Exploring the Lower and Upper Solutions Approach for ABC-Fractional Derivative Differential Equations

• Published in: International journal of applied and computational mathematics 2024, Vol.10 (6), Article 170
• Main Authors: Talib, Imran, Riaz, M. Bilal, Batool, Asmat, Tunç, Cemil
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Boundary conditions; Boundary value problems; Computational Science and Engineering; Derivatives; Differential equations; Fixed points (mathematics); Fractional calculus; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Operators (mathematics); Original Paper; Theoretical
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-024-01803-8

The lower and upper solution approach has been widely employed in the literature to ensure the existence of solutions for integer-order boundary value problems. Therefore, in this proposed study, our primary objective is to extend this method to establish the existence results for Atangna-Baleanu-Caputo (ABC) fractional differential equations of order 0 < γ < 1 , with generalized nonlinear boundary conditions. We propose a generalized approach that unifies the existence criteria for certain specific boundary value problems formulated using the ABC fractional-order derivative operator, particularly addressing periodic and anti-periodic cases as special instances. The framework of the proposed generalized approach relies heavily on the concept of coupled lower and upper solutions together with certain fixed point results, including Arzela-Ascoli and Schauder’s fixed point theorems. By means of the generalized approach, we first define appropriate lower and upper solutions that bound the potential solution. We then construct a modified problem that incorporates these bounding solutions, ensuring the existence of a solution to the original problems without relying on iterative techniques. This approach involves verifying that the lower solution is less than or equal to the upper solution, and that both satisfy the given boundary conditions, thus guaranteeing the existence of a solution within the specified bounds. The inclusion of the specific examples with periodic and anti-periodic boundary conditions further reinforces the validity and relevance of our theoretical results.