In this paper, the fractional pantograph equations in quantum calculus are studied. We propose a q-fractional variant of the nonlinear pantograph equation with a given initial condition and a nonlinear neutral q-fractional pantograph equation. Based on the fixed-point theory, the main theorems provide some new results regarding the existence and uniqueness of solutions for q-fractional pantograph equations.

中文翻译：

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量子分数受电弓方程的解的存在唯一性

本文研究了量子微积分中的缩放比例方程。我们提出了具有给定初始条件的非线性受电弓方程的q分数变体和非线性中立的q分数受电弓方程。基于定点理论，主要定理提供了一些有关q分数缩放弓方程解的存在性和唯一性的新结果。