New fixed points of mixed monotone operators with applications to nonlinear integral equations

Area of Science:

• Analysis
• Functional Analysis
• Nonlinear Equations

Background:

• Basic results on cones and partial orders in Banach spaces are reviewed.
• Existing literature on fixed point theorems often relies on compactness or continuity conditions.

Purpose of the Study:

• To establish new fixed point theorems for mixed monotone operators.
• To apply these theorems to nonlinear functional integral equations.
• To generalize and extend existing results in the field.

Main Methods:

• Utilizing properties of cones and partial orders within Banach spaces.
• Employing monotone iterative techniques.
• Developing theorems for operators with concavity and convexity properties.

Main Results:

• New fixed point theorems for mixed monotone operators are derived.
• These theorems are proven without assumptions of compactness or continuity.
• The results are successfully applied to nonlinear functional integral equations on unbounded regions.

Conclusions:

• The established fixed point theorems offer a broader applicability.
• The findings extend and generalize previous research on monotone operators and integral equations.
• This work provides a foundation for further investigation into nonlinear analysis.