In this article, we study a modified maximum principle approach under condition on the weight of the delay term in the feedback and the weight of the term without delay. On that basis, we prove the existence of global solutions for a quasilinear Schrödinger equation in an unbounded domain with a general nonlinear nonlinear optimal control condition in the weakly nonlinear internal feedback. The equation includes many special cases such as classical Schrödinger equations, fractional Schrödinger equations, and relativistic Schrödinger equations, etc. Our results are established by means of the fixed point theory associated with the Schrödinger operator in suitable b-metric spaces. Moreover, we establish general stability estimates by using some properties of Schrödinger convex functions.

中文翻译：

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具有一般非线性最优控制条件的拟线性Schrödinger方程整体解的存在性。

在本文中，我们研究了在反馈中延迟项的权重和无延迟项的权重的条件下的改进的最大原理方法。在此基础上，我们证明了在弱非线性内部反馈中具有一般非线性非线性最优控制条件的无界域中的拟线性Schrödinger方程的整体解的存在性。该方程式包括许多特殊情况，例如经典Schrödinger方程，分数Schrödinger方程和相对论Schrödinger方程等。我们的结果是通过与Schrödinger算子相关的不动点理论在合适的b-度量空间中建立的。此外，我们通过使用Schrödinger凸函数的某些属性来建立一般的稳定性估计。