#### Songnian He, Lili Liu, Xiaolong Qin, A self-adaptive hybrid steepest descent algorithm for solving a class of variational inequalities, Vol. 2018 (2018), Article ID 49, pp. 1-9

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DOI: 10.23952/jnfa.2018.49

Received October 4, 2018; Accepted December 21, 2019; Published December 31, 2018

Abstract. Let $\mathcal{H}$ be a real Hilbert space. In this paper, we propose a new self-adaptive hybrid steepest descent algorithm for solving a variational inequality problem VI$(Fix(T), F)$, were $F: \mathcal{H}\rightarrow \mathcal{H}$ is a boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of $\mathcal{H}$) and strongly monotone operator and $T: \mathcal{H}\rightarrow \mathcal{H}$ is a nonexpansive mapping with a nonempty fixed point set $Fix(T)$. The strong convergence of our proposed algorithm is proved and the convergence rate estimation is also obtained. The advantage of our algorithm is that it does not require a priori knowledge of the Lipschitz constant of $F$ on any bounded subset of $\mathcal{H}$ and also the strong monotone coefficient.