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University of North Texas

1. Farmer, Matthew Ray. Applications in Fixed Point Theory.

Degree: 2005, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc4971/

Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, GΓΆhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.
*Advisors/Committee Members: Bator, Elizabeth M., Lewis, Paul, Jackson, Stephen C..*

Subjects/Keywords: Fixed point theory.; Banach spaces.; metric space; Banach spaces; non-expansive maps; contraction maps; fixed points; uniformly convex Banach spaces

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Farmer, M. R. (2005). Applications in Fixed Point Theory. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc4971/

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Farmer, Matthew Ray. “Applications in Fixed Point Theory.” 2005. Thesis, University of North Texas. Accessed August 09, 2020. https://digital.library.unt.edu/ark:/67531/metadc4971/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Farmer, Matthew Ray. “Applications in Fixed Point Theory.” 2005. Web. 09 Aug 2020.

Vancouver:

Farmer MR. Applications in Fixed Point Theory. [Internet] [Thesis]. University of North Texas; 2005. [cited 2020 Aug 09]. Available from: https://digital.library.unt.edu/ark:/67531/metadc4971/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Farmer MR. Applications in Fixed Point Theory. [Thesis]. University of North Texas; 2005. Available from: https://digital.library.unt.edu/ark:/67531/metadc4971/

Not specified: Masters Thesis or Doctoral Dissertation

University of North Texas

2. Huggins, Mark C. (Mark Christopher). A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema.

Degree: 1993, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc500353/

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function π which is the uniform limit of a sequence of sawtooth functions πβ : [0, 1] β [0, 1] with increasingly sharp teeth. Let π = [0, 1] x [0, 1] and πΉ(π) be the Hausdorff metric space determined by π. We define contraction maps π€β , π€β , π€β on π. These maps define a contraction map π€ on πΉ(π) via π€(π΄) = π€β(π΄) β π€β(π΄) β π€β(π΄). The iteration under π€ of the diagonal in π defines a sequence of graphs of continuous functions πβ. Since π€ is a contraction map in the compact metric space πΉ(π), π€ has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function π. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in πΆ[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set of proper local extrema is residual in πΆ[0,1]. In the fourth and last chapter we actually construct our function and prove it is continuous, nowhere-differentiable and has a dense set of proper local extrema. Lastly we iterate the set {(0,0), (1,1)} under π€ and plot its points. Any terms not defined in Chapters 2 through 4 may be found in [2,4]. The same applies to the basic properties of metric spaces which have not been explicitly stated. Throughout, we will let π© and π½ denote the natural numbers and the real numbers, respectively.
*Advisors/Committee Members: Bator, Elizabeth M., Lewis, Paul Weldon, Jackson, Steve, 1957-.*

Subjects/Keywords: contraction maps; contractive maps; continuous nowhere differentiable functions; continuous functions; Baire Category Theorem; Functions, Continuous.

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Huggins, M. C. (. C. (1993). A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc500353/

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Huggins, Mark C (Mark Christopher). “A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema.” 1993. Thesis, University of North Texas. Accessed August 09, 2020. https://digital.library.unt.edu/ark:/67531/metadc500353/.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Huggins, Mark C (Mark Christopher). “A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema.” 1993. Web. 09 Aug 2020.

Vancouver:

Huggins MC(C. A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema. [Internet] [Thesis]. University of North Texas; 1993. [cited 2020 Aug 09]. Available from: https://digital.library.unt.edu/ark:/67531/metadc500353/.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Huggins MC(C. A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema. [Thesis]. University of North Texas; 1993. Available from: https://digital.library.unt.edu/ark:/67531/metadc500353/

Not specified: Masters Thesis or Doctoral Dissertation