1. Introduction

The need for fast computational methods in different areas of application, such as weather prediction, fluid dynamics, image and signal processing, optimization problems, large-scale interconnected systems, etc., has always been of great interest (see, e.g. [Reference Bertsekas and Tsitsiklis12]). This is simply due to the large numerical computations that are involved. Most of the problems that arise in these areas of application can be decomposed in such a way that the computations can be handled simultaneously by more powerful processors. This further led to the continuation of research on the development of parallel algorithms and high-efficient computing machines with multiprocessors, see, for example, [Reference Geer29].

The convex feasibility problem (CFP): find $x\in C=\cap _{i=1}^{n} F(T_i)$, where $T_i:H\to H$, for each $i$, is an operator defined on a real Hilbert space $H$, is a central task in mathematics. Such problems arise in different areas of application, such as road design [Reference Bauschke and Koch10], medical image reconstruction [Reference Elsner, Koltracht and Neumann28], signal processing, optimization problems, radiation therapy [Reference Censor, Elfving, Kopf and Bortfeld20], to mention but a few. There are many algorithms in the literature for approximating solutions of CFPs. One of the most commonly used algorithms is the projection algorithms that employ projections onto the underlying set, see, e.g. [Reference Bauschke and Borwein8, Reference Bauschke and Combettes9, Reference Cegielski14, Reference Cegielski, Reich and Zalas15, Reference Censor and Cegielski17, Reference Censor and Zenios18, Reference Censor, Chen, Combettes, Davidi and Herman21, Reference Combettes24, Reference Combettes25, Reference Herman32, Reference Masad and Reich40] and the references therein.The Krasnoselkii–Mann algorithm is often used when the underlying set is a finite intersection of fixed point sets of some non-expansive-type mappings, see, e.g. [Reference Browder and Petryshyn13, Reference Halpern30, Reference Kim and Xu33, Reference Lions34, Reference Mann37, Reference Marino and Xu38, Reference Reich42, Reference Reich43]. In an attempt to speed up the computation of algorithms, several authors have developed parallel algorithms for CFPs, see, e.g. [Reference Aharoni and Censor2, Reference Bauschke, Combettes and Kruk11, Reference Censor16, Reference Censor, Gordon and Gordon19]. The parallel algorithms, which can be run in a multiprocessor machine is, without doubt, better than the non-parallel algorithms that run in a machine with a single processor. This is simply because, for multiprocessor machines, the task can be distributed among the processors. However, in a case where the output of the processors needs to be synchronized before computing the next iterate, the master processor has to wait for the slowest processor to provide its output. A typical case of this is when the convex combination of the output from each processor is to be taken before moving to the next step. In essence, in synchronous parallel iterative algorithms, some of the processors will be idle for some time at each step. Moreover, if there is a failure of transmission from one of the processors, then the master processor will have to resend the last output for recomputation thereby delaying other processors. One way of preventing the problems associated with the synchronous method is via asynchronization. In asynchronous-parallel iterative algorithms, there is no need to wait for other processors in order to compute the next iterate. This method allows flexibility in such a way that a processor can use out-of-date information to compute the next output, see e.g., [Reference Zhimin, Yangyang, Ming and Wotao47]. The general method of approximating CFPs via synchronous parallel iterative algorithm can be put in the following form

(1.1)\begin{equation} x^{n+1}=F_n(x^{n}), \quad x_0\in H, \ n\geq 1, \end{equation}

where $F_n$ is a sequence of operators. However, the asynchronous version of 1.1 takes, in general, the following form:

(1.2)\begin{equation} x^{n+1}=F_n(\tilde{x}^{n}), \quad x_0\in H, \ n\geq 1, \end{equation}

where $\tilde {x}^{n}$ is either $x^{n}$ or $x^{n-j}$ for some $j\in \mathbb {N}$, where $j$ is the delay amount of some processor. To fix ideas, suppose we have $p$ processors and we set

\[ J^{n}=(j_1^{n}, j_2^{n}, j_3^{n}, \ldots, j_p^{n})\in {\mathbb{N}}^{p} \]

to be the delay vector at each iteration step. Then, the $i^{th}$ component of the vector $J^{n}$ gives the delay amount of the $i^{th}$ processor. Moreover, suppose $\{i_n\}_{n\geq 1}\subset \mathbb {N}$, where $1\leq i_n\leq p$, is an index sequence that gives information on whose output will be used to compute $x^{n+1}$. Then $\tilde {x}^{n}$ is the last iterate sent to the $i_n^{th}$ processor, i.e., $\tilde {x}^{n}=x^{n-j_{i_n}^{n}}$. It is, however, worth mentioning that, while the asynchronous-parallel iterative algorithms have advantages over synchronous parallel iterative algorithms, they also have their own disadvantages. For instance, the asynchronous-parallel iterative algorithms are not easy to handle when it comes to convergence analysis and implementation compared to synchronous parallel iterative algorithms.

The asynchronous-parallel iterative algorithm was first introduced by Chazan and Miranker [Reference Chazan and Miranker23] in 1969 to solve system of linear equations. Since then, many authors have successfully apply the method to solve problems in different areas, such as optimization [Reference Liu and Wright35, Reference Liu, Wright, Ré, Bittorf and Sridhar36, Reference Tai and Tseng44, Reference Zhang and Kwok46], nonlinear problems [Reference Bahi, Miellou and Rhofir4, Reference Baudet7], differential equations [Reference Aharoni and Barak1, Reference Amitai, Averbuch, Israeli and Itzikowitz3, Reference Chau, Spiteri, Guivarch and Boisson22, Reference Donzis and Aditya27] to mention but a few. Baudet [Reference Baudet7], in 1978, used totally asynchronous parallel iterative algorithm (in which arbitrary amount of delay is allowed without any bound) for $p$-contractions (see, e.g. [Reference Zhimin, Yangyang, Ming and Wotao47]) for definition of $p$-contractions). In [Reference Tseng, Bertsekas and Tsitsiklis45], Tseng et al. considered partially asynchronous parallel iterative algorithm (where the delay vectors are bounded from above by some non-negative number, say, $\rho$) for quasi-non-expansive mappings (see preliminaries below for definition). Recently, Zhimin et al. introduced ARock algorithm for approximation of fixed points of non-expansive mappings in Hilbert spaces. As opposed to the methods in [Reference Baudet7] and [Reference Tseng, Bertsekas and Tsitsiklis45] which assign coordinates in a deterministic manner, the ARock method is stochastic. More precisely, the ARock algorithm is following:

Algorithm 1:

ARock: a framework for async-parallel coordinate updates

where $\mathcal {H}=H_1\times H_2\times H_3\ldots \times H_m$, $S_{i_k}{x}_k=(0,\,\ldots,\,0,\,(Sx)_{i_k},\,0,\,\ldots,\,0)$.

More recently, Heaton and Censor [Reference Heaton and Censor31] introduced the asynchronous inertial algorithm (ASI) for non-expansive mappings and proved weak convergence result via the monotonicity of a sequence that includes classical error added to the error introduced from using out-of-date iterates. Consider a collection of $m$ non-expansive operators $\{T_i\}_{i=1}^{m}$ on H with a common fixed point. Consider the CFP with $C=\cap _{i=1}^{m}F(T_i)$, $i\in \{1,\,2,\,3,\,\ldots,\,m\}$. For each $i\in \{1,\,2,\,3,\,\ldots,\,m\}$, set $S_i=I-T_i$, then the ASI algorithm is as follows:

Algorithm 2:

Asynchronous Inertial Algorithm

They proved the following theorem:

Theorem 1.2 Heaton and Censor [Reference Heaton and Censor31]

Let $\{x^{k}\}_{k\in \mathbb {N}}$ be a sequence generated by the ASI algorithm and suppose Assumption $1$ holds. If there is an $\epsilon >0$ such that

\[ 0<\epsilon\leq \lambda_k\leq \frac{1}{2\tau+1+\epsilon}, \ \ k\in \mathbb{N}, \]

then the sequence $\{x^{k}\}_{k\in \mathbb {N}}$ converges weakly to a common fixed point $x^{*}$ of the family $\{T_i\}_{i=1}^{m},$ i.e.,

\[ x^{k}\rightharpoonup x^{*}\in C=\cap_{i=1}^{m}F(T_i). \]

In this paper, we prove weak and strong convergence theorem (under additional mild condition) for the class of strict pseudo-contractions which properly contains the class of non-expansive operators considered in [Reference Heaton and Censor31]. More precisely, we prove the following theorem:

Theorem Let $T_i:H\longrightarrow H,\, \ i=1,\,2,\,3,\,\ldots,\,m$ be $k_i$-strictly pseudo-contractive mappings. Let $\{x_n\}$ be a sequence defined by $x^{0}\in H,$

\[ x^{n+1}= \begin{cases} x^{n}, & \leq \sup_{n\in \mathbb{N}}\|d^{n}\|_{\infty},\\ (1-\alpha_n)x^{n}-\alpha_n(T_{i_n}\tilde{x}^{n}+(x^{n}-\tilde{x}^{n})), & \text{otherwise}. \end{cases} \]

Suppose $\sup _{n\in \mathbb {N}}\|d^{n}\|_{\infty }\leq \rho$ for some $\rho \geq 1$ and that

\[ 0<\epsilon_0\leq \alpha_n\leq \displaystyle\frac{1-k}{2\rho+1+\epsilon_0}<1-k, \quad \forall \ n\in \mathbb{N}, \]

for some $\epsilon _0>0,$ where $k=\displaystyle \max _{1\leq i\leq m}k_i$. Then, the sequence $\{x_n\}$ converges weakly to some point $x^{*}\in C=\cap _{i=1}^{n}F(T_i)\neq \emptyset$.

This class of operators allows some flexibility when data come with some noise in which the operator involved may fail to be non-expansive. Furthermore, we present the application of the ASI algorithm to solving convex minimization problems and Hammerstein integral equations.