Strong convergence of an new iterative method for a zero of accretive operator and nonexpansive mapping

Abstract

Let E be a Banach space and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence {x _n } by the algorithm $x_{n+1}={\alpha }_n\gamma \varphi \left(x_n\right)+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n$, where {a _n } and {r _n } are two sequences satisfying certain conditions, $J_{r_n}$ denotes the resolvent (I + r _n A)^-1 for r _n > 0, F be a strongly positive bounded linear operator on E is $0<\gamma <\bar{\gamma }$, and ϕ be a MKC on E. Strong convergence of the algorithm {x _n } is proved assuming E either has a weakly continuous duality map or is uniformly smooth.

MSC: 47H09; 47H10

1 Introduction

Let E be a real Banach space, C a nonempty closed convex subset of E, and T : C → C a mapping. Recall that T is nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥ for all x, y ∈ C. A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T, that is, F(T) = {x ∈ C, Tx = x}.

It is assumed throughout the paper that T is a nonexpansive mapping such that $F\left(t\right)\ne \varnothing$. The normalized duality mapping J from a Banach space E into $2^{E^{*}}$ is given by J(x) = {f ∈ E* : 〈x, f〉 = ∥x∥² = ∥f∥²}, x ∈ E, where E* denotes the dual space of E and 〈.,.〉 denotes the generalized duality pairing.

Theorem 1.1. (Banach [1]). Let (X, d) be a complete metric space and let f be a contraction on X, that is, there exists r ∈ (0, 1) such that d(f(x), f(y)) ≤ rd(x, y) for all x, y ∈ X. Then f has a unique fixed point.

Theorem 1.2. (Meir and Keeler [2]). Let (X, d) be a complete metric space and let ϕ be a Meir-Keeler contraction (MKC, for short) on X, that is, for every ε > 0, there exists δ > 0 such that d(x, y) < ε + δ implies d(ϕ(x), ϕ(y)) < ε for all x, y ∈ X. Then ϕ has a unique fixed point.

This theorem is one of generalizations of Theorem 1.1, because contractions are Meir-Keeler contractions.

Let F be a strongly positive bounded linear operator on E, that is, there exists a constant $\tilde{\gamma }>0$ such that

$$⟨Fx,J\left(x\right)⟩\ge \tilde{\gamma }{∥x∥}^2,∥aI-bF∥=\underset{∥x∥\le 1}{\text{sup}}\left\{\left|⟨\left(aI-bF\right)x,J\left(x\right)⟩\right|:a\in \left[0,1\right],b\in \left[0,1\right]\right\},$$

where I is the identity mapping and J is the normalized duality mapping.

Let D be a subset of C. Then Q : C → D is called a retraction from C onto D if Q(x) = x for all x ∈ D. A retraction Q : C → D is said to be sunny if Q(x + t(x - Q(x))) = Q(x) for all x ∈ C and t ≥ 0 whenever x + t(x - Q(x)) ∈ C. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D. In a smooth Banach space E, it is known (cf. [[3], p. 48]) that Q : C → D is a sunny nonexpansive retraction if and only if the following condition holds:

$$⟨x-Q\left(x\right),J\left(z-Q\left(x\right)\right)⟩\le 0,x\in C,z\in D.$$

(1.1)

Recall that an operator A with domain D(A) and range R(A) in E is said to be accretive, if for each x _i ∈ D(A) and y _i ∈ Ax _i , i = 1, 2, there is a j ∈ J(x₂ - x₁) such that

$$⟨y_2-y_1,j⟩\ge 0.$$

An accretive operator A is m-accretive if R(I + λA) = E for all λ > 0. Denote by N(A) the zero set of A; i.e.,

$$N\left(A\right):=A^{-1}0=\left\{x\in D\left(A\right):Ax=0\right\}.$$

Throughout the rest of this paper it is always assumed that A is m-accretive and N(A) is nonempty. Denote by J _r the resolvent of A for r > 0:

$$J_r={\left(I+rA\right)}^{-1}.$$

Note that if A is m-accretive, then J _r : E → E is nonexpansive and F(J _r ) = N(A) for all r > 0. We also denote by A _r the Yosida approximation of A, i.e., $A_r=\frac{1}{r}\left(I-J_r\right)$. It is well known that J _r is a nonexpansive mapping from E to C := D(A).

Recall that a gauge is a continuous strictly increasing function φ : [0, ∞) → [0, ∞) such that φ(0) = 0 and φ(t) → ∞ as t → ∞. Associated to a gauge φ is the duality mapping J _φ : E → E* defined by

$$J_{\phi }\left(x\right)=\left\{x^{*}\in E^{*}:⟨x,x^{*}⟩=∥x∥\phi \left(∥x∥\right),∥x^{*}∥=\phi \left(∥x∥\right)\right\},x\in E.$$

Following Browder [4], we say that a Banach space E has a weakly continuous duality map if there exists a gauge φ for which the duality map J _φ is single-valued and weak-to-weak* sequentially continuous(i.e., if {x _n } is a sequence in E weakly convergent to a point x, then the sequence J _φ (x _n ) converges weakly* to J _φ (x)). It is known that l^phas a weakly continuous duality map for all 1 < p < ∞, with gauge φ(t) = t^p-1. Set

$$\Phi \left(t\right)=\underset{0}{\overset{t}{\int }}\phi \left(\tau \right)d\tau ,t\ge 0.$$

(1.2)

Then

$$J_{\phi }\left(x\right)=\partial \Phi \left(∥x∥\right),x\in E,$$

where ∂ denotes the subdifferential in the sense of convex analysis.

Recently, Hong-Kun Xu [5] introduced the following iterative scheme: for x₁ = x ∈ C,

$$x_{n+1}={\alpha }_{n}u+\left(1+{\alpha }_n\right)J_{r_n}{x}_n,\forall n\ge 1,$$

(1.3)

where {a _n } and {r _n } are two sequences satisfying certain conditions, and $J_{r_n}$ denotes the resolvent (I + r _n A)^-1 for r _n > 0. He proved the strong convergence of the algorithm {x _n } assuming E either has a weakly continuous duality map or is uniformly smooth.

Motivated and inspired by the results of Hong-Kun Xu, we introduce the following iterative scheme: for any x₀ ∈ E,

$$x_{n+1}={\alpha }_n\gamma \varphi \left(x_n\right)+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n,\forall n\ge 0,$$

(1.4)

where {a _n } and {r _n } are two sequences satisfying certain conditions, $J_{r_n}$ denotes the resolvent (I + r _n A)^-1 for r _n > 0, F be a strongly positive bounded linear operator on E is $0<\gamma <\bar{\gamma }$, and ϕ be a MKC on E. Strong convergence of the algorithm {x _n } is proved assuming E either has a weakly continuous duality map or is uniformly smooth. Our results extend and improve the corresponding results of Hong-Kun Xu [5] and many others.

2 Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1. [5]. Assume that E has a weakly continuous duality map J _φ with gauge φ,

• (i) For all x, y ∈ E, there holds the inequality

  $$\Phi \left(∥x+y∥\right)\le \Phi \left(∥x∥\right)+⟨y,J_{\phi }\left(x+y\right)⟩.$$

• (ii) Assume a sequence {x _n } in E is weakly convergent to a point x, then there holds the equality

  $$\underset{n\to \infty }{\text{lim}\text{sup}}\Phi \left(∥x_n-y∥\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\Phi \left(∥x_n-x∥\right)+\Phi \left(∥y-x∥\right),x,y\in E.$$

Lemma 2.2. [6, 7]. Let {s _n } be a sequence of nonnegative real numbers satisfying

$$s_{n+1}\le \left(1-{\lambda }_n\right)s_n+{\lambda }_n{\delta }_n+{\gamma }_n,n\ge 0,$$

where {λ _n }, {δ _n } and {γ _n } satisfy the following conditions:

• (i) {λ _n } ⊂ [0,1] and ${\sum }_{n=0}^{\infty }{\lambda }_n=\infty$,

• (ii) lim sup_n→∞δ _n ≤ 0 or ${\sum }_{n=0}^{\infty }{\lambda }_n{\delta }_n<\infty$ (iii) ${\gamma }_n\ge 0\left(n\ge 0\right),{\sum }_{n=0}^{\infty }{\gamma }_n<\infty$. Then lim_n→∞s _n = 0.

Lemma 2.3. (The Resolvent Identity [8, 9]). For λ > 0 and ν > 0 and x ∈ E,

$$J_{\lambda }x=J_{\nu }\left(\frac{\nu }{\lambda }+\left(1-\frac{\nu }{\lambda }\right)J_{\lambda }x\right).$$

Lemma 2.4. (see [ [10], Lemma 2.3]). Assume that F is a strongly positive linear bounded operator on a smooth Banach space E with coefficient $\bar{\gamma }>0$ and 0 < ρ ≤ ∥F∥^-1. Then,

$$∥I-\rho F∥\le 1-\rho \bar{\gamma }.$$

Lemma 2.5. (see [ [11], Lemma 2.3]). Let ϕ be a MKC on a convex subset C of a Banach space E. Then for each ε > 0, there exists r ∈ (0,1) such that

$$∥x-y∥\ge \epsilon \mathsf{\text{implies}}∥\varphi x-\varphi y∥\le r∥x-y∥\forall x,y\in C.$$

Lemma 2.6. Let E be a reflexive Banach space which admits a weakly continuous duality map J _φ with gauge φ. Let T : E → E be a nonexpansive mapping. Now given ϕ : E → E be a MKC, F be a strongly positive linear bounded operator with coefficient $\bar{\gamma }>0$. Assume that $0<\gamma <\bar{\gamma }$, the sequence {x _t } defined by x _t = tγϕ(x _t ) + (I - tF)Tx _t . Then T has a fixed point if and only if {x _t } remains bounded as t → 0⁺, and in this case, {x _t } converges as t → 0⁺ strongly to a fixed point of T. If $\tilde{x}:=\underset{t\to 0}{\text{lim}}{x}_t$, then $\tilde{x}$ uniquely solves the variational inequality

$$⟨\left(F-\gamma \varphi \right)\tilde{x},J\left(\tilde{x}-p\right)⟩\le 0,P\in F\left(T\right).$$

Proof. The definition of {x _t } is well defined. Indeed, from the definition of MKC, we can see MKC is also a nonexpansive mapping. Consider a mapping S _t on E defined by

$$S_t\left(x\right)=t\gamma \varphi \left(x\right)+\left(I-tF\right)Tx,x\in E.$$

It is easy to see that S _t is a contraction. Indeed, by Lemma 2.4, we have

$$\begin{array}{ll}\hfill ∥S_{t}x-S_{t}y∥& \le t\gamma ∥\varphi \left(x\right)-\varphi \left(y\right)∥+∥\left(I-tF\right)\left(Tx-Ty\right)∥\\ \le t\gamma ∥x-y∥+\left(1-t\bar{\gamma }\right)∥Tx-Ty∥\\ \le \left[1-t\left(\bar{\gamma }-\gamma \right)\right]∥x-y∥,\end{array}$$

for all x, y ∈ E. Hence S _t has a unique fixed point, denoted as x _t , which uniquely solves the fixed point equation

$$x_t=t\gamma \varphi \left(x_t\right)+\left(I-tF\right)T{x}_t,x_t\in E.$$

(2.1)

We next show the sequence {x _t } is bounded. Indeed, we may assume $F\left(t\right)\ne \varnothing$ and with no loss of generality t < ∥F∥^-1. Take p ∈ F(T) to deduce that, for t ∈ (0, 1),

$$\begin{array}{ll}\hfill ∥x_t-p∥& =∥t\gamma \varphi \left(x_t\right)+\left(I-tF\right)T{x}_t-p∥\\ =∥t\left(\gamma \varphi \left(x_t\right)-Fp\right)+\left(I-tF\right)\left(T{x}_t-p\right)∥\\ \le \left(1-t\bar{\gamma }\right)∥x_t-p∥+t\gamma ∥x_t-p∥+t∥\gamma \varphi \left(p\right)-Fp∥\\ \le \left[1-t\left(\bar{\gamma }-\gamma \right)\right]∥x_t-p∥+t∥\gamma \varphi \left(p\right)-Fp∥.\end{array}$$

Hence

$$∥x_t-p∥\le \frac{1}{\bar{\gamma }-\gamma }∥\gamma \varphi \left(p\right)-Fp∥$$

and {x _t } is bounded.

Next assume that {x _t } is bounded as t → 0⁺. Assume t _n → 0⁺ and $\left\{x_{t_n}\right\}$ is bounded. Since E is reflexive, we may assume that $x_{t_n}\rightharpoonup z$ for some z ∈ E. Since J _φ is weakly continuous, we have by Lemma 2.1,

$$\underset{n\to \infty }{\text{lim}\text{sup}}\Phi \left(∥x_{t_n}-x∥\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\Phi \left(∥x_{t_n}-z∥\right)+\Phi \left(∥x-z∥\right),\forall x\in E.$$

Put

$$f\left(x\right)=\underset{n\to \infty }{\text{lim}\text{sup}}\Phi \left(∥x_{t_n}-x∥\right),x\in E.$$

It follows that

$$f\left(x\right)=f\left(z\right)+\Phi \left(∥x-z∥\right),x\in E.$$

Since

$$∥x_{t_n}-T{x}_{t_n}∥=t_n∥\gamma \varphi \left(x_{t_n}\right)-FT{x}_{t_n}∥\to 0,$$

we obtain

$$\begin{array}{ll}\hfill f\left(Tz\right)& =\text{lim }{\text{sup}}_{n\to \infty }\Phi \left(∥x_{t_n}-Tz∥\right)=\text{lim }{\text{sup}}_{n\to \infty }\Phi \left(∥T{x}_{t_n}-Tz∥\right)\\ \le \text{lim }{\text{sup}}_{n\to \infty }\Phi \left(∥x_{t_n}-z∥\right)=f\left(z\right).\end{array}$$

(2.2)

On the other hand, however,

$$f\left(Tz\right)=f\left(z\right)+\Phi \left(∥Tz-z∥\right)$$

(2.3)

Combining Equations (2.2) and (2.3) yields

$$\Phi \left(∥Tz-z∥\right)\le 0.$$

Hence, Tz = z and z ∈ F(T).

Finally, we prove that {x _t } converges strongly to a fixed point of T provided it remains bounded when t → 0.

Let {t _n } be a sequence in (0, 1) such that t _n → 0 and $x_{t_n}\rightharpoonup z$ as n → ∞. Then the argument above shows that z ∈ F(T). We next show that $x_{t_n}\to z$. By contradiction, there is a number ε₀ > 0 such that $∥x_{t_n}-z∥\ge {\epsilon }_0$. Then by Lemma 2.8, there is a number r ∈ (0, 1) such that

$$\begin{array}{ll}\hfill ∥\varphi \left(x_{t_n}\right)-\varphi \left(z\right)∥& \le r∥x_{t_n}-z∥,\\ \hfill ∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)& =⟨x_{t_n}-z,J_{\phi }\left(x_{t_n}-z\right)⟩\\ =⟨t_n\left(\gamma \varphi \left(x_{t_n}\right)-Fz\right)+\left(I-t_{n}F\right)\left(T{x}_{t_n}-z\right),J_{\phi }\left(x_{t_n}-z\right)⟩\\ \le t_n⟨\gamma \varphi \left(x_{t_n}\right)-Fz,J_{\phi }\left(x_n-z\right)⟩+∥\left(I-t_{n}F\right)\left(T{x}_{t_n}-z\right)∥\phi \left(∥x_{t_n}-z∥\right)\\ \le \left(1-t_n\bar{\gamma }\right)∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)+t_n⟨\gamma \varphi \left(x_{t_n}\right)-Fz,J_{\phi }\left(x_n-z\right)⟩.\end{array}$$

It follows that

$$\begin{array}{ll}\hfill ∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)& \le \frac{1}{\bar{\gamma }}⟨\gamma \varphi \left(x_{t_n}\right)-Fz,J_{\phi }\left(x_{t_n}-z\right)⟩\\ =\frac{1}{\bar{\gamma }}\left[⟨\gamma \varphi \left(x_{t_n}\right)-\gamma \varphi \left(z\right),J_{\phi }\left(x_{t_n}-z\right)⟩+⟨\gamma \varphi \left(z\right)-Fz,J_{\phi }\left(x_{t_n}-z\right)⟩\right]\\ \left.\le \frac{\gamma }{\bar{\gamma }}r∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)+\frac{1}{\bar{\gamma }}⟨\gamma \varphi \left(z\right)-Fz,J_{\phi }\left(x_{t_n}-z\right)⟩\right].\end{array}$$

Therefore,

$$∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)\le \frac{1}{\bar{\gamma }-\gamma r}⟨\gamma \varphi \left(z\right)-Fz,J_{\varphi }\left(x_{t_n}-z\right)⟩.$$

Now observing that $x_{t_n}\rightharpoonup z$ implies $J_{\phi }\left(x_{t_n}-z\right)\rightharpoonup 0$, we conclude from the last inequality that

$$\underset{n\to \infty }{\text{lim}}∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)=0.$$

It contradicts $∥x_{t_n}-z∥\phi \left(∥x_{t_n}-z∥\right)\ge {\epsilon }_0\phi \left({\epsilon }_0\right)>0$. Hence $x_{t_n}\to z$.

We finally prove that the entire net {x _t } converges strongly. Towards this end, we assume that two null sequences {t _n } and {s _n } in (0, 1) are such that

$$t_n\to 0,x_{t_n}\to z\mathsf{\text{and}}{s}_n\to 0,x_{s_n}\to ź.$$

We have to show $z=ź$. Indeed, for p ∈ F(T). Since

$$x_t=t\gamma \varphi \left(x_t\right)+\left(I-tf\right)T{x}_t,$$

we derive that

$$\left(F-\gamma \varphi \right)x_t=-\frac{1}{t}\left(I-tF\right)\left(I-T\right)x_t.$$

(2.4)

Notice

$$\begin{array}{ll}\hfill ⟨\left(I-T\right)x_t-\left(I-T\right)p,J_{\phi }\left(x_t-p\right)⟩& =∥x_t-p∥\phi \left(∥x_t-p∥\right)+⟨Tp-T{x}_t,J_{\phi }\left(x_t-p\right)⟩\\ \ge ∥x_t-p∥\phi \left(∥x_t-p∥\right)-∥Tp-T{x}_t∥∥J_{\phi }\left(x_t-p\right)∥\\ \ge ∥x_t-p∥\left[\phi \left(∥x_t-p∥\right)-\phi \left(∥x_t-p∥\right)\right]\\ =0.\end{array}$$

It follows that,

$$\begin{array}{c}〈(F-\gamma \varphi )x_t,J_{\phi }(x_t-p)〉=-\frac{1}{t}〈(I-tF)(I-T)x_t,J_{\phi }(x_t-p)〉\\ \le 〈(F(I-T)x_t,J_{\phi }(x_t-p)〉.\end{array}$$

(2.5)

Now replacing t in (2.5) with t _n and letting n → ∞, noticing $\left(I-T\right)x_{t_n}\to \left(I-T\right)z=0$ for z ∈ F(T), we obtain 〈(F - γϕ)z, J _φ (z - p)〉 ≤ 0. In the same way, we have $⟨\left(F-\gamma \varphi \right)ź,J_{\phi }\left(ź-p\right)⟩\le 0$.

Thus, we have

$$⟨\left(F-\gamma \varphi \right)z,J_{\phi }\left(z-ź\right)⟩\le 0\mathsf{\text{and}}⟨\left(F-\gamma \varphi \right)ź,J_{\phi }\left(ź-z\right)⟩\le 0.$$

(2.6)

Adding up (2.6) gets

$$⟨\left(F-\gamma \varphi \right)z-\left(F-\gamma \varphi \right)ź,J_{\phi }\left(z-ź\right)⟩\le 0.$$

On the other hand, without loss of generality, we may assume there is a number ε such that $∥z-ź∥\ge \epsilon$, then by Lemma 2.5 there is a number r₁ such that $∥\varphi \left(z\right)-\varphi \left(ź\right)∥\le r_1∥z-ź∥$. Noticing that

$$\begin{array}{ll}\hfill J_{\phi }& =\left(\phi \left(∥x∥\right)/∥x∥\right)J\left(x\right)x\ne 0,\\ \hfill ⟨\left(F-\gamma \varphi \right)z-\left(F-\gamma \varphi \right)ź,J_{\phi }\left(z-ź\right)⟩& =⟨F\left(z-ź\right),J_{\phi }\left(z-ź\right)⟩-⟨\gamma \varphi z-\gamma \varphi ź,J_{\phi }\left(z-ź\right)⟩\\ \ge \bar{\gamma }∥z-ź∥∥J_{\phi }\left(z-ź\right)∥-\gamma r_1∥z-ź∥∥J_{\phi }\left(z-ź\right)∥\\ \ge \left(\bar{\gamma }-\gamma r_1\right)∥z-ź∥\phi \left(∥z-ź∥\right)\\ >0.\end{array}$$

Hence $z=ź$ and {x _t } converges strongly. Thus we may assume $x_t\to \tilde{x}$. Since we have proved that, for all t ∈ (0, 1) and p ∈ F(T),

$$〈(F-\gamma \varphi )x_t,J_{\phi }(x_t-p)〉\le 〈(F(I-T)x_t,J_{\phi }(x_t-p)〉,$$

letting t → 0, we obtain that

$$⟨\left(F-\gamma \varphi \right)\tilde{x},J_{\phi }\left(\tilde{x}-p\right)⟩\le 0.$$

This implies that

$$⟨\left(F-\gamma \varphi \right)\tilde{x},J\left(\tilde{x}-p\right)⟩\le 0.$$

Lemma 2.7. (see [12]). Assume that C₂ ≥ C₁ > 0. Then $∥J_{C_1}x-x∥\le 2∥J_{C_2}x-x∥$ for all x ∈ E.

Lemma 2.8. [13]. Let C be a nonempty closed convex subset of a reflexive Banach space E which satisfies Opial's condition, and suppose T : C → E is a nonexpansive mapping. Then the mapping I - T is demiclosed at zero, that is x _n ⇀ x and ∥x _n - Tx _n ∥ → 0, then x = Tx.

Lemma 2.9. In a smooth Banach space E there holds the inequality

$${∥x+y∥}^2\le {∥x∥}^2+2⟨y,J\left(x+y\right)⟩,x,y\in E.$$

3 Main result

Theorem 3.1. Suppose that E is reflexive which admits a weakly continuous duality map J _φ with gauge φ and A is an m-accretive operator in E such that $F^{*}=N\left(A\right)\ne \varnothing$. Now given ϕ : E → E be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient $\bar{\gamma }>0,0<\gamma <\bar{\gamma }$. Assume

• (i) ${\text{lim}}_{n\to \infty }{\alpha }_n=0,{\sum }_{n=0}^{\infty }{\alpha }_n=\infty$;

• (ii) r _n → ∞.

Then {x _n } defined by (1.4) converges strongly to a point in F*.

Proof. First notice that {x _n } is bounded. Indeed, take p ∈ F* to get

$$\begin{array}{ll}\hfill ∥x_{n+1}-p∥& =∥{\alpha }_n\varphi \left(x_n\right)+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n-p∥\\ =∥{\alpha }_n\gamma \varphi \left(x_n\right)-{\alpha }_{n}Fp+{\alpha }_{n}Fp+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n-p∥\\ =∥{\alpha }_n\left(\gamma \varphi \left(x_n\right)-Fp\right)+\left(I-{\alpha }_{n}F\right)\left(J_{r_n}{x}_n-p\right)∥\\ \le \left(1-{\alpha }_n\bar{\gamma }\right)∥x_n-p∥+{\alpha }_n\gamma ∥x_n-p∥+{\alpha }_n∥\gamma \varphi \left(p\right)-Fp∥\\ \le \left[1-{\alpha }_n\left(\bar{\gamma }-\gamma \right)\right]∥x_n-p∥+{\alpha }_n∥\gamma \varphi \left(p\right)-Fp∥\\ \le \left[1-{\alpha }_n\left(\bar{\gamma }-\gamma \right)\right]∥x_n-p∥+{\alpha }_n\left(\bar{\gamma }-\gamma \right)\frac{∥\gamma \varphi \left(p\right)-Fp∥}{\bar{\gamma }-\gamma }\\ \le \text{max}\left\{∥x_n-p∥,\frac{∥\gamma \varphi \left(p\right)-Fp∥}{\bar{\gamma }-\gamma }\right\},n\ge 0.\end{array}$$

By induction, we have

$$∥x_n-p∥\le \text{max}\left\{∥x_0-p∥,\frac{∥\gamma \varphi \left(p\right)-Fp∥}{\bar{\gamma }-\gamma }\right\},n\ge 0.$$

This implies that {x _n } is bounded and hence

$$\begin{array}{ll}\hfill ∥x_{n+1}-J_{r_n}{x}_n∥& =∥{\alpha }_n\gamma \varphi \left(x_n\right)+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n-J_{r_n}{x}_n∥\\ ={\alpha }_n∥\gamma \varphi \left(x_n\right)-F{J}_{r_n}{x}_n∥\to 0.\end{array}$$

We next prove that

lim sup_n→∞〈γϕ(p) - Fp, J _φ (x _n - p)〉 ≤ 0, where p = lim_t→0x _t with $x_t=t\gamma \varphi \left(x_t\right)+\left(I-tF\right)J_{r_n}{x}_t$.

Since {x _n } is bounded, take a subsequence $\left\{x_{n_k}\right\}$ of {x _n } such that

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨\gamma \varphi \left(p\right)-Fp,J_{\varphi }\left(x_n-p\right)⟩=\underset{k\to \infty }{\text{lim}}⟨\gamma \varphi \left(p\right)-Fp,J_{\phi }\left(x_{n_k}-p\right)⟩.$$

(3.1)

Since E is reflexive, we may further assume that $x_{n_k}\rightharpoonup \tilde{x}$. Moreover, since

$$∥x_{n+1}-J_{r_n}{x}_n∥\to 0,$$

we obtain

$$J_{r_{n_k}-1}{x}_{n_k}-1\rightharpoonup \tilde{x}.$$

Taking the limit as k → ∞ in the relation

$$\left[J_{r_{n_k}-1}{x}_{n_k-1},A_{r_{n_k}-1}{x}_{n_k}-1\right]\in A,$$

we get $\left[\tilde{x},0\right]\in A$. That is, $\tilde{x}\in F^{*}$. Hence by (3.1) and Lemma 2.6 we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨\gamma \varphi \left(p\right)-Fp,J_{\phi }\left(x_n-p\right)⟩=\underset{k\to \infty }{\text{lim}}⟨\gamma \varphi \left(p\right)-Fp,J_{\phi }\left(x_{n_k}-p\right)⟩=⟨\gamma \varphi \left(p\right)-Fp,J_{\phi }\left(\tilde{x}-p\right)⟩\le 0.$$

Finally to prove that x _n → p, we apply Lemma 2.1 to get

$$\begin{array}{ll}\hfill \Phi \left(∥x_{n+1}-p∥\right)& =\Phi \left(∥{\alpha }_n\gamma \varphi \left(x_n\right)+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n-p∥\right)\\ =\Phi \left(∥\left(I-{\alpha }_{n}F\right)\left(J_{r_n}{x}_n-p\right)+{\alpha }_n\left(\gamma \varphi \left(x_n\right)-Fp\right)∥\right)\\ =\Phi \left(∥\left(I-{\alpha }_{n}F\right)\left(J_{r_n}{x}_n-p\right)+{\alpha }_n\left(\gamma \varphi \left(x_n\right)-\gamma \varphi \left(p\right)\right)+{\alpha }_n\left(\gamma \varphi \left(p\right)-Fp\right)∥\right)\\ \le \Phi \left.\left(∥\left(I-{\alpha }_{n}F\right)\left(J_{r_n}{x}_n-p\right)+{\alpha }_n\left(\gamma \varphi \left(x_n\right)-\gamma \varphi \left(p\right)\right)∥\right)\right)+{\alpha }_n⟨\gamma \varphi \left(p\right)-Fp,J_{\phi }\left(x_{n+1}-p\right)⟩\\ \le \left[1-{\alpha }_n\left(\bar{\gamma }-\gamma \right)\right]\Phi \left(∥x_n-p∥\right)+{\alpha }_n⟨\gamma \varphi \left(p\right)-Fp,J_{\phi }\left(x_{n+1}-p\right)⟩.\end{array}$$

An application of Lemma 2.2 yields that Φ(∥x _n - p∥) → 0. That is, ∥x _n - p∥ → 0, i.e., x _n → p. The proof is complete.

Theorem 3.2. Suppose that E is reflexive which admits a weakly continuous duality map J _φ with gauge φ and A is an m-accretive operator in E such that $F^{*}=N\left(A\right)\ne \varnothing$. Now given ϕ : E → E be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient $\bar{\gamma }>0,0<\gamma <\bar{\gamma }$. Assume

• (i) ${\text{lim}}_{n\to \infty }{\alpha }_n=0,{\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, and ${\sum }_{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty \left(e.g.,{\alpha }_n=\frac{1}{n}\right)$;

• (ii) r _n ≥ ε for all n and ${\sum }_{n=1}^{\infty }\left|r_{n+1}-r_n\right|<\infty \left(e.g.,r_n=1+\frac{1}{n}\right)$.

Then {x _n } defined by (1.4) converges strongly to a point in F*.

Proof. We only include the differences. We have

$$x_{n+1}={\alpha }_n\gamma \varphi \left(x_n\right)+\left(I-{\alpha }_{n}F\right)J_{r_n}{x}_n,x_n={\alpha }_{n-1}\gamma \varphi \left(x_{n-1}\right)+\left(I-{\alpha }_{n-1}F\right)J_{r_{n-1}}{x}_{n-1}.$$

Thus,

$$x_{n+1}-x_n=\left(I-{\alpha }_{n}F\right)\left(J_{r_n}{x}_n-J_{r_{n-1}}{x}_{n-1}\right)+{\alpha }_n\gamma \varphi \left(x_n\right)+{\alpha }_{n-1}\gamma \varphi \left(x_{n-1}\right)+\left({\alpha }_n-{\alpha }_{n-1}\right)F{J}_{r_{n-1}}{x}_{n-1}.$$

(3.2)

If r_n-1≤ r _n , using the resolvent identity

$$J_{r_n}{x}_n=J_{r_{n-1}}\left(\frac{r_{n-1}}{r_n}{x}_n+\left(1-\frac{r_{n-1}}{r_n}\right)J_{r_n}{x}_n\right),$$

we obtain

$$\begin{array}{ll}\hfill ∥J_{r_n}{x}_n-J_{r_{n-1}}{x}_{n-1}∥& \le \frac{r_{n-1}}{r_n}∥x_n-x_{n-1}∥+\left(1-\frac{r_{n-1}}{r_n}\right)∥J_{r_n}{x}_n-x_{n-1}∥\\ \le ∥x_n-x_{n-1}∥+\left(\frac{r_n-r_{n-1}}{r_n}\right)∥J_{r_n}{x}_n-x_{n-1}∥\\ \le ∥x_n-x_{n-1}∥+\frac{1}{\epsilon }\left|r_{n-1}-r_n\right|∥J_{r_n}{x}_n-x_{n-1}∥.\end{array}$$

(3.2a)

It follows from (3.2) that

$$∥x_{n+1}-x_n∥\le \left(1-{\alpha }_n\left(\bar{\gamma }-\gamma \right)\right)∥x_n-x_{n-1}∥+M\left(\left|{\alpha }_n-{\alpha }_{n-1}\right|+\left|r_{n-1}-r_n\right|\right),$$

(3.3)

where M > 0 is some appropriate constant. Similarly we can prove (3.3) if r_n-1≥ r _n . By assumptions (i) and (ii) and Lemma 2.2, we conclude that

$$∥x_{n+1}-x_n∥\to 0.$$

This implies that

$$∥x_n-J_{r_n}{x}_n∥\le ∥x_{n+1}-x_n∥+∥x_{n+1}-J_{r_n}{x}_n∥\to 0,$$

(3.4)

since $∥x_{n+1}-J_{r_n}{x}_n∥={\alpha }_n∥\gamma \varphi \left(x_n\right)-F{J}_{r_n}{x}_n∥\to 0$. It follows that

$$∥A_{r_n}{x}_n∥=\frac{1}{r_n}∥x_n-J_{r_n}{x}_n∥\le \frac{1}{\epsilon }∥x_n-J_{r_n}{x}_n∥\to 0.$$

Now if $\left\{x_{n_k}\right\}$ is a subsequence of {x _n } converging weakly to a point $\tilde{x}$, then taking the limit as k → ∞ in the relation

$$\left[J_{r_{n_k}}{x}_{n_k},A_{r_{n_k}}{x}_{n_k}\right]\in A,$$

we get $\left[\tilde{x},0\right]\in A$; i.e., $\tilde{x}\in F^{*}$. We therefore conclude that all weak limit points of {x _n } are zeros of A.

The rest of the proof follows that of Theorem 3.1.

Finally, we consider the framework of uniformly smooth Banach spaces. Assume r _n ≥ ε for some ε > 0 (not necessarily r _n → ∞), A is an m-accretive operator in E. Moreover let ϕ : E → E be a MKC and F be a strongly positive linear bounded operator on E. Since $J_{r_n}$ is nonexpansive, the map $S:x\in E\mapsto t\gamma \varphi \left(x\right)+\left(I-tF\right)J_{r_n}x$ is a contraction and for each integer n ≥ 1 it has a unique fixed z _t,n ∈ E. Hence the scheme

$$z_{t,n}=t\gamma \varphi \left(z_{t,n}\right)+\left(I-tF\right)J_{r_n}{z}_{t,n}$$

(3.5)

is well defined.

Note that {z _t,n } is uniformly bounded; indeed, $∥z_{t,n}-p∥\le \frac{1}{\bar{\gamma }-\gamma }∥\gamma \varphi \left(p\right)-Fp∥$ for all t ∈ (0, 1), n ≥ 1 and p ∈ F*. A key component of the proof of the next theorem is the following lemma.

Lemma 3.1. The limit $ẑ={\text{lim}}_{t\to 0}{z}_{t,n}$ is uniform for all n ≥ 1.

Proof. It suffices to show that for any positive integer n _t (which may depend on t ∈ (0, 1)), if $z_{t,n_t}\in E$ is the unique point in E that satisfies the property

$$z_{t,n_t}=t\gamma \varphi \left(z_{t,n_t}\right)+\left(I-tF\right)J_{r_{n_t}}{z}_{t,n_t},$$

(3.6)

then $\left\{z_{t,n_t}\right\}$ converges as t → 0 to a point in F*. For simplicity put

$$w_t=z_{t,n_t}\mathsf{\text{and}}{V}_t=J_{r_{n_t}}.$$

It follows that

$$w_t=t\gamma \varphi \left(w_t\right)+\left(I-tF\right)V_t{w}_t.$$

(3.7)

Note that Fix(V _t ) = F* for all t. Note also that {w _t } is bounded; indeed, we have $∥w_t-p∥\le \frac{1}{\bar{\gamma }-\gamma }∥\gamma \varphi \left(p\right)-Fp∥$ for all t ∈ (0, 1) and p ∈ F*. Since {V _t w _t } is bounded, it is easy to see that

$$∥w_t-V_t{w}_t∥=t∥\gamma \varphi \left(w_t\right)-F{V}_t{w}_t∥\to 0,\mathsf{\text{as}}t\to 0.$$

Since r _n ≥ ε for all n, by Lemma 2.7, we have

$$∥w_t-J_{\epsilon }{w}_t∥\le 2∥w_t-J_{r_{n_t}}{w}_t∥=2∥w_t-V_t{w}_t∥\to 0.$$

(3.8)

Let {t _k } be a sequence in (0,1) such that t _k → 0 as k → ∞. Define a function f on E by

$$f\left(w\right)=\mathsf{\text{LI}}{\mathsf{\text{M}}}_k\frac{1}{2}{∥w_{t_k}-w∥}^2,w\in E,$$

where LIM denotes a Banach limit on l^∞. Let

$$K:=\left\{w\in E:f\left(w\right)=\text{min}\left\{f\left(y\right):y\in E\right\}\right\}.$$

Then K is a nonempty closed convex bounded subset of E. We claim that K is also invariant under the nonexpansive mapping J _ε . Indeed, noting (3.8), we have for w ∈ K,

$$\begin{array}{ll}\hfill f\left(J_{\epsilon }w\right)& =\mathsf{\text{LI}}{\mathsf{\text{M}}}_k\frac{1}{2}{∥w_{t_k}-J_{\epsilon }w∥}^2=\mathsf{\text{LI}}{\mathsf{\text{M}}}_k\frac{1}{2}{∥J_{\epsilon }{w}_{t_k}-J_{\epsilon }w∥}^2\\ \le \mathsf{\text{LI}}{\mathsf{\text{M}}}_k\frac{1}{2}{∥w_{t_k}-w∥}^2=f\left(w\right).\end{array}$$

Since a uniformly smooth Banach space has the fixed point property for nonexpansive mappings and since J _ε is a nonexpansive self-mapping of E, J _ε has a fixed point in K, say w'. Now since w' is also a minimizer of f over E, it follows that, for w ∈ E,

$$\begin{array}{ll}\hfill 0& \le \frac{f\left(w^{\prime }+\lambda \left(w-w^{\prime }\right)\right)-f\left(w^{\prime }\right)}{\lambda }\\ =\mathsf{\text{LI}}{\mathsf{\text{M}}}_k\frac{\frac{1}{2}{∥w_{t_k}-w^{\prime }+\lambda \left(w^{\prime }-w\right)∥}^2-\frac{1}{2}{∥w_{t_k}-w^{\prime }∥}^2}{\lambda }.\end{array}$$

Since E is uniformly smooth, the duality map J is uniformly continuous on bounded sets, letting λ → 0⁺ in the last equation yields

$$0\le \mathsf{\text{LI}}{\mathsf{\text{M}}}_k⟨w^{\prime }-w,J\left(w_{t_k}-w^{\prime }\right)⟩,w\in E.$$

(3.9)

Since

$$w_{t_k}-w^{\prime }=\left(I-t_{k}F\right)\left(V_{t_k}{w}_{t_k}-w^{\prime }\right)+t_k\left(\gamma \varphi \left(w_{t_k}\right)-F{w}^{\prime }\right),$$

we obtain

$$\begin{array}{ll}\hfill {∥w_{t_n}-w^{\prime }∥}^2& =t_k⟨\gamma \varphi \left(w_{t_k}\right)-F{w}^{\prime },J\left(w_{t_k}-w^{\prime }\right)⟩+⟨\left(I-t_{k}F\right)\left(V_{t_k}{w}_{t_k}-w^{\prime }\right),J\left(w_{t_k}-w^{\prime }\right)⟩\\ \le t_k⟨\gamma \varphi \left(w_{t_k}\right)-F{w}^{\prime },J\left(w_{t_k}-w^{\prime }\right)⟩+\left(1-t_k\bar{\gamma }\right){∥w_{t_k}-w^{\prime }∥}^2\\ \le t_k⟨\gamma \varphi \left(w_{t_k}\right)-\gamma \varphi \left(w^{\prime }\right),J\left(w_{t_k}-w^{\prime }\right)⟩+t_k⟨\gamma \varphi \left(w^{\prime }\right)-F{w}^{\prime },J\left(w_{t_k}-w^{\prime }\right)⟩+\left(1+t_k\bar{\gamma }\right){∥w_{t_k}-w^{\prime }∥}^2\\ \le \left[1-t_k\left(\bar{\gamma }-\gamma \right)\right]{∥w_{t_k}-w^{\prime }∥}^2+t_k⟨\gamma \varphi \left(w^{\prime }\right)-F{w}^{\prime },J\left(w_{t_k}-w^{\prime }\right)⟩.\end{array}$$

It follows that

$${∥w_{t_k}-w^{\prime }∥}^2\le \frac{1}{\bar{\gamma }-\gamma }⟨\gamma \varphi \left(w^{\prime }\right)-F{w}^{\prime },J\left(w_{t_k}-w^{\prime }\right)⟩.$$

(3.10)

Upon letting w = γϕ(w') - Fw' + w' in (3.9), we see that the last equation implies

$$\mathsf{\text{LI}}{\mathsf{\text{M}}}_k{∥w_{t_k}-w^{\prime }∥}^2\le 0.$$

(3.11)

Therefore, $\left\{w_{t_k}\right\}$ contains a subsequence, still denoted $\left\{w_{t_k}\right\}$, converging strongly to w₁ (say). By virtue of (3.8), w₁ is a fixed point of J _ε ; i.e., a point in F*.

To prove that the entire net {w _t } converges strongly, assume {s _k } is another null subsequence in (0, 1) such that $w_{s_k}\to w_2$ strongly. Then w₂ ∈ F*.

Repeating the argument of (3.10) we obtain

$${∥w_t-w^{\prime }∥}^2\le \frac{1}{\bar{\gamma }-\gamma }⟨\gamma \varphi \left(w^{\prime }\right)-F{w}^{\prime },J\left(w_t-w^{\prime }\right)⟩,\forall w^{\prime }\in F^{*}.$$

In particular,

$${∥w_1-w_2∥}^2\le \frac{1}{\bar{\gamma }-\gamma }⟨\gamma \varphi \left(w_1\right)-F{w}_1,J\left(w_2-w_1\right)⟩$$

(3.12)

and

$${∥w_2-w_1∥}^2\le \frac{1}{\bar{\gamma }-\gamma }⟨\gamma \varphi \left(w_2\right)-F{w}_2,J\left(w_1-w_2\right)⟩.$$

(3.13)

Adding up the last two equations gives

$${∥w_1-w_2∥}^2\le 0.$$

That is, w₁ = w₂. This concludes the proof.

Theorem 3.3. Suppose that E is a uniformly smooth Banach space and A is an m-accretive operator in E such that $F^{*}=N\left(A\right)\ne \varnothing$. Now given ϕ : E → E be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient $\bar{\gamma }>0,0<\gamma <\bar{\gamma }$. Assume

• (i) ${\text{lim}}_{n\to \infty }{\alpha }_n=0,{\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, and ${\sum }_{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty \left(e.g.,{\alpha }_n=\frac{1}{n}\right)$;

• (ii) lim_n→∞= r _n = r,r ∈ R⁺, r _n ≥ ε for all n and ${\sum }_{n=1}^{\infty }\left|r_{n+1}-r_n\right|<\infty \left(e.g.,r_n=1+\frac{1}{n}\right)$.

Then {x _n } defined by (1.4) converges strongly to a point in F*.

Proof. Since

$$\begin{array}{ll}\hfill ∥J_{r_n}{x}_n-J_r{x}_n∥& \le ∥J_r\left(\frac{r}{r_n}{x}_n+\left(1-\frac{r}{r_n}\right)J_{r_n}{x}_n\right)-J_r{x}_n∥\\ \le ∥\left(\frac{r}{r_n}{x}_n+\left(1-\frac{r}{r_n}\right)J_{r_n}{x}_n\right)-x_n∥\\ \le \left|1-\frac{r}{r_n}\right|∥J_{r_n}{x}_n-x_n∥\to 0\mathsf{\text{as}}\left(n\to \infty \right).\end{array}$$

(3.14)

Thus

$$∥x_n-J_r{x}_n∥\le ∥x_n-J_{r_n}{x}_n∥+∥J_{r_n}{x}_n-J_r{x}_n∥\to 0.$$

(3.15)

We next claim that $\text{lim }{\text{sup}}_{n\to \infty }⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_n-ẑ\right)⟩\le 0$, where $ẑ={\text{lim}}_{t\to 0}{z}_{t,n}$ with z _t,n = tγϕ(z _t,n ) + (I - tF)J _r z _t,n .

For this purpose, let $\left\{x_{n_k}\right\}$ be a subsequence chosen in such a way that $\text{lim}\underset{n\to \infty }{\text{sup}}⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_n-ẑ\right)⟩=\underset{k\to \infty }{\text{lim}}⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_{n_k}-ẑ\right)⟩$ and $x_{n_k}\rightharpoonup \tilde{x}$. Moreover, since ∥x _n - J _r x _n ∥ → 0, using Lemma 2.8, we know $\tilde{x}\in F\left(J_r\right)$. Hence by Lemma 2.6, we have

$$\underset{n\to \infty }{\lim \sup }〈\gamma \varphi (\widehat{z})-F\widehat{z},J(x_n-\widehat{z})〉=\underset{k\to \infty }{\lim }〈\gamma \varphi (\widehat{z})-F\widehat{z},J(x_{n_k}-\widehat{z})〉=〈\gamma \varphi (\widehat{z})-F\widehat{z},\tilde{x}-\widehat{z})〉\le 0.$$

(3.16)

Finally to prove that $x_n\to ẑ$ strongly, we write

$$x_{n+1}-ẑ=\left(I-{\alpha }_{n}F\right)\left(J_{r_n}{x}_n-ẑ\right)+{\alpha }_n\left(\gamma \varphi \left(x_n\right)-Fẑ\right).$$

Apply Lemma 2.9 to get

$$\begin{array}{ll}\hfill {∥x_{n+1}-ẑ∥}^2& \le {\left(1-{\alpha }_n\bar{\gamma }\right)}^2{∥x_n-ẑ∥}^2+2{\alpha }_n⟨\gamma \varphi \left(x_n\right)-Fẑ,J\left(x_{n+1}-ẑ\right)⟩\\ \le {\left(1-{\alpha }_n\bar{\gamma }\right)}^2{∥x_n-ẑ∥}^2+2{\alpha }_n⟨\gamma \varphi \left(x_n\right)-\gamma \varphi \left(ẑ\right),J\left(x_{n+1}-ẑ\right)⟩\\ +2{\alpha }_n⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_{n+1}-ẑ\right)⟩\\ \le {\left(1-{\alpha }_n\bar{\gamma }\right)}^2{∥x_n-ẑ∥}^2+{\alpha }_n\gamma \left({∥x_n-ẑ∥}^2+{∥x_{n+1}-ẑ∥}^2\right)\\ +2{\alpha }_n⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_{n+1}-ẑ\right)⟩\end{array}$$

It follows that

$$\begin{array}{ll}\hfill {∥x_{n+1}-ẑ∥}^2& \le \frac{{\left(1-{\alpha }_n\bar{\gamma }\right)}^2+{\alpha }_n\gamma }{1-{\alpha }_n\gamma }{∥x_n-ẑ∥}^2+\frac{2{\alpha }_n}{1-{\alpha }_n\gamma }⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_{n+1}-ẑ\right)⟩\\ \le \left(1-\frac{2{\alpha }_n\left(\bar{\gamma }-\gamma \right)}{1-{\alpha }_n\gamma }\right){∥x_n-ẑ∥}^2+\frac{2{\alpha }_n\left(\bar{\gamma }-\gamma \right)}{1-{\alpha }_n\gamma }\left[\frac{1}{\bar{\gamma }-\gamma }\right.⟨\gamma \varphi \left(ẑ\right)-Fẑ,J\left(x_{n+1}-ẑ\right)⟩\\ \left.+\frac{{\alpha }_n{\bar{\gamma }}^2}{2\left(\bar{\gamma }-\gamma \right)}{M}_1\right],\end{array}$$

where $M_1={\text{sup}}_{n\ge 1}{∥x_n-ẑ∥}^2$. By Lemma 2.2 and (3.16), we see that $x_n\to ẑ$.

Remark 3.4. If γ = 1, F is the identity operator and ϕ(x _n ) = u in our results, we can obtain Theorems 3.1, 4.1, 4.2, 4.4 and Lemma 4.3 of Hong-Kun Xu [5].