The Proximal Point Method for Infinite Families of Maximal Monotone Operators and Set-Valued Mappings

Abstract

In the present paper we use the proximal point method in order to find an approximate common zero of an infinite collection of maximal monotone maps in a real Hilbert space under the presence of computational errors. We prove that the inexact proximal point method generates an approximate solution if these errors are sufficiently small.

1. Preliminaries and the First Main Result

Proximal point method is an important tool in solving optimization problems [1,2,3,4,5]. It is also used for solving variational inequalities with monotone operators [6,7,8,9,10,11,12,13,14,15] which is an important topic of nonlinear analysis and optimization. In the present paper we use a proximal point method in order to find an approximate common zero of an infinite collection of maximal monotone operators in a real Hilbert space under the presence of computational errors. Usually in the study of variational inequalities the main issue is the existence of their solutions [10,13,15]. In the present paper we are interested in their approximate solutions. We prove that the inexact proximal point method produces an approximate solution if computational errors are summable. Also we show that if the computational errors are small enough, then the inexact proximal point method generates approximate solutions. Our results extend the results of [16] obtained for finite families of monotone operators.

Assume that $(X,\langle ·,·\rangle )$ is a real a Hilbert space endowed with an inner product $\langle ·,·\rangle$. This inner product induces the norm $\parallel ·\parallel$.

A set-valued mapping $T:X\to 2^X$ is called a monotone operator only if the following is valid:

$$\langle z-z^{\prime },w-w^{\prime }\rangle \ge 0\text{for all points}z,z^{\prime },w,w^{\prime }\in X$$

$$\mathrm{satisfying}w\in T\left(z\right) \mathrm{and} w^{\prime }\in T\left(z^{\prime }\right).$$

(1)

It is called maximal monotone if, in addition, the graph, calculated as follows:

$$\left\{\right(z,w)\in X\times X:w\in T(z\left)\right\}$$

is not properly contained in the graph of any other monotone mapping.

Assume that $T:X\to 2^X$ is a maximal monotone mapping. The proximal point algorithm produces, for an arbitrary sequence of positive numbers and an arbitrary initial point in the Hilbert space, a sequence of iterates and the goal is to establish the convergence of this sequence of iterates. It should be mentioned that that in a general infinite-dimensional Hilbert space this convergence is usually weak. The proximal algorithm used in order to solve the inclusion $0\in T\left(z\right)$ is strongly based on the fundamental result obtained by Minty [17], who proved that, for an arbitrary point $z\in X$ and an arbitrary positive number c, there exists a unique point $u\in \mathcal{X}$ satisfying the following:

$$z\in (I+clT)\left(u\right),$$

where $I:X\to X$ is the identity mapping ($Ix=x$ for every point $x\in X$).

The mapping, calculated as follows:

$$P_{c,T}:={(I+cT)}^{-1}$$

(2)

is therefore a single-valued self-mapping of X (where c is an arbitrary positive number). Moreover, this mapping is nonexpansive, written as follows:

$$\parallel P_{c,\mathcal{T}}\left(z\right)-P_{c,T}\left(z^{\prime }\right)\parallel \le \parallel z-z^{\prime }\parallel \text{for all points}z,z^{\prime }\in X$$

(3)

and

$$P_{c,T}\left(z\right)=z\text{if and only if}0\in T\left(z\right).$$

(4)

According to [18] $P_{c,T}$ is called the proximal mapping associated with $cT$.

The proximal point algorithm produces, for an arbitrary sequence ${\left\{c_k\right\}}_{k=0}^{\infty }$ of positive numbers and an arbitrary initial point $z^0\in X$, a sequence of iterates ${\left\{z^k\right\}}_{k=0}^{\infty }\subset X$ such that the equation is written as follows:

$$z^{k+1}:=P_{c_k,T}\left(z^k\right),k=0,1,\dots$$

It is clear that the following formula:

$$\mathrm{graph}\left(T\right):=\left\{\right(x,w)\in X\times X:w\in T(x\left)\right\}$$

is closed in the norm topology of the product space $X\times X$.

Define the following:

$$F\left(T\right)=\{z\in X:0\in T(z\left)\right\}.$$

(5)

Usually algorithms studied in the literature produce iterates converging weakly to a point of of $F\left(T\right)$. In this paper, for a given positive number $ϵ$, we are interested to find a point $x\in X$ such that there is $y\in T\left(x\right)$ satisfying $\parallel y\parallel \le ϵ$.

For each point $x\in X$ and each nonempty set $A\subset X$ write the following:

$$\rho (x,A):=\inf \{\parallel x-y\parallel :y\in A\}.$$

For each $x\in X$ and each $r>0$ set the following:

$$B(x,r)=\{y\in X:\parallel x-y\parallel \le r\}.$$

We denote by Card $\left(A\right)$ the cardinality of the set A. For every number $x\in R^1=(-\infty ,\infty )$ define $\lfloor x\rfloor =\max \{i \text{is an integer}:i\le x\}.$

We apply the proximal point algorithm in order to generate an appropriate approximation of a point which is a common zero of an infinite collection of maximal monotone maps and a fixed point of a set-valued mapping.

It is easy to see that the following lemma holds (see [11]):

Lemma 1.

Let $z,x_0,x_1\in X$. Then calculate the following:

$$2^{-1}{\parallel z-x_0\parallel }^2-2^{-1}{\parallel z-x_1\parallel }^2-2^{-1}{\parallel x_0-x_1\parallel }^2=\langle x_0-x_1,x_1-z\rangle .$$

In the sequel we use the following lemma [16].

Lemma 2.

Let $A:X\to 2^X$ be a maximal monotone mapping and calculate the following:

$$z\in F\left(A\right).$$

Then obtain the following:

$${\parallel z-x\parallel }^2-{\parallel z-A\left(x\right)\parallel }^2-{\parallel x-A\left(x\right)\parallel }^2\ge 0.$$

It should be mentioned that variational inequalities, besides their great theoretical importance, have many important applications in many areas, including large-scale optimization, image processing, and convex feasibility problems. In our work, we establish results on the behavior of inexact iterates under the presence of computational error, which is always present in practice. The main difference between the results of this paper and the results known in the literature [6,7,8,9,10,11,12,13,14,15,19,20] is that here we deal with infinite families of nonlinear operators.

Example 1.

Assume that $C_{\alpha }$, $\alpha \in \mathcal{A}$ is a family of nonempty, closed convex sets in a Hilbert space X which have a nonempty intersection. For each $\alpha \in \mathcal{A}$, define a convex function $f_{\alpha }\left(x\right)=\inf \{\parallel x-y\parallel :y\in C_{\alpha }\}$, $x\in X$ and a maximal monotone operator, written as follows:

$$T_{\alpha }\left(x\right)=\partial f_{\alpha }\left(x\right),x\in X.$$

It is well-known [11,12] that a common zero point of this family of operators belongs to the intersection of sets $C_{\alpha }$, $\alpha \in \mathcal{A}$. In other words, it is a solution to our original feasibility problem.

2. Infinite Families of Variational Inequalities

Now we extend the results of [16] to the case of an infinite family of variational inequalities. We consider a problem of finding a point that is a fixed point of a set-valued mapping (T in the sequel) and also a common zero point of a family of maximal monotone operators ($T_{\alpha }$, $\alpha \in A$ in the sequel). More precisely, we prove two main results that show that the inexact proximal point method generates an approximate solution if these errors are sufficiently small. In the first result we consider the inexact iterates with summable errors, while the second one deals with nonsummable errors.

Let $(X,\langle ·,·\rangle )$ be a Hilbert space equipped with an inner product $\langle ·,·\rangle$, which induces the norm $\parallel ·\parallel$.

Assume the following:

$$T:X\to 2^X\backslash \{\varnothing \},\overline{c}\in (0,1],$$

(6)

$$F_T=\{z\in X:T\left(z\right)=\left\{z\right\}\}\ne \varnothing .$$

(7)

Assume the following for each $z\in F_T$, each $x\in X$, and each $y\in T\left(x\right)$:

$$\parallel z-y\parallel \le \parallel z-x\parallel .$$

(8)

We use the following assumption:

(A) For each $x\in X$, each $y\in T\left(x\right)$ and each $z\in F_T$, assume the following:

$${\parallel z-x\parallel }^2\ge {\parallel z-y\parallel }^2+\overline{c}{\parallel x-y\parallel }^2.$$

Assumption (A) is well-known in the literature for single-valued operators. See [11,12] and the references mentioned therein. In particular it holds with $\overline{c}=1$ for projection operators and firmly for nonexpansive operators.

Assume that $\mathcal{A}$ is a nonempty set and that for each $\alpha \in \mathcal{A}$, $T_{\alpha }:X\to 2^X$ is a maximal monotone operator. We suppose the following for each $\alpha \in \mathcal{A}$:

$$F\left(T_{\alpha }\right)=\{z\in X:0\in T_{\alpha }\left(z\right)\}\ne \varnothing .$$

Set the following:

$$F=F_T\cap {\cap }_{\alpha \in \mathcal{A}}F\left(T_{\alpha }\right).$$

(9)

For each $ϵ\in (0,1]$ and each $\alpha \in \mathcal{A}$, define the following:

$$F\left(T_{\alpha ,ϵ}\right)=B(0,ϵ)+\{x\in X:T_{\alpha }\left(x\right)\cap B(0,ϵ)\ne \varnothing \},$$

$$F_{ϵ}=\{x\in X:T\left(x\right)\subset B(x,ϵ)\}\cap {\cap }_{\alpha \in \mathcal{A}}F\left(T_{\alpha ,ϵ}\right).$$

(10)

Let $\overline{\lambda }>0$ and let $\overline{\lambda }=\infty$ and ${\overline{\lambda }}^{-1}=0$, if $\mathcal{A}=\varnothing$.

For each $x\in X$ set, define the following:

$$\tilde{T}\left(x\right)=T\left(x\right)\cup \{P_{c,T_{\alpha }}\left(x\right):\alpha \in \mathcal{A},c\in [\overline{\lambda },\infty )\}.$$

(11)

The following result shows that in the case of inexact iterates under the presence of summable computational errors, most of the iterates are approximate solutions of our problem. Namely, it is shown that for a given $ϵ>0$ the number of the inexact iterates that do not belong to $F_{ϵ}$ does not exceed a constant depending on $ϵ$. The value of this constant is obtained.

Theorem 1.

Let (A) hold, $M>1$, $ϵ\in (0,1)$, ${\left\{r_i\right\}}_{i=0}^{\infty }\subset (0,1)$, ${\left\{{∆}_i\right\}}_{i=0}^{\infty }\subset (0,\infty )$ satisfy the following:

$$r=\sum _{i=0}^{\infty }{r}_i<\infty ,\underset{i\to \infty }{lim}{∆}_i=0,$$

(12)

$$B(0,M)\cap F\ne \varnothing ,$$

(13)

$$\gamma =2^{-1}min\{1,\overline{\lambda }\}ϵ,$$

(14)

$n_0$ be a natural number such that the following is valid:

$${∆}_i<\gamma /4,r_i<\gamma /4 \mathit{for} \mathit{each} \mathit{integer} i\ge n_0.$$

(15)

Assume that ${\left\{x_t\right\}}_{t=0}^{\infty }\subset X$, written as follows:

$$\parallel x_0\parallel \le M$$

(16)

and that for each integer $k\ge 0$, the following is valid:

$$B(x_{k+1},r_k)\cap \{\xi \in \tilde{T}\left(x_k\right):$$

$$\parallel x_k-\xi \parallel \ge \parallel x_k-\eta \parallel -{∆}_k,\eta \in T\left(x_k\right)$$

$$and for each \alpha \in \mathcal{A} there exists c_{\alpha }\ge \overline{\lambda } such that$$

$$\parallel x_k-\xi \parallel \ge \parallel x_k-P_{c_{\alpha },T_{\alpha }}\left(x_k\right)\parallel -{∆}_k\}.$$

(17)

Then calculate the following:

$$Card\left(\right\{n\in \{0,1,\dots ,\}:\parallel x_n-x_{n+1}\parallel \ge \gamma \left\}\right)$$

$$\le {\overline{c}}^{-1}{\gamma }^{-2}(4M^2+8Mr+6r^2)$$

and if $k\ge n_0$ is an integer and $\parallel x_k-x_{k+1}\parallel \le \gamma$, then $x_k\in F_{ϵ}$.

Proof. 

In view of (13) the following exists:

$$z_{*}\in B(0,M)\cap F.$$

(18)

By (17), for each integer $k\ge 0$ the following exists:

$${\xi }_k\in B(x_{k+1},r_k)\cap \tilde{T}\left(x_k\right)$$

(19)

such that the following is calculated:

$$\parallel x_k-{\xi }_k\parallel \ge \parallel x_k-\eta \parallel -{∆}_k,\eta \in T\left(x_k\right)$$

(20)

and for each $\alpha \in \mathcal{A}$ there exists $c_{k,\alpha }\ge \overline{\lambda }$ such that the following is calculated:

$$\parallel x_k-{\xi }_k\parallel \ge \parallel x_k-P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\parallel -{∆}_k.$$

(21)

Let $k\ge 0$ be an integer. In view of (11) and (19), there are two cases, written as follows:

$${\xi }_k\in T\left(x_k\right);$$

(22)

there exists ${\alpha }_k\in \mathcal{A}$, $c_k\ge \overline{\lambda }$ such that the following is calculated:

$${\xi }_k=P_{c_k,T_{{\alpha }_k}}\left(x_k\right).$$

(23)

Assume that (22) holds. Assumption (A) and (18), (22) imply the following:

$${\parallel z_{*}-x_k\parallel }^2\ge {\parallel z_{*}-{\xi }_k\parallel }^2+\overline{c}{\parallel x_k-{\xi }_k\parallel }^2.$$

(24)

Assume that (23) hold. Lemma 2 and (9), (18), (23) imply that (24) holds. Thus (24) holds in both cases. By (19) and (24), calculate the following:

$$\parallel {\xi }_k-z_{*}\parallel \le \parallel z_{*}-x_k\parallel ,$$

(25)

$$\parallel x_{k+1}-z_{*}\parallel \le \parallel x_{k+1}-{\xi }_k\parallel +\parallel {\xi }_k-z_{*}\parallel \le r_k+\parallel x_k-z_{*}\parallel .$$

(26)

By (12), (16), (18), and (26), for each integer $k\ge 0$, calculate the following:

$$\parallel z_{*}-x_k\parallel \le \parallel z_{*}-x_0\parallel +\sum _{i=0}^{\infty }{r}_i\le 2M+r.$$

(27)

Let $k\ge 0$ be an integer. By (19), (25), (27), and the triangle inequality, calculate the following:

$$|{\parallel x_k-x_{k+1}\parallel }^2-{\parallel x_k-{\xi }_k\parallel }^2|$$

$$\le |\parallel x_k-x_{k+1}\parallel -\parallel x_k-{\xi }_k\parallel |(\parallel x_k-x_{k+1}\parallel +\parallel x_k-{\xi }_k\parallel )$$

$$\le \parallel x_{k+1}-{\xi }_k\parallel (2\parallel x_k-{\xi }_k\parallel +\parallel x_{k+1}-{\xi }_k\parallel )$$

$$\le r_k(4M+2r+r_k)\le r_k(4M+3r).$$

(28)

By (19) and (27), calculate the following:

$$|{\parallel x_{k+1}-z_{*}\parallel }^2-{\parallel {\xi }_k-z_{*}\parallel }^2|$$

$$\le |\parallel x_{k+1}-z_{*}\parallel -\parallel {\xi }_k-z_{*}\parallel |(\parallel x_{k+1}-z_{*}\parallel +\parallel {\xi }_k-z_{*}\parallel )$$

$$\le \parallel x_{k+1}-{\xi }_k\parallel (2\parallel z_{*}-x_{k+1}\parallel +\parallel x_{k+1}-{\xi }_k\parallel )$$

$$\le r_k(4M+3r).$$

(29)

Equations (24), (28), and (29) imply the following:

$${\parallel z_{*}-x_k\parallel }^2\ge {\parallel z_{*}-{\xi }_k\parallel }^2+\overline{c}{\parallel x_k-{\xi }_k\parallel }^2$$

$$\ge {\parallel x_{k+1}-z_{*}\parallel }^2-r_k(4M+3r)$$

$$+\overline{c}{\parallel x_k-x_{k+1}\parallel }^2-r_k(4M+3r)$$

$$\ge {\parallel x_{k+1}-z_{*}\parallel }^2+\overline{c}{\parallel x_k-x_{k+1}\parallel }^2-2r_k(4M+3r).$$

(30)

Set the following:

$$E=\{n\in \{0,1,\dots \}:\parallel x_n-x_{n+1}\parallel \ge \gamma \}.$$

(31)

Let Q be a natural number and calculated as follows:

$$E_Q=\{0,\dots ,Q-1\}\cap E.$$

(32)

It follows from (16), (18), and (30) the following is calculated:

$$4M^2\ge {\parallel x_0-z_{*}\parallel }^2\ge {\parallel x_0-z_{*}\parallel }^2-{\parallel x_Q-z_{*}\parallel }^2$$

$$=\sum _{k=0}^{Q-1}[{\parallel x_k-z_{*}\parallel }^2-{\parallel x_{k+1}-z_{*}\parallel }^2]$$

$$\ge \sum _{k=0}^{Q-1}(\overline{c}{\parallel x_k-x_{k+1}\parallel }^2-r_k(4M+3r)).$$

By the relation above and (31), the following is calculated:

$${\overline{c}}^{-1}(4M^2+2r(4M+3r))\ge \sum _{k=0}^{Q-1}{\parallel x_k-x_{k+1}\parallel }^2\ge {\gamma }^2(\mathrm{Card}\left(E_Q\right),$$

$$\mathrm{Card}\left(E_Q\right)\le {\overline{c}}^{-1}{\gamma }^{-2}(4M^2+8Mr+6r^2).$$

Since Q is any natural number we conclude the following:

$$\mathrm{Card}\left(E\right)\le {\overline{c}}^{-1}{\gamma }^{-2}(4M^2+8Mr+6r^2).$$

Assume that the following:

$$k\ge n_0$$

(33)

is an integer and assume the following:

$$\parallel x_k-x_{k+1}\parallel \le \gamma .$$

(34)

By (19) and (34), obtain the following:

$$\parallel x_k-{\xi }_k\parallel \le \parallel x_k-x_{k+1}\parallel +\parallel x_{k+1}-{\xi }_k\parallel \le {\gamma }_k+r_k.$$

(35)

It follows from (14), (15), (20), (33), and (35) that for each $\eta \in T\left(x_k\right)$ the following is calculated:

$$\parallel x_k-\eta \parallel \le \parallel x_k-{\xi }_k\parallel +{∆}_k\le \gamma +r_k+{∆}_k,$$

$$T\left(x_k\right)\subset B(x_k,\gamma +r_k+{∆}_k)\subset B(x_K,2\gamma )\subset B(x_k,ϵ).$$

(36)

Let $\alpha \in \mathcal{A}$. By (15), (21), (33), and (35), the following is calculated:

$$\parallel x_k-P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\parallel \le \parallel x_k-{\xi }_k\parallel +{∆}_k\le \gamma +r_k+{∆}_k\le 2\gamma .$$

(37)

By (2), the following is calculated:

$$x_k\in (I+c_{k,\alpha }{T}_{\alpha })\left(P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\right).$$

(38)

By (38), the following is calculated:

$$x_k-P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\in c_{k,\alpha }{T}_{\alpha }\left(P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\right),$$

$$c_{k,\alpha }^{-1}(x_k-P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right))\in T_{\alpha }\left(P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\right).$$

(39)

By (14), (37), and (39), the following is calculated:

$$c_{k,\alpha }^{-1}(x_k-P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right))\le 2{\overline{\lambda }}^{-1}\gamma \le ϵ$$

$$P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\in F_{ϵ}\left(T_{\alpha }\right).$$

Together with (37) this implies the following:

$$B(x_k,ϵ)\cap F_{ϵ}\left(T_{\alpha }\right)\ne \varnothing ,\alpha \in \mathcal{A}.$$

Combined with (6) this implies the following:

$$x_k\in F_{ϵ}.$$

Theorem 1 is proved. □

Our next result shows that in the case of inexact iterates an $ϵ$-approximate fixed point can be obtained under the presence of computational errors ${\delta }_0,{\delta }_1$ after $n_0$ iterates and these constants depend on $ϵ$. The estimations for these constants are obtained.

Theorem 2.

Let (A) hold, $M>1$, $ϵ\in (0,1)$, calculated as follows:

$$B(0,M)\cap F\ne \varnothing ,$$

(40)

$$\gamma =2^{-1}min\{1,\overline{\lambda }\}ϵ,$$

(41)

positive numbers ${\delta }_1,{\delta }_2$ satisfy the following:

$${\delta }_1,{\delta }_2\le \gamma /4,{\delta }_1\le {(12M+8)}^{-1}\overline{c}{\gamma }^2/2,$$

(42)

and

$$n_0=\lfloor 2{(2M+1)}^2{\overline{c}}^{-1}{\gamma }^{-2}\rfloor +2.$$

(43)

Assume that ${\left\{x_t\right\}}_{t=0}^{\infty }\subset X$, written as follows:

$$\parallel x_0\parallel \le M$$

(44)

and that for each integer $k\ge 0$, the following is calculated:

$$B(x_{k+1},{\delta }_1)\cap \{\xi \in \tilde{T}\left(x_k\right):$$

$$\parallel x_k-\xi \parallel \ge \parallel x_k-\eta \parallel -{\delta }_0,\eta \in T\left(x_k\right)$$

$$and for each \alpha \in \mathcal{A} there exists c_{\alpha }\ge \overline{\lambda } such that$$

$$\parallel x_k-\xi \parallel \ge \parallel x_k-P_{c_{\alpha },T_{\alpha }}\left(x_k\right)\parallel -{\delta }_0\}.$$

(45)

Then there exists an integer $q\in [1,n_0]$ such that the following is calculated:

$$\parallel x_i\parallel \le 3M+1,i\in \{1,\dots ,q\}$$

and

$$\parallel x_q-x_{q+1}\parallel \le \gamma .$$

Moreover, if an integer $q\ge 0$ and $\parallel x_q-x_{q+1}\parallel \le \gamma$, then the following is calculated:

$$x_q\in F_{ϵ}.$$

Proof. 

In view of (40) the following exists:

$$z_{*}\in B(0,M)\cap F.$$

(46)

By (45), for each integer $k\ge 0$ there exists

$${\xi }_k\in B(x_{k+1},{\delta }_1)\cap \tilde{T}\left(x_k\right)$$

(47)

such that the following is obtained:

$$\parallel x_k-{\xi }_k\parallel \ge \parallel x_k-\eta \parallel -{\delta }_0,\eta \in T\left(x_k\right)$$

(48)

and such that for each $\alpha \in \mathcal{A}$ there exists $c_{k,\alpha }\ge \overline{\lambda }$ such that the following is obtained:

$$\parallel x_k-\xi \parallel \ge \parallel x_k-P_{c_{k,\alpha },T_{\alpha }}\left(x_k\right)\parallel -{\delta }_0.$$

(49)

Let $k\ge 0$ be an integer. In view of (41) and (47), there are two cases, calculated as follows:

$${\xi }_k\in T\left(x_k\right);$$

(50)

there exists ${\alpha }_k\in \mathcal{A}$, $c_k\ge \overline{\lambda }$ such that the following is calculated:

$${\xi }_k=P_{c_k,T_{{\alpha }_k}}\left(x_k\right)$$

(51)

Assume that (50) holds. Assuming (A) and (46), (50) implies the following:

$${\parallel z_{*}-x_k\parallel }^2\ge {\parallel z_{*}-{\xi }_k\parallel }^2+\overline{c}{\parallel x_k-{\xi }_k\parallel }^2.$$

(52)

Assume that (51) holds. Lemma 2 and (9), (46) imply that (52) holds. Thus (52) holds in both cases. By (47) and (52), the following is obtained:

$$\parallel {\xi }_k-z_{*}\parallel \le \parallel z_{*}-x_k\parallel ,$$

(53)

$$\parallel x_{k+1}-z_{*}\parallel \le \parallel x_{k+1}-{\xi }_k\parallel +\parallel {\xi }_k-z_{*}\parallel \le {\delta }_1+\parallel x_k-z_{*}\parallel .$$

(54)

By (44), (46) and (54), the following is obtained:

$$\parallel z_{*}-x_1\parallel \le \parallel z_{*}-x_0\parallel +{\delta }_1\le 2M+1.$$

(55)

Let $k\ge 0$ be an integer. By (47), (53), and the triangle inequality, the following is obtained:

$${|{\parallel x_k-x_{k+1}\parallel }^2-\parallel x_k-{\xi }_k\parallel |}^2$$

$$\le |\parallel x_k-x_{k+1}\parallel -\parallel x_k-{\xi }_k\parallel |(\parallel x_k-x_{k+1}\parallel +\parallel x_k-{\xi }_k)\parallel )$$

$$\le \parallel x_{k+1}-{\xi }_k\parallel (2\parallel x_k-{\xi }_k\parallel +\parallel x_{k+1}-{\xi }_k\parallel )$$

$$\le {\delta }_1(2\parallel x_k-z_{*}\parallel +2\parallel z_{*}-{\xi }_k\parallel +{\delta }_1)\le {\delta }_1(4\parallel x_k-z_{*}\parallel +1).$$

(56)

By (47), (57), and the triangle inequality, the following is obtained:

$$|{\parallel x_{k+1}-z_{*}\parallel }^2-{\parallel {\xi }_k-z_{*}\parallel }^2|$$

$$\le |\parallel x_{k+1}-z_{*}\parallel -\parallel {\xi }_k-z_{*}\parallel |(\parallel x_{k+1}-z_{*}\parallel +\parallel {\xi }_k-z_{*}\parallel )$$

$$\le \parallel x_{k+1}-{\xi }_k\parallel (2\parallel z_{*}-{\xi }_k\parallel +\parallel x_{k+1}-{\xi }_k\parallel )$$

$$\le {\delta }_1(2\parallel z_{*}-x_k\parallel +1).$$

(57)

Equations (52), (56), and (57) imply the following:

$${\parallel z_{*}-x_k\parallel }^2\ge {\parallel z_{*}-{\xi }_k\parallel }^2+\overline{c}{\parallel x_k-{\xi }_k\parallel }^2$$

$$\ge {\parallel x_{k+1}-z_{*}\parallel }^2-{\delta }_1(2\parallel (z_{*}-x_k\parallel +1)+\overline{c}{\parallel x_k-x_{k+1}\parallel }^2-{\delta }_1(4\parallel (z_{*}-x_k\parallel +1)$$

$$\ge {\parallel x_{k+1}-z_{*}\parallel }^2+\overline{c}{\parallel x_k-x_{k+1}\parallel }^2-{\delta }_1(6\parallel (z_{*}-x_k\parallel +2).$$

(58)

Assume that s is a natural number and that for each integer $k\in [1,s]$, the following is calculated:

$$\parallel x_k-x_{k+1}\parallel >\gamma .$$

(59)

Assume the following:

$$k\in \{1,\dots ,s\}$$

and

$$\parallel x_k-z_{*}\parallel \le 2M+1.$$

(60)

(In view of (55), Equation (60) holds for $k=1$.) In view of (58)–(60), calculate the following:

$${\parallel z_{*}-x_k\parallel }^2\ge {\parallel z_{*}-x_{k+1}\parallel }^2+\overline{c}{\gamma }^2-{\delta }_1(12M+8).$$

(61)

It follows from (42), that (60) and (61) result in the following:

$${\delta }_1(12M+8)\le \overline{c}{\gamma }^2/2,$$

$${\parallel z_{*}-x_k\parallel }^2-{\parallel z_{*}-x_{k+1}\parallel }^2\ge \overline{c}{\gamma }^2/2,$$

(62)

$$\parallel z_{*}-x_{k+1}\parallel \le 2M+1.$$

(63)

Thus we showed that if $k\in \{1,\dots ,s\}$ and (66) holds, then (62) and (63) hold. By induction this implies that for all $k=1,\dots ,s+1$, the following is obtained:

$$\parallel z_{*}-x_k\parallel \le 2M+1$$

and for all $k=1,\dots ,s$, (62) holds.

It follows from (43), (55), and (62) that the following is obtained:

$${(2M+1)}^2\ge {\parallel z_{*}-x_1\parallel }^2)\ge {\parallel z_{*}-x_1\parallel }^2-{\parallel z_{*}-x_{s+1}\parallel }^2$$

$$=\sum _{k=1}^s({\parallel z_{*}-x_k\parallel }^2-{\parallel z_{*}-x_{k+1}\parallel }^2)\ge 2^{-1}\overline{c}{\gamma }^{2}s$$

and

$$s\le 2{(2M+1)}^2{\overline{c}}^{-1}{\gamma }^{-2}\le n_0-1.$$

Thus we have shown that the following property holds:

If $s\ge 1$ is an integer and for each $k\in \{1,\dots ,s\}$ (59) holds, then $s\le n_0-1$ and

$$\parallel x_k\parallel \le 3M+1,k=1\dots ,s+1.$$

This implies that there exists an integer $q\in \{1,\dots ,n_0\}$ such that the following is obtained:

$$\parallel x_k\parallel \le 3M+1,k\in \{0,\dots ,q\},$$

$$\parallel x_q-x_{q+1}\parallel \le \gamma .$$

Assume that $q\ge 1$ is an integer and that

$$\parallel x_q-x_{q+1}\parallel \le \gamma .$$

(64)

By (47) and (64), the following is obtained:

$$\parallel x_q-{\xi }_q\parallel \le \parallel x_q-x_{q+1}\parallel +\parallel x_{q+1}-{\xi }_q\parallel \le \gamma +{\delta }_1.$$

(65)

It follows from (41), (42), and (48) that for each $\eta \in T\left(x_q\right)$, the following is calculated:

$$\parallel x_q-\eta \parallel \le \parallel x_q-{\xi }_q\parallel +{\delta }_0\le \gamma +{\delta }_1+{\delta }_0\le 2\gamma ,$$

$$T\left(x_q\right)\subset B(x_q,2\gamma )\subset B(x_q,ϵ).$$

(66)

Let $\alpha \in \mathcal{A}$. By (42), (49), and (65), the following is obtained:

$$\parallel x_q-P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right)\parallel \le \parallel x_q-{\xi }_q\parallel +{\delta }_0\le \gamma +{\delta }_0+{\delta }_1\le 2\gamma .$$

(67)

By (2), the following is obtained:

$$x_q\in (I+c_{q,\alpha }{T}_{\alpha })\left(P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right)\right),$$

$$x_q-P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right)\in c_{q,\alpha }{T}_{\alpha }\left(P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right)\right),$$

$$c_{q,\alpha }^{-1}(x_q-P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right))\in T_{\alpha }\left(P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right)\right).$$

(68)

By (17), (41), (67), and (68), the following is obtained:

$$\parallel c_{q,\alpha }^{-1}(x_q-P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right))\parallel \le 2{\overline{\lambda }}^{-1}\gamma \le ϵ$$

$$P_{c_{q,\alpha },T_{\alpha }}\left(x_q\right)\in F_{ϵ}\left(T_{\alpha }\right),$$

$$B(x_q,ϵ)\cap F_{ϵ}\left(T_{\alpha }\right)\ne \varnothing ,\alpha \in \mathcal{A}.$$

Combined with (66), this implies the following:

$$x_q\in {\tilde{F}}_{ϵ}.$$

Theorem 2 is proved. □

Example 2.

Assume that X is a Hilbert space, $f_{\alpha }:X\to (-\infty ,\infty ]$, $\alpha \in \mathcal{A}$ is a family of proper, convex, lower semicontinuous functions with a common minimizer which should be found. Such problems arise in the study of convex feasibility problems. For each $\alpha \in \mathcal{A}$, define the following:

$$T_{\alpha }\left(x\right)=\partial f_{\alpha }\left(x\right),x\in X$$

which is a maximal monotone operator. It is well-known [5] that a common zero point of this family of operators is a common minimizer of our family of convex functions. In this case, all the conditions of our main results hold and they can be applied in order to construct approximations of solutions.

3. Conclusions

In the paper we study the proximal point algorithm in order to find a point that is an approximate common zero of an infinite collection of maximal monotone maps in a real Hilbert space and an approximate fixed point of set-valued mapping, under the presence of computational errors. We prove two main results that show that the inexact proximal point method generates an approximate solution if these errors are sufficiently small. In the first result we consider the inexact iterates with summable errors, while the second one deals with nonsummable errors.