Existence and Uniqueness of Positive Solution for Discrete Multipoint Boundary Value Problems

Area of Science:

• Mathematics
• Applied Mathematics
• Numerical Analysis

Background:

• Difference equations are fundamental in modeling discrete dynamical systems.
• Understanding the behavior of solutions, particularly positive ones, is crucial for applications.
• Boundary value problems for difference equations present unique analytical challenges.

Purpose of the Study:

• To investigate the existence and uniqueness of positive solutions for a second-order nonlinear difference equation.
• To analyze the equation under two distinct sets of boundary conditions.
• To develop an iterative approach for constructing approximate positive solutions.

Main Methods:

• The core methodology relies on a fixed-point theorem applied to a sum operator.
• Analysis involves techniques for second-order difference equations.
• Iterative methods are employed for solution approximation.

Main Results:

• Existence and uniqueness of at least one positive solution are established.
• The study demonstrates the convergence of an iterative series to the solution.
• The findings are valid for specific ranges of parameters α and β, and integer η.

Conclusions:

• The paper successfully establishes the existence and uniqueness of positive solutions for the given difference equation.
• A constructive iterative method is provided for approximating these solutions.
• This work contributes to the analytical and numerical understanding of discrete boundary value problems.