Solvability of a boundary value problem at resonance.

Area of Science:

• Differential Equations
• Mathematical Analysis
• Nonlinear Analysis

Background:

• Boundary value problems are fundamental in modeling physical phenomena.
• Fractional calculus extends classical calculus concepts to non-integer orders.
• Resonance conditions in boundary value problems present unique solvability challenges.

Purpose of the Study:

• To investigate the solvability of a nonlinear fractional boundary value problem.
• To address the specific challenges posed by resonance conditions.
• To establish the existence of solutions for the defined problem.

Main Methods:

• Application of fixed point theorems.
• Utilizing analytical techniques for nonlinear problems.
• Perturbation methods to simplify and solve the problem.

Main Results:

• The perturbed nonlinear fractional boundary value problem is shown to have a solution.
• The original problem is proven to be solvable under the given conditions.
• A specific example illustrates the theoretical findings.

Conclusions:

• The employed fixed point theorems and analytical methods are effective for this class of problems.
• The study contributes to the understanding of solvability for nonlinear fractional boundary value problems at resonance.
• The findings provide a basis for further research in fractional differential equations.