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

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Fixed points of a new type of contractive mappings in complete metric spaces

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.

Some notes on the paper "Cone metric spaces and fixed point theorems

Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings"

Common fixed point results for noncommuting mappings without continuity in cone metric spaces

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.

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On the Mathematical Properties of the Structural Similarity Index

Cone metric spaces and fixed point theorems of contractive mappings

New Ball Campanato-Type Function Spaces and Their Applications

This book explores properties of generalized Herz spaces and establishes a real-variable theory of Hardy spaces associated with generalized Herz spaces. 

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Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko

Let A be a general expansive matrix on $$\mathbb {R}^n$$ and $$H_A^{p(\cdot ),q}(\mathbb R^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A, where $$p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty ]$$ denotes a variable exponent function satisfying the globally log-Hölder continuous condition and $$q\in (0,\infty )$$ . In this article, the authors give the appropriate dual space of $$H_A^{p(\cdot ),q}(\mathbb {R}^n)$$ with full range $$p(\cdot )$$ by introducing some anisotropic variable $$\eta $$ -type Campanato spaces. As an application, the Carleson measure characterizations of these $$\eta $$ -type Campanato spaces are established. To achieve these, the authors first deduce several equivalent characterizations of the anisotropic variable $$\eta $$ -type Campanato space, and then introduce the anisotropic variable tent-Lorentz spaces and establish their atomic decomposition. All these results are new even for the isotropic Hardy–Lorentz space $$H^{p,q}(\mathbb {R}^n)$$ and the isotropic variable Hardy–Lorentz space $$H^{p(\cdot ),q}(\mathbb {R}^n)$$ . 

Long-Guang Huang

Papers

Cone metric spaces and fixed point theorems of contractive mappings

New Ball Campanato-Type Function Spaces and Their Applications

New Ball Campanato-Type Function Spaces and Their Applications

Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko

Dual Spaces of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications