Approximate Solutions of a Fixed-Point Problem with an Algorithm Based on Unions of Nonexpansive Mappings

Abstract

In this paper, we study a fixed-point problem with a set-valued mapping by using an algorithm based on unions of nonexpansive mappings. We show that an approximate solution is reached after a finite number of iterations in the presence of computational errors. This result is an extension of the results known in the literature.

1. Introduction

The study of fixed-point problems is an important topic in nonlinear analysis [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. These problems have various applications in mathematical analysis, optimization theory, engineering, medicine, and the natural sciences [14,15,16,17,18,19,20]. In particular, in [21], a novel framework for the investigation of iterative algorithms was introduced. This framework was given in terms of a certain nonlinear set-valued map T defined on a space X. For every $x\in X$, $T\left(x\right)$ is a finite union of values of single-valued paracontracting operators. Tam [21] established a convergence for this algorithm. Note that his result was a generalization of the result attained by Bauschke and Noll [22]. In our recent paper [23], we obtained an extension of a result of [21]. It should be mentioned that in [21], X is a finite-dimensional Euclidean space, while in [23] and in the present paper, X is an arbitrary metric space. The main result of [23] was obtained for inexact iterations of operators under the assumption that the common fixed-point problem has a solution. In the present paper, we prove an extension of this result in a case in which the common fixed-point problem has only an approximated solution.

2. Preliminaries

Assume that $(X,\rho )$ is a metric space endowed with a metric $\rho$ and that $C\subset X$ is its nonempty closed set. For every $u\in X$ and every $\Delta \in (0,\infty )$, we set

$$B(u,\Delta )=\{v\in X:\rho (u,v)\le \Delta \}.$$

For every map $A:C\to C$, we define

$$\mathrm{Fix}\left(A\right)=\{u\in C:A(u)=u\}.$$

Assume that $T_i:C\to C$, $i=1,\dots ,m$, where $m\ge 1$ is an integer, $0<\overline{c}\le 1$, and that for every $j\in \{1,\dots ,m\}$, every $u\in \mathrm{Fix}\left(T_j\right)$, and every $v\in C$,

$$\rho {(u,v)}^2-\rho {(u,T_j\left(v\right))}^2\ge \overline{c}\rho {(v,T_j\left(v\right))}^2.$$

(1)

It should be mentioned that inequality (1) is true for many nonlinear operators [14,15].

Assume that

$$\varphi :X\to 2^{\{1,\dots ,m\}}\backslash \{\varnothing \}.$$

(2)

We set

$$T\left(u\right)=\{T_j\left(u\right):j\in \varphi \left(u\right)\}.$$

(3)

for each $u\in C$ and

$$F\left(T\right)=\{u\in C:u\in T(u\left)\right\}.$$

In this paper, we study the fixed-point problem

$$\mathrm{Find}x\in X\mathrm{such} \mathrm{that}x\in T\left(x\right).$$

(4)

This problem was introduced and studied in [21]. It should be mentioned that in [21], X was a finite-dimensional Euclidean space, and the mappings $T_i$, $i=1,\dots ,m$ were paracontracting. Tam [21] considered a sequence of iterations ${\left\{x_k\right\}}_{k=0}^{\infty }\subset X$ satisfying $x_{k+1}\in T\left(x_k\right)$ for every integer $k\ge 0$ and established its convergence under the assumption that the mappings $T_i$, $i=1,\dots ,m$ had a common fixed point. In [21], this convergent result was applied to sparsity-constrained minimization. Note that the result in [21] was a generalization of the result attained by Bauschke and Noll [22]. In our recent paper [23], we considered mappings acting on a general metric space and obtained two extensions of the result from [21]. In the first result, we studied exact iterations of the set-valued mapping, while in the second one, we dealt with its inexact iterations while taking computational errors into account. More precisely, in [23], for a given computational error $\delta >0$, we considered a sequence ${\left\{x_k\right\}}_{k=0}^{\infty }\subset X$ satisfying $B(x_{k+1},\delta )\cap T\left(x_k\right)\ne \varnothing$ for every integer $k\ge 0$ and analyzed its behavior. This result was also obtained under the assumption that the mappings $T_i$, $i=1,\dots ,m$ had a common fixed point. In the present paper, we generalize this result. Instead of assuming the existence of a common fixed point, we suppose that there exists an approximate common fixed point z such that

$$B(z,\gamma )\cap \mathrm{Fix}\left(T_i\right)\ne \varnothing ,i=1,\dots ,m,$$

where $\gamma$ is a given small positive constant. In other words, a small neighborhood of z contains a fixed point of every mapping.

We fix

$$\theta \in C.$$

For any $u\in R^1$, we set

$$\lfloor u\rfloor =\max \{j:j\mathrm{as} \mathrm{an} \mathrm{integer} \mathrm{and}j\le u\}.$$

We prove the following theorem in the presence of computational errors. This theorem shows that after a certain number of iterations, we obtain an approximate solution to our fixed-point problem. The number of iterations depends on the computational error.

Theorem 1. 

Let $M>0$, $ϵ\in (0,1]$,

$$\gamma \in (0,{\left(18\right)}^{-1}{(4M+4)}^{-1}{ϵ}^2\overline{c}),$$

(5)

$$z\in B(\theta ,M)$$

(6)

satisfy

$$B(z,\gamma )\cap \mathrm{Fix}\left(T_i\right)\ne \varnothing ,i=1,\dots ,m,$$

(7)

$$Q=\lfloor 8{ϵ}^{-2}{M}^2{\overline{c}}^{-1}\rfloor +1,$$

(8)

and $\delta \in (0,\gamma ).$ Assume that ${\left\{x_k\right\}}_{k=0}^{\infty }\subset C$,

$$\rho (\theta ,x_0)\le M$$

(9)

and that

$$B(x_{k+1},\delta )\cap T\left(x_k\right)\ne \varnothing ,k=0,1,\dots .$$

(10)

Then, there is a nonnegative integer $p<Q$ for which

$$B(x_p,ϵ)\cap T\left(x_p\right)\ne \varnothing .$$

(11)

In the theorem above, we assume the existence of a point z that satisfies (7), which means that z is an approximate fixed point for all of the mappings $T_i$, $i=1,\dots ,m$. This result has a prototype in [23], which was obtained under the assumption that z is a common fixed point for all $T_k$, $k=1,\dots ,m$.

3. Proof of Theorem 1

Proof. 

Assume that for every nonnegative integer $k<Q$, relation (11) is not true. Then, for every nonnegative integer $k<Q$,

$$B(x_k,ϵ)\cap T\left(x_k\right)=\varnothing .$$

(12)

We set

$$M_0=2M+1.$$

(13)

According to (7), for every $k\in \{1,\dots ,m\}$, there is

$$z_k\in \mathrm{Fix}\left(T_k\right)$$

(14)

such that

$$\rho (z,z_k)\le \gamma .$$

(15)

According to (6) and (9),

$$\rho (x_0,z)\le 2M.$$

(16)

Let $i\in [0,Q-1]$ be an integer. According to (10), there is

$${\widehat{x}}_{i+1}\in T\left(x_i\right)$$

(17)

for which

$$\rho (x_{i+1},{\widehat{x}}_{i+1})\le \delta .$$

(18)

Equations (3) and (17) imply that there is an integer $j\in [1,m]$ for which

$${\widehat{x}}_{i+1}=T_j\left(x_i\right).$$

(19)

It follows from (1) and (19) that

$$\rho {(z_j,x_i)}^2\ge \rho {(z_j,{\widehat{x}}_{i+1})}^2+\overline{c}\rho {(x_i,{\widehat{x}}_{i+1})}^2.$$

(20)

According to (12) and (19),

$$\rho (x_i,{\widehat{x}}_{i+1})>ϵ.$$

(21)

In view of (20) and (21),

$$\rho {(z_j,x_i)}^2\ge \rho {(z_j,{\widehat{x}}_{i+1})}^2+\overline{c}{ϵ}^2.$$

(22)

Assume that

$$\rho (z,x_i)\le M_0.$$

(23)

(In view of (13) and (16), Equation (23) holds for $i=0$). Equations (15) and (23) imply that

$$\rho (z_j,x_i)\le \rho (z_j,z)+\rho (z,x_i)\le M_0+\gamma .$$

(24)

It follows from (5), (13), (22), and (24) that

$$\rho {(z_j,{\widehat{x}}_{i+1})}^2\le \rho {(z_j,x_i)}^2-{ϵ}^2\overline{c}\le {(M_0+\gamma )}^2-{ϵ}^2\overline{c}$$

$$=M_0^2+\gamma (\gamma +2M_0)-{ϵ}^2\overline{c}\le M_0^2+\gamma (1+2M_0)-{ϵ}^2\overline{c}$$

$$\le M_0^2-7\gamma (1+2M_0)\le {(M_0-2\gamma )}^2$$

and

$$\rho (z_j,{\widehat{x}}_{i+1})\le M_0-2\gamma .$$

(25)

According to (15) and (25),

$$\rho (z,x_{i+1})\le \rho (z,z_j)+\rho (z_j,{\widehat{x}}_{t+1})+\rho ({\widehat{x}}_{i+1},x_{i+1})$$

$$\le M_0-2\gamma +\gamma +\gamma \le M_0$$

and

$$\rho (z,x_{i+1})\le M_0.$$

(26)

According to (22),

$$\rho {(z_j,{\widehat{x}}_{i+1})}^2\le \rho {(z_j,x_i)}^2-{ϵ}^2\overline{c}.$$

(27)

Equations (15) and (23) imply that

$$|\rho {(x_i,z)}^2-\rho {(x_i,z_j)}^2|$$

$$\le (\rho (x_i,z)+\rho (x_i,z_j))|\rho (x_i,z)-\rho (x_i,z_j)|$$

$$\le (\rho (x_i,z)+\rho (x_i,z)+\gamma )\rho (z_j,z)\le \gamma (2M_0+1).$$

(28)

It follows from (15), (18), and (26) that

$$|\rho {(x_{i+1},z)}^2-\rho {({\widehat{x}}_{i+1},z_j)}^2|$$

$$\le (\rho (x_{i+1},z)+\rho ({\widehat{x}}_{i+1},z_j))|\rho (x_{i+1},z)-\rho ({\widehat{x}}_{i+1},z_j)|$$

$$\le (2M_0+\gamma +\delta )(\rho (z_j,z)+\rho ({\widehat{x}}_{i+1},x_{i+1}))\le (2M_0+2)(\gamma +\delta ).$$

(29)

By (5), (13), (22), and (29),

$$\rho {(x_{i+1},z)}^2\le \rho {(z_j,{\widehat{x}}_{i+1})}^2+2\gamma (2M_0+2)$$

$$\le \rho {(z_j,x_i)}^2-\overline{c}{ϵ}^2+2\gamma (2M_0+2)$$

$$\le \rho {(x_i,z)}^2-{ϵ}^2\overline{c}+\gamma (2M_0+1)+2\gamma (2M_0+2)$$

$$\rho {(x_i,z)}^2-{ϵ}^2\overline{c}+3\gamma (2M_0+1)$$

$$\le \rho {(x_i,z)}^2-{ϵ}^2\overline{c}/2.$$

(30)

Thus, we have shown by induction that (23) and (30) hold for $i=0,\dots ,Q-1$. By (16) and (30),

$$4M^2\ge \rho (z,x_0){)}^2$$

$$\ge \rho {(z,x_0)}^2-\rho {(z,x_Q)}^2$$

$$=\sum _{i=0}^{Q-1}(\rho {(z,x_i)}^2-\rho {(z,x_{i+1})}^2)\ge Q\overline{c}{ϵ}^2/2,$$

and

$$Q\le 8M^2{\overline{c}}^{-1}{ϵ}^{-2}.$$

This contradicts (8). The contradiction that we have reached proves Theorem 1. □

4. Extensions

We use the notation and definitions introduced in Section 2.

Lemma 1. 

Assume that $M_0>0$,

$$z\in B(\theta ,M_0),$$

(31)

$$B(z,1)\cap \mathrm{Fix}\left(T_i\right)\ne \varnothing ,i=1,\dots ,m,$$

(32)

$$x_0\in B(\theta ,M_0),$$

(33)

$x_1\in C$, and

$$B(x_1,1)\cap T\left(x_0\right)\ne \varnothing .$$

(34)

Then,

$$\rho (x_1,\theta )\le 3M_0+3.$$

Proof. 

According to (3), there is an integer $j\in [1,m]$ for which

$$\rho (x_1,T_j\left(x_0\right))\le 1.$$

(35)

According to (32), there is

$$z_j\in \mathrm{Fix}\left(T_j\right)$$

(36)

for which

$$\rho (z,z_j)\le 1.$$

(37)

Equations (1), (31), (33), and (35)–(37) imply that

$$\rho (x_1,\theta )\le \rho (\theta ,T_j\left(x_0\right))+\rho (T_j\left(x_0\right),x_1)$$

$$\le 1+\rho (\theta ,z)+\rho (z,z_j)+\rho (z_j,T_j\left(x_0\right))$$

$$\le 1+M_0+1+\rho (z_j,x_0)$$

$$\le 2+M_0+\rho (\theta ,x_0)+\rho (\theta ,z)+\rho (z,z_j)$$

$$\le 3+3M_0.$$

Lemma 1 is proved. □

Theorem 2. 

Let $M>0$, $ϵ\in (0,1]$,

$$\gamma \in (0,{\left(18\right)}^{-1}{(12M+12)}^{-1}{ϵ}^2\overline{c}),$$

$$z\in B(\theta ,M)$$

satisfy

$$B(z,\gamma )\cap \mathrm{Fix}\left(T_i\right)\ne \varnothing ,i=1,\dots ,m,$$

$$Q=\lfloor 8{ϵ}^{-2}{(3M+3)}^2{\overline{c}}^{-1}\rfloor +1,$$

and $\delta \in (0,\gamma ).$

Assume that ${\left\{x_k\right\}}_{k=0}^{\infty }\subset C$,

$$\rho (\theta ,x_0)\le M,$$

and that

$$B(x_{k+1},\delta )\cap T\left(x_k\right)\ne \varnothing ,k=0,1,\dots .$$

Then, there is $j\in \{1,\dots ,Q\}$ for which

$$B(x_j,ϵ)\cap T\left(x_j\right)\ne \varnothing .$$

Proof. 

Lemma 1 implies that

$$\rho (x_1,\theta )\le 3M+3.$$

The application of Theorem 1 to the sequence ${\left\{x_{i+1}\right\}}_{i=0}^{\infty }$ implies our result. □

Theorem 3. 

Let $M>0$, $ϵ\in (0,1]$,

$$\{\xi \in C:B(\xi ,ϵ)\cap T(\xi )\ne \varnothing \}\subset B(\theta ,M),$$

(38)

$$\gamma \in (0,{\left(18\right)}^{-1}{(12M+12)}^{-1}{ϵ}^2\overline{c}),$$

$$z\in B(\theta ,M)$$

satisfy

$$B(z,\gamma )\cap \mathrm{Fix}\left(T_i\right)\ne \varnothing ,i=1,\dots ,m,$$

$$Q=\lfloor 8{ϵ}^{-2}{(3M+3)}^2{\overline{c}}^{-1}\rfloor +1,$$

and $\delta \in (0,\gamma ).$

Assume that ${\left\{x_k\right\}}_{k=0}^{\infty }\subset C$,

$$\rho (\theta ,x_0)\le M,$$

and that

$$B(x_{k+1},\delta )\cap T\left(x_k\right)\ne \varnothing ,k=0,1,\dots .$$

Then, there exists a strictly increasing sequence of natural numbers ${\left\{q_j\right\}}_{j=1}^{\infty }$ such that

$$1\le q_0\le Q,$$

(39)

and for each integer $j\ge 0$,

$$q_{j+1}-q_j\le Q$$

(40)

$$B(x_{q_j},ϵ)\cap T\left(x_{q_j}\right)\ne \varnothing .$$

(41)

Proof. 

Theorem 2 implies that there exists $q_0\in \{1,\dots ,Q\}$ for which

$$B(x_{q_0},ϵ)\cap T\left(x_{q_0}\right)\ne \varnothing .$$

Assume that $p\in \{0,1,\dots \}$, $q_j$, $j=0,\dots ,p$ are natural numbers such that for any integer j satisfying $0\le j<p$, (40) holds, and assume that (41) is true for all $j=0,\dots ,p$. We set

$$y_i=x_{i+q_p},i=0,1,\dots .$$

According to (38) and (41),

$$\rho (\theta ,y_0)\le M.$$

Clearly, all of the assumptions of Theorem 2 hold with $x_i=y_i$, $i=0,1,\dots$, and Theorem 2 implies that there is $j\in \{1,\dots ,Q\}$ for which

$$B(y_j,ϵ)\cap T\left(y_j\right)\ne \varnothing .$$

We set

$$q_{p+1}=q_p+j.$$

Clearly,

$$B(x_{q_{p+1}},ϵ)\cap T\left(x_{q_{p+1}}\right)\ne \varnothing .$$

Thus, by induction, we have constructed the sequence of natural numbers ${\left\{q_j\right\}}_{j=1}^{\infty }$ and proved Theorem 3. □