Projection methods for quasi-nonexpansive multivalued mappings in Hilbert spaces

1.   Introduction

Let $\mathbb{K}$ be a nonempty closed and convex subset of real Hilbert space $\mathbb{H}$. Define $S: \mathbb{K}\to\mathbb{K}$ to be a continuous mapping. A point $\bar{u}\in \mathbb{K}$ is said to be a fixed point of $S$ if $S(\bar{u}) = \bar{u}$. Also, the $F(S)$ represents the set of all fixed points of $S$. Several authors have investigated the existence of fixed points for theorems of single-valued nonexpansive mappings (for example, ^[1,2,3,4,5]).

Mann ^[6] proposed the following method in 1953 for approximating the fixed point of a nonexpansive mapping $S$ in a Hilbert space $\mathbb{H}:$

where $\{a_n\}$ is a sequence in $[0, 1].$

Ishikawa ^[7] generalized Mann's iterative algorithm (1.1) in 1974 by introducing the iteration:

where $\{a_n\}$ and $\{b_n\}$ are sequences in $[0, 1].$

Noor ^[8] introduced and generalized Ishikawa's iterative algorithm (1.2) in 2000 by introducing the following iterative procedure for solving the fixed point problem of a single-valued nonlinear mapping:

where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}$ are sequences in $[0, 1].$

Yildirim and Özdemir ^[9] introduced a new iteration process in 2009 which is an n-step for finding the common fixed points. It is produced by the following processes:

where $\{a_{jn}\}$ be a sequence in $[\epsilon, 1-\epsilon]$ for some $\epsilon\in(0, 1)$, for each $j\in\{1, 2, \, \dots, \, r\}.$

Sainuan ^[10] developed a new iteration called P-iteration in 2015. The P-iteration is defined as:

where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}$ are sequences in $[0, 1].$

The D-iteration was introduced in 2018 by Daengsaen and Khemphet ^[11], who used the Sainuan's iteration concept. It is produced by the following processes:

where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}$ are sequences in $[0, 1].$

The heavy ball method, which was studied in ^[12,13] for maximal monotone operators by the proximal point algorithm, was used by Alvarez and Attouch ^[19]. This algorithm is known as the inertial proximal point algorithm, and it is written as follows:

where $I$ is the identity mapping. It was proved that if $\{\gamma_n\}$ is non-decreasing and $\{\lambda_n\}\subset[ 0, 1)$ with

then algorithm (1.7) converges weakly to a zero of $B.$

Nakajo and Takahashi ^[18] proposed modifying Mann's iteration method (1.1) to obtain a strong convergence theorem in Hilbert spaces $\mathbb{H}:$

where $\{a_n\}\subseteq[0, a ]$ for some $a\in [ 0, 1).$ They proved that the sequence $\{u_n\}$ converges strongly to $P_{F(S)}u_0.$

In 2021, Chaolamjiak et al. ^[14] proposed modifying SP iteration method (1.4) to obtain a strong convergence theorem in Hilbert spaces $\mathbb{H}:$

for all $n\geq1,$ where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}\subset(0, 1).$ They proved that the sequence $\{u_n\}$ converges strongly to a common fixed point of $S_1, S_2$ and $S_3.$

The results ^{[11,18,19,21]} provide incentive. In order to locate a common fixed point of three quasi-nonexpansive multivalued mappings, we introduce the D-iterative approach with the inertial technical term. We can prove strong convergence theorems by combining shrinking projection methods with inertial D-iteration. Finally, we compare our inertial projection method to the traditional projection method and conduct numerical tests to support our major findings with different choices of the initial values $x_0$ and $x_1$ in 4 case.

2.   Preliminaries

Let $CB(\mathbb{K})$ and $K(\mathbb{K})$ denote the families of nonempty closed bounded, and compact, respectively.

The Hausdorff metric on $CB(\mathbb{K})$ is defined by:

where $d(u, Q) = \inf_{\alpha\in Q}\{\|u-\alpha\|\}.$

A single-valued mapping $S : \mathbb{K}\to \mathbb{K}$ is said to be nonexpansive if

A multivalued mapping $S : \mathbb{K}\to CB(\mathbb{K})$ if $\bar{u}\in S\bar{u}$ and

Then $S$ is said to be quasi-nonexpansive.

Condition (A). Let $\mathbb{H}$ be a Hilbert space and $\mathbb{K}$ be a subset of $\mathbb{H}.$ A multivalued mapping $S : \mathbb{K}\to CB(\mathbb{K})$ is said to satisfy Condition (A) if $\|u-\bar{u}\| = d(u, S\bar{u})$ for all $u\in \mathbb{H}$ and $\bar{u}\in F(S).$

We now give the example of quasi-nonexpansive multivalued mapping $S$ which satisfies Condition (A) and the fixed point set $F(S)$ contains more than one element.

Example. In Euclidean space $\mathbb{R},$ let $\mathbb{K}\ = [0, 2]$ and $S : \mathbb{K}\to CB(\mathbb{K})$ be defined by

It is easy to see that $F(S) = \{0, 2\}.$

Lemma 2.1. ^[14] Let $\mathbb{H}$ be a real Hilbert space. Let $S : \mathbb{H}\to CB(\mathbb{H})$ be a quasi-nonexpansive mapping with $F(S)\neq\emptyset.$ Then, $F(S)$ is closed, and if $S$ satisfies Condition (A), then $F(S)$ is convex.

A multivalued mapping $S : \mathbb{K}\to CB(\mathbb{K})$ is said to be hybrid if

Lemma 2.2. ^[15] Let $\mathbb{K}$ be a closed convex subset of a real Hilbert space $\mathbb{H}.$ Let $S : \mathbb{K}\to K(\mathbb{K})$ be a hybrid multivalued mapping. Let $\{u_n\}$ be a sequence in $\mathbb{K}$ such that $u_n\to\bar{u}$ and $\lim_{n\to\infty}\|u_n-x_n\| = 0$ for some $x_n\in Su_n.$ Then, $\bar{u}\in S\bar{u}.$

Lemma 2.3. ^[16] Let $\mathbb{X}$ be a Banach space satisfying Opial's condition and let $\{u_n\}$ be a sequence in $\mathbb{X}.$ Let $x, y\in \mathbb{X}$ be such that $\lim_{n\to\infty}\|u_n-x\|$ and $\lim_{n\to\infty}\|u_n-y\|$ exist. If $\{u_{n_k}\}$ and $\{u_{m_k}\}$ are subsequences of $\{u_n\}$ which converge weakly to $x$ and $y$, respectively, then $x = y.$

Lemma 2.4. ^[17] Let $\mathbb{K}$ be a nonempty closed convex subset of a real Hilbert space $\mathbb{H}.$ For each $x, y\in \mathbb{H}$ and $v\in\mathbb{R},$ the set

is closed and convex.

Lemma 2.5. ^[18] Let $\mathbb{K}$ be a nonempty closed convex subset of a real Hilbert space $\mathbb{H}$ and $P_\mathbb{K} : \mathbb{H}\to \mathbb{K}$ be the metric projection from $\mathbb{H}$ onto $\mathbb{K}$. Then

for all $u\in \mathbb{H}$ and $v\in \mathbb{K}.$

Lemma 2.6. ^[19] Let $\{\alpha_n\}, \, \{\beta_n\}$ and $\{\gamma_n\}$ be the sequences in $[ 0, \infty)$ such that

for all $n\geq1, \, \sum_{n = 1}^\infty \gamma_n < \infty,$ and there exists a real number $\beta$ with $0 \leq \beta_n\leq \beta < 1$ for all $n\geq1.$ Then, the followings hold

$(a)$ $\sum_{n\geq1} [ \alpha_n-\alpha_{n-1}]_+ < \infty,$ where $[ t ]_+ = \max\{t, 0 \};$

$(b)$ there exists $\alpha^*\in[0, \infty)$ such that $\lim_{n\to\infty}\alpha_n = \alpha^*.$

Lemma 2.7. ^[20] Let $\mathbb{H}$ be a real Hilbert space. Then, for each $u, v\in \mathbb{H}$ and $t\in [0, 1]$

$(a)$ $\|u-v\|^2\leq \|u\|^2+\|v\|^2-2\langle u, v\rangle;$

$(b)$ $\|tu-(1-t)v\|^2 = t\|u\|^2+ (1-t)\|v\|^2-t(1-t)\|u-v\|^2;$

$(c)$ If $\{u_n\}$ is a sequence in $\mathbb{H}$ such that $u_n\rightharpoonup u$, then

3.   Main results

Theorem 3.1. Let $\mathbb{K}$ be a closed convex subset of a real Hilbert space $\mathbb{H}$ and $S_1, S_2, S_3: \mathbb{H} \to CB(\mathbb{K})$ be quasi-nonexpansive multivalued mappings with $\Upsilon: = F(S_1)\cap F(S_2)\cap F(S_3)\neq\emptyset$ and $I-S_i$ is demiclosed at $0$ for all $i\in \{1, 2, 3\}$. Let $\{u_n\}$ be a sequence generated by

for all $n\geq1,$ where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}\subset(0, 1).$ Assume that the following conditions hold

$(a)$ $\sum_{n = 1}^\infty\lambda_n\|u_n-u_{n-1}\| < \infty;$

$(b)$ $0 < \liminf_{n\to\infty} a_n < \limsup_{n\to\infty}a_n < 1;$

$(c)$ $0 < \liminf_{n\to\infty} b_n < \limsup_{n\to\infty}b_n < 1;$

$(d)$ $0 < \liminf_{n\to\infty} c_n < \limsup_{n\to\infty}c_n < 1.$

If $S_1, \, S_2$ and $S_3$ satisfy Condition (A), then the sequence $\{u_n\}$ converges weakly to a common fixed point of $S_1, \, S_2$ and $S_3$.

Proof. Let $\bar{u}\in\Upsilon.$ From $S_1, \, S_2$ and $S_3$ satisfy Condition (A), for $x_n\in S_1t_n, \, y_n\in S_2v_n, z_n\in S_3\rho_n$ and using (3.1), we obtain

and

Using Lemma 2.6, (3.5) and the assumption $(a),$ we have $\lim_{n\to\infty}\|u_n-\bar{u}\|$ exists. Thus, $\{u_n\}$ is bounded and also $\{\rho_n\}, \, \{v_n\}$ and $\{t_n\}.$ From Lemma 2.7$(b),$ we get

and

Combination (3.6)–(3.8), we get

The inequality (3.9) implies that

Using conditions $(a)$–$(d)$, $\lim_{n\to\infty}\|u_n-\bar{u}\|$ exists and (3.10), we obtain

This implies that

Because $\{u_n\}$ is bounded, there exists a subsequence $\{u_{n_k}\}$ of $\{u_n\}$ such that $u_{n_k}\rightharpoonup \bar{u}$ some $\bar{u}\in \mathbb{K}.$ From (3.12), we have $t_{n_k}\rightharpoonup \bar{u}.$ Because $I-S_1$ is demiclosed at $0$ and (3.11), we obtain $\bar{u}\in S_1\bar{u}.$ From (3.13), we have $v_{n_k}\rightharpoonup \bar{u}.$ Because $I-S_2$ is demiclosed at $0$ and (3.11), we obtain $\bar{u}\in S_2\bar{u}.$ It follows from (3.14) that $\rho_{n_k}\rightharpoonup \bar{u}.$ Again, because $I-S_3$ is demiclosed at $0$ and (3.11), we have $\bar{u}\in S_3\bar{u}.$ This implies that $\bar{u}\in \Upsilon.$ Now, we show that $\{u_n\}$ converges weakly to $\bar{u}$. We take another subsequence $\{u_{m_k}\}$ of $\{u_n\}$ converging weakly to some $u^*\in \Upsilon.$ Because $\lim_{n\to\infty}\|u_n-\bar{u}\|$ exists and Lemma 2.3.Thus, we have $\bar{u} = u^*.$

Theorem 3.2. Let $\mathbb{K}$ be a nonempty closed convex subset of a real Hilbert space $\mathbb{H}$ and $S_1, \, S_2, \, S_3 : \mathbb{K}\to CB(\mathbb{K})$ be quasi-nonexpansive multivalued mappings with $\Upsilon: = F(S_1)\cap F(S_2)\cap F(S_3)\neq\emptyset$ and $I-S_i$ is demiclosed at $0$ for all $i\in \{1, 2, 3\}$. Let $\{u_n\}$ be a sequence generated by

for all $n\geq1,$ where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}\subset(0, 1).$ Assume that the following conditions hold

$(a)$ $\sum_{n = 1}^\infty\lambda_n\|u_n-u_{n-1}\| < \infty;$

$(b)$ $0 < \liminf_{n\to\infty} a_n < \limsup_{n\to\infty}a_n < 1;$

$(c)$ $0 < \liminf_{n\to\infty} b_n < \limsup_{n\to\infty}b_n < 1;$

$(d)$ $0 < \liminf_{n\to\infty} c_n < \limsup_{n\to\infty}c_n < 1.$

If $S_1, \, S_2$ and $S_3$ satisfy Condition (A), then the sequence $\{u_n\}$ converges strongly to a common fixed point of $S_1, \, S_2$ and $S_3$.

Proof. Step Ⅰ. Show that $\{u_n\}$ is well defined. Using $S_1, \, S_2$ and $S_3$ satisfy Condition (A), Lemma 2.1, $\Upsilon$ is closed and convex. Firstly, we show that $\mathbb{K}_n$ is closed and convex for all $n\geq1.$ Since induction on $n$ that $\mathbb{K}_n$ is closed and convex. For $n = 1, \, \mathbb{K}_1 = \mathbb{K}$ is closed and convex. Suppose that $\mathbb{K}_n$ is closed and convex for some $n\geq1.$ Using the definition $\mathbb{K}_{n + 1}$ and Lemma 2.4, we have that $\mathbb{K}_{n + 1}$ is closed and convex. Thus, $\mathbb{K}_n$ is closed and convex for all $n\geq1$. Next, we show that $\Upsilon\subseteq \mathbb{K}_n$ for each $n\geq1.$ From Lemma 2.6$(b)$ and $S_1, \, S_2$ and $S_3$ satisfy Condition (A), let $\bar{u}\in\Upsilon$ for $x_n\in S_1t_n, \, y_n\in S_2v_n, z_n\in S_3\rho_n$ and using (3.15), we obtain

and

Therefore, from (3.18), $\bar{u}\in \mathbb{K}_n, \, n\geq1.$ This implies that $\Upsilon\subseteq \mathbb{K}_n$ for each $n\geq1,$ and hence, $\mathbb{K}_n\neq\emptyset.$ Thus, $\{u_n\}$ is well defined.

Step Ⅱ. Show that $u_n\to u\in \mathbb{K}$ as $n\to\infty.$ Since $u_n\in P_{\mathbb{K}_{n}}u_1, \, \mathbb{K}_{n + 1}\subseteq \mathbb{K}_n,$ and $u_{n+1}\in \mathbb{K}_n,$ we obtain

Since $\Upsilon\subseteq \mathbb{K}_n,$ we obtain

for all $x\in\Upsilon.$ The inequalities (3.19) and (3.20) imply that the sequence $\{u_n-u_1\}$ is bounded and non-decreasing. Therefore, $\lim_{n\to\infty}\|u_n-u_1\|$ exists.

For $m > n,$ from the definition of $\mathbb{K}_n,$ we obtain $u_m\in P_{\mathbb{K}_m}u_1\in \mathbb{K}_m\subseteq \mathbb{K}_n.$ Using Lemma 2.5, we have

From $\lim_{n\to\infty}\|u_n-u_1\|$ exists and follows (3.21), we have that $\lim_{n\to\infty}\|u_n-u_m\| = 0.$ Therefore, $\{u_n\}$ is a Cauchy sequence in $\mathbb{K},$ and so $u_n\to u \in \mathbb{K}$ as $n\to\infty.$

Step Ⅲ. Show that $\lim_{n\to\infty}\|t_n-x_n\| = \lim_{n\to\infty}\|x_n-y_n\| = \lim_{n\to\infty}\|y_n-z_n\| = 0,$ where $x_n\in S_1t_n, \, y_n\in S_2v_n$ and $z_n\in S_3\rho_n.$ From Step Ⅱ, we obtain $\lim_{n\to\infty}\|u_{n+1}-u_n\| = 0.$ Because $u_{n+1}\in \mathbb{K}_n,$ we have that

Using the assumption $(a)$ and (3.22), we have

Because $S_1$ satisfies condition (A) and using Lemma 2.7, we obtain

Using (3.6), (3.7) and (3.24), we have

The inequality (3.25) implies that

From conditions $(a)$–$(d),$ (3.23) and (3.25), we have (3.11). From (3.13), (3.14) and the same proof in Theorem 3.1, we have

From Step Ⅱ, we know that $u_n\to u \in \mathbb{K}.$ It follows (3.27), we obtain that $t_n\to u.$ Because $I-S_1$ is demiclosed at $0,$ we have $u\in F(S_1).$ In the same way, we have that $u\in F(S_2)$ and $u\in F(S_3).$ This implies that $u\in\Upsilon.$

Step Ⅳ. Show that $u = P_{\Upsilon}u_1.$ From $u\in \Upsilon$ and (3.19), we obtain

Using the definition of the projection operator, we can conclude that $u = P_{\Upsilon}u_1.$

Theorem 3.3. Let $\mathbb{K}$ be a nonempty closed convex subset of a real Hilbert space $\mathbb{H}$ and $S_1, \, S_2, \, S_3 : \mathbb{K}\to CB(\mathbb{K})$ be quasi-nonexpansive multivalued mappings with $\Upsilon: = F(S_1)\cap F(S_2)\cap F(S_3)\neq\emptyset$ and $I-S_i$ is demiclosed at $0$ for all $i\in\{1, 2, 3\}.$ Let $\{u_n\}$ be a sequence generated by

for all $n\geq1,$ where $\{a_n\}, \, \{b_n\}$ and $\{c_n\}\subset(0, 1).$ Assume that the following conditions hold

$(a)$ $\sum_{n = 1}^\infty\lambda_n\|u_n-u_{n-1}\| < \infty;$

$(b)$ $0 < \liminf_{n\to\infty} a_n < \limsup_{n\to\infty}a_n < 1;$

$(c)$ $0 < \liminf_{n\to\infty} b_n < \limsup_{n\to\infty}b_n < 1;$

$(d)$ $0 < \liminf_{n\to\infty} c_n < \limsup_{n\to\infty}c_n < 1.$

If $S_1, \, S_2$ and $S_3$ satisfy Condition (A), then the sequence $\{u_n\}$ converges strongly to a common fixed point of $S_1, \, S_2$ and $S_3$.

Proof. From the same method of Theorem 3.2 step by step, we can conclude the proof by replacing $\mathbb{K}_{n + 1}$ by $\mathbb{K}_n,$ expect in Step 1. Showing that $\Upsilon\subseteq \mathbb{K}_n$ for each $n\geq1.$ Next, we show that $\Upsilon\subseteq R_n$ for all $n\geq1.$ Indeed, by mathematical induction, for $n = 1,$ we obtain $\Upsilon\subseteq \mathbb{K} = R_1$. Suppose that $\Upsilon\subseteq R_n$ for all $n\geq1.$ Because $u_{n + 1}$ is the projection of $u_1$ onto $\mathbb{K}_n\cap R_n,$ we obtain

Therefore, $\Upsilon\subseteq \mathbb{K}_{n + 1}.$ Hence, $\Upsilon\subseteq \mathbb{K}_n\cap R_n.$ This implies that $\{u_n\}$ is well defined.

Next, we show that $u_n\to q\in \mathbb{K}$ as $n\to\infty.$ Using the definition of $R_n,$ we obtain $u_n = P_{R_n}u_1.$ Because $u_{n+1}\in R_n,$ we have the inequality (3.19) and

From (3.19) and (3.29) we have the sequence $\{u_n-u_1\}$ is bounded and non-decreasing, and so $\lim_{n\to\infty}\|u_n-u_1\|$ exists. For $m > n,$ by definition of $R_n,$ we have $u_m = P_{R_m}u_1\in R_m\subseteq R_n.$ Using Lemma 2.5, we have (3.21). Because $\lim_{n\to\infty}\|u_n-u_1\|$ exists, it follows (3.21), we obtain $\lim_{n\to\infty}\|u_m-u_n\| = 0.$ Therefore, $\{u_n\}$ is a Cauchy sequence in $\mathbb{K},$ and hence $u_n\to q\in \mathbb{K}$ as $n\to\infty.$ In fact, we have $\lim_{n\to\infty}\|u_{n+1}-u_n\| = 0.$ Using the same proof of Steps 3 and 4 in Theorem 3.2, we obtain $q = P_{\Upsilon}u_1.$

4.   Numerical results

It is commonly known that computing the projection of a point on an intersection is quite difficult. However, this can also be stated as the following optimization problem for computing purposes

where $\mathbb{K}^* = \mathbb{K}_n\cap R_n.$ See ^[22] for a list of several more approaches to handle projection onto intersection of sets computationally.

The set $\mathbb{K}_{n+1}$ can be found by $\mathbb{K}_n\cap R_n,$ where

The projection can be thought of as the following optimization problem by point (4.2):

where $\mathbb{K}_{n+1} = \mathbb{K}_{n}\cap R_n.$

Example. Let $\mathbb{H} = \mathbb{R}^3$ and $\mathbb{K} = [2, 5]^3.$

Let $\mathbb{K}_1 = \{u = (u_1, u_2, u_3)\in\mathbb{R}^3 : \sqrt{(u_1-5)^2+ (u_2-5)^2+(u_3-5)^2}\leq 2\}.$ We defined $S_1, \, S_2, \, S_3 : \mathbb{R}^3\to CB(\mathbb{R}^3)$ as:

We see that $S_1, S_2$ and $S_3$ are quasi-nonexpansive and $F(S_1)\cap F(S_2)\cap F(S_3) = \{(5, 5, 5)\}.$ Let $a_n = \frac{n+4}{5n+5}, \, b_n = \frac{n+2}{7n+2}, \, c_n = \frac{7n+7}{9n+9}$ and

We compare a numerical test between our inertial method defined in Theorem 3.3 and method (1.10). The stopping criterion is defined by $\|u_{n + 1}-u_n\| < 10^{-10}.$ We make different choices of the initial values $x_0$ and $x_1$ as follows (see Figure 1 and Table 1)

Case 1: $x_0 = (2.6816, 2.4389, 2.8891)$ and $x_1 = (2.7733, 2.2146, 2.2555).$

Case 2: $x_0 = (3.9587, 4.6121, 2.9779)$ and $x_1 = (3.7123, 3.4894, 4.8867).$

Case 3: $x_0 = (4.9401, 3.9274, 2.8475)$ and $x_1 = (3.8675, 3.7185, 4.7133).$

Case 4: $x_0 = (3.9933, 4.9899, 4.2187)$ and $x_1 = (3.7722, 3.6190, 2.9152).$

5.   Conclusions

In this paper, we proved convergence theorems in Hilbert spaces using a modified D-iteration.We proved the weak and strong convergence of the iterative algorithms to the common fixed point under some suitable assumptions.

Acknowledgments

Firstly, Anantachai Padcharoen (anantachai.p@rbru.ac.th) would like to thank the Research and Development Institute of Rambhai Barni Rajabhat University. Finally, Kritsana Sokhuma would like to thank Phranakhon Rajabhat University.

Conflict of interest

The authors declare that they have no competing interests.