# New Iterative Scheme Involving Self-Adaptive Method for Solving Mixed Variational Inequalities

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma 1.**

**Proof.**

**Definition 1**

**.**We can define the resolvent operator involving a maximal monotone operator A on H, for a given constant $\rho >0,$ such as:

**Remark 1.**

**Lemma 2**

**.**For a given $f\in E,$ and $z\in H,$ we have

**Lemma 3**

**.**Given a function $f\in H$ as a solution of the inequality (1), then we have

**Lemma 4.**

## 3. Main Results

Algorithm 1 Self-adaptive Iterative Scheme |

Step 0: Given $\rho >0,\phantom{\rule{4pt}{0ex}}\u03f5>0,\phantom{\rule{4pt}{0ex}}\mu \in (0,\phantom{\rule{4pt}{0ex}}1),\phantom{\rule{4pt}{0ex}}\gamma \in [1,\phantom{\rule{4pt}{0ex}}2),\phantom{\rule{4pt}{0ex}}{\delta}_{0},\phantom{\rule{4pt}{0ex}}\delta \in (0,\phantom{\rule{4pt}{0ex}}1)$ and ${f}^{0}\in H,$ set $n=0.$ Step 1: Stopping criteria: Set ${\rho}_{n}=\rho .$ If $\u2225R\left({f}^{n}\right)\u2225<\u03f5,\phantom{\rule{4pt}{0ex}}$ otherwise, satisfying
$$\u2225{\rho}_{n}(T\left({f}^{n}\right)-T\left({h}^{n}\right))\u2225<\delta \u2225R\left({f}^{n}\right)\u2225,$$
where
$${h}^{n}={J}_{\varphi}[{f}^{n}-\gamma D\left({f}^{n}\right)-\gamma T\left({f}^{n}\right)],$$
Step 2: Compute
$$D\left({f}^{n}\right)=R\left({f}^{n}\right)-\rho T\left({f}^{n}\right)+\rho T{J}_{\varphi}[{f}^{n}-\rho T\left({f}^{n}\right)],$$
where
$$R\left({f}^{n}\right):={f}^{n}-{J}_{\varphi}[{f}^{n}-\rho T\left({f}^{n}\right)].$$
Step 3: Get the next iterate
$$\begin{array}{ccc}\hfill {h}^{n+1}& =& {J}_{\varphi}[{f}^{n}-\gamma D\left({f}^{n}\right)-\gamma T\left({f}^{n}\right)],\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {f}^{n+1}& =& {J}_{\varphi}[g\left({h}^{n+1}\right)-\rho T\left({h}^{n+1}\right)],\hfill \end{array}$$
then set $\rho =\frac{{\rho}_{n}}{\mu},$ else set $\rho ={\rho}_{n}.$ Repeat step 1 by substituting $n=n+1$. |

**Theorem 1.**

**Proof.**

**f**${}^{n})$ give the following expression:

## 4. Numerical Results

**Example 1.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Order of Matrix | Algorithm 1 |
---|---|

n | No. It. |

100 | 42 |

200 | 54 |

300 | 46 |

500 | 31 |

700 | 41 |

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**MDPI and ACS Style**

Mukheimer, A.; Ullah, S.; Bux, M.; Arif, M.S.; Abodayeh, K.
New Iterative Scheme Involving Self-Adaptive Method for Solving Mixed Variational Inequalities. *Axioms* **2023**, *12*, 310.
https://doi.org/10.3390/axioms12030310

**AMA Style**

Mukheimer A, Ullah S, Bux M, Arif MS, Abodayeh K.
New Iterative Scheme Involving Self-Adaptive Method for Solving Mixed Variational Inequalities. *Axioms*. 2023; 12(3):310.
https://doi.org/10.3390/axioms12030310

**Chicago/Turabian Style**

Mukheimer, Aiman, Saleem Ullah, Muhammad Bux, Muhammad Shoaib Arif, and Kamaleldin Abodayeh.
2023. "New Iterative Scheme Involving Self-Adaptive Method for Solving Mixed Variational Inequalities" *Axioms* 12, no. 3: 310.
https://doi.org/10.3390/axioms12030310