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. (1)

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

  2. (2)

    if E is reflexive, then J is surjective;

  3. (3)

    if E is smooth, then the duality mapping J is single valued.

  4. (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. (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. (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. (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. (2)

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

  3. (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. (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

  1. (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 \(\alpha\in[0,1]\);

  2. (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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (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. (1)

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

  2. (2)

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

  3. (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. □