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Hyperbolas

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse...

Geometry of Hyperbolas

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A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...

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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...

Hyperbolic Functions

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A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...

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Cross-Modal Multivariate Pattern Analysis

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Convergence theorems for generalized nonexpansive multivalued mappings in hyperbolic spaces.

Jong Kyu Kim¹, Ramesh Prasad Pathak², Samir Dashputre³

¹Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 51767 Korea.

|July 8, 2016

Summary

This study proves the existence of fixed points for generalized nonexpansive multivalued mappings in hyperbolic spaces. It also establishes convergence theorems for an iterative scheme approximating these fixed points.

Keywords:

Generalized nonexpansive multivalued mappings Hyperbolic spaces Iteration process Strong and [Formula: see text]-convergence theorems

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Area of Science:

Mathematics
Nonlinear Analysis
Metric Geometry

Background:

Fixed-point theory is crucial for solving equations in various mathematical fields.
Multivalued mappings and hyperbolic spaces present unique challenges in convergence analysis.
Existing iterative schemes require further refinement for generalized nonexpansive mappings.

Purpose of the Study:

To establish the existence of fixed points for generalized nonexpansive multivalued mappings in hyperbolic spaces.
To prove convergence theorems for a specific iterative scheme.
To extend and improve existing results in the literature.

Main Methods:

Utilizing concepts of fixed-point theory in the context of hyperbolic spaces.
Applying iterative schemes to approximate fixed points of multivalued mappings.
Developing and proving convergence theorems under generalized nonexpansive conditions.

Main Results:

Existence of a fixed point for generalized nonexpansive multivalued mappings is established.
Convergence theorems (both [Formula: see text]-convergence and strong convergence) are proven for the iterative scheme.
The study extends and improves upon recent findings in multivalued fixed-point theory.

Conclusions:

The proposed iterative scheme is effective for approximating fixed points of generalized nonexpansive multivalued mappings in hyperbolic spaces.
The findings contribute to the advancement of fixed-point theory in non-Euclidean geometries.
This research offers enhanced theoretical tools for problems involving such mappings.