Approximation of Solutions of an Equilibrium Problem in a Banach Space

Abstract

An equilibrium problem is investigated based on a hybrid projection iterative algorithm. Strong convergence theorems for solutions of the equilibrium problem are established in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property.

1. Introduction

Equilibrium problems which were introduced by Fan [1] and Blum and Oettli [2] have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. It has been shown [3–8] that equilibrium, problems include variational inequalities, fixed point, the Nash equilibrium, and game theory as special cases. A number of iterative algorithms have recently been studying for fixed point and equilibrium problems, see [9–26] and the references therein. However, there were few results established in the framework of the Banach spaces. In this paper, we suggest and analyze a projection iterative algorithm for finding solutions of equilibrium in a Banach space.

2. Preliminaries

In what follows, we always assume that 𝐸 is a Banach space with the dual space 𝐸^∗. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. We use the symbol 𝐽 to stand for the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by𝐽𝑥=𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²=‖𝑓^∗‖²,∀𝑥∈𝐸,(2.1)

where ⟨⋅,⋅⟩ denotes the generalized duality pairing of elements between 𝐸 and 𝐸^∗.

Let 𝑈_𝐸={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝑈_𝐸 with 𝑥≠𝑦. It is said to be uniformly convex if for any 𝜖∈(0,2] there exists 𝛿>0 such that for any 𝑥,𝑦∈𝑈_𝐸,‖𝑥−𝑦‖≥𝜖implies‖‖‖𝑥+𝑦2‖‖‖≤1−𝛿.(2.2)

It is known that a uniformly convex Banach space is reflexive and strictly convex; for details see [27] and the references therein.

Recall that a Banach space 𝐸 is said to have the Kadec-Klee property if a sequence {𝑥_𝑛} of 𝐸 satisfies that 𝑥_𝑛⇀𝑥∈𝐶, where ⇀ denotes the weak convergence, and ‖𝑥_𝑛‖→‖𝑥‖, where → denotes the strong convergence, and then 𝑥_𝑛→𝑥. It is known that if 𝐸 is uniformly convex, then 𝐸 enjoys the Kadec-Klee property; for details see [26] and the references therein.

𝐸 is said to be smooth provided lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for all 𝑥,𝑦∈𝑈_𝐸. It is also said to be uniformly smooth if the limit is attained uniformly for all 𝑥,𝑦∈𝑈_𝐸.

It is well known that if 𝐸^∗ is strictly convex, then 𝐽 is single valued; if 𝐸^∗ is reflexive, and smooth, then 𝐽 is single valued and demicontinuous; for more details see [27, 28] and the references therein.

It is also well known that if 𝐷 is a nonempty, closed, and convex subset of a Hilbert space 𝐻, and 𝑃_𝐷∶𝐻→𝐷 is the metric projection from 𝐻 onto 𝐷, then 𝑃_𝐷 is nonexpansive. This fact actually characterizes the Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [29] introduced a generalized projection operator Π_𝐷 in the Banach spaces which is an analogue of the metric projection in the Hilbert spaces.

Let 𝐸 be a smooth Banach space. Consider the functional defined by𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(2.3) Notice that, in a Hilbert space 𝐻, (2.3) is reduced to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² for all 𝑥,𝑦∈𝐻. The generalized projection Π_𝐶∶𝐸→𝐶 is a mapping that is assigned to an arbitrary point 𝑥∈𝐸, the minimum point of the functional 𝜙(𝑥,𝑦); that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the following minimization problem:𝜙𝑥,𝑥=min_𝑦∈𝐶𝜙(𝑦,𝑥).(2.4)

The existence and uniqueness of the operator Π_𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and the strict monotonicity of the mapping 𝐽; see, for example, [27, 28]. In the Hilbert spaces, Π_𝐶=𝑃_𝐶. It is obvious from the definition of the function 𝜙 that𝜙(𝑥,(𝑦‖)𝑦=‖𝜙−(‖𝑥𝑥,‖𝑧))²+≤𝜙𝜙(𝑧(,𝑦𝑦,)𝑥+)≤2⟨(𝑥‖𝑦−‖𝑧+,𝐽‖𝑧𝑥−‖)𝐽²𝑦,⟩,∀𝑥∀,𝑦𝑥,∈𝑦,𝐸𝑧,∈𝐸.((22..56))

Let 𝑇∶𝐶→𝐶 be a mapping. Recall that a point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). 𝑇 is said to be relatively nonexpansive if𝐹(𝑇)=𝐹(𝑇)≠∅,𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇).(2.7)

The asymptotic behavior of a relatively nonexpansive mapping was studied in [27, 29, 30].

Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ, where ℝ denotes the set of real numbers. In this paper, we consider the following equilibrium problem. Find 𝑝∈𝐶 such that𝑓(𝑝,𝑦)≥0,∀𝑦∈𝐶.(2.8) We use EP(𝑓) to denote the solution set of the equilibrium problem (2.3). That is,EP(𝑓)={𝑝∈𝐶∶𝑓(𝑝,𝑦)≥0,∀𝑦∈𝐶}.(2.9)

Given a mapping 𝑄∶𝐶→𝐸^∗, let𝑓(𝑥,𝑦)=⟨𝑄𝑥,𝑦−𝑥⟩,∀𝑥,𝑦∈𝐶.(2.10)

Then 𝑝∈EP(𝑓) if and only if 𝑝 is a solution of the following variational inequality. Find 𝑝 such that⟨𝑄𝑝,𝑦−𝑝⟩≥0,∀𝑦∈𝐶.(2.11)

To study the equilibrium problem (2.8), we may assume that 𝑓 satisfies the following conditions:

(A1) 𝑓(𝑥,𝑥)=0,forall𝑥∈𝐶;

(A2) 𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶;

(A3)limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦),∀𝑥,𝑦,𝑧∈𝐶;(2.12)

(A4) for each 𝑥∈𝐶, 𝑦↦𝑓(𝑥,𝑦) is convex and weakly lower semicontinuous.

In this paper, we study the problem of approximating solutions of equilibrium problem (2.8) based on a hybrid projection iterative algorithm in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property. To prove our main results, we need the following lemmas.

Lemma 2.1. Let 𝐸 be a strictly convex and uniformly smooth Banach space and 𝐶 a nonempty, closed, and convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐸. Then (a)(see [2]). There exists 𝑧∈𝐶 such that𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.13)(b)(see [31]). Define a mapping 𝑇^𝑓𝑟∶𝐸→𝐶 by𝑇^𝑓𝑟𝑥=𝑧∈𝐶∶𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩,∀𝑦∈𝐶.(2.14) Then the following conclusions hold:(1)𝑇^𝑓𝑟 is single valued;(2)𝑇^𝑓𝑟 is a firmly nonexpansive-type mapping; that is, for all 𝑥,𝑦∈𝐸,𝑇^𝑓𝑟𝑥−𝑇^𝑓𝑟𝑦,𝐽𝑇^𝑓𝑟𝑥−𝐽𝑇^𝑓𝑟𝑦≤𝑇^𝑓𝑟𝑥−𝑇^𝑓𝑟𝑦,𝐽𝑥−𝐽𝑦;(2.15)(3)𝐹(𝑇^𝑓𝑟)=EP(𝑓); (4)EP(𝑓) is closed and convex;(5)𝑇^𝑓𝑟 is relatively nonexpansive.

Lemma 2.2 (see [29]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝐶 a nonempty, closed, and convex subset of 𝐸. Let 𝑥∈𝐸, and 𝑥₀∈𝐶. Then 𝑥₀=Π_𝐶𝑥 if and only if

⟨𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀⟩≥0,∀𝑦∈𝐶.(2.16)

Lemma 2.3 (see [29]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝐶 a nonempty, closed, and convex subset of 𝐸, and 𝑥∈𝐸. Then

𝜙𝑦,Π_𝐶𝑥+𝜙Π_𝐶𝑥,𝑥≤𝜙(𝑦,𝑥),∀𝑦∈𝐶.(2.17)

Lemma 2.4 (see [27]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space. Then one has the following

𝜙(𝑥,𝑦)=0⟺𝑥=𝑦,∀𝑥,𝑦∈𝐸.(2.18)

3. Main Results

Theorem 3.1. Let 𝐸 be a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property and 𝐶 a nonempty, closed, and convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) such that EP(𝑓)≠∅. Let {𝑥_𝑛} be a sequence generated by the following manner: 𝑥₀∈𝐸cChosenarbitrarily,₁𝑥=𝐶,₁𝑦_𝑛=Π_𝐶1𝑥₀,∈𝐶,suchthat𝑓𝑦_𝑛,𝑢+1𝑟_𝑛𝐶⟨𝑢−𝑦_𝑛,𝐽𝑦_𝑛−𝐽𝑥_𝑛⟩≥0,∀𝑢∈𝐶,_𝑛+1=𝑢∈𝐶𝑥_𝑛∶2⟨𝑥_𝑛−𝑢,𝐽𝑥_𝑛−𝐽𝑦_𝑛⟩≥𝜙𝑥_𝑛,𝑦_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1,(3.1) where {𝑟_𝑛} is a real number sequence in [𝑟,∞), where 𝑟 is some positive real number. Then the sequence {𝑥_𝑛} converges strongly to 𝑥=Π_EP(𝑓)𝑥₀.

Proof. In view of Lemma 2.1, we see that EP(𝑓) is closed and convex. Next, we show that 𝐶_𝑛 is closed and convex. It is not hard to see that 𝐶_𝑛 is closed. Therefore, we only show that 𝐶_𝑛 is convex. It is obvious that 𝐶₁=𝐶 is convex. Suppose that 𝐶_ℎ is convex for some ℎ∈ℕ. Next, we show that 𝐶_ℎ+1 is also convex for the same ℎ. Let 𝑎,𝑏∈𝐶_ℎ+1 and 𝑐=𝑡𝑎+(1−𝑡)𝑏, where 𝑡∈(0,1). It follows that 𝜙𝑥_ℎ,𝑦_ℎ≤2⟨𝑥_ℎ−𝑎,𝐽𝑥_ℎ−𝐽𝑦_ℎ⟩,𝜙𝑥_ℎ,𝑦_ℎ≤2⟨𝑥_ℎ−𝑏,𝐽𝑥_ℎ−𝐽𝑦_ℎ⟩,(3.2)where 𝑎,𝑏∈𝐶_ℎ. From the above two inequalities, we can get that 𝜙𝑥_ℎ,𝑦_ℎ≤2⟨𝑥_ℎ−𝑐,𝐽𝑥_ℎ−𝐽𝑦_ℎ⟩,(3.3)where 𝑐∈𝐶_ℎ. It follows that 𝐶_ℎ+1 is closed and convex. This completes the proof that 𝐶_𝑛 is closed, and convex.
Next, we show that EP(𝑓)⊂𝐶_𝑛. It is obvious that EP(𝑓)⊂𝐶=𝐶₁. Suppose that EP(𝑓)⊂𝐶_ℎ for some ℎ∈ℕ. For any 𝑧∈EP(𝑓)⊂𝐶_ℎ, we see from Lemma 2.1 that𝜙𝑧,𝑦_ℎ≤𝜙𝑧,𝑥_ℎ.(3.4) On the other hand, we obtain from (2.6) that 𝜙𝑧,𝑦_ℎ=𝜙𝑧,𝑥_ℎ+𝜙𝑥_ℎ,𝑦_ℎ+2⟨𝑧−𝑥_ℎ,𝐽𝑥_ℎ−𝐽𝑦_ℎ⟩.(3.5) Combining (3.4) with (3.5), we arrive at 2⟨𝑥_ℎ−𝑧,𝐽𝑥_ℎ−𝐽𝑦_ℎ⟩≥𝜙𝑥_ℎ,𝑦_ℎ(3.6)which implies that 𝑧∈𝐶_ℎ+1. This shows that EP(𝑓)⊂𝐶_ℎ+1. This completes the proof that EP(𝑓)⊂𝐶_𝑛.
Next, we show that {𝑥_𝑛} is a convergent sequence and strongly converges to 𝑥, where 𝑥∈EP(𝑓). Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we see from Lemma 2.2 that⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑧∈𝐶_𝑛.(3.7) It follows from EP(𝑓)⊂𝐶_𝑛 that ⟨𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑤∈EP(𝑓).(3.8) By virtue of Lemma 2.3, we obtain that 𝜙𝑥_𝑛,𝑥₀=≤𝜙𝜙ΠΠ_𝐶𝑛𝑥_0EP(𝑓),𝑥₀≤𝜙Π_EP(𝑓)𝑥₀,𝑥₀−𝜙Π_EP(𝑓)𝑥₀,𝑥_𝑛𝑥₀,𝑥₀.(3.9) This implies that the sequence {𝜙(𝑥_𝑛,𝑥₀)} is bounded. It follows from (2.5) that the sequence {𝑥_𝑛} is also bounded. Since the space is reflexive, we may assume that 𝑥_𝑛⇀𝑥. Since 𝐶_𝑛 is closed and convex, we see that 𝑥∈𝐶_𝑛. On the other hand, we see from the weakly lower semicontinuity of the norm that 𝜙𝑥,𝑥₀=‖‖𝑥‖‖²−2𝑥,𝐽𝑥₀+‖‖𝑥₀‖‖²≤=lliimmiinnf_𝑛→∞f_𝑛→∞‖‖𝑥^𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑥₀⟩+‖‖𝑥₀‖‖²≤limsup_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝜙𝑥_𝑛,𝑥₀𝑥,𝑥₀,(3.10)which implies that 𝜙(𝑥_𝑛,𝑥₀)→𝜙(𝑥,𝑥₀) as 𝑛→∞. Hence, ‖𝑥_𝑛‖→‖𝑥‖ as 𝑛→∞. In view of the Kadec-Klee property of 𝐸, we see that 𝑥_𝑛→𝑥 as 𝑛→∞. Notice that 𝑥_𝑛+1=Π_EP(𝑓)𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛. It follows that 𝜙𝑥_𝑛+1,𝑥_𝑛=≤𝜙𝜙𝑥𝑥_𝑛+1𝑛+1,Π_𝐶𝑛𝑥₀=𝜙𝑥_𝑛+1,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀,𝑥₀−𝜙𝑥_𝑛,𝑥₀.(3.11) Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛, we arrive at 𝜙(𝑥_𝑛,𝑥₀)≤𝜙(𝑥_𝑛+1,𝑥₀). This shows that {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. It follows from the boundedness that lim_𝑛→∞𝜙(𝑥_,𝑥₀) exists. It follows that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.12) By virtue of 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, we find that 𝜙𝑥_𝑛,𝑦_𝑛≤2𝑥_𝑛−𝑥_𝑛+1,𝐽𝑥_𝑛−𝐽𝑦_𝑛.(3.13) It follows that lim_𝑛→∞𝜙𝑥_𝑛,𝑦_𝑛=0.(3.14) In view of (2.5), we see that lim_𝑛→∞‖‖𝑥_𝑛‖‖−‖‖𝑦_𝑛‖‖=0.(3.15) Since 𝑥_𝑛→𝑥, we find that lim_𝑛→∞‖‖𝑦_𝑛‖‖=‖‖𝑥‖‖.(3.16) It follows that lim_𝑛→∞‖‖𝐽𝑦_𝑛‖‖=‖‖𝐽𝑥‖‖.(3.17) This implies that {𝐽𝑦_𝑛} is bounded. Note that both 𝐸 and 𝐸^∗ are reflexive. We may assume that 𝐽𝑦_𝑛⇀𝑦^∗∈𝐸^∗. In view of the reflexivity of 𝐸, we see that there exists an element 𝑦∈𝐸 such that 𝐽𝑦=𝑦^∗. It follows that 𝜙𝑥_𝑛,𝑦_𝑛=‖‖𝑥_𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑦_𝑛⟩+‖‖𝑦_𝑛‖‖²=‖‖𝑥_𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑦_𝑛⟩+‖‖𝐽𝑦_𝑛‖‖².(3.18) Taking liminf_𝑛→∞ on the both sides of the equality above yields that 0≥‖‖𝑥‖‖²−2⟨𝑥,𝑦^∗⟩+‖𝑦^∗‖²=‖‖𝑥‖‖²−2⟨𝑥,𝐽𝑦⟩+‖𝐽𝑦‖²=‖‖𝑥‖‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²=𝜙𝑥,𝑦.(3.19) That is, 𝑥=𝑦, which in turn implies that 𝑦^∗=𝐽𝑥. It follows that 𝐽𝑦_𝑛⇀𝐽𝑥∈𝐸^∗. Since 𝐸^∗ enjoys the Kadec-Klee property, we obtain from (3.17) that lim_𝑛→∞𝐽𝑦_𝑛=𝐽𝑥. Since 𝐽⁻¹∶𝐸^∗→𝐸 is demicontinuous, we find that 𝑦_𝑛⇀𝑥. This implies from (3.16) and the Kadec-Klee property of 𝐸 that lim_𝑛→∞𝑦_𝑛=𝑥. This in turn implies that lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0. Since 𝐽 is uniformly norm-to-norm continuous on any bounded sets, we find that lim_𝑛→∞‖‖𝐽𝑦_𝑛−𝐽𝑥_𝑛‖‖=0.(3.20) Next, we show that 𝑥∈𝐸𝐹(𝑓). In view of Lemma 2.1, we find from 𝑦_𝑛=𝑇^𝑓𝑟_𝑛𝑥_𝑛 that 𝑓𝑦_𝑛,𝑢+1𝑟_𝑛⟨𝑢−𝑦_𝑛,𝐽𝑦_𝑛−𝐽𝑥_𝑛⟩≥0,∀𝑢∈𝐶.(3.21) It follows from condition (A2) and (3.20) that 1𝑟_𝑛‖‖𝑢−𝑦_𝑛‖‖‖‖𝐽𝑦_𝑛−𝐽𝑥_𝑛‖‖≥𝑓𝑢,𝑦_𝑛,∀𝑢∈𝐶.(3.22) In view of condition (A4), we obtain from (3.17) that 𝑓𝑢,𝑥≤0,∀𝑢∈𝐶.(3.23) For 0<𝑡<1 and 𝑢∈𝐶, define 𝑢_𝑡=𝑡𝑢+(1−𝑡)𝑥. It follows that 𝑢_𝑡∈𝐶, which yields that 𝑓(𝑢_𝑡,𝑥)≤0. It follows from conditions (A1) and (A4) that 0=𝑓𝑢_𝑡,𝑢_𝑡≤𝑡𝑓𝑢_𝑡,𝑢+(1−𝑡)𝑓𝑢_𝑡,𝑥≤𝑡𝑓𝑢_𝑡,𝑢.(3.24) That is, 𝑓𝑢_𝑡,𝑢≥0.(3.25) Letting 𝑡↓0, we find from condition (A3) that 𝑓(𝑥,𝑢)≥0, forall𝑢∈𝐶. This implies that 𝑥∈EP(𝑓). This shows that 𝑥∈EP(𝑓).
Finally, we prove that 𝑥=Π_EP(𝑓)𝑥₀. Letting 𝑛→∞ in (3.8), we see that𝑥−𝑤,𝐽𝑥₀−𝐽𝑥≥0,∀𝑤∈EP(𝑓).(3.26) In view of Lemma 2.2, we can obtain that 𝑥=Π_EP(𝑓)𝑥₀. This completes the proof.

In the framework of the Hilbert spaces, we have the following.

Corollary 3.2. Let 𝐸 be a Hilbert space and 𝐶 a nonempty, closed, and convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) such that EP(𝑓)≠∅. Let {𝑥_𝑛} be a sequence generated by the following manner: 𝑥₀∈𝐸c𝐶hosenarbitrarily,₁𝑥=𝐶,₁𝑦_𝑛=𝑃_𝐶1𝑥₀,∈𝐶,suchthat𝑓𝑦_𝑛,𝑢+1𝑟_𝑛𝐶⟨𝑢−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛⟩≥0,∀𝑢∈𝐶,_𝑛+1=𝑢∈𝐶𝑥^𝑛∶_𝑛+12⟨𝑥_𝑛−𝑢,𝑥_𝑛−𝑦_𝑛⟩≥‖‖𝑥_𝑛−𝑦_𝑛‖‖²,=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥1,(3.27) where {𝑟_𝑛} is a real number sequence in [𝑟,∞), where 𝑟 is some positive real number. Then the sequence {𝑥_𝑛} converges strongly to 𝑥=𝑃_EP(𝑓)𝑥₀.