In this paper, by concept of Γ -function which is define on q-G-m (quasi-G-metric) space, we establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above. As application we prove generalized flower petal theorem in q-G-m.Keywords: Γ -Function, q-G-m space, generalized EVP, lower semicontinuous from above function, generalized Caristi's (common) fixed point theorem, nonconvex minimax theorem, generalized flower petal theorem.

( ‫ﺧﻮرﻣﻮﺳﻲ‬ ‫در‬‫ﻫﺎﺷﻤﻲ‬

By using the concept of Γ-distance, we prove EVP (Ekeland's variational principle) on quasi-F-metric (q-F-m) spaces. We apply EVP to get the existence of the solution to EP (equilibrium problem) in complete q-F-m spaces with Γ-distances. Also, we generalize Nadler's fixed point theorem.

Abstract:We consider a distance function on generalized metric spaces and we get a generalization of Ekeland Variational Principle (EVP). Next, we prove that EVP is equivalent to Caristi-Kirk fixed point theorem and minimization Takahashi's theorem.

Introduction: Periodontal disease is an inflammatory disease of the supporting tissues of the teeth caused by certain microorganisms and leads to progressive destruction of the periodontal ligament, alveolar bone with pocket formation or the recession, or both. Periodontal disease is usually based on clinical parameters diagnosed and recorded. Other advanced diagnostic techniques of periodontal disease is evaluation of host response, which includes the study of specific or non-specific mediators such as albumin, globulin and creatine phosphokinase enzyme by biochemical methods or immunological response that as part of the individual periodontal infections are known. Methods: The study population consisted of 30 patients. After observing isolation by cotton rolls, 50 to 100 ml GCF via 30 paper point and wetting with a height of 5 mm of the area with more involvement in terms of standard of patients with moderate periodontitis referred to Periodontics ward of Dental School in Mashhad pre-treatment phase 1 (scaling and root planning) and 4 weeks after the treatment is provided. Then changes in albumin, total protein, globulin and CPK enzyme activity in the GCF using biochemical kits measured. Result: In our study, mean CPK in patients with chronic periodontitis before treatment was further after treatment but not significantly different, but the average amount of albumin, total protein and globulin chronic periodontitis patients before treatment lesser than after treatment but there was no significant difference either. Conclusion: In this study, we showed that albumin, total protein and globulin was significantly increased after treatment of periodontitis patients.