Course Name Code Semester T+U Hours Credit ECTS
Fixed Point Theory and Its Applications I MAT 625 0 3 + 0 3 6
Precondition Courses
Recommended Optional Courses <p>1. Topology</p> <p>2. Functional Analysis</p>
Course Language Turkish
Course Level Doctorate Degree
Course Type Optional
Course Coordinator Doç.Dr. MAHPEYKER ÖZTÜRK
Course Lecturers Doç.Dr. MAHPEYKER ÖZTÜRK,
Course Assistants
Course Category Available Basic Education in the Field
Course Objective

To understand the concept of fixed point.
To learn Banach fixed point theorem and further extensions of this theorem in metric spaces. To learn the applications of fixed point theorem in metric spaces.

Course Content

Fixed Point, Contraction Mapping, single-valued and multivalued mappings in metric spaces, Banach Fixed Point Theorem, Extensions of Banach Fixed Point Theorem, Caristis Theorem and its Equivalents, Set valued Contractios, Generalized Contractions, Structure and Properties of the Fixed Point Sets,Picards Theorem, Iteration Methods, Cauchy Theorem, Applications of Banach Fixed Point Theorem to Reel Analysis, Linear Equation Systems, Differantial and Integral Equations.

# Course Learning Outcomes Teaching Methods Assessment Methods
1 He\she comprehends different contraction mappings and understand the concept of fixed point Lecture, Question-Answer, Problem Solving, Testing, Homework,
2 He\she explains the single valued and multivalued maooings in metric spaces. Lecture, Question-Answer, Problem Solving, Testing, Homework,
3 He\she analysis the Banach fixed point theorem and some extensions of Banach fixed point theorem Lecture, Question-Answer, Problem Solving, Testing, Homework,
4 He\she explains the Caristi-Ekeland Theorem and its equivalence Lecture, Question-Answer, Problem Solving, Testing, Homework,
5 He\she explains the set-valued mappings and Generalized contraction mappings Lecture, Question-Answer, Problem Solving, Testing, Homework,
6 He\she comments the structure of fixed point sets and properties of these sets. Lecture, Question-Answer, Problem Solving, Testing, Homework,
7 He\she explains the Picard Theorem,iteration method and Cauchy Problem Lecture, Question-Answer, Problem Solving, Testing, Homework,
8 He\she explains the applications of Banach fixed point theorem to Linear Equation Systems, Differantial and Integral Equations Lecture, Question-Answer, Problem Solving, Testing, Homework,
Week Course Topics Preliminary Preparation
1 The concept of fixed point, Different Contraction Mappings
2 The single valued and multivalued mappings in metric spaces
3 The Banach fixed point theorem
4 The extensions of Banach fixed point theorem
5 The Caristi-Ekeland Theorem and its equivalences
6 Set valued mappings
7 Genaralized Contraction mappings
8 The structure and properties of fixed point sets
9 Mid-term exam
10 The Pickard´s theorem, iterations methods and Cauchy problem
11 The applications of Banach fixed point theorem to real analysis
12 The applications of Banach fixed point theorem to linear equation systems
13 The applications of Banach fixed point theorem to Differantial Equations
14 The applications of Banach fixed point theorem to Integral Equations
Resources
Course Notes
Course Resources

1. The Computation of Fixed Points and Its Applications,1976
2. Topics in Metric Fixed Point Theory, 1990
3. Handbook of Metric Fixed Point Theory,2001
4. An Introduction to Metric Spaces and Fixed Point Theory, 2001
5. Fixed Point Theory, 2003
6. Homotopy Methods in Topological Fixed and Periodic Points Theory,2006
7. Fixed Point Theory for Lipschitzian-type Mappings with Applications,2009
8. Fixed Point Theory in Ordered Sets and Applications,2010

Order Program Outcomes Level of Contribution
1 2 3 4 5
1 At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application. X
2 Student completes the missing or limited knowledge by using the scientific methods. X
3 Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result. X
4 Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field. X
5 Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches. X
6 At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity. X
7 Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters. X
8 Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work. X
9 Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad. X
10 Student follows the necessary technological developments in his/her field, and s/he uses them. X
11 Student looks out for the scientific and ethic values while gathering, interpreting and publishing data. X
Evaluation System
Semester Studies Contribution Rate
1. Ara Sınav 80
1. Ödev 10
2. Ödev 10
Total 100
1. Yıl İçinin Başarıya 50
1. Final 50
Total 100
ECTS - Workload Activity Quantity Time (Hours) Total Workload (Hours)
Course Duration (Including the exam week: 16x Total course hours) 16 3 48
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 20 20
Assignment 2 5 10
Final examination 1 25 25
Performance Task (Seminar) 1 10 10
Total Workload 161
Total Workload / 25 (Hours) 6.44
dersAKTSKredisi 6