Discussion on the fixed point problems with constraint inequalities

Area of Science:

• Mathematical Analysis
• Topology

Background:

• Fixed-point theory is crucial in various mathematical fields.
• Existing theorems often require complete metric spaces, limiting their application.

Purpose of the Study:

• Introduce comparable complete metric spaces.
• Establish new fixed-point theorems for incomplete metric spaces.
• Generalize existing fixed-point theorems.

Main Methods:

• Developing the concept of comparable complete metric spaces.
• Applying fixed-point theorems in the context of incomplete metric spaces.
• Proving new theorems and deriving corollaries.

Main Results:

• New fixed-point theorems for mappings in incomplete metric spaces.
• Achieved results with weaker conditions than prior studies (e.g., Ansari et al., 2018).
• Demonstrated generalization of existing literature through examples.

Conclusions:

• The introduced concept and theorems expand the scope of fixed-point theory.
• The results offer a more generalized framework applicable to a wider range of spaces.
• This work provides a foundation for further research in metric spaces and fixed-point theory.