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.

/pdf/on-the-mathematical-properties-of-the-structural-similarity-l94e02oe66.pdf

On the Mathematical Properties of the Structural Similarity Index

Cone metric spaces and fixed point theorems of contractive mappings

Common fixed point theorems for some new generalized contractive type mappings

Abstract This paper is concerned with the nonlinear Klein–Gordon–Maxwell systems. Unlike all known results in the literature, the Schrödinger operator $-\Delta +V$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> </mml:math> 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. 

https://boundaryvalueproblems.springeropen.com/counter/pdf/10.1186/s13661-022-01664-4

Xian Zhang

Papers

Cone metric spaces and fixed point theorems of contractive mappings

Common fixed point theorems for some new generalized contractive type mappings

Nontrivial solutions for Klein–Gordon–Maxwell systems with sign-changing potentials