Abstract

We introduce an iterative process for finding an element in the common fixed point sets of two continuous pseudocontractive mappings. As a consequence, we provide an approximation method for a common fixed point of a finite family of pseudocontractive mappings. Furthermore, our convergence theorem is applied to a convex minimization problem. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

1. Introduction

Let 𝐻 be a real Hilbert space. A mapping 𝑇 with domain 𝐷(𝑇)βŠ‚π» and range 𝑅(𝑇) in 𝐻 is called pseudocontractive if for each π‘₯,π‘¦βˆˆπ·(𝑇) we haveβŸ¨π‘‡π‘₯βˆ’π‘‡π‘¦,π‘₯βˆ’π‘¦βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2.(1.1)𝑇 is called strongly pseudocontractive if there exists π‘˜βˆˆ(0,1) such that ⟨π‘₯βˆ’π‘¦,𝑇π‘₯βˆ’π‘‡π‘¦βŸ©β‰€π‘˜β€–π‘₯βˆ’π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπ·(𝑇),(1.2) and 𝑇 is said to be π‘˜-strict pseudocontractive if there exists a constant 0β‰€π‘˜<1 such that⟨π‘₯βˆ’π‘¦,𝑇π‘₯βˆ’π‘‡π‘¦βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2βˆ’π‘˜β€–(πΌβˆ’π‘‡)π‘₯βˆ’(πΌβˆ’π‘‡)𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπ·(𝑇).(1.3) The operator 𝑇 is called Lipschitzian if there exists 𝐿β‰₯0 such that ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€πΏβ€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπ·(𝑇). If 𝐿=1, then 𝑇 is called nonexpansive, and if 𝐿∈[0,1), then 𝑇 is called a contraction. As a result of Kato [1], it follows from inequality (1.1) that 𝑇 is pseudocontractive if and only if the inequalityβ€–β€–π‘₯βˆ’π‘¦β€–β‰€β€–(1+𝑑)(π‘₯βˆ’π‘¦)βˆ’π‘‘(𝑇π‘₯βˆ’π‘‡π‘¦)(1.4) holds for each π‘₯,π‘¦βˆˆπ·(𝑇) and for all 𝑑>0.

Apart from being an important generalization of nonexpansive, strongly pseudocontractive and π‘˜-strict pseudocontractive mappings, interest in pseudocontractive mappings stems mainly from their firm connection with the important class of nonlinear accretive operators, where a mapping 𝐴 with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝐻 is called accretive if the inequalityβ€–β€–π‘₯βˆ’π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦+𝑠(𝐴π‘₯βˆ’π΄π‘¦)(1.5) holds for every π‘₯,π‘¦βˆˆπ·(𝐴) and for all 𝑠>0. We observe that 𝐴 is accretive if and only if π‘‡βˆΆ=πΌβˆ’π΄ is pseudocontractive, and thus a zero of 𝐴, 𝑁(𝐴)∢={π‘₯∈𝐷(𝐴)∢𝐴π‘₯=0}, is a fixed point of 𝑇, 𝐹(𝑇)∢={π‘₯∈𝐷(𝑇)βˆΆπ‘‡π‘₯=π‘₯}. It is now well known that if 𝐴 is accretive then the solutions of the equation 𝐴π‘₯=0 correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts have been devoted to iterative methods for approximating fixed points of 𝑇 when 𝑇 is pseudocontractive (see, e.g., [2–4] and the references contained therein).

Construction of fixed points of nonexpansive mappings via Mann's algorithm [5] has extensively been investigated recently in the literature (see, e.g., [6, 7] and references therein). Related works can also be found in [7–18]. Mann's algorithm is defined by π‘₯0∈𝐾 andπ‘₯𝑛+1=𝛼𝑛π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0,(1.6) where {𝛼𝑛} is a real control sequence in the interval (0,1). If 𝑇 is a nonexpansive mapping with a fixed point and if the control sequence {𝛼𝑛} is chosen so that βˆ‘βˆžπ‘›=0𝛼𝑛(1βˆ’π›Όπ‘›)=∞, then the sequence {π‘₯𝑛} generated by Mann's algorithm (1.6) converges weakly to a fixed point of 𝑇 (this is indeed true in a uniformly convex Banach space with a FrΓ©chet differentiable norm [7]). However, this convergence is in general not strong (see the counterexample in [19]; see also [20]).

For a sequence {𝛼𝑛} of real numbers in (0,1) and an arbitrary π‘’βˆˆπΆ, let the sequence {π‘₯𝑛} in 𝐾 be iteratively defined by π‘₯0∈𝐾 andπ‘₯𝑛+1∢=𝛼𝑛+1𝑒+1βˆ’π›Όπ‘›+1𝑇π‘₯𝑛,𝑛β‰₯0,(1.7) where 𝑇 is a nonexpansive mapping of 𝐢 into itself. Halpern [11] was the first to study the convergence of Algorithm (1.7) in the framework of Hilbert spaces. Lions [14] and Wittmann [21] improved the result of Halpern by proving strong convergence of {π‘₯𝑛} to a fixed point of 𝑇 if the real sequence {𝛼𝑛} satisfies certain conditions. Reich [22], Shioji and Takahashi [16], and Zegeye and Shahzad [23] extend the result of Wittmann [21] to the case of Banach space.

In 2000, Moudafi [24] introduced viscosity approximation method and proved that if 𝐻 is a real Hilbert space, for given π‘₯0∈𝐢, the sequence {π‘₯𝑛} generated by the algorithmπ‘₯𝑛+1∢=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π‘₯𝑛,𝑛β‰₯0,(1.8) where π‘“βˆΆπΆβ†’πΆ is a contraction mapping and {𝛼𝑛}βŠ‚(0,1) satisfies certain conditions, converges strongly to a common fixed point of 𝑇. Moudafi [24] generalizes Halpern’s theorems in the direction of viscosity approximations. In [25], Zegeye et al. extended Moudafi's result to the class of Lipschitz pseudocontractive mappings in Banach spaces more general than Hilbert spaces. Viscosity approximations are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations.

Our concern now is the following. Is it possible to construct a viscosity approximation sequence that converges strongly to a fixed point of pseudocontractive mappings more general than nonexpansive mappings?

In this paper, motivated and inspired by the work of Halpern [11], Moudafi [24], and the methods of Takahashi and Zembayashi [26], we introduce a viscosity approximation method for finding a common fixed point of two continuous pseudocontractive mappings. As a consequence, we provide an approximation method for a common fixed point of finite family of pseudocontractive mappings. This provides affirmative answer to the above concern. Furthermore, we apply our convergence theorem to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators.

2. Preliminaries

Let 𝐢 be closed and convex subset of a real Hilbert space 𝐻. For every point π‘₯∈𝐻, there exists a unique nearest point in 𝐢, denoted by 𝑃𝐢π‘₯, such that β€–β€–π‘₯βˆ’π‘ƒπΆπ‘₯‖‖≀‖π‘₯βˆ’π‘¦β€–,βˆ€π‘¦βˆˆπΆ.(2.1)𝑃𝐢 is called the metric projection of 𝐻 onto 𝐢. We know that 𝑃𝐢 is a nonexpansive mapping of 𝐻 onto 𝐢. In connection with metric projection, we have the following lemma.

Lemma 2.1. Let 𝐢 be a nonempty convex subset of a Hilbert space 𝐻. Let π‘₯∈𝐻 and π‘₯0∈𝐢. Then, π‘₯0=𝑃𝐢π‘₯ if and only if βŸ¨π‘§βˆ’π‘₯0,π‘₯0βˆ’π‘₯⟩β‰₯0,βˆ€π‘§βˆˆπΆ.(2.2)

Lemma 2.2 (see [27]). Let {π‘Žπ‘›} be a sequence of nonnegative real numbers satisfying the following relation: π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+πœŽπ‘›,𝑛β‰₯0,(2.3) where (i)  {𝛾𝑛}βŠ‚[0,1], βˆ‘π›Ύπ‘›=∞ and (ii)  limsupπ‘›β†’βˆžπœŽπ‘›/𝛾𝑛≀0 or βˆ‘|πœŽπ‘›|<∞. Then, π‘Žπ‘›β†’0 as π‘›β†’βˆž.

By a similar argument in [28], we have the following lemma.

Lemma 2.3. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space H. Let π΄βˆΆπΆβ†’π» be a continuous accretive mapping. Then, for π‘Ÿ>0 and π‘₯∈𝐻, there exists π‘§βˆˆπΆ such that 1βŸ¨π‘¦βˆ’π‘§,π΄π‘§βŸ©+π‘ŸβŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.4)

Moreover, by a similar argument of the proof of Lemmas 2.8 and 2.9 of [26], we get the following lemma.

Lemma 2.4. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π΄βˆΆπΆβ†’π» be a continuous accretive mapping. For π‘Ÿ>0 and π‘₯∈𝐻, define a mapping π‘‡π‘ŸβˆΆπ»β†’πΆ as follows: π‘‡π‘Ÿξ‚†1π‘₯∢=π‘§βˆˆπΆβˆΆβŸ¨π‘¦βˆ’π‘§,π΄π‘§βŸ©+π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(2.5) for all π‘₯∈𝐻. Then, the following hold: (1)π‘‡π‘Ÿ is single valued;(2)π‘‡π‘Ÿ is firmly nonexpansive type mapping, that is, for all π‘₯,π‘¦βˆˆπ», β€–β€–π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦β€–β€–2β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π‘₯βˆ’π‘¦βŸ©;(2.6)(3)𝐹(π‘‡π‘Ÿ)=VI(𝐢,𝐴);(4)VI(𝐢,𝐴) is closed and convex.

3. Main Results

In the sequel, we will make use of the following lemmas.

Lemma 3.1. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space H. Let π‘‡βˆΆπΆβ†’π» be a continuous pseudocontractive mapping. Then, for π‘Ÿ>0 and π‘₯∈𝐻, there exists π‘§βˆˆπΆ such that 1βŸ¨π‘¦βˆ’π‘§,π‘‡π‘§βŸ©βˆ’π‘ŸβŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿ)π‘§βˆ’π‘₯βŸ©β‰€0,βˆ€π‘¦βˆˆπΆ.(3.1)

Proof. Let π‘₯∈𝐻 and π‘Ÿ>0. Let 𝐴∢=πΌβˆ’π‘‡, where 𝐼 is the identity mapping on 𝐢. Then, clearly 𝐴 is continuous accretive mapping. Thus, by Lemma 2.3, there exists π‘§βˆˆπΆ such that βŸ¨π‘¦βˆ’π‘§,π΄π‘§βŸ©+(1/π‘Ÿ)βŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0, for all π‘¦βˆˆπΆ. But this is equivalent to βŸ¨π‘¦βˆ’π‘§,π‘‡π‘§βŸ©βˆ’(1/π‘Ÿ)βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿ)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπΆ. Hence, the lemma holds.

Lemma 3.2. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π‘‡βˆΆπΆβ†’πΆ be continuous pseudocontractive mapping. For π‘Ÿ>0 and π‘₯∈𝐻, define a mapping πΉπ‘ŸβˆΆπ»β†’πΆ as follows: πΉπ‘Ÿξ‚†1π‘₯∢=π‘§βˆˆπΆβˆΆβŸ¨π‘¦βˆ’π‘§,π‘‡π‘§βŸ©βˆ’π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿ)π‘§βˆ’π‘₯βŸ©β‰€0,βˆ€π‘¦βˆˆπΆ(3.2) for all π‘₯∈𝐻. Then, the following hold: (1)πΉπ‘Ÿ is single valued;(2)πΉπ‘Ÿ is firmly nonexpansive type mapping, that is, for all π‘₯,π‘¦βˆˆπ», β€–β€–πΉπ‘Ÿπ‘₯βˆ’πΉπ‘Ÿπ‘¦β€–β€–2β‰€βŸ¨πΉπ‘Ÿπ‘₯βˆ’πΉπ‘Ÿπ‘¦,π‘₯βˆ’π‘¦βŸ©;(3.3)(3)𝐹(πΉπ‘Ÿ)=𝐹(𝑇);(4)𝐹(T) is closed and convex.

Proof. We note that βŸ¨π‘¦βˆ’π‘§,π‘‡π‘§βŸ©βˆ’(1/π‘Ÿ)βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿ)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπΆ, is equivalent to βŸ¨π‘¦βˆ’π‘§,π΄π‘§βŸ©+(1/π‘Ÿ)βŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0, for all π‘¦βˆˆπΆ, where 𝐴∢=πΌβˆ’π‘‡ is continuous accretive mapping and 𝐼 the identity mapping on 𝐢. Moreover, as 𝑇 is self-map, we have that VI(𝐢,𝐴)=𝐹(𝑇). Thus, by Lemma 2.4, the conclusions of (1)–(4) hold.

Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π‘‡π‘–βˆΆπΆβ†’πΆ, for 𝑖=1,2, be continuous pseudocontractive mappings. Then, in what follows, π‘‡π‘Ÿπ‘›,πΉπ‘Ÿπ‘›βˆΆπ»β†’πΆ are defined as follows. For π‘₯∈𝐻 and {π‘Ÿπ‘›}βŠ‚(0,∞), define π‘‡π‘Ÿπ‘›ξ‚»π‘₯∢=π‘§βˆˆπΆβˆΆβŸ¨π‘¦βˆ’π‘§,𝑇11π‘§βŸ©βˆ’π‘Ÿπ‘›ξ«ξ€·π‘¦βˆ’π‘§,1+π‘Ÿπ‘›ξ€Έξ¬ξ‚Ό,πΉπ‘§βˆ’π‘₯≀0,βˆ€π‘¦βˆˆπΆπ‘Ÿπ‘›ξ‚»π‘₯∢=π‘§βˆˆπΆβˆΆβŸ¨π‘¦βˆ’π‘§,𝑇21π‘§βŸ©βˆ’π‘Ÿπ‘›ξ«ξ€·π‘¦βˆ’π‘§,1+π‘Ÿπ‘›ξ€Έξ¬ξ‚Ό.π‘§βˆ’π‘₯≀0,βˆ€π‘¦βˆˆπΆ(3.4) Now, we prove our main convergence theorem.

Theorem 3.3. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π‘‡π‘–βˆΆπΆβ†’πΆ, for 𝑖=1,2, be continuous pseudocontractive mappings such that β‹‚πΉβˆΆ=2𝑖=1𝐹(𝑇𝑖)β‰ βˆ…. Let 𝑓 be a contraction of 𝐢 into itself, and let {π‘₯𝑛} be a sequence generated by π‘₯1∈𝐢 and π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘Ÿπ‘›πΉπ‘Ÿπ‘›π‘₯𝑛,(3.5) where {𝛼𝑛}βŠ‚[0,1] and {π‘Ÿπ‘›}βŠ‚(0,∞) such that limπ‘›β†’βˆžπ›Όπ‘›=0, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, and βˆ‘βˆžπ‘›=1|π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›|<∞. Then, the sequence {π‘₯𝑛}𝑛β‰₯1 converges strongly to π‘§βˆˆπΉ, where 𝑧=𝑃𝐹𝑓(𝑧).

Proof. Let 𝑄=𝑃𝐹. Then, 𝑄𝑓 is a contraction of 𝐢 into 𝐢. In fact, we have that ‖𝑄𝑓(π‘₯)βˆ’π‘„π‘“(𝑦)‖≀‖𝑓(π‘₯)βˆ’π‘“(𝑦)‖≀𝛽‖π‘₯βˆ’π‘¦β€–,(3.6) for all π‘₯,π‘¦βˆˆπΆ, where 𝛽 is contraction constant of 𝑓. So 𝑄𝑓 is a contraction of 𝐢 into itself. Since 𝐢 is closed subset of 𝐻, there exists a unique element 𝑧 of 𝐢 such that 𝑧=𝑄𝑓(𝑧).
Let π‘£βˆˆπΉ, and let π‘’π‘›βˆΆ=π‘‡π‘Ÿπ‘›π‘€π‘›, where π‘€π‘›βˆΆ=πΉπ‘Ÿπ‘›π‘₯𝑛. Then, we have from Lemma 3.2 that ‖‖𝑒𝑛‖‖=β€–β€–π‘‡βˆ’π‘£π‘Ÿπ‘›π‘€π‘›βˆ’π‘‡π‘Ÿπ‘›π‘£β€–β€–β‰€β€–β€–π‘€π‘›β€–β€–βˆ’π‘£=β€–πΉπ‘Ÿπ‘›π‘₯π‘›βˆ’πΉπ‘Ÿπ‘›β€–β€–π‘₯𝑣‖≀𝑛‖‖.βˆ’π‘£(3.7) Moreover, from (3.5) and (3.7), we get that β€–β€–π‘₯𝑛+1β€–β€–=β€–β€–π›Όβˆ’π‘£π‘›ξ€·π‘“ξ€·π‘₯𝑛+ξ€·βˆ’π‘£1βˆ’π›Όπ‘›π‘‡ξ€Έξ€·π‘Ÿπ‘›πΉπ‘Ÿπ‘›π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘£β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯𝑛‖‖+ξ€·βˆ’π‘£1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘’π‘›β€–β€–βˆ’π‘£β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯𝑛‖‖+ξ€·βˆ’π‘£1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘£β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘“(𝑣)+𝛼𝑛‖𝑓(𝑣)βˆ’π‘£β€–+1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘£β‰€π›Όπ‘›π›½β€–β€–π‘₯π‘›β€–β€–βˆ’π‘£+𝛼𝑛‖𝑓(𝑣)βˆ’π‘£β€–+1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯𝑛‖‖=ξ€·βˆ’π‘£1βˆ’(1βˆ’π›½)𝛼𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘£+(1βˆ’π›½)𝛼𝑛1ξ‚Άξ‚»β€–β€–π‘₯1βˆ’π›½β€–π‘“(𝑣)βˆ’π‘£β€–β‰€max𝑛‖‖,1βˆ’π‘£ξ‚Ό.1βˆ’π›½β€–π‘“(𝑣)βˆ’π‘£β€–(3.8) By induction, we get that β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘£β‰€max1β€–β€–,1βˆ’π‘£ξ‚Ό1βˆ’π›½β€–π‘“(𝑣)βˆ’π‘£β€–,𝑛β‰₯1.(3.9) Therefore, {π‘₯𝑛} is bounded. Consequently, we get that {𝑀𝑛}, {π‘‡π‘Ÿπ‘›π‘€π‘›}, {πΉπ‘Ÿπ‘›π‘₯𝑛}, and {𝑓(π‘₯𝑛)} are bounded. Next, we show that β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0. But from (3.5) we have that β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=‖‖𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘’π‘›βˆ’π›Όπ‘›βˆ’1𝑓π‘₯π‘›βˆ’1ξ€Έβˆ’ξ€·1βˆ’π›Όπ‘›βˆ’1ξ€Έπ‘’π‘›βˆ’1‖‖≀‖‖𝛼𝑛𝑓π‘₯π‘›ξ€Έβˆ’π›Όπ‘›π‘“ξ€·π‘₯π‘›βˆ’1ξ€Έ+𝛼𝑛𝑓π‘₯π‘›βˆ’1ξ€Έβˆ’π›Όπ‘›βˆ’1𝑓π‘₯π‘›βˆ’1ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘’π‘›βˆ’ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘’π‘›βˆ’1+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘’π‘›βˆ’1βˆ’ξ€·1βˆ’π›Όπ‘›βˆ’1ξ€Έπ‘’π‘›βˆ’1‖‖≀𝛼𝑛𝛽‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝐾+1βˆ’π›Όπ‘›ξ€Έβ‹…β€–β€–π‘’π‘›βˆ’π‘’π‘›βˆ’1‖‖≀𝛼𝑛𝛽‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝐾+1βˆ’π›Όπ‘›ξ€Έβ‹…β€–β€–π‘€π‘›βˆ’π‘€π‘›βˆ’1β€–β€–,(3.10) where 𝐾=2sup{‖𝑓(π‘₯𝑛)β€–+β€–π‘’π‘›β€–βˆΆπ‘›βˆˆN}. Moreover, since 𝑀𝑛=πΉπ‘Ÿπ‘›π‘₯𝑛 and 𝑀𝑛+1=πΉπ‘Ÿπ‘›+1π‘₯𝑛+1, we get that βŸ¨π‘¦βˆ’π‘€π‘›,𝑇2𝑀𝑛1βŸ©βˆ’π‘Ÿπ‘›ξ«π‘¦βˆ’π‘€π‘›,ξ€·1+π‘Ÿπ‘›ξ€Έπ‘€π‘›βˆ’π‘₯𝑛≀0,βˆ€π‘¦βˆˆπΆ,(3.11)ξ«π‘¦βˆ’π‘€π‘›+1,𝑇2𝑀𝑛+1ξ¬βˆ’1π‘Ÿπ‘›+1ξ«π‘¦βˆ’π‘€π‘›+1,ξ€·1+π‘Ÿπ‘›+1𝑀𝑛+1βˆ’π‘₯𝑛+1≀0,βˆ€π‘¦βˆˆπΆ.(3.12) Putting π‘¦βˆΆ=𝑀𝑛+1 in (3.11) and π‘¦βˆΆ=𝑀𝑛 in (3.12), we get that 𝑀𝑛+1βˆ’π‘€π‘›,𝑇2π‘€π‘›ξ¬βˆ’1π‘Ÿπ‘›ξ«π‘€π‘›+1βˆ’π‘€π‘›,ξ€·1+π‘Ÿπ‘›ξ€Έπ‘€π‘›βˆ’π‘₯𝑛≀0,(3.13)ξ«π‘€π‘›βˆ’π‘€π‘›+1,𝑇2𝑀𝑛+1ξ¬βˆ’1π‘Ÿπ‘›+1ξ«π‘€π‘›βˆ’π‘€π‘›+1,ξ€·1+π‘Ÿπ‘›+1𝑀𝑛+1βˆ’π‘₯𝑛+1≀0.(3.14) Adding (3.13) and (3.14), we have βŸ¨π‘€π‘›+1βˆ’π‘€π‘›,𝑇2π‘€π‘›βˆ’π‘‡2𝑀𝑛+1ξ„”π‘€βŸ©βˆ’π‘›+1βˆ’π‘€π‘›,ξ€·1+π‘Ÿπ‘›ξ€Έπ‘€π‘›βˆ’π‘₯π‘›π‘Ÿπ‘›βˆ’ξ€·1+π‘Ÿπ‘›+1𝑀𝑛+1βˆ’π‘₯𝑛+1π‘Ÿπ‘›+1≀0,(3.15) which implies that 𝑀𝑛+1βˆ’π‘€π‘›,𝑀𝑛+1βˆ’π‘‡2𝑀𝑛+1ξ€Έβˆ’ξ€·π‘€π‘›βˆ’π‘‡2π‘€π‘›βˆ’ξƒ‘π‘€ξ€Έξ¬π‘›+1βˆ’π‘€π‘›,π‘€π‘›βˆ’π‘₯π‘›π‘Ÿπ‘›βˆ’π‘€π‘›+1βˆ’π‘₯𝑛+1π‘Ÿπ‘›+1≀0.(3.16) Now, using the fact that 𝑇2 is pseudocontractive, we get that 𝑀𝑛+1βˆ’π‘€π‘›,π‘€π‘›βˆ’π‘₯π‘›π‘Ÿπ‘›βˆ’π‘€π‘›+1βˆ’π‘₯𝑛+1π‘Ÿπ‘›+1ξƒ’β‰₯0,(3.17) and hence 𝑀𝑛+1βˆ’π‘€π‘›,π‘€π‘›βˆ’π‘€π‘›+1+𝑀𝑛+1βˆ’π‘₯π‘›βˆ’π‘Ÿπ‘›π‘Ÿπ‘›+1𝑀𝑛+1βˆ’π‘₯𝑛+1ξ€Έξƒ’β‰₯0.(3.18) Without loss of generality, let us assume that there exists a real number 𝑏 such that π‘Ÿπ‘›>𝑏>0 for all π‘›βˆˆN. Then, we have ‖‖𝑀𝑛+1βˆ’π‘€π‘›β€–β€–2≀𝑀𝑛+1βˆ’π‘€π‘›,π‘₯𝑛+1βˆ’π‘₯𝑛+ξ‚΅π‘Ÿ1βˆ’π‘›π‘Ÿπ‘›+1𝑀𝑛+1βˆ’π‘₯𝑛+1≀‖‖𝑀𝑛+1βˆ’π‘€π‘›β€–β€–ξƒ―β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||ξ‚΅π‘Ÿ1βˆ’π‘›π‘Ÿπ‘›+1ξ‚Ά||||⋅‖‖𝑀𝑛+1βˆ’π‘₯𝑛+1β€–β€–ξƒ°,(3.19) and hence from (3.19) we obtain that ‖‖𝑀𝑛+1βˆ’π‘€π‘›β€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+1π‘Ÿπ‘›+1||π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›||⋅‖‖𝑀𝑛+1βˆ’π‘₯𝑛+1‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+1𝑏||π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›||𝐿,(3.20) where 𝐿=sup{β€–π‘€π‘›βˆ’π‘₯π‘›β€–βˆΆπ‘›βˆˆN}. Furthermore, from (3.10) and (3.20), we have that β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀𝛼𝑛𝛽‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝐾+ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+1𝑏||π‘Ÿπ‘›βˆ’π‘Ÿπ‘›βˆ’1||𝐿=ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯(1βˆ’π›½)π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–||𝛼+πΎπ‘›βˆ’π›Όπ‘›βˆ’1||+ξ€·1βˆ’π›Όπ‘›ξ€ΈπΏπ‘||π‘Ÿπ‘›βˆ’π‘Ÿπ‘›βˆ’1||.(3.21) Now, using conditions of {𝛼𝑛}, {π‘Ÿπ‘›} and Lemma 2.2, we have that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=0.(3.22) Consequently, from (3.20) and (3.22), we obtain that limπ‘›β†’βˆžβ€–β€–π‘€π‘›+1βˆ’π‘€π‘›β€–β€–=0.(3.23) Similarly, taking 𝑒𝑛=π‘‡π‘Ÿπ‘›π‘€π‘› and 𝑒𝑛+1=π‘‡π‘Ÿπ‘›+1𝑀𝑛+1 and following the method used for 𝑀𝑛, we get that limπ‘›β†’βˆžβ€–π‘’π‘›+1βˆ’π‘’π‘›β€–=0. Furthermore, since π‘₯𝑛=π›Όπ‘›βˆ’1𝑓(π‘₯π‘›βˆ’1)+(1βˆ’π›Όπ‘›βˆ’1)π‘’π‘›βˆ’1, we have that β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘’π‘›βˆ’1β€–β€–+β€–β€–π‘’π‘›βˆ’1βˆ’π‘’π‘›β€–β€–β‰€π›Όπ‘›βˆ’1‖‖𝑓π‘₯π‘›βˆ’1ξ€Έβˆ’π‘’π‘›βˆ’1β€–β€–+β€–β€–π‘’π‘›βˆ’1βˆ’π‘’π‘›β€–β€–.(3.24) Thus, since 𝛼𝑛→0, we obtain that β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–βŸΆ0.(3.25) Moreover, for π‘£βˆˆπΉ, using Lemma 3.2, we get that β€–β€–π‘€π‘›β€–β€–βˆ’π‘£2=β€–β€–πΉπ‘Ÿπ‘›π‘₯π‘›βˆ’πΉπ‘Ÿπ‘›π‘£β€–β€–2β‰€ξ«πΉπ‘Ÿπ‘›π‘₯π‘›βˆ’πΉπ‘Ÿπ‘›π‘£,π‘₯π‘›ξ¬βˆ’π‘£=βŸ¨π‘€π‘›βˆ’π‘£,π‘₯𝑛=1βˆ’π‘£βŸ©2ξ‚€β€–β€–π‘€π‘›β€–β€–βˆ’π‘£2+β€–β€–π‘₯π‘›β€–β€–βˆ’π‘£2βˆ’β€–β€–π‘₯π‘›βˆ’π‘€π‘›β€–β€–2,(3.26) and hence β€–β€–π‘€π‘›β€–β€–βˆ’π‘£2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘£2βˆ’β€–β€–π‘₯π‘›βˆ’π‘€π‘›β€–β€–2.(3.27) Therefore, from (3.5), the convexity of β€–β‹…β€–2, (3.7) and (3.27) we get that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘£2=‖‖𝛼𝑛𝑓(π‘₯𝑛)+(1βˆ’π›Όπ‘›)π‘’π‘›β€–β€–βˆ’π‘£2≀𝛼𝑛‖‖𝑓(π‘₯𝑛‖‖)βˆ’π‘£2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘’π‘›β€–β€–βˆ’π‘£2≀𝛼𝑛‖‖𝑓(π‘₯𝑛‖‖)βˆ’π‘£2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘€π‘›β€–β€–βˆ’π‘£2≀𝛼𝑛‖‖𝑓(π‘₯𝑛‖‖)βˆ’π‘£2+ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘£2βˆ’β€–β€–π‘₯π‘›βˆ’π‘€π‘›β€–β€–2≀𝛼𝑛‖‖𝑓(π‘₯𝑛‖‖)βˆ’π‘£2+β€–β€–π‘₯π‘›β€–β€–βˆ’π‘£2βˆ’ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘€π‘›β€–β€–2,(3.28) and hence ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘€π‘›β€–β€–2≀𝛼𝑛‖‖𝑓(π‘₯𝑛‖‖)βˆ’π‘£2+β€–β€–π‘₯π‘›β€–β€–βˆ’π‘£2βˆ’β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘£2≀𝛼𝑛‖‖𝑓(π‘₯𝑛‖‖)βˆ’π‘£2+β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–ξ€·β€–β€–π‘₯𝑛‖‖+β€–β€–π‘₯βˆ’π‘£π‘›+1β€–β€–ξ€Έ.βˆ’v(3.29) So we have β€–π‘₯π‘›βˆ’π‘€π‘›β€–β†’0 as π‘›β†’βˆž. This implies with (3.25) that β€–π‘’π‘›βˆ’π‘€π‘›β€–β‰€β€–π‘’π‘›βˆ’π‘₯𝑛‖+β€–π‘₯π‘›βˆ’π‘€π‘›β€–β†’0 as π‘›β†’βˆž.
Next, we show that limsupπ‘›β†’βˆžβŸ¨π‘“(𝑧)βˆ’π‘§,π‘₯π‘›βˆ’π‘§βŸ©β‰€0,(3.30) where 𝑧=𝑃𝐹𝑓(𝑧). To show this inequality, we choose a subsequence {π‘₯𝑛𝑖} of {π‘₯𝑛} such that limsupπ‘›β†’βˆžβŸ¨π‘“(𝑧)βˆ’π‘§,π‘₯π‘›βˆ’π‘§βŸ©=limπ‘–β†’βˆžξ«π‘“(𝑧)βˆ’π‘§,π‘₯𝑛𝑖.βˆ’π‘§(3.31) Since {π‘₯𝑛𝑖} is bounded, there exists a subsequence {π‘₯𝑛𝑖𝑗} of {π‘₯𝑛𝑖} and π‘€βˆˆπ» such that π‘₯𝑛𝑖𝑗⇀𝑀. Without loss of generality, we may assume that π‘₯𝑛𝑖⇀𝑀. Since {π‘₯𝑛𝑖}βŠ‚πΆ and 𝐢 is convex and closed, we get that π‘€βˆˆπΆ. Moreover, since π‘₯π‘›βˆ’π‘€π‘›β†’0 as π‘›β†’βˆž, we have that 𝑀𝑛𝑖⇀𝑀. Now, we show that π‘€βˆˆπΉ. Note that, from the definition of 𝑀𝑛𝑖, we have ξ«π‘¦βˆ’π‘€π‘›π‘–,𝑇2π‘€π‘›π‘–ξ¬βˆ’1π‘Ÿπ‘›iξ«π‘¦βˆ’π‘€π‘›π‘–,ξ€·π‘Ÿπ‘›π‘–ξ€Έπ‘€+1π‘›π‘–βˆ’π‘₯𝑛𝑖≀0,βˆ€π‘¦βˆˆπΆ.(3.32) Put 𝑧𝑑=𝑑𝑣+(1βˆ’π‘‘)𝑀 for all π‘‘βˆˆ(0,1] and π‘£βˆˆπΆ. Consequently, we get that π‘§π‘‘βˆˆπΆ. From (3.32) and pseudocontractivity of 𝑇2, it follows that βŸ¨π‘€π‘›π‘–βˆ’π‘§π‘‘,𝑇2π‘§π‘‘ξ«π‘€βŸ©β‰₯π‘›π‘–βˆ’π‘§π‘‘,𝑇2𝑧𝑑+zπ‘‘βˆ’π‘€π‘›π‘–,𝑇2π‘€π‘›π‘–ξ¬βˆ’1π‘Ÿπ‘›π‘–ξ«π‘§π‘‘βˆ’π‘€π‘›π‘–,ξ€·1+π‘Ÿπ‘›π‘–ξ€Έπ‘€π‘›π‘–βˆ’π‘₯𝑛𝑖𝑧=βˆ’π‘‘βˆ’π‘€π‘›π‘–,𝑇2π‘§π‘‘βˆ’π‘‡2π‘€π‘›π‘–ξ¬βˆ’1π‘Ÿπ‘›π‘–ξ«π‘§π‘‘βˆ’π‘€π‘›π‘–,π‘€π‘›π‘–βˆ’π‘₯π‘›π‘–ξ¬βˆ’ξ«π‘§π‘‘βˆ’π‘€π‘›π‘–,𝑀𝑛𝑖‖‖𝑧β‰₯βˆ’π‘‘βˆ’π‘€π‘›π‘–β€–β€–2βˆ’1π‘Ÿπ‘›π‘–ξ«π‘§π‘‘βˆ’π‘€π‘›π‘–,π‘€π‘›π‘–βˆ’π‘₯π‘›π‘–ξ¬βˆ’ξ«π‘§π‘‘βˆ’π‘€π‘›π‘–,𝑀𝑛=ξ«π‘€π‘›π‘–βˆ’π‘§π‘‘,π‘§π‘‘ξ¬βˆ’ξ„”π‘§π‘‘βˆ’π‘€π‘›π‘–,π‘€π‘›π‘–βˆ’π‘₯π‘›π‘–π‘Ÿπ‘›π‘–ξ„•.(3.33) Then, since π‘€π‘›βˆ’π‘₯𝑛→0, as π‘›β†’βˆž, we obtain that (π‘€π‘›π‘–βˆ’π‘₯𝑛𝑖)/π‘Ÿπ‘›π‘–β†’0 as π‘–β†’βˆž. Thus, as π‘–β†’βˆž, it follows that βŸ¨π‘€βˆ’π‘§π‘‘,𝑇2π‘§π‘‘βŸ©β‰₯βŸ¨π‘€βˆ’π‘§π‘‘,π‘§π‘‘βŸ©,(3.34) and hence βˆ’βŸ¨π‘£βˆ’π‘€,𝑇2π‘§π‘‘βŸ©β‰₯βˆ’βŸ¨π‘£βˆ’π‘€,π‘§π‘‘βŸ©βˆ€π‘£βˆˆπΆ.(3.35) Letting 𝑑→0 and using the fact that 𝑇2 is continuous, we obtain that βˆ’βŸ¨π‘£βˆ’π‘€,𝑇2π‘€βŸ©β‰₯βˆ’βŸ¨π‘£βˆ’π‘€,π‘€βŸ©βˆ€π‘£βˆˆπΆ.(3.36) Now, let 𝑣=𝑇2𝑀. Then, we obtain that 𝑀=𝑇2𝑀, and hence π‘€βˆˆπΉ(𝑇2). Furthermore, the fact that π‘’π‘›βˆ’π‘€π‘›β†’0 and 𝑀𝑛𝑖⇀𝑀 imply that 𝑒𝑛𝑖⇀𝑀, following the method used for 𝑀𝑛, we obtain that π‘€βˆˆπΉ(𝑇1), and hence β‹‚π‘€βˆˆ2𝑖=1𝐹(𝑇𝑖). Therefore, since 𝑧=𝑃𝐹𝑓(𝑧), by Lemma 2.1, we have limsupπ‘›β†’βˆžβŸ¨π‘“(𝑧)βˆ’π‘§,π‘₯π‘›βˆ’π‘§βŸ©=limπ‘–β†’βˆžβŸ¨π‘“(𝑧)βˆ’π‘§,π‘₯π‘›π‘–βˆ’π‘§βŸ©=βŸ¨π‘“(𝑧)βˆ’π‘§,π‘€βˆ’π‘§βŸ©β‰€0.(3.37) Now, we show that π‘₯𝑛→𝑧 as π‘›β†’βˆž. From π‘₯𝑛+1βˆ’π‘§=𝛼𝑛(𝑓(π‘₯𝑛)βˆ’π‘§)+(1βˆ’π›Όπ‘›)(π‘’π‘›βˆ’π‘§), we have that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2≀1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘’π‘›β€–β€–βˆ’π‘§2+2𝛼𝑛𝑓π‘₯π‘›ξ€Έβˆ’π‘§,π‘₯𝑛+1=ξ€·βˆ’π‘§1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘’π‘›β€–β€–βˆ’π‘§2+2𝛼𝑛𝑓π‘₯π‘›ξ€Έβˆ’π‘“(𝑧),π‘₯𝑛+1ξ¬βˆ’π‘§+2π›Όπ‘›βŸ¨π‘“(𝑧)βˆ’π‘§,π‘₯𝑛+1β‰€ξ€·βˆ’π‘§βŸ©1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+2𝛼𝑛𝛽‖‖π‘₯𝑛‖‖⋅‖‖π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§+2𝛼𝑛𝑓(𝑧)βˆ’π‘§,π‘₯𝑛+1ξ¬β‰€ξ€·βˆ’π‘§1βˆ’π›Όπ‘›ξ€Έ2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+𝛼𝑛𝛽‖‖π‘₯π‘›β€–β€–βˆ’π‘§2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2+2𝛼𝑛𝑓(𝑧)βˆ’π‘§,π‘₯𝑛+1.βˆ’π‘§(3.38) This implies that, β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2≀1βˆ’π›Όπ‘›ξ€Έ2+𝛼𝑛𝛽1βˆ’π›Όπ‘›π›½β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+2𝛼𝑛1βˆ’π›Όπ‘›π›½ξ«π‘“(𝑧)βˆ’π‘§,π‘₯𝑛+1=βˆ’π‘§1βˆ’2𝛼𝑛+𝛼𝑛𝛽1βˆ’π›Όπ‘›π›½β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+𝛼2𝑛1βˆ’π›Όπ‘›π›½β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+2𝛼𝑛1βˆ’π›Όπ‘›π›½ξ«π‘“(𝑧)βˆ’π‘§,π‘₯𝑛+1ξ¬β‰€ξ€·βˆ’π‘§1βˆ’π›Ύπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+πœŽπ‘›,(3.39) where π›Ύπ‘›βˆΆ=2(1βˆ’π›½)𝛼𝑛/(1βˆ’π›Όπ‘›π›½), πœŽπ‘›βˆΆ=(2(1βˆ’π›½)𝛼𝑛/(1βˆ’π›Όπ‘›π›½)){𝛼𝑛𝑀/2(1βˆ’π›½)+(1/(1βˆ’π›½))βŸ¨π‘“(𝑧)βˆ’π‘§,π‘₯𝑛+1βˆ’π‘§βŸ©}, for 𝑀=sup{β€–π‘₯π‘›βˆ’π‘§β€–2βˆΆπ‘›βˆˆN}. But note that βˆ‘βˆžπ‘›=1𝛾𝑛=∞, limπ‘›β†’βˆžπ›Ύπ‘›=0, and limsupπ‘›β†’βˆžπœŽπ‘›/𝛾𝑛≀0. Therefore, by Lemma 2.2, we conclude that {π‘₯𝑛} converges to π‘§βˆˆπΉ, where 𝑧=𝑃𝐹𝑓(𝑧). This completes the proof.

If, in Theorem 3.3, 𝑓=π‘’βˆˆπΆ is a constant mapping, then we get 𝑧=𝑃𝐹(𝑒). In fact, we have the following corollary.

Corollary 3.4. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space H. Let π‘‡π‘–βˆΆπΆβ†’πΆ, for 𝑖=1,2, be continuous pseudocontractive mappings such that β‹‚πΉβˆΆ=2𝑖=1𝐹(𝑇𝑖)β‰ βˆ…. Let {π‘₯𝑛} be a sequence generated by π‘₯1,π‘’βˆˆπΆ and π‘₯𝑛+1=𝛼𝑛𝑒+1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘Ÿπ‘›πΉπ‘Ÿπ‘›π‘₯𝑛,(3.40) where {𝛼𝑛}βŠ‚[0,1] and {π‘Ÿπ‘›}βŠ‚(0,∞) such that limπ‘›β†’βˆžπ›Όπ‘›=0, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, and βˆ‘βˆžπ‘›=1|π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›|<∞. Then, the sequence {π‘₯𝑛}𝑛β‰₯1 converges strongly to π‘§βˆˆπΉ, where 𝑧=𝑃𝐹(𝑒).

If, in Theorem 3.3, we have that 𝑇2≑𝐼, identity mapping on 𝐢, then we obtain the following corollary.

Corollary 3.5. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space H. Let 𝑇1βˆΆπΆβ†’πΆ be continuous pseudocontractive mapping such that 𝐹(𝑇1)β‰ βˆ…. Let 𝑓 be a contraction of 𝐢 into itself, and let {π‘₯𝑛} be a sequence generated by π‘₯1∈𝐢 and π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘Ÿπ‘›π‘₯𝑛,(3.41) where {𝛼𝑛}βŠ‚[0,1] and {π‘Ÿπ‘›}βŠ‚(0,∞) such that limπ‘›β†’βˆžπ›Όπ‘›=0, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, and βˆ‘βˆžπ‘›=1|π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›|<∞. Then, the sequence {π‘₯𝑛}𝑛β‰₯1 converges strongly to π‘§βˆˆπΉ, where 𝑧=𝑃𝐹(𝑇1)𝑓(𝑧).

Let 𝐻 be a real Hilbert space. Let π΄π‘–βˆΆπ»β†’π», for 𝑖=1,2, be accretive mappings. Let π‘‡ξ…žπ‘Ÿπ‘›π‘₯∢={π‘§βˆˆπ»βˆΆβŸ¨π‘¦βˆ’π‘§,(πΌβˆ’π΄1)π‘§βŸ©βˆ’(1/π‘Ÿπ‘›)βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿπ‘›)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπ»}, πΉξ…žπ‘Ÿπ‘›π‘₯∢={π‘§βˆˆπ»βˆΆβŸ¨π‘¦βˆ’π‘§,(πΌβˆ’π΄2)π‘§βŸ©βˆ’(1/π‘Ÿπ‘›)βŸ¨π‘¦βˆ’z,(1+π‘Ÿπ‘›)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπ»}. Then we have the following convergence theorem for a zero of two accretive mappings.

Theorem 3.6. Let 𝐻 be a real Hilbert space. Let π΄π‘–βˆΆπ»β†’π», for 𝑖=1,2, be continuous accretive mappings such that β‹‚N∢=2𝑖=1𝑁(𝐴𝑖)β‰ βˆ…. Let 𝑓 be a contraction of 𝐻 into itself, and let {π‘₯𝑛} be a sequence generated by π‘₯1∈𝐻 and π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ…žπ‘Ÿπ‘›πΉξ…žπ‘Ÿπ‘›π‘₯𝑛,(3.42) where {𝛼𝑛}βŠ‚[0,1] and {π‘Ÿπ‘›}βŠ‚(0,∞) such that limπ‘›β†’βˆžπ›Όπ‘›=0, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, and βˆ‘βˆžπ‘›=1|π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›|<∞. Then, the sequence {π‘₯𝑛}𝑛β‰₯1 converges strongly to π‘§βˆˆπ‘, where 𝑧=𝑃𝑁(𝑓(𝑧)).

Proof. Let π‘‡π‘–βˆΆ=(πΌβˆ’π΄π‘–), for 𝑖=1,2. Then, we get that 𝑇𝑖, for 𝑖=1,2, are continuous pseudocontractive mappings with β‹‚2𝑖=1𝑁(𝐴𝑖⋂)=2𝑖=1𝐹(𝑇𝑖). Thus, the conclusion follows from Theorem 3.3.

The proof of the following theorem can be easily obtained from the method of proof of Theorem 3.3.

Theorem 3.7. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space H. Let π‘‡π‘–βˆΆπΆβ†’πΆ, for 𝑖=1,2,…,𝐿, be continuous pseudocontractive mappings such that β‹‚πΉβˆΆ=𝐿𝑖=1𝐹(𝑇𝑖)β‰ βˆ…. Let 𝑓 be a contraction of 𝐢 into itself, and let {π‘₯𝑛} be a sequence generated by π‘₯1∈𝐢 and π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€ΈπΎ1,π‘Ÿπ‘›πΎ2,π‘Ÿπ‘›,…,𝐾𝑁,π‘Ÿπ‘›π‘₯𝑛,(3.43) where 𝐾𝑖,π‘Ÿπ‘›π‘₯∢={π‘§βˆˆπΆβˆΆβŸ¨π‘¦βˆ’π‘§,π‘‡π‘–π‘§βŸ©βˆ’(1/π‘Ÿπ‘›)βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿπ‘›)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπΆ}, for 𝑖=1,2,…,𝐿, and {𝛼𝑛}βŠ‚[0,1] and {π‘Ÿπ‘›}βŠ‚(0,∞) such that limπ‘›β†’βˆžπ›Όπ‘›=0, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, and βˆ‘βˆžπ‘›=1|π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›|<∞. Then, the sequence {π‘₯𝑛}𝑛β‰₯1 converges strongly to π‘§βˆˆπΉ, where 𝑧=𝑃𝐹(𝑓(𝑧)).

4. Application

In this section, we study the problem of finding a minimizer of a continuously FrΓ©chet differentiable convex functional in Hilbert spaces. Let β„Ž and 𝑔 be continuously FrΓ©chet differentiable convex functionals such that the gradient of β„Ž, (βˆ‡β„Ž) and the gradient of 𝑔, (βˆ‡π‘”) are continuous and accretive. For 𝛾>0 and π‘₯∈𝐻, let π‘‡π‘Ÿξ…žξ…žπ‘›π‘₯∢={π‘§βˆˆπ»βˆΆβŸ¨π‘¦βˆ’π‘§,(πΌβˆ’(βˆ‡β„Ž))π‘§βŸ©βˆ’(1/π‘Ÿπ‘›)βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿπ‘›)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπ»} and πΉπ‘Ÿξ…žξ…žπ‘›π‘₯∢={π‘§βˆˆπ»βˆΆβŸ¨π‘¦βˆ’π‘§,(πΌβˆ’(βˆ‡π‘”))π‘§βŸ©βˆ’(1/π‘Ÿπ‘›)βŸ¨π‘¦βˆ’π‘§,(1+π‘Ÿπ‘›)π‘§βˆ’π‘₯βŸ©β‰€0, for all π‘¦βˆˆπ»} for all π‘₯∈𝐻. Then, the following theorem holds.

Theorem 4.1. Let 𝐻 be a real Hilbert space. Let β„Ž and 𝑔 be continuously FrΓ©chet differentiable convex functionals such that the gradient of β„Ž, (βˆ‡β„Ž) and the gradient of 𝑔, (βˆ‡π‘”) are continuous and accretive such that π‘βˆΆ=𝑁(βˆ‡β„Ž)βˆ©π‘(βˆ‡π‘”)β‰ βˆ…. Let 𝑓 be a contraction of 𝐻 into itself, and let {π‘₯𝑛} be a sequence generated by π‘₯1∈𝐻 and π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘Ÿξ…žξ…žπ‘›πΉπ‘Ÿξ…žξ…žπ‘›π‘₯𝑛,(4.1) where {𝛼𝑛}βŠ‚[0,1] and {r𝑛}βŠ‚(0,∞) such that limπ‘›β†’βˆžπ›Όπ‘›=0, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, and βˆ‘βˆžπ‘›=1|π‘Ÿπ‘›+1βˆ’π‘Ÿπ‘›|<∞. Then, the sequence {π‘₯𝑛}𝑛β‰₯1 converges strongly to π‘§βˆˆπΉ, where 𝑧=𝑃𝑁(𝑓(𝑧)).

Proof. The conclusion follows from Theorem 3.6. We note that from the convexity and FrΓ©chet differentiability of β„Ž and 𝑔 we have 𝑁(βˆ‡β„Ž)=argminπ‘¦βˆˆπΆβ„Ž(𝑦) and 𝑁(βˆ‡π‘”)=argminπ‘¦βˆˆπΆπ‘”(𝑦).

Remark 4.2. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators. In particular, Theorem 3.3 extends Theorem 2.2 of Moudafi [24] and Theorem 4.1 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudocontractive mappings. Moreover, this provides affirmative answer to the concern raised.