Browsing by Author "Batool, A."
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Article Exploring the Lower and Upper Solutions Approach for Abc-Fractional Derivative Differential Equations(Springer, 2024) Talib, I.; Riaz, M.B.; Batool, A.; Tunç, C.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. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2024.Article Extension of Lower and Upper Solutions Approach for Generalized Nonlinear Fractional Boundary Value Problems(Taylor and Francis Ltd., 2022) Batool, A.; Talib, I.; Riaz, M.B.; Tunç, C.Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results. © 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
