝑘𝜌𝑓(𝑥)−𝑓𝑝,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (20)

Definition 8. A mapping 𝑇∶𝐶→𝐶 is said to be closed if, for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, 𝑥=𝑦.

Example 9. Let 𝑇∶𝐻→𝐻 be a mapping defined by 𝑇𝑥=(3/2)𝑥, for all 𝑥∈𝐻. Then, it is easy to see that 𝐹(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}={0}. Moreover, it is found that ‖0−𝑇𝑥‖²=32𝑥²=94‖𝑥‖²=‖𝑥‖²+54‖𝑥‖²≤‖𝑥‖²+32‖𝑥‖²=‖𝑥‖²+16+86‖𝑥‖²=‖0−𝑥‖²+233𝑥−2𝑥²2+23(1)‖𝑥‖²−‖0‖²=‖0−𝑥‖²+𝑘‖𝑥−𝑇𝑥‖²+2𝑘𝜌𝑓(𝑥)−𝑓(0), (21) for all 𝑥∈𝐻, where 𝑘=2/3, 𝜌=1, and 𝑓=‖⋅‖². Furthermore, if {𝑥_𝑛}⊂𝐻 such that 𝑥_𝑛→𝑥, then we have 𝑇𝑥_𝑛=(3/2)𝑥_𝑛→(3/2)𝑥 and 𝑇𝑥=(3/2)𝑥. This means that 𝑇 is closed and quasi-strict 𝑓-pseudocontraction.

For solving the equilibrium problem for a bifunction Θ∶𝐶×𝐶→ℝ, let us assume that Θ satisfies the following conditions:(A1)Θ(𝑥,𝑥)=0, for all 𝑥∈𝐶; (A2)Θ is monotone; that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶;(A3) for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0Θ(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤Θ(𝑥,𝑦);(A4) for each 𝑥∈𝐶, 𝑦→Θ(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 10 (see [27]). Let 𝐶 be a nonempty closed convex subset of 𝐻, and let Θ be a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑟> 0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that Θ+1𝑧,𝑦𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶. (22)

Lemma 11 (see [32]). Let 𝐶 be a closed convex subset of 𝐻 and let Θ be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟+1𝑥=𝑧∈𝐶∶Θ𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (23) for all 𝑥∈𝐶. Then, the following hold: (i)𝑇_𝑟 is single-valued; (ii)𝑇_𝑟 is firmly nonexpansive-type mapping; that is, for any 𝑥,𝑦∈𝐻, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑇_𝑟𝑥−𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(iii)𝐹(𝑇_𝑟)=𝐸𝑃(Θ); (iv)𝐸𝑃(Θ) is closed and convex;(v)‖𝑝−𝑇_𝑟𝑥‖²+‖𝑇_𝑟𝑥−𝑥‖²≤‖𝑝−𝑥‖², for all 𝑥∈𝐻,𝑝∈𝐹(𝑇_𝑟).

Lemma 12 (see [42]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Hilbert space 𝐻. Let 𝐴∶𝐶→𝐶 be a continuous and monotone mapping, let 𝜑∶𝐶→ℝ be a lower semicontinuous and convex function, and let Θ be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, then, there exists 𝑢∈𝐶 such that Θ+𝑦(𝑢)+1𝑢,𝑦𝐴𝑢,𝑦−𝑢+𝜑−𝜑𝑟𝑦−𝑢,𝑢−𝑥≥0,∀𝑦∈𝐶. (24) Define a mapping 𝐾_𝑟∶𝐶→𝐶 as follows: 𝐾_𝑟(𝑥)=+𝑦(𝑢)+1𝑢∈𝐶∶Θ𝑢,𝑦𝐴𝑢,𝑦−𝑢+𝜑−𝜑𝑟𝑦−𝑢,𝑢−𝑥≥0,∀𝑦∈𝐶, (25) for all 𝑥∈𝐶. Then, the following conclusions hold: (1)𝐾_𝑟 is single-valued; (2)𝐾_𝑟 is firmly nonexpansive type; that is, for all 𝑥,𝑦∈𝐸, ⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐾_𝑟𝑥−𝐾_𝑟𝑦⟩≤⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝑥−𝑦⟩;(3)𝐹(𝐾_𝑟)=𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑); (4)𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑) is closed and convex;(5)‖𝑝−𝐾_𝑟𝑥‖²+‖𝐾_𝑟𝑥−𝑥‖²≤‖𝑝−𝑥‖², for all 𝑥∈𝐻,𝑝∈𝐹(𝐾_𝑟).

Lemma 13. Let 𝐻 be a real Hilbert space, and let 𝐾∶𝐷(𝐾)→𝑅(𝐾) be a mapping. Then, the following are equivalent:(i)𝐾 is firmly nonexpansive (i.e.,‖𝐾𝑥−𝐾𝑦‖²≤⟨𝐾𝑥−𝐾𝑦,𝑥−𝑦⟩, for all 𝑥,𝑦∈𝐷(𝐾));(ii)‖𝐾𝑥−𝐾𝑦‖²≤‖𝑥−𝑦‖²−‖(𝐼−𝐾)𝑥−(𝐼−𝐾)𝑦‖², for all 𝑥,𝑦∈𝐷(𝐾).

Proof. For each 𝑥,𝑦∈𝐷(𝐾), we notice that 𝐾𝑥−𝐾𝑦²≤⟺𝐾𝑥−𝐾𝑦,𝑥−𝑦𝐾𝑥−𝐾𝑦²≤𝑥−𝑦²−𝑥−𝑦²++−𝑥−𝑦,𝐾𝑥−𝐾𝑦𝐾𝑥−𝐾𝑦,𝑥−𝑦𝐾𝑥−𝐾𝑦²⟺𝐾𝑥−𝐾𝑦²≤𝑥−𝑦²−()()𝑦+()()𝑦⟺𝑥−𝑦,𝐼−𝐾𝑥−𝐼−𝐾𝐾𝑥−𝐾𝑦,𝐼−𝐾𝑥−𝐼−𝐾𝐾𝑥−𝐾𝑦²≤𝑥−𝑦²−)()()()𝑦⟺(𝐼−𝐾𝑥−𝐼−𝐾𝑦,𝐼−𝐾𝑥−𝐼−𝐾𝐾𝑥−𝐾𝑦²≤𝑥−𝑦²−(𝐼−𝐾)𝑥−(𝐼−𝐾)𝑦². (26) The proof is complete.

The following lemma is important since it provides the significant inequality related to quasi-strict 𝑓-pseudocontractions and firmly nonexpansive mappings within the framework of Hilbert spaces.

Lemma 14. Let 𝐶 be a nonempty, closed, convex subset of real Hilbert spaces 𝐻. Let 𝑇∶𝐶→𝐶 be a quasi-strict 𝑓-pseudocontraction, and let 𝐾∶𝐶→𝐶 be a firmly nonexpansive mapping such that Ω∶=𝐹(𝑇)∩𝐹(𝐾)≠∅. Then, ‖𝑥−𝐾𝑇𝑥‖²+‖𝐾𝑇𝑥−𝑇𝑥‖²≤2+1−𝑘𝑥−𝑝,𝑥−𝑇𝑥+2𝑥−𝑝,𝑇𝑥−𝐾𝑇𝑥2𝑘𝜌𝑓(𝑥)𝑝,1−𝑘−𝑓 (27) for all 𝑥∈𝐶 and 𝑝∈Ω.

Proof. Let 𝑥∈𝐶 and 𝑝∈Ω. By the quasi-strict 𝑓-pseudocontractility of 𝑇, we have that 𝑝,𝑇𝑥_𝑓⩽𝑝,𝑥_𝑓+𝑘(𝑥,𝑇𝑥)_𝑓𝑝⟺−2𝜌𝑓𝑝−𝑇𝑥²≤𝑝−𝑥²+𝑘‖𝑥−𝑇𝑥‖²𝑓(𝑥)𝑝⟺+2𝑘𝜌−𝑓𝑝−𝑥²+2𝑝−𝑥,𝑥−𝑇𝑥+‖𝑥−𝑇𝑥‖²≤𝑝−𝑥²+𝑘‖𝑥−𝑇𝑥‖²𝑓(𝑥)𝑝⟺(‖+2𝑘𝜌−𝑓1−𝑘)‖𝑥−𝑇𝑥²𝑓(𝑥)𝑝⟺‖‖≤2𝑥−𝑝,𝑥−𝑇𝑥+2𝑘𝜌−𝑓𝑥−𝑇𝑥²≤2+1−𝑘𝑥−𝑝,𝑥−𝑇𝑥2𝑘𝜌𝑓(𝑥)𝑝.1−𝑘−𝑓 (28) It follows from Lemma 13 and (28) that 𝑝−𝑥²+‖𝑥−𝐾𝑇𝑥‖²=+2𝑝−𝑥,𝑥−𝐾𝑇𝑥𝑝−𝐾𝑇𝑥²≤𝑝−𝑇𝑥²−‖𝐾𝑇𝑥−𝑇𝑥‖²=𝑝,𝑇𝑥_𝑓−2𝜌𝑓𝑝−‖𝐾𝑇𝑥−𝑇𝑥‖²≤𝑝,𝑥_𝑓+𝑘(𝑥,𝑇𝑥)_𝑓𝑝𝑝−‖‖−2𝜌𝑓−2𝜌𝑓𝐾𝑇𝑥−𝑇𝑥²=𝑝,𝑥_𝑓−2𝜌𝑓𝑝+𝑘‖𝑥−𝑇𝑥‖²+2𝑘𝜌𝑓(𝑥)−𝑓𝑝−‖𝐾𝑇𝑥−𝑇𝑥‖²≤𝑝−𝑥²2+𝑘+1−𝑘𝑥−𝑝,𝑥−𝑇𝑥2𝑘𝜌𝑓(𝑥)𝑝𝑓(𝑥)𝑝−‖‖1−𝑘−𝑓+2𝑘𝜌−𝑓𝐾𝑇𝑥−𝑇𝑥²≤𝑝−𝑥²+2𝑘+1−𝑘𝑥−𝑝,𝑥−𝑇𝑥2𝑘²𝜌𝑓(𝑥)𝑝𝑓(𝑥)𝑝−‖‖1−𝑘−𝑓+2𝑘𝜌−𝑓𝐾𝑇𝑥−𝑇𝑥², (29) and, then, ‖𝑥−𝐾𝑇𝑥‖²+‖𝐾𝑇𝑥−𝑇𝑥‖²≤2𝑘+1−𝑘𝑥−𝑝,𝑥−𝑇𝑥+2𝑥−𝑝,𝑥−𝐾𝑇𝑥2𝑘²𝜌𝑓(𝑥)𝑝=1−𝑘+2𝑘𝜌−𝑓2𝑘𝑘1−𝑘𝑥−𝑝,𝑥−𝑇𝑥+2𝑥−𝑝,𝑥−𝑇𝑥+2𝑥−𝑝,𝑇𝑥−𝐾𝑇𝑥+2𝑘𝜌𝑓(𝑥)𝑝=21−𝑘+1−𝑓+1−𝑘𝑥−𝑝,𝑥−𝑇𝑥+2𝑥−𝑝,𝑇𝑥−𝐾𝑇𝑥2𝑘𝜌𝑓(𝑥)𝑝.1−𝑘−𝑓 (30) This completes the proof.

3. Main Result

In this section, some available properties of a quasi-strict 𝑓-pseudocontraction 𝑇 are used to prove that the set of fixed points 𝐹(𝑇) is closed and convex. An iterative shrinking metric 𝑓-projection method is provided in order to find a common fixed point of a quasi-strict 𝑓-pseudocontraction and a countable family of firmly nonexpansive mappings.

Lemma 15. Let 𝐶 be a nonempty, closed, convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a quasi-strict 𝑓-pseudocontraction. Then, the fixed point set 𝐹(𝑇) of 𝑇 is closed and convex.

Proof. Let {𝑝_𝑛} be a sequence in 𝐹(𝑇) such that 𝑝_𝑛→𝑝∈𝐶 as 𝑛→∞. It follows from (28) that 𝑝_𝑛,𝑇𝑝_𝑓⩽𝑝_𝑛,𝑝_𝑓+𝑘𝑝,𝑇𝑝_𝑓−2𝜌𝑓𝑝_𝑛⟺𝑝−𝑇𝑝²≤21−𝑘𝑝−𝑝_𝑛+,𝑝−𝑇𝑝2𝑘𝜌𝑓𝑝𝑝1−𝑘−𝑓_𝑛. (31) Taking limsup_𝑛→∞on both sides of (31), so we have that 𝑝−𝑇𝑝²=limsup_𝑛→∞𝑝−𝑇𝑝²≤limsup_𝑛→∞21−𝑘⟨𝑝−𝑝_𝑛+,𝑝−𝑇𝑝⟩2𝑘𝜌𝑓𝑝𝑝1−𝑘−𝑓_𝑛≤21−𝑘limsup_𝑛→∞⟨𝑝−𝑝_𝑛+,𝑝−𝑇𝑝⟩2𝑘𝜌1−𝑘limsup_𝑛→∞𝑓𝑝−𝑓𝑝_𝑛≤2𝑘𝜌1−𝑘limsup_𝑛→∞𝑓𝑝+limsup_𝑛→∞−𝑓𝑝_𝑛=2𝑘𝜌𝑓𝑝1−𝑘−liminf_𝑛→∞𝑓𝑝_𝑛≤0. (32) This means that 𝑝=𝑇𝑝.
We next show that 𝐹(𝑇) is convex. For arbitrary 𝑝₁,𝑝₂∈𝐹(𝑇) and 𝑡∈(0,1), we let 𝑝_𝑡=𝑡𝑝₁+(1−𝑡)𝑝₂. By the definition of 𝑇, we have that 𝑝₁,𝑇𝑝_𝑡𝑓≤𝑝₁,𝑝_𝑡𝑓+𝑘𝑝_𝑡,𝑇𝑝_𝑡𝑓−2𝜌𝑓𝑝₁,𝑝₂,𝑇𝑝_𝑡𝑓≤𝑝₂,𝑝_𝑡𝑓+𝑘𝑝_𝑡,𝑇𝑝_𝑡𝑓−2𝜌𝑓𝑝₂. (33) By (28), it is easy to see that (33) are equivalent to 𝑝_𝑡−𝑇𝑝_𝑡²≤2𝑝1−𝑘^𝑡−𝑝₁,𝑝_𝑡−𝑇𝑝_𝑡+2𝑘𝜌𝑓𝑝1−𝑘^𝑡−𝑓𝑝₁,𝑝_𝑡−𝑇𝑝_𝑡²≤2𝑝1−𝑘^𝑡−𝑝₂,𝑝_𝑡−𝑇𝑝_𝑡+2𝑘𝜌𝑓𝑝1−𝑘^𝑡−𝑓𝑝₂, (34)(35) respectively. Multiplying by 𝑡 and (1−𝑡) on both sides of (34) and (35), respectively, and adding the two inequalities, we have that 𝑝_𝑡−𝑇𝑝_𝑡²≤2𝑝1−𝑘^𝑡−𝑡𝑝₁+(1−𝑡)𝑝₂,𝑝_𝑡−𝑇𝑝_𝑡+2𝑘𝜌𝑓𝑝1−𝑘^𝑡−𝑡𝑓𝑝₁−(1−𝑡)𝑓𝑝₂=21−𝑘⟨𝑝_𝑡−𝑝_𝑡,𝑝_𝑡−𝑇𝑝_𝑡⟩+2𝑘𝜌𝑓1−𝑘𝑡𝑝₁+(1−𝑡)𝑝₂−𝑡𝑓𝑝₁−(1−𝑡)𝑓𝑝₂≤2𝑘𝜌𝑝1−𝑘𝑡𝑓₁+(1−𝑡)𝑓𝑝₂−𝑡𝑓𝑝₁−(1−𝑡)𝑓𝑝₂=0. (36) Hence, 𝑇𝑝_𝑡=𝑝_𝑡. This complete the proof.

Theorem 16. Let 𝐻 be a real Hilbert space, 𝐶 a nonempty, closed, be convex subset of 𝐸, let 𝑇 be a closed and quasi-strict 𝑓-pseudocontraction from 𝐶 into itself, and let {𝐾_𝑛}^∞_𝑛=1 be a countable family of firmly nonexpansive mappings from 𝐶 into itself which satisfies the 𝑁𝑆𝑇-condition such that Ω∶=𝐹(𝑇)∩⋂^∞_𝑛=1𝐹(𝐾_𝑛)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝐶∈𝐻,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃^𝑓_𝐶1𝑥₀,𝐶_𝑛+1=𝑧∈𝐶_𝑛∣𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛²+𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛²≤2𝑥1−𝑘^𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛+2𝑥_𝑛−𝑧,𝑇𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+2𝑘𝜌𝑓𝑥1−𝑘^𝑛(𝑧)𝑥−𝑓_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥₀. (37) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃^𝑓_Ω𝑥₀.

Proof. The proof is divided into six steps.
Step 1. Show that 𝐶_𝑛 is closed and convex, for all 𝑛≥1.
For 𝑛=1, 𝐶₁=𝐶 is closed and convex. Assume that 𝐶_𝑖 is closed and convex for some 𝑖∈ℕ. For 𝑧∈𝐶_𝑖+1, we have that 𝑥_𝑖−𝐾_𝑖𝑇𝑥_𝑖²+𝐾_𝑖𝑇𝑥_𝑖−𝑇𝑥_𝑖²≤2𝑥1−𝑘^𝑖−𝑧,𝑥_𝑖−𝑇𝑥_𝑖+2𝑥_𝑖−𝑧,𝑇𝑥_𝑖−𝐾𝑇𝑥_𝑖+2𝑘𝜌𝑓𝑥1−𝑘^𝑖−𝑓(𝑧). (38) It is not hard to see that the continuity and linearity of ⟨⋅,𝑥_𝑖−𝑇𝑥_𝑖⟩ and ⟨⋅,𝑇𝑥_𝑖−𝐾𝑇𝑥_𝑖⟩ together with the lower semicontinuity and convexity of 𝑓 allow 𝐶_𝑖+1 to be closed and convex. Then, for all 𝑛≥1, 𝐶_𝑛 is closed and convex.
Step 2. Show that Ω⊂⋂^∞_𝑛=1𝐶_𝑛∶=𝐷.
It is obvious that Ω∶=𝐹(𝑇)∩⋂^∞_𝑛=1𝐹(𝐾_𝑛)⊂𝐶=𝐶₁. Suppose that Ω⊂𝐶_𝑖 for some 𝑖∈ℕ. For any 𝑝∈Ω, we have 𝑝∈𝐶_𝑖, and by Lemma 14 we obtain that 𝑥_𝑖−𝐾_𝑖𝑇𝑥_𝑖²+𝐾_𝑖𝑇𝑥_𝑖−𝑇𝑥_𝑖²≤2𝑥1−𝑘^𝑖−𝑝,𝑥_𝑖−𝑇𝑥_𝑖+2𝑥_𝑖−𝑝,𝑇𝑥_𝑖−𝐾𝑇𝑥_𝑖+2𝑘𝜌𝑓𝑥1−𝑘^𝑖−𝑓𝑝. (39) This means that 𝑝∈𝐶_𝑖+1. By mathematical induction, Ω⊂𝐶_𝑛 for all 𝑛≥1. Therefore, Ω⊂⋂^∞_𝑛=1𝐶_𝑛∶=𝐷≠∅.
Step 3. Show that {𝑥_𝑛} is bounded and that the lim_𝑛→∞(𝑥_𝑛,𝑥₀)_𝑓 exists.
Since 𝑓∶𝑋→ℝ is a convex and lower semicontinuous mapping, applying Lemma 6, we see that there exist 𝑧∈𝐻 and 𝛼∈ℝ such that 𝑓𝑦≥𝑦,𝑧+𝛼,∀𝑦∈𝐻. (40) It follows that 𝑥_𝑛,𝑥_0𝑓=𝑥_𝑛²−2⟨𝑥_𝑛,𝑥₀⟩+𝑥₀²+2𝜌𝑓𝑥_𝑛≥𝑥_𝑛²−2𝑥_𝑛,𝑥₀+𝑥₀²+2𝜌𝑥_𝑛=𝑥,𝑧+2𝜌𝛼_𝑛2−2𝑥_𝑛,𝑥₀−𝜌𝑧+𝑥₀²≥𝑥+2𝜌𝛼_𝑛2−2𝑥₀−𝜌𝑧𝑥_𝑛+𝑥₀²=𝑥+2𝜌𝛼_𝑛−𝑥₀−𝜌𝑧²+𝑥₀²−𝑥₀−𝜌𝑧²+2𝜌𝛼. (41) Since 𝑥_𝑛=𝑃^𝑓_𝐶𝑛𝑥₀, it follows from (41) that 𝑥₀²−𝑥₀≤𝑥−𝜌𝑧+2𝜌𝛼_𝑛−𝑥₀−𝜌𝑧²+𝑥₀²−𝑥₀≤𝑥−𝜌𝑧+2𝜌𝛼_𝑛,𝑥_0𝑓=𝑃^𝑓_𝐶𝑛(𝑥₀),𝑥_0𝑓=inf_{𝜉∈𝐶𝑛}𝜉,𝑥_0𝑓≤𝑢,𝑥_0𝑓 (42) for each 𝑢∈Ω. This implies that {𝑥_𝑛} and (𝑥_𝑛,𝑥₀)_𝑓 are bounded. By the fact that 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛 and Lemma 5, we obtain that 𝑥_𝑛+1−𝑥_𝑛²+𝑥_𝑛,𝑥_0𝑓≤𝑥_𝑛+1,𝑥_0𝑓. (43) Since ‖𝑥_𝑛+1−𝑥_𝑛‖²≥0, {(𝑥_𝑛,𝑥₀)_𝑓} is nondecreasing. Therefore, the limit of {(𝑥_𝑛,𝑥₀)_𝑓} exists.
Step 4. Show that 𝑥_𝑛→𝑝 as 𝑛→∞, where 𝑝=𝑃^𝑓_𝐷𝑥₀.
Let {𝑥_𝑛𝑖}⊂{𝑥_𝑛}. From the boundedness of {𝑥_𝑛𝑖}, there exists {𝑥_𝑛𝑖𝑗}⊂{𝑥_𝑛𝑖} such that 𝑥_𝑛𝑖𝑗⇀𝑝as𝑗⟶∞. (44) Writing 𝑥_𝑗∶=𝑥_𝑛𝑖𝑗, it is easy to see that 𝑝∈𝐶_𝑗, where 𝐶_𝑗∶=𝐶_𝑛𝑖𝑗. Note that 𝑥_𝑗,𝑥_0𝑓=𝑃^𝑓𝐶_𝑗(𝑥₀),𝑥_0𝑓=min_𝜉∈𝐶_𝑗𝜉,𝑥_0𝑓≤𝑝,𝑥_0𝑓. (45) On the other hand, since 𝑥_𝑗⇀𝑝, so 𝑥_𝑗−𝑥₀⇀𝑝−𝑥₀, and, then, by weakly lower semicontinuity of ‖⋅‖² and 𝑓, we obtain that 𝑝−𝑥₀²≤liminf_𝑗→∞𝑥_𝑗−𝑥₀²,𝑓𝑝≤liminf_𝑗→∞𝑓𝑥_𝑗. (46)(47) Combining (46) and (47), we obtain that 𝑝,𝑥_0𝑓=𝑝−𝑥₀²+2𝜌𝑓𝑝≤liminf_𝑗→∞𝑥_𝑗−𝑥₀²+2𝜌liminf_𝑗→∞𝑓𝑥_𝑗≤liminf_𝑗→∞𝑥_𝑗−𝑥₀²+2𝜌𝑓𝑥_𝑗=liminf_𝑗→∞𝑥_𝑗,𝑥_0𝑓. (48) It follows from (45) and (48), that 𝑝,𝑥_0𝑓≤liminf_𝑗→∞𝑥_𝑗,𝑥_0𝑓≤limsup_𝑗→∞𝑥_𝑗,𝑥_0𝑓≤𝑝,𝑥_0𝑓, (49) and, then, lim_𝑗→∞𝑥_𝑗,𝑥_0𝑓=𝑝,𝑥_0𝑓. (50) Next, we consider that limsup_𝑗→∞𝑥_𝑗−𝑥₀²=limsup_𝑗→∞𝑥_𝑗,𝑥_0𝑓−2𝜌𝑓𝑥_𝑗≤limsup_𝑗→∞𝑥_𝑗,𝑥_0𝑓+limsup_𝑗→∞−2𝜌𝑓𝑥_𝑗=𝑝,𝑥_0𝑓−2𝜌liminf_𝑗→∞𝑓𝑥_𝑗≤𝑝,𝑥_0𝑓−2𝜌𝑓𝑝=𝑝−𝑥₀². (51) Combining (46) and (51), we obtain that 𝑝−𝑥₀²≤liminf_𝑗→∞𝑥_𝑗−𝑥₀²≤limsup_𝑗→∞𝑥_𝑗−𝑥₀²≤𝑝−𝑥₀², (52) and, then, lim_𝑗→∞𝑥_𝑗−𝑥₀²=𝑝−𝑥₀². (53) Note that 𝑓𝑥_𝑗=1𝑥2𝜌^𝑗,𝑥_0𝑓−𝑥_𝑗−𝑥₀². (54) Then, we have that limsup_𝑗→∞𝑓𝑥_𝑗=12𝜌limsup_𝑗→∞𝑥_𝑗,𝑥_0𝑓−𝑥_𝑗−𝑥₀²=12𝜌𝑝,𝑥_0𝑓−𝑝−𝑥₀²=𝑓𝑝. (55) Combining (47) and (55), we obtain that 𝑓𝑝≤liminf_𝑗→∞𝑓𝑥_𝑗≤limsup_𝑗→∞𝑓𝑥_𝑗=𝑓𝑝, (56) and, then, lim_𝑗→∞𝑓𝑥_𝑛𝑖𝑗=lim_𝑗→∞𝑓𝑥_𝑗=𝑓𝑝. (57) By Lemma 1, this implies that lim_𝑛→∞𝑓𝑥_𝑛=𝑓𝑝. (58) On the other hand, we note that 𝑥_𝑗−𝑝²=𝑥_𝑗−𝑥₀−𝑝−𝑥₀²=𝑥_𝑗−𝑥₀²−2𝑥_𝑗−𝑥₀,𝑝−𝑥₀+𝑝−𝑥₀². (59) It follows from (44) and (53) that lim_𝑗→∞𝑥_𝑗−𝑝²=lim_𝑗→∞𝑥_𝑗−𝑥₀²−2𝑥_𝑗−𝑥₀,𝑝−𝑥₀+𝑝−𝑥₀²=lim_𝑗→∞𝑥_𝑗−𝑥₀²−2lim_𝑗→∞𝑥_𝑗−𝑥₀,𝑝−𝑥₀+𝑝−𝑥₀²=𝑝−𝑥₀²−2𝑝−𝑥₀,𝑝−𝑥₀+𝑝−𝑥₀²=0. (60) Therefore, 𝑥_𝑛𝑖𝑗=𝑥_𝑗→𝑝 as 𝑗→∞. This implies by Lemma 1 that 𝑥_𝑛⟶𝑝as𝑛⟶∞. (61) Thus, 𝜔_𝑤𝑥_𝑛=𝑝. (62) It is easy to show that 𝑝∈𝐶_𝑛, for all 𝑛≥1. Hence, 𝑝∈⋂^∞_𝑛=1𝐶_𝑛=∶𝐷. Since 𝑥_𝑛=𝑃^𝑓_𝐶𝑛𝑥₀, so, by Lemma 4, we have that 𝑥_𝑛−𝑦,𝑥₀−𝑥_𝑛+𝜌𝑓𝑦−𝜌𝑓𝑥_𝑛≥0,∀𝑦∈𝐷. (63) Letting 𝑛→∞, so we obtain that ⟨𝑝−𝑦,𝑥₀−𝑝⟩+𝜌𝑓𝑦−𝜌𝑓𝑝≥0,∀𝑦∈𝐷, (64) which implies that 𝑝=𝑃^𝑓_𝐷𝑥₀.
Step 5. Show that 𝑝∈Ω.
Firstly, we prove that {𝑇𝑥_𝑛} and {𝐾_𝑛𝑇𝑥_𝑛} are bounded. Indeed, taking 𝑣∈Ω=𝐹(𝑇)∩⋂^∞_𝑛=1𝐹(𝐾_𝑛)and then by (28), we have that ‖𝑣‖²−2‖𝑣‖𝑇𝑥_𝑛+𝑇𝑥_𝑛²=‖𝑣‖−𝑇𝑥_𝑛²≤𝑣−𝑇𝑥_𝑛²≤𝑣−𝑥_𝑛²+𝑘𝑥_𝑛−𝑇𝑥_𝑛²+2𝑘𝜌𝑓𝑥_𝑛(𝑣)≤−𝑓𝑣−𝑥_𝑛²2+𝑘𝑥1−𝑘^𝑛−𝑣,𝑥_𝑛−𝑇𝑥_𝑛+2𝑘𝜌𝑓𝑥1−𝑘^𝑛(𝑣)𝑓𝑥−𝑓+2𝑘𝜌_𝑛(𝑣)≤−𝑓𝑣−𝑥_𝑛²+2𝑘𝑥1−𝑘^𝑛−𝑣,𝑥_𝑛−𝑇𝑥_𝑛𝑘+2𝑘𝜌𝑓𝑥1−𝑘+1_𝑛(𝑣)≤−𝑓𝑣−𝑥_𝑛²+2𝑘𝑥1−𝑘^𝑛−𝑣𝑥_𝑛+2𝑘𝑥1−𝑘^𝑛−𝑣𝑇𝑥_𝑛+2𝑘𝜌𝑓𝑥1−𝑘^𝑛−𝑓(𝑣). (65) By a simple calculation, we get that 𝑇𝑥_𝑛²≤𝑣−𝑥_𝑛²+2𝑘𝑥1−𝑘^𝑛−𝑣𝑥_𝑛−‖𝑣‖²+2𝑘𝜌𝑓𝑥1−𝑘^𝑛(𝑣)+−𝑓2𝑘𝑥1−𝑘^𝑛−𝑣+2‖𝑣‖𝑇𝑥_𝑛≤𝑀+𝑀𝑇𝑥_𝑛1=𝑀+22𝑀𝑇𝑥_𝑛1≤𝑀+2𝑀²+𝑇𝑥_𝑛²1=𝑀+2𝑀²+12𝑇𝑥_𝑛², (66) where 𝑀∶=sup{‖𝑣−𝑥_𝑛‖²+(2𝑘/1−𝑘)‖𝑥_𝑛−𝑣‖×‖𝑥_𝑛‖−‖𝑣‖²+(2𝑘𝜌/1−𝑘)(𝑓(𝑥_𝑛)−𝑓(𝑣))∣𝑛∈ℕ} and 𝑀∶=sup{(2𝑘/1−𝑘)‖𝑥_𝑛−𝑣‖+2‖𝑣‖∣𝑛∈ℕ}. So we have that 𝑇𝑥_𝑛²≤2𝑀+𝑀², (67) for all 𝑛∈ℕ. Therefore, {𝑇𝑥_𝑛} is bounded. Notice that, for each 𝑣∈Ω, 𝑣−𝐾_𝑛𝑇𝑥_𝑛≤𝑣−𝑇𝑥_𝑛, (68) for all 𝑛∈ℕ. Therefore, {𝐾_𝑛𝑇𝑥_𝑛} is also bounded. Moreover, we note that 𝑥_𝑛+1−𝑥_𝑛≤𝑥_𝑛+1−𝑝+𝑝−𝑥_𝑛⟶0as𝑛⟶∞. (69) Thus, by the fact that 𝑥_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1 and by (58), we obtain that 𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛²+𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛²≤2𝑥1−𝑘^𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑇𝑥_𝑛+2𝑥_𝑛−𝑥_𝑛+1,𝑇𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+2𝑘𝜌𝑓𝑥1−𝑘^𝑛−𝑓𝑥_𝑛+1⟶0as𝑛⟶∞. (70) This means that 𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛⟶0,𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛⟶0as𝑛⟶∞. (71) For this reason, we have that 𝑥_𝑛−𝑇𝑥_𝑛=𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛⟶0as𝑛⟶∞. (72) Next, we have that 𝑇𝑥_𝑛−𝑝≤𝑇𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝⟶0as𝑛⟶∞. (73) That is, 𝑇𝑥_𝑛→𝑝 as 𝑛→∞. It follows from the closed mapping of 𝑇, that 𝑇𝑝=𝑝; thus, 𝑝∈𝐹(𝑇).
On the other hand, let us consider that 𝑥_𝑛−𝐾_𝑛𝑥_𝑛≤𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+𝐾_𝑛𝑇𝑥_𝑛−𝐾_𝑛𝑥_𝑛≤𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑥_𝑛⟶0as𝑛⟶∞. (74) It follows from the NST-condition of {𝐾_𝑛} and (62), that 𝑝∈⋂^∞_𝑛=1𝐹(𝐾_𝑛). Therefore, 𝑥_𝑛→𝑝∈𝐹(𝑇)∩⋂^∞_𝑛=1𝐹(𝐾_𝑛)=Ω.
Step 6. Show that 𝑝=𝑃^𝑓_Ω𝑥₀.
Notice by Step 2 that Ω⊂𝐷; so we have that 𝑃^𝑓_Ω𝑥₀∈𝐷, and then by Step 5 it yields that 𝑝,𝑥_0𝑓=𝑃^𝑓_𝐷𝑥₀,𝑥_0𝑓=inf_𝜉∈𝐷𝜉,𝑥_0𝑓≤𝑃^𝑓_Ω𝑥₀,𝑥_0𝑓=inf_𝜁∈Ω𝜁,𝑥_0𝑓≤𝑝,𝑥_0𝑓. (75) This shows that (𝑃^𝑓_Ω𝑥₀,𝑥₀)_𝑓=(𝑝,𝑥₀)_𝑓. It follows from the uniqueness 𝑝=𝑃^𝑓_Ω𝑥₀. Then, {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_Ω𝑥₀. This completes the proof.

4. Deduced Theorems and Applications

In this section, some applications of the main theorem are provided in order to find some common solutions of problems in a Hilbert space.

If 𝑓=‖⋅‖², then (𝑥,𝑦)_‖⋅‖²=‖𝑥−𝑦‖²+2𝜌‖𝑥‖², for all (𝑥,𝑦)∈𝐶×𝐻, and 𝑃^𝑓_𝐶𝑛𝑥₀=𝑃^‖⋅‖2_𝐶𝑛𝑥₀, for all 𝑛∈ℕ; then, by Theorem 16, we obtain the following corollary.

Corollary 17. Let 𝐶, 𝐻, {𝐾_𝑛} be the same as in Theorem 16, and 𝑇 be a closed and quasi-strict ‖⋅‖²-pseudocontraction from 𝐶 into itself such that Ω∶=𝐹(𝑇)∩⋂^∞_𝑛=1𝐹(𝐾_𝑛)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝐶∈𝐻,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃^‖⋅‖2_𝐶1𝑥₀,𝐶_𝑛+1=𝑧∈𝐶_𝑛∣𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛²+𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛²≤2𝑥1−𝑘^𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛+2𝑥_𝑛−𝑧,𝑇𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛+2𝑘𝜌𝑥1−𝑘^𝑛2−‖𝑧‖²,𝑥_𝑛+1=𝑃^‖⋅‖2_𝐶𝑛+1𝑥₀. (76) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃^‖⋅‖2_Ω𝑥₀.

If 𝑓 is a constant function, saying that 𝑓=𝑎∈ℝ, then (𝑥,𝑦)_𝑎=‖𝑥−𝑦‖²+2𝜌𝑎, and it is not hard to see that 𝑇 coincides with a quasi-strict pseudocontraction. Thus, by Theorem 16, we obtain the following corollary.

Corollary 18. Let 𝐶, 𝐻, {𝐾_𝑛} be the same as in Theorem 16, and let 𝑇 be a closed and quasi-strict pseudocontraction from 𝐶 into itself such that Ω∶=𝐹(𝑇)∩⋂^∞_𝑛=1𝐹(𝐾_𝑛)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝐶∈𝐻,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃^𝑎_𝐶1𝑥₀,𝐶_𝑛+1=𝑧∈𝐶_𝑛∣𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛²+𝐾_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛²≤2𝑥1−𝑘^𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛+2𝑥_𝑛−𝑧,𝑇𝑥_𝑛−𝐾_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝑃^𝑎_𝐶𝑛+1𝑥₀. (77) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃^𝑎_Ω𝑥₀.

Let 𝐾∶𝐶→𝐶 be a firmly nonexpansive mapping and 𝐾_𝑛=𝐾 for all 𝑛∈ℕ. Then, it is not hard to verify that a family {𝐾_𝑛} satisfies NST-condition. Therefore, if 𝑓=𝑎=0, then Corollary 18 reduces to the following corollary.

Corollary 19. Let 𝐶, 𝐻, 𝑇 be the same as in Corollary 18, and let 𝐾 be a firmly nonexpansive mapping from 𝐶 into itself such that Ω∶=𝐹(𝑇)∩𝐹(𝐾)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝐶∈𝐻,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃_𝐶1𝑥₀,𝐶_𝑛+1=𝑧∈𝐶_𝑛∣𝑥_𝑛−𝐾𝑇𝑥_𝑛²+𝐾𝑇𝑥_𝑛−𝑇𝑥_𝑛²≤2𝑥1−𝑘^𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛+2𝑥_𝑛−𝑧,𝑇𝑥_𝑛−𝐾𝑇𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀. (78) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑥₀.

Finally, we provide some applications of the main theorem to find a common solution of fixed point problems of a closed and quasi-strict 𝑓-pseudocontraction and generalized mixed equilibrium problems via an iterative shrinking metric 𝑓-projection method within the framework of Hilbert spaces.

Theorem 20. Let 𝐶, 𝐻, 𝑇 be the same as in Theorem 16, let Θ be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), let 𝜑 be a lower semicontinuous and convex function, and let 𝐴∶𝐶→𝐻 be a continuous and monotone mapping such that Ω∶=𝐹(𝑇)∩𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝐶∈𝐻,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃^𝑓_𝐶1𝑥₀,𝑢_𝑛∈𝐶suchthatΘ𝑢_𝑛,𝑦+⟨𝐴𝑢_𝑛,𝑦−𝑢_𝑛𝑦𝑢⟩+𝜑−𝜑_𝑛+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑇𝑥_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝐶_𝑛∣𝑥_𝑛−𝑢_𝑛²+𝑢_𝑛−𝑇𝑥_𝑛²≤2𝑥1−𝑘^𝑛−𝑧,𝑥_𝑛−𝑇𝑥_𝑛+2𝑥_𝑛−𝑧,𝑇𝑥_𝑛−𝑢_𝑛+2𝑘𝜌𝑓𝑥1−𝑘^𝑛(𝑧),𝑥−𝑓_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥₀, (79) where 𝑘∈[0,1) and 𝑟_𝑛>0, for all 𝑛∈ℕ, with liminf_𝑛→∞𝑟_𝑛>0. Then, {𝑥_𝑛} converges strongly to 𝑃^𝑓_Ω(𝑥₀).

Proof. By Lemma 12(3) it is not hard to see that ⋂^∞_𝑛=1𝐹(𝐾_𝑟𝑛)=𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑). From the definition of {𝑥_𝑛}, it is easy to see that 𝑢_𝑛=𝐾_𝑟𝑛𝑇𝑥_𝑛, and by (62) we have that 𝜔_𝑤(𝑥_𝑛)={𝑝}. Next, we will show that a countable family {𝐾_𝑟𝑛} satisfies the NST-condition. It is sufficient to show that 𝑝∈𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑). It follows from (61) and (71) that 𝑢_𝑛=𝐾_𝑟𝑛𝑇𝑥_𝑛→𝑝 as 𝑛→∞. Define Φ∶𝐶×𝐶→ℝ by Φ(𝑥,𝑦)=Θ(𝑥,𝑦)+⟨𝐴𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥), for all 𝑥,𝑦∈𝐶. It is not hard to verify that Φ satisfies conditions (A1)–(A4). By (A2), we have that 1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥Φ𝑦,𝑢_𝑛,∀𝑦∈𝐶. (80) By using (A4) and liminf_𝑛→∞𝑟_𝑛>0, we obtain, that 0≥Φ(𝑦,𝑝), for all 𝑦∈𝐶. For 𝑡∈(0,1] and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. So, from (A1) and (A4), we have that 0=Φ𝑦_𝑡,𝑦_𝑡=Φ𝑦_𝑡()𝑝𝑦,𝑡𝑦+1−𝑡≤𝑡Φ_𝑡,𝑦+(1−𝑡)Φ𝑦_𝑡,𝑝≤𝑡Φ𝑦_𝑡,𝑦. (81) Dividing by 𝑡, we have that Φ𝑦_𝑡,𝑦≥0,∀𝑦∈𝐶. (82) From (A3), we have that 0≤lim_𝑡→0Φ(𝑦_𝑡,𝑦)=lim_𝑡→0Φ(𝑡𝑦+(1−𝑡)𝑝,𝑦)≤Φ(𝑝,𝑦), for all 𝑦∈𝐶, and, hence, 𝑝∈𝐺𝑀𝐸𝑃(Θ,𝐴,𝜑)=⋂^∞_𝑛=1𝐹(𝐾_𝑟𝑛), so 𝜔_𝑤(𝑥_𝑛)⊂⋂^∞_𝑛=1𝐹(𝐾_𝑟𝑛). Therefore, {𝐾_𝑟𝑛}^∞_𝑛=1 satisfies NST-condition. Applying Theorem 16, we conclude that 𝑥_𝑛→𝑃^𝑓_Ω𝑥₀.

Acknowledgment

The authors would like to thank the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand, for supporting this paper.