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Some existence results on nonlinear fractional differential equations
Dumitru Baleanu1, Shahram Rezapour, Hakimeh Mohammadi
1Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey. dumitru@cankaya.edu.tr
Summary
This study proves the existence and uniqueness of solutions for nonlinear fractional differential equations using fixed-point methods. The research addresses specific boundary conditions, advancing fractional calculus applications.
Area of Science:
- Mathematics
- Applied Mathematics
- Fractional Calculus
Background:
- Fractional differential equations (FDEs) are increasingly used to model complex phenomena.
- Boundary-value problems (BVPs) are crucial for analyzing FDE solutions.
- Fixed-point theorems provide powerful tools for proving solution existence and uniqueness.
Purpose of the Study:
- To investigate the existence and uniqueness of solutions for a nonlinear fractional differential equation.
- To analyze the problem with a Riemann-Liouville fractional derivative.
- To consider two distinct types of boundary conditions: periodic and three-point.
Main Methods:
- Application of fixed-point theorems.
- Analysis of nonlinear fractional differential equations.
- Utilizing Riemann-Liouville fractional derivatives.
- Solving boundary-value problems with specific conditions.
Main Results:
- Demonstrated the existence and uniqueness of a solution for the specified nonlinear fractional differential equation boundary-value problem.
- Established conditions for the solvability of the FDE under different boundary constraints.
- Provided a theoretical framework for analyzing such problems.
Conclusions:
- The fixed-point approach is effective for guaranteeing unique solutions in nonlinear fractional differential equations.
- The findings contribute to the theoretical understanding of fractional boundary-value problems.
- This work has implications for modeling systems described by fractional dynamics.
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