Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory

1.   Introduction

Mathematical modelling provides a systematic formalism for the understanding of the corresponding real-world problem. Moreover, adequate mathematical tools for the analysis of the translated real-world problem are at our disposal. Fixed point theory (FPT), an important branch of nonlinear functional analysis, is prominent for modelling a variety of real-world problems. It is worth mentioning that the real-world phenomenon can be translated into well known existential as well as computational FPP.

The EP theory provides an other systematic formalism for modelling the real-world problems with possible applications in optimization theory, variational inequality theory and game theory ^{[7,10,13,17,18,19,21,25,28,31,32]}. In 1994, Blum and Oettli ^[13] proposed the (monotone-) EP in Hilbert spaces. Since then various classical iterative algorithms are employed to compute the optimal solution of the (monotone-) EP and the FPP. It is remarked that the convergence characteristic and the speed of convergence are the principal attributes of an iterative algorithm. All the classical iterative algorithms from FPT or EP theory have a common shortcoming that the convergence characteristic occurs with respect to the weak topology. In order to enforce the strong convergence characteristic, one has to assume stronger assumptions on the domain and/or constraints. Moreover, strong convergence characteristic of an iterative algorithm is often more desirable than weak convergence characteristic in an infinite dimensional framework.

The efficiency of an iterative algorithm can be improved by employing the inertial extrapolation technique ^[29]. This technique has successfully been combined with the different classical iterative algorithms; see e.g., ^{[2,3,4,5,6,8,9,14,15,16,23,27]}. On the other hand, the parallel architecture of the algorithm helps to reduce the computational cost.

In 2006, Tada and Takahashi ^[33] suggested a hybrid framework for the analysis of monotone EP and FPP in Hilbert spaces. On the other hand, the iterative algorithm proposed in ^[33] fails for the case of pseudomonotone EP. In order to address this issue, Anh ^[1] suggested a hybrid extragradient method, based on the seminal work of Korpelevich ^[24], to address the pseudomonotone EP together with the FPP. Inspired by the work of Anh ^[1], Hieu et al. ^[21] suggested a parallel hybrid extragradient framework to address the pseudomonotone EP together with the FPP associated with nonexpansive operators.

Inspired and motivated by the ongoing research, it is natural to study the pseudomonotone EP together with the FPP associated with the class of an $\eta$-demimetric operators. We therefore, suggest some variants of the classical Mann iterative algorithm ^[26] and the Halpern iterative algorithm ^[20] in Hilbert spaces. We formulate these variants endowed with the inertial extrapolation technique and parallel hybrid architecture for speedy strong convergence results in Hilbert spaces.

The rest of the paper is organized as follows. We present some relevant preliminary concepts and useful results regarding the pseudomonotone EP and FPP in Section 2. Section 3 comprises strong convergence results of the proposed variants of the parallel hybrid extragradient algorithm as well as Halpern iterative algorithm under suitable set of constraints. In Section 4, we provide detailed numerical results for the demonstration of the main results in Section 3 as well as the viability of the proposed variants with respect to various real-world applications.

2.   Preliminaries

Throughout this section, the triplet $(\mathcal{H}, < \cdot, \cdot > , \|\cdot\|)$ denotes the real Hilbert space, the inner product and the induced norm, respectively. The symbolic representation of the weak and strong convergence characteristic are $\rightharpoonup$ and $\rightarrow$, respectively. Recall that a Hilbert space satisfies the Opial's condition, i.e., for a sequence $(p_{k})\subset \mathcal{H}$ with $p_{k}\rightharpoonup \nu$ then the inequality $\liminf_{k \rightarrow \infty}\Vert p_{k}-\nu\Vert < \liminf_{k \rightarrow \infty}\Vert p_{k}-\mu\Vert$ holds for all $\mu \in \mathcal{H}$ with $\nu \neq \mu$. Moreover, $\mathcal{H}$ satisfies the the Kadec-Klee property, i.e., if $p_{k}\rightharpoonup \nu$ and $\Vert p_{k}\Vert \rightarrow \Vert \nu \Vert$ as $k \rightarrow \infty$, then $\Vert p_{k}-\nu\Vert\rightarrow 0$ as $k \rightarrow \infty$.

For a nonempty closed and convex subset $K\subseteq \mathcal{H}$, the metric projection operator $\Pi^{\mathcal{H}}_{K}:\mathcal{H}\rightarrow K$ is defined as $\Pi^{\mathcal{H}}_{K}(\mu) = argmin_{\nu \in K}\Vert \mu-\nu\Vert$. If $T:\mathcal{H} \rightarrow \mathcal{H}$ is an operator then $Fix(T) = \{\nu \in \mathcal{H}|\nu = T\nu\}$ represents the set of fixed points of the operator $T$. Recall that the operator $T$ is called $\eta$-demimetric (see ^[35]) where $\eta \in (-\infty, 1)$, if $Fix(T) \neq \emptyset$ and

The above definition is equivalently represented as

Recall also that a bifunction $g:K\times K\rightarrow \mathbb{R}\cup \{+\infty \}$ is coined as (ⅰ) monotone if $g(\mu, \nu)+g(\nu, \mu)\leq 0, \ {\rm{ for\ all }}\ \mu, \nu\in K;$ and (ⅱ) strongly pseudomonotone if $g(\mu, \nu)\geq 0\Rightarrow g(\nu, \mu)\leq -\alpha \Vert \mu-\nu\Vert ^{2}, \; \; {\rm{for\ all }}\ \mu, \nu\in K$, where $\alpha > 0$. It is worth mentioning that the monotonicity of a bifunction implies the pseudo-monotonicity, but the converse is not true. Recall the EP associated with the bifunction $g$ is to find $\mu \in K$ such that $g(\mu, \nu)\geq 0$ for all $\nu \in K$. The set of solutions of the equilibrium problem is denoted by $EP(g)$.

Assumption 2.1. ^[12,13] Let $g:K\times K\rightarrow \mathbb{R}\cup \{+\infty \}$ bea bifunction satisfying the following assumptions:

(A1) $g$ is pseudomonotone, i.e., $g(\mu, \nu)\geq 0\Rightarrow g(\mu, \nu)\leq 0, \ {{for\ all}}\; \mu, \nu\in K$;

(A2) $g$ is Lipschitz-type continuous, i.e., there exist two nonnegativeconstants $d_{1}, d_{2}$ such that

(A3) $g$ is weakly continuous on $K\times K$ imply that, if $\mu, \nu \in K$ and $(p_{k})$, $(q_{k})$ are two sequences in $K$ such that $p_{k}\rightharpoonup \mu$ and $q_{k}\rightharpoonup \nu$ respectively, then $f(p_{k}, q_{k}) \rightarrow f(\mu, \nu)$;

(A4) For each fixed $\mu\in K$, $g(\mu, \cdot)$ is convex and subdifferentiable on $K$.

In view of the Assumption 2.1, $EP(g)$ associated with the bifunction $g$ is weakly closed and convex.

Let $g_{i}:K\times K\rightarrow \mathbb{R}\cup \{+\infty \}$ be a finite family of bifunctions satisfying Assumption 2.1. Then for all $i \in \{1, 2, \cdots, M\}$, we can compute the same Lipschitz coefficients $(d_{1}, d_{2})$ for the family of bifunctions $g_{i}$ by employing the condition (A2) as

where $d_{1} = \max_{1\leq i \leq M}\{d_{1, i}\}$ and $d_{2} = \max_{1\leq i \leq M}\{d_{2, i}\}$. Therefore, $g_{i}(\mu, \nu)+g_{i}(\nu, \xi)\geq g_{i}(\mu, \xi)-d_{1}\Vert \mu-\nu \Vert^{2}-d_{2}\Vert \nu-\xi\Vert^{2}$. In addition, we assume $T_{j}:\mathcal{H} \rightarrow \mathcal{H}$ to be a finite family of $\eta$-demimetric operators such that $\Gamma: = \big(\bigcap^{M}_{i = 1}EP(g_{i})\big)\cap \big(\bigcap^{N}_{j = 1}Fix(T_{j})\big) \neq \emptyset$. Then we are interested in the following problem:

Lemma 2.2. ^[11] Let $\mu, \nu\in \mathcal{H}$ and $\beta \in \mathbb{R}$ then

(1) $\Vert \mu+\nu\Vert^{2} \leq \Vert \mu \Vert^{2}+2 \langle \nu, \mu+\nu \rangle$;

(2) $\Vert \mu-\nu\Vert^{2} = \Vert \mu \Vert^{2}-\Vert \nu \Vert^{2}-2 \langle\mu-\nu, \nu \rangle$;

(3) $\|\beta\mu+(1-\beta)\nu\|^{2} = \beta\|\mu\|^{2}+(1-\beta)\|\nu\|^{2}-\beta(1-\beta)\|\mu-\nu\|^{2}.$

Lemma 2.3. ^[35] Let $T:K\rightarrow \mathcal{H}$ be an $\eta$-demimetric operator defined on a nonempty, closed and convex subset $K$ of a Hilbert space $\mathcal{H}$ with $\eta \in (-\infty, 1)$. Then $Fix(T)$ is closed and convex.

Lemma 2.4. ^[36] Let $T:K\rightarrow \mathcal{H}$ be an $\eta$-demimetric operator defined on a nonempty, closed and convex subset $K$ of a Hilbert space $\mathcal{H}$ with $\eta \in (-\infty, 1)$. Then the operator $L = (1-\gamma)Id+\gamma T$ is quasi-nonexpansive provided that $Fix(T)\neq \emptyset$ and $0 < \gamma < 1-\eta$.

Lemma 2.5. ^[11] Let $T:K \rightarrow K$ be a nonexpansive operator defined on a nonempty closed convex subset $K$ of a real Hilbert space$\mathcal{H}$ and let $(p_{k})$ be a sequence in $K$. If $p_{k}\rightharpoonup x$ and if $(Id-T)p_{k}\rightarrow 0$, then $x \in Fix(T)$.

Lemma 2.6. ^[37] Let $h:K\rightarrow \mathbb{R}$ be aconvex and subdifferentiable function on nonempty closed and convex subset $K$ of a real Hilbert space $\mathcal{H}$. Then, $p_{\ast}$ solves the $\min \{h(q):q\in K\}$, if and only if $0\in\partial h(p_{\ast})+N_{K}(p_{\ast})$, where $\partial h(\cdot)$ denotes the subdifferential of $h$ and $N_{K}(\bar{p})$ is the normal cone of $K$ at $\bar{p}$.

3.   Algorithm and convergence analysis

Our main iterative algorithm of this section has the following architecture:

Theorem 3.1. Let the following conditions:

${\rm(C1)}$ $\sum^{\infty}_{k = 1}\xi_{k}\|p_{k}-p_{k-1}\| < \infty$;

${\rm(C2)}$ $0 < a^{\ast} \leq \gamma_{k} \leq min\{1-\eta_{1}, \cdots, 1-\eta_{N}\}$,

hold. Then Algorithm 1 solves the problem 2.1.

The following result is crucial for the strong convergence result of the Algorithm 1.

Lemma 3.2. ^[1,30] Suppose that $\nu_{\ast} \in EP(g_{i})$, and $p_{k}$, $e_{k}$, $u_{i, k}$, $v_{i, k}$, $i \in \{1, 2, \cdots, M\}$ are defined in Step 1 of the Algorithm 1. Then we have

Proof of Theorem 3.1.

Step 1. The Algorithm 1 is stable.

Observe the following representation of the set $C_{k+1}$:

This infers that $C_{k+1}$ is closed and convex for all $k \geq 1$. It is well-known that $EP(g_{i})$ and $Fix(T_{j})$ (from the Assumption 2.1 and Lemma 2.3, respectively) are closed and convex. Hence $\Gamma$ is nonempty, closed and convex. For any $p_{\ast}\in \Gamma$, it follows from Algorithm 1 that

From (3.1) and recalling Lemma 2.4, we obtain

Now recalling Lemma 3.2, the above estimate implies that

The above estimate (3.2) infers that $\Gamma \subset C_{k+1}$. It is now clear from these facts that the Algorithm 1 is well-defined.

Step 2. The limit $\lim_{k\rightarrow \infty }\Vert p_{k}-p_{1}\Vert$ exists.

From $p_{k+1} = \Pi^{\mathcal{H}}_{C_{k+1}}p_{1},$ we have $\langle p_{k+1}-p_{1}, p_{k+1}-\nu \rangle \leq 0$ for each $\nu \in C_{k+1}$. In particular, we have $\langle p_{k+1}-p_{1}, p_{k+1}-p_{\ast} \rangle \leq 0$ for each $p_{\ast}\in \Gamma$. This proves that the sequence $(\Vert p_{k}-p_{1}\Vert)$ is bounded. However, from $p_{k} = \Pi^{\mathcal{H}_{1}}_{C_{k}}p_{1}$ and $p_{k+1} = \Pi^{\mathcal{H}_{1}}_{C_{k+1}}p_{1} \in C_{k+1},$ we have that

This infers that $(\Vert p_{k}-p_{1}\Vert)$ is nondecreasing and hence

Step 3. $\tilde{p_{\ast}}\in \Gamma.$

Compute

Utilizing (3.3), the above estimate infers that

Recalling the definition of $(e_{k})$ and the condition (${\rm{C1}}$), we have

Recalling (3.4) and (3.5), the following relation

infers that

Note that $p_{k+1}\in C_{k+1}$, therefore the following relation

infers, on employing (3.4) and the condition (C1), that

Again, recalling (3.4) and (3.7), the following relation

infers that

In view of the condition (C2), observe the variant of (3.2)

Recalling (3.8) and condition (C1), we get

The above estimate (3.9) implies that

Reasoning as above, recalling (3.5), (3.8) and (3.10), we have

● $\Vert \bar{v}_{k}-e_{k}\Vert \leq\Vert \bar{v}_{k}-u_{i_{k}, k}\Vert+\Vert u_{i_{k}, k}-e_{k}\Vert \rightarrow 0$;

● $\Vert \bar{v}_{k}-p_{k}\Vert \leq\Vert \bar{v}_{k}-e_{k}\Vert+\Vert e_{k}-p_{k}\Vert \rightarrow 0$;

● $\Vert {w}_{k}-e_{k}\Vert \leq\Vert {w}_{k}-p_{k}\Vert+\Vert p_{k}-e_{k}\Vert \rightarrow 0$;

● $\Vert {w}_{k}-\bar{v}_{k}\Vert \leq\Vert {w}_{k}-e_{k}\Vert+\Vert e_{k}-\bar{v}_{k}\Vert \rightarrow 0$.

In view of the estimate $\lim_{k \rightarrow \infty}\Vert {w}_{k}-\bar{v}_{k}\Vert = 0$, we have

Next, we show that $\tilde{p_{\ast}} \in \bigcap^{M}_{i = 1}EP(g_{i})$.

Observe that

Recalling Lemma 2.6, we get

This implies the existence of $\tilde{x} \in \partial_{2} g_{i}(e_{k}, u_{i, k})$ and $\tilde{x_{*}} \in N_{K}(u_{i, k})$ such that

Since $\tilde{x_{*}} \in N_{K}(u_{i, k})$ and $\langle \tilde{x_{*}}, \nu-u_{i, k} \rangle \leq 0$ for all $\nu \in K$. Therefore recalling (3.12), we have

Since $\tilde{x} \in \partial_{2}g_{i}(e_{k}, u_{i, k})$,

Therefore recalling (3.13) and (3.14), we obtain

Observe from the fact that $(p_{k})$ is bounded then $p_{k_{t}} \rightharpoonup \tilde{p_{\ast}} \in \mathcal{H}$ as $t \rightarrow \infty$ for a subsequence $(p_{k_{t}})$ of $(p_{k})$. This also infers that $\bar{w}_{k_{t}} \rightharpoonup \tilde{p_{\ast}}$, $\bar{v}_{k_{t}} \rightharpoonup\tilde{p_{\ast}}$ and $b_{k_{t}} \rightharpoonup \tilde{p_{\ast}}$ as $t \rightarrow \infty$. Since $e_{k} \rightharpoonup \tilde{p_{\ast}}$ and $\Vert e_{k}-u_{i, k}\Vert \rightarrow 0$ as $k \rightarrow \infty$, this implies $u_{i, k} \rightharpoonup \tilde{p_{\ast}}$. Recalling the assumption (A3) and (3.15), we deduce that $g_{i}(\tilde{p_{\ast}}, \nu) \geq 0$ for all $\nu \in K$ and $i \in \{1, 2, \cdots, M\}$. Therefore, $\tilde{p_{\ast}} \in \bigcap^{M}_{i = 1}EP(g_{i})$. Moreover, recall that $\bar{v}_{k_{t}} \rightharpoonup \tilde{p_{\ast}}$ as $t \rightarrow \infty$ and (3.11) we have $\tilde{p_{\ast}} \in \bigcap^{N}_{j = 1}Fix(T_{j})$. Hence $\tilde{p_{\ast}} \in \Gamma$.

Step 4. $p_{k}\rightarrow p_{\ast} = \Pi^{\mathcal{H}}_{\Gamma }p_{1}.$

Since $p_{\ast} = \Pi^{\mathcal{H}}_{\Gamma}p_{1}$ and $\tilde{p_{\ast}} \in \Gamma$, therefore we have $p_{k+1} = \Pi^{\mathcal{H}}_{C_{k+1}}p_{1}$ and $p_{\ast}\in \Gamma \subset C_{k+1}$. This implies that

By recalling the weak lower semicontinuity of the norm, we have

Recalling the uniqueness of the metric projection operator yields that $\tilde{p_{\ast}} = p_{\ast} = \Pi^{\mathcal{H}}_{\Gamma }p_{1}$. Also $\lim_{t \rightarrow \infty}\Vert p_{k_{t}}-p_{1}\Vert = \Vert p_{\ast}-p_{1}\Vert = \Vert \tilde{p_{\ast}}-p_{1}\Vert$. Moreover, recalling the Kadec-Klee property of $\mathcal{H}$ with the fact that $p_{k_{t}}-p_{1}\rightharpoonup \tilde{p_{\ast}}-p_{1}$, we have $p_{k_{t}}-p_{1}\rightarrow \tilde{p_{\ast}}-p_{1}$ and hence $p_{k_{t}}\rightarrow \tilde{p_{\ast}}$. This completes the proof.

Corollary 3.3. Let $K \subseteq \mathcal{H}$ be a nonempty closed and convex subset of a real Hilbert space $\mathcal{H}$. For all $i \in \{1, 2, \cdots, M\}$, let $g_{i}:K\times K\rightarrow \mathbb{R}\cup \{+\infty \}$ be a finite family of bifunctions satisfying Assumption 2.1. Assume that $\Gamma: = \bigcap^{M}_{i = 1}EP(g_{i}) \neq \emptyset$, such that

Assume that the condition (C1) holds, then the sequence $(p_{k})$ generated by (3.16) strongly converges to a point in $\Gamma$.

We now propose an other variant of the hybrid iterative algorithm embedded with the Halpern iterative algorithm ^[20].

Remark 3.4. Note for the Algorithm 2 that the claim $p_{k}$ is a common solution of the EP and FPP provided that $p_{k+1} = p_{k}$, in general is not true. So intrinsically a stopping criterion is implemented for $k > k_{max}$ for some chosen sufficiently large number $k_{max}$.

Theorem 3.5. Let $\Gamma \neq \emptyset$ and the following conditions:

${\rm(C1)}$ $\sum^{\infty}_{k = 1}\xi_{k}\|p_{k}-p_{k-1}\| < \infty$;

${\rm(C2)}$ $0 < a^{\ast} \leq \gamma_{k} \leq min\{1-\eta_{1}, \cdots, 1-\eta_{N}\}$ and $\lim_{k \rightarrow \infty}\beta_{k} = 0$,

hold. Then the Algorithm 2 solves the problem 2.1.

Proof. Observe that the set $C_{k+1}$ can be expressed in the following form:

Recalling the proof of Theorem 3.1, we deduce that the sets $\Gamma \; {\rm{and }}\ C_{k+1}$ are closed and convex satisfying $\Gamma \subset C_{k+1}$ for all $k \geq 0$. Further, $(p_{k})$ is bounded and

Since $p_{k+1} = \Pi^{\mathcal{H}}_{C_{k+1}}(q) \in C_{k+1}$, we have

Recalling the estimate (3.17) and the conditions (C1) and (C2), we obtain

Reasoning as above, we get

The rest of the proof of Theorem 3.5 follows from the proof of Theorem 3.1 and is therefore omitted.

The following remark elaborate how to align condition (C1) in a computer-assisted iterative algorithm.

Remark 3.6. We remark here that the condition ${\rm(C1)}$ can easily be aligned in a computer-assisted iterative algorithm since the value of $\|p_{k}-p_{k-1}\|$ is quantified before choosing $\xi_{k}$ such that $0 \leq \xi_{k} \leq \widehat{\xi_{k}}$ with

Here $\{ \sigma_{k}\}$ denotes a sequence of positives $\sum^{\infty}_{k = 1}\sigma_{k} < \infty$ and $\xi \in [0, 1)$.

As a direct application of Theorem 3.1, we have the following variant of the problem 2.1, namely the generalized split variational inequality problem associated with a finite family of single-valued monotone and hemicontinuous operators $A_{j}: K \rightarrow \mathcal{H}$ defined on a nonempty closed convex subset $K$ of a real Hilbert space $\mathcal{H}$ for each $j \in \{1, 2, \cdots, N\}$. The set $VI(K, A)$ represents all the solutions of the following variational inequality problem $\langle A\mu, \nu-\mu \rangle \geq 0\; \; \forall \; \; \nu\; \in \; \; C$.

Theorem 3.7. Assume that $\Gamma = \bigcap^{M}_{i = 1}VI(C, A_{i})\cap \bigcap^{N}_{j = 1}Fix(T_{j}) \neq \emptyset$ and the conditions (C1)–(C4) hold. Then the sequence $(p_{k})$

generated by (3.18) solves the problem 2.1.

Proof. Observe that, if we set $g_{i}(\bar{\mu}, \bar{\nu}) = \langle A_{i}(\bar{\mu}), \bar{\nu}-\bar{\mu} \rangle$ for all $\bar{\mu}, \bar{\nu} \in K$, then each $A_{i}$ being $L$-Lipschitz continuous infers that $g_{i}$ is Lipschitz-type continuous with $d_{1} = d_{2} = \frac{L}{2}$. Moreover, the pseudo-monotonicity of $A_{i}$ ensures the pseudo-monotonicity of $g_{i}$. Recalling the assumptions (A3)–(A4) and the Algorithm 1, note that

can be transformed into

Hence recalling $g_{i}(\bar{\mu}, \bar{\nu}) = \langle A_{i}(\bar{\mu}), \bar{\nu}-\bar{\mu} \rangle$ for all $\bar{\mu}, \bar{\nu} \in K$ and for all $i \in \{1, 2, \cdots, M\}$ in Theorem 3.1, we have the desired result.

4.   Numerical experiment and results

This section provides the effective viability of the algorithm via a suitable numerical experiment.

Example 4.1. Let $\mathcal{H} = \mathbb{R}$ be the set of all real numbers with the inner product defined by $\langle p, q\rangle = pq,$ for all $p, q \in \mathbb{R}$ and the induced usual norm $|\cdot|$. For each $i = \{1, 2, \cdots, M\}$, let the family of pseudomonotone bifunctions $g_{i}(p, q): K\times K \rightarrow \mathbb{R}$ on $K = [0, 1] \subset \mathcal{H}$, is defined by $g_{i}(p, q) = S_{i}(p)(q-p)$, where

where $0 < \lambda_{1} < \lambda_{2} < ... < \lambda_{M} < 1$. Note that $EP(g_{i}) = [0, \lambda_{i}]$ if and only if $0 \leq p \leq \lambda_{i}$ and $q \in [0, 1]$. Consequently, $\bigcap^{M}_{i = 1}EP(g_{i}) = [0, \lambda_{1}]$. For each $j \in \{1, 2, \cdots, N\}$, let the family of operators $T_{j}: \mathbb{R} \rightarrow \mathbb{R}$ be defined by

Clearly, $T_{j}$ defines a finite family of $\eta$-demimetric operators with $\bigcap^{N}_{j = 1}Fix(T_{j}) = \{0\}$. Hence $\Gamma = (\bigcap^{M}_{i = 1}EP(g_{i})) \cap (\bigcap^{N}_{j = 1}Fix(T_{j})) = 0$. In order to compute the numerical values of the Algorithm 1, we choose $\xi = 0.5$, $\alpha_{k} = \frac{1}{100k+1}$, $\mu = \frac{1}{7}$, $\lambda_{i} = \frac{i}{(M+1)}$, $M = 2 \times 10^{5}$ and $N = 3 \times 10^{5}$. Since

Observe that the expression

in the Algorithm 1 is equivalent to the following relation $u_{i, k} = e_{k}-\mu S_{i}(e_{k}), \; \; {for\ all}\; \; i \in \{1, 2, \cdots, M\}.$ Similarly $v_{i, k} = e_{k}-\mu S_{i}(u{i, k}), \; \; {for\ all}\; \; i \in \{1, 2, \cdots, M\}. \label{4.1}$ Hence, we can compute the intermediate approximation $\bar{v}_{k}$ which is farthest from $e_{k}$ among $v_{i, k}$, for all $i \in \{1, 2, \cdots, M\}$. Generally, at the $k^{th}$ step if $E_{k} = \|p_{k}-p_{k-1}\| = 0$ then $p_{k}\in \Gamma$ implies that $p_{k}$ is the required solution of the problem. The terminating criteria is set as $E_{k} < 10^{-6}$. The values of the Algorithm 1 and its variant are listed in the following table (see Table 1):

The values of the non-inertial and non-parallel variant of the Algorithm 1 referred as Alg.$1^{\ast}$ are listed in the following table (see Table 2):

The error plotting $E_{k}$ against the Algorithm 1 and its variants for each choices in Tables 1 and 2 are illustrated in Figure 1.

Example 4.2. Let $\mathcal{H} = \mathbb{R}^{n}$ with the induced norm $\Vert p \Vert = \sqrt{\sum^{n}_{i = 1}\vert p_{i} \vert^{2}}$ and the inner product $\langle p, q \rangle = \sum^{n}_{i = 1}p_{i}q_{i}$, for all $p = (p_{1}, p_{2}, \cdots, p_{n}) \in \mathbb{R}^{n}$ and $q = (q_{1}, q_{2}, \cdots, q_{n}) \in \mathbb{R}^{n}$. The set $K$ is given by $K = \{p \in \mathbb{R}^{n}_{+}:\vert p_{k}\vert \leq 1\}$, where $k = \{1, 2, \cdots, n\}$. Consider the following problem:

where $g_{i}:K \times K \rightarrow \mathbb{R}$ is defined by:

where $S_{i, k} \in (0, 1)$ is randomly generated for all $i = \{1, 2, \cdots, M\}$ and $k = \{1, 2, \cdots, n\}$. For each $j \in \{1, 2, \cdots, N\}$, let the family of operators $T_{j}: \mathcal{H} \rightarrow \mathcal{H}$ be defined by

for all $p \in \mathcal{H}$. It is easy to observe that $\Gamma = \bigcap^{M}_{i = 1}EP(g_{i}) \cap \bigcap^{N}_{j = 1}Fix(T_{j}) = 0$. The values of the Algorithm 1 and its non-inertial variant are listed in the following table (see Table 3):

The values of the non-inertial and non-parallel variant of the Algorithm 1 referred as Alg.$1^{\ast}$ are listed in the following table (see Table 4):

The error plotting $E_{k}\leq 10^{-6}$ against the Algorithm 1 and its variants for each choices in Tables 3 and 4 are illustrated in Figure 2.

Example 4.3. Let $L^{2}([0, 1]) = \mathcal{H}$ with induced norm $\Vert p \Vert = (\int^{1}_{0}\vert p(s)\vert^{2} ds)^{\frac{1}{2}}$ and the inner product $\langle p, q \rangle = \int^{1}_{0}p(s)q(s)ds$, for all $p, q \in L^{2}([0, 1])$ and $s \in [0, 1]$. The feasible set $K$ is given by: $K = \{p \in L^{2}([0, 1]): \Vert p \Vert \leq 1\}$. Consider the following problem:

where $g_{i}(p, q)$ is defined as $\langle S_{i}p, q-p\rangle$ with the operator $S_{i}:L^{2}([0, 1]) \rightarrow L^{2}([0, 1])$ given by

Since each $g_{i}$ is monotone and hence pseudomonotone on $C$. For each $j \in \{1, 2, \cdots, N\}$, let the family of operators $T_{j}: \mathcal{H} \rightarrow \mathcal{H}$ be defined by

Then $T_{j}$ is a finite family of $\eta$-demimetric operators. It is easy to observe that $\Gamma = \bigcap^{M}_{i = 1}EP(g_{i}) \cap \bigcap^{N}_{j = 1}Fix(T_{j}) = 0$. Choose $M = 50$ and $N = 100$. The values of the Algorithm 1 and its non-inertial variant have been computed for different choices of $p_{0}$ and $p_{1}$ in the following table (see Table 5):

The values of the non-inertial and non-parallel variant of the Algorithm 1 referred as Alg.$1^{\ast}$ have been computed for different choices of $p_{0}$ and $p_{1}$ in the following table (see Table 6):

The error plotting $E_{k} = < 10^{-4}$ against the Algorithm 1 and its variants for each choices in Tables 5 and 6 are illustrated in Figure 3.

We can see from Tables 1–6 and Figures 1–3 that the Algorithm 1 out performs its variants with respect to the reduction in the error, time consumption and the number of iterations required for the convergence towards the common solution.

5.   Conclusions

In this paper, we have constructed some variants of the classical extragradient algorithm that are embedded with the inertial extrapolation and hybrid projection techniques. We have shown that the algorithm strongly converges towards the common solution of the problem 2.1. A useful instance of the main result, that is, Theorem 3.1, as well as an appropriate example for the viability of the algorithm, have also been incorporated. It is worth mentioning that the problem 2.1 is a natural mathematical model for various real-world problems. As a consequence, our theoretical framework constitutes an important topic of future research.

Conflict of interest

The authors declare that they have no competing interests.

Acknowledgments

The authors wish to thank the anonymous referees for their comments and suggestions.

The author Yasir Arfat acknowledge the support via the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi, Thailand (Grant No.16/2562).

The authors Y. Arfat, P. Kumam, W. Kumam and K. Sitthithakerngkiet acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this reserch was funded by King Mongkut's University of Technology North Bangkok, Contract No. KMUTNB-65-KNOW-28.