Two new extragradient methods for solving equilibrium problems

Abstract

In this paper, we are concern with the classical equilibrium problem in real Hilbert spaces and introduce two new extragradient variants for it. By taking into account several fixed point theory techniques, we obtain simple structure methods that converge strongly and hence demonstrate the theoretical advantage of our methods. Moreover, our convergence assumptions are weaker than those assumed in related works in the literature. Primary numerical examples with comparisons illustrate the behaviour of our proposed scheme and show its advantages.

1 Introduction

In this work we study the classical equilibrium problem originally introduced by Muu and Oettli [27] and has been elaborated further by Blum and Oettli [4] (see also [11]). Given a non-empty, close and convex subset $\mathbb {K}$ of a real Hilbert space $\mathbb {E}$ and let $f : \mathbb {E} \times \mathbb {E} \rightarrow \mathbb {R}$ be a bifunction such that $f (y, y) = 0,$ for all $y \in \mathbb {K}$. With this data, the equilibrium problem is formulated as follows.

We denote the solution set of (EP) by $\Omega .$ Equilibrium problems attract much interest due to their generality in unifying various mathematical problems such as fixed point problems, vector and scalar minimization problems, variational inequalities, complementarity problems, and many more, see e.g., [3, 4, 7, 12, 13, 27]. An important historical remark is that (EP) is also acknowledged as the well-known Ky Fan inequality due to his contribution [11].

The study of equilibrium problems is divided roughly into two parts, the first is theoretical research of the existence of solutions to (EP) and the second is concern with the development of iterative methods for finding such solutions. Regarding the second direction the interested reader is refereed to the many existing results, see e.g., [8,9,10, 15,16,17, 26, 28, 30, 33, 38]. Moreover, techniques for non-monotone problems can be found in [19, 20, 32, 34].

For our purposes we recall Tran et al. [37] iterative scheme that is formulated as follows.

$$\begin{aligned} \left\{ \begin{array}{l} x_{n} \in \mathbb {K}, \\ y_{n} = \underset{y \in \mathbb {K}}{\arg \min } \{ \zeta f(x_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \}, \\ x_{n+1} = \underset{y \in \mathbb {K}}{\arg \min } \{ \zeta f(y_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \}, \\ \end{array} \right. \end{aligned}$$

(1)

where $\zeta$ is some constant depending on the Lipschitz constant of the involved bifunction. This method is known as the two-step extragradient scheme taking its name from the work of Korpelevich [21] focus on saddle points. This method holds two major drawback, the first is the constant step size that require the knowledge or approximation of the Lipschitz constant of the involved bifunction and it only converges weakly in Hilbert spaces. In most cases, the Lipschitz constants are unknown or difficult to compute because it is difficult to check for every three elements in the underlying abstract space [1, 29]. From the computational point of view it might be difficult to estimate the Lipschitz constant a-priori, and hence the convergence rate and applicability of the method could be effected.

Hence, a natural question arises:

Is it possible to introduce a strong convergent extragradient algorithm with adaptive stepsize rule for solving pseudomonotone equilibrium problems (EP)?

Motivated by the above, as well the works in [6, 24, 37], we answer the above question by introducing two strong convergence extragradient-type methods for solving pseudomonotone equilibrium problems in real Hilbert spaces. Moreover, we avoid the need to know the Lipschitz constant of the involved bifunction by using an adaptive stepsize rule.

The outline of our work is as follows. In Sect. 2 we recall some basic results and definitions. Then in Sect. 3 we introduce and analyse our new methods and afterwards in Sect. 4 we present some mathematical applications of our main results and finally in Sect. 5 we illustrate and compare the behaviour of our algorithms.

2 Preliminaries

Let $\mathbb {K}$ be a non-empty, close and convex subset of a real Hilbert space $\mathbb {E}$. The metric projection $P_{\mathbb {K}}(x)$ of $x \in \mathbb {E}$ onto a closed and convex subset $\mathbb {K}$ of $\mathbb {E}$ is defined by

$$\begin{aligned} P_{\mathbb {K}}(x) = \underset{y \in \mathbb {K}}{\arg \min } \Vert y - x\Vert . \end{aligned}$$

(2)

Some useful properties of the metric projection are given next.

Lemma 2.1

e.g., [22] The metric projection $P_{\mathbb {K}} : \mathbb {E} \rightarrow \mathbb {K}$ satisfy the following.

• (i) $\Vert y_{1} - P_{\mathbb {K}}(y_{2}) \Vert ^{2} + \Vert P_{\mathbb {K}}(y_{2}) - y_{2} \Vert ^{2} \le \Vert y_{1} - y_{2} \Vert ^{2}, \,\, y_{1} \in \mathbb {K}, y_{2} \in \mathbb {E}.$

• (ii) $y_{3} = P_{\mathbb {K}}(y_{1})$ if and only if $\langle y_{1} - y_{3}, y_{2} - y_{3} \rangle \le 0, \,\, \forall \, y_{2} \in \mathbb {K}.$

• (iii) $\Vert y_{1} - P_{\mathbb {K}}(y_{1}) \Vert \le \Vert y_{1} - y_{2} \Vert , \,\, y_{2} \in \mathbb {K}, y_{1} \in \mathbb {E}.$

Definition 2.2

Let $\mathbb {K}$ be a subset of a real Hilbert space $\mathbb {E}$ and $\chi : \mathbb {K} \rightarrow \mathbb {R}$ a given convex function.

• (1) The subdifferential of set $\chi$ at $x \in \mathbb {K}$ is defined by

  $$\begin{aligned} \partial \chi (x) = \{ z \in \mathbb {E} : \chi (y) - \chi (x) \ge \langle z, y - x \rangle ,\,\forall \, y \in \mathbb {K} \}. \end{aligned}$$

  (3)

• (2) The normal cone at $x \in \mathbb {K}$ is defined by

  $$\begin{aligned} N_{\mathbb {K}}(x) = \{ z \in \mathbb {E} : \langle z, y - x \rangle \le 0, \,\forall \, y \in \mathbb {K} \}. \end{aligned}$$

  (4)

Lemma 2.3

[36] Let $\chi : \mathbb {K} \rightarrow \mathbb {R}$ be a sub-differentiable, lower semi-continuous and function on $\mathbb {K}$. An element $x \in \mathbb {K}$ is a minimizer of a function $\chi$ iff $0 \in \partial \chi (x) + N_{\mathbb {K}}(x)$, where $\partial \chi (x)$ stands for the sub-differential of $\chi$ at $x \in \mathbb {K}$ and $N_{\mathbb {K}}(x)$ the normal cone of $\mathbb {K}$ at x.

Lemma 2.4

[40] Assume that $\{\gamma _{n}\} \subset (0, +\infty )$ is a sequence satisfying $\gamma _{n+1} \le (1 - \tau _{n}) \gamma _{n} + \tau _{n} \delta _{n}, \,\, \text {for all} \, n \in \mathbb {N}.$ Moreover, $\{\tau _{n}\} \subset (0, 1)$ and $\{\delta _{n}\} \subset \mathbb {R}$ are sequences such that $\lim _{n \rightarrow \infty } \tau _{n} = 0, \,\, \sum _{n=1}^{\infty } \tau _{n} = \infty \,\, \text {and} \,\, \limsup _{n \rightarrow \infty } \delta _{n} \le 0.$ Therefore, $\lim _{n \rightarrow \infty } \gamma _{n} = 0.$

Lemma 2.5

[23] Assume that $\{\gamma _{n}\} \subset \mathbb {R}$ be a sequence and there exists a subsequence $\{n_{i}\}$ of $\{n\}$ such that $\gamma _{n_{i}} < \gamma _{n_{i+1}}$ for all $i \in \mathbb {N}.$ Then, there is a non decreasing sequence $m_{k} \subset \mathbb {N}$ such as $m_{k} \rightarrow \infty$ as $k \rightarrow \infty ,$ and the subsequent conditions are fulfilled by all (sufficiently large) numbers $k \in \mathbb {N}$:

$$\begin{aligned} \gamma _{m_{k}} \le \gamma _{m_{k+1}} \,\, \text {and} \,\, \gamma _{k} \le \gamma _{m_{k+1}}. \end{aligned}$$

Indeed, $m_{k} = \max \{ j \le k: \gamma _{j} \le \gamma _{j+1} \}.$

Lemma 2.6

[2] For all $y_{1}, y_{2} \in \mathbb {E}$ and $\delta \in \mathbb {R},$ then subsequent relationship hold.

• (i) $\Vert \delta y_{1} + (1 - \delta ) y_{2} \Vert ^{2} = \delta \Vert y_{1}\Vert ^{2} + (1 - \delta )\Vert y_{2} \Vert ^{2} - \delta (1 - \delta )\Vert y_{1} - y_{2}\Vert ^{2}.$

• (ii) $\Vert y_{1} + y_{2} \Vert ^{2} \le \Vert y_{1}\Vert ^{2} + 2 \langle y_{2}, y_{1} + y_{2} \rangle$.

3 Main results

In this section, we introduce and analysed our two extragradient-type methods for solving pseudomonotone equilibrium problems in real Hilbert spaces. For the convergence theorems of the methods we assume the following conditions.

• (f1) The bifunction $f : \mathbb {E} \times \mathbb {E} \rightarrow \mathbb {R}$ is pseudomonotone on $\mathbb {K}$, that is

  $$\begin{aligned}&f (y_{1}, y_{2}) \ge 0 \Longrightarrow f(y_{2}, y_{1}) \le 0,\nonumber \\&\quad \forall \, y_{1}, y_{2} \in \mathbb {K}. \end{aligned}$$

  (5)

• (f2) The bifunction $f : \mathbb {E} \times \mathbb {E} \rightarrow \mathbb {R}$ is Lipschitz-type continuity [25] on $\mathbb {K}$, that is, if there exist constants $c_{1}, c_{2} > 0$ such that

  $$\begin{aligned} f(y_{1}, y_{3}) \le f(y_{1}, y_{2}) + f(y_{2}, y_{3}) + c_{1}\Vert y_{1} - y_{2}\Vert ^{2} + c_{2}\Vert y_{2} - y_{3}\Vert ^{2},\quad \forall \, y_{1}, y_{2}, y_{3} \in \mathbb {K}. \end{aligned}$$

  (6)

• (f3) $\limsup \limits _{n\rightarrow \infty } f(y_{n}, y) \le f(y^{*}, y)$ for all $y \in \mathbb {K}$ and $\{y_{n}\} \subset \mathbb {K}$ satisfy $y_{n} \rightharpoonup y^{*}.$

• (f4) $f (x,\cdot )$ is convex and sub-differentiable on $\mathbb {E}$ for each fixed $x \in \mathbb {E}.$

Theorem 3.1

Suppose that condition (f1)–(f4) are satisfied and solution set $\Omega$ is non-empty. Then, any sequence $\{x_{n}\}$ generated by Algorithm 1 converges strongly to $\wp ^{*} = P_{\Omega } (0).$

Proof

We start by proving the boundedness of the sequence $\{x_{n}\}$. By using Lemma 2.3, we get

$$\begin{aligned} 0 \in \partial _{2} \Big (\zeta f(y_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \Big )(p_{n}) + N_{\mathbb {E}_{n}}(p_{n}). \end{aligned}$$

Thus, there is $\omega \in \partial f(y_{n}, p_{n})$ and $\overline{\omega } \in N_{\mathbb {E}_{n}}(p_{n})$ such that $\zeta \omega + p_{n} - x_{n} + \overline{\omega } = 0.$ Thus, we have

$$\begin{aligned} \langle x_{n} - p_{n}, y - p_{n} \rangle = \zeta \langle \omega , y - p_{n} \rangle + \langle \overline{\omega }, y - p_{n} \rangle , \,\, \forall \, y \in \mathbb {E}_{n}. \end{aligned}$$

Given that $\overline{\omega } \in N_{\mathbb {E}_{n}}(p_{n})$ and $\langle \overline{\omega }, y - p_{n} \rangle \le 0,$ for all $y \in \mathbb {E}_{n}.$ Therefore, we have

$$\begin{aligned} \zeta \langle \omega , y - p_{n} \rangle \ge \langle x_{n} - p_{n}, y - p_{n} \rangle , \quad \forall \, y \in \mathbb {E}_{n}. \end{aligned}$$

(7)

Since $\omega \in \partial f(y_{n}, p_{n})$, then

$$\begin{aligned} f(y_{n}, y) - f(y_{n}, p_{n}) \ge \langle \omega , y - p_{n} \rangle , \quad \forall \, y \in \mathbb {E}. \end{aligned}$$

(8)

Combining expressions (7) and (8), we obtain

$$\begin{aligned} \zeta f(y_{n}, y) - \zeta f(y_{n}, p_{n}) \ge \langle x_{n} - p_{n}, y - p_{n} \rangle , \quad \forall \, y \in \mathbb {E}_{n}. \end{aligned}$$

(9)

Substituting $y=\wp ^{*}$ in (9), we obtain

$$\begin{aligned} \zeta f(y_{n}, \wp ^{*}) - \zeta f(y_{n}, p_{n}) \ge \langle x_{n} - p_{n}, \wp ^{*} - p_{n} \rangle . \end{aligned}$$

(10)

Since $\wp ^{*} \in \Omega$, then $f(\wp ^{*}, y_{n}) \ge 0$ implies that $f(y_{n}, \wp ^{*}) \le 0$ and together with Assumption (f1), we obtain

$$\begin{aligned} \langle x_{n} - p_{n}, p_{n} - \wp ^{*}\rangle \ge \zeta f(y_{n}, p_{n}). \end{aligned}$$

(11)

Following Assumption (f2), we have

$$\begin{aligned} f(x_{n}, p_{n}) \le f (x_{n}, y_{n}) + f(y_{n}, p_{n}) + c_{1} \Vert x_{n} - y_{n} \Vert ^{2} + c_{2} \Vert y_{n} - p_{n}\Vert ^{2}. \end{aligned}$$

(12)

Combining (11) and (12), we get that

$$\begin{aligned} \begin{aligned} \langle x_{n} - p_{n}, p_{n} - \wp ^{*}\rangle&\ge \zeta \big \{f(x_{n}, p_{n}) - f (x_{n}, y_{n}) \big \} - c_{1} \zeta \Vert x_{n} - y_{n} \Vert ^{2} - c_{2} \zeta \Vert y_{n} - p_{n}\Vert ^{2}. \end{aligned} \end{aligned}$$

(13)

By the definition of $\mathbb {E}_{n}$ and the fact that $p_{n} \in \mathbb {E}_{n}$, we get $\langle x_{n} - \zeta \omega _{n} - y_{n}, p_{n} - y_{n} \rangle \le 0,$ which implies that

$$\begin{aligned} \langle x_{n} - y_{n}, p_{n} - y_{n} \rangle \le \zeta \langle \omega _{n}, p_{n} - y_{n} \rangle . \end{aligned}$$

(14)

Since $\omega _{n} \in \partial _{2} f(x_{n}, y_{n}),$ we obtain

$$\begin{aligned} f(x_{n}, y) - f(x_{n}, y_{n}) \ge \langle \omega _{n}, y - y_{n} \rangle ,\quad \forall \, y \in \mathbb {E}. \end{aligned}$$

By replacing $y = p_{n}$, we obtain

$$\begin{aligned} f(x_{n}, p_{n}) - f(x_{n}, y_{n}) \ge \langle \omega _{n}, p_{n} - y_{n} \rangle . \end{aligned}$$

(15)

It follows from inequalities (14) and (15) that

$$\begin{aligned} \zeta \big \{f(x_{n}, p_{n}) - f(x_{n}, y_{n}) \big \} \ge \langle x_{n} - y_{n}, p_{n} - y_{n} \rangle . \end{aligned}$$

(16)

From (13) and (16), we have

$$\begin{aligned} \begin{aligned} \langle x_{n} - p_{n}, p_{n} - \wp ^{*}\rangle&\ge \langle x_{n} - y_{n}, p_{n} - y_{n} \rangle - c_{1} \zeta \Vert x_{n} - y_{n} \Vert ^{2} - c_{2} \zeta \Vert y_{n} - p_{n}\Vert ^{2}. \end{aligned} \end{aligned}$$

(17)

Now we obtain the following equalities.

$$\begin{aligned} 2 \langle x_{n} - p_{n}, p_{n} - \wp ^{*}\rangle = \Vert x_{n} - \wp ^{*}\Vert ^{2} - \Vert p_{n} - x_{n}\Vert ^{2} - \Vert p_{n} - \wp ^{*}\Vert ^{2} \end{aligned}$$

and

$$\begin{aligned} 2 \langle y_{n} - x_{n}, y_{n} - p_{n} \rangle = \Vert x_{n} - y_{n}\Vert ^{2} + \Vert p_{n} - y_{n}\Vert ^{2} - \Vert x_{n} - p_{n}\Vert ^{2}. \end{aligned}$$

The above together with (17), imply that

$$\begin{aligned} \Vert p_{n} - \wp ^{*}\Vert ^{2} \le \Vert x_{n} - \wp ^{*}\Vert ^{2} - (1 - 2c_{1}\zeta ) \Vert x_{n} - y_{n}\Vert ^{2} -(1 - 2c_{2} \zeta ) \Vert p_{n} - y_{n}\Vert ^{2}. \end{aligned}$$

(18)

Since $0< \zeta < \min \{\frac{1}{2c_{1}}, \frac{1}{2c_{2}} \}$ with expression (18) implies that

$$\begin{aligned} \Vert p_{n} - \wp ^{*}\Vert ^{2} \le \Vert x_{n} - \wp ^{*}\Vert ^{2}. \end{aligned}$$

(19)

Since $\wp ^{*} \in \Omega$, we get

$$\begin{aligned} \big \Vert x_{n+1} - \wp ^{*} \big \Vert&= \big \Vert (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n} - \wp ^{*} \big \Vert \nonumber \\&= \big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) - \varphi _{n} \wp ^{*} \big \Vert \nonumber \\&\le \big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) \big \Vert + \varphi _{n} \big \Vert \wp ^{*} \big \Vert . \end{aligned}$$

(20)

Next, we compute the following:

$$\begin{aligned}&\big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) \big \Vert ^{2} \nonumber \\&\quad = (1 - \phi _{n} - \varphi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n}^{2} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} + 2 \big \langle (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}), \phi _{n} (p_{n} - \wp ^{*}) \big \rangle \nonumber \\&\quad \le (1 - \phi _{n} - \varphi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n}^{2} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} + 2 \phi _{n} (1 - \phi _{n} - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert \big \Vert p_{n} - \wp ^{*} \big \Vert \end{aligned}$$

(21)

$$\begin{aligned}&\quad \le (1 - \phi _{n} - \varphi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n}^{2} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\qquad + \phi _{n} (1 - \phi _{n} - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} (1 - \phi _{n} - \varphi _{n}) \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad \le (1 - \phi _{n} - \varphi _{n}) (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} (1 - \varphi _{n}) \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} \end{aligned}$$

(22)

$$\begin{aligned}&\quad \le (1 - \phi _{n} - \varphi _{n}) (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = (1 - \varphi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2}. \end{aligned}$$

(23)

Thus, we have

$$\begin{aligned} \big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) \big \Vert \le (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert . \end{aligned}$$

(24)

Combining (20) and (24), we get

$$\begin{aligned} \big \Vert x_{n+1} - \wp ^{*} \big \Vert&\le (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert + \varphi _{n} \big \Vert \wp ^{*} \big \Vert \nonumber \\&\le \max \Big \{ \big \Vert x_{n} - \wp ^{*} \big \Vert , \big \Vert \wp ^{*} \big \Vert \Big \} \nonumber \\&\le \max \Big \{ \big \Vert x_{0} - \wp ^{*} \big \Vert , \big \Vert \wp ^{*} \big \Vert \Big \} \end{aligned}$$

(25)

and the boundedness of $\{x_{n}\}$ is obtained. Now we continue with the strong convergence of the sequence $\{x_{n}\}$. Indeed, by using definition of $\{x_{n+1}\},$ we have

$$\begin{aligned} \big \Vert x_{n+1} - \wp ^{*} \big \Vert ^{2}&= \big \Vert (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&= \big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) - \varphi _{n} \wp ^{*} \big \Vert ^{2} \nonumber \\&= \big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) \big \Vert ^{2} + \varphi _{n}^{2} \big \Vert \wp ^{*} \big \Vert ^{2} \nonumber \\&\qquad - 2 \big \langle (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}), \, \varphi _{n} \wp ^{*} \big \rangle . \end{aligned}$$

(26)

From (22), we have

$$\begin{aligned}&\big \Vert (1 - \phi _{n} - \varphi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) \big \Vert ^{2} \nonumber \\&\quad \le (1 - \phi _{n} - \varphi _{n}) (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} (1 - \varphi _{n}) \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2}. \end{aligned}$$

(27)

Combining (26) and (27) (for some $K_{2}>0$), we obtain

$$\begin{aligned}&\big \Vert x_{n+1} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad \le (1 - \phi _{n} - \varphi _{n}) (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} (1 - \varphi _{n}) \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} + \varphi _{n} K_{2} \nonumber \\&\quad \le (1 - \phi _{n} - \varphi _{n}) (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \varphi _{n} K_{2} \nonumber \\&\qquad + \phi _{n} (1 - \varphi _{n}) \big [ \Vert x_{n} - \wp ^{*}\Vert ^{2} - (1 - 2c_{1}\zeta ) \Vert x_{n} - y_{n}\Vert ^{2} -(1 - 2c_{2} \zeta ) \Vert p_{n} - y_{n}\Vert ^{2} \big ] \nonumber \\&\quad = (1 - \varphi _{n})^{2} \Vert x_{n} - \wp ^{*}\Vert ^{2} + \varphi _{n} K_{2} \nonumber \\&\qquad - \phi _{n} (1 - \varphi _{n}) \big [ (1 - 2c_{1}\zeta ) \Vert x_{n} - y_{n}\Vert ^{2} + (1 - 2c_{2} \zeta ) \Vert p_{n} - y_{n}\Vert ^{2} \big ] \nonumber \\&\quad \le \Vert x_{n} - \wp ^{*}\Vert ^{2} + \varphi _{n} K_{2} \nonumber \\&\qquad - \phi _{n} (1 - \varphi _{n}) \Big [ (1 - 2c_{1}\zeta ) \Vert x_{n} - y_{n}\Vert ^{2} + (1 - 2c_{2} \zeta ) \Vert p_{n} - y_{n}\Vert ^{2} \Big ]. \end{aligned}$$

(28)

By following the conditions (f1) and (f2), the solution set $\Omega$ is a closed and convex set, see for example, [14, 37]). Given that $\wp ^{*} = P_{\Omega }(0),$ and by Lemma 2.1 (ii), we have

$$\begin{aligned} \langle 0 - \wp ^{*}, y - \wp ^{*} \rangle \le 0, \,\, \forall \, y \in \Omega . \end{aligned}$$

(29)

Now we divide the rest of the proof into the following two parts:

Case 1: Suppose that there is a fixed number $N_{1} \in \mathbb {N}$ such that

$$\begin{aligned} \Vert x_{n+1} - \wp ^{*} \Vert \le \Vert x_{n} - \wp ^{*} \Vert , \,\, \forall \, n \ge N_{1}. \end{aligned}$$

(30)

Then $\lim _{n \rightarrow \infty } \Vert x_{n} - \wp ^{*}\Vert$ exists. From (28), we have

$$\begin{aligned}&\phi _{n} (1 - \varphi _{n}) \Big [ (1 - 2c_{1}\zeta ) \Vert x_{n} - y_{n}\Vert ^{2} + (1 - 2c_{2} \zeta ) \Vert p_{n} - y_{n}\Vert ^{2} \Big ] \nonumber \\&\quad \le \Vert x_{n} - \wp ^{*}\Vert ^{2} + \varphi _{n} K_{2} - \Vert x_{n+1} - \wp ^{*} \Vert ^{2}. \end{aligned}$$

(31)

The existence of $\lim _{n \rightarrow \infty } \Vert x_{n} - \wp ^{*}\Vert$ and $\varphi _{n} \rightarrow 0$, we infer that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert x_{n} - y_{n}\Vert = \lim _{n \rightarrow \infty } \Vert p_{n} - y_{n}\Vert = 0. \end{aligned}$$

(32)

It follows that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert x_{n} - p_{n}\Vert \le \lim _{n \rightarrow \infty } \Vert x_{n} - y_{n}\Vert + \lim _{n \rightarrow \infty } \Vert y_{n} - p_{n}\Vert = 0. \end{aligned}$$

(33)

It follows from (33) and $\varphi _{n} \rightarrow 0$, that

$$\begin{aligned} \big \Vert x_{n+1} - x_{n} \big \Vert&= \big \Vert (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n} - x_{n} \big \Vert \nonumber \\&= \big \Vert x_{n} - \varphi _{n} x_{n} + \phi _{n} p_{n} - \phi _{n} x_{n} - x_{n} \big \Vert \nonumber \\&\le \phi _{n} \big \Vert p_{n} - x_{n} \big \Vert + \varphi _{n} \big \Vert x_{n} \big \Vert , \end{aligned}$$

(34)

which gives that

$$\begin{aligned} \Vert x_{n+1} - x_{n}\Vert \rightarrow 0 \quad \text {as} \quad n \rightarrow \infty . \end{aligned}$$

(35)

We deduce that $\{y_{n}\}$ and $\{p_{n}\}$ are bounded. The reflexivity of $\mathbb {E}$ and the boundedness of $\{x_{n}\}$ guarantee that there is a subsequence $\{x_{n_{k}}\}$ such that $\{x_{n_{k}}\} \rightharpoonup \hat{x} \in \mathbb {E}$ as $k \rightarrow \infty .$ Next, we need to show that $\hat{x} \in \Omega .$ Due to (9), the Lipschitz-type continuous of f and (16), we get

$$\begin{aligned} \zeta f(y_{n_{k}}, y)&\ge \zeta f(y_{n_{k}}, p_{n_{k}}) + \langle x_{n_{k}} - p_{n_{k}}, y - p_{n_{k}} \rangle \nonumber \\&\ge \zeta f(x_{n_{k}}, p_{n_{k}}) - \zeta f(x_{n_{k}}, y_{n_{k}}) - c_{1}\zeta \Vert x_{n_{k}} - y_{n_{k}}\Vert ^{2} \nonumber \\&\quad - c_{2} \zeta \Vert y_{n_{k}} - p_{n_{k}} \Vert ^{2} + \langle x_{n_{k}} - p_{n_{k}}, y - p_{n_{k}} \rangle \nonumber \\&\ge \langle x_{n_{k}} - y_{n_{k}}, p_{n_{k}} - y_{n_{k}} \rangle - c_{1}\zeta \Vert x_{n_{k}} - y_{n_{k}}\Vert ^{2} \nonumber \\&\quad - c_{2} \zeta \Vert y_{n_{k}} - p_{n_{k}} \Vert ^{2} + \langle x_{n_{k}} - p_{n_{k}}, y - p_{n_{k}} \rangle , \end{aligned}$$

(36)

where y is an arbitrary point in $\mathbb {E}_{n}$. The boundedness of $\{x_{n}\}$ and from (32), (33) right-hand side converge to zero. Since $\zeta > 0,$ Assumption (f3) and $y_{n_{k}} \rightharpoonup z$, we have

$$\begin{aligned} 0 \le \limsup _{k \rightarrow \infty } f(y_{n_{k}}, y) \le f(\hat{x}, y), \,\, \forall \, y \in \mathbb {E}_{n}, \end{aligned}$$

(37)

which implies that $f(\hat{x}, y) \ge 0,$ $\forall y \in \mathbb {K}$, and hence $\hat{x} \in \Omega .$ Next, we consider

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \langle \wp ^{*}, \wp ^{*} - x_{n} \rangle \nonumber \\&\quad = \limsup _{k \rightarrow \infty } \langle \wp ^{*}, \wp ^{*} - x_{n_k} \rangle = \langle \wp ^{*}, \wp ^{*} - \hat{x} \rangle \le 0. \end{aligned}$$

(38)

We have $\lim _{n \rightarrow \infty } \big \Vert x_{n+1} - x_{n} \big \Vert = 0.$ We can infer that

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \langle \wp ^{*}, \wp ^{*} - x_{n+1} \rangle \nonumber \\&\quad \le \limsup _{n \rightarrow \infty } \langle \wp ^{*}, \wp ^{*} - x_{n} \rangle + \limsup _{n \rightarrow \infty } \langle \wp ^{*}, x_{n} - x_{n+1} \rangle \le 0. \end{aligned}$$

(39)

Next, assume that $t_{n} = (1 - \phi _{n}) x_{n} + \phi _{n} p_{n}.$ Thus, we get

$$\begin{aligned} x_{n+1} = t_{n} - \varphi _{n} x_{n} = (1 - \varphi _{n}) t_{n} - \varphi _{n} (x_{n} - t_{n}) = (1 - \varphi _{n}) t_{n} - \varphi _{n} \phi _{n} (x_{n} - p_{n}). \end{aligned}$$

(40)

where $x_{n} - t_{n} = x_{n} - (1 - \phi _{n}) x_{n} - \phi _{n} p_{n} = \phi _{n} (x_{n} - p_{n}).$ Therefore, we have

$$\begin{aligned}&\big \Vert x_{n+1} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = \big \Vert (1 - \varphi _{n}) t_{n} + \phi _{n} \varphi _{n} (p_{n} - x_{n}) - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = \big \Vert (1 - \varphi _{n}) (t_{n} - \wp ^{*}) + \big [ \phi _{n} \varphi _{n} (p_{n} - x_{n}) - \varphi _{n} \wp ^{*} \big ] \big \Vert ^{2} \nonumber \\&\quad \le (1 - \varphi _{n})^{2} \big \Vert t_{n} - \wp ^{*} \big \Vert ^{2} + 2 \big \langle \phi _{n} \varphi _{n} (p_{n} - x_{n}) - \varphi _{n} \wp ^{*}, \nonumber \\&\qquad (1 - \varphi _{n}) (t_{n} - \wp ^{*}) + \phi _{n} \varphi _{n} (p_{n} - x_{n}) - \varphi _{n} \wp ^{*} \big \rangle \nonumber \\&\quad = (1 - \varphi _{n})^{2} \big \Vert t_{n} - \wp ^{*} \big \Vert ^{2} + 2 \big \langle \phi _{n} \varphi _{n} (p_{n} - x_{n}) - \varphi _{n} \wp ^{*}, \, t_{n} - \varphi _{n} t_{n} - \varphi _{n} (x_{n} - t_{n}) - \wp ^{*} \big \rangle \nonumber \\&\quad = (1 - \varphi _{n}) \big \Vert t_{n} - \wp ^{*} \big \Vert ^{2} + 2 \phi _{n} \varphi _{n} \big \langle p_{n} - x_{n}, \, x_{n+1} - \wp ^{*} \big \rangle + 2 \varphi _{n} \big \langle \wp ^{*}, \, \wp ^{*} - x_{n+1} \big \rangle \nonumber \\&\quad \le (1 - \varphi _{n}) \big \Vert t_{n} - \wp ^{*} \big \Vert ^{2} + 2 \phi _{n} \varphi _{n} \big \Vert p_{n} - x_{n} \big \Vert \big \Vert x_{n+1} - \wp ^{*} \big \Vert + 2 \varphi _{n} \big \langle \wp ^{*}, \, \wp ^{*} - x_{n+1} \big \rangle . \end{aligned}$$

(41)

We next evaluate

$$\begin{aligned}&\big \Vert t_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = \big \Vert (1 - \phi _{n}) x_{n} + \phi _{n} p_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = \big \Vert (1 - \phi _{n}) (x_{n} - \wp ^{*}) + \phi _{n} (p_{n} - \wp ^{*}) \big \Vert ^{2} \nonumber \\&\quad = (1 - \phi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n}^{2} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} + 2 \big \langle (1 - \phi _{n}) (x_{n} - \wp ^{*}), \phi _{n} (p_{n} - \wp ^{*}) \big \rangle \nonumber \\&\quad \le (1 - \phi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n}^{2} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} + 2 \phi _{n} (1 - \phi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert \big \Vert p_{n} - \wp ^{*} \big \Vert \nonumber \\&\quad \le (1 - \phi _{n})^{2} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n}^{2} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} (1 - \phi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2}\nonumber \\&\qquad + \phi _{n} (1 - \phi _{n}) \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = (1 - \phi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} \big \Vert p_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad \le (1 - \phi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \phi _{n} \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad = \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2}. \end{aligned}$$

(42)

Combining (41) and (42) yields

$$\begin{aligned}&\big \Vert x_{n+1} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad \le (1 - \varphi _{n}) \big \Vert x_{n} - \wp ^{*} \big \Vert ^{2} + \varphi _{n} \Big [ 2 \phi _{n} \big \Vert p_{n} - x_{n} \big \Vert \big \Vert x_{n+1} - \wp ^{*} \big \Vert + 2 \varphi _{n} \big \langle \wp ^{*}, \, \wp ^{*} - x_{n+1} \big \rangle \Big ]. \end{aligned}$$

(43)

Due to (39), (43) and Lemma 2.4, we deduce that $\big \Vert x_{n} - \wp ^{*} \big \Vert \rightarrow 0$ as $n \rightarrow \infty .$

Case 2: Assume that there is a subsequence $\{n_{i}\}$ of $\{n\}$ such that

$$\begin{aligned} \Vert x_{n_i} - \wp ^{*} \Vert \le \Vert x_{n_{i+1}} - \wp ^{*} \Vert , \,\, \forall i \, \in \mathbb {N}. \end{aligned}$$

By Lemma 2.5, there is a sequence $\{m_{k}\} \subset \mathbb {N}$ ($\{m_{k}\} \rightarrow \infty$), such that

$$\begin{aligned} \Vert x_{m_{k}} - \wp ^{*} \Vert \le \Vert x_{m_{k+1}} - \wp ^{*} \Vert \quad \text {and} \quad \Vert x_{k} - \wp ^{*} \Vert \le \Vert x_{m_{k+1}} - \wp ^{*} \Vert , \,\, \forall \, k \in \mathbb {N}. \end{aligned}$$

(44)

From (31), we have

$$\begin{aligned}&\phi _{m_{k}} (1 - \varphi _{m_{k}}) \Big [ (1 - 2c_{1}\zeta ) \Vert x_{m_{k}} - y_{m_{k}}\Vert ^{2} + (1 - 2c_{2} \zeta ) \Vert p_{m_{k}} - y_{m_{k}}\Vert ^{2} \Big ] \nonumber \\&\quad \le \Vert x_{m_{k}} - \wp ^{*}\Vert ^{2} + \varphi _{m_{k}} K_{2} - \Vert x_{m_{k}+1} - \wp ^{*} \Vert ^{2}. \end{aligned}$$

(45)

Due to $\varphi _{m_{k}} \rightarrow 0$, we deduce the following:

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert x_{m_{k}} - y_{m_{k}}\Vert = \lim _{n \rightarrow \infty } \Vert p_{m_{k}} - y_{m_{k}}\Vert = 0. \end{aligned}$$

(46)

It follows that

$$\begin{aligned} \big \Vert x_{m_{k}+1} - x_{m_{k}} \big \Vert&= \big \Vert (1 - \phi _{m_{k}} - \varphi _{m_{k}} ) x_{m_{k}} + \phi _{m_{k}} p_{m_{k}} - x_{m_{k}} \big \Vert \nonumber \\&= \big \Vert x_{m_{k}} - \varphi _{m_{k}} x_{m_{k}} + \phi _{m_{k}} p_{m_{k}} - \phi _{m_{k}} x_{m_{k}} - x_{m_{k}} \big \Vert \nonumber \\&\le \phi _{m_{k}} \big \Vert p_{m_{k}} - x_{m_{k}} \big \Vert + \varphi _{m_{k}} \big \Vert x_{m_{k}} \big \Vert \longrightarrow 0. \end{aligned}$$

(47)

Using similar argument as in Case 1, we get

$$\begin{aligned} \limsup _{k \rightarrow \infty } \langle \wp ^{*}, x_{m_{k}+1} - \wp ^{*} \rangle \le 0. \end{aligned}$$

(48)

Now, using (43) and (44), we have

$$\begin{aligned} \big \Vert x_{m_{k}+1} - \wp ^{*} \big \Vert ^{2}&\le (1 - \varphi _{m_{k}}) \big \Vert x_{m_{k}} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad + \varphi _{m_{k}} \Big [ 2 \phi _{m_{k}} \big \Vert p_{m_{k}} - x_{m_{k}} \big \Vert \big \Vert x_{m_{k}+1} - \wp ^{*} \big \Vert + 2 \varphi _{m_{k}} \big \langle \wp ^{*}, \, \wp ^{*} - x_{m_{k}+1} \big \rangle \Big ] \nonumber \\&\le (1 - \varphi _{m_{k}}) \big \Vert x_{m_{k+1}} - \wp ^{*} \big \Vert ^{2} \nonumber \\&\quad + \varphi _{m_{k}} \Big [ 2 \phi _{m_{k}} \big \Vert p_{m_{k}} - x_{m_{k}} \big \Vert \big \Vert x_{m_{k}+1} - \wp ^{*} \big \Vert + 2 \varphi _{m_{k}} \big \langle \wp ^{*}, \, \wp ^{*} - x_{m_{k}+1} \big \rangle \Big ]. \end{aligned}$$

(49)

It follows that

$$\begin{aligned} \big \Vert x_{m_{k}+1} - \wp ^{*} \big \Vert ^{2}&\le 2 \phi _{m_{k}} \big \Vert p_{m_{k}} - x_{m_{k}} \big \Vert \big \Vert x_{m_{k}+1} - \wp ^{*} \big \Vert + 2 \varphi _{m_{k}} \big \langle \wp ^{*}, \, \wp ^{*} - x_{m_{k}+1} \big \rangle . \end{aligned}$$

(50)

Since $\varphi _{m_{k}} \rightarrow 0$ and $\big \Vert x_{m_{k}} - \wp ^{*} \big \Vert$ is bounded, (48) and (50), yield

$$\begin{aligned} \Vert x_{m_{k}+1} - \wp ^{*}\Vert ^{2} \rightarrow 0, \,\, \text {as} \,\, k \rightarrow \infty . \end{aligned}$$

(51)

The above implies that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert x_{k} - \wp ^{*} \Vert ^{2} \le \lim _{n \rightarrow \infty } \Vert x_{m_{k}+1} - \wp ^{*}\Vert ^{2} \le 0. \end{aligned}$$

(52)

Consequently, $x_{n} \rightarrow \wp ^{*}$ and the desired result is obtained. $\square$

Next, we introduce a variant of Algorithm 1 in which the constant step size $\zeta$ is chosen adaptively and thus yield a sequence $\zeta _{n}$ that does not require the knowledge of the Lipschitz-like parameters of the bifunction f.

We start by a simple result concerning the sequence $\{\zeta _{n}\}$.

Lemma 3.2

The sequence $\{\zeta _{n} \}$ generated according to Algorithm 2 is monotonically decreasing with the lower bound $\min \big \{\frac{\eta }{2 \max \{c_{1}, c_{2}\}}, \zeta _{0} \big \}.$

Proof

Assuming that $f(x_{n}, p_{n}) - f(x_{n}, y_{n}) - f(y_{n}, p_{n}) > 0$, such that

$$\begin{aligned} \frac{\eta (\Vert x_{n} - y_{n}\Vert ^{2} + \Vert p_{n} - y_{n}\Vert ^{2})}{2 [f(x_{n}, p_{n}) - f(x_{n}, y_{n}) - f(y_{n}, p_{n})]}&\ge \frac{\eta (\Vert x_{n} - y_{n}\Vert ^{2} + \Vert p_{n} - y_{n}\Vert ^{2})}{2 [c_{1}\Vert x_{n} - y_{n}\Vert ^{2} + c_{2} \Vert p_{n} - y_{n}\Vert ^{2}]} \nonumber \\&\ge \frac{\eta }{2 \max \{c_{1}, c_{2}\}}. \end{aligned}$$

(53)

$\square$

Lemma 3.3

[31] Assume that the bifunction $f: \mathbb {E} \times \mathbb {E} \rightarrow \mathbb {R}$ satisfies the conditions (f1)–(f4); then for every $\wp ^{*} \in \Omega \ne \emptyset$, we have

$$\begin{aligned} \Vert p_{n} - \wp ^{*}\Vert ^{2} \le \Vert x_{n} - \wp ^{*}\Vert ^{2} - \Big (1 - \frac{\eta \zeta _{n}}{\zeta _{n+1}} \Big )\Vert x_{n} - y_{n}\Vert ^{2} - \Big (1 - \frac{\eta \zeta _{n}}{\zeta _{n+1}} \Big ) \Vert p_{n} - y_{n}\Vert ^{2}. \end{aligned}$$

Theorem 3.4

Suppose that condition (f1)–(f4) are hold and solution set $\Omega$ is non-empty. Then, any sequence $\{x_{n}\}$ generated by Algorithm 2 converges strongly to $\wp ^{*} = P_{\Omega } (0).$

Proof

By the definition of $\zeta _{n}$, there is a number $N_{0} \in \mathbb {N}$ such that

$$\begin{aligned} \Vert p_{n} - \wp ^{*}\Vert ^{2} \le \Vert x_{n} - \wp ^{*}\Vert ^{2}, \quad \forall \, n \ge N_{0}. \end{aligned}$$

(54)

The rest of the proof follows from similar arguments in the proof of Theorem 3.1 and hence omitted. $\square$

4 Applications

In this section we consider two mathematical applications, resolve variational inequalities and fixed point problem, and translates our methods for solving these problems.

Given a operator $\mathcal {T}: \mathbb {E} \rightarrow \mathbb {E}$, the fixed point problem is formulated as follows:

Assume that the above operator $\mathcal {T}$ fulfils the following conditions. We assume that following requirements:

($\mathcal {T}$1).:

The operator $\mathcal {T}$ is $\kappa$-strict pseudo-contraction (see e.g., [5]) on $\mathbb {K}$ if

$$\begin{aligned} \Vert Ty_{1} - Ty_{2}\Vert ^{2} \le \Vert y_{1} - y_{2}\Vert ^{2} + \kappa \Vert (y_{1} - Ty_{1}) - (y_{2} - Ty_{2})\Vert ^{2}, \,\, \forall \, y_{1}, y_{2} \in \mathbb {K}; \end{aligned}$$

($\mathcal {T}$2).:

The operator $\mathcal {T}$ is weakly sequentially continuous on $\mathbb {K}$ if

$$\begin{aligned} \mathcal {T}(y_{n}) \rightharpoonup \mathcal {T}(y^{*}) \,\, \text {for any sequence in} \,\, \mathbb {K} \,\, \text {satisfying} \,\, y_{n} \rightharpoonup y^{*}. \end{aligned}$$

For such $\mathcal {T}$ we define the bifunction $f(x, y) = \langle x - \mathcal {T} x, y - x \rangle$, and it can be easily proven that f satisfies Assumptions (f1)–(f4) with $2 c_{1} = 2 c_{2} = \frac{3 - 2 \kappa }{1 - \kappa }$, see for example [39]. With this data Algorithm 1 translates to:

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = \underset{y \in \mathbb {K}}{\arg \min } \{ \zeta f(x_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \} = P_{\mathbb {K}} \big [ x_{n} - \zeta (x_{n} - \mathcal {T} (x_{n})) \big ], \\ p_{n} = \underset{y \in \mathbb {E}_{n}}{\arg \min } \{ \zeta f(y_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \} = P_{\mathbb {E}_{n}} \big [ x_{n} - \zeta (y_{n} - \mathcal {T} (y_{n})) \big ]. \end{array}\right. } \end{aligned}$$

(55)

Corollary 4.1

Let $\mathcal {T} : \mathbb {K} \rightarrow \mathbb {K}$ be a mapping satisfying ($\mathcal {T}$1)–($\mathcal {T}$2) and $Fix(\mathcal {T}) \ne \emptyset .$ Let $x_{0} \in \mathbb {K},$ $0< \zeta < \frac{1 - \kappa }{3 - 2 \kappa }$, $\{\phi _{n}\} \subset (a, b) \subset (0, 1-\varphi _{n})$ and $\{\varphi _{n}\} \subset (0, 1)$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \varphi _{n} = 0 \,\,\text {and} \,\, \sum _{n=1}^{\infty } \varphi _{n} = +\infty . \end{aligned}$$

Consider the iterative update:

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = P_{\mathbb {K}} \big [ x_{n} - \zeta (x_{n} - \mathcal {T} (x_{n})) \big ], \\ p_{n} = P_{\mathbb {E}_{n}} \big [ x_{n} - \zeta (y_{n} - \mathcal {T}(y_{n})) \big ], \\ x_{n+1} = (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n}, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}_{n} = \{z \in \mathbb {E} : \langle (1 - \zeta ) x_{n} + \zeta \mathcal {T}(x_{n}) - y_{n}, z - y_{n} \rangle \le 0 \}. \end{aligned}$$

Then, $\{x_{n}\}$ converges strongly to $\wp ^{*} \in Fix(\mathcal {T}).$

Corollary 4.2

Let $\mathcal {T} : \mathbb {K} \rightarrow \mathbb {K}$ be a mapping satisfying ($\mathcal {T}$1)–($\mathcal {T}$2) and $Fix(\mathcal {T}) \ne \emptyset .$ Let $x_{0} \in \mathbb {K},$ $\eta \in (0, 1),$ $\zeta _{0} > 0$, $\{\phi _{n}\} \subset (a, b) \subset (0, 1-\varphi _{n})$ and $\{\varphi _{n}\} \subset (0, 1)$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \varphi _{n} = 0 \,\,\text {and} \,\, \sum _{n=1}^{\infty } \varphi _{n} = +\infty . \end{aligned}$$

Consider the iterative update:

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = P_{\mathbb {K}} \big [ x_{n} - \zeta _{n} (x_{n} - \mathcal {T} (x_{n})) \big ], \\ p_{n} = P_{\mathbb {E}_{n}} \big [ x_{n} - \zeta _{n} (y_{n} - \mathcal {T}(y_{n})) \big ], \\ x_{n+1} = (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n}, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}_{n} = \{z \in \mathbb {E} : \langle (1 - \zeta _{n}) x_{n} + \zeta _{n} \mathcal {T}(x_{n}) - y_{n}, z - y_{n} \rangle \le 0 \}. \end{aligned}$$

Compute

$$\begin{aligned} \zeta _{n+1} = {\left\{ \begin{array}{ll} \min \bigg \{ \zeta _{n}, \,\, \frac{\eta \Vert x_{n} - y_{n}\Vert ^{2} + \eta \Vert p_{n} - y_{n}\Vert ^{2}}{2 \big [\big \langle (x_{n} - y_{n}) - [T(x_{n}) - T(y_{n})], p_{n} - y_{n} \big \rangle \big ]} \bigg \} \\ \text {if} \quad \big \langle (x_{n} - y_{n}) - [T(x_{n}) - T(y_{n})], p_{n} - y_{n} \big \rangle > 0, \\ \\ \zeta _{n}, otherwise. \\ \end{array}\right. } \end{aligned}$$

Then, $\{x_{n}\}$ converges strongly to $\wp ^{*} \in Fix(\mathcal {T}).$

Next, we apply our results to the classical variational inequalities (VI) problem [18, 35]. Given a set $\mathbb {K}$ and an operator $\mathcal {G} : \mathbb {E} \rightarrow \mathbb {E}$, the (VI) is formulated as follows:

For our purposes we assume that the following requirements are fulfilled.

($\mathcal {G}$1).:

The problem (VI) has solution set, represented by $VI(\mathcal {G}, \mathbb {K})$ is non-empty.

($\mathcal {G}$2).:

The operator $\mathcal {G}$ is pseudo-monotone, that is

$$\begin{aligned} \big \langle \mathcal {G}(y_{1}), y_{2} - y_{1} \big \rangle \ge 0 \Longrightarrow \big \langle \mathcal {G}(y_{2}), y_{1} - y_{2} \big \rangle \le 0, \,\, \forall \, y_{1}, y_{2} \in \mathbb {K}. \end{aligned}$$

($\mathcal {G}$3).:

The operator $\mathcal {G}$ is Lipschitz continuous, i.e., there is $L >0$ such that

$$\begin{aligned} \Vert \mathcal {G}(y_{1}) - \mathcal {G}(y_{2}) \Vert \le L \Vert y_{1} - y_{2}\Vert , \,\, \forall \, y_{1}, y_{2} \in \mathbb {K}; \end{aligned}$$

($\mathcal {G}$4).:

$\limsup \limits _{n\rightarrow \infty } \langle G(x_{n}), y - x_{n} \rangle \le \langle G(p), y - p \rangle ,$ $\forall \, y \in \mathcal {C}$ and $\{x_{n}\} \subset \mathcal {C}$ satisfying $x_{n} \rightharpoonup p.$

For the above $\mathcal {G}$ we define $f(x, y) := \big \langle \mathcal {G}(x), y - x \big \rangle$ for all $x, y \in \mathbb {K}$. Thus (EP) translates to the above variational inequality with $L = 2c_{1} = 2 c_{2}.$ Observe that with such bifunction f, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = \underset{y \in \mathbb {K}}{\arg \min } \{ \zeta f(x_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \} = P_{\mathbb {K}}(x_{n} - \zeta \mathcal {G}(x_{n})), \\ p_{n} = \underset{y \in \mathbb {E}_{n}}{\arg \min } \{ \zeta f(y_{n}, y) + \frac{1}{2} \Vert x_{n} - y\Vert ^{2} \} = P_{\mathbb {E}_{n}} (x_{n} - \zeta \mathcal {G}(y_{n})). \end{array}\right. } \end{aligned}$$

(56)

Corollary 4.3

Let the operator $\mathcal {G} : \mathbb {K} \rightarrow \mathbb {E}$ satisfy ($\mathcal {G}$1)–($\mathcal {G}$4). Let $x_{0} \in \mathbb {K},$ $0< \zeta < \frac{1}{L}$, $\{\phi _{n}\} \subset (a, b) \subset (0, 1-\varphi _{n})$ and $\{\varphi _{n}\} \subset (0, 1)$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \varphi _{n} = 0 \,\,\text {and} \,\, \sum _{n=1}^{\infty } \varphi _{n} = +\infty . \end{aligned}$$

Consider the iterative update

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = P_{\mathbb {K}}(x_{n} - \zeta \mathcal {G}(x_{n})), \\ p_{n} = P_{\mathbb {E}_{n}} (x_{n} - \zeta \mathcal {G}(y_{n})), \\ x_{n+1} = (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n}, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}_{n} = \{z \in \mathbb {E} : \langle x_{n} - \zeta \mathcal {G}(x_{n}) - y_{n}, z - y_{n} \rangle \le 0 \}. \end{aligned}$$

Then the sequence $\{x_{n}\}$ converges strongly to $\wp ^{*} \in VI(\mathcal {G}, \mathbb {K}).$

Corollary 4.4

Let the operator $\mathcal {G} : \mathbb {K} \rightarrow \mathbb {E}$ satisfy ($\mathcal {G}$1)–($\mathcal {G}$4). Let $x_{0} \in \mathbb {K},$ $\eta \in (0, 1),$ $\zeta _{0} > 0$, $\{\phi _{n}\} \subset (a, b) \subset (0, 1-\varphi _{n})$ and $\{\varphi _{n}\} \subset (0, 1)$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \varphi _{n} = 0 \,\,\text {and} \,\, \sum _{n=1}^{\infty } \varphi _{n} = +\infty . \end{aligned}$$

Consider the iterative update

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = P_{\mathbb {K}}(x_{n} - \zeta _{n} \mathcal {G}(x_{n})), \\ p_{n} = P_{\mathbb {E}_{n}} (x_{n} - \zeta _{n} \mathcal {G}(y_{n})), \\ x_{n+1} = (1 - \phi _{n} - \varphi _{n} ) x_{n} + \phi _{n} p_{n}, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}_{n} = \{z \in \mathbb {E} : \langle x_{n} - \zeta _{n} \mathcal {G}(x_{n}) - y_{n}, z - y_{n} \rangle \le 0 \}. \end{aligned}$$

Compute

$$\begin{aligned} \zeta _{n+1} = {\left\{ \begin{array}{ll} \min \bigg \{ \zeta _{n}, \,\, \frac{\eta \Vert x_{n} - y_{n}\Vert ^{2} + \eta \Vert p_{n} - y_{n}\Vert ^{2}}{2 \big [ \big \langle \mathcal {G}(x_{n})- \mathcal {G}(y_{n}), p_{n} - y_{n} \big \rangle \big ]} \bigg \} \quad \text {if} \quad \big \langle \mathcal {G}(x_{n})- \mathcal {G}(y_{n}), p_{n} - y_{n} \big \rangle > 0, \\ \\ \zeta _{n}, else. \\ \end{array}\right. } \end{aligned}$$

Then $\{x_{n}\}$ strongly converges to $\wp ^{*} \in VI(\mathcal {G}, \mathbb {K}).$

Remark 4.5

Note that Assumption ($\mathcal {G}$4) could be exempted in case that $\mathcal {G}$ is monotone. Indeed, this condition is a particular case of (f3) and is used to prove (37). In the absence of condition ($\mathcal {G}$4), the inequality (36) can be accomplished by imposing the monotonicity condition on $\mathcal {G}$. In that case, we get

$$\begin{aligned} \langle \mathcal {G}(y), y - y_{n} \rangle \ge \langle \mathcal {G}(y_{n}), y - y_{n} \rangle ,\quad \forall \, y \in \mathbb {K}. \end{aligned}$$

(57)

From (36), we obtain

$$\begin{aligned} \limsup _{k \rightarrow \infty } \langle \mathcal {G}(y_{n_{k}}), y - y_{n_{k}} \rangle \ge 0,\,\, \forall \, y \in \mathbb {E}_{n}. \end{aligned}$$

(58)

By (57) and (58), we conclude that

$$\begin{aligned} \limsup _{k \rightarrow \infty } \langle \mathcal {G}(y), y - y_{n_{k}} \rangle \ge 0,\quad \forall \, y \in \mathbb {K}. \end{aligned}$$

(59)

Let $y_{s} = (1 - s)z + s y$, for $s \in [0, 1].$ It is clear that $y_{s} \in \mathbb {K}$ for every $s \in (0, 1).$ Since $y_{n_{k}} \rightharpoonup z \in \mathbb {K}$ and $\langle \mathcal {G}(y), y - z \rangle \ge 0$ for every $y \in \mathbb {K}$, we have

$$\begin{aligned} 0 \le \langle \mathcal {G}(y_{s}), y_{s} - z \rangle = s \langle \mathcal {G}(y_{s}), y - z \rangle . \end{aligned}$$

(60)

Therefore, $\langle \mathcal {G}(y_{s}), y - z \rangle \ge 0,$ for $s\in (0, 1).$ Since $y_{s} \rightarrow z$ as $s \rightarrow 0$ and due to continuity of $\mathcal {G}$, we have $\langle \mathcal {G}(z), y - z \rangle \ge 0,$ for $y \in \mathbb {K},$ gives that $z \in VI(\mathcal {G}, \mathbb {K}).$

Remark 4.6

From the above discussion, it can be concluded that Corollaries 4.3 and 4.4 still hold, even if we remove the condition ($\mathcal {G}$4) in case that the bifunction are monotone.

5 Numerical illustrations

In this section, we include three numerical test problems and illustrate the behaviour of our methods with comparisons to some related works in the literature.

Example 5.1

Consider the set (box)

$$\begin{aligned} \mathbb {K} := \{ x \in \mathbb {R}^{m} : -5 \le x_{i} \le 5 \}, \end{aligned}$$

and $f: \mathbb {K} \times \mathbb {K} \rightarrow \mathbb {R}$ is

$$\begin{aligned} f(x, y) = \langle A x + B y + d, y - x \rangle , \quad \forall \, x, y \in \mathbb {K}, \end{aligned}$$

where $d \in \mathbb {R}^{m}$ and A, B are matrices of order m. The matrix A is symmetric positive semi-definite and the matrix and a symmetric negative semi-definite matrix $B - A$ through Lipschitz-type constants are $c_{1} = c_{2} = \frac{1}{2}\Vert A - B\Vert$ (see [37] for details). Two matrices A, B are define by

$$\begin{aligned} A = \begin{pmatrix} 3.1 &{} 2 &{} 0 &{} 0 &{} 0 \\ 2 &{} 3.6 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 3.5 &{} 2 &{} 0 \\ 0 &{} 0 &{} 2 &{} 3.3 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 3 \end{pmatrix}, \quad B = \begin{pmatrix} 1.6 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1.6 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1.5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1.5 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 2 \end{pmatrix}, \quad d = \begin{pmatrix} 1 \\ -2 \\ -1 \\ 2 \\ -1 \end{pmatrix}. \end{aligned}$$

The numerical and graphical results are presented in Figs. 1, 2, 3, 4 and Table 1 by considering different initial points and $TOL=10^{-4}.$ The control parameters are taken in the following way:

• (i) $\zeta = \frac{1}{3c_{1}}$ and $\phi _{n} = \frac{1}{60(n + 2)}$ for Algorithm 2 (H-EgA) in [14].

• (ii) $\zeta = \frac{1}{3c_{1}},$ $\varphi _{n} = \frac{1}{60(n + 2)}$ and $\phi _{n} = \frac{7}{10}(1 - \varphi _{n})$ for Algorithm 1 (MT-EgA).

• (iii) $\zeta _{0} = 0.7,$ $\eta = 0.9,$ $\varphi _{n} = \frac{1}{60(n + 2)}$ and $\phi _{n} = \frac{7}{10}(1 - \varphi _{n})$ for Algorithm 2 (EMT-EgA).

Fig. 1

Comparison of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] with $x_{0}=(0,0,0,0,0)^{T}$

Fig. 2

Comparison of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] with $x_{0}=(1,1,1,1,1)^{T}$

Fig. 3

Comparison of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] with $x_{0}=(1,0,-1,2,1)^{T}$

Fig. 4

Comparison of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] with $x_{0}=(2,-1,3,-4,5)^{T}$

Table 1 Numerical data for Figs. 1, 2, 3 and 4

Example 5.2

Assume that set $\mathbb {K} \subset \mathbb {R}^{5}$ is defined by

$$\begin{aligned} \mathbb {K} = \big \{ (x_{1}, \ldots , x_{5}) : x_{1} \ge -1, x_{i} \ge 1, i = 2, \cdots , 5 \big \}. \end{aligned}$$

Let $f : \mathbb {K} \times \mathbb {K} \rightarrow \mathbb {R}$ is defined in the following way:

$$\begin{aligned} f(x, y) = \sum _{i=2}^{5} (y_{i} - x_{i}) \Vert x\Vert , \,\, \forall x, y \in \mathbb {R}^{5}. \end{aligned}$$

Then, f is Lipschitz-type continuous with $c_{1} = c_{2} = 2,$ and satisfy the items (f1)–(f4). The solution set of the equilibrium problem is $EP(f, \mathbb {K}) = \{(x_{1}, 1, 1, 1, 1) : x_{1} > - 1 \}$ (for more details see [39]). All numerical results are reported in Figs. 5, 6 and Table 2, 3, 4, 5, 6 and 7 by assuming different initial points and $TOL=10^{-3}.$ The control parameters are taken in the following way:

• (i) $\zeta = \frac{1}{4c_{1}}$ and $\phi _{n} = \frac{1}{40(n + 2)}$ for Algorithm 2 (H-EgA) in [14];

• (ii) $\zeta = \frac{1}{4c_{1}},$ $\varphi _{n} = \frac{1}{40(n + 2)}$ and $\phi _{n} = \frac{6}{10}(1 - \varphi _{n})$ for Algorithm 1 (MT-EgA).

• (iii) $\zeta _{0} = 0.7,$ $\eta = 0.85,$ $\varphi _{n} = \frac{1}{40(n + 2)}$ and $\phi _{n} = \frac{6}{10}(1 - \varphi _{n})$ for Algorithm 2 (EMT-EgA).

Fig. 5

Comparison of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] with $x_{0}=(2,3,2,5,2)^{T}$

Fig. 6

Comparison of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] with $x_{0}=(5,6,3,10,8)^{T}$

Table 2 Example 5.2: numerical study of Algorithm 3.2 in [14] and $x_{0}=(2,3,2,5,2)^{T}$

Table 3 Example 5.2: numerical study of Algorithm 1 and $x_{0}=(2,3,2,5,2)^{T}$

Table 4 Example 5.2: numerical study of Algorithm 2 and $x_{0}=(2,3,2,5,2)^{T}$

Table 5 Example 5.2: numerical study of Algorithm 3.2 in [14] and $x_{0}=(5,6,3,10,8)^{T}$

Table 6 Example 5.2: numerical study of Algorithm 1 and $x_{0}=(5,6,3,10,8)^{T}$

Table 7 Example 5.2: numerical study of Algorithm 2 and $x_{0}=(5,6,3,10,8)^{T}$

Example 5.3

Consider the set

$$\begin{aligned} \mathbb {K} := \{ x \in L^{2}([0, 1]): \Vert x\Vert \le 1 \}. \end{aligned}$$

Let us define an operator $\mathcal {G} : \mathbb {K} \rightarrow \mathbb {E}$ such that

$$\begin{aligned} \mathcal {G} (x)(t) = \int _{0}^{1} \big [ x(t) - H(t, s) f(x(s)) \big ] ds + g(t), \end{aligned}$$

where

$$\begin{aligned} H(t, s) = \frac{2ts e^{(t+s)}}{e \sqrt{e^{2}-1}}, \quad f(x) = \cos (x), \quad g(t) = \frac{2t e^{t}}{e \sqrt{e^{2}-1}}. \end{aligned}$$

In the above $\mathbb {E} = L^{2}([0, 1])$ is a Hilbert space through inner product $\langle x, y \rangle = \int _{0}^{1} x(t) y(t) dt, \,\, \forall x, y \in \mathbb {E}$ and the induced norm is $\Vert x\Vert = \sqrt{\int _{0}^{1} |x(t)|^{2} dt}.$ Numerical and graphical results of three methods are shown in Figs. 7, 8, 9, 10 and Table 8 by considering different initial points and $TOL=10^{-3}.$ The control parameters are taken in the following way:

• (i) $\zeta = \frac{1}{5c_{1}}$ and $\phi _{n} = \frac{1}{100(n + 2)}$ for Algorithm 2 (H-EgA) in [14];

• (ii) $\zeta = \frac{1}{5c_{1}},$ $\varphi _{n} = \frac{1}{100(n + 2)}$ and $\phi _{n} = \frac{3}{10}(1 - \varphi _{n})$ for Algorithm 1 (MT-EgA);

• (iii) $\zeta _{0} = 0.50,$ $\eta = 0.50,$ $\varphi _{n} = \frac{1}{100(n + 2)}$ and $\phi _{n} = \frac{3}{10}(1 - \varphi _{n})$ for Algorithm 2 (EMT-EgA).

6 Conclusion

This study established two techniques to figure out the problems of equilibrium. The initial method is a strong convergence through a Mann-type scheme and fixed step size, based on the Lipschitz coefficients. The second method includes a key edge over the initial method due to the self-adapting step size rule. Numerical conclusions have been mentioned to show the numerical effectiveness of proposed methods compared to other methods. Such numerical studies indicate that the Mann-type scheme normally helps in increasing the efficiency of the iterative sequence compared to the Halpern method.

Fig. 7

Numerical inspection of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] when $x_{0}=2t^{2}$

Fig. 8

Numerical inspection of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] when $x_{0}=6t^{3}$

Fig. 9

Numerical inspection of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] when $x_{0}=t^{2}\cos (t)$

Fig. 10

Numerical inspection of Algorithm 1 with Algorithm 2 and Algorithm 3.2 in [14] when $x_{0}=t^{2}\sin (t)$

Table 8 Numerical data for Figs. 7, 8, 9 and 10