1 Introduction

Let \({{\mathbb {X}}}\) be a Hilbert space and \(K\subset {{\mathbb {X}}}\) be a nonempty closed convex bounded subset. For \(i=1, 2, 3\), suppose that \({\Phi _i}: K\rightarrow K\) is a nonexpansive-type operator with Lipschitz coefficient \(\theta _i\in [0, 1]\), that is, \(\left\| \Phi _i(x)-\Phi _i(y)\right\| \le \theta _i\left\| x-y\right\| \) for each \(x, y\in K\). In this paper, based on the iterative scheme (in short, JF) due to Ali et al. (2020), we introduce and investigate the following general semi-implicit (also named as implicit midpoint rule) approximation (in short, JFESD) with errors for \({\Phi _i}\) \((i=1, 2, 3)\):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\varrho _{n + 1}} = {\Phi _1} \left( \left( 1 - {r_n} \right) {\varsigma _n} + {r_n}{\Phi _1}(\varsigma _n) + {r_n}{d_n} \right) , \\ \displaystyle {\varsigma _n} = {\Phi _2} \left( \frac{{{\vartheta _n} + {\varsigma _n}}}{2} \right) + {e_n}, \\ \displaystyle {\vartheta _n}={\Phi _3} \left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi _3} \left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) +{h_n}, \end{array} \right. \end{aligned}$$
(1.1)

where \(\{r_n\},\{s_n\}\subseteq [0,1]\) are two real number sequences, and \(\{d_n\}\), \(\{e_n\}\) and \(\{h_n\}\) are three errors to take into account some possible inexact computations of the nonexpansive-type operator points, which satisfy hypothesis (H):

  1. (i)

    \(d_n=d_n^\prime +d_n^{\prime \prime }\) with \(\lim _{n\rightarrow \infty }\Vert d_n^\prime \Vert =0\) and \(\sum _{n=0}^\infty \Vert d_n^{\prime \prime }\Vert <\infty \);

  2. (ii)

    \(\lim _{n\rightarrow \infty } \Vert e_n\Vert =0 \);   (iii) \(\sum ^{\infty }_{n=0} \Vert h_n\Vert < \infty \).

Some special cases of the JFESD (1.1) can be found as follows:

  • (Case 1) While \(d_n, e_n, h_n\equiv 0\) for all \(n\in {{\mathbb {N}}}\), the JFESD (1.1) reduces to the following semi-implicit iteration (in short, JFSD) for three different nonexpansive-type operators:

    $$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \displaystyle {\varrho _{n + 1}} = {\Phi _1} \left( \left( 1-{r_n} \right) {\varsigma _n} + {r_n}{\Phi _1}(\varsigma _n) \right) , \\ \displaystyle {\varsigma _n} = {\Phi _2} \left( \frac{{{\vartheta _n} + {\varsigma _n}}}{2} \right) , \\ \displaystyle {\vartheta _n}={\Phi _3} \left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi _3} \left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) . \end{array} \right. \end{aligned}$$
    (1.2)

  • (Case 2) If \(\Phi _1=\Phi _2=\Phi _3=\Phi \), then the JFESD (1.1) is equivalent to a class of semi-implicit scheme (in short, JFES) with errors for a nonexpansive-type operator as follows:

    $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\varrho _{n + 1}} = {\Phi } \left( \left( 1 - {r_n} \right) {\varsigma _n} + {r_n}{\Phi }(\varsigma _n) + {r_n}{d_n} \right) , \\ \displaystyle {\varsigma _n} = {\Phi } \left( \frac{{{\vartheta _n} + {\varsigma _n}}}{2} \right) + {e_n}, \\ \displaystyle {\vartheta _n}={\Phi } \left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi } \left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) +{h_n}. \end{array} \right. \end{aligned}$$
    (1.3)

  • (Case 3) When \(d_n=e_n=h_n\equiv 0\) for each \(n\in {{\mathbb {N}}}\), the JFES (1.3) becomes the following new semi-implicit iterative process (in short, JFS) for a nonexpansive-type operator:

    $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\varrho _{n + 1}} = {\Phi } \left( \left( 1 - {r_n} \right) {\varsigma _n} + {r_n}{\Phi }(\varsigma _n) \right) , \\ \displaystyle {\varsigma _n} = {\Phi } \left( \frac{{{\vartheta _n} + {\varsigma _n}}}{2} \right) , \\ \displaystyle {\vartheta _n}={\Phi } \left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi } \left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) . \end{array} \right. \end{aligned}$$
    (1.4)

Remark 1.1

(i) In regard to nonexpansive-type operator \(\Phi _i\) for \(i=1,2,3\) in (1.1)–(1.4), one can see that \(\Phi _i\) is nonexpansive if the Lipschitz coefficient \(\theta _i=1\); and when \(\theta _i\in [0, 1)\), \(\Phi _i\) becomes a contraction operator. Furthermore, if \(\theta _i>0\) for \(i=1,2,3\), then \(\Phi _i\) is called \(\theta _i\)-Lipschitzian continuous. In other words, for \(i=1,2,3\), nonexpansive-type operator with Lipschitz coefficient \(\theta _i\) is said to be \(\theta _i\)-Lipschitz continuous operator, here \(\theta _i\le 1\).

(ii) We note that the above iterative processes (1.1)–(1.4) are brand new and not reported in the literature.

Example 1.1

It is well known that the projection operator \(P_K: {{\mathbb {R}}}^n\rightarrow {{\mathbb {R}}}^n\) defined by

$$\begin{aligned} P_K(x)=\arg \min _{z\in K} \Vert z-x\Vert _2^2, \end{aligned}$$
(1.5)

is a nonexpansive-type operator with Lipschitz coefficient 1 and it is widely used to solve variational inequality, which is an important branch of optimization and was introduced in the early sixties by the study of mechanics.

Remark 1.2

For some \(i=1,2,3\), when \(\Phi _i=P_K\), the projection operator in Example 1.1, it is easy to see that the iterations (1.1)–(1.4) are still different from Algorithms 2.2–2.6 in Noor (2007).

As we all know, research of nonexpansive-type operators has a long history, one of the most important fields is to find fixed points via applying fixed point theory. In fact, many problems can be formulated as fixed point models or can be solved by fixed point theory. For example, the whole world has been profoundly impacted by the novel coronavirus since 2019 and it is imperative to depict the spread of the coronavirus. Panda et al. (2021) applied fractional derivatives to improve the 2019-nCoV/SARS-CoV-2 models, and by means of fixed point theory, existence and uniqueness of solutions of the models were proved. More applications of fixed point theory, such as equations, image process and other optimization problems, one can refer to Cacciapaglia and Sannino (2021); Ali et al. (2022); Panda et al. (2020); Harker and Pang (1990); Ali et al. (2020); Maldar (2021); Hanjing and Suantai (2020) and the reference therein.

In 1922, S. Banach first used the Picard iteration method to create a fixed point theorem in metric spaces, that is the famous Banach contraction principle. After that, many scholars introduced a lot of iteration methods, such as Mann iteration (in short, MANN) (Mann 1953), Ishikawa iterative process (in short, ISHIKAWA) (Ishikawa 1974), Noor three-step iterative approximation scheme (in short, NOOR) (Noor 2007; Noor and Yao 2007), novel fixed point algorithm (in short, SAKURAI) (Sakurai and Iiduka 2014) and so on, to approximate fixed points of various nonexpansive-type operators and obtained others fixed point theorems, see (Thakur et al. 2016) and the reference therein. Recently, to answer the question: Is it possible to define a new iterative scheme, whose rate of convergence is better than that of some known and leading iterative schemes, for generalized nonexpansive-type operators due to Hardy and Rogers? Ali et al. (2020) considered the JF as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle x_{n+1} = \Phi \left( (1-r_n)y _n + r_n \Phi (y_n)\right) , \\ \displaystyle y_n = \Phi (z _n), \\ \displaystyle z_n = \Phi \left( (1-s_n)x_n + s_n \Phi (x_n)\right) , \end{array} \right. \end{aligned}$$
(1.6)

where \(r_n,s_n \in (0,1)\) are two sequences, and studied the conditions of weak and strong convergence for (1.6) to a generalized self-map \(\Phi \) on a nonempty closed convex subset of a Banach space. The numerical examples in Ali et al. (2020) indicate that the JF (1.6) is much faster than some existing algorithms.

Remark 1.3

It is worth noting that the semi-implicit terms in the second and third equations of the JFS (1.4) replace the corresponding terms of the JF (1.6).

On the one hand, to solve nonlinear problems arising in mechanics, economics, optimization, differential equations and others mathematics and engineering problems, variational inequalities mentioned in Example 1.1 have become efficient tools and have been paid great attention by many scholars. See, for example, Daniele et al. (2003); Harker and Pang (1990); Noor and Yao (2007); Phannipa and Atid (2021); Noor (2007) and the reference therein.

In 1964, Stampacchia (1964) considered the following variational inequality problem of finding a point \(u \in K\) such that

$$\begin{aligned} \langle F(u), v-u \rangle \ge 0, \quad \forall v\in K, \end{aligned}$$
(1.7)

where K is a nonempty, closed and convex set of a real Hilbert space \({\mathbb {X}}\), and \(\langle \cdot ,\cdot \rangle \) is the inner product in \({\mathbb {X}}\), \(F: K \rightarrow K\) is a specific nonlinear operator. The set of solutions for the variational inequality (1.7) is denoted by VI(K, F). As everyone knows that the problem (1.7) is equivalent to the following fixed point problem:

$$\begin{aligned} u=P_K(u -\kappa F(u)), \end{aligned}$$
(1.8)

here \(\kappa \) is a positive constant and \(P_K\) is the projection operator formulated as in (1.5). There are many methods to solve the inequality problems of the form (1.7) (see Harker and Pang 1990), which include excellent numerical behaviors based on projection and contraction operator techniques.

For seeking common fixed points of nonexpansive-type operators and variational inequalities, Noor and Yao (2007) introduced some iterative schemes with projection operators and proved convergence of the iterative algorithms. To get strong convergence in more weak conditions, Phannipa and Atid (2021) introduced an iteration scheme and showed that the iteration scheme strongly converges to a common fixed point of nonexpansive-type operators and variational inequalities in Hilbert spaces. Very recently, based on the fundamental relation between variational inequality and fixed point problem presented in (1.8), Maldar (2021) reformulated the NOOR, JF (1.6) and some other iterative algorithms to approximate the common fixed points of generalized nonexpansive-type mappings and solutions of variational inequality problems.

On the other hand, the semi-implicit rule is a powerful numerical method for solving ordinary differential equations (Deuflhard 1985), differential algebraic equations (Schneider 1993). In addition, as an approximation method, some authors applied semi-implicit rule to iterative scheme of nonexpansive-type operators. Alghamdi et al. (2014) established a semi-implicit rule for nonexpansive-type operators and proved weak convergence of the iteration in Hilbert spaces, and used the algorithm to solve a nonlinear time-dependent evolution equation. Based on the semi-implicit rule, Luo and Cai (2017) and Aibinu et al. (2018) developed viscosity algorithms for nonexpansive-type operators and stated that these algorithms can strongly converge to a fixed point in smooth Banach spaces.

While calculating every iteration, errors will occur naturally, so it is worth of studying convergence of algorithms with errors. Liu (1995) first established ISHIKAWA and MANN with errors and proved that the iterations strongly converge to the unique solution of the accretive operator equations in Banach spaces. Following the work of Liu (1995), Chang et al. (2003) studied ISHIKAWA with mixed errors of nonexpansive-type operators, and found necessary and sufficient conditions for the iterative sequence to strongly converge to a fixed point in Banach spaces. Ni and Yao (2015) constructed a modified ISHIKAWA with errors for nonexpansive-type operator, and obtained strong convergence of the iterative algorithm in reflexive Banach spaces under suitable conditions.

By the importance of stable and unstable equilibria are really different in dynamics systems, the stability of equilibrium points should be investigated seriously. Lemaire (1996) may be the first one to investigate the stability of the iteration method for nonexpansive-type operators. In 2017, Alber (2017) proved weak and strong convergence and stability of some iteration schemes with perturbations in a Banach space. Lately, Li and Lan (2019) proposed convergence and stability analysis of new Picard–Mann iteration processes with errors for the semi-implicit rules of two different nonexpansive-type operators, and proved iterative approximation of solutions for a class of optimal control problems with elliptic boundary value constraint. Furthermore, Li and Lan (2019) gave numerical examples to show that the new Picard–Mann iteration process with mixed errors is more effective than the Picard–Mann semi-implicit iteration (in short, PMMI) with mixed errors, the Picard–Mann semi-implicit iterative process (in short, PMI) and other related iterative processes for a nonexpansive-type operator. And then, Ali et al. (2021) defined Picard’s three-step iteration for approximating fixed points of Zamfirescu operators which contain contraction mappings, and proved the iteration is almost T-stable and compared convergence of the three-step iteration with that of some leading iterative processes.

On the basis of Ali et al. (2020), Li and Lan (2019) and other work mentioned above, to obtain a fast and stable iterative scheme of nonexpansive-type operators and to solve the variational inequality (1.7), we explored the JFESD (1.1) and its special case (1.2)–(1.4). Then, we analyze convergence and stability of the sequences generated by (1.1)–(1.4) in Hilbert spaces, and obtain convergence rate of the schemes (1.1)–(1.4). Finally, we provide some numerical examples to illustrate that our schemes are more effective than other aforesaid known iterative schemes and apply the general semi-implicit approximation (1.1) to solve Stampacchia variational inequality (1.7).

2 Main results

In this section, we introduce a useful definition and an important lemma, and prove convergence and stability of the JFESD (1.1). Furthermore, using our iteration scheme JFESD (1.1), an integral equation as a numerical example shall be solved.

Definition 2.1

Let \(({{\mathbb {X}}},d)\) be a metric space, \(\Phi : {{\mathbb {X}}} \rightarrow {{\mathbb {X}}}\) be a given operator and \(\varrho _0 \in {{\mathbb {X}}}\). Assume that the iteration scheme \(\{\varrho _n\}\) generated by

$$\begin{aligned} \varrho _{n+1}=f(\Phi ,\varrho _n), \end{aligned}$$

converges to a fixed point u of \(\Phi \), i.e., \(u\in Fix(\Phi ): =\left\{ x\in {{\mathbb {X}}}\vert \, \Phi (x)=x\right\} \). Taking \(\{\varsigma _n\}\) be an arbitrary sequence in \({{\mathbb {X}}}\) and setting \(\epsilon _n = d\left( \varsigma _{n+1}, f(\Phi ,\varsigma _n)\right) \) for \(n\in {{\mathbb {N}}}\), then the fixed point iteration procedure \(\{\varrho _n\}\) is called \(\Phi \)-stable or stable with respect to \(\Phi \), and \(\lim _{n \rightarrow \infty } \epsilon _n= 0\) if and only if \(\lim _{n \rightarrow \infty } \varsigma _n = u\).

Lemma 2.1

(Liu 1995) Let \(\{a_n\},\{b_n\}\) and \(\{c_n\}\) be three nonnegative real sequences meeting:

$$\begin{aligned} a_{n+1} \le (1-t_n)a_n + b_n + c_n \end{aligned}$$

with \(t_n\in [0,1]\), \(\sum ^\infty _{n=0}t_n=\infty \), \(b_n=o(t_n)\), and \(\sum ^\infty _{n=0}c_n<\infty \). Then \(\lim _{n \rightarrow \infty }a_n=0\).

In the sequel, we give convergence and stability analysis of the JFESD (1.1) via using Lemma 2.1 and Definition 2.1.

Theorem 2.1

Let K be a nonempty closed convex bounded subset of Hilbert space \({{\mathbb {X}}}\). If for \(i=1,2,3\), \(\Phi _i: K \rightarrow K\) is a nonexpansive-type operator with Lipschitz coefficient \(\theta _i\in [0,1]\) satisfying \(\sum _{i=1}^{3}\theta _i < 3\) and \(Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3) \ne \emptyset \),

then the following statements hold:

  • (i) The iterative sequence \(\{\varrho _n\}\) generated by JFESD (1.1) converges to \(\varrho ^* \in Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3)\) with convergence rate for every step (i.e., \(n\in {\mathbb {N}}\)):

    $$\begin{aligned} \rho _n= {\theta _1}{\theta _2}{\theta _3}\cdot \frac{\left[ 1-{r_n}(1-{\theta _1})\right] \left[ 1-{s_n}(1-{\theta _3})\right] }{(2-{\theta _2})\left\{ 2-{\theta _3}\left[ 1-{s_n}(1-{\theta _3}) \right] \right\} }, \end{aligned}$$

    and there exists a constant \(\rho \in (0, 1]\) such that \(\lim _{n \rightarrow \infty }\rho _n<\rho \).

  • (ii) For any sequence \(\{w_n\} \subset {{\mathbb {X}}}\), \(\lim _{n \rightarrow \infty }w_n = \varrho ^*\) if and only if \(\lim _{n \rightarrow \infty } \epsilon _n = 0\), here \(\{\epsilon _n\}\) is generated by

    $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \epsilon _n = \left\| w_{n+1} - {\Phi _1} \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi _1} {\eta _n} +{r_n} {d_n}\right) \right\| , \\ \displaystyle {\eta _n} = {\Phi _2}\left( \frac{\eta _n + \xi _n}{2}\right) +e_n,\\ \displaystyle {\xi _n} = {\Phi _3}\left( \left( 1-s_n\right) \frac{\xi _n+w_n}{2} + s_n \Phi _3\left( \frac{\xi _n+w_n}{2}\right) \right) +h_n. \end{array} \right. \end{aligned}$$
    (2.1)

    Scilicet, the JFESD (1.1) is \(\Phi \)-stable.

Proof

First, we prove convergence of the iterative scheme JFESD (1.1). Let \(\varrho ^* \in Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3)\). Then by (1.1), one has

$$\begin{aligned} \left\| {\vartheta _n}-\varrho ^* \right\|\le & {} \left\| {\Phi _3}\left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi _3}\left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) -{\Phi _3}\left( \varrho ^*\right) \right\| +\left\| {h_n}\right\| \nonumber \\\le & {} \theta _3\left\| \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi _3}\left( \frac{{\vartheta _n}+{\varrho _n}}{2}\right) -\varrho ^* \right\| +\left\| {h_n}\right\| \nonumber \\\le & {} \theta _3\left( 1-{s_n}\right) \left\| \frac{{\vartheta _n}+{\varrho _n}}{2}-\varrho ^* \right\| +\theta _3^2{s_n}\left\| \frac{{\vartheta _n}+{\varrho _n}}{2}-\varrho ^* \right\| +\left\| {h_n}\right\| \nonumber \\\le & {} \frac{\iota _n}{2} \left( \left\| {\vartheta _n}-\varrho ^* \right\| +\left\| {\varrho _n}-\varrho ^* \right\| \right) +\left\| {h_n}\right\| , \end{aligned}$$

where \(\iota _n=\theta _3\left( 1-{s_n}+\theta _3{s_n}\right) \), this indicates that

$$\begin{aligned} \left\| {\vartheta _n}-\varrho ^*\right\| \le \frac{\iota _n}{2-\iota _n}\left\| {\varrho _n}-\varrho ^* \right\| +\frac{2}{2-\iota _n}\left\| {h_n}\right\| . \end{aligned}$$
(2.2)

Furthermore, it follows from the second formulation of (1.1) that

$$\begin{aligned} \left\| {\varsigma _n}-\varrho ^* \right\|\le & {} {\theta _2}\left\| \frac{{\vartheta _n}+{\varsigma _n}}{2}-\varrho ^* \right\| +\left\| {e_n} \right\| \nonumber \\\le & {} \frac{{\theta _2}}{2} \left( \left\| {\vartheta _n}-\varrho ^* \right\| +\left\| {\varsigma _n}-\varrho ^* \right\| \right) +\left\| {e_n} \right\| , \end{aligned}$$

this implies with (2.2) and \(\iota _n\le \theta _3\) for \(n\in {{\mathbb {N}}}\) that

$$\begin{aligned} \left\| {\varsigma _n}-\varrho ^* \right\|\le & {} \frac{\theta _2}{2-\theta _2} \left\| {\vartheta _n}-\varrho ^* \right\| +\frac{2}{2-\theta _2}\left\| {e_n} \right\| \nonumber \\\le & {} \frac{\theta _2\iota _n}{\left( 2-\theta _2\right) \left( 2-\iota _n\right) } \left\| {\varrho _n}-\varrho ^*\right\| \nonumber \\&+\frac{2\theta _2}{\left( 2-\theta _2\right) \left( 2-\theta _3\right) }\left\| {h_n}\right\| +\frac{2}{2-\theta _2}\left\| {e_n}\right\| . \end{aligned}$$
(2.3)

Taking \(\tau _n={\theta _1} \left( 1-{r_n}+{\theta _1{r_n}}\right) \), then \(\tau _n\le \theta _1\) for \(n\in {{\mathbb {N}}}\) and by (1.1) and (2.3), and \(\{r_n\}, \{s_n\}\subseteq [0,1]\), now we know that for each \(n\in {{\mathbb {N}}}\):

$$\begin{aligned} \left\| {\varrho _{n+1}}-{\varrho ^*} \right\| \le&\,{\theta _1}\left\| \left( 1-{r_n} \right) {\varsigma _n}+{r_n}{\Phi _1}{\varsigma _n}+{r_n}{d_n}-\varrho ^* \right\| \nonumber \\ \le&\,{\theta _1} \left( 1-{r_n} \right) \left\| {\varsigma _n}-\varrho ^* \right\| +{\theta _1}^2{r_n}\left\| {\varsigma _n}-\varrho ^* \right\| +{\theta _1}\left\| {r_n}{d_n} \right\| \nonumber \\ \le&\,\tau _n\left\| {\varsigma _n}-\varrho ^* \right\| +{\theta _1}{r_n}(\left\| d_n^\prime \right\| + \left\| d_n^{\prime \prime }\right\| ) \nonumber \\ \le&\,\left( 1-t_n\right) \left\| {\varrho _n}-\varrho ^*\right\| +\frac{2\theta _2\theta _1}{\left( 2-\theta _2\right) \left( 2-\theta _3\right) } \left\| {h_n}\right\| \nonumber \\&+\,\frac{2\theta _1}{2-\theta _2}\left\| {e_n}\right\| +{\theta _1}{r_n}(\left\| d_n^\prime \right\| +\left\| d_n^{\prime \prime }\right\| ), \end{aligned}$$
(2.4)

where

$$\begin{aligned} t_n =&\, \frac{(2-\theta _2)(2-\iota _n)-\theta _2 \tau _n\iota _n}{(2-\theta _2)(2-\iota _n)} \nonumber \\ \ge&\,\frac{1}{4} \left\{ \left[ 1+(1-\theta _2) \right] \left[ 1+(1-\theta _3)\right] -\theta _1\theta _2\theta _3\right\} \nonumber \\ >&\,0, \\ 1-t_n =\,&\dfrac{\theta _2\tau _n\iota _n}{\left( 2-\theta _2\right) \left( 2-\iota _n\right) }\ge 0, \end{aligned}$$

which is because \(\tau _n \in [0,\theta _1]\), \(\iota _n \in [0,\theta _3]\), \(\sum ^{3}_{i=1} \theta _i < 3\), \(\theta _i\in [0,1]\) for \(i=1,2,3\), and there exists \(i \in \{1,2,3\}\) such that \(\theta _i<1\), i.e., \(\theta _1 \theta _2 \theta _3<1\). Therefore, one has a lower bound \(t_{*}>0\) of \(\{t_n\} \subseteq [0,1]\), that is \(t_*=\lim \inf \nolimits _{n \rightarrow \infty } t_n \in (0,1]\). Furthermore, we can easily know that \(\sum ^{\infty }_{n=0}t_n=\infty \). Thus it follows from (2.4) that

$$\begin{aligned} \left\| {\varrho _{n+1}}-{\varrho ^*} \right\|\le & {} \left( 1-t_n\right) \left\| {\varrho _n}-\varrho ^*\right\| +t_n \left[ \frac{2\theta _1}{t_*(2-\theta _2)}\left\| {e_n}\right\| +\frac{{\theta _1{r_n}}}{t_*}\left\| d_n^\prime \right\| \right] \nonumber \\&+ \left[ \frac{2\theta _1\theta _2}{\left( 2-\theta _2\right) \left( 2-\theta _3\right) }\left\| {h_n}\right\| +{\theta _1}{r_n}\left\| d_n^{\prime \prime }\right\| \right] . \end{aligned}$$
(2.5)

By Lemma 2.1, the inequality (2.5) yields with the boundedness of \(\{r_n\}\) and the condition (H) that \(\left\| {\varrho _n}-\varrho ^*\right\| \) as \(n \rightarrow \infty \), i.e., the sequence \(\{\varrho _n\}\) generated by the JFESD (1.1) strongly converges to \(\varrho ^*\) with convergence rate in every step:

$$\begin{aligned} \rho _n= {\theta _1}{\theta _2}{\theta _3}\cdot \frac{\left[ 1-{r_n}(1-{\theta _1})\right] \left[ 1-{s_n}(1-{\theta _3})\right] }{(2-{\theta _2})\left\{ 2-{\theta _3}\left[ 1-{s_n}(1-{\theta _3}) \right] \right\} }. \end{aligned}$$

Next, we prove the iterative scheme JFESD (1.1) is \(\Phi \)-stable. First, we show that if \(\lim _{n \rightarrow \infty }\epsilon _n \rightarrow 0\), then \(\lim _{n \rightarrow \infty } w_n \rightarrow 0\). In fact, it follows from (2.1) that

$$\begin{aligned} \epsilon _n= & {} \left\| w_{n+1} - {\Phi _1} \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi _1} {\eta _n} +{r_n} {d_n}\right) \right\| \nonumber \\\ge & {} \left\| w_{n+1} - w^*\right\| - \left\| {\Phi _1} \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi _1} {\eta _n} +{r_n} {d_n}\right) - w^*\right\| , \end{aligned}$$

which implies with (2.5) that

$$\begin{aligned} \left\| w_{n+1} - w^*\right\|\le & {} \epsilon _n + \left\| {\Phi _1} \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi _1} {\eta _n} +{r_n} {d_n}\right) - w^*\right\| \nonumber \\\le & {} \left( 1-t_n\right) \left\| {w_n}-w^*\right\| +t_n \left[ \frac{1}{t_*}\epsilon _n+\frac{2\theta _1}{t_*(2-\theta _2)}\left\| {e_n}\right\| +\frac{{\theta _1{r_n}}}{t_*}\left\| d_n^\prime \right\| \right] \nonumber \\&+ \left[ \frac{2\theta _1\theta _2}{\left( 2-\theta _2\right) \left( 2-\theta _3\right) }\left\| {h_n}\right\| +{\theta _1}{r_n}\left\| d_n^{\prime \prime }\right\| \right] . \end{aligned}$$

Hence, if \(\lim _{n \rightarrow \infty } \epsilon _n \rightarrow 0\), then from Lemma 2.1 and the given conditions, we get \(\lim _{n \rightarrow \infty }\left\| w_n - w^*\right\| \rightarrow 0\) and \(\lim _{n \rightarrow \infty } w_n = w^*\).

Whereas, if \(\lim _{n \rightarrow \infty } w_n=w^*\), by (2.4) and the assumptions, one knows that

$$\begin{aligned} \epsilon _n= & {} \left\| w_{n+1} - {\Phi _1} \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi _1} {\eta _n} +{r_n} {d_n}\right) \right\| \nonumber \\\le & {} \left( 1-t_n\right) \left\| {w_n}-w^*\right\| +\frac{2\theta _1}{2-\theta _2}\left\| {e_n}\right\| +{\theta _1}{r_n}\left\| d_n^\prime \right\| \\&+\frac{2\theta _2\theta _1}{\left( 2-\theta _2\right) \left( 2-\theta _3\right) } \left\| {h_n}\right\| +{\theta _1}{r_n}\left\| d_n^{\prime \prime }\right\| \rightarrow 0, \end{aligned}$$

this shows that \(\lim _{n \rightarrow \infty }\epsilon _n \rightarrow 0\). The proof is completed. \(\square \)

From Theorem 2.1, it is easy to obtain the following results to the special cases (1.2)–(1.4) of the JFESD (1.1).

Corollary 2.1

Suppose that \({{\mathbb {X}}}\), K and operator \(\Phi _i\) \((i=1, 2, 3)\) are the same as in Theorem 2.1. Then the sequence \(\{\varrho _n\}\) generated by the JFSD (1.2) converges to \(\varrho ^* \in Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3)\) with convergence rate for \(n \in {\mathbb {N}}\):

$$\begin{aligned} \rho _n= {\theta _1}{\theta _2}{\theta _3}\cdot \frac{\left[ 1-{r_n}(1-{\theta _1})\right] \left[ 1-{s_n}(1-{\theta _3})\right] }{(2-{\theta _2})\left\{ 2-{\theta _3}\left[ 1-{s_n}(1-{\theta _3}) \right] \right\} }, \end{aligned}$$

and there exists a positive constant \(\rho \) such that \(\rho _n<\rho \le 1\) for any \(n\in {{\mathbb {N}}}\). Moreover, for any sequence \(\{w_n\} \subset {{\mathbb {X}}}\), \(\lim \limits _{n \rightarrow \infty }w_n = w^*\) if and only if \(\lim \limits _{n \rightarrow \infty } \epsilon _n = 0\), here \(\epsilon _n\) is defined by

$$\begin{aligned} \left\{ \begin{aligned}&\epsilon _n = \left\| w_{n+1} - {\Phi _1} \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi _1} {\eta _n} \right) \right\| ,\\&{\eta _n} = {\Phi _2}\left( \frac{\eta _n + \xi _n}{2}\right) , \\&{\xi _n} = {\Phi _3}\left( \left( 1-s_n\right) \frac{\xi _n+w_n}{2} + s_n \Phi _3\left( \frac{\xi _n+w_n}{2}\right) \right) , \end{aligned} \right. \end{aligned}$$

i.e., the iteration scheme JFSD (1.2) is \(\Phi \)-stable.

Corollary 2.2

Let \({{\mathbb {X}}}\) and K be the same as in Theorem 2.1. If \(\Phi _1=\Phi _2=\Phi _3=\Phi : K \rightarrow K\) is nonexpansive-type operator with Lipschitz coefficient \(\theta \in [0,1)\) and \(Fix(\Phi ) \ne \emptyset \), then the sequence \(\{\varrho _n\}\) generated by the JFES (1.3) converges to \(\varrho ^*\in Fix(\Phi )\) with convergence rate:

$$\begin{aligned} \rho _n= \frac{\theta ^3\left[ 1-{r_n}(1-{\theta })\right] \left[ 1-{s_n}(1-{\theta })\right] }{(2-{\theta })\left\{ 2-{\theta }\left[ 1-{s_n}(1-{\theta }) \right] \right\} } \end{aligned}$$
(2.6)

for every \(n\in {{\mathbb {N}}}\) and there exists a constant \(\rho \in (0,1]\) such that \(\rho _n<\rho \) for \(n\in {{\mathbb {N}}}\). Furthermore, letting a sequence \(\{\epsilon _n\}\) be decided by

$$\begin{aligned} \left\{ \begin{aligned}&\epsilon _n = \left\| w_{n+1} - {\Phi } \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi } {\eta _n} +{r_n} {d_n}\right) \right\| , \\&{\eta _n} = {\Phi }\left( \frac{\eta _n + \xi _n}{2}\right) +e_n, \\&{\xi _n} = {\Phi }\left( \left( 1-s_n\right) \frac{\xi _n+w_n}{2} + s_n \Phi \left( \frac{\xi _n+w_n}{2}\right) \right) + h_n, \end{aligned} \right. \end{aligned}$$

then for any sequence \(\{w_n\} \subset {{\mathbb {X}}}\), \(\lim _{n \rightarrow \infty }w_n =w^*\) if and only if \(\lim \limits _{n \rightarrow \infty } \epsilon _n = 0\) and the iteration scheme JFES (1.3) is called \(\Phi \)-stable.

Corollary 2.3

Assume that \({{\mathbb {X}}}\), K and \(\Phi \) are the same as in Corollary  2.2. then the sequence \(\{x_n\}\) determined by the JFS (1.4) converges to \(\varrho ^*\in Fix(\Phi )\) with convergence rate \(\rho _n\) of (2.6) in every step. Furthermore, for any sequence \(\{w_n\}\subset {{\mathbb {X}}}\), \(\lim _{n \rightarrow \infty }w_n = w^*\) if and only if \(\lim _{n \rightarrow \infty }\epsilon _n = 0\), where \(\epsilon _n\) is confirmed as follows:

$$\begin{aligned} \left\{ \begin{aligned}&\epsilon _n = \left\| w_{n+1} - {\Phi } \left( \left( 1-{r_n}\right) {\eta _n}+{r_n} {\Phi } {\eta _n}\right) \right\| , \\&{\eta _n} = {\Phi }\left( \frac{\eta _n + \xi _n}{2}\right) , \\&{\xi _n} = {\Phi }\left( \left( 1-s_n\right) \frac{\xi _n+w_n}{2} + s_n \Phi \left( \frac{\xi _n+w_n}{2}\right) \right) , \end{aligned} \right. \end{aligned}$$

that is, the iteration scheme JFS (1.4) is \(\Phi \)-stable.

To verify our results, we give a numerical example as follows.

Example 2.1

We consider a known integral equation, which often arises in many physical problems, defined as

$$\begin{aligned} f(\varrho )= a+\int _{0}^{\varrho }k(\varrho ,\varsigma )f(\varsigma )d\varsigma , \quad \forall \varrho \in [0,l], \end{aligned}$$
(2.7)

where l and \(a=f(0)\) are real constants, and \(k: {{\mathbb {R}}}\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is a given continuous function.

From Eq. (2.7), one can easily decide an operator \(\Phi : {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) by

$$\begin{aligned} \Phi (f): =\Phi (f(\varrho ))= a + \int _{0}^{\varrho }k(\varrho ,\varsigma )f(\varsigma )d\varsigma ,\quad \forall \varrho \in [0,l], f\in {{\mathbb {R}}}, \end{aligned}$$
(2.8)

and knows that if \(\sup _{\varrho \in [0,l]} \int _{0}^{\varrho }\left\| k(\varrho ,\varsigma )\right\| d\varsigma <1\), then \(\Phi \) is a nonexpansive-type operator. In fact, for any \(f, g\in {{\mathbb {R}}}\), we have

$$\begin{aligned} \Vert \Phi (f)- \Phi (g)\Vert= & {} \sup _{\varrho \in [0,l]} \left\| \int _{0}^{\varrho }k(\varrho ,\varsigma )\cdot \left[ f(\varsigma )-g(\varsigma )\right] d\varsigma \right\| \nonumber \\\le & {} \left( \sup _{\varrho \in [0,l]} \int _{0}^{\varrho }\left\| k(\varrho ,\varsigma )\right\| d\varsigma \right) \Vert f-g\Vert . \end{aligned}$$

By Banach contraction mapping principle, the operator \(\Phi \) has a fixed point, which is the solution of Eq. (2.7).

With regard to (2.7) and (2.8), taking \(a=1\), \(l=\frac{9}{10}\) and \(k(\cdot ,\cdot )\equiv 1\), then by simple calculation, one can easily see that \(\sup _{\varrho \in [0,l]} \int _{0}^{\varrho }\left\| k(\varrho ,\varsigma )\right\| d\varsigma =\frac{9}{10}<1\) and an exact solution of the following example of (2.7):

$$\begin{aligned} f(\varrho ) = 1 + \int _{0}^{\varrho } f(\varsigma )d(\varsigma ) \end{aligned}$$
(2.9)

is \(f(\varrho )=e^\varrho \) for every \(\varrho \in \left[ 0,\frac{9}{10}\right] \), which is a fixed point of special operator \(\Phi \) to (2.9). In subsequent work, the exact solution of the particular case (2.9) shall be numerically approximated via using our new iterative schemes JFESD (1.1) and JFSD (1.2). Assume \(\Phi _1=\Phi \), \(\Phi _2=\Phi _3=I\), the identity operator, \(r_n = 0.3\), \(s_n=0.5\), \(d_n^\prime =\frac{n^2}{10^n}\), \(d_n^{\prime \prime }=\frac{n}{10^n}\), \(e_n=\frac{1}{5^n}\), \(h_n=\frac{10}{8^n}\) for each \(n\in {{\mathbb {N}}}\), and the initial function \(f(\varrho )=\varrho \) is given. Then the numerical solutions after some steps are displayed in Figs. 1 and 2, and mean square errors (in short, MSE) of JFESD (1.1) and JFSD (1.2) have been computed in Fig. 3. It is light to see that the iteration schemes JFESD (1.1) and JFSD (1.2) can fastly converge to the exact solution of (2.9), respectively, and although the convergence speed to JFESD (1.1) and JFSD (1.2) are not the same, the numbers of iteration (in short, Iter.) for converging to the precise solution are no more than twenty . These indicate the validity of Theorem 2.1 and Corollary 2.1.

Fig. 1
figure 1

Approximating solutions of (2.9) by the JFESD (1.1)

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Fig. 2
figure 2

Approximating solutions of (2.9) by the JFSD (1.2)

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3 Simulation and applications

In this section, using the new iterative schemes presented in this paper and based on Matlab 2020b and R 4.1, we will propose two numerical simulation examples for approximating fixed points of nonexpansive-type operators and to check on the efficiency of our methods. Furthermore, we apply the new scheme JFESD (1.1) to solve the variational inequality (1.7).

Example 3.1

Let \(K=[0,+\infty ]\), \(\Phi (\varrho )=\sqrt{\varrho ^2-6 \varrho + 30}\), and for any \(n\in {{\mathbb {N}}}\), \(r_n=s_n =\frac{1}{2}\), \(d_n^\prime =\frac{n^2}{10^n}\), \(d_n^{\prime \prime }=\frac{n}{10^n}\), \(e_n=\frac{1}{5^n}\) and \(h_n=\frac{10}{8^n}\).

In Li and Lan (2019), Li and Lan proved the operator \(\Phi \) is a nonexpansive-type operator and showed that one fixed point of \(\Phi \) is 5. Now we will use our new iteration schemes to approximate the fixed points of \(\Phi \). To make clear that our schemes are better than others, we generate 1000 random initial points between 0 and 20, and set the stop condition as \(\Vert \varrho _{n+1} - \varrho _n\Vert \le \varepsilon \), here \(\varepsilon = 10^{-5}, 10^{-10}\). The numerical simulation results for two special cases of the JFESD (1.1): JFES (1.3) and JFS (1.4), and some others known schemes JF, PMMI, PMI, MANN, ISHIKAWA, NOOR and SAKURAI are appeared in Figs. 4 and 5.

Fig. 3
figure 3

MSE of the JFESD (1.1) and the JFSD (1.2) for (2.9)

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Fig. 4
figure 4

Comparison of Iter. with random initial values and different tolerances for Example 3.1

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Fig. 5
figure 5

Final approximations with different initial values and tolerances for Example 3.1

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In Fig. 4, horizontal axis means different iterative schemes, vertical axis is the Iter. of iterative schemes with different stopped conditions. From Fig. 4, one can easily see that the JFS (1.4) converges faster than other schemes under given stopped conditions, and they are nearly not affected by initial points. In Fig. 5, horizontal axis means different iterative schemes, vertical axis means the final approximated value, it is clear that the JFES (1.2) stably converges to a fixed point for the given nonexpansive-type operator.

Example 3.2

Take \(K=[0,+\infty ]\), \(r_n=s_n=\frac{1}{2}\), \(d_n^\prime =\frac{n^2}{10^n}\), \(d_n^{\prime \prime }=\frac{n}{10^n}\), \(e_n = \frac{1}{5^n}\) and \(h_n=\frac{10}{8^n}\) for all \(n\in {{\mathbb {N}}}\), and define \(\Phi _i=\Phi \) for \(i=1, 2, 3\) as

$$\begin{aligned}&\Phi (\varrho ) = \frac{1}{2}\left( \sin \varrho +\cos \varrho +\frac{1}{2}\varrho \right) + \frac{3\pi +2}{4}, \end{aligned}$$
(3.1)

where \(\varrho \in [0, \infty )\).

From (3.1), we know that one fixed point of \(\Phi \) is \(\pi \) and

$$\begin{aligned} \vert \Phi (\varrho )-\Phi (\varsigma )\vert\le & {} \frac{1}{4}\left| \varrho -\varsigma \right| + \frac{1}{2}\left| [\sin (\varrho ) - \sin (\varsigma )] + [\cos (\varrho ) - \cos (\varsigma )]\right| \nonumber \\= & {} \frac{1}{4}\left| \varrho -\varsigma \right| + \left| \cos \left( \frac{\varrho +\varsigma }{2}\right) - \sin \left( \frac{\varrho +\varsigma }{2}\right) \right| \cdot \left| \sin \left( \frac{\varrho -\varsigma }{2}\right) \right| \nonumber \\\le & {} \frac{3}{4}\vert \varrho -\varsigma \vert , \end{aligned}$$

which shows that \(\Phi \) is a nonexpansive-type operator. It follows from Theorem 2.1 that the JFESD (1.1) and its some special cases can approximate the fixed point of \(\Phi \). Indeed, taking \(\varrho _0=18.5\), absolute errors of the iterative schemes JFES, JFS, JF, PMMI, MANN, ISHIKAWA, NOOR and SAKURAI for Example 3.2 are listed in Table 1, and one can readily see that the absolute error of our new iteration scheme JFS (1.4) becomes to 0 at the soonest. This implies that the JFS (1.4) is the best one of the compared eight schemes.

Table 1 Absolute errors of some iterative schemes for Example 3.2
Full size table

We note that from the fixed point problem (1.8), if the second \(\varsigma _n\) in the first equation of the JFESD (1.1) is substituted by \(P_K(\varsigma _n-\kappa F(\varsigma _n))\), where the projection operator \(P_K\) is the same as in (1.5), F is a nonlinear operator on K and \(\kappa \) is a positive constant, then the JFESD (1.1) is equivalent to the following iteration:

$$\begin{aligned} \left\{ \begin{aligned}&{\varrho _{n + 1}} = {\Phi _1} \left( \left( 1 - {r_n} \right) {\varsigma _n} + {r_n}{\Phi _1}(P_K (\varsigma _n-\kappa F( \varsigma _n))) + {r_n}{d_n} \right) , \\&{\varsigma _n} = {\Phi _2} \left( \frac{{{\vartheta _n} + {\varsigma _n}}}{2} \right) + {e_n}, \\&{\vartheta _n}={\Phi _3} \left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}{\Phi _3} \left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) +{h_n}, \end{aligned} \right. \end{aligned}$$
(3.2)

where \(\{r_n\}\) , \(\{s_n\} \) , \(\{d_n\}\), \(\{e_n\}\) and \(\{h_n\}\) are the same sequences as in (1.1). Thus, the iterative process (3.2), as an application of the new iteration scheme JFESD (1.1), can be deemed to find common fixed points of operator \(\Phi _i\) for \(i=1, 2, 3\) and solutions of Stampacchia variation inequality (1.7). This shall be proposed at the end of this section.

Theorem 3.1

Let \({{\mathbb {X}}}\) be a Hilbert space endowed with the norm \(\Vert \cdot \Vert \) and an inner product \(\langle \cdot , \cdot \rangle \), and \(K\subset {{\mathbb {X}}}\) be a nonempty closed convex bounded set. Suppose that the following conditions hold:

  • \((C_1)\) For \(i=1,2,3\), \(\Phi _i: K \rightarrow K\) is nonexpansive-type operator with Lipschitz coefficient \(\theta _i\in [0,1]\) and \(Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3) \cap VI(K,F) \ne \emptyset \).

  • \((C_2)\) \(r_n\in (0,1]\) for all \(n\in {{\mathbb {N}}}\) and \(\lim _{n\rightarrow \infty }r_n\ne 0\) or \(\lim _{n\rightarrow \infty }r_n\) does not exist .

  • \((C_3)\) \(F: K\rightarrow K\) is a \(\mu \)-Lipschitzian continuous and \(\sigma \)-strongly monotone operator, i.e., if there exists a constant \(\sigma >0\) such that \(\langle F(x) - F(y), x-y\rangle \ge \sigma \Vert x-y\Vert ^2\) for all \(x, y\in K\).

  • \((C_4)\) The constant \(\kappa \), which is the same constant as in (1.8), satisfies

    $$\begin{aligned} \kappa \in \left\{ \begin{aligned}&(0, \infty ), \quad&\text{ if }\quad \theta _1=0. \\&\left( 0, \frac{\sigma +\sqrt{\sigma ^2+\mu ^2\theta _1^{-1}\left( 1-\theta _1\right) }}{\mu ^2}\right) , \quad&\text{ if }\quad \theta _1 \ne 0, \end{aligned} \right. \end{aligned}$$
    (3.3)

Then convergence and stability of the iteration (3.2) can be analyzed as follows:

  • (i) The iterative sequence \(\left\{ \varrho _n\right\} \) defined by (3.2) converges to \(Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3) \cap VI(K,F)\) with convergence rate:

    $$\begin{aligned} {\hat{\rho }}_n = {\theta _1}{\theta _2}{\theta _3} \cdot \frac{[1-{r_n}(1-\upsilon \theta _1)][1-{s_n}(1-{\theta _3})]}{(2-{\theta _2})\{2-{\theta _3} [1-{s_n}(1-{\theta _3}) ]\}} \end{aligned}$$

    for any \(n\in {\mathbb {N}}\), where \(\upsilon = \sqrt{1-2\kappa \sigma +\kappa ^2\mu ^2}\), and there exists a constant \({\hat{\rho }}\in (0, 1]\) such that \({\hat{\rho }}_n<{\hat{\rho }}\).

  • (ii) For any sequence \(\{w_n\}\subset {{\mathbb {X}}}\), \(\lim _{n \rightarrow \infty }w_n = w^*\) if and only if \(\lim _{n\rightarrow \infty }\epsilon _n=0\), here \(\epsilon _n\) decided by

    $$\begin{aligned} \left\{ \begin{aligned}&\epsilon _n = \left\| w_{n+1} - {\Phi _1} \left( \left( 1 - {r_n} \right) {\eta _n} + {r_n}{\Phi _1}\left( P_K \left( \eta _n-\kappa F(\eta _n)\right) \right) + {r_n}{d_n} \right) \right\| , \\&{\eta _n} = {\Phi _2}\left( \frac{\eta _n + \xi _n}{2}\right) +e_n, \\&{\xi _n} = {\Phi _3}\left( \left( 1-s_n\right) \frac{\xi _n+w_n}{2} + s_n \Phi _3\left( \frac{\xi _n+w_n}{2}\right) \right) + h_n, \end{aligned} \right. \end{aligned}$$
    (3.4)

    namely, the iteration scheme (3.2) is \(\Phi \)-stable.

Proof

By the condition \((C_3)\), one gets

$$\begin{aligned} \mu \Vert x-y\Vert ^2\ge \Vert F(x)-F(y)\Vert \cdot \Vert x-y\Vert \ge \langle F(x) - F(y), x-y\rangle \ge \sigma \Vert x-y\Vert ^2 \end{aligned}$$

for any \(x, y\in K\), i.e., \(\mu \ge \sigma \) and knows that \(1-2\kappa \sigma +\kappa ^2 \mu ^2\ge 0\). Setting \(\varrho ^*\in Fix(\Phi _1 \cap \Phi _2 \cap \Phi _3) \cap VI(K,F)\), that is, \(\varrho ^*\in K\) is a common fixed point of operator \(\Phi _i\) for \(i=1, 2, 3\) and a solution of Stampacchia variational inequality (1.7), then it follows that

$$\begin{aligned}&\left\| \varsigma _n-\varrho ^*-\kappa \left[ F (\varsigma _n)-F(\varrho ^*)\right] \right\| ^2\\&\quad =\left\| \varsigma _n - \varrho ^*\right\| ^2 -2 \left\langle \varsigma _n - \varrho ^*,\kappa \left[ F (\varsigma _n)-F(\varrho ^*)\right] \right\rangle + \left\| \kappa \left[ F (\varsigma _n)-F(\varrho ^*)\right] \right\| ^2\\&\quad \le \left( 1-2\kappa \sigma +\kappa ^2 \mu ^2\right) \left\| \varsigma _n - \varrho ^*\right\| ^2, \end{aligned}$$

and so

$$\begin{aligned} \left\| \varsigma _n-\varrho ^*-\kappa \left[ F (\varsigma _n)-F(\varrho ^*)\right] \right\| \le \upsilon \left\| \varsigma _n - \varrho ^*\right\| , \end{aligned}$$
(3.5)

here \(\upsilon =\sqrt{1-2\kappa \sigma +\kappa ^2\mu ^2}\). By (3.2), Example 1.1 and (3.5), now we know that

$$\begin{aligned} \left\| \varrho _{n+1}-\varrho ^*\right\|= & {} \left\| {\Phi _1} \left( \left( 1-{r_n} \right) {\varsigma _n} + {r_n}{\Phi _1}\left( P_K (\varsigma _n-\kappa F( \varsigma _n))\right) + {r_n}{d_n} \right) - \varrho ^*\right\| \nonumber \\\le & {} \theta _1\Big [ (1-r_n)\left\| \varsigma _n-\varrho ^*\right\| +\left\| r_nd_n\right\| \nonumber \\&+ r_n \left\| \Phi _1\left( P_K (\varsigma _n-\kappa F(\varsigma _n))\right) - \Phi _1\left( P_K (\varrho ^*-\kappa F(\varrho ^*))\right) \right\| \Big ]\nonumber \\\le & {} \theta _1(1-r_n)\left\| \varsigma _n-\varrho ^*\right\| +\theta _1 r_n(\left\| d_n^\prime \right\| +\left\| d_n^{\prime \prime }\right\| ) \nonumber \\&+ r_n\theta _1^2 \left\| \varsigma _n-\varrho ^*-\kappa \left( F(\varsigma _n)-F(\varrho ^*)\right) \right\| \nonumber \\\le & {} {\hat{\tau }}_n\left\| {\varsigma _n}-\varrho ^* \right\| +{\theta _1}{r_n}(\left\| d_n^\prime \right\| + \left\| d_n^{\prime \prime }\right\| ), \end{aligned}$$
(3.6)

where \({\hat{\tau }}_n= \theta _1\left[ (1-r_n) + \upsilon \theta _1 r_n \right] \), as the condition \((C_4)\) intends that \({\hat{\tau }}_n=0<1\) is always right if \(\theta _1=0\). Otherwise, it is easy to see that \(\upsilon <\theta _1^{-1}\) when \(\theta _1 \ne 0\) and \({\hat{\tau }}_n<\theta _1\) by the condition \((C_2)\). Thus, it follows from the condition \((C_1)\) that there exists a constant \({\hat{t}}_*=\lim \inf \nolimits _{n \rightarrow \infty } {\hat{t}}_n \in (0,1]\), where \({\hat{t}}_n=\frac{(4-2\theta _2-2\iota _n) + \theta _2 \iota _n (1-{\hat{\tau }}_n)}{(2-\theta _2)(2-\iota _n)}\). Hence, for each \(n\in {{\mathbb {N}}}\), replacing \(\tau _n\), \(t_*\) and \(1-t_n\) in (2.4) by \({\hat{\tau }}_n\), \({\hat{t}}_*\) and \(1-{\hat{t}}_n=\frac{\theta _2{\hat{\tau }}_n\iota _n}{\left( 2-\theta _2\right) \left( 2-\iota _n\right) }\) in several, where \(\iota _n\) is the same as in (2.2), then similar to the proof of Theorem 2.1, it follows from (2.3), (3.6), \({\hat{\tau }}_n < \theta _1\) and Lemma 2.1 that \(\left\| {\varrho _n}-\varrho ^*\right\| \) as \(n \rightarrow \infty \), i.e., the sequence \(\{\varrho _n\}\) strongly converges to \(\varrho ^*\) with the following convergence rate in every step:

$$\begin{aligned} {\hat{\rho }}_n = {\theta _1}{\theta _2}{\theta _3} \cdot \frac{[1-{r_n}(1-\upsilon \theta _1)][1-{s_n}(1-{\theta _3})]}{(2-{\theta _2})\{2-{\theta _3} [1-{s_n}(1-{\theta _3})]\}}. \end{aligned}$$

The rest proof can be immediately obtained from the proof of Theorem 2.1 and so it is omitted. \(\square \)

Remark 3.1

(i) One can easily see that the difference between the JFESD (1.1) and the iteration (3.2) is only the second \(\varsigma _n\) in the first equation of (1.1), which depends on the equivalence of Stampacchia variational inequality (1.7) and the fixed point problem (1.8). That is to say, solving the inequality (1.7) is equivalent to finding common fixed points of nonexpansive-type operators \(\Phi _1\), \(\Phi _2\), \(\Phi _3\) and \(P_K\).

(ii) As projection operator is a nonexpansive-type operator, and the conditions \((C_1)\) and \((C_2)\) in Theorem 3.1 separately mean \(\theta _i=1\) for \(i=1, 2, 3\) and the lower bound of \({\hat{t}}_n\) exists, the operator \(\Phi _i\) with Lipschitz coefficient \(\theta _i\in [0,1]\) in (3.2) can be replaced by projection operator for each \(i=1,2,3\), and so one can get a new iteration scheme with errors for the variational inequality (1.7) as follows:

$$\begin{aligned} \left\{ \begin{aligned}&{\varrho _{n + 1}} = P_K \left( \left( 1 - {r_n} \right) {\varsigma _n} + {r_n}P_K(P_K (\varsigma _n-\kappa F \varsigma _n)) + {r_n}{d_n} \right) ,\\&{\varsigma _n} = P_K \left( \frac{{{\vartheta _n} + {\varsigma _n}}}{2} \right) + {e_n}, \\&{\vartheta _n}=P_K \left( \left( 1-{s_n} \right) \frac{{\vartheta _n}+{\varrho _n}}{2}+{s_n}P_K \left( \frac{{\vartheta _n}+{\varrho _n}}{2} \right) \right) +{h_n}, \end{aligned} \right. \end{aligned}$$
(3.7)

and one can achieve convergence and stability of the scheme (3.7) under some suitable conditions.

(iii) Furthermore, some variational inequalities of the form (1.7), for example, Noor (general) variational inequality introduced and studied by Noor (2007), can be solved via the new approximation method of the form JFESD (1.1) or its special cases (3.2) and (3.7). However, it follows from Remark 1.2, Figs. 4 and 5 in Example 3.1 and Table 1 of Example 3.2 that the convergence and stable of our approximation methods presented in this paper are better than those of the Noor (2007).

4 Conclusions

In this paper, we introduced a class of general semi-implicit approximations with errors and proved that the general iterative approximations converge to common fixed points of three different nonexpansive-type operators in Hilbert spaces. In addition, we studied stability of the iterative approximations, obtained convergence and validated our iterative schemes based on some numerical examples, which show that the new iterative methods presented in this paper have better convergence rate and stability. Finally, to solve Stampacchia variational inequality (1.7), we applied the new iterative methods to approximate common fixed points of nonexpansive-type operators and the projection operator associated with the inequality (1.7).

However, the following two open questions are worthy of future research:

  • (1) Nonexpansive-type operators include various forms, does the general iteration scheme JFESD (1.1) also converge to fixed points of generalized nonexpansive operators (Thakur et al. 2016; Ali et al. 2020), asymptotically quasi-nonexpansive type operators (Chang et al. 2003), totally quasi-D-asymptotically nonexpansive operator (Ni and Yao 2015), and other nonexpansive operators?

  • (2) Many iteration schemes are composed of Picard and Mann iteration schemes, so whether can all iteration schemes based on Picard and Mann iterations converge to fixed points of nonexpansive-type operators? And how about are convergence and stability when errors and semi-implicit rule are considered?