A new class of inertial algorithms with monotonic step sizes for solving fixed point and variational inequalities

• Published in: Mathematical methods in the applied sciences Vol. 45; no. 16; pp. 9061 - 9088
• Main Authors: Rehman, Habib ur, Kumam, Poom, Kumam, Wiyada, Sombut, Kamonrat
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.11.2022
• Subjects: Adaptive algorithms; fixed point problem; Fixed points (mathematics); Hilbert space; Inequalities; inertial algorithms; Iterative algorithms; Iterative methods; Mathematical analysis; strong convergence theorems; subgradient extragradient algorithm; Tseng's extragradient algorithm; variational inequalities
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8293

This study presents two inertial type extragradient algorithms for finding a common solution to the monotone variational inequalities and fixed point problems for ρ$$ \rho $$‐demicontractive mapping in real Hilbert spaces. We provide inertial type iterative algorithms with self‐adaptive variable step size rules that do not require prior knowledge of the operator value. Our algorithms employ a basic step size rule, which is derived by certain computations at each iteration. Without previous knowledge of the operators Lipschitz constant, two strong convergence theorems were obtained. Finally, we present a number of numerical experiments to evaluate the efficacy and applicability of the proposed algorithms. The conclusions of this study on variational inequality and fixed point problems support and extend previous findings.