Fixed point method for the existence of solutions to antiperiodic boundary value problems in fractional differential equations within hexagonal suprametric spaces - Sumaiya Tasneem Zubair; Thabet Abdeljawad; Manar A. Alqudah; Aiman Mukheimer (Boundary Value Problems)
Abstract The present article investigates the domain of metric spaces, going beyond traditional bounds by presenting hexagonal suprametric spaces with the aim of extending upon the idea of hexagonal metric spaces (Tiwari and Sharma in Ann. Math. Comput. Sci. 6:35–48, 2022) and Branciari suprametric spaces (Tasneem Zubair in Results Nonlinear Anal. 7(3):80–93, 2024). By means of meticulous analysis and clarification, we illuminate the nuances of this recently established metric space and its elongated counterparts. The newly introduced metric is demonstrated through several illustrations, and its topology is examined. Through the application of well-known fixed point theorems to the framework of theorems about hexagonal suprametric spaces, we reveal a corollary that leads to symmetry requirements, which are required for the existence and uniqueness of fixed points with respect to self-operators in such a space. Eventually, by applying the theoretically proven fixed point theorems, this research examines the existence of solutions for the following nonlinear fractional differential equations with antiperiodic boundary conditions of order z ∈ ( 3 , 4 ] $\mathfrak{z} \in (3, 4]$ : { D z c ϰ ( t ) = ϱ ( t , ϰ ( t ) ) , t ∈ [ 0 , T ] , T > 0 , ϰ ( 0 ) = − ϰ ( T ) , ϰ ′ ( 0 ) = − ϰ ′ ( T ) , ϰ ″ ( 0 ) = − ϰ ″ ( T ) , ϰ ‴ ( 0 ) = − ϰ ‴ ( T ) , $$ \left \{ \begin{aligned} & ^{\mathfrak{c}}\mathsf{D}^{\mathfrak{z}}\varkappa (\mathfrak{t}) = \varrho (\mathfrak{t},\varkappa (\mathfrak{t})),\;\mathfrak{t}\in \mathsf{[0,T]}, \mathsf{T}>0, \\ &\varkappa (0) = -\varkappa (\mathsf{T}),\ \varkappa '(0) = - \varkappa '(\mathsf{T}),\ \varkappa ''(0) = -\varkappa ''(\mathsf{T}), \ \varkappa '''(0) = -\varkappa '''(\mathsf{T}), \end{aligned} \right . $$ in which ϱ is a specified continuous function and D z c ${}^{\mathfrak{c}}\mathsf{D}^{\mathfrak{z}}$ is the Caputo fractional derivative of order z $\mathfrak{z}$ . The theoretical result is demonstrated through an illustrative example presented in the concluding section of the article.