Infinitely many homoclinic solutions for second order nonlinear difference equations with p-Laplacian.

Area of Science:

• Mathematical analysis
• Differential equations
• Discrete mathematics

Background:

• Discrete p-Laplacian equations are crucial in modeling various physical phenomena.
• Understanding homoclinic solutions is vital for analyzing dynamical systems.
• Existing methods often rely on restrictive conditions like periodicity or the Ambrosetti-Rabinowitz condition.

Purpose of the Study:

• To investigate the existence and multiplicity of nontrivial homoclinic solutions for discrete p-Laplacian equations.
• To develop a theoretical framework applicable without the Ambrosetti-Rabinowitz condition.
• To analyze equations with coercive weight functions and superlinear nonlinearities.

Main Methods:

• Application of Nehari manifold methods.
• Utilizing critical point theory.
• Analysis of discrete p-Laplacian operators.

Main Results:

• Established the existence of at least one nontrivial homoclinic solution.
• Proved multiplicity results, indicating the presence of multiple solutions.
• Demonstrated the effectiveness of the methods without standard assumptions.

Conclusions:

• The study successfully proves the existence and multiplicity of nontrivial homoclinic solutions for the studied discrete p-Laplacian equations.
• The employed methods offer a more flexible approach, bypassing common limitations in the field.
• This work contributes to a deeper understanding of discrete dynamical systems and their solutions.