Fixed point results for G-α-contractive maps with application to boundary value problems.

Area of Science:

• Mathematical Analysis
• Topology
• Fixed Point Theory

Background:

• Existing metric spaces like G-metric, metric-like, and b-metric have limitations.
• A unified framework is needed to explore broader topological and structural properties.
• Fixed point theory is crucial for solving various mathematical problems.

Purpose of the Study:

• To introduce and investigate a novel concept: the generalized b-metric-like space.
• To establish new fixed-point theorems for specific contractive mappings within these spaces.
• To demonstrate the practical utility of the new framework through examples and an application.

Main Methods:

• Unification of G-metric, metric-like, and b-metric concepts.
• Definition and analysis of topological and structural properties of the new space.
• Development of fixed-point theorems for G-α-admissible contractive mappings.
• Application to first-order periodic boundary value problems.

Main Results:

• A new class of spaces, generalized b-metric-like spaces, has been defined.
• New fixed-point theorems for G-α-admissible contractive mappings have been established.
• The existence of solutions for periodic boundary value problems is demonstrated.

Conclusions:

• The generalized b-metric-like space provides a unified and powerful framework.
• The derived fixed-point theorems offer valuable tools for theoretical advancements.
• The application highlights the practical relevance and applicability of the new results.