In this paper we have established some fixed point theorems in complete and compact metric spaces.

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Underground utilities are important assets that provide basic services for society’s daily life. They are generally very complex and remain unnoticed until they fail due to any particular reason. The stakeholders involved in the design, construction, and maintenance of utility infrastructure face many problems due to the traditional underground utility management system, resulting in injuries, loss of life, disruptions, project delays, and financial loss. The problem with the traditional system is that it uses 2D drawings and keeps unreliable information or a lack of updated information, which makes it an inefficient utility management system. With the advancement in construction information technology, we can address this effectively by integrating BIM and GIS. In this paper, a novel integrated BIM-GIS framework for underground utility management systems was developed on the basis of IFC to CityGML mapping. It provides an effective underground utility management system that facilitates designers in optimization of the design, assists in the excavator’s operator by providing real-time three-dimensional spatial information during the construction process, and acts as an as-built information database for utility facility management. For validation, a real-time project case study indicated that the proposed system can effectively provide comprehensive underground utility information at different project stages.