We introduce the notion of λ-spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang 2007. We establish some critical point theorems in the setting of λ-spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al. 1983 and the results given by Khanh and Quy 2010 to λ-spaces and cone metric spaces. As applications of our results, we characterize the completeness of λ-space cone metric spaces and quasimetric spaces are special cases of λ-space and studying the Ekeland type variational principle for single variable vector-valued functions as well as for multivalued bifunctions in the setting of cone metric spaces.

Yttrium oxyfluoride (YOF) protective materials were fabricated on sputter-deposited yttrium oxide (Y2O3) by high-density (sulfur fluoride) SF6 plasma irradiation. The structures, compositions, and fluorocarbon-plasma etching behaviors of these films were systematically characterized by various techniques. After exposure to SF6 plasma, the Y2O3 film surface was fluorinated significantly to form a YOF film with an approximate average thickness of 30 nm. X-ray photoelectron spectroscopy revealed few changes in the elemental and chemical compositions of the surface layer after fluorination, confirming the chemical stability of the YOF/Y2O3 sample. Transmission electron microscopy confirmed a complete lattice pattern on the YOF/Y2O3 structure after fluorocarbon plasma exposure. These results indicate that the SF6 plasma-treated Y2O3 film is more erosion resistant than the commercial Y2O3 coating, and thus accumulates fewer contamination particles.

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We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.

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