Approximation of a zero point of monotone operators with nonsummable errors

Abstract

In this paper, we study an iterative scheme for two different types of resolvents of a monotone operator defined on a Banach space. These resolvents are generalizations of resolvents of a monotone operator in a Hilbert space. We obtain iterative approximations of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications.

1 Introduction

Let H be a real Hilbert space and let $A \subset H \times H$ be a maximal monotone operator. Then the zero point problem is to find $u \in H$ such that

$$ 0 \in A u. $$

(1.1)

Such a $u \in H$ is called a zero point (or a zero) of A. The set of zero points of A is denoted by $A^{-1}0$. This problem is connected with many problems in Nonlinear Analysis and Optimization, that is, convex minimization problems, variational inequality problems, equilibrium problems and so on. A well-known method for solving (1.1) is the proximal point algorithm: $x_{1} \in H$ and

$$ x_{n+1}=J_{r_{n}}x_{n}, \quad n=1,2, \ldots, $$

(1.2)

where $\{r_{n}\} \subset\mathopen]0, \infty\mathclose[$ and $J_{r_{n}}=(I+r_{n}A)^{-1}$. This algorithm was first introduced by Martinet [1]. In 1976, Rockafellar [2] proved that if $\liminf_{n} r_{n} > 0$ and $A^{-1}0 \ne\emptyset$, then the sequence $\{x_{n}\}$ defined by (1.2) converges weakly to a solution of the zero point problem. Later, many researchers have studied this problem; see [3–9] and others.

On the other hand, Kimura [10] introduced the following iterative scheme for finding a fixed point of nonexpansive mappings by the shrinking projection method with error in a Hilbert space:

Theorem 1.1

(Kimura [10])

Let C be a bounded closed convex subset of a Hilbert space H with $D= \operatorname {diam}C =\sup_{x,y\in C}\Vert x-y\Vert < \infty$, and let $T:C\to H$ be a nonexpansive mapping having a fixed point. Let $\{\epsilon_{n}\}$ be a nonnegative real sequence such that $\epsilon_{0}=\limsup_{n} \epsilon_{n} < \infty$. For a given point $u\in H$, generate an iterative sequence $\{x_{n}\}$ as follows: $x_{1} \in C$ such that $\Vert x_{1}-u\Vert <\epsilon_{1}$, $C_{1} =C$,

$$\begin{aligned} & C_{n+1} = \bigl\{ z \in C : \Vert z-Tx_{n}\Vert \leq \Vert z-x_{n}\Vert \bigr\} \cap C_{n}, \\ & x_{n+1} \in C_{n+1} \quad \textit{such that}\quad \Vert u-x_{n+1} \Vert ^{2} \leq d(u,C_{n+1})^{2}+ \epsilon^{2}_{n+1} \end{aligned}$$

for all $n \in \mathbb {N}$. Then

$$\limsup_{n\to\infty} \Vert x_{n}-Tx_{n}\Vert \leq2\epsilon_{0}. $$

Further, if $\epsilon_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{F(T)}u \in F(T)$.

We remark that the original result of the theorem above deals with a family of nonexpansive mappings, and the shrinking projection method was first introduced by Takahashi et al. [11]. This result was extended to more general Banach spaces by Kimura [12] (see also Ibaraki and Kimura [13]).

In this paper, we study the shrinking projection method with error introduced by Kimura [10] (see also [12, 14]). We obtain an iterative approximation of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications.

2 Preliminaries

Let E be a real Banach space with its dual $E^{*}$. The normalized duality mapping J from E into $E^{*}$ is defined by

$$Jx=\bigl\{ x^{*} \in E^{*}: \bigl\langle x, x^{*} \bigr\rangle = \Vert x \Vert ^{2} = \bigl\Vert x^{*}\bigr\Vert ^{2}\bigr\} $$

for each $x \in E$. We also know the following properties: see [15, 16] for more details.

• (1) $Jx \ne\emptyset$ for each $x \in E$;

• (2) if E is reflexive, then J is surjective;

• (3) if E is smooth, then the duality mapping J is single valued.

• (4) if E is strictly convex, then J is one-to-one and satisfies that $\langle x-y, x^{*}-y^{*} \rangle> 0$ for each $x,y \in E$ with $x \neq y$, $x^{*} \in Jx$ and $y^{*} \in Jy$;

• (5) if E is reflexive, smooth, and strictly convex, then the duality mapping $J_{*}: E^{*} \to E$ is the inverse of J, that is, $J_{*} = J^{-1}$;

• (6) if E uniformly smooth, then the duality mapping J is uniformly norm to norm continuous on each bounded set of E.

Let E be a reflexive and strictly convex Banach space and let C be a nonempty closed convex subset of E. It is well known that for each $x\in E$ there exists a unique point $z \in C$ such that $\Vert x-z\Vert =\min\{\Vert x-y\Vert : y \in C\}$. Such a point z is denoted by $P_{C} x$ and $P_{C}$ is called the metric projection of E onto C. The following result is well known; see, for instance, [16].

Lemma 2.1

Let E be a reflexive, smooth, and strictly convex Banach space, let C be a nonempty closed convex subset of E, let $P_{C}$ be the metric projection of E onto C, let $x \in E$ and let $x_{0} \in C$. Then $x_{0} = P_{C} x$ if and only if

$$\bigl\langle x_{0}-y, J(x-x_{0}) \bigr\rangle \geq0 $$

for all $y \in C$.

Let C be a nonempty closed convex subset of a smooth Banach space E. A mapping $T: C \to E$ is said to be of type (P) [17] if

$$\bigl\langle Tx-Ty, J(x-Tx)-J(y-Ty) \bigr\rangle \geq0 $$

for each $x,y \in C$. A mapping $T: C \to E$ is said to be of type (Q) [17, 18] if

$$\bigl\langle Tx-Ty, (Jx-JTx)-(Jy-JTy) \bigr\rangle \geq0 $$

for each $x,y \in C$. We denote by $F(T)$ the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T if C contains a sequence $\{x_{n}\}$ such that $x_{n}\rightharpoonup p$ and $x_{n} -Tx_{n} \to0$. The set of all asymptotic fixed points of T is denoted by $\hat{F}(T)$. It is clear that if $T: C \to E$ is of type (P) and $F(T)$ is nonempty, then

$$ \bigl\langle Tx-p, J(x-Tx) \bigr\rangle \geq0 $$

(2.1)

for each $x \in C$ and $p \in F(T)$. Let E be a reflexive, smooth, and strictly convex Banach space and let C be a nonempty closed convex subset of E. It is well known that the metric projection $P_{C}$ of E onto C is a mapping of type (P). We also know that if $T: C \to E$ is of type (Q) and $F(T)$ is nonempty, then

$$ \langle Tx-p, Jx-JTx \rangle\geq0 $$

(2.2)

for each $x \in C$ and $p \in F(T)$.

The following results describe the relation between the set of fixed points and that of asymptotic fixed points for each type of mapping.

Lemma 2.2

(Aoyama-Kohsaka-Takahashi [19])

Let E be a smooth Banach space, let C be a nonempty closed convex subset of E and let $T: C \to E$ be a mapping of type (P). If $F(T)$ is nonempty, then $F(T)$ is closed and convex and $F(T)=\hat{F}(T)$.

Lemma 2.3

(Kohsaka-Takahashi [18])

Let E be a strictly convex Banach space whose norm is uniformly Gâteaux differentiable, let C be a nonempty closed convex subset of E and let $T: C \to E$ be a mapping of type (Q). If $F(T)$ is nonempty, then $F(T)$ is closed and convex and $F(T)=\hat{F}(T)$.

In 1984, Tsukada [20] proved the following theorem for the metric projections in a Banach space. For the exact definition of Mosco limit $\mathrm {M}\text{-}\!\lim _{n} C_{n}$, see [21].

Theorem 2.4

(Tsukada [20])

Let E be a reflexive and strictly convex Banach space and let $\{C_{n}\}$ be a sequence of nonempty closed convex subsets of E. If $C_{0} =\mathrm {M}\text{-}\!\lim _{n} C_{n}$ exists and is nonempty, then for each $x \in E$, $\{P_{C_{n}}x\}$ converges weakly to $P_{C_{0}}x$, where $P_{C_{n}}$ is the metric projection of E onto $C_{n}$. Moreover, if E has the Kadec-Klee property, the convergence is in the strong topology.

One of the simplest example of the sequence $\{C_{n}\}$ satisfying the condition in this theorem above is a decreasing sequence with respect to inclusion; $C_{n+1}\subset C_{n}$ for each $n\in \mathbb {N}$. In this case, $\mathrm {M}\text{-}\!\lim C_{n} =\bigcap_{n=1}^{\infty} C_{n}$ (see [7, 12, 21, 22] for more details).

Let E be a smooth Banach space and consider the following function $V: E \times E \to \mathbb {R}$ defined by

$$ V(x, y) = \Vert x\Vert ^{2} - 2\langle x, Jy \rangle+ \Vert y\Vert ^{2} $$

(2.3)

for each $x,y \in E$. We know the following properties:

• (1) $(\Vert x\Vert -\Vert y\Vert )^{2} \leq V(x,y) \leq(\Vert x\Vert +\Vert y\Vert )^{2}$ for each $x,y \in E$;

• (2) $V(x,y) + V(y,x) = 2 \langle x-y, Jx-Jy \rangle$ for each $x,y \in E$;

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

• (4) if E is additionally assumed to be strictly convex, then $V(x,y)=0$ if and only if $x=y$.

Lemma 2.5

(Kamimura-Takahashi [23])

Let E be a smooth and uniformly convex Banach space and let $\{x_{n}\}$ and $\{y_{n}\}$ be sequences in E such that either $\{x_{n}\}$ or $\{y_{n}\}$ is bounded. If $\lim_{n} V(x_{n}, y_{n})=0$, then $\lim_{n} \Vert x_{n}-y_{n} \Vert =0$.

The following results show the existence of mappings $\underline{g}_{r}$ and $\overline{g}_{r}$, related to the convex structures of a Banach space E. These mappings play important roles in our result.

Theorem 2.6

(Xu [24])

Let E be a Banach space, $r\in\mathopen]0, \infty\mathclose[$ and $B_{r}= \{x\in E : \Vert x \Vert \leq r\}$. Then

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

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

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

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

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

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

Theorem 2.7

(Kimura [12])

Let E be a uniformly smooth and uniformly convex Banach space and let $r>0$. Then the function $\underline{g}_{r}$ and $\overline{g}_{r}$ in Theorem  2.6 satisfies

$$\underline{g}_{r}\bigl(\Vert x-y\Vert \bigr)\leq V(x,y)\leq \overline{g}_{r}\bigl(\Vert x-y\Vert \bigr) $$

for all $x,y\in B_{r}$.

3 Approximation theorem for the resolvents of type (P)

In this section, we discuss an iterative scheme of resolvents of a monotone operator defined on a Banach space. Let E be a reflexive, smooth, and strictly convex Banach space. An operator $A \subset E \times E^{*}$ with domain $D(A)=\{ x \in E: Ax \ne\emptyset\}$ and range $R(A)=\bigcup\{Ax: x \in D(A)\}$ is said to be monotone if $\langle x-y, x^{*}-y^{*} \rangle\geq0$ for any $(x, x^{*}), (y, y^{*}) \in A$. A monotone operator A is said to be maximal if $A=B$ whenever $B \subset E \times E^{*}$ is a monotone operator such that $A \subset B$. We denote by $A^{-1}0$ the set $\{z\in D(A): 0\in Az\}$.

Let C be a nonempty closed convex subset of E, let $r>0$ and let $A\subset E\times E^{*}$ be a monotone operator satisfying

$$ D(A) \subset C \subset R\bigl(I+rJ^{-1}A\bigr) $$

(3.1)

for $r>0$. It is well known that if A is maximal monotone operator, then $R(I+rJ^{-1}A)=E$; see [25–27]. Hence, if A is maximal monotone, then (3.1) holds for $C=\overline{D(A)}$. We also know that $\overline{D(A)}$ is convex; see [28]. If A satisfies (3.1) for $r>0$, we can define the resolvent (of type (P)) $P_{r}:C\to D(A)$ of A by

$$ P_{r} x =\bigl\{ z\in E : 0\in J(z-x)+rAz \bigr\} $$

(3.2)

for all $x\in C$. In other words, $P_{r}x=(I+rJ^{-1}A)^{-1}x$ for all $x\in C$. The Yosida approximation $A_{r}:C\to E^{*}$ is also defined $A_{r}x=J(x-P_{r}x)/r$ for all $x\in C$. We know the following; see, for instance, [15, 17, 19]:

• (1) $P_{r}$ is mapping of type (P) from C into $D(A)$;

• (2) $(P_{r} x, A_{r}x)\in A$ for all $x \in C$;

• (3) $\Vert A_{r}x \Vert \leq \vert Ax \vert :=\inf\{\Vert x^{*}\Vert : x^{*} \in Ax\}$ for all $x \in D(A)$;

• (4) $F(P_{r})=A^{-1}0$.

We obtain an approximation theorem for a zero point of a monotone operator in a smooth and uniformly convex Banach space by using the resolvent of type (P).

Theorem 3.1

Let E be a smooth and uniformly convex Banach space and let $A\subset E\times E^{*}$ be a monotone operator with $A^{-1}0 \ne \emptyset$. Let $\{r_{n}\}$ be a positive real sequence such that $\liminf_{n} r_{n} >0$, let C be a nonempty bounded closed convex subset of E satisfying

$$D(A) \subset C \subset R\bigl(I+r_{n}J^{-1}A\bigr) $$

for all $n\in \mathbb {N}$ and let $r\in\mathopen]0,\infty\mathclose[$ such that $C \subset B_{r}$. Let $\{\delta_{n}\}$ be a nonnegative real sequence and let $\delta_{0}=\limsup_{n} \delta_{n}$. For a given point $u\in E$, generate a sequence $\{x_{n}\}$ by $x_{1} = x \in C$, $C_{1} =C$, and

$$\begin{aligned} & y_{n} = P_{r_{n}} x_{n}, \\ & C_{n+1} = \bigl\{ z \in C : \bigl\langle y_{n} - z, J(x_{n} - y_{n}) \bigr\rangle \geq0\bigr\} \cap C_{n}, \\ & x_{n+1} \in\bigl\{ z \in C : \Vert u-z\Vert ^{2} \leq d(u,C_{n+1})^{2}+\delta _{n+1}\bigr\} \cap C_{n+1}, \end{aligned}$$

for all $n \in \mathbb {N}$. Then

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\underline {g}_{r}^{-1}(\delta_{0}). $$

Moreover, if $\delta_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{A^{-1}0}u$.

Proof

Since $C_{n}$ includes $A^{-1}0\ne\emptyset$ for all $n\in \mathbb {N}$, $\{C_{n}\}$ is a sequence of nonempty closed convex subsets and, by definition, it is decreasing with respect to inclusion. Let $p_{n}=P_{C_{n}}u$ for all $n\in \mathbb {N}$. Then, by Theorem 2.4, we see that $\{p_{n}\}$ converges strongly to $p_{0}=P_{C_{0}}u$, where $C_{0}=\bigcap_{n=1}^{\infty}C_{n}$. Since $x_{n}\in C_{n}$ and $d(u,C_{n})=\Vert u-p_{n} \Vert$, we see that

$$\Vert u-x_{n} \Vert ^{2} \leq \Vert u-p_{n} \Vert ^{2} +\delta_{n} $$

for every $n\in \mathbb {N}\setminus\{1\}$. From Theorem 2.6(i), we see that for $\alpha\in \mathopen]0,1\mathclose[$,

$$\begin{aligned} \Vert p_{n} -u \Vert ^{2} &\leq\bigl\Vert \alpha p_{n} +(1-\alpha)x_{n}-u \bigr\Vert ^{2} \\ &\leq\alpha \Vert p_{n} -u \Vert ^{2} +(1-\alpha)\Vert x_{n}-u \Vert ^{2} - \alpha(1-\alpha)\underline{g}_{r} \bigl(\Vert p_{n} -x_{n} \Vert \bigr) \end{aligned}$$

and thus

$$\alpha\underline{g}_{r} \bigl(\Vert p_{n} -x_{n} \Vert \bigr) \leq \Vert x_{n}-u \Vert ^{2} - \Vert p_{n} -u \Vert ^{2} \leq \delta_{n}. $$

As $\alpha\to1$, we see that $\underline{g}_{r} (\Vert p_{n} -x_{n} \Vert )\leq\delta_{n}$ and thus $\Vert p_{n} -x_{n} \Vert \leq\underline{g}_{r}^{-1}(\delta_{n})$. Using the definition of $p_{n}$, we see that $p_{n+1}\in C_{n+1}$ and thus

$$\bigl\langle y_{n} - p_{n+1}, J(x_{n} - y_{n}) \bigr\rangle \geq0, $$

or equivalently,

$$\bigl\langle x_{n} - p_{n+1}, J(x_{n} - y_{n}) \bigr\rangle \geq \Vert x_{n}-y_{n}\Vert ^{2}. $$

Hence we obtain

$$\Vert x_{n}-y_{n}\Vert \leq \Vert x_{n}-p_{n+1} \Vert \leq \Vert x_{n}-p_{n}\Vert + \Vert p_{n}-p_{n+1}\Vert \leq\underline{g}_{r}^{-1}( \delta_{n}) + \Vert p_{n}-p_{n+1}\Vert $$

for every $n\in \mathbb {N}\setminus\{1\}$. Since $\lim_{n} p_{n}=p_{0}$ and $\limsup_{n} \delta_{n}=\delta_{0}$, we see that

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\underline {g}_{r}^{-1}(\delta_{0}). $$

For the latter part of the theorem, suppose that $\delta_{0}=0$. Then we see that

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\underline{g}_{r}^{-1}(0)=0 $$

and

$$\limsup_{n\to\infty}\underline{g}_{r}\bigl(\Vert x_{n}-p_{n}\Vert \bigr) \leq\limsup_{n\to\infty} \delta_{n}=0. $$

Therefore, we obtain

$$\lim_{n\to\infty} \Vert x_{n}-y_{n}\Vert =0 \quad \text{and}\quad \lim_{n\to\infty} \Vert x_{n}-p_{n} \Vert =0. $$

Hence, we also obtain

$$ \lim_{n\to\infty} x_{n}=p_{0} \quad \text{and} \quad \lim_{n\to\infty} y_{n}=p_{0}. $$

(3.3)

So, from

$$\Vert y_{n} - P_{r_{1}} y_{n} \Vert = r_{1}\Vert A_{r_{1}} y_{n} \Vert \leq r_{1}\vert A y_{n} \vert \leq r_{1}\biggl\Vert \frac{J(x_{n} - y_{n})}{r_{n}}\biggr\Vert = r_{1}\biggl\Vert \frac{x_{n} - y_{n}}{r_{n}}\biggr\Vert . $$

and $\liminf_{n} r_{n} > 0$, we see that $\lim_{n}\Vert y_{n}-P_{r_{1}}y_{n}\Vert =0$. Then, by Lemma 2.2 and (3.3), we obtain $x_{n}\to p_{0} \in\hat{F}(P_{r_{1}})=F(P_{r_{1}})=A^{-1}0$. Since $A^{-1}0\subset C_{0}$, we get $p_{0}=P_{C_{0}}u=P_{A^{-1}0}u$, which completes the proof. □

4 Approximation theorem for the resolvents of type (Q)

We next consider an iterative scheme of resolvents of a monotone operator which is different type of Section 3, in a Banach space. Let C be a nonempty closed convex subset of a reflexive, smooth, and strictly convex Banach space E, let $r>0$ and let $A\subset E\times E^{*}$ be a monotone operator satisfying

$$ D(A) \subset C \subset J^{-1}R(J+rA) $$

(4.1)

for $r>0$. It is well known that if A is maximal monotone operator, then $J^{-1}R(J+rA)=E$; see [25–27]. Hence, if A is maximal monotone, then (4.1) holds for $C=\overline{D(A)}$. We also know that $\overline{D(A)}$ is convex; see [28]. If A satisfies (4.1) for $r>0$, then we can define the resolvent (of type (Q)) $Q_{r}:C\to D(A)$ of A by

$$ Q_{r} x =\{z\in E : Jx\in Jz+rAz \} $$

(4.2)

for all $x\in C$. In other words, $Q_{r}x=(J+rA)^{-1}Jx$ for all $x\in C$. We know the following; see, for instance, [17, 18]:

• (1) $Q_{r}$ is mapping of type (Q) from C into $D(A)$;

• (2) $(Jx-JQ_{r}x)/r \in AQ_{r} x$ for all $x \in C$;

• (3) $F(Q_{r})=A^{-1}0$.

Before our result, we need the following lemma.

Lemma 4.1

Let E be a reflexive, smooth, and strictly convex Banach space, and let $A\subset E\times E^{*}$ be a monotone operator. Let $r>0$ and C be a closed convex subset of E satisfying (4.1) for $r>0$. Then the following holds:

$$V(x,Q_{r}x)+V(Q_{r}x,x)\leq2r\bigl\langle x-Q_{r}x, x^{*}\bigr\rangle $$

for all $(x,x^{*})\in A$.

Proof

Let $(x,x^{*})\in A$. Since $(Jx-JQ_{r} x)/r \in AQ_{r} x$, we see that

$$\begin{aligned}& 0 \leq \biggl\langle x-Q_{r} x, x^{*}-\frac{Jx-JQ_{r} x}{r} \biggr\rangle , \\& \biggl\langle x-Q_{r} x,\frac{Jx-JQ_{r} x}{r} \biggr\rangle \leq \bigl\langle x-Q_{r} x, x^{*}\bigr\rangle , \\& \langle x-Q_{r} x,Jx-JQ_{r} x\rangle \leq r\bigl\langle x-Q_{r} x, x^{*}\bigr\rangle . \end{aligned}$$

From the property of V, we see that

$$V(x,Q_{r}x)+V(Q_{r}x,x) = 2\langle x-Q_{r}x, Jx-JQ_{r}x\rangle \leq2r\bigl\langle x-Q_{r} x, x^{*}\bigr\rangle $$

for all $(x,x^{*})\in A$. □

We obtain an approximation theorem for a zero point of a monotone operator in a smooth and uniformly convex Banach space by using the resolvent of type (Q).

Theorem 4.2

Let E be a uniformly smooth and uniformly convex Banach space and let $A\subset E\times E^{*}$ be a monotone operator with $A^{-1}0\ne \emptyset$. Let $\{r_{n}\}$ be a positive real numbers such that $\liminf_{n} r_{n} >0$, let C be a nonempty bounded closed convex subset of E satisfying

$$D(A) \subset C \subset J^{-1}R(J+r_{n}A) $$

for all $n\in \mathbb {N}$ and let $r\in\mathopen]0,\infty\mathclose[$ such that $C \subset B_{r}$. Let $\{\delta_{n}\}$ be a nonnegative real sequence and let $\delta_{0}=\limsup_{n} \delta_{n}$. For a given point $u\in E$, generate a sequence $\{x_{n}\}$ by $x_{1} = x \in C$, $C_{1} =C$, and

$$\begin{aligned} &y_{n}=Q_{r_{n}}x_{n}, \\ &C_{n+1} = \bigl\{ z \in C : \langle y_{n} - z, Jx_{n} - Jy_{n} \rangle\geq0\bigr\} \cap C_{n}, \\ &x_{n+1} \in\bigl\{ z \in C : \Vert u-z\Vert ^{2} \leq d(u,C_{n+1})^{2}+\delta _{n+1}\bigr\} \cap C_{n+1}, \end{aligned}$$

for all $n \in \mathbb {N}$. Then

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\underline{g}_{r}^{-1}\bigl(\overline{g}_{r} \bigl(\underline{g}_{r}^{-1}(\delta_{0})\bigr) \bigr). $$

Moreover, if $\delta_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{A^{-1}0}u$.

Proof

Since $C_{n}$ includes $A^{-1}0\ne\emptyset$ for all $n\in \mathbb {N}$, $\{C_{n}\}$ is a sequence of nonempty closed convex subsets and, by definition, it is decreasing with respect to inclusion. Let $p_{n}=P_{C_{n}}u$ for all $n\in \mathbb {N}$. Then, by Theorem 2.4, we see that $\{p_{n}\}$ converges strongly to $p_{0}=P_{C_{0}}u$, where $C_{0}=\bigcap_{n=1}^{\infty}C_{n}$. Since $x_{n}\in C_{n}$ and $d(u,C_{n})=\Vert u-p_{n} \Vert$, we see that

$$\Vert u-x_{n} \Vert ^{2} \leq \Vert u-p_{n} \Vert ^{2} +\delta_{n} $$

for every $n\in \mathbb {N}\setminus\{1\}$. From Theorem 2.6(i), we see that for $\alpha\in\mathopen]0,1\mathclose[$,

$$\begin{aligned} \Vert p_{n} -u \Vert ^{2} &\leq\bigl\Vert \alpha p_{n} +(1-\alpha)x_{n}-u \bigr\Vert ^{2} \\ &\leq\alpha \Vert p_{n} -u \Vert ^{2} +(1-\alpha)\Vert x_{n}-u \Vert ^{2} - \alpha(1-\alpha)\underline{g}_{r} \bigl(\Vert p_{n} -x_{n} \Vert \bigr) \end{aligned}$$

and thus

$$\alpha\underline{g}_{r} \bigl(\Vert p_{n} -x_{n} \Vert \bigr) \leq \Vert x_{n}-u \Vert ^{2} - \Vert p_{n} -u \Vert ^{2} \leq \delta_{n}. $$

As $\alpha\to1$, we see that $\underline{g}_{r} (\Vert p_{n} -x_{n} \Vert )\leq\delta_{n}$ and thus $\Vert p_{n} -x_{n} \Vert \leq\underline{g}_{r}^{-1}(\delta_{n})$. Using the definition of $p_{n}$, we see that $p_{n+1}\in C_{n+1}$ and thus

$$\langle y_{n} - p_{n+1}, Jx_{n} - Jy_{n} \rangle\geq0. $$

From the property of the function V, we see that

$$\begin{aligned} 0 &\leq2\langle y_{n} - p_{n+1}, Jx_{n} - Jy_{n} \rangle \\ &= 2\langle p_{n+1} - y_{n}, Jy_{n} - Jx_{n} \rangle \\ &= V(p_{n+1}, x_{n})-V(p_{n+1}, y_{n})-V(y_{n}, x_{n}) \\ &\leq V(p_{n+1}, x_{n})-V(y_{n}, x_{n}). \end{aligned}$$

By Theorem 2.7, we obtain

$$\begin{aligned} V(y_{n}, x_{n}) &\leq V(p_{n+1}, x_{n}) \\ &= V(p_{n+1}, p_{n})+V(p_{n}, x_{n})+2 \langle p_{n+1}-p_{n}, Jp_{n}-Jx_{n}\rangle \\ &\leq V(p_{n+1}, p_{n})+\overline{g}_{r}\bigl( \Vert p_{n}-x_{n}\Vert \bigr) +2\langle p_{n+1}-p_{n}, Jp_{n}-Jx_{n}\rangle \\ &\leq V(p_{n+1}, p_{n})+\overline{g}_{r}\bigl( \underline{g}_{r}^{-1}(\delta_{n})\bigr) +2\langle p_{n+1}-p_{n}, Jp_{n}-Jx_{n}\rangle. \end{aligned}$$

Since $\limsup_{n}\delta_{n}=\delta_{0}$ and $p_{n} \to p_{0}$, we see that

$$\limsup_{n\to\infty} V(y_{n}, x_{n}) \leq \overline{g}_{r}\bigl(\underline {g}_{r}^{-1}( \delta_{0})\bigr). $$

Therefore, by Theorem 2.7, we see that

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\limsup_{n\to\infty} \underline{g}_{r}^{-1} \bigl(V(y_{n}, x_{n})\bigr) \leq\underline{g}_{r}^{-1} \bigl(\overline{g}_{r}\bigl(\underline{g}_{r}^{-1}( \delta_{0})\bigr)\bigr). $$

For the latter part of the theorem, suppose that $\delta_{0}=0$. Then we see that

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq \underline{g}_{r}^{-1}\bigl(\overline{g}_{r} \bigl(\underline{g}_{r}^{-1}(0)\bigr)\bigr)=0 $$

and

$$\limsup_{n\to\infty}\underline{g}_{r}\bigl(\Vert x_{n}-p_{n}\Vert \bigr) \leq\limsup_{n\to\infty} \delta_{n}=0. $$

Therefore, we obtain

$$\lim_{n\to\infty} \Vert x_{n}-y_{n}\Vert =0 \quad \text{and}\quad \lim_{n\to\infty} \Vert x_{n}-p_{n} \Vert =0. $$

Hence, we also obtain

$$ \lim_{n\to\infty} x_{n} = p_{0} \quad \text{and}\quad \lim_{n\to\infty} y_{n} = p_{0}. $$

(4.3)

Since E is uniformly smooth, the duality mapping J is uniformly norm-to-norm continuous on each bounded subset on E. Therefore, we obtain

$$ \lim_{n\to\infty} \Vert Jx_{n}-Jy_{n} \Vert =0. $$

(4.4)

From Lemma 4.1 we see that

$$V(y_{n}, Q_{r_{1}}y_{n}) \leq V(y_{n}, Q_{r_{1}}y_{n})+V(Q_{r_{1}}y_{n}, y_{n}) \leq2r_{1}\bigl\langle y_{n}-Q_{r_{1}}y_{n}, x^{*}\bigr\rangle $$

for all $x^{*}\in Ay_{n}$. From $y_{n}$, $Q_{r_{1}}y_{n} \in D(A) \subset C \subset B_{r}$ and $(Jx_{n}-Jy_{n})/r_{n} \in Ay_{n}$, we see that

$$\begin{aligned} V(y_{n}, Q_{r_{1}}y_{n}) \leq& 2r_{1} \biggl\langle y_{n}-Q_{r_{1}}y_{n}, \frac{Jx_{n}-Jy_{n}}{r_{n}} \biggr\rangle \\ \leq& 2r_{1}\Vert y_{n}-Q_{r_{1}}y_{n} \Vert \biggl\Vert \frac{Jx_{n}-Jy_{n}}{r_{n}}\biggr\Vert \\ \leq& 2r_{1} \bigl(\Vert y_{n}\Vert +\Vert Q_{r_{1}}y_{n} \Vert \bigr) \biggl\Vert \frac{Jx_{n}-Jy_{n}}{r_{n}} \biggr\Vert \\ = & 4r_{1} r \biggl\Vert \frac{Jx_{n}-Jy_{n}}{r_{n}}\biggr\Vert . \end{aligned}$$

Since $\liminf_{n} r_{n} > 0$ and (4.4), we obtain

$$\limsup_{n\to\infty} V(y_{n},Q_{r_{1}}y_{n}) \leq0. $$

This implies $\lim_{n} V(y_{n},Q_{r_{1}}y_{n}) = 0$. From Theorem 2.5, we see that

$$\lim_{n\to\infty} \Vert y_{n} -Q_{r_{1}}y_{n} \Vert =0. $$

Then, by Lemma 2.3 and (4.3), we see that $x_{n}\to p_{0} \in\hat{F}(Q_{r_{1}})=F(Q_{r_{1}})=A^{-1}0$. Since $A^{-1}0\subset C_{0}$, we get $p_{0}=P_{C_{0}}u=P_{A^{-1}0}u$, which completes the proof. □

5 Applications

In this section, we give some applications of Theorems 3.1 and 4.2. We first study the convex minimization problem: Let E be a reflexive, smooth, and strictly convex Banach space with its dual $E^{*}$ and let $f:E \to\mathopen]-\infty, \infty\mathclose]$ be a proper lower semicontinuous convex function. Then the subdifferential ∂f of f is defined as follows:

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

for all $x \in E$. By Rockafellar’s theorem [29, 30], the subdifferential $\partial f \subset E \times E^{*}$ is maximal monotone. It is easy to see that $(\partial f)^{-1}0=\mathop{\mathrm{argmin}}\{f(x):x \in E\}$. It is also known that, see, for instance, [15, 27, 28],

$$ D(\partial f)\subset D(f) \subset\overline{D(\partial f)} . $$

(5.1)

As a direct consequence of Theorems 3.1 and 4.2, we can show the following corollaries.

Corollary 5.1

Let E be a smooth and uniformly convex Banach space, let $f:E \to\mathopen]-\infty, \infty\mathclose]$ be a proper lower semicontinuous convex function with $D(f)$ being bounded, and let $r\in\mathopen]0,\infty\mathclose[$ such that $D(f) \subset B_{r}$. Let $\{\delta_{n}\}$ be a nonnegative real sequence and let $\delta_{0}=\limsup_{n} \delta_{n}$. For a given point $u\in E$, generate a sequence $\{x_{n}\}$ by $x_{1} = x \in\overline{D(f)}$, $C_{1} =\overline{D(f)}$, and

$$\begin{aligned} & y_{n} = \mathop{\mathrm{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2r_{n}} \Vert y-x_{n}\Vert ^{2} \biggr\} , \\ & C_{n+1} = \bigl\{ z \in\overline{D(f)} : \bigl\langle y_{n} - z, J(x_{n} - y_{n}) \bigr\rangle \geq0\bigr\} \cap C_{n}, \\ & x_{n+1} \in\bigl\{ z \in\overline{D(f)} : \Vert u-z\Vert ^{2} \leq d(u,C_{n+1})^{2}+\delta_{n+1} \bigr\} \cap C_{n+1}, \end{aligned}$$

for all $n \in \mathbb {N}$, where $\{r_{n}\}\subset\mathopen]0,\infty\mathclose[$ such that $\liminf_{n} r_{n} >0$. If $(\partial f)^{-1}0$ is nonempty, then

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\underline {g}_{r}^{-1}(\delta_{0}). $$

Moreover, if $\delta_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{(\partial f)^{-1}0}u$.

Proof

Put $C=\overline{D(f)}$. Since the subdifferential $\partial f \subset E \times E^{*}$ is maximal monotone, we have $E=R(I+r\partial f)$ for all $r>0$ and hence, from (5.1), we see that

$$D(\partial f)\subset\overline{D(\partial f)} =\overline{D(f)} =C \subset E= R(I+r\partial f) $$

for all $r>0$.

Fix $r>0$ and $z\in C$. Let $P_{r}$ be the resolvent (of type (P)) of ∂f, then we also know that

$$P_{r} z = \mathop{\mathrm{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2r} \Vert y-z\Vert ^{2} \biggr\} . $$

Therefore, we obtain the desired result by Theorem 3.1. □

Corollary 5.2

Let E be a uniformly smooth and uniformly convex Banach space, let $f:E \to\mathopen]-\infty, \infty\mathclose]$ be a proper lower semicontinuous convex function with $D(f)$ being bounded and let $r\in\mathopen]0,\infty\mathclose[$ such that $D(f) \subset B_{r}$. Let $\{\delta_{n}\}$ be a nonnegative real sequence and let $\delta_{0}=\limsup_{n} \delta_{n}$. For a given point $u\in E$, generate a sequence $\{x_{n}\}$ by $x_{1} = x \in\overline{D(f)}$, $C_{1} =\overline{D(f)}$, and

$$\begin{aligned} & y_{n}=\mathop{\mathrm{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2r_{n}} \Vert y\Vert ^{2} -\frac{1}{r_{n}}\langle y, Jx_{n} \rangle \biggr\} , \\ & C_{n+1} = \bigl\{ z \in\overline{D(f)} : \langle y_{n} - z, Jx_{n} - Jy_{n} \rangle\geq0\bigr\} \cap C_{n}, \\ & x_{n+1} \in\bigl\{ z \in\overline{D(f)} : \Vert u-z\Vert ^{2} \leq d(u,C_{n+1})^{2}+\delta_{n+1} \bigr\} \cap C_{n+1}, \end{aligned}$$

for all $n \in \mathbb {N}$, where $\{r_{n}\}\subset\mathopen]0,\infty\mathclose[$ such that $\liminf_{n} r_{n} >0$. If $(\partial f)^{-1}0$ is nonempty, then

$$\limsup_{n\to\infty} \Vert x_{n}-y_{n}\Vert \leq\underline{g}_{r}^{-1}\bigl(\overline{g}_{r} \bigl(\underline{g}_{r}^{-1}(\delta_{0})\bigr) \bigr). $$

Moreover, if $\delta_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{(\partial f)^{-1}0}u$.

Proof

Fix $r>0$ and $z\in C$. Let $Q_{r}$ be the resolvent (of type (Q)) of ∂f, then we also know that

$$Q_{r} z = \mathop{\mathrm{argmin}}_{y \in E} \biggl\{ f(y) + \frac {1}{2r} \Vert y\Vert ^{2} -\frac{1}{r}\langle y, Jz \rangle \biggr\} . $$

In the same way as Corollary 5.1, we obtain the desired result by Theorem 4.2. □

Next, we study the approximation of fixed points for mappings of type (P) and (Q). Before show our applications, we need the following results.

Lemma 5.3

([17])

Let E be a reflexive, smooth, and strictly convex Banach space, let C be a nonempty subset of E, let $T:C\to E$ be a mapping, and let $A_{T}\subset E\times E^{*}$ be an operator defined by $A_{T}=J(T^{-1}-I)$. Then T is of mapping of type (P) if and only if $A_{T}$ is monotone. In this case $T=(I+J^{-1}A_{T})^{-1}$.

Lemma 5.4

([31])

Let E be a reflexive, smooth, and strictly convex Banach space, let C be a nonempty subset of E and let $T:C\to E$ be a mapping, and let $A_{T}\subset E\times E^{*}$ be an operator defined by $A_{T}=JT^{-1}-J$. Then T is a mapping of type (Q) if and only if $A_{T}$ is monotone. In this case $T=(J+A_{T})^{-1}J$.

As a direct consequence of Theorems 3.1 and 4.2, we can show the following corollaries.

Corollary 5.5

Let E be a smooth and uniformly convex Banach space, let C be a bounded closed convex subset of E. Let $T:C \to C$ be a mapping of type (P) with $F(T)$ being nonempty and let $r\in\mathopen]0,\infty\mathclose[$ such that $C \subset B_{r}$. Let $\{\delta_{n}\}$ be a nonnegative real sequence and let $\delta_{0}=\limsup_{n} \delta_{n}$. For a given point $u\in E$, generate a sequence $\{x_{n}\}$ by $x_{1} = x \in C$, $C_{1} =C$, and

$$\begin{aligned} &C_{n+1} = \bigl\{ z \in C : \bigl\langle Tx_{n} - z, J(x_{n} - Tx_{n}) \bigr\rangle \geq0\bigr\} \cap C_{n}, \\ &x_{n+1} \in\bigl\{ z \in C : \Vert u-z\Vert ^{2} \leq d(u,C_{n+1})^{2}+\delta _{n+1}\bigr\} \cap C_{n+1}, \end{aligned}$$

for all $n \in \mathbb {N}$, where $\{r_{n}\}\subset(0,\infty)$ such that $\liminf_{n} r_{n} >0$. Then

$$\limsup_{n\to\infty} \Vert x_{n}-Tx_{n}\Vert \leq\underline {g}_{r}^{-1}(\delta_{0}). $$

Moreover, if $\delta_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{F(T)}u$.

Proof

Put $A_{T}=J(T^{-1}-I)$ and $r_{n}=1$ for all $n\in \mathbb {N}$. From Lemma 5.3, we see that T is the resolvent (of type (P)) of $A_{T}$ for 1 and

$$D(A_{T}) =R(T) \subset C=D(T)=R\bigl(I+J^{-1}A_{T} \bigr). $$

Therefore, we obtain the desired result by Theorem 3.1. □

Corollary 5.6

Let E be a uniformly smooth and uniformly convex Banach space, let C be a bounded closed convex subset of E. Let $T:C \to C$ be a mapping of type (Q) with $F(T)$ being nonempty and let $r\in\mathopen]0,\infty\mathclose[$ such that $C \subset B_{r}$. Let $\{\delta_{n}\}$ be a nonnegative real sequence and let $\delta_{0}=\limsup_{n} \delta_{n}$. For a given point $u\in E$, generate a sequence $\{x_{n}\}$ by $x_{1} = x \in C$, $C_{1} =C$, and

$$\begin{aligned} &C_{n+1} = \bigl\{ z \in C : \langle Tx_{n} - z, Jx_{n} - JTx_{n} \rangle\geq0\bigr\} \cap C_{n}, \\ &x_{n+1} \in\bigl\{ z \in C : \Vert u-z\Vert ^{2} \leq d(u,C_{n+1})^{2}+\delta _{n+1}\bigr\} \cap C_{n+1}, \end{aligned}$$

for all $n \in \mathbb {N}$. Then

$$\limsup_{n\to\infty} \Vert x_{n}-Tx_{n}\Vert \leq\underline{g}_{r}^{-1}\bigl(\overline{g}_{r} \bigl(\underline{g}_{r}^{-1}(\delta_{0})\bigr) \bigr). $$

Moreover, if $\delta_{0}=0$, then $\{x_{n}\}$ converges strongly to $P_{F(T)}u$.

Proof

In the same way as Corollary 5.5, we obtain the desired result by Lemma 5.4 and Theorem 4.2. □