The Proximal Point Method with Remotest Set Control for Maximal Monotone Operators and Quasi-Nonexpansive Mappings

Abstract

In the present paper, we use the proximal point method with remotest set control for find an approximate common zero of a finite 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 summable. Also, we show that if the computational errors are small enough, then the inexact proximal point method generates approximate solutions

1. Preliminaries and the First Main Result

The 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 with remotest set control to find an approximate common zero of a finite 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 the computational errors are summable. Also, we show that if the computational errors are small enough, then the inexact proximal point method generates approximate solutions.

Asssume that $(\mathcal{X},⟨·,·⟩)$ is a real Hilbert space endowed with an inner product $⟨·,·⟩$. This inner product induces the norm $\parallel ·\parallel$.

A set-valued mapping $\mathcal{T}:\mathcal{X}\to 2^{\mathcal{X}}$ is called a monotone operator if and only if

$$⟨z-z^{\prime },w-w^{\prime }⟩\ge 0\mathrm{for}\mathrm{all}\mathrm{points}z,z^{\prime },w,w^{\prime }\in \mathcal{X}$$

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

(1)

It is called maximal monotone if, in addition, the graph

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

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

Assume that $\mathcal{T}:\mathcal{X}\to 2^{\mathcal{X}}$ 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 $0\in \mathcal{T}(z)$ is strongly based on the fundamental result obtained by Minty [16], who proved that, for an arbitrary point $z\in \mathcal{X}$ and an arbitrary positive number c, there exists a unique point $u\in \mathcal{X}$ satisfying

$$z\in (\mathcal{I}+c\mathcal{T})(u),$$

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

The mapping

$${\mathcal{P}}_{c,\mathcal{T}}:={(\mathcal{I}+c\mathcal{T})}^{-1}$$

(2)

is therefore a single-valued self-mapping of $\mathcal{X}$ (where c is an arbitrary positive number). Moreover, this mapping is nonexpansive:

$$\parallel {\mathcal{P}}_{c,\mathcal{T}}(z)-{\mathcal{P}}_{c,\mathcal{T}}(z^{\prime })\parallel \le \parallel z-z^{\prime }\parallel \mathrm{for}\mathrm{all}\mathrm{points} z,z^{\prime }\in \mathcal{X}$$

(3)

and

$${\mathcal{P}}_{c,\mathcal{T}}(z)=z \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if} 0\in \mathcal{T}(z).$$

(4)

According to [17], ${\mathcal{P}}_{c,\mathcal{T}}$ is called the proximal mapping associated with $c\mathcal{T}$.

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

$$z^{k+1}:={\mathcal{P}}_{c_k,\mathcal{T}}(z^k),k=0,1,\dots$$

It is clear that the

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

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

Define

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

(5)

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

For each point $x\in \mathcal{X}$ and each nonempty set $\mathcal{A}\subset \mathcal{X}$, put

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

For each $x\in \mathcal{X}$ and each $r>0$ set

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

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

We apply the proximal point algorithm with remotest set control in order to generate an appropriate approximation of a point which is a common zero of a finite collection of maximal monotone maps and a common fixed point of a finite collection of quasi-nonexpansive mappings (see (8)).

Let ${\mathcal{L}}_1$ be a finite set of maximal monotone operators $\mathcal{T}:\mathcal{X}\to 2^{\mathcal{X}}$ and ${\mathcal{L}}_2$ be a finite set of mappings $\mathcal{T}:\mathcal{X}\to \mathcal{X}$. We suppose that the set ${\mathcal{L}}_1\cup {\mathcal{L}}_2$ is nonempty. (Note that one of the sets ${\mathcal{L}}_1$ or ${\mathcal{L}}_2$ may be empty.)

Let $\overline{c}\in (0,1]$ and let $\overline{c}=1$, if ${\mathcal{L}}_2=\varnothing$.

We suppose that

$$\mathcal{F}(\mathcal{T})\ne \varnothing \mathrm{for}\mathrm{every}\mathrm{mapping} \mathcal{T}\in {\mathcal{L}}_1$$

(6)

and that for every operator $\mathcal{T}\in {\mathcal{L}}_2$, we have

$$\mathrm{Fix}(\mathcal{T}):=\{z\in \mathcal{X}:\mathcal{T}(z)=z\}\ne \varnothing ,$$

(7)

$${\parallel z-x\parallel }^2\ge {\parallel z-\mathcal{T}(x)\parallel }^2+\overline{c}{\parallel x-\mathcal{T}(x)\parallel }^2$$

$$\mathrm{for}\mathrm{every}\mathrm{point}x\in \mathcal{X} \mathrm{and}\mathrm{every}\mathrm{point} z\in \mathrm{Fix}(\mathcal{T}).$$

(8)

Let $\overline{\lambda }>0$ and let $\overline{\lambda }=\infty$ and ${\overline{\lambda }}^{-1}=0$, if ${\mathcal{L}}_1=\varnothing$.

Define

$$\mathcal{F}:=({\cap }_{\mathcal{T}\in {\mathcal{L}}_1}\mathcal{F}(\mathcal{T}))\cap ({\cap }_{Q\in {\mathcal{L}}_2}\mathrm{Fix}(\mathcal{Q})).$$

(9)

Let $ϵ>0$. For every monotone operator $\mathcal{T}\in {\mathcal{L}}_1$, define

$${\mathcal{F}}_{ϵ}(\mathcal{T})=\{x\in \mathcal{X}:\mathcal{T}(x)\cap \mathcal{B}(0,ϵ)\ne \varnothing \}$$

(10)

and for every mapping $\mathcal{T}\in {\mathcal{L}}_2$, set

$${\mathrm{Fix}}_{ϵ}(\mathcal{T})=\{x\in \mathcal{X}:\parallel \mathcal{T}(x)-x\parallel \le ϵ\}.$$

(11)

Define

$${\mathcal{F}}_{ϵ}=({\cap }_{\mathcal{T}\in {\mathcal{L}}_1}{\mathcal{F}}_{ϵ}(\mathcal{T}))\cap ({\cap }_{\mathcal{Q}\in {\mathcal{L}}_2}{\mathrm{Fix}}_{ϵ}(\mathcal{Q})),$$

$${\tilde{\mathcal{F}}}_{ϵ}=({\cap }_{\mathcal{T}\in {\mathcal{L}}_1}(\mathcal{B}(0,ϵ)+{\mathcal{F}}_{ϵ}(\mathcal{T})))\cap ({\cap }_{\mathcal{Q}\in {\mathcal{L}}_2}{\mathrm{Fix}}_{ϵ}(\mathcal{Q})).$$

(12)

We are interested in finding approximate solutions belonging to the set ${\tilde{\mathcal{F}}}_{ϵ}$ where $ϵ$ is sufficiently small.

Set

$$\mathcal{L}={\mathcal{L}}_2\cup \{{\mathcal{P}}_{c,\mathcal{T}}:\mathcal{T}\in {\mathcal{L}}_1,c\in [\overline{\lambda },\infty )\}.$$

(13)

Usually, in the literature, the convergence of some iterative processes to common fixed points is under consideration. In infinite-dimensional spaces, this convergence is usually weak. Here, instead of obtaining such convergence, we show that our iterative process generates approximate common fixed points, elements of $F_{ϵ}$ with some small positive $ϵ$. For this purpose, we use the so-called proximal point method with remotest set control [18]. Such results are interesting and important because in practice only a finite number of iterations are produced.

Below is our first result on the proximal point algorithm.

Theorem 1.

Let $\mathcal{M}>0$, $ϵ>0$,

$$\mathcal{B}(0,\mathcal{M})\cap \mathcal{F}\ne \varnothing .$$

(14)

Assume that ${\left\{x_k\right\}}_{k=0}^{\infty }\subset \mathcal{X},$${\left\{{\mathcal{S}}_k\right\}}_{k=0}^{\infty }\subset \mathcal{L}$,

$$\parallel x_0\parallel \le \mathcal{M}$$

(15)

and that for each integer $k\ge 0$,

$$\parallel x_k-{\mathcal{S}}_k(x_k)\parallel \ge \parallel x_k-\mathcal{S}(x_k)\parallel ,\mathcal{S}\in {\mathcal{L}}_2,$$

(16)

and for each $\mathcal{T}\in {\mathcal{L}}_1$ there is $c_{\mathcal{T}}\ge \overline{\lambda }$ such that

$$\parallel x_k-{\mathcal{S}}_k(x_k)\parallel \ge \parallel x_k-{\mathcal{P}}_{c_{\mathcal{T}},\mathcal{T}}(x_k)\parallel$$

(17)

and that

$$x_{k+1}={\mathcal{S}}_k(x_k).$$

(18)

Then,

$$Card(\{i\in \{0,1,2,\dots \}:x_i\notin {\tilde{\mathcal{F}}}_{ϵ}\})\le 4{\mathcal{M}}^2{\overline{c}}^{-1}{ϵ}^{-2}{(min\{1,\overline{\lambda }\})}^{-2}.$$

2. Auxiliary Results

It is easy to see that the following lemma holds.

Lemma 1.

Let $z,x_0,x_1\in \mathcal{X}$. Then,

$$2^{-1}{\parallel z-x_0\parallel }^2-2^{-1}{\parallel z-x_1\parallel }^2-2^{-1}{\parallel x_0-x_1\parallel }^2=⟨x_0-x_1,x_1-z⟩.$$

Lemma 2.

Assume that

$$z\in \mathcal{F},$$

the integers $p,q$ satisfy $0\le p<q$, $\mathcal{S}(k)\in \mathcal{L}$, $k=p,\dots ,q-1$,

$${\left\{{ϵ}_k\right\}}_{k=p}^{q-1}\subset (0,\infty ),{\left\{x_k\right\}}_{k=p}^q\subset \mathcal{X}$$

and that for all integers $k\in \{p,\dots ,q-1\}$,

$$\parallel x_{k+1}-\left[\mathcal{S}\left(k\right)\right]\left(x_k\right)\parallel \le {ϵ}_k.$$

(19)

Then, for every integer $k\in \{p+1\dots ,q\}$, the following inequality holds:

$$\parallel z-x_k\parallel \le \parallel z-x_p\parallel +\sum _{i=p}^{k-1}{ϵ}_i.$$

Proof. 

Let an integer $k\in \{p,\dots ,q-1\}$. By (3)–(5), (7), (9), (13), and (19),

$$\parallel z-x_{k+1}\parallel \le \parallel z-\left[\mathcal{S}(k)\right](x_k)\parallel +\parallel \mathcal{S}\left(k\right)\left(x_k\right)-x_{k+1}\parallel \le \parallel z-x_k\parallel +{ϵ}_k.$$

This implies the validity of Lemma 2. □

Lemma 3.

Let

$$\mathcal{A}\in \mathcal{L},x\in \mathcal{X},$$

$$z\in \mathcal{F}.$$

(20)

Then,

$${\parallel z-x\parallel }^2{-\parallel z-\mathcal{A}(x)\parallel }^2-\overline{c}{\parallel x-\mathcal{A}(x)\parallel }^2\ge 0.$$

(21)

Proof. 

There are two cases:

(i) $\mathcal{T}\in {\mathcal{L}}_2$;

(ii) There exists a mapping $\mathcal{T}\in {\mathcal{L}}_1$ and a number $c\in [\overline{\lambda },\infty )$ such that $\mathcal{A}={\mathcal{P}}_{c,\mathcal{T}}$.

If (i) holds, then (21) follows from (8) and (20). Assume that (ii) holds. Then, by Lemma 2,

$$2^{-1}{\parallel z-x\parallel }^2-2^{-1}{\parallel z-\mathcal{A}(x)\parallel }^2-2^{-1}{\parallel x-\mathcal{A}(x)\parallel }^2=⟨x-\mathcal{A}(x),\mathcal{A}(x)-z⟩.$$

(22)

By (ii) and (2), there are $\mathcal{T}\in {\mathcal{L}}_1$, $c\ge {\overline{\lambda }}_1$ such that

$$\mathcal{A}(x)={\mathcal{P}}_{c,\mathcal{T}}(x) \mathrm{and} x\in (\mathcal{I}+c\mathcal{T})(\mathcal{A}(x)),$$

$$x-\mathcal{A}(x)\in c\mathcal{T}(\mathcal{A}(x)).$$

(23)

By (1), (5), (20), (22), and (23), Equation (29) holds. Lemma 3 is proved. □

3. Proof of Theorem 1

Put

$$\gamma =min\{ϵ,ϵ\overline{\lambda }\}.$$

(24)

In view of (14), there exists a point

$$z\in \mathcal{B}(0,\mathcal{M})\cap \mathcal{F}.$$

(25)

By (15) and (25), we have

$$\parallel z-x_0\parallel \le 2\mathcal{M}.$$

(26)

It follows from Lemma 3, (18), (25) and (26) that for all integers $k\ge 0$,

$${\parallel z-x_k\parallel }^2-{\parallel z-x_{k+1}\parallel }^2-\overline{c}{\parallel x_k-x_{k+1}\parallel }^2\ge 0,$$

(27)

$$\parallel z-x_{k+1}\parallel \le \parallel z-x_k\parallel ,$$

(28)

$$\parallel z-x_k\parallel \le 2\mathcal{M}.$$

(29)

Define

$$\mathcal{E}=\{i\in \{0,1,\dots ,\}:\parallel x_{i+1}-x_i\parallel \ge \gamma \}.$$

(30)

Let $\mathcal{Q}$ be a natural number. In view of (26), (27), and (30), we have

$$4{\mathcal{M}}^2\ge {\parallel x_0-z\parallel }^2\ge {\parallel x_0-z\parallel }^2-{\parallel x_{\mathcal{Q}}-z\parallel }^2$$

$$=\sum _{i=0}^{\mathcal{Q}-1}({\parallel x_i-z\parallel }^2-{\parallel x_{i+1}-z\parallel }^2)\ge \sum _{i=0}^{n-1}\overline{c}{\parallel x_i-x_{i+1}\parallel }^2$$

$$\ge \overline{c}{\gamma }^2\mathrm{Card}(\{i\in \{0,\dots ,\mathcal{Q}-1\}:\parallel x_{i+1}-x_i\parallel \ge \gamma \}),$$

$$\mathrm{Card}(\mathcal{E}\cap \{0,\dots ,\mathcal{Q}-1\}\})\le 4{\mathcal{M}}^2{\gamma }^{-2}{\overline{c}}^{-1}.$$

Since the relation above holds for every natural number $\mathcal{Q}$, we conclude that

$$\mathrm{Card}(\mathcal{E})\le 4{\mathcal{M}}^2{\gamma }^{-2}{\overline{c}}^{-1}=4{\mathcal{M}}^2{ϵ}^{-2}{\overline{c}}^{-1}min{\{1,\overline{\lambda }\}}^{-2}.$$

(31)

Let

$$n\in \{0,1,\dots \}\setminus \mathcal{E}.$$

(32)

In view of (18), (30), and (32),

$$\parallel x_n-\mathcal{S}(n)(x_n)\parallel =\parallel x_n-x_{n+1}\parallel \le \gamma .$$

(33)

By (11), (16), (24), and (33), for each $\mathcal{T}\in {\mathcal{L}}_2$,

$$\parallel x_n-\mathcal{T}(x_n)\parallel \le \gamma ,x_n\in {\mathrm{Fix}}_{ϵ}(\mathcal{T}),\mathcal{T}\in {\mathcal{L}}_2,.$$

(34)

Assume that $\mathcal{T}\in {\mathcal{L}}_1$. By (17) and (33), there exists $c\ge \overline{\lambda }$ such that

$$\parallel x_n-{\mathcal{P}}_{c,\mathcal{T}}\left(x_n\right)\parallel =\parallel x_n-{\mathcal{S}}_n(x_n)\parallel \le \gamma .$$

(35)

Clearly,

$$x_n\in (\mathcal{I}+c\mathcal{T})({\mathcal{P}}_{c,\mathcal{T}}(x_n)),$$

$$x_n-{\mathcal{P}}_{c,\mathcal{T}}(x_n)\in c\mathcal{T}({\mathcal{P}}_{c,\mathcal{T}}(x_n))$$

and by (24) and (35),

$$\parallel c^{-1}(x_n-{\mathcal{P}}_{c,\mathcal{T}}\left(x_n\right))\parallel \le {\overline{\lambda }}^{-1}\gamma \le ϵ,$$

$$c^{-1}(x_n-{\mathcal{P}}_{c,\mathcal{T}}(x_n))\in \mathcal{T}({\mathcal{P}}_{c,\mathcal{T}}(x_n)),$$

$${\mathcal{P}}_{c,\mathcal{T}}(x_n)\in {\mathcal{F}}_{ϵ}(\mathcal{T}).$$

Together with (35), this implies that

$$x_n\in {\mathcal{F}}_{ϵ}(\mathcal{T})+\mathcal{B}(0,ϵ),\mathcal{T}\in {\mathcal{L}}_1.$$

Combined with (34), this implies that

$$x_n\in {\tilde{\mathcal{F}}}_{ϵ}$$

for $n\in \{0,1,\dots ,\}\setminus \mathcal{E}.$ Together with (31), this implies that

$$\{n\in \{0,1,2,\dots \}:x_n\notin {\tilde{\mathcal{F}}}_{ϵ}\}\subset \mathcal{E},$$

$$\mathrm{Card}(\{i\in \{0,1,\dots \}:x_i\notin {\tilde{\mathcal{F}}}_{ϵ}\})$$

$$\le \mathrm{Card}(\mathcal{E})\le 4{\mathcal{M}}^2{\overline{c}}^{-1}{ϵ}^{-2}{(min\{1,\overline{\lambda }\})}^{-2}.$$

Theorem 1 is proved.

4. Approximate Solutions Under the Presence of Summable Errors

In this section, we prove an extension of Theorem 1 which shows that the inexact proximal point method with the remotest set control generates an approximate solution if perturbations are summable.

Theorem 2.

Let $\mathcal{M}>0$, $ϵ\in (0,1)$, ${\left\{{\Delta }_i\right\}}_{i=0}^{\infty }\subset [0,\infty )$ satisfy

$$\Delta :=\sum _{i=0}^{\infty }{\Delta }_i<\infty ,$$

(36)

$$\mathcal{B}(0,\mathcal{M})\cap \mathcal{F}\ne \varnothing .$$

(37)

Assume that ${\left\{x_k\right\}}_{k=0}^{\infty }\subset \mathcal{X},$${\left\{{\mathcal{S}}_k\right\}}_{k=0}^{\infty }\subset \mathcal{L}$,

$$\parallel x_0\parallel \le \mathcal{M}$$

(38)

and that for each integer $k\ge 0$,

$$\parallel x_k-{\mathcal{S}}_k\left(x_k\right)\parallel \ge \parallel x_k-\mathcal{S}(x_k)\parallel ,\mathcal{S}\in {\mathcal{L}}_2$$

(39)

and for each $\mathcal{T}\in {\mathcal{L}}_1$ there is $c_{\mathcal{T}}\ge \overline{\lambda }$ such that

$$\parallel x_k-{\mathcal{S}}_k\left(x_k\right)\parallel \ge \parallel x_k-{\mathcal{P}}_{c_{\mathcal{T}},\mathcal{T}}\left(x_k\right)\parallel$$

(40)

and that

$$\parallel x_{k+1}-{\mathcal{S}}_k(x_k)\parallel \le {\Delta }_k.$$

(41)

Then,

$$Card(\{i\in \{0,1,\dots \}:x_i\notin {\tilde{\mathcal{F}}}_{ϵ}\})\le (4{\mathcal{M}}^2+\Delta (4\mathcal{M}+2)){\overline{c}}^{-1}{ϵ}^{-2}min{\{1,\overline{\lambda }\}}^{-2}.$$

Proof. 

In view of (37), there exists a point

$$z\in \mathcal{B}(0,\mathcal{M})\cap \mathcal{F}.$$

(42)

By (38) and (42),

$$\parallel z-x_0\parallel \le 2\mathcal{M}.$$

(43)

It follows from Lemma 2 and (41)–(43) that for all integers $i\ge 0$,

$$\parallel z-x_i\parallel \le 2\mathcal{M}+\sum _{j=0}^i{\Delta }_j.$$

(44)

Put

$$\gamma =min\{ϵ,ϵ\overline{\lambda }\}.$$

(45)

Let $i\ge 0$ be an integer. Lemma 3 and (42) imply that

$${\parallel z-x_i\parallel }^2-{\parallel z-{\mathcal{S}}_i\left(x_i\right)\parallel }^2\ge \overline{c}{\parallel x_i-{\mathcal{S}}_i(x_i)\parallel }^2.$$

(46)

By (41), (44), and (46),

$$|{\parallel z-x_{i+1}\parallel }^2-{\parallel z-{\mathcal{S}}_i\left(x_i\right)\parallel }^2|$$

$$\le |\parallel z-x_{i+1}\parallel -\parallel z-{\mathcal{S}}_i\left(x_i\right)\parallel |(\parallel z-x_{i+1}\parallel +\parallel z-{\mathcal{S}}_i\left(x_i\right)\parallel )$$

$$\le 2\parallel x_{i+1}-{\mathcal{S}}_i\left(x_i\right)\parallel ((2\mathcal{M}+\Lambda )$$

$$\le 2{\Delta }_i(2\mathcal{M}+\Lambda ).$$

(47)

It follows from (44), (46), and (47) that

$$\overline{c}{\parallel x_i-{\mathcal{S}}_i\left(x_i\right)\parallel }^2$$

$$\le {\parallel z-x_{i+1}\parallel }^2-{\parallel z-{\mathcal{S}}_i(x_i)\parallel }^2$$

$$\le {\parallel z-x_i\parallel }^2-{\parallel z-x_{i+1}\parallel }^2+2{\Delta }_i(2\mathcal{M}+\Delta ).$$

(48)

By (43) and (48), for each natural number $\mathcal{Q}$,

$$4{\mathcal{M}}^2\ge {\parallel x_0-z\parallel }^2$$

$$\ge \parallel x_0{-z\parallel }^2-{\parallel x_{\mathcal{Q}}-z\parallel }^2$$

$$=\sum _{n=0}^{\mathcal{Q}-1}(\parallel x_n{-z\parallel }^2-\parallel x_{n+1}-z{\parallel }^2)$$

$$\ge \sum _{n=0}^{\mathcal{Q}-1}(\overline{c}{\parallel x_n-\mathcal{S}(n)(x_n)\parallel }^2-2{\Delta }_n(2\mathcal{M}+\Lambda ))$$

and in view of (36),

$$4{\mathcal{M}}^2+\Delta (4\mathcal{M}+2\Delta )={(2\mathcal{M})}^2+2(2\mathcal{M}+\Delta )\sum _{j=0}^{\infty }{\Delta }_j$$

$$\ge \sum _{n=0}^{\mathcal{Q}-1}(\overline{c}{\parallel x_n-{\mathcal{S}}_n(x_n)\parallel }^2)$$

$$\ge \overline{c}{\gamma }^{2}Card(\{n\in \{0,\dots ,\mathcal{Q}-1\}:\parallel x_n-{\mathcal{S}}_n(x_n)\parallel \ge \gamma \}).$$

Since the relation above holds for every natural number $\mathcal{Q}$, we conclude that

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

$$\le {\overline{c}}^{-1}{\gamma }^{-2}(4{\mathcal{M}}^2+(4\mathcal{M}+2\Delta )\Delta ).$$

(49)

Define

$$\mathcal{E}=\{n\in \{0,1,\dots ,\}:\parallel x_n-{\mathcal{S}}_n\left(x_n\right)\parallel \ge \gamma \}.$$

(50)

In view of (49) and (50), we have

$$\mathrm{Card}(\mathcal{E})\le {\overline{c}}^{-1}{\gamma }^{-2}(4{\mathcal{M}}^2+2(2\mathcal{M}+\Delta )\Delta ).$$

(51)

Let an integer $n\ge 0$ satisfy

$$n\notin \mathcal{E}.$$

(52)

In view of (50) and (52),

$$\parallel x_n-{\mathcal{S}}_n(x_n)\parallel <\gamma .$$

(53)

By (39), (45), and (53), for each $\mathcal{T}\in {\mathcal{L}}_2$,

$$ϵ\ge \gamma \ge \parallel x_n-{\mathcal{S}}_n\left(x_n\right)\parallel \ge \parallel x_n-\mathcal{T}\left(x_n\right)\parallel ,$$

$$x_n\in {\mathrm{Fix}}_{ϵ}(\mathcal{T}),\mathcal{T}\in {\mathcal{L}}_2.$$

(54)

Assume that $\mathcal{T}\in {\mathcal{L}}_1$. By (40) and (53), there exists $c\ge \overline{\lambda }$ such that

$$\parallel x_n-{\mathcal{P}}_{c,\mathcal{T}}(x_n)\parallel \le \parallel x_n-{\mathcal{S}}_n\left(x_n\right)\parallel \le \gamma .$$

(55)

Clearly, in view of (2),

$$x_n\in (\mathcal{I}+c\mathcal{T})({\mathcal{P}}_{c,\mathcal{T}}(x_n)),$$

$$x_n-{\mathcal{P}}_{c,\mathcal{T}}(x_n)\in c\mathcal{T}({\mathcal{P}}_{c,\mathcal{T}}(x_n)),$$

$$c^{-1}(x_n-{\mathcal{P}}_{c,\mathcal{T}}(x_n))\in \mathcal{T}({\mathcal{P}}_{c,\mathcal{T}}(x_n))$$

and by (45),

$$\parallel c^{-1}(x_n-{\mathcal{P}}_{c,\mathcal{T}}\left(x_n\right))\parallel \le {\overline{\lambda }}^{-1}\gamma \le ϵ.$$

Thus,

$${\mathcal{P}}_{c,\mathcal{T}}(x_n)\in {\mathcal{F}}_{ϵ}(\mathcal{T}),\mathcal{T}\in {\mathcal{L}}_1.$$

Together with (54) and (55), this implies that

$$x_n\in {\tilde{\mathcal{F}}}_{ϵ}$$

for each $n\in \{0,1,\dots ,\}\setminus \mathcal{E}.$ Thus,

$$\{n\in \{0,1,2,\dots \}:x_n\notin {\tilde{\mathcal{F}}}_{ϵ}\}\subset \mathcal{E}$$

and in view of (51),

$$\mathrm{Card}(\{i\in \{0,1,\dots \}:x_i\notin {\tilde{\mathcal{F}}}_{ϵ}\})$$

$$\le \mathrm{Card}(\mathcal{E})\le (4{\mathcal{M}}^2+\Delta (4\mathcal{M}+2\Delta )){\overline{c}}^{-1}{\gamma }^{-2}.$$

Theorem 2 is proved. □

5. Approximate Solutions Under the Presence of Nonsummable Errors

We show that if the perturbations are sufficiently small, then the the inexact proximal point method produces approximate solutions.

Theorem 3.

Let $\mathcal{M}>0$, $ϵ\in (0,1]$ be such that

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

(56)

δ is a positive number satisfying

$$\delta \le {16}^{-1}{(4\mathcal{M}+1)}^{-1}\overline{c}{ϵ}^{2}min{\{\overline{\lambda },1\}}^2$$

(57)

and $n_0$ is a natural number satisfying

$$n_0>2+32{\mathcal{M}}^2{\overline{c}}^{-1}{ϵ}^{-2}min{\{\overline{\lambda },1\}}^{-2}.$$

(58)

Assume that

$${\left\{{\mathcal{S}}_k\right\}}_{k=0}^{\infty }\subset \mathcal{L},{\left\{x_k\right\}}_{k=0}^{\infty }\subset \mathcal{X},$$

$$\parallel x_0\parallel \le \mathcal{M},$$

(59)

for each integer $k\ge 0$,

$$\parallel x_k-{\mathcal{S}}_k\left(x_k\right)\parallel \ge \parallel x_k-\mathcal{S}\left(x_k\right)\parallel ,\mathcal{S}\in {\mathcal{L}}_2$$

(60)

and for each $\mathcal{T}\in {\mathcal{L}}_1$ there is $c_{\mathcal{T}}\ge \overline{\lambda }$ such that

$$\parallel x_k-{\mathcal{S}}_k\left(x_k\right)\parallel \ge \parallel x_k-{\mathcal{P}}_{c,\mathcal{T}}(x_k)\parallel$$

(61)

and that

$$\parallel x_{k+1}-{\mathcal{S}}_k\left(x_k\right)\parallel \le \delta .$$

(62)

Then, there exists an integer $q\in [0,n_0-1]$ such that

$$\parallel x_k\parallel \le 3\mathcal{M} for all integers k=0,\dots ,q+1,$$

$$\parallel x_q-x_{q+1}\parallel \le 2^{-1}ϵmin\{\overline{\lambda },1\}.$$

(63)

Moreover, if an integer $q\ge 0$ and (63) holds, then

$$x_q\in {\tilde{\mathcal{F}}}_{ϵ}.$$

Proof. 

Fix

$${ϵ}_0=2^{-1}ϵmin\{\overline{\lambda },1\}$$

(64)

and a point

$$z\in \mathcal{B}(0,\mathcal{M})\cap \mathcal{F}.$$

(65)

Assume that q is a natural number such that for each integer $p\in [0,q-1]$,

$$\parallel x_p-x_{p+1}\parallel >{ϵ}_0.$$

(66)

Clearly,

$$\parallel x_0-z\parallel \le 2\mathcal{M}.$$

Assume that an integer $p\in [0,q-1]$ and that

$$\parallel x_p-z\parallel \le 2\mathcal{M}.$$

(67)

By (65) and Lemma 3,

$${\parallel z-x_p\parallel }^2-{\parallel z-{\mathcal{S}}_p\left(x_p\right)\parallel }^2\ge \overline{c}{\parallel x_p-{\mathcal{S}}_p(x_p)\parallel }^2.$$

(68)

In view of (62), (67), and (68),

$$|{\parallel z-x_{p+1}\parallel }^2-{\parallel z-{\mathcal{S}}_p(x_p)\parallel }^2|$$

$$\le |\parallel z-x_{p+1}\parallel -\parallel z-{\mathcal{S}}_p\left(x_p\right)\parallel |(\parallel z-x_{p+1}\parallel +\parallel z-{\mathcal{S}}_p\left(x_p\right)\parallel )$$

$$\le \parallel x_{p+1}-{\mathcal{S}}_p\left(x_p\right)\parallel (2\parallel z-{\mathcal{S}}_p\left(x_p\right)\parallel +\parallel x_{p+1}-{\mathcal{S}}_p\left(x_p\right)\parallel )$$

$$\le \delta (2\parallel z-{\mathcal{S}}_p\left(x_p\right)\parallel +\delta )\le \delta (4\mathcal{M}+1).$$

(69)

It follows from (57), (62), (64), and (66) that

$$\parallel x_p-{\mathcal{S}}_p\left(x_p\right)\parallel \ge \parallel x_{p+1}-x_p\parallel -\parallel {\mathcal{S}}_p(x_p)-x_{p+1}\parallel$$

$$>{ϵ}_0-\delta >{ϵ}_0/2.$$

(70)

It follows from (57), (64), and (68)–(70) that

$${\parallel z-x_p\parallel }^2-{\parallel z-x_{p+1}\parallel }^2\ge {\parallel z-x_p\parallel }^2-{\parallel z-{\mathcal{S}}_p(x_p)\parallel }^2-\delta (4\mathcal{M}+1)$$

$$\ge \overline{c}{\parallel x_p-{\mathcal{S}}_p(x_p)\parallel }^2-\delta (4\mathcal{M}+1)\ge \overline{c}{ϵ}_0^2/4-\delta (4\mathcal{M}+1)\ge \overline{c}{ϵ}_0^2/8.$$

By (77) and the relation above,

$$\parallel x_{p+1}-z\parallel \le 2\mathcal{M},$$

$${\parallel z-x_p\parallel }^2-{\parallel z-x_{p+1}\parallel }^2\ge \overline{c}{ϵ}_0^2/8.$$

(71)

By induction and using (71), we conclude that for all integers $p\in [0,q]$,

$$\parallel x_p-z\parallel \le 2\mathcal{M}$$

and (71) holds each $p\in \{0,\dots ,q-1\}$. Together with (58), (64), and (66), this implies that

$$4{\mathcal{M}}^2\ge {\parallel z-x_0\parallel }^2\ge {\parallel z-x_0\parallel }^2-{\parallel z-x_q\parallel }^2$$

$$=\sum _{i=0}^{q-1}(\parallel z-x_i{\parallel }^2-{\parallel z-x_{i+1}\parallel }^2\ge q\overline{c}{ϵ}_0^2/8$$

and

$$q\le 32{\mathcal{M}}^2{\overline{c}}^{-1}{ϵ}_0^{-2}\le n_0-1.$$

Thus, if $q\ge 1$ is an integer such that for each integer $p\in [0,q-1]$ (66) holds, then

$$q\le n_0-1.$$

This implies that there exists an integer $q\in [0,n_0]$ such that for every integer $p\in [0,q]$,

$$\parallel x_p\parallel \le 3\mathcal{M},$$

$$\parallel x_q-x_{q+1}\parallel \le {ϵ}_0.$$

(72)

Assume that $q\ge 0$ is an integer and that (72) holds. In view of (62) and (72),

$$\parallel x_q-{\mathcal{S}}_q\left(x_q\right)\parallel \le \parallel x_q-x_{q+1}\parallel +\parallel x_{q+1}-{\mathcal{S}}_q(x_q)\parallel \le {ϵ}_0+\delta .$$

(73)

It follows from (57), (60), (64), and (73) that

$$ϵ\ge {ϵ}_0+\delta \ge \parallel x_q-{\mathcal{S}}_q\left(x_n\right)\parallel \ge \parallel x_q-\mathcal{T}\left(x_q\right)\parallel ,$$

$$x_q\in {\mathrm{Fix}}_{ϵ}(\mathcal{T}).$$

Assume that $\mathcal{T}\in {\mathcal{L}}_1$. By (43), there exists $c\ge \overline{\lambda }$ such that

$${ϵ}_0+\delta \ge \parallel x_q-{\mathcal{S}}_q\left(x_q\right)\parallel \ge \parallel x_q-{\mathcal{P}}_{c,\mathcal{T}}\left(x_q\right)\parallel ,$$

$$x_q\in (\mathcal{I}+c\mathcal{T})({\mathcal{P}}_{c,\mathcal{T}}(x_q)),$$

$$x_q-{\mathcal{P}}_{c,\mathcal{T}}(x_q)\in c\mathcal{T}({\mathcal{P}}_{c,\mathcal{T}}(x_q)),$$

$$c^{-1}(x_q-{\mathcal{P}}_{c,\mathcal{T}}(x_q))\in \mathcal{T}({\mathcal{P}}_{c,\mathcal{T}}(x_q))$$

and

$$\parallel c^{-1}(x_q-{\mathcal{P}}_{c,\mathcal{T}}\left(x_q\right))\parallel \le {\overline{\lambda }}^{-1}({ϵ}_0+\delta )\le ϵ.$$

Thus,

$${\mathcal{P}}_{c,\mathcal{T}}(x_q))\in {\mathcal{F}}_{ϵ}(\mathcal{T}),x_q\in \mathcal{B}(0,ϵ)+{\mathcal{F}}_{ϵ}(\mathcal{T}),$$

$$x_q\in {\tilde{\mathcal{F}}}_{ϵ}$$

Theorem 3 is proved. □