This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.

中文翻译：

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自反Banach空间中广义非线性似变分不等式的可解性和算法

本文讨论了自反Banach空间中一类新的广义非线性似变分不等式的可解性和算法。通过使用Banach不动点定理，Schauder不动点定理和FanKKM定理，我们获得了充分的条件，从而保证了广义非线性似变分不等式解的存在。我们还介绍了一种辅助变分不等式，并利用极大极小不等式，得到了该辅助变分不等式解的存在性和唯一性，用于提出一种求解广义非线性变分不等式的迭代算法。在某些情况下，借助辅助原理技术，我们都建立了广义非线性似变分不等式解的存在性和唯一性，并讨论了由迭代算法生成的迭代序列的收敛性。我们的结果扩展，改进和统一了文献中的几种已知结果。