On shrinking projection method for cutter type mappings with nonsummable errors

Abstract

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space.

1 Introduction

Iterative methods play an important role in approximation theory. Several problems can be transformed into a problem of finding a fixed point of certain mappings. Many iterative methods have been proposed and analyzed (for example, see [3, 4, 6, 18, 19, 25, 28]). In this paper, we are interested in one of the promising methods, namely, the shrinking projection method. It was proposed by Takahashi et al. [24] who proved that a sequence generated by this method converges strongly to a fixed point of a certain mapping in the Hilbert space setting. Kimura [15] modified this iterative scheme in the sense that the inexact value of the projection is allowed, while the asymptotic behavior of the iterative sequence performs well. Our paper concerns cutter type mappings in Banach spaces with certain geometric properties, and it can be regarded as an extension and a supplement to the recent results of Ibaraki and Kimura [12] and of Ibaraki and Kajiba [11].

Let $E:=(E,\|\cdot \|)$ be a real Banach space with the dual space $E^{*}$. The strong and weak convergence in E are denoted by → and ⇀, respectively. The normalized duality mapping $J:E\to 2^{E^{*}}$ is defined by

$$ Jx:=\bigl\{ x^{*}\in E^{*}:\bigl\langle x,x^{*}\bigr\rangle = \Vert x \Vert \bigl\Vert x^{*} \bigr\Vert = \Vert x \Vert ^{2} \bigr\} \quad \text{for all $x\in E$}. $$

Recall that E is

• smooth if $\lim_{t\to 0}\frac{1}{t}(\|x+ty\|-\|x\|)$ exists for all $x,y\in E$;

• strictly convex if $\frac{1}{2}\|x+y\|<1$ for all $x,y\in E$ with $\|x\|=\|y\|=1$ and $x\neq y$.

If E is smooth, strictly convex, and reflexive, then J is single-valued and surjective. In this case, we can treat Jx as an element of $E^{*}$. For more details on the duality mappings, we refer to [5].

Let C be a nonempty subset of a smooth Banach space E. In this paper, we are interested in the following three different generalizations of cutter mappings in the Banach space setting [2, 16]. A mapping $T:C\to E$ with a nonempty fixed point set $\operatorname{Fix}(T):=\{z\in C:z=Tz\}$ is said to be

• a cutter mapping of type (P) if $\langle Tx-z,J(Tx-x)\rangle \le 0$ for all $x\in C$ and $z\in \operatorname{Fix}(T)$;

• a cutter mapping of type (Q) if $\langle Tx-z,JTx-Jx\rangle \le 0$ for all $x\in C$ and $z\in \operatorname{Fix}(T)$;

• a cutter mapping of type (R) if $\langle JTx-Jz,Tx-x\rangle \le 0$ for all $x\in C$ and $z\in \operatorname{Fix}(T)$.

Suppose that $V:E\times E\to [0,\infty )$ is a function defined by

$$ V(x,y):= \Vert x \Vert ^{2}-2\langle x,Jy\rangle + \Vert y \Vert ^{2} \quad \text{for all $x,y\in E$}. $$

It is known that

• $(\|x\|-\|y\|)^{2}\le V(x,y)\le (\|x\|+\|y\|)^{2}$ for all $x,y\in E$;

• $V(x,y)=V(x,z)+V(z,y)+2\langle x-z,Jz-Jy\rangle$ for all $x,y,z\in E$.

Lemma 1

Suppose that E is a smooth Banach space and C is a nonempty closed and convex subset of E. The following statements are true:

• (a) If $T:C\to E$ is a cutter mapping of type (P), then $\operatorname{Fix}(T)$ is closed and convex;

• (b) Suppose that E is strictly convex. If $T:C\to E$ is a cutter mapping of type (Q), then $\operatorname{Fix}(T)$ is closed and convex.

Proof

(a) Suppose that T is a cutter mapping of type (P) where C is closed and convex. We show that $\operatorname{Fix}(T)$ is closed. Assume that $\{z_{n}\}$ is a sequence in $\operatorname{Fix}(T)$ such that $z_{n}\to z\in C$; then $\langle Tz-z_{n},J(Tz-z)\rangle \le 0$ for all $n\ge 1$. In particular, $\|Tz-z\|^{2}=\langle Tz-z,J(Tz-z)\rangle \le 0$, that is, $z\in \operatorname{Fix}(T)$. Hence, $\operatorname{Fix}(T)$ is closed. Next, we prove that $\operatorname{Fix}(T)$ is convex. To see this, let $z,z'\in \operatorname{Fix}(T)$ and $\lambda \in [0,1]$. We write $w:=\lambda z+(1-\lambda )z'$. Obviously, $w\in C$. Moreover, $\langle Tw-z,J(Tw-w)\rangle \le 0$ and $\langle Tw-z',J(Tw-w)\rangle \le 0$. This implies that $\|Tw-w\|^{2}=\langle Tw-w,J(Tw-w)\rangle \le 0$, that is, $w\in \operatorname{Fix}(T)$.

(b) Suppose that E is strictly convex and $T:C\to E$ is a cutter mapping of type (Q), where C is closed and convex. We show that $\operatorname{Fix}(T)$ is closed. Assume that $\{z_{n}\}$ is a sequence in $\operatorname{Fix}(T)$ such that $z_{n}\to z\in C$; then $\langle Tz-z_{n},JTz-Jz\rangle \le 0$ for all $n\ge 1$. In particular, $\langle Tz-z,JTz-Jz\rangle \le 0$. It follows from the strict convexity of E that $z\in \operatorname{Fix}(T)$. Hence, $\operatorname{Fix}(T)$ is closed. Next, we prove that $\operatorname{Fix}(T)$ is convex. To see this, let $z,z'\in \operatorname{Fix}(T)$ and $\lambda \in [0,1]$. We write $w:=\lambda z+(1-\lambda )z'$. Obviously, $w\in C$. Moreover, $\langle Tw-z,JTw-Jw\rangle \le 0$ and $\langle Tw-z',JTw-Jw\rangle \le 0$. This implies that $\langle Tw-w,JTw-Jw\rangle \le 0$ and hence $w\in \operatorname{Fix}(T)$. □

In this paper, we also consider the following geometric properties. A Banach space E is

• uniformly convex if $\lim_{n}\|x_{n}-y_{n}\|=0$ whenever $\{x_{n}\}$ and $\{y_{n}\}$ are sequences in E satisfying $\lim_{n}\|x_{n}\|=\lim_{n}\|y_{n}\|=\lim_{n}\frac{1}{2}\|x_{n}+y_{n} \|=1$;

• uniformly smooth if $\lim_{t\to 0}\frac{1}{t}(\|x+ty\|-\|x\|)$ exists uniformly for all $x,y\in E$ with $\|x\|=\|y\|=1$.

Note that every uniformly convex (uniformly smooth, respectively) space is reflexive and strictly convex (smooth, respectively). Moreover, uniform convexity and uniform smoothness are dual to each other, that is, E is uniformly convex (uniformly smooth, respectively) if and only if $E^{*}$ is uniformly smooth (uniformly convex, respectively).

Lemma 2

([5])

If E is a uniformly smooth Banach space, then $J:E\to E^{*}$ is norm-to-norm uniformly continuous on bounded sets.

Lemma 3

([14])

Suppose that E is a uniformly convex and smooth Banach space and suppose that $\{x_{n}\}$ and $\{y_{n}\}$ are sequences in E. If $\lim_{n}V(x_{n},y_{n})=0$ and either $\{x_{n}\}$ or $\{y_{n}\}$ is bounded, then $\lim_{n}\|x_{n}-y_{n}\|=0$.

In each subsection, we present some prototypes of cutter type mappings of types (P), (Q), and (R) together with their properties.

1.1 Metric projections

Suppose that E is a strictly convex and reflexive Banach space. Let C be a closed convex subset of E and $x\in E$. It is known that there exists a unique element $z\in C$ such

$$ \Vert z-x \Vert =\min \bigl\{ \Vert y-x \Vert :y\in C\bigr\} . $$

Such an element z is denoted by $P_{C}x$. Now, we call $P_{C}$ the metric projection of E onto C. It is easy to see that $\operatorname{Fix}(P_{C})=C$ and $P_{C}$ is a cutter mapping of type (P). The following easy result is also needed in our study.

Lemma 4

Suppose that C is a closed convex subset of a strictly convex and reflexive Banach space E and $u\in E$. If F is a nonempty closed convex subset of C such that $P_{C}u\in F$, then $P_{F}u=P_{C}u$.

Proof

It follows from the definition of $P_{C}$ that $\|P_{C}u-u\|\le \|y-u\|$ for all $y\in C$. Since $F\subset C$, we have $\|P_{C}u-u\|\le \|y-u\|$ for all $y\in F$. It follows from $P_{C}u\in F$ that $P_{C}u=P_{F}u$. □

The following lemma (see [15, Lemma 2.1]) is easily deduced from the result of Tsukada [26]. Recall that E satisfies the Kadec–Klee property if $x_{n}\to x$ whenever $\{x_{n}\}$ is a sequence in E such that $x_{n}\rightharpoonup x\in E$ and $\|x_{n}\|\to \|x\|$. It is known that every uniformly convex space satisfies the Kadec–Klee property.

Lemma 5

([15])

Suppose that E is a strictly convex reflexive Banach space and E satisfies the Kadec–Klee property. If $\{C_{n}\}$ is a sequence of nonempty closed convex subsets of E such that $C_{n+1}\subset C_{n}$ for all $n\ge 1$ and $C_{0}:=\bigcap_{n=1}^{\infty }C_{n}$ is nonempty, then $\{P_{C_{n}}x\}$ converges strongly to $P_{C_{0}}x$ for all $x\in E$.

1.2 Generalized projections

Suppose that E is a smooth, strictly convex, and reflexive Banach space. Let C be a closed convex subset of E and $x\in E$. It is known that there exists a unique element $z\in C$ such

$$ V(z,x)=\min \bigl\{ V(y,x):y\in C\bigr\} :=V(C,x). $$

Such an element z is denoted by $\varPi _{C}x$. Now, we call $\varPi _{C}$ the generalized projection of E onto C. It is easy to see that $\operatorname{Fix}(\varPi _{C})=C$ and $\varPi _{C}$ is a cutter mapping of type (Q). Moreover, we have the following result.

Lemma 6

([1])

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Suppose that $x\in E$ and $z\in C$. Then the following statements are equivalent:

• $z=\varPi _{C}x$;

• $\langle y-z,Jz-Jx\rangle \ge 0$ for all $y\in C$;

• $V(y,z)+V(z,x)\le V(y,x)$ for all $y\in C$.

Lemma 7

Suppose that C is a closed convex subset of a smooth, strictly convex, and reflexive Banach space E and $u\in E$. If F is a nonempty closed convex subset of C such that $\varPi _{C}u\in F$, then $\varPi _{F}u=\varPi _{C}u$.

Proof

It follows from the definition of $\varPi _{C}$ that $V(\varPi _{C}u,u)\le V(y,u)$ for all $y\in C$. Since $F\subset C$, we have $V(\varPi _{C}u,u)\le V(y,u)$ for all $y\in F$. It follows from $\varPi _{C}u\in F$ that $\varPi _{C}u=\varPi _{F}u$. □

The following lemma is easily deduced from the result of Ibaraki et al. [13].

Lemma 8

([13])

Suppose that E is a strictly convex, smooth, and reflexive Banach space and E satisfies the Kadec–Klee property. If $\{C_{n}\}$ is a sequence of nonempty closed convex subsets of E such that $C_{n+1}\subset C_{n}$ for all $n\ge 1$ and $C_{0}:=\bigcap_{n=1}^{\infty }C_{n}$ is nonempty, then $\{\varPi _{C_{n}}x\}$ converges strongly to $\varPi _{C_{0}}x$ for all $x\in E$.

1.3 Sunny generalized nonexpansive retractions

Suppose that E is a smooth Banach space and $C\subset E$. A mapping $R:C\to E$ is a generalized nonexpansive if $\operatorname{Fix}(R)\neq \varnothing$ and $V(Rx,p)\le V(x,p)$ for all $x\in C$ and for all $p\in \operatorname{Fix}(R)$. A mapping $R:E\to C$ is

• a retraction if $R^{2}=R$;

• sunny if $R(Rx+t(x-Rx))=Rx$ for all $x\in E$ and for all $t>0$.

If $R:E\to C$ is a sunny generalized nonexpansive retraction from E onto C, then $\operatorname{Fix}(R)=C$ and R is a cutter mapping of type (R).

Kohsaka and Takahashi [17] proved the following result.

Lemma 9

([17])

Suppose that E is a smooth, strictly convex, and reflexive Banach space, and suppose that $C^{*}$ is a nonempty closed convex subset of $E^{*}$. Then $R:=J^{-1}\varPi _{C^{*}}J$ is a sunny generalized nonexpansive retraction from E onto $J^{-1}C^{*}$, where $\varPi _{C^{*}}$ is the generalized projection of $E^{*}$ onto $C^{*}$.

The purpose of this paper is to present some asymptotic behavior of an iterative sequence generated by the shrinking projection method for cutter type mappings.

2 Preliminaries

In this section, we collect some auxiliary results used in our main results. For a Banach space $E:=(E,\|\cdot \|)$ and $r>0$, we let $B_{r}:=\{x\in E:\|x\|\le r\}$.

Lemma 10

([27])

Suppose that E is a Banach space and $r>0$. Then the following statements are true:

• (a) If E is uniformly convex, then there is a continuous, strictly increasing, and convex function $\underline{g}_{r}:[0,2r]\to [0,\infty )$ such that $\underline{g}_{r}(0)=0$ and

  $$ \bigl\Vert (1-\lambda )x+\lambda y \bigr\Vert ^{2}\le (1-\lambda ) \Vert x \Vert ^{2}+\lambda \Vert y \Vert ^{2}-(1-\lambda )\lambda \underline{g}_{r}\bigl( \Vert x-y \Vert \bigr) $$

  for all $x,y\in B_{r}$ and for all $\lambda \in [0,1]$;

• (b) If E is uniformly smooth, then there is a continuous, strictly increasing, and convex function $\overline{g}_{r}:[0,2r]\to [0,\infty )$ such that $\overline{g}_{r}(0)=0$ and

  $$ \bigl\Vert (1-\lambda )x+\lambda y \bigr\Vert ^{2}\ge (1-\lambda ) \Vert x \Vert ^{2}+\lambda \Vert y \Vert ^{2}-(1-\lambda )\lambda \overline{g}_{r}\bigl( \Vert x-y \Vert \bigr) $$

  for all $x,y\in B_{r}$ and for all $\lambda \in [0,1]$.

Lemma 11

([15])

Suppose that E is a Banach space and $r>0$. Then the following statements are true:

• (a) If E is uniformly convex and smooth, then the function $\underline{g}_{r}$ in Lemma 10(a) satisfies $\underline{g}_{r}(\|x-y\|)\le V(x,y)$ for all $x,y\in B_{r}$;

• (b) If E is uniformly smooth, then the function $\overline{g}_{r}$ in Lemma 10(b) satisfies $\overline{g}_{r}(\|x-y\|)\ge V(x,y)$ for all $x,y\in B_{r}$.

3 Main results

The main results are presented according to the types of cutter mappings. The results of each subsection are given with respect to the functionals $\|\cdot \|^{2}$ and $V(\cdot ,\cdot )$. The corresponding remark about the related results is presented. Our results are some extensions of the recent ones proved by Ibaraki and Kajiba [11] for mappings of types (P), (Q), and (R). Recall that for a nonempty subset C of a smooth, strictly convex, and reflexive Banach space E, a mapping $T:C\to E$ is of

• type (P) if $\langle Tx-Ty,J(x-Tx)-J(y-Ty)\rangle \ge 0$ for all $x,y\in C$;

• type (Q) if $\langle Tx-Ty,(Jx-JTx)-(Jy-JTy)\rangle \ge 0$ for all $x,y\in C$;

• type (R) if $\langle JTx-JTy,(x-Tx)-(y-Ty)\rangle \ge 0$ for all $x,y\in C$.

It is clear that every mapping of type (X) with a fixed point is a cutter mapping of type (X) where X = P, Q, R. Moreover, it is not hard to see that if $T:C\to E$ is a mapping of type (X) where X = P, Q, R, then $I-T$ is strongly closed at zero, that is, $p\in \operatorname{Fix}(T)$ whenever $\{x_{n}\}$ is a sequence in C such that $x_{n}\to p\in C$ and $Tx_{n}\to p$.

3.1 Two key inequalities for metric projections and generalized projections

We now prove two key inequalities of this paper. The first result is a Banach space version of the result in [23] concerning the generalized projection. The second one is for the metric projection.

Lemma 12

Suppose that E is a smooth, strictly convex, and reflexive Banach space and C is a closed convex subset of E. Suppose that $u\in E$ and $\delta \ge 0$. If $x\in C$ satisfies

$$ V(x,u)\le V(C,u)+\delta , $$

then $V(x,\varPi _{C}u)\le \delta$.

Proof

Note that $V(x,\varPi _{C}u)+V(C,u)=V(x,\varPi _{C}u)+V(\varPi _{C}u,u)\le V(x,u) \le V(C,u)+\delta$. Specifically, $V(x,\varPi _{C}u)\le \delta$. □

Lemma 13

Suppose that E is a uniformly convex Banach space, $u\in E$, and $\delta \ge 0$. Suppose that C is a closed convex subset of E such that $C-u\subset B_{r}$ for some $r>0$. If $x\in C$ satisfies

$$ \Vert u-x \Vert ^{2}\le d^{2}(u,C)+\delta , $$

then $\underline{g}_{r}(\|x-P_{C}u\|)\le \delta$, where $\underline{g}_{r}$ is the function defined in Lemma 10(a).

Proof

Let $p:=P_{C}u$. Then $\|u-x\|^{2}\le \|u-p\|^{2}+\delta$. Let $\lambda \in (0,1)$. It follows that $\lambda p+(1-\lambda )x\in C$. Note that $\{p-u,x-u\}\subset C-u\subset B_{r}$, and we make use of the function $\underline{g}_{r}$ in Lemma 10(a) to estimate the term $\|\lambda (p-u)+(1-\lambda )(x-u)\|^{2}$. Hence,

$$\begin{aligned} \Vert p-u \Vert ^{2} &\le \bigl\Vert \lambda p+(1-\lambda )x-u \bigr\Vert ^{2} \\ &= \bigl\Vert \lambda (p-u)+(1-\lambda ) (x-u) \bigr\Vert ^{2} \\ &\le \lambda \Vert p-u \Vert ^{2}+(1-\lambda ) \Vert x-u \Vert ^{2}-\lambda (1-\lambda ) \underline{g}_{r}\bigl( \Vert p-x \Vert \bigr) \\ &\le \Vert p-u \Vert ^{2}+(1-\lambda )\delta -\lambda (1-\lambda ) \underline{g}_{r}\bigl( \Vert p-x \Vert \bigr). \end{aligned}$$

In particular, $\lambda \underline{g}_{r}(\|p-x\|)\le \delta$. Letting $\lambda \uparrow 1$ gives the result. □

3.2 Cutter mappings of type (P)

Theorem 14

Suppose that E is a smooth and uniformly convex Banach space and C is a closed convex subset of E. Suppose that $T:C\to E$ is a cutter mapping of type (P). Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. For given $u\in E$, a sequence $\{x_{n}\}\subset C$ is generated as follows: $x_{1}\in C$, $C_{1}:=C$, and

$$\begin{aligned}& C_{n+1}:=\bigl\{ z\in C_{n}:\bigl\langle Tx_{n}-z,J(x_{n}-Tx_{n})\bigr\rangle \ge 0 \bigr\} ; \\& x_{n+1}\in C_{n+1}\textit{ satisfies } \Vert x_{n+1}-u \Vert ^{2}\le d^{2}(u,C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $C-u\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}_{r}^{-1}(\delta _{0})$, where $\underline{g}_{r}$ is defined by Lemma 10;

• (2) If $\delta _{0}=0$ and $I-T$ is strongly closed at zero, then $x_{n}\to P_{\operatorname{Fix}(T)}u$.

Proof

First, we note that each $C_{n}$ is nonempty because $\operatorname{Fix}(T)\subset C_{n}$. Moreover, it is clear that each $C_{n}$ is closed and convex. For convenience, we write $p_{n}:=P_{C_{n}}u$ and $p:=P_{\bigcap _{n=1}^{\infty }C_{n}}u$. Note that $p_{n}\to p$ (see Lemma 5). It follows from $p\in C_{n+1}$ that

$$ \bigl\langle Tx_{n}-p,J(x_{n}-Tx_{n}) \bigr\rangle \ge 0. $$

In particular,

$$ \bigl\langle x_{n}-p,J(x_{n}-Tx_{n})\bigr\rangle \ge \Vert x_{n}-Tx_{n} \Vert ^{2}. $$

(1) Suppose that $C-u\subset B_{r}$ for some $r>0$. It follows from Lemma 13 that

$$ \Vert x_{n}-Tx_{n} \Vert \le \Vert x_{n}-p \Vert \le \Vert x_{n}-p_{n} \Vert + \Vert p_{n}-p \Vert \le \underline{g}_{r}^{-1}( \delta _{n})+ \Vert p_{n}-p \Vert . $$

Hence, $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\delta _{0})$.

(2) We assume that $\delta _{0}=0$ and $I-T$ is strongly closed at zero. Note that $\|x_{n}-u\|^{2}\le \|p_{n}-u\|^{2}+\delta _{n}$. In particular, $\{x_{n}\}$ is bounded. We prove that $x_{n}\rightharpoonup p$. Suppose that $\{x_{n_{k}}\}$ is a subsequence of $\{x_{n}\}$ such that $x_{n_{k}}\rightharpoonup q$ for some $q\in E$. It follows that $q\in \bigcap_{n=1}^{\infty }C_{n}$ and $\|q-u\|^{2}\le \liminf_{k}\|x_{n_{k}}-u\|^{2}\le \limsup_{k}\|x_{n_{k}}-u \|^{2}\le \lim_{k}\|p_{n_{k}}-u\|^{2}=\|p-u\|^{2}$. Hence, $q=p$. Moreover, we have $\lim_{n}\|x_{n}-u\|=\|p-u\|$. It follows from the Kadec–Klee property and $x_{n}-u\rightharpoonup p-u$ that $x_{n}-u\to p-u$ and hence $x_{n}\to p$. It follows from a part of (1) that $\lim_{n}\|x_{n}-Tx_{n}\|=0$. Since $I-T$ is strongly closed at zero, we have $p\in \operatorname{Fix}(T)\subset \bigcap_{n=1}^{\infty }C_{n}$. Note that $\operatorname{Fix}(T)$ is closed and convex (see Lemma 1). It follows from Lemma 4 that $x_{n}\to p=P_{\operatorname{Fix}(T)}u$. □

Theorem 15

Suppose that E is a smooth and uniformly convex Banach space and C is a closed convex subset of E. Suppose that $T:C\to E$ is a cutter mapping of type (P). Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. For given $u\in E$, a sequence $\{x_{n}\}\subset C$ is generated as follows: $x_{1}\in C$, $C_{1}:=C$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\bigl\langle Tx_{n}-z,J(x_{n}-Tx_{n})\bigr\rangle \ge 0 \bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies }V(x_{n+1},u) \le V(C_{n+1},u)+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $C\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}_{r}^{-1}(\delta _{0})$, where $\underline{g}_{r}$ is defined by Lemma 10.

• (2) If $\delta _{0}=0$ and $I-T$ is strongly closed at zero, then $x_{n}\to \varPi _{\operatorname{Fix}(T)}u$.

Proof

First, we note that each $C_{n}$ is nonempty because $\operatorname{Fix}(T)\subset C_{n}$. Moreover, it is clear that each $C_{n}$ is closed and convex. For convenience, we write $\pi _{n}:=\varPi _{C_{n}}u$ and $\pi :=\varPi _{\bigcap _{n=1}^{\infty }C_{n}}u$. Note that $\pi _{n}\to \pi$ (see Lemma 8). It follows from $\pi \in C_{n+1}$ that

$$ \bigl\langle Tx_{n}-\pi ,J(x_{n}-Tx_{n}) \bigr\rangle \ge 0. $$

In particular,

$$ \Vert x_{n}-Tx_{n} \Vert ^{2}\le \bigl\langle x_{n}-\pi ,J(x_{n}-Tx_{n})\bigr\rangle \le \Vert x_{n}-\pi \Vert \Vert x_{n}-Tx_{n} \Vert . $$

Note that $V(x_{n},\pi _{n})\le \delta _{n}$ (see Lemma 12).

(1) Suppose that $C\subset B_{r}$ for some $r>0$. It follows from Lemma 11 that $\|x_{n}-\pi _{n}\|\le \underline{g}^{-1}_{r}(V(x_{n},\pi _{n}))\le \underline{g}^{-1}_{r}(\delta _{n})$, which implies that

$$\begin{aligned} \Vert x_{n}-Tx_{n} \Vert &\le \Vert x_{n}-\pi \Vert \\ &\le \Vert x_{n}-\pi _{n} \Vert + \Vert \pi _{n}-\pi \Vert \\ &\le \underline{g}^{-1}_{r}(\delta _{n})+ \Vert \pi _{n}-\pi \Vert . \end{aligned}$$

Hence, $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\delta _{0})$.

(2) We assume that $\delta _{0}=0$ and $I-T$ is strongly closed at zero. Since $\delta _{0}=0$, we have $\lim_{n}V(x_{n},\pi _{n})=0$. It follows from Lemma 3 that $\lim_{n}\|x_{n}-\pi _{n}\|=0$. In particular, $\lim_{n}\|x_{n}-Tx_{n}\|\le \lim_{n}\|x_{n}-\pi \|=\lim_{n}\|x_{n}- \pi _{n}\|=0$. Since $I-T$ is strongly closed at zero, we have $\pi \in \operatorname{Fix}(T)\subset \bigcap_{n=1}^{\infty }C_{n}$. Note that $\operatorname{Fix}(T)$ is closed and convex (see Lemma 1). It follows from Lemma 7 that $x_{n}\to \pi =\varPi _{\operatorname{Fix}(T)}u$. □

Remark 16

Our Theorem 14 and Theorem 15 generalize Theorem 3.1 of [12] and Theorem 3.1 of [11], respectively. In fact, the mapping of type (P) in [11, 12] is replaced by the cutter mapping of type (P).

3.3 Cutter mappings of type (Q)

Theorem 17

Suppose that E is a uniformly smooth and uniformly convex Banach space and C is a closed convex subset of E. Suppose that $T:C\to E$ is a cutter mapping of type (Q). Suppose that $\{\delta _{n}\}$ is a sequence in $[0,\infty )$ with $\delta _{0}:=\limsup_{n}\delta _{n}$. For given $u\in E$, a sequence $\{x_{n}\}\subset C$ is generated as follows: $x_{1}\in C$, $C_{1}:=C$ and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle Tx_{n}-z,Jx_{n}-JTx_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies } \Vert x_{n+1}-u \Vert ^{2}\le d^{2}(u,C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $(C-u)\cup C\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\overline{g}_{r}( \underline{g}_{r}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ and $\overline{g}_{r}$ are defined by Lemma 10;

• (2) If $\delta _{0}=0$ and $I-T$ is strongly closed at zero, then $x_{n}\to P_{\operatorname{Fix}(T)}u$.

Proof

First, we note that each $C_{n}$ is nonempty because $\operatorname{Fix}(T)\subset C_{n}$. Moreover, it is clear that each $C_{n}$ is closed and convex. For convenience, we write $p_{n}:=P_{C_{n}}u$ and $p:=P_{\bigcap _{n=1}^{\infty }C_{n}}u$. Note that $p_{n}\to p$ (see Lemma 8). It follows from $p \in C_{n+1}$ that $\langle Tx_{n}-p,Jx_{n}-JTx_{n}\rangle \ge 0$ and hence

$$ V(Tx_{n},x_{n})\le V(p,Tx_{n})+V(Tx_{n},x_{n}) \le V(p,x_{n}). $$

(1) Assume that $(C-u)\cup C\subset B_{r}$ for some $r>0$. It follows from Lemma 13 that $\underline{g}_{r}(\|p_{n}-x_{n}\|)\le \delta _{n}$. By Lemma 11, we have $V(p,x_{n})\le \overline{g}_{r}(\|p-x_{n}\|)$. This implies that

$$\begin{aligned} V(Tx_{n},x_{n}) &\le V(p,x_{n}) \\ &\le \overline{g}_{r}\bigl( \Vert p-x_{n} \Vert \bigr) \\ &\le \overline{g}_{r}\bigl( \Vert p-p_{n} \Vert + \Vert p_{n}-x_{n} \Vert \bigr) \\ &\le \overline{g}_{r}\bigl( \Vert p-p_{n} \Vert + \underline{g}_{r}^{-1}(\delta _{n})\bigr). \end{aligned}$$

This implies that

$$ \limsup_{n} V(Tx_{n},x_{n})\le \overline{g}_{r}\bigl(\underline{g}_{r}^{-1}( \delta _{0})\bigr). $$

In particular, $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\overline{g}_{r}( \underline{g}_{r}^{-1}(\delta _{0})))$.

(2) We assume that $\delta _{0}=0$ and $I-T$ is strongly closed at zero. Note that $\|x_{n}-u\|^{2}\le \|p_{n}-u\|^{2}+\delta _{n}$. In particular, $\{x_{n}\}$ is bounded. We prove that $x_{n}\rightharpoonup p$. Suppose that $\{x_{n_{k}}\}$ is a subsequence of $\{x_{n}\}$ such that $x_{n_{k}}\rightharpoonup q$ for some $q\in E$. It follows that $q\in \bigcap_{n=1}^{\infty }C_{n}$ and $\|q-u\|^{2}\le \liminf_{k}\|x_{n_{k}}-u\|^{2}\le \limsup_{k}\|x_{n_{k}}-u \|^{2}\le \lim_{k}\|p_{n_{k}}-u\|^{2}=\|p-u\|^{2}$. Hence, $q=p$. Moreover, we have $\lim_{n}\|x_{n}-u\|=\|p-u\|$. It follows from the Kadec–Klee property and $x_{n}-u\rightharpoonup p-u$ that $x_{n}-u\to p-u$ and hence $x_{n}\to p$. It follows from the uniform convexity of E and $\lim_{n}V(p,x_{n})=0$ that $\lim_{n}\|x_{n}-Tx_{n}\|=0$. Since $I-T$ is strongly closed at zero, we have $p\in \operatorname{Fix}(T)\subset \bigcap_{n=1}^{\infty }C_{n}$. Note that $\operatorname{Fix}(T)$ is closed and convex (see Lemma 1). It follows from Lemma 4 that $x_{n}\to p=P_{\operatorname{Fix}(T)}u$. □

Theorem 18

Suppose that E is a uniformly smooth and uniformly convex Banach space and C is a closed convex subset of E. Suppose that $T:C\to E$ is a cutter mapping of type (Q). Suppose that $\{\delta _{n}\}$ is a sequence in $[0,\infty )$ with $\delta _{0}:=\limsup_{n}\delta _{n}$. For given $u\in E$, a sequence $\{x_{n}\}\subset C$ is generated as follows: $x_{1}\in C$, $C_{1}:=C$ and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle Tx_{n}-z,Jx_{n}-JTx_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies }V(x_{n+1},u) \le V(C_{n+1},u)+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $C\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\overline{g}_{r}( \underline{g}_{r}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ and $\overline{g}_{r}$ are defined by Lemma 10;

• (2) If $\delta _{0}=0$ and $I-T$ is strongly closed at zero, then $x_{n}\to \varPi _{\operatorname{Fix}(T)}u$.

Proof

First, we note that each $C_{n}$ is nonempty because $\operatorname{Fix}(T)\subset C_{n}$. Moreover, it is clear that each $C_{n}$ is closed and convex. For convenience, we write $\pi _{n}:=\varPi _{C_{n}}u$ and $\pi :=\varPi _{\bigcap _{n=1}^{\infty }C_{n}}u$. It follows from $\pi \in C_{n+1}$ that

$$ \langle Tx_{n}-\pi ,Jx_{n}-JTx_{n}\rangle \ge 0, $$

and hence

$$ V(Tx_{n},x_{n})\le V(\pi ,Tx_{n})+V(Tx_{n},x_{n}) \le V(\pi ,x_{n}). $$

Note that $\pi _{n}\to \pi$ (see Lemma 8). By Lemma 12, we have $V(x_{n},\pi _{n})\le \delta _{n}$.

(1) Suppose that $C\subset B_{r}$ for some $r>0$. It follows from Lemma 11 that $V(\pi ,x_{n})\le \overline{g}_{r}(\|\pi -x_{n}\|)$ and $\underline{g}_{r}(\|\pi -x_{n}\|)\le V(x_{n},\pi _{n})\le \delta _{n}$. In particular,

$$\begin{aligned} V(Tx_{n},x_{n})&\le V(\pi ,x_{n}) \\ &\le \overline{g}_{r}\bigl( \Vert \pi -x_{n} \Vert \bigr) \\ &\le \overline{g}_{r}\bigl( \Vert \pi -\pi _{n} \Vert + \Vert \pi _{n}-x_{n} \Vert \bigr) \\ &\le \overline{g}_{r}\bigl( \Vert \pi -\pi _{n} \Vert +\underline{g}_{r}^{-1}( \delta _{n}) \bigr) . \end{aligned}$$

This implies that

$$ \limsup_{n} V(Tx_{n},x_{n})\le \overline{g}_{r}\bigl(\underline{g}_{r}^{-1}( \delta _{0})\bigr). $$

In particular, $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\overline{g}_{r}( \underline{g}_{r}^{-1}(\delta _{0})))$.

(2) We assume that $\delta _{0}=0$ and $I-T$ is strongly closed at zero. Since $\delta _{0}=0$, we have $\lim_{n}V(x_{n},\pi _{n})=0$. It follows from Lemma 3 that $\lim_{n}\|x_{n}-\pi _{n}\|=0$. In particular, $\lim_{n}\|x_{n}-\pi \|=0$. Now, we have $\lim_{n}V(Tx_{n},x_{n})\le \lim_{n}V(\pi ,x_{n})=0$. It follows from the uniform convexity of E that $\lim_{n}\|x_{n}-Tx_{n}\|=0$. Since $I-T$ is strongly closed at zero, we have $\pi \in \operatorname{Fix}(T)\subset \bigcap_{n=1}^{\infty }C_{n}$. Note that $\operatorname{Fix}(T)$ is closed and convex (see Lemma 1). It follows from Lemma 7 that $x_{n}\to \pi =\varPi _{\operatorname{Fix}(T)}u$. □

Remark 19

Our Theorem 17 and Theorem 18 generalize Theorem 4.1 of [12] and Theorem 4.1 of [11], respectively. In fact, the mapping of type (Q) in [11, 12] is replaced by the cutter mapping of type (Q).

3.4 Cutter mappings of type (R)

Suppose that $V^{*}:E^{*}\times E^{*}\to [0,\infty )$ is defined by

$$ V^{*}\bigl(x^{*},y^{*}\bigr):= \bigl\Vert x^{*} \bigr\Vert ^{2}-2\bigl\langle x^{*},J^{*}y^{*}\bigr\rangle + \bigl\Vert y^{*} \bigr\Vert ^{2}\quad \text{for all $x^{*},y^{*}\in E^{*}$}. $$

Note that for $x,y\in E$ we have

$$\begin{aligned} V^{*}(Jy,Jx)&= \Vert Jy \Vert ^{2}-2\bigl\langle Jy,J^{*}Jx\bigr\rangle + \Vert Jx \Vert ^{2} \\ &= \Vert y \Vert ^{2}-2\langle x,Jy\rangle + \Vert x \Vert ^{2} \\ &=V(x,y). \end{aligned}$$

Inspired by the work of Honda et al. [7], we obtain the following results.

Lemma 20

Suppose that C is a nonempty subset of a smooth, strictly convex, and reflexive Banach space E. Suppose that $T:C\to E$ is a mapping, and we define $T^{*}:JC\to E^{*}$ by

$$ T^{*}x^{*}:=JTJ^{-1}x^{*}\quad \textit{for all $x^{*}\in JC$}. $$

Then the following statements are true:

• (1) $J\operatorname{Fix}(T)=\operatorname{Fix}(T^{*})$;

• (2) T is a cutter mapping of type (Q) if and only if $T^{*}$ is a cutter mapping of type (R);

• (3) T is a cutter mapping of type (R) if and only if $T^{*}$ is a cutter mapping of type (Q).

Proof

(1) Suppose that $p\in C$ and $p^{*}:=Jp$. If $p=Tp$, then $T^{*}p^{*}=JTJ^{-1}(Jp)=Jp=p^{*}$. If $p^{*}=T^{*}p$, then $Tp=J^{-1}T^{*}Jp=J^{-1}T^{*}p^{*}=J^{-1}p^{*}=p$.

(2) Suppose that T is a cutter mapping of type (Q). To see that $T^{*}$ is a cutter mapping of type (R), let $x^{*}\in JC$ and $p^{*}\in \operatorname{Fix}(T^{*})$. Then $x^{*}=Jx$ and $p^{*}=Jp$ for some $(x,p)\in C\times \operatorname{Fix}(T)$. It follows that

$$ \bigl\langle J^{*}p^{*}-J^{*}T^{*}x^{*},T^{*}x^{*}-x^{*} \bigr\rangle = \langle p-Tx,JTx-Jx \rangle \ge 0. $$

On the other hand, we assume that $T^{*}$ is a cutter mapping of type (R). To see that T is a cutter mapping of type (Q), let $(x,p)\in C\times \operatorname{Fix}(T)$. We write $x^{*}=Jx$ and $p^{*}=Jp$. It follows that $x^{*}\in JC$ and $p^{*}\in \operatorname{Fix}(T^{*})$. Moreover, we have

$$ \langle p-Tx,JTx-Jx\rangle =\bigl\langle J^{*}p^{*}-J^{*}T^{*}x^{*},T^{*}x^{*}-x^{*} \bigr\rangle \ge 0. $$

(3) follows similarly. □

Based on Lemma 20 and Theorem 17, we obtain the following result.

Theorem 21

Suppose that E is a uniformly smooth and uniformly convex Banach space. Suppose that C is a subset of E such that JC is closed and convex. Suppose that $T:C\to E$ is a cutter mapping of type (R). Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. For given $u\in E$, a sequence $\{x_{n}\}\subset C$ is generated as follows: $x_{1}\in C$, $C_{1}:=C$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle JTx_{n}-Jz,x_{n}-Tx_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies } \Vert Ju-Jx_{n+1} \Vert ^{2}\le d^{2}(Ju,JC_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $(JC-Ju)\cup JC\subset B_{r}^{*}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\overline{g}^{*}_{r}({ \underline{g}^{*}_{r}}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ is defined by Lemma 10 (as E is uniformly convex) and $\underline{g}^{*}_{r}$ and $\overline{g}^{*}_{r}$ are defined by Lemma 10 (as $E^{*}$ is uniformly convex and uniformly smooth, respectively). Here, $B_{r}^{*}:=\{x^{*}\in E^{*}:\|x^{*}\|\le r\}$.

• (2) If $\delta _{0}=0$ and $I-T$ is strongly closed at zero, then $x_{n}\to J^{-1}P^{*}_{J\operatorname{Fix}(T)}Ju$, where $P^{*}_{J\operatorname{Fix}(T)}$ is a metric projection of $E^{*}$ onto $J\operatorname{Fix}(T)$.

Proof

Set $T^{*}:=JTJ^{-1}$. Then $T^{*}:JC\to E^{*}$ is a cutter mapping of type (Q). Let $u^{*}=Ju$. Define two sequences $\{x_{n}^{*}\}\subset JC$ and $\{C^{*}_{n}\}$ by

$$ x_{n}^{*}:=Jx_{n}\quad \text{and} \quad C_{n}^{*}:=JC_{n}\quad \text{for all $n\ge 1$}. $$

It follows that

$$ C_{n+1}^{*}=\bigl\{ z^{*}\in C_{n}^{*}:\bigl\langle T^{*}x^{*}_{n}-z^{*},Jx_{n}^{*}-JT^{*}x_{n}^{*} \bigr\rangle \ge 0\bigr\} $$

and

$$ x^{*}_{n+1}\in C^{*}_{n+1}\text{ satisfies } \bigl\Vert x^{*}_{n+1}-u^{*} \bigr\Vert ^{2} \le d^{2}\bigl(u^{*},C^{*}_{n+1} \bigr)+\delta _{n+1}. $$

(1) Suppose that $(JC-Ju)\cup JC\subset B_{r}^{*}$ for some $r>0$. Using Theorem 17 for cutter mappings of type (Q) gives

$$ \limsup_{n}V(x_{n},Tx_{n})= \limsup_{n} V^{*}\bigl(T^{*}x_{n},x^{*}_{n} \bigr) \le \overline{g}^{*}_{r}\bigl({ \underline{g}^{*}_{r}}^{-1}(\delta _{0})\bigr). $$

This implies that

$$ \limsup_{n} \Vert x_{n}-Tx_{n} \Vert \le \underline{g}^{-1}_{r}\bigl( \overline{g}^{*}_{r}\bigl({ \underline{g}^{*}_{r}}^{-1}( \delta _{0})\bigr)\bigr). $$

(2) We assume that $\delta _{0}=0$ and $I-T$ is strongly closed at zero. It follows from the uniform convexity and uniform smoothness of E that $I^{*}-T^{*}$ is strongly closed at zero, where $I^{*}$ is an identity mapping of $E^{*}$. Using Theorem 17 for cutter mappings of type (Q) gives . Hence, $x_{n}=J^{-1}x^{*}_{n}\to J^{-1}P^{*}_{J\operatorname{Fix}(T)}Ju$. □

Based on Lemma 20 and Theorem 18, we obtain the following result.

Theorem 22

Suppose that E is a uniformly smooth and uniformly convex Banach space. Suppose that C is a subset of E such that JC is closed and convex. Suppose that $T:C\to E$ is a cutter mapping of type (R). Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. For given $u\in E$, a sequence $\{x_{n}\}\subset C$ is generated as follows: $x_{1}\in C$, $C_{1}:=C$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle JTx_{n}-Jz,x_{n}-Tx_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies }V(u,x_{n+1}) \le V(u,C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $C\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Tx_{n}\|\le \underline{g}^{-1}_{r}(\overline{g}^{*}_{r}({ \underline{g}^{*}_{r}}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ is defined by Lemma 10 (as E is uniformly convex) and $\underline{g}^{*}_{r}$ and $\overline{g}^{*}_{r}$ are defined by Lemma 10 (as $E^{*}$ is uniformly convex and uniformly smooth, respectively);

• (2) If $\delta _{0}=0$ and $I-T$ is strongly closed at zero, then $x_{n}\to R_{\operatorname{Fix}(T)}u$.

Proof

Set $T^{*}:=JTJ^{-1}$. Then $T^{*}:JC\to E^{*}$ is a cutter mapping of type (Q). Let $u^{*}=Ju$. Define two sequences $\{x_{n}^{*}\}\subset JC$ and $\{C^{*}_{n}\}$ by

$$ x_{n}^{*}:=Jx_{n}\quad \text{and} \quad C_{n}^{*}:=JC_{n}\quad \text{for all $n\ge 1$}. $$

It follows that

$$ C_{n+1}^{*}=\bigl\{ z^{*}\in C_{n}^{*}:\bigl\langle T^{*}x^{*}_{n}-z^{*},Jx_{n}^{*}-JT^{*}x_{n}^{*} \bigr\rangle \ge 0\bigr\} $$

and

$$ x^{*}_{n+1}\in C^{*}_{n+1}\text{ satisfies } V^{*}\bigl(x^{*}_{n+1},u^{*}\bigr) \le V^{*}\bigl(C^{*}_{n+1},u^{*} \bigr)+\delta _{n+1}. $$

(1) Suppose that $C\subset B_{r}$ for some $r>0$. Using Theorem 18 for cutter mappings of type (Q) gives

$$ \limsup_{n}V(x_{n},Tx_{n})= \limsup_{n} V^{*}\bigl(T^{*}x_{n},x^{*}_{n} \bigr) \le \overline{g}^{*}_{r}\bigl({ \underline{g}^{*}_{r}}^{-1}(\delta _{0})\bigr). $$

This implies that

$$ \limsup_{n} \Vert x_{n}-Tx_{n} \Vert \le \underline{g}^{-1}_{r}\bigl( \overline{g}^{*}_{r}\bigl({ \underline{g}^{*}_{r}}^{-1}( \delta _{0})\bigr)\bigr). $$

(2) We assume that $\delta _{0}=0$ and $I-T$ is strongly closed at zero. Note that $I^{*}-T^{*}$ is strongly closed at zero, where $I^{*}$ is the identity mapping of $E^{*}$. Using Theorem 18 for cutter mappings of type (Q) gives $x_{n}^{*}\to \varPi ^{*}_{\operatorname{Fix}(T^{*})}u^{*}$, where $\varPi ^{*}_{\operatorname{Fix}(T^{*})}$ is the generalized projection from $E^{*}$ onto $\operatorname{Fix}(T^{*})$. It follows from Lemma 9 that

$$ x_{n}=J^{-1}Jx_{n}\to J^{-1} \varPi ^{*}_{\operatorname{Fix}(T^{*})}u^{*}=J^{-1} \varPi ^{*}_{\operatorname{Fix}(T^{*})}Ju=R_{\operatorname{Fix}(T)}u. $$

 □

Remark 23

Our Theorem 21 and Theorem 22 generalize Theorem 5.1 of [12] and Theorem 5.1 of [11], respectively. In fact, the mapping of type (R) in [11, 12] is replaced by the cutter mapping of type (R). It is worth mentioning that the bound of the limit superior $\limsup_{n}\|x_{n}-Tx_{n}\|$ in [11, 12] is $\underline{g}_{r}^{-1}(\overline{g}^{*}_{r}(\underline{g}^{*-1}_{r}( \overline{g}^{*}_{r}({\underline{g}^{*}_{r}}^{-1}(\delta _{0})))))$.

4 Deduced results for maximal monotone operators

We now discuss the problem of finding a zero of maximal monotone operators in a Banach space ([8–10]). Suppose that E is a smooth, strictly convex, and reflexive Banach space. An operator $A\subset E\times E^{*}$ with domain $\operatorname{dom}(A):=\{x\in E:Ax\neq \varnothing \}$ and range $\operatorname{ran}(A):=\bigcup \{Ax:x\in E\}$ is monotone if $\langle x-y,x^{*}-y^{*}\rangle \ge 0$ for all $(x,x^{*}),(y,y^{*})\in A$. A monotone operator $A\subset E\times E^{*}$ is maximal if $A'=A$ whenever $A'\subset E\times E^{*}$ is monotone and $A\subset A'$. We are interested in finding a zero of a maximal monotone operator, that is, an element $u\in E$ such that $(u,0)\in A$. The set of zeros of A is denoted by $\operatorname{Zer}(A)$.

Suppose that $A\subset E\times E^{*}$ and $B\subset E^{*}\times E(=E^{*}\times (E^{*})^{*})$ are maximal monotone. It is known that $\overline{\operatorname{dom}(A)}$ and $\overline{\operatorname{dom}(B)}$ are convex; and

$$ \operatorname{ran}\bigl(I+\lambda J^{-1}A\bigr)=J^{-1} \bigl(\operatorname{ran}(J+ \lambda A)\bigr)=\operatorname{ran}(I+\lambda BJ)=E $$

for all $\lambda >0$. Here, I is the identity operator. In particular, the following three single-valued operators are well defined:

$$\begin{aligned}& P_{\lambda } :=\bigl(I+\lambda J^{-1}A\bigr)^{-1}: \overline{\operatorname{dom}(A)}\to E; \\& Q_{\lambda } :=(J+\lambda A)^{-1}J:\overline{ \operatorname{dom}(A)} \to E; \\& R_{\lambda } :=(I+\lambda BJ)^{-1}:J^{-1} \overline{ \operatorname{dom}(B)}\to E. \end{aligned}$$

It is known that

• (a) $\operatorname{Fix}(P_{\lambda})=\operatorname{Fix}(Q_{\lambda})= \operatorname{Zer}(A)$ and $\operatorname{Fix}(R_{\lambda})=\operatorname{Zer}(BJ)$.

• (b) $P_{\lambda}$, $Q_{\lambda}$, and $Q_{\lambda}$ are mappings of types (P), (Q), and (R), respectively.

• (c) If $\operatorname{Zer}(A)\neq \varnothing$, then $P_{\lambda}$ and $Q_{\lambda}$ are cutter mappings of types (P) and (Q), respectively, and both $I-P_{\lambda}$ and $I-Q_{\lambda}$ are strongly closed at zero.

• (d) If $\operatorname{Zer}(BJ)\neq \varnothing$, then $R_{\lambda}$ is a cutter mapping of type (R), and $I-R_{\lambda}$ is strongly closed at zero.

We immediately obtain the following corollaries.

4.1 Results for the resolvent $P_{\lambda}$

Corollary 24

Suppose that E is a smooth and uniformly convex Banach space and $A\subset E\times E^{*}$ is a maximal monotone operator such that $\operatorname{Zer}(A)\neq \varnothing$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. Construct a sequence $\{x_{n}\}\subset \overline{\operatorname{dom}(A)}$ as follows: $x_{1}\in \overline{\operatorname{dom}(A)}$, $C_{1}:=\overline{\operatorname{dom}(A)}$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\bigl\langle P_{\lambda }x_{n}-z,J(x_{n}-P_{ \lambda }x_{n}) \bigr\rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies } \Vert x_{n+1}-u \Vert ^{2}\le d^{2}(u,C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(A)}-u\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-P_{\lambda }x_{n}\|\le \underline{g}_{r}^{-1}( \delta _{0})$, where $\underline{g}_{r}$ is defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to P_{\operatorname{Zer}(A)}u$.

Corollary 25

Suppose that E is a smooth and uniformly convex Banach space and $A\subset E\times E^{*}$ is a maximal monotone operator such that $\operatorname{Zer}(A)\neq \varnothing$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. Construct a sequence $\{x_{n}\}\subset \overline{\operatorname{dom}(A)}$ as follows: $x_{1}\in \overline{\operatorname{dom}(A)}$, $C_{1}:=\overline{\operatorname{dom}(A)}$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\bigl\langle P_{\lambda }x_{n}-z,J(x_{n}-P_{ \lambda }x_{n}) \bigr\rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies }V(x_{n+1},u) \le V(C_{n+1},u)+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(A)}\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-P_{\lambda }x_{n}\|\le \underline{g}_{r}^{-1}( \delta _{0})$, where $\underline{g}_{r}$ is defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to \varPi _{\operatorname{Zer}(A)}u$.

4.2 Results for the resolvent $Q_{\lambda}$

Corollary 26

Suppose that E is a uniformly smooth and uniformly convex Banach space and that $A\subset E\times E^{*}$ is a maximal monotone operator such that $\operatorname{Zer}(A)\neq \varnothing$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. Construct a sequence $\{x_{n}\}\subset \overline{\operatorname{dom}(A)}$ as follows: $x_{1}\in \overline{\operatorname{dom}(A)}$, $C_{1}:=\overline{\operatorname{dom}(A)}$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle Q_{\lambda }x_{n}-z,Jx_{n}-JQ_{ \lambda }x_{n} \rangle \ge 0\bigr\} \\& x_{n+1} \in C_{n+1}\textit{ satisfies } \Vert x_{n+1}-u \Vert \le d(u,C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $(\overline{\operatorname{dom}(A)}-u)\cup \overline{\operatorname{dom}(A)} \subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Q_{\lambda }x_{n}\|\le \underline{g}^{-1}_{r}( \overline{g}_{r}(\underline{g}_{r}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ and $\overline{g}_{r}$ are defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to P_{\operatorname{Zer}(A)}u$.

Corollary 27

Suppose that E is a uniformly smooth and uniformly convex Banach space and that $A\subset E\times E^{*}$ is a maximal monotone operator such that $\operatorname{Zer}(A)\neq \varnothing$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. Construct a sequence $\{x_{n}\}\subset \overline{\operatorname{dom}(A)}$ as follows: $x_{1}\in \overline{\operatorname{dom}(A)}$, $C_{1}:=\overline{\operatorname{dom}(A)}$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle Q_{\lambda }x_{n}-z,Jx_{n}-JQ_{ \lambda }x_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies }V(x_{n+1},u) \le V(C_{n+1},u)+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(A)}\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-Q_{\lambda }x_{n}\|\le \underline{g}^{-1}_{r}( \overline{g}_{r}(\underline{g}_{r}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ and $\overline{g}_{r}$ are defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to \varPi _{\operatorname{Zer}(A)}u$.

4.3 Results for the resolvent $R_{\lambda}$

Corollary 28

Suppose that E is a uniformly smooth and uniformly convex Banach space and $B\subset E^{*}\times E$ such that $\operatorname{Zer}(BJ)\neq \varnothing$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. Construct a sequence $\{x_{n}\}\subset \overline{\operatorname{dom}(BJ)}$ as follows: $x_{1}\in \overline{\operatorname{dom}(BJ)}$, $C_{1}:=\overline{\operatorname{dom}(BJ)}$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle JR_{\lambda }x_{n}-Jz,x_{n}-R_{ \lambda }x_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies } \Vert Ju-Jx_{n+1} \Vert ^{2}\le d^{2}(Ju,JC_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $(J\overline{\operatorname{dom}(BJ)}-Ju)\cup J \overline{\operatorname{dom}(BJ)} \subset B_{r}^{*}$ for some $r>0$, then $\limsup_{n}\|x_{n}-R_{\lambda }x_{n}\|\le \underline{g}^{-1}_{r}( \overline{g}^{*}_{r}({\underline{g}^{*}_{r}}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ is defined by Lemma 10 (as E is uniformly convex) and $\underline{g}^{*}_{r}$ and $\overline{g}^{*}_{r}$ are defined by Lemma 10 (as $E^{*}$ is uniformly convex and uniformly smooth, respectively);

• (2) If $\delta _{0}=0$, then $x_{n}\to J^{-1}P^{*}_{J\operatorname{Zer}(BJ)}Ju$, where $P^{*}_{J\operatorname{Zer}(BJ)}$ is a metric projection of $E^{*}$ onto $J\operatorname{Zer}(BJ)$.

Corollary 29

Suppose that E is a uniformly smooth and uniformly convex Banach space and $B\subset E^{*}\times E$ such that $\operatorname{Zer}(BJ)\neq \varnothing$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a sequence of nonnegative real numbers such that $\delta _{0}:=\limsup_{n}\delta _{n}$. Construct a sequence $\{x_{n}\}\subset \overline{\operatorname{dom}(BJ)}$ as follows: $x_{1}\in \overline{\operatorname{dom}(BJ)}$, $C_{1}:=\overline{\operatorname{dom}(BJ)}$, and

$$\begin{aligned}& C_{n+1} :=\bigl\{ z\in C_{n}:\langle JR_{\lambda }x_{n}-Jz,x_{n}-R_{ \lambda }x_{n} \rangle \ge 0\bigr\} ; \\& x_{n+1} \in C_{n+1}\textit{ satisfies }V(u,x_{n+1}) \le V(u,C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n\ge 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(BJ)}\subset B_{r}$ for some $r>0$, then $\limsup_{n}\|x_{n}-R_{\lambda }x_{n}\|\le \underline{g}^{-1}_{r}( \overline{g}^{*}_{r}({\underline{g}^{*}_{r}}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ is defined by Lemma 10 (as E is uniformly convex) and $\underline{g}^{*}_{r}$ and $\overline{g}^{*}_{r}$ are defined by Lemma 10 (as $E^{*}$ is uniformly convex and uniformly smooth, respectively);

• (2) If $\delta _{0}=0$, then $x_{n}\to R_{\operatorname{Zer}(BJ)}u$.

4.4 Applications to convex minimization problems

We discuss the convex minimization problem in a Banach space. This problem is to find a minimizer of a proper lower semicontinuous convex function in a Banach space. Suppose that E is a reflexive, smooth, and strictly convex Banach space with its dual $E^{*}$ and $f:E \to (-\infty , \infty ]$ and $f^{*}:E^{*} \to (-\infty , \infty ]$ are proper lower semicontinuous convex functions. Then the subdifferentials of f and $f^{*}$ are defined as follows:

$$\begin{aligned} &\partial f (x) = \bigl\{ x^{*} \in E^{*}: f (x) + \bigl\langle y-x, x^{*} \bigr\rangle \leq f(y), \ \forall y \in E\bigr\} \quad (\forall x \in E), \\ &\partial f^{*} \bigl(x^{*}\bigr) = \bigl\{ x \in E: f^{*}\bigl(x^{*}\bigr) + \bigl\langle x, y^{*}-x^{*} \bigr\rangle \leq f^{*} \bigl(y^{*}\bigr), \ \forall y^{*} \in E^{*} \bigr\} \quad \bigl(\forall x^{*} \in E^{*}\bigr). \end{aligned}$$

By Rockafellar’s theorem [20, 21], the subdifferentials $\partial f \subset E \times E^{*}$ and $\partial f^{*} \subset E^{*} \times E$ are maximal monotone. It is easy to see that $\operatorname{Zer}(\partial f)=\operatorname{argmin} \{f(x):x \in E \}$ and $\operatorname{Zer}(\partial f^{*})=\operatorname{argmin} \{f^{*}(x^{*}):x^{*} \in E^{*}\}$.

Fix $\lambda >0$ and $z\in E$. Let $P_{\lambda}$ and $Q_{\lambda}$ be the resolvent of ∂f, and let $R_{\lambda}$ be the resolvent of $\partial f^{*}$, then we know that

$$\begin{aligned} &P_{\lambda} z = \bigl(I+\lambda J^{-1}\partial f \bigr)^{-1} = \mathop{ \operatorname{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2\lambda} \Vert y-z \Vert ^{2} \biggr\} , \\ &Q_{\lambda} z = (J+\lambda \partial f)^{-1}J = \mathop{ \operatorname{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2\lambda} \Vert y \Vert ^{2} -\frac {1}{\lambda}\langle y, Jz \rangle \biggr\} , \\ &R_{\lambda} z = \bigl(I+\lambda \partial f^{*} \bigr)^{-1}= J^{-1}\mathop{ \operatorname{argmin}}_{y^{*} \in E^{*}} \biggl\{ f^{*}\bigl(y^{*}\bigr) + \frac{1}{2\lambda} \bigl\Vert y^{*} \bigr\Vert ^{2} - \frac {1}{\lambda}\bigl\langle z, y^{*} \bigr\rangle \biggr\} . \end{aligned}$$

See, for instance, [8, 10, 22]. As a direct consequence of our theorems, we can show the following applications.

Corollary 30

Suppose that E is a smooth and uniformly convex Banach space and $f:E \to (-\infty , \infty ]$ is a proper lower semicontinuous convex function such that $\operatorname{Zer}(\partial f)\ne \emptyset$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a nonnegative real sequence such that $\delta _{0}:=\limsup_{n} \delta _{n}$. Construct a sequence $\{x_{n}\} \subset \overline{\operatorname{dom}(\partial f)}$ as follows: $x_{1} \in \overline{\operatorname{dom}(\partial f)}$, $C_{1} =\overline{\operatorname{dom}(\partial f)}$, and

$$\begin{aligned} & y_{n} := \mathop{\operatorname{argmin}}_{y \in E} \biggl\{ f(y) + \frac{1}{2\lambda} \Vert y-x_{n} \Vert ^{2} \biggr\} , \\ & C_{n+1} := \bigl\{ z\in C_{n} : \bigl\langle y_{n} - z, J(x_{n} - y_{n}) \bigr\rangle \geq 0\bigr\} , \\ & x_{n+1} \in C_{n+1} \textit{ satisfies } \Vert x_{n+1}-u \Vert ^{2} \leq d(u,C_{n+1})^{2}+ \delta _{n+1} \end{aligned}$$

for all $n \geq 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(\partial f)}-u\subset B_{r}$ for some $r>0$, then $\limsup_{n}\Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}( \delta _{0})$, where $\underline{g}_{r}$ is defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to P_{\operatorname{Zer}(\partial f)}u$.

Corollary 31

Suppose that E is a smooth and uniformly convex Banach space and $f:E \to (-\infty , \infty ]$ is a proper lower semicontinuous convex function such that $\operatorname{Zer}(\partial f)\ne \emptyset$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a nonnegative real sequence such that $\delta _{0}:=\limsup_{n} \delta _{n}$. Construct a sequence $\{x_{n}\} \subset \overline{\operatorname{dom}(\partial f)}$ as follows: $x_{1} \in \overline{\operatorname{dom}(\partial f)}$, $C_{1} =\overline{\operatorname{dom}(\partial f)}$, and

$$\begin{aligned} & y_{n} := \mathop{\operatorname{argmin}}_{y \in E} \biggl\{ f(y) + \frac{1}{2\lambda} \Vert y-x_{n} \Vert ^{2} \biggr\} , \\ & C_{n+1} := \bigl\{ z\in C_{n} : \bigl\langle y_{n} - z, J(x_{n} - y_{n}) \bigr\rangle \geq 0\bigr\} , \\ & x_{n+1} \in C_{n+1} \textit{ satisfies } V(x_{n+1},u) \leq V(C_{n+1},u)+ \delta _{n+1} \end{aligned}$$

for all $n \geq 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(\partial f)}\subset B_{r}$ for some $r>0$, then $\limsup_{n}\Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}( \delta _{0})$, where $\underline{g}_{r}$ is defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to \varPi _{\operatorname{Zer}(\partial f)}u$.

Corollary 32

Suppose that E is a uniformly smooth and uniformly convex Banach space and $f:E \to (-\infty , \infty ]$ is a proper lower semicontinuous convex function such that $\operatorname{Zer}(\partial f)\ne \emptyset$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a nonnegative real sequence such that $\delta _{0}:=\limsup_{n} \delta _{n}$. Construct a sequence $\{x_{n}\} \subset \overline{\operatorname{dom}(\partial f)}$ as follows: $x_{1} \in \overline{\operatorname{dom}(\partial f)}$, $C_{1} =\overline{\operatorname{dom}(\partial f)}$, and

$$\begin{aligned} & y_{n} := \mathop{\operatorname{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2\lambda} \Vert y \Vert ^{2} - \frac {1}{\lambda}\langle y, Jx_{n} \rangle \biggr\} , \\ & C_{n+1} := \bigl\{ z\in C_{n} : \langle y_{n} - z, Jx_{n} - Jy_{n} \rangle \geq 0 \bigr\} , \\ & x_{n+1} \in C_{n+1} \textit{ satisfies } \Vert x_{n+1}-u \Vert ^{2} \leq d(u,C_{n+1})^{2}+ \delta _{n+1} \end{aligned}$$

for all $n \geq 1$. Then the following statements are true:

• (1) If $(\overline{\operatorname{dom}(\partial f)}-u)\cup \overline{\operatorname{dom}(\partial f)} \subset B_{r}$ for some $r>0$, then $\limsup_{n}\Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}( \overline{g}_{r}(\underline{g}_{r}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ and $\overline{g}_{r}$ are defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to P_{\operatorname{Zer}(\partial f)}u$.

Corollary 33

Suppose that E is a uniformly smooth and uniformly convex Banach space and $f:E \to (-\infty , \infty ]$ is a proper lower semicontinuous convex function such that $\operatorname{Zer}(\partial f)\ne \emptyset$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a nonnegative real sequence such that $\delta _{0}:=\limsup_{n} \delta _{n}$. Construct a sequence $\{x_{n}\} \subset \overline{\operatorname{dom}(\partial f)}$ as follows: $x_{1} \in \overline{\operatorname{dom}(\partial f)}$, $C_{1} =\overline{\operatorname{dom}(\partial f)}$, and

$$\begin{aligned} & y_{n} := \mathop{\operatorname{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2\lambda} \Vert y \Vert ^{2} - \frac {1}{\lambda}\langle y, Jx_{n} \rangle \biggr\} , \\ & C_{n+1} := \bigl\{ z\in C_{n} : \langle y_{n} - z, Jx_{n} - Jy_{n} \rangle \geq 0 \bigr\} , \\ & x_{n+1} \in C_{n+1} \textit{ satisfies } V(x_{n+1},u) \leq V(C_{n+1},u)+ \delta _{n+1} \end{aligned}$$

for all $n \geq 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(\partial f)}\subset B_{r}$ for some $r>0$, then $\limsup_{n}\Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}( \overline{g}_{r}(\underline{g}_{r}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ and $\overline{g}_{r}$ are defined by Lemma 10;

• (2) If $\delta _{0}=0$, then $x_{n}\to \varPi _{\operatorname{Zer}(\partial f)}u$.

Corollary 34

Suppose that E is a uniformly smooth and uniformly convex Banach space and $f^{*}:E^{*} \to (-\infty , \infty ]$ is a proper lower semicontinuous convex function such that $\operatorname{Zer}(\partial f^{*})\ne \emptyset$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a nonnegative real sequence such that $\delta _{0}:=\limsup_{n} \delta _{n}$. Construct a sequence $\{x_{n}\} \subset \overline{\operatorname{dom}(\partial f^{*}J)}$ as follows: $x_{1} \in \overline{\operatorname{dom}(\partial f^{*}J)}$, $C_{1} =\overline{\operatorname{dom}(\partial f^{*}J)}$, and

$$\begin{aligned} & y_{n} := J^{-1}\mathop{\operatorname{argmin}}_{y^{*} \in E^{*}} \biggl\{ f^{*}\bigl(y^{*}\bigr) + \frac{1}{2\lambda} \bigl\Vert y^{*} \bigr\Vert ^{2} - \frac {1}{\lambda}\bigl\langle x_{n}, y^{*}\bigr\rangle \biggr\} , \\ & C_{n+1} := \bigl\{ z\in C_{n} : \langle Jy_{n} - Jz, x_{n} - y_{n} \rangle \geq 0 \bigr\} , \\ & x_{n+1} \in C_{n+1} \textit{ satisfies } \Vert Ju-Jx_{n+1} \Vert ^{2} \leq d(u,C_{n+1})^{2}+ \delta _{n+1} \end{aligned}$$

for all $n \geq 1$. Then the following statements are true:

• (1) If $(J\overline{\operatorname{dom}(\partial f^{*}J)}-Ju)\cup J \overline{\operatorname{dom}(\partial f^{*}J)}\subset B^{*}_{r}$ for some $r>0$, then $\limsup_{n}\Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}( \overline{g}^{*}_{r}({\underline{g}^{*}_{r}}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ is defined by Lemma 10 (as E is uniformly convex) and; $\underline{g}^{*}_{r}$ and $\overline{g}^{*}_{r}$ are defined by Lemma 10 (as $E^{*}$ is uniformly convex and uniformly smooth, respectively);

• (2) If $\delta _{0}=0$, then $x_{n}\to J^{-1}P^{*}_{J\operatorname{Zer}(\partial f^{*}J)}Ju$, where $P^{*}_{J\operatorname{Zer}(\partial f^{*}J)}$ is a metric projection of $E^{*}$ onto $J\operatorname{Zer}(\partial f^{*}J)$.

Corollary 35

Suppose that E is a uniformly smooth and uniformly convex Banach space and $f^{*}:E^{*} \to (-\infty , \infty ]$ is a proper lower semicontinuous convex function such that $\operatorname{Zer}(\partial f^{*})\ne \emptyset$. Suppose that $u\in E$ and $\lambda \in (0,\infty )$. Suppose that $\{\delta _{n}\}$ is a nonnegative real sequence such that $\delta _{0}:=\limsup_{n} \delta _{n}$. Construct a sequence $\{x_{n}\} \subset \overline{\operatorname{dom}(\partial f^{*}J)}$ as follows: $x_{1} \in \overline{\operatorname{dom}(\partial f^{*}J)}$, $C_{1} =\overline{\operatorname{dom}(\partial f^{*}J)}$, and

$$\begin{aligned} & y_{n} := J^{-1}\mathop{\operatorname{argmin}}_{y^{*} \in E^{*}} \biggl\{ f^{*}\bigl(y^{*}\bigr) + \frac{1}{2\lambda} \bigl\Vert y^{*} \bigr\Vert ^{2} - \frac {1}{\lambda}\bigl\langle x_{n}, y^{*}\bigr\rangle \biggr\} , \\ & C_{n+1} := \bigl\{ z\in C_{n} : \langle Jy_{n} - Jz, x_{n} - y_{n} \rangle \geq 0 \bigr\} , \\ & x_{n+1} \in C_{n+1} \textit{ satisfies } V(u, x_{n+1}) \leq V(u, C_{n+1})+ \delta _{n+1} \end{aligned}$$

for all $n \geq 1$. Then the following statements are true:

• (1) If $\overline{\operatorname{dom}(\partial f^{*}J)}\subset B_{r}$ for some $r>0$, then $\limsup_{n}\Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}( \overline{g}^{*}_{r}({\underline{g}^{*}_{r}}^{-1}(\delta _{0})))$, where $\underline{g}_{r}$ is defined by Lemma 10 (as E is uniformly convex) and; $\underline{g}^{*}_{r}$ and $\overline{g}^{*}_{r}$ are defined by Lemma 10 (as $E^{*}$ is uniformly convex and uniformly smooth, respectively);

• (2) If $\delta _{0}=0$, then $x_{n}\to R_{\operatorname{Zer}(\partial f^{*}J)}u$.

5 Conclusions

In this paper, we prove two key inequalities for metric projections and generalized projections in a certain Banach space. Using them, we obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. [24] for cutter mappings of types (P), (Q), and (R). The mappings studied in this paper are more general than the ones in [8, 10]. In particular, the results in [8, 10] are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex function defined on a Banach space. It would be interesting to extend our work to the class of quasinonexpansive mappings of Bregman type.