We study the extragradient method for solving quasi-equilibrium problems in Banach spaces, which generalizes the extragradient method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem assuming neither any monotonicity assumption on the bifunction nor any weak continuity assumption of f in its arguments that in the many well-known methods have been used. Also, we give a necessary and sufficient condition for the solution set of the quasi-equilibrium problem to be nonempty and we show that, in this case, this iterative sequence converges strongly to a solution of the quasi-equilibrium problem. In other words, we prove strong convergence of the generated sequence to a solution of the quasi-equilibrium problem without assuming existence of a solution of the problem. Finally, we give an application of our main result to a generalized Nash equilibrium problem.

中文翻译：

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BANACH空间中准平衡问题的超梯度方法

我们研究了求解Banach空间中拟平衡问题的外梯度法，它推广了求解平衡问题和拟变分不等式的外梯度法。我们提出了一个正则化过程，它确保生成的序列强收敛到准平衡问题的解，在该问题的标准假设下，假设双函数既没有任何单调性假设，也没有任何弱连续性假设F在它的论点中，已经使用了许多众所周知的方法。此外，我们给出了准平衡问题的解集是非空的充分必要条件，并且我们证明了在这种情况下，这个迭代序列强烈地收敛到准平衡问题的解。换句话说，我们证明了生成的序列对准平衡问题的解决方案具有很强的收敛性，而无需假设存在问题的解决方案。最后，我们将我们的主要结果应用于广义纳什均衡问题。