In this paper, we study the solution of fractal energy integral equation for one-dimensional compressible flows without body force using measure of noncompactness. We also discuss the solution of the local fractal equation of losing energy system using the notion of local fractal differential idea. For this, a new notion of $\chi$-$\mathrm{\Delta }$-set contraction condition under simulation function is defined and two main fixed point and coupled fixed point results are obtained.

中文翻译：

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使用非紧致度的解无体力的分形能量积分算子

在本文中，我们利用非紧致性的度量研究了一维可压缩流动而没有体力的分形能量积分方程的解。我们还使用局部分形微分思想的概念来讨论能量损失系统的局部分形方程的解。为此，一个新的概念$\chi$--$\mathrm{\Delta }$定义了模拟函数下的设定收缩条件，得到了两个主要的固定点和耦合的固定点结果。