In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.

中文翻译：

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在Caputo整合环境中集电弓包含问题的新结构

在这项工作中，我们首次在Caputo和Riemann-Liouville设置中同时应用了新定义的顺应算子，从而重新制定并研究了著名的受电弓微分方程。实际上，我们得出了符合Riemann-Liouville整合积分条件的三点Caputo整合受电弓BVP的包含版本所对应的解的存在性准则。为了实现此目标，我们在某些情况下建立了主要结果，包括下半连续，上半连续和Lipschitz集值映射。最终，本研究的最后一部分致力于提出两个数值模拟示例，以证实我们的发现的一致性。