On some fixed point results using simulation functions via w-distance with applications

Area of Science:

• Mathematical Analysis
• Fixed-Point Theory
• Metric Spaces

Background:

• The study builds upon the established notion of simulation functions in mathematical analysis.
• It addresses the need for new contraction principles within the framework of w-distance mappings.

Purpose of the Study:

• To introduce and define a novel concept termed $$-$$-contraction.
• To establish new fixed-point theorems in complete metric spaces using this concept.
• To demonstrate the applicability of the new results to nonlinear fractional differential equations, integral equations, and spring-mass systems.

Main Methods:

• Development of the $$-$$-contraction concept based on simulation functions and w-distance mappings.
• Formulation and rigorous proof of fixed-point theorems in complete metric spaces.
• Application of the theoretical results to solve specific types of nonlinear equations and physical systems.

Main Results:

• Several new fixed-point results are established for $$-$$-contractions in complete metric spaces.
• Concrete examples are provided to illustrate the practical utility and validity of the derived theorems.
• The research demonstrates the existence of unique solutions for nonlinear fractional differential equations, nonlinear integral equations, and spring-mass systems.

Conclusions:

• The newly introduced $$-$$-contraction concept offers a powerful tool for fixed-point theory.
• The findings significantly generalize and extend existing results in the field.
• The study highlights the broad applicability of the new framework to diverse mathematical and applied problems.