On Graphical Symmetric Spaces, Fixed-Point Theorems and the Existence of Positive Solution of Fractional Periodic Boundary Value Problems

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H-(& thetasym;,phi)-contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems.