Endpoint results are presented for multi{valued CYCLIC CONTRACTION mappings on complete metric spaces (X; d). Our results extend previous results given by Nadler (1969), Da er{Kaneko (1995), Harandi (2010), Moradi and Kojasteh (2012) and Karapinar (2011).

In this paper, we establish and prove the existence of best proximity points for multivalued CYCLIC FF-CONTRACTION mappings in complete metric spaces. Our results improve and extend various results in literature.

The purpose of this paper is to  nd of the theoretical results of  xed point theorems for a mixed monotone mapping in a metric space endowed with partially order by using the GENERALIZED Kanann type contractivity of assumption. Also, as an application, we prove the existence and uniqueness of solution for a  rst{order ordinary di erential equation with periodic boundary conditions admitting only the existence of a mixed -solution.

The aim of this paper is to prove the existence and uniqueness of fixed point for (j-f)-weak CONTRACTION mappings and (y−f)-weak CONTRACTION mappings in a complete and Hausdorff GENERALIZED metric space.

In this paper, we first introduce a GENERALIZED version of open shop scheduling (OSS), called GENERALIZED CYCLIC open shop scheduling (GCOSS) and then develop a hybrid method of metaheuristic to solve this problem. Open shop scheduling is concerned with processing n jobs on m machines, where each job has exactly m operations and operation i of each job has to be processed on machine i . However, in our proposed model of GCOSS, processing each operation needs more than one machine (or other resources) simultaneously. Furthermore, the schedule is repeated more than once. It is known that OSS is NP-hard. Therefore, for obtaining a good solution for GCOSS, which is obviously NP-hard, a hybrid algorithm is also developed. This method is constructed by hybridizing ant colony optimization (ACO), beam search and linear programming (LP). To verify the accuracy of the method, we also compare the results of this algorithm with the optimal solution for some special problems.

Let, and be nonempty subsets of a matric space. Then the mapping is called CYCLIC if, and. Consider the following optimization problem Let, certainly if the condition be true for some then it is best answer for optimization problem, that we called it triple best proximity point of. In this paper, first we introduce the notion of 3-CYCLIC summing Meir-Keeler CONTRACTIONs as a generalization of 3-CYCLIC summing CONTRACTIONs, then we obtain the conditions for the existence of a triple best proximity point for these class of mappings in the metric spaces with property UC. Our results in this paper are true for a n-CYCLIC summing Meir-Keeler CONTRACTION just we work with order 3 for the simplicity of proofs. Note that, our results are generalizations of some existing theorems with shorter and simpler proofs. Note that, our results are generalizations of some existing theorems with shorter and simpler proofs.

Background: Anti-CYCLIC citrullinated peptide (CCP) antibodies are highly specific and predictive for rheumatoid arthritis (RA). Recently, existence of anti-CCP in several other inflammatory conditions from psoriatic arthritis to neurodegenerative disorder like Alzheimer’s disease (AD) has been reported. We compared the level of anti-CCP titer between the patients with mild cognitive impairment (MCI) and Alzheimer’s disease; then, we compared it between the patients with mild rheumatoid arthritis and Alzheimer’s disease.Methods: In this cross-sectional study, 40 patients with rheumatoid arthritis, 29 patients with Alzheimer’s disease, 30 patients with mild cognitive impairment and 30 healthy controls were enrolled. Anti-CCP titer was measured by solid phase enzyme immunoassay kit. Statistical analysis was performed using one-way analysis of variance and post-hoc Tukey tests.Findings: Level of anti-CCP titer (mean±SD) was higher significantly in patients with Alzheimer’ s disease (13.6±1.6) was higher than healthy subjects (4.2±1.1) and patients with mild cognitive impairment (4.8±1.3) (P=0.0100 for both). Anti-CCP serum level in 32.1% of patients with Alzheimer’s disease and 95.5% of patients with mild rheumatoid arthritis was higher than normal range. In addition, the antibody level was significantly higher in patients with rheumatoid arthritis (54.7±4.8) compared to patients with Alzheimer’s disease (P=0.0001).Conclusion: We obviously observed differences in Anti-CCP titer between patients with Alzheimer’s disease and those with mild rheumatoid arthritis. It was expected since anti-CCP has a high specificity for rheumatoid arthritis and become positive in early stages of disease with a high predictive value. However, we found that in early stages of cognitive impairment, we could not use this biomarker as a diagnostic test but during disease progression, a significant difference was observed between patients with mild cognitive impairment and those with Alzheimer’s disease. It needs more studies to confirm the role of anti-CCP antibody production in patients with Alzheimer’s disease.

We introduce the notion of quasi-CYCLIC-nonCYCLIC pair and its relevant new notion of coincidence quasi-best proximity points in a con-vex metric space. In this way we generalize the notion of coincidence-best proximity point already introduced by M. Gabeleh et al [14]. It turns out that under some circumstances this new class of mappings contains the class of CYCLIC-nonCYCLIC mappings as a subclass. The existence and con-vergence of coincidence-best and coincidence quasi-best proximity points in the setting of convex metric spaces are investigated.

Bentea and Tarnauceanu (An. Stiint. Univ. Al. I. Cuza Ias, Ser. Noua, Mat., 54 (1) (2008), 209-220) proposed the following problem: Find an explicit formula for the number of fuzzy subgroups of a finite hamiltonian group of type Q8´Zn where Q8 is the quaternion group of order 8 and n is an arbitrary odd integer. In this paper we consider more general group: the direct product of a GENERALIZED quaternion group of any even order and a CYCLIC group of any odd order. For this group we give an explicit formula for the number of fuzzy subgroups.