Author
Xian Zhang
Bio: Xian Zhang is an academic researcher from Jimei University. The author has contributed to research in topics: Metric space & Metric (mathematics). The author has an hindex of 2, co-authored 2 publications receiving 1141 citations.
Papers
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TL;DR: In this paper, the authors introduce cone metric spaces and prove fixed point theorems of contractive mappings on these spaces, and prove some fixed point properties of the mappings.
1,171 citations
TL;DR: In this paper, some new generalized contractive type conditions for a pair of mappings in metric space are defined, and some common fixed point results for these mappings are presented.
64 citations
TL;DR: In this paper , the Schrödinger operator was shown to admit a nontrivial solution for the nonlinear Klein-Gordon-Maxwell system, and the weaker superlinear conditions were imposed instead of the common 4-superlinear conditions on f .
Abstract: Abstract This paper is concerned with the nonlinear Klein–Gordon–Maxwell systems. Unlike all known results in the literature, the Schrödinger operator $-\Delta +V$ − Δ + V is allowed to be indefinite and the weaker superlinear conditions are imposed instead of the common 4-superlinear conditions on f . By combining a local linking argument and Morse theory, we obtain that the system admits a nontrivial solution.
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TL;DR: In this article, a new concept of contraction was introduced and a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature was proved.
Abstract: In the article, we introduce a new concept of contraction and prove a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature. The article includes an example which shows the validity of our results, additionally there is delivered numerical data which illustrates the provided example. MSC: 47H10; 54E50
661 citations
01 Jan 2008
TL;DR: In this paper, Huang et al. showed that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normalized constant M > k, and that for non-normal cones and omitting the assumption of normality in some results of Huang and Zhang, they obtained generalizations of the results.
Abstract: Abstract Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normal constant M > k . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results.
553 citations
TL;DR: In this paper, Huang et al. showed that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normalized constant M > k.
519 citations
TL;DR: The existence of common fixed points for mappings satisfying certain contractive conditions, without appealing to continuity, in a cone metric space is established in this paper, which generalizes several well-known comparable results in the literature.
472 citations
TL;DR: A series of normalized and generalized metrics based on the important ingredients of SSIM are constructed and it is shown that such modified measures are valid distance metrics and have many useful properties, among which the most significant ones include quasi-convexity, a region of convexity around the minimizer, and distance preservation under orthogonal or unitary transformations.
Abstract: Since its introduction in 2004, the structural similarity (SSIM) index has gained widespread popularity as a tool to assess the quality of images and to evaluate the performance of image processing algorithms and systems. There has been also a growing interest of using SSIM as an objective function in optimization problems in a variety of image processing applications. One major issue that could strongly impede the progress of such efforts is the lack of understanding of the mathematical properties of the SSIM measure. For example, some highly desirable properties such as convexity and triangular inequality that are possessed by the mean squared error may not hold. In this paper, we first construct a series of normalized and generalized (vector-valued) metrics based on the important ingredients of SSIM. We then show that such modified measures are valid distance metrics and have many useful properties, among which the most significant ones include quasi-convexity, a region of convexity around the minimizer, and distance preservation under orthogonal or unitary transformations. The groundwork laid here extends the potentials of SSIM in both theoretical development and practical applications.
362 citations