Fixed points of multivalued convex contractions with application.

Area of Science:

• Fixed-point theory
• Nonlinear analysis
• Metric spaces

Background:

• Fixed-point theory is crucial for solving equations.
• B-metric spaces generalize metric spaces, offering broader applicability.
• Convex contractions are a significant class of mappings in analysis.

Purpose of the Study:

• To establish fixed-point theorems for single-valued convex contraction mappings in b-metric spaces.
• To extend these results to multivalued and F-convex contractions.
• To investigate applications in solving integral equations.

Main Methods:

• Utilizing concepts of convex contraction mappings.
• Working within the framework of b-metric spaces.
• Developing and extending fixed-point theorems.

Main Results:

• Established fixed-point results for single-valued convex contractions in b-metric spaces.
• Extended theorems for multivalued convex contractions and F-convex contractions.
• Obtained an analogue of Nadler's fixed-point theorem for multivalued convex contractions.
• Demonstrated an application to solving nonlinear Fredholm integral equations.

Conclusions:

• The study expands the scope of fixed-point theory in generalized metric spaces.
• The findings provide new tools for analyzing nonlinear equations.
• The research offers insights into the relationships between different types of contractions.