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Inequalities express mathematical relationships where two values are not equal and are compared using symbols such as <, >, ≤, or ≥. These expressions define a range of possible solutions rather than a single value. Interval notation provides a concise way to express these solution sets, especially when the variable spans a continuous range. An open interval, written as (a, b), excludes the endpoints, while a closed interval [a, b] includes them. There are also half-open...
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A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality:  can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the...
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The graph of the equation where y equals x squared forms a curve known as a parabola. This curve acts as a boundary in the coordinate plane, dividing it into distinct regions based on the relative position of points.When the equality sign in the equation is replaced with an inequality—such as greater than, less than, greater than or equal to, or less than or equal to—the graphical representation changes from a single curve into a broader shaded area that signifies the set of all...
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Basic inequality on a b-metric space and its applications.

Tomonari Suzuki1

  • 1Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu, 804-8550 Japan.

Journal of Inequalities and Applications
|October 31, 2017
PubMed
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Researchers proved fundamental inequalities and fixed-point theorems in b-metric spaces. They explored conditions impacting sequence Cauchyness, distinguishing between those that imply it and those that do not.

Keywords:
Cauchy sequenceb-metric spacefixed point theorem

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Area of Science:

  • Mathematical Analysis
  • Topology

Background:

  • b-metric spaces are a generalization of metric spaces with a constant 'b' greater than or equal to 1.
  • Inequalities and fixed-point theorems are foundational concepts in mathematical analysis with wide applications.

Purpose of the Study:

  • To establish basic inequalities within the framework of b-metric spaces.
  • To investigate and prove fixed-point theorems in these spaces.
  • To analyze conditions related to the convergence of sequences in b-metric spaces.

Main Methods:

  • Proving fundamental inequalities specific to b-metric spaces.
  • Applying fixed-point theory techniques to b-metric spaces.
  • Analyzing sequence properties under different conditions.

Main Results:

  • Establishment of a basic inequality in b-metric spaces.
  • Demonstration of several fixed-point theorems.
  • Identification of two distinct conditions affecting sequence Cauchyness.

Conclusions:

  • The study contributes foundational results to the theory of b-metric spaces.
  • The findings offer new insights into convergence properties of sequences.
  • The distinction between the two conditions provides a nuanced understanding of Cauchyness.