We introduce and study the concept of φ-metric modular space and, then define φ-α-Meir-Keeler contraction on it and explore its fixed point. Further, we define the Hausdorff distance between two non-empty compact subsets of the considered space. Some topological properties of φ-metric modular space are also explored. Additionally, we prove the existence of the attractor (fractal) of the IFS consisting of φ-α-Meir-Keeler contractions.

中文翻译：

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φ-度量模空间中的不动点定理和迭代函数系统

介绍并研究了φ-度量模空间的概念，并在其上定义了φ-α-Meir-Keeler收缩并探讨了其不动点。此外，我们定义了所考虑空间的两个非空紧子集之间的豪斯多夫距离。还探讨了 φ 度量模空间的一些拓扑性质。此外，我们证明了由 φ-α-Meir-Keeler 收缩组成的 IFS 吸引子（分形）的存在。