On multivalued contractions in cone metric spaces without normality

Area of Science:

• Mathematics
• Functional Analysis
• Metric Spaces

Background:

• Cone metric spaces generalize metric spaces using cones.
• Previous work by Wardowski (2011) established fixed-point theorems for set-valued contractions in normal cone metric spaces.
• Janković et al. (2011) highlighted limitations of fixed-point problems in nonnormal cone metric spaces.

Purpose of the Study:

• To extend Wardowski's fixed-point results to a more general setting.
• To prove fixed-point theorems for set-valued contractions without the normality assumption on the cone.
• To establish a new cone metric H : A × A → E for families of subsets.

Main Methods:

• Introduction of a new cone metric H : A × A → E.
• Application of fixed-point theory for set-valued contractions.
• Development of proofs that do not rely on the normality of the underlying cone.

Main Results:

• A new cone metric H was established for families of subsets in a cone metric space.
• Fixed-point theorems for set-valued contractions were proven without the normality assumption on cones.
• The results generalize and improve upon existing theorems in the field.

Conclusions:

• The normality assumption on cones is not necessary for establishing fixed-point theorems for set-valued contractions.
• The study expands the applicability of fixed-point theory in cone metric spaces.
• This work provides a foundation for further research in generalized metric spaces.