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Fixed Point of Surface Transformation and a Fixed Point Theorem for Triangular Surface Mapping

Şükrü İLGÜN

Department of Mathematics Education, Faculty of Education, Kafkas University, Kars, Turkey

Abstract: In this study, it was shown that the existence of fixed points of some surface transformations was defined as an example according to the theorems in functional analysis in differential geometry. It was shown that fi real-valued coordinate functions of F triangular space mapping defined from En to Em has a single fixed point, if

Keywords: fixed point, triangular mapping, triangular space mapping, surface transformation

Received June 25, 2018; Revised August 13, 2018; Accepted September 21, 2018

Citation: Şükrü İLGÜN, Fixed Point of Surface Transformation and a Fixed Point Theorem for Triangular Surface Mapping. Turkish Journal of Analysis and Number Theory, 6 (5). 132-135. January 2018. doi:10.12691/tjant-6-5-2

1. Introduction

In this study, we defined the triangular mapping in addition to surface mapping which exist in differential geometry. We proof that this triangular surface mapping has a unique fixed point and we examine the condition of convergence to this fixed point. I define a triangular surface mapping and examine the conditions of existence and uniqueness of this surface map’s fixed point by originating the study “A Fixed Point Theorem For Triangular Mappings” made by F. Sart [2].

2. Definitions and Theorems

2.1. Definition: Let be any space and a map of , or of a subset of into . .A point is called a fixed point for if . The set of all fixed point of is denoted by Fix (f) [5].

2.2. Definition: Let be a normed space, letbe a non-empty subset of and be a transformation. If and for and

Then, F is called a quasi-convergence transformation [7].

2.1. Example:

Let be a sphere surface. Let’s define a surface transformation at In this case, since and, then becomes a fixed point at the transformation of (See Figure 1).

Figure 1. Sphere Surface

F surface transformation is a quasi-convergence transformation because for.

Theorem 2.1: Let be a Banach space, let be a non-empty, compact convex subset of and let be a continuous transformation. In such case, F has at least one fixed point [6].

2.2. Example: Let be a transformation defined on a saddle surface .

This transformation has at least one fixed point according to the Theorem 2.1.

For. Under the transformation, is a fixed point and F has infinite fixed points.

Moreover, F surface transformation is a quasi-convergence transformation because for

2.3. Definition: Giving a function let denote the real-valued functions on such that

for all points in These functions are called the Euclidean coordinate functions of and we write

The function is differentiable provided its coordinate functions are differentiable in the unusal sense. A differentiable function is called a mapping from to [1].

2.4. Definition: Let : → be a mapping. If is a tangent vector to at P, let be a initial velocity of the curve t → (p + t). The resulting function send tangent vectors to to tangent vectors to and is called the tangent map of [1].

2.5. Definition: A function : → from one surface to another is differentiable provided that for each patch in and in the composite function is Euclidean differentiable (and defined on an open set of R2). is then called a mapping of surfaces [1].

2.6. Definition: Let be a open-convex subset of and C be nonempty compact subset of Let be real-valued and differentiable functions. If mapping is defined with and

then is called triangular mapping.

2.3. Example:

If the Jakobian matrix of function is stated as following:

Lets take . In this case,

2.7. Definition: A square matrix is called upper triangular if all etries below the main diagonal are zones. Similarly, a square matrix is called lower triangular if all the entries above the main diagonal are zeros. A matrix that is either upper or lower triangular is called triangular [4].

Theorem 2.2. Let be an open convex set of , and a non-empty compact subset Let be a triangular mapping of , i.e. of a type

which has the following properties:

1. F is a self-map on ,

2. F is continuous and differentiable on

3. F has a bounded derivative on such that

Then;

Ÿ F has a unique fixed point in

Ÿ For any in the iterative sequence converges to the unique fixed point.) [2].

Uniquenses proof. Suppose that X and Y are two different fixed points. By using fixed point definition

On the other hand, by the mean-value theorem (Lagrange’s theorem), there is an n-tuble of points such that

where

W₁, W₂, … , Wn being each on the line segment joining X and Y. Therefore

and thus 1 is eigenvalue of the matrix . As it is triangular 1 belongs to diagonal, which contradicts the assumption on those elements [2].

2.4. Example: Lets state the Jakobiyen matrix of transformation as following

From here,

The number of fixed points of F transformation is one according to Theorem 2.2. This fixed point is the point of

Then,

2.5. Example: For and

From here,

Then,

If

for the function has infinite fixed-points according to Theorem 2.2.

Theorem 2.3. If a function is continuously differentible in an open set of containing the points and and the line segment connecting them, then an equation

Where (a, b, c) an interior point of the line segment, is valid. (PlaneMath: mean value theorem for several variables=3) [3]

3. A New Definition and Theorem

Following definition 3.1 is triangular space mapping which was defined by using definition 2.1 and 2.5, and 2.6.

Definition 3.1: Let and be surfaces in Euclid space, and let be a mapping of surface. In addition, if is a triangular mapping, then is called triangular space mapping.

Theorem 3.1: Let be a surface and, let be a triangular space mapping. If including the transformation of to a single fixed point

where are real-valued coordinate functions of .

Proof: Let be triangular space mapping which has a single fixed point. Then, from Theorem 2.2

Determinant value of this matrix can be 1. If this value is 1, then the determinant value belongs to the diagonal of this matrix. Because, this matrix is the lower triangular matrix. At that rate,

3.1. Example: Let’s define a triangular surface transformation as ,

and

Figure 2. Triangular Surface Transformation

For Jakobiyen matrix it is obvious that is true for

Because each component is within the interval of

It can be told that JF.

Then, according to Theorem 4.2.2 transformation of has limited derivative at

In this case;

1) has one fixed point at

2) For any Picard iteration progression of converges to this fixed point.

This point is

The Picard iteration progression of

converges to the point of .

In this case Then is a fixed point of transformation of.

That means

References

[1]   | O’neill B. 1997. Elementary Differential Geometry, Second Edition, Academic Press, San Diego, USA.
[2]   | Sart F., 2007. A Fixed Point Theorem For Triangular Mappings, Journal of Applied Analysis, Vol.13, No.1, 2007, 77-81.
[3]   | PlaneMath: mean value theorem for several variables, http://www.planemath.com.
[4]   | Anton H. 1977. Elementary Linear Algebra, Second Edition, Quinn & Boden Com, USA.
[5]   | Granas A. Dugundji J. 2003. Fixed Point Theory, Springer-Verlag New York, USA.
[6]   | Khamsi, M. A. and Kirk W. A., 2001. An Introduction to Metric Spaces and Fixed Point Theory.
[7]   | Petryshyn, W. V. and Williamson, T. E., 1973. Strong and weak convergence of thesequence of successive approximations for quasi-nonexpansive mappings, J.Math. Anal. Appl., 43, 459-497.