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Fixed points of contractive mappings in b-metric-like spaces.

TheScientificWorldJournal·2014
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Fixed point results for G-α-contractive maps with application to boundary value problems.

Nawab Hussain1, Vahid Parvaneh2, Jamal Rezaei Roshan3

  • 1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.

Thescientificworldjournal
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PubMed
Summary

This study introduces a new generalized b-metric-like space, unifying existing metric concepts. It establishes fixed-point theorems for G-α-admissible mappings, with applications to boundary value problems.

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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.