Iterative algorithm for solving monotone inclusion and fixed point problem of a finite family of demimetric mappings

1.   Introduction

Monotone inclusion problem (MIP) is the problem to identify a zero of the sum of two operators and it is described as

where $A:H \rightarrow H$ is an $M$-cocoercive operator, $M$ is a linear, bounded operator on a real Hilbert space $H$ and $B:H\rightarrow 2^{H}$ is a maximal monotone operator. Monotone inclusion problem incorporates various problems such as convex optimization, machine learning, statistical regression, signal and image processing, see ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]}. Lions and Mercier ^[18] defined the forward-backward splitting algorithm which is one of the most effective method for solving the problem (1.1) and it is defined by

where $A$ is monotone, $B$ is $1/L$-cocoercive operator and $\lambda_{n}\in (0, 2/L)$ is a step size parameter. But the algorithm defined in (1.2) converges weakly to a solution of the monotone inclusion problem. Later, Tseng ^[15] improved this forward-backward splitting algorithm and also proved weak convergence of it. After that, Gibali and Thong ^[19] proposed a modified version of Tseng's splitting algorithm and proved strong convergence of the proposed algorithm. To speed up the convergence rate of the algorithms, Moudafi and Oliny ^[20] introduced the following algorithm which includes the inertial parameter.

where $\epsilon_{n}\in [0, 1)$ is inertial parameter. They proved the weak convergence of above algorithm (1.3) assuming the conditions $\sum_{n = 1} ^{\infty} \epsilon_{n} \|\zeta_{n}-\zeta_{n-1}\|^{2} < \infty$ and $\lambda_{n} < 2/L$, where $L$ is Lipschitz constant of operator $B$. Later on, the following preconditioning algorithm was defined by Lorenz and Pock ^[10] for solving the monotone inclusion problem and proved weak convergence of it.

It is clear that the algorithm (1.4) reduces to forward-backward algorithm (1.2) if we take $\epsilon_{n} = 0$ and $M = I$. Dixit et al.^[4], in $2021,$ introduced accelerated preconditioning forward-backward normal S-iteration and proved weak convergence under few assumptions in a real Hilbert space $H$.

where $J_{\lambda, M}^{A, B} = (I+\lambda M^{-1}A)^{-1} (I-\lambda M^{-1}B), \gamma_{n}\in (0, 1), \epsilon_{n}\in [0, 1)$ and $\lambda\in[0, 1).$ Next, Altiparmak and Karahan ^[21] proposed a new preconditioning forward-backward splitting algorithm and proved strong convergence of it.

where ${\epsilon_{n}}\subset [0, \theta]$ with $\theta\in [0, 1)$ and ${\beta_{n}}, {\gamma_{n}}\in (0, 1)$ and $f:H \rightarrow H$ is a $k$- contraction mapping with respect to $M$-norm. Recently, in 2021, the same authors ^[22] proved strong convergence of a modified preconditioning algorithm for solving the monotone inclusion problem in the following manner:

where $J_{\lambda, M}^{A, B} = (I+\lambda M^{-1}B)^{-1} (I-\lambda M^{-1}A), \alpha_{n}, \beta_{n}, \gamma_{n} \in (0, 1), {\epsilon_{n}}\subset [0, \theta]$ with $\theta\in [0, 1)$ and $h:H\rightarrow H$ is $k$-contraction with respect to $M$-norm.

On the other hand, the theory of fixed points has been an appealing topic of research. Several researchers have worked in this direction, see ^{[23,24,25,26,27]}.

We have considered the Monotone inclusion problem and fixed point problem of a finite family of demimetric mappings in the context of Hilbert space in this paper as we defined the demimetric operator in the notion of inner product with respect to M-norm, where M is the linear, self-adjoint, positive and bounded operator. In the setting of a Hilbert space, that is, in the complete inner product space, the existence of a solution to the monotone inclusion problem is guaranteed under certain conditions. The authors defined contraction, nonexpansive, quasi-nonexpansive operators with respect to $M$-norm in these algorithms and proved weak and strong convergence of those algorithms under suitable assumptions. Our main contributions to this research are as follows:

● We introduce an operator namely, a demimetric operator with respect to $M$-norm;

● We define a new algorithm which is the combination of forward-backward and Tseng method for solving the monotone inclusion problem together with a new step for solving the fixed point problem of the finite family of demimetric operators;

● We also utilize the inertial extrapolation step strategy due to Polyak ^[29] to enhance the convergence rate of the proposed algorithm.

2.   Preliminaries

In this section, we list some definitions and lemmas which contribute significantly to our main result. Throughout the study, suppose $H$ is a real Hilbert space, $Q$ is a nonempty, closed and convex subset of $H$, $M$ is a linear, self-adjoint, positive and bounded operator on $H$ and $Fix(U)$ denotes the set of all fixed points of the mapping $U$.

Definition 2.1. ^[30] Assume that $A: H\rightarrow 2^{H}$ is a set-valued operator. It is called monotone if

The operator $A$ is called maximal monotone if the graph of the operator $A$ is not properly contained in the graph of any other monotone operator.

Definition 2.2. ^[31] Assume that $Q$ is nonempty subset of $H$ and $\zeta\in H.$ If for any $\upsilon\in H$, there exists a unique point $\varphi\in Q$ such that $\|\varphi-\zeta\|\le \|\upsilon-\zeta\| \; {\rm{ for\; all }}\; \upsilon\in H$ then $\varphi$ is called metric projection of $\zeta$ onto $Q.$ It is symbolized by $\varphi = P_{Q}\zeta.$ If $P_{Q}\zeta$ exists and it can be uniquely obtained for all $\zeta\in H$, then $P_{Q}:H\rightarrow Q$ is called metric projection operator. The operator $P_{Q}$ is nonexpansive and it satisfies the following inequality

Definition 2.3. ^[32] Assume that $M$ is a bounded and linear operator on $H$. The operator $M$ is said to be self-adjoint if $M^{*} = M$ where $M^{*}$ denotes the adjoint of $M$. If $\langle M(\zeta), \zeta)\rangle > 0$ for all $\zeta (\ne 0)\in H$, then $M$ is called positive definite operator. The $M$-inner product is given by $\langle\zeta, \psi\rangle_{M} = \langle \zeta, M(\psi)\rangle$ for all $\zeta, \psi \in H$ and the corresponding M-norm is also given by $\|\zeta\|^{2}_{M} = \langle \zeta, M(\zeta)\rangle$ for all $\zeta\in H$ by using the self-adjoint, linear and bounded operator $M$.

Definition 2.4. ^[4] Assume that $Q$ is a nonempty subset of $H$, $M$ is a positive definite operator on $H$ and $U: Q \rightarrow H$ is an operator. Then $U$ is called

(i) $M$-cocoercive operator if

(ii) nonexpansive operator with respect to $M$ -norm if

(iii) quasi-nonexpansive operator with respect to $M$ -norm if

(iv) $h$-contraction with respect to $M$-norm if there exists $h\in [0, 1)$ such that

Lemma 2.1. ^[30] For $\zeta, \psi, \omega, \iota\in H$ and $\alpha, \beta, \gamma\in [0, 1],$ where $\alpha+\beta+\gamma = 1,$ we have

(i) $\|\zeta+\psi\|^{2}\le \|\zeta\|^{2}+ 2 \langle \psi, \zeta+\psi\rangle,$

(ii) $\|\beta\zeta+(1-\beta)\psi\|^{2} = \beta\|\zeta\|^{2}+(1-\beta)\|\psi\|^{2}-\beta(1-\beta)\|\zeta-\psi\|^{2}$,

(iii) $\|\alpha\zeta+\beta\psi+\gamma\omega\|^{2} = \alpha \|\zeta\|^{2}+ \beta\|\psi\|^{2}+ \gamma\|\omega\|^{2}- \alpha\beta \|\zeta-\psi\|^{2}-\alpha \gamma\|\zeta-\omega\|^{2}-\beta\gamma\|\psi-\omega\|^{2},$

(iv) $2 \langle \zeta-\psi, \omega-\iota\rangle = \|\zeta-\iota\|^{2}+\|\psi-\omega\|^{2}-\|\zeta-\omega\|^{2}-\|\psi-\iota\|^{2},$

(v) $\|\zeta\pm\psi\|^{2} = \|\zeta\|^{2}\pm 2 \langle \zeta, \psi\rangle +\|\psi\|^{2}.$

Lemma 2.2. ^[33] Let $U: Q \rightarrow H$ be the nonexpansive operator with $Fix(U)\ne \phi.$ Then the mapping $I-U$ is demiclosed at origin, that is, for any sequence $\{\zeta_{n}\}\in H$ such that $\zeta_{n}\rightarrow \zeta \in H$ and $\|\zeta_{n}-U\zeta_{n}\|\rightarrow 0$ as $n\rightarrow \infty$, we have $\zeta\in Fix(U).$

Lemma 2.3. ^[34] Assume that $S:Q\rightarrow H$ is $\xi$-demimetric operator with $\xi\in (-\infty, 1).$ Then, $Fix(S)$ is closed and convex.

Lemma 2.4. ^[4] Suppose that $A$ is $M$-cocoercive operator on $H,$ $B:H\rightarrow 2^{H}$ is maximal monotone operator. Then, $J_{\lambda, M}^{A, B} = (I+\lambda M^{-1}B)^{-1}(I-\lambda M^{-1}A)$ is nonexpansive with respect to $M$-norm for $\lambda\in (0, 1].$

Lemma 2.5. ^[4] Suppose that $A$ is $M$-cocoercive operator on $H,$ $B: H\rightarrow 2^{H}$ is a maximal monotone operator and $\lambda$ is a non-negative real number. Then, $\zeta\in H$ is a solution of monotone inclusion problem (1.1) if and only if $(I+\lambda M^{-1}B)^{-1}(I-\lambda M^{-1}A)(\zeta) = \zeta.$

Lemma 2.6. ^[35] Suppose that $\{t_{n}\} \subset [0, \infty)$ is a sequence of real numbers. Let

where $\{p_{n}\} \subset [0, 1]$ and $\{t_{n}\}\subset (-\infty, \infty)$ satisfying the following assumptions:

(1) $\sum_{n = 1}^{\infty}p_{n} = \infty,$

(2) $\limsup\limits_{n\rightarrow \infty}t_{n}\le 0.$

Then $\lim_{n \to \infty} s_{n} = 0.$

Lemma 2.7. ^[36] Assume that $\{\zeta_{n}\} \subseteq [0, \infty)$ such that there exists a subsequence $\{\zeta_{n_{j}}\}$ of $\{\zeta_{n}\}$ such that $\zeta_{n_{j}} < \zeta_{n_{j+1}}.$ Then, there exists a nondecreasing sequence $\{n_{k}\}$ of natural numbers satisfying $\lim_{k \to \infty} n_{k} = \infty$, $\zeta_{n_{k}} \le \zeta_{n_{k+1}}$ and $\zeta_{k} \le \zeta_{n_{k+1}}$ for all $k\in \mathbb{N}.$ Also, $n_{k}$ is the greatest number $n$ in the set $\{1, 2, ..., k\}$ satisfying $\zeta_{n} < \zeta_{n+1}.$

3.   Main result

In this part, we will prove the strong convergence of the following algorithm for finding a common solution of monotone inclusion and fixed point problem of a finite family of demimetric operators in the setting of a real Hilbert space $H$. Assume that $A$ is $M$-cocoercive operator on $H$, $B: H \rightarrow 2^{H}$ is a maximally monotone operator. Suppose $S_{i}$ is a finite family of $\xi$-demimetric operators with $\xi \in (-\infty, 1)$ such that $I-S_{i}$ is demiclosed at origin for all $i = 0, 1, 2, ...N-1$, $h:H \rightarrow H$ is a contraction mapping with respect to $M$-norm with constant $k \in(0, 1].$

Algorithm 3.1. Let $\zeta_{0}, \zeta_{1} \in H$. Compute $\{\varphi_{n}\}, \{\upsilon_{n}\}, \{\kappa_{n}\} \; {\rm{ and }}\; \{\zeta_{n}\}$ using

where $S_{n}=\frac{1}{N} \sum_{i=0} ^{N-1} (1-q_{n})I+q_{n}S_{i}$

Definition 3.1. A mapping $U:Q\rightarrow H$, where $Q$ is closed, convex and nonempty subset of $H$ is called $\xi$-demimetric with respect to $M$-norm, where $\xi\in (-\infty, 1)$ if $Fix(U)\ne \phi$ such that

Example 3.1. Let $H = \mathbb{R}^{3}, Q = H$ and $M(\zeta) = (5\zeta_{1}, 4 \zeta_{2}, 5\zeta_{3})$ for all $\zeta = (\zeta_{1}, \zeta_{2}, \zeta_{3})\in H.$ The mapping $U:H \rightarrow H$ defined by $U(\zeta) = -2 \zeta$ for all $\zeta = (\zeta_{1}, \zeta_{2}, \zeta_{3})$ is $\frac{1}{3}$-demimetric mapping with respect to M-norm.

Now, we prove a result for $\xi$-demimetric operator with respect to $M$-norm which is motivated by Lemma (2.2) of ^[37].

Lemma 3.1. Suppose $S:Q\rightarrow H$ is $\xi$-demimetric operator with respect to $M$-norm, where $\xi\in (-\infty, 1)$ and $Fix(S)$ is nonempty. Let $P = (1-\gamma)I+\gamma S,$ where $\gamma\in (-\infty, \infty)$ with $\gamma\in (0, 1-\xi],$ then $P:Q\rightarrow H$ is a quasi-nonexpansive operator.

Proof. It is direct that $Fix(S) = Fix(P).$ Now, since $S:Q\rightarrow H$ is demimetric operator, we have for any $\zeta\in Q$ and $y\in Fix(P),$

This implies that $P$ is 0-demimetric mapping. Also, for $\zeta\in Q$, $y\in Fix(P)$ and using Lemma (2.1), we deduce

Hence, $P$ is a quasi-nonexpansive operator.

Lemma 3.2. The mapping $S_{n}$ defined by $S_{n} = \frac{1}{N} \sum_{i = 0} ^{N-1} (1-q_{n})I+q_{n}S_{i}$ is quasi-nonexpansive. \proof Let $\zeta^{*}\in \Omega.$

Consider

Hence, $S_{n}$ is quasi-nonexpansive.

Lemma 3.3. Assume that $\{\zeta_{n}\}$ is bounded sequence of real numbers and $\zeta^{*}\in \Omega = (A+B)^{-1}(0)\cap \bigcap\limits_{i = 0}^{N-1} Fix(S_{i})$. If $lim_{n\to\infty}\|J_{\lambda, M}^{A, B}\zeta_{n}-\zeta_{n}\|_{M} = 0$ and $lim_{n\to\infty}\|\kappa_{n}-S_{n}\kappa_{n}\|_{M} = 0$. Then $\limsup\limits_{n\rightarrow \infty} \langle h(\zeta^{*})-\zeta^{*}, \zeta_{n+1}-\zeta^{*}\rangle_{M}\le 0.$

Proof. Since the sequence $\{\zeta_{n}\}$ is bounded, so there exists a subsequence $\{\zeta_{n_{k}}\}$ of $\{\zeta_{n}\}$ such that $\zeta_{n_{k}} \rightharpoonup \zeta$

Consider

Using the condition $\|J_{\lambda, M}^{A, B}\zeta_{n}-\zeta_{n}\|_{M}\rightarrow0$ as $n \rightarrow \infty$ and by using Lemma (2.5), we deduce that $\zeta \in (A+B)^{-1}(0).$ Also, since $lim_{n\to\infty}\|\zeta_{n}-\kappa_{n}\| = 0$ and $\zeta_{n_{k}}\rightharpoonup \zeta,$ so we obtain $\kappa_{n_{k}}\rightharpoonup \zeta.$ By using the condition $lim_{n\to\infty}\|\kappa_{n}-S_{n}\kappa_{n}\| = 0$ and using Lemma (2.2), we obtain that $\zeta\in Fix(S_{i}).$ Thus $\zeta\in \Omega.$

From Eq (3.2) and by the property of metric projection, we have

Hence the lemma is proved.

Theorem 3.1. Suppose that the solution set $\Omega = (A+B)^{-1}(0)\; \cap \bigcap\limits_{i = 0}^{N-1} Fix(S_{i})$ is non-empty and the sequence $\{\zeta_{n}\}$ is generated by algorithm (3.1), where $\{\epsilon_{n}\}\subset[0, \theta]$ with $\theta\in [0, 1), \{\alpha_{n}\}, \{\beta_{n}\}, \{\gamma_{n}\}, \{\delta_{n}\}\in (0, 1)$ such that the following conditions hold:

(i) $0 < a\le \alpha_{n}\le b < 1 \; {\rm{ for \; some }}\; a, b\in \mathbb{R}$,

(ii) $0 < c\le \beta_{n}\le d < 1 \; {\rm{ for \; some }}\; c, d\in \mathbb{R}$,

(iii) $lim_{n\to\infty}\gamma_{n} = 0, \sum_{n = 1} ^{\infty}\gamma_{n} = \infty,$

(iv) $\sum_{n = 1} ^{\infty}\epsilon_{n}\|\zeta_{n}-\zeta_{n-1}\| < \infty,$

(v) for any $n\in \mathbb{N}, 0 < a^{*} < \liminf\limits_{n\rightarrow \infty}\delta_{n} \le \limsup\limits_{n\rightarrow \infty}\delta_{n} < b^{*} < 1-\gamma_{n},$ where $a^{*}, b^{*}\in \mathbb{R}^{+}.$

Then the sequence $\{\zeta_{n}\}$ converges strongly to a point $\zeta^{*}\in \Omega = P_{\Omega}h(\zeta^{*}).$

Proof. First we prove that the sequence $\{\zeta_{n}\}$ is bounded. Let $\zeta^{*}\in \Omega.$

Consider

Since, $J_{\lambda, M}^{A, B}$ is nonexpansive with respect to $M$-norm, so using nonexpansiveness of $J_{\lambda, M}^{A, B}$, we have

and

From Eqs (3.1)–(3.5) and using the fact that $h$ is $k$-contraction and $S_{n}$ is quasi-nonexpansive with respect to $M$-norm, we have

From assumptions $(iii)$ and $(iv)$, we have $\lim _{n \to \infty}\frac{\epsilon_{n}}{\gamma_{n}}\|\zeta_{n}-\zeta_{n-1}\|_{M} = 0.$ So, there exists a positive integer $N_{1} > 0$ such that $\frac{\epsilon_{n}}{\gamma_{n}}\|\zeta_{n}-\zeta_{n-1}\| \le N_{1}.$ By using Eq (3.6), we obtain

Continuing like this,

Thus the sequence $\{\zeta_{n}\}$ is bounded and therefore, the sequences $\{\varphi_{n}\}, \{\upsilon_{n}\}, \{\kappa_{n}\}$ are also bounded.

Now, we show that $\zeta_{n}\rightarrow \zeta^{*}.$ Using Lemma (2.1), we obtain

and

Using Eqs $(3.7)$–$(3.9)$, Lemma (2.1) and using the fact that $J_{\lambda, M}^{A, B}$ is nonexpansive and $S_{n}$ is quasi-nonexpansive, we obtain

which implies that

Equation (3.11) is equivalent to

where, $s_{n} = \|\zeta_{n}-\zeta^{*}\|^{2}_{M}$, $p_{n} = \gamma_{n}(1-k)$ and $t_{n} = \frac{2}{1-k}\langle h(\zeta^{*})-\zeta^{*}, \zeta_{n+1}-\zeta^{*}\rangle+ \frac{2\epsilon_{n}}{\gamma_{n}(1-k)}\|\zeta_{n}-\zeta^{*}\|_{M}\|\zeta_{n}-\zeta_{n-1}\|_{M}+\frac{\epsilon_{n}^{2}}{\gamma_{n}(1-k)}\|\zeta_{n}-\zeta_{n-1}\|^{2}_{M}$

Now, we consider two possible cases on the sequence $\{\|\zeta_{n}-\zeta^{*}\|_{M}\}.$

Case 1. Suppose that there exists some positive integer $n_{0}$ such that the sequence $\{\|\zeta_{n}-\zeta^{*}\|_{M}\}$ is nonincreasing sequence for any $n\ge n_{0}.$ Also, the sequence $\{\|\zeta_{n}-\zeta^{*}\|_{M}\}$ is bounded below by zero. So, it is convergent.

By using Lemma (2.1), Eqs (3.7) and (3.9), we have

and

From Eqs (3.13) and (3.14), we obtain

which implies that

Since, $\sum_{n = 1} ^{\infty}\epsilon_{n}\|\zeta_{n}-\zeta_{n-1}\| < \infty,$ so we have

Using the condition $\lim _{n \to \infty }\gamma_{n} = 0,$ Eq (3.16) and taking limit $n\rightarrow \infty$ in Eq (3.15), we get

Now, consider

Taking limit $n \rightarrow \infty$ in Eq (3.18) and using Eq (3.17), we obtain

By using triangle inequality, we have

Taking limit $n\rightarrow \infty$ in Eq (3.20) and using Eqs (3.17) and (3.19), we deduce

Also,

Taking limit $n\rightarrow \infty$ in Eq (3.22) and using Eqs (3.19) and (3.21) in Eq (3.22), we have

and

Taking limit $n \rightarrow \infty$ in Eq (3.24) and using Eqs (3.17), (3.19) and (3.23), we obtain

Again using Eq (3.1) and Lemma (2.1), we deduce that

Using nonexpansiveness of $J_{\lambda, M}^{A, B}$, quasi-nonexpansiveness of $S_{n}$, Eqs (3.7)–(3.9), we have

Taking limit $n\rightarrow \infty$ in Eq (3.26) and using Eq (3.16), condition $\lim _{n \to \infty }\gamma_{n} = 0,$ we obtain

By using triangle inequality, we have

Taking limit $n\rightarrow \infty$ in Eq (3.28) and using Eqs (3.17) and (3.25), we get

Also,

Taking limit $n\rightarrow \infty$ in Eq (3.30) and using Eqs (3.27) and (3.29), we obtain

From Eqs (3.19) and (3.25) and using triangle inequality, we obtain

From Eq (3.1),

Taking limit $n\rightarrow \infty$ in Eq (3.33) and using Eq (3.16), we get

Similarly, we can prove

Now, from Eqs (3.31) and (3.35), we can prove

which implies

Again consider

Taking limit $n\rightarrow \infty$ in Eq (3.37) and using the condition $\lim _{n \to \infty } \gamma_{n} = 0$, Eqs (3.27) and (3.36), we obtain

Also, using Eqs (3.34) and (3.38), we have

From Eqs (3.17) and (3.34), we can prove

Taking limit $n \rightarrow \infty$ in above equation, we have

Since, the sequence $\{\zeta_{n}\}$ is a bounded sequence of real numbers, so using Lemma (3.3), we obtain

Thus, by making use of Eq (3.16), we have $\limsup\limits_{n\rightarrow \infty} t_{n} \le 0$. Hence from Lemma (2.6), $\zeta_{n} \rightarrow \zeta^{*}$ as $n\rightarrow \infty.$

Case 2. There exists a subsequence $\{\|\zeta_{n_{j}}-\zeta^{*}\|^{2}_{M}\}$ of $\{\|\zeta_{n}-\zeta^{*}\|^{2}_{M}\}$ such that $\|\zeta_{n_{j}}-\zeta^{*}\|^{2}_{M}\le \|\zeta_{n_{j+1}}-\zeta^{*}\|^{2}_{M}$ for any $j\in N.$ By using Lemma (2.7), we see that there exists a nondecreasing sequence $\{\zeta_{n_{k}}\}$ of $\mathbb{N}$ such that $\lim _{k \to \infty } n_{k} = \infty$ and the following inequalities hold for all $k\in \mathbb{N}.$

and

Similar to Eq (3.15), we have

Using Eqs (3.16), (3.42) and condition $\lim _{k \to \infty } \gamma_{n_{k}} = 0$ in Eq (3.44), we obtain

Similarly as in Case 1, we have

Using Eq (3.12), we have

Using Eq (3.42) in Eq (3.46), we obtain

As $p_{n_{k}} > 0,$ so Eq (3.47) implies $s_{n_{k}}\le t_{n_{k}},$ that is, we have

Similar to Case 1, we obtain $\limsup\limits_{n\rightarrow \infty} t_{n_{k}}\le 0.$ So, we have by using Lemma (2.6),

From Eq (3.43), we get $\zeta_{k} \rightarrow \zeta^{*}$ as $k\rightarrow \infty.$ This completes the proof.

4.   Numerical example

In this section, we provide a numerical example to show the numerical efficiency of the proposed algorithm. We performed all numerical experiments on a Dell computer equipped with a 3.20 GHz Intel (R) Core (TM) i5-3470 CPU and 8 GB of memory. The MATLAB R 2022 a platform was used as the implementation environment.

Let $H = \mathbb{R}^{3}.$ Define the operators $h, A, B, M, S_{i}: H\rightarrow H$ as $h(\zeta) = \frac{\zeta}{2}, A(\zeta) = (5\zeta_{1}, 4 \zeta_{2}, 5\zeta_{3}), B = (4\zeta_{1}, 4\zeta_{2}, 4\zeta_{3}), M = A, S_{i} \zeta = -3\zeta$ for $i = 0, 1.$ for all $\zeta = (\zeta_{1}, \zeta_{2}, \zeta_{3})\in H.$ Clearly, $M$ is self-adjoint, bounded and positive operator, $A$ is $M$-cocoercive operator, $B$ is maximally monotone operator, $h$ is $\frac{1}{2}$-contraction operator with respect to $M$-norm and $S_{i}$ is a finite family of $\frac{1}{2}$-demimetric operators. To find the numerical values of $\zeta_{n},$ we choose $\alpha_{n} = \frac{1}{2n}, \lambda = 0.5, q_{n} = 0.1, \beta_{n} = \frac{1}{2n}, \gamma_{n} = \frac{1}{n}, \delta_{n} = \frac{n}{2(n+1)}, \epsilon_{n} = 0.8.$ We compare our algorithm (3.1) with algorithms (1.4) and (1.5) with different choices of $\zeta_{0}$ and $\zeta_{1}.$ We consider three cases:

Case 1. $\zeta_{0} = [20, 20, 20]$ and $\zeta_{1} = [10, 10, 10]$.

Case 2. $\zeta_{0} = [10, 10, 10]$ and $\zeta_{1} = [5, 5, 5]$.

Case 3. $\zeta_{0} = [-1, -1, -1]$ and $\zeta_{1} = [-2, -2, -2].$

The values of iteration number (n) for various choices of $\zeta_{0}$ and $\zeta_{1}$ are given in the Tables 1–3. Comparison of Algorithm (3.1) by taking $E_{n} = \|\zeta_{n}-\zeta_{n-1}\| < 10^{-4}$ with the algorithms Dixit et al. (1.4) and Lorenz and Pock (1.5) for Cases 1–3 are shown in Figures 1–3 respectively.

5.   Conclusions

We introduced an algorithm for finding the common solution of monotone inclusion and fixed point problem of a finite family of demimetric mappings in a real Hilbert space and showed that our algorithm has strong convergence under some conditions. Moreover, we proved the algorithm has a better rate of convergence by giving a numerical example.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author Anjali delightedly acknowledge the University Grants Commission (UGC), New Delhi and the authors S. Haque and N. Mlaiki would like to thank the Prince Sultan University for paying the publication fees for this work through TAS LAB.

Conflict of interest

The authors declare no conflicts of interest.