This paper presents a fixed point theorem for cyclic relatively nonexpansive mappings in generalized semimetric spaces. The author uses a geometric notion of seminormal structure to establish the result and then applies it to uniformly convex Banach spaces. The paper also discusses the stability of seminormal structure in generalized semimetric spaces. The paper begins with an introduction to normal structure in Banach spaces and the fixed point theorem of Kirk. It then extends this to cyclic mappings in metric spaces, leading to the study of best proximity points. The author then introduces the concept of cyclic relatively nonexpansive mappings and proves a fixed point theorem for such mappings in generalized semimetric spaces. The paper defines generalized semimetric spaces and discusses the properties of admissible subsets. It introduces the concept of seminormal structure and proves a theorem that shows that if a pair of subsets in a generalized semimetric space has seminormal structure, then a cyclic relatively nonexpansive mapping has a fixed point in the intersection of the subsets. The paper also discusses the stability of seminormal structure in generalized semimetric spaces and proves a theorem that shows that if a cyclic relatively nonexpansive mapping is defined on a pair of subsets with seminormal structure, then there exists a point in the union of the subsets such that the distance from the point to its image under the mapping is bounded by a given value. The paper concludes with a corollary that applies the main result to uniformly convex Banach spaces and provides an example to illustrate the results. The author also acknowledges the support of a grant from IPM.This paper presents a fixed point theorem for cyclic relatively nonexpansive mappings in generalized semimetric spaces. The author uses a geometric notion of seminormal structure to establish the result and then applies it to uniformly convex Banach spaces. The paper also discusses the stability of seminormal structure in generalized semimetric spaces. The paper begins with an introduction to normal structure in Banach spaces and the fixed point theorem of Kirk. It then extends this to cyclic mappings in metric spaces, leading to the study of best proximity points. The author then introduces the concept of cyclic relatively nonexpansive mappings and proves a fixed point theorem for such mappings in generalized semimetric spaces. The paper defines generalized semimetric spaces and discusses the properties of admissible subsets. It introduces the concept of seminormal structure and proves a theorem that shows that if a pair of subsets in a generalized semimetric space has seminormal structure, then a cyclic relatively nonexpansive mapping has a fixed point in the intersection of the subsets. The paper also discusses the stability of seminormal structure in generalized semimetric spaces and proves a theorem that shows that if a cyclic relatively nonexpansive mapping is defined on a pair of subsets with seminormal structure, then there exists a point in the union of the subsets such that the distance from the point to its image under the mapping is bounded by a given value. The paper concludes with a corollary that applies the main result to uniformly convex Banach spaces and provides an example to illustrate the results. The author also acknowledges the support of a grant from IPM.