ECTS @ IUE ECTS @ IUE ECTS @ IUE ECTS @ IUE ECTS @ IUE ECTS @ IUE ECTS @ IUE

Syllabus ( MATH 681 )

   Basic information
Course title: Theory of Set Differential Equations in Metric Spaces
Course code: MATH 681
Lecturer: Prof. Dr. Coşkun YAKAR
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Third Cycle (Doctoral)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: Math 652
Professional practice: No
Purpose of the course: To present the stability results and applications of the Theory of Set Differential Equations in metric spaces.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Explain the basic concepts of Theory of Set Differential Equations in Metric Spaces.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Gain original, independent and critical thinking, and develop theoretical concepts and tools,
    4. Question and find out innovative approaches.
    5. Acquire scientific knowledge and work independently,
    6. Develop mathematical, communicative, problem-solving, brainstorming skills.
    7. Support his/her ideas with various arguments and present them clearly to a range of audience, formally and informally through a variety of techniques
    8. Write progress reports clearly on the basis of published documents, thesis, etc
    9. Effectively express his/her research ideas and findings both orally and in writing

    Method of assessment

    1. Written exam
    2. Oral exam
    3. Homework assignment
    4. Seminar/presentation
    5. Term paper
  2. Explain the Fundamental Definitions, Concepts, Theorems , Stability and Applications of Theory of Set Differential Equations in Metric Spaces.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Gain original, independent and critical thinking, and develop theoretical concepts and tools,
    3. Question and find out innovative approaches.
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  3. Generilaze, Empasize and Apply the concept of Theory of Set Differential Equations in Metric Spaces, Impulsive Differential Equations and Hybrit Systems to the Theory of Set Differential Equations in Metric Spaces.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Gain original, independent and critical thinking, and develop theoretical concepts and tools,
    4. Understand relevant research methodologies and techniques and their appropriate application within his/her research field,
    5. Acquire scientific knowledge and work independently,
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Effectively express his/her research ideas and findings both orally and in writing

    Method of assessment

    1. Oral exam
    2. Seminar/presentation
    3. Term paper
  4. Distinguish the difference between Ordinary Differential Equations, Fuzz Differential Equation and Theory of Set Differential Equations in Metric Spaces.

    Contribution to Program Outcomes

    1. Gain original, independent and critical thinking, and develop theoretical concepts and tools,
    2. Question and find out innovative approaches.
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
    2. Oral exam
  5. Develop awareness for the Theory of Set Differential Equations in Metric Spaces.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics in a specialized way
    2. Gain original, independent and critical thinking, and develop theoretical concepts and tools,
    3. Understand relevant research methodologies and techniques and their appropriate application within his/her research field,
    4. Analyze critically and evaluate his/her findings and those of others,
    5. Question and find out innovative approaches.
    6. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
    2. Seminar/presentation
    3. Term paper
   Contents Up
Week 1: Basic Definition and Theorems; Compact Convex Subsets of R^n, The Hausdorff Metric
Week 2: Support Functions, Continuity and Measurability
Week 3: Differentiation, Integration, Subsets of Banach Spaces.
Week 4: Basic Theory; Comparison Principles, Local Existence and Uniqueness, Local Existence and Extremal Solutions
Week 5: Monotone Iterative Technique.
Week 6: The Method of Quasilinearization. Midterm Exam I.
Week 7: Global Existence, Approximate Solutions, Existence of Euler Solutions
Week 8: Proximal Normal and Flow Invariance, Existence, Upper Semicontinuous Case.
Week 9: Stability Theory; Lyapunov-like Functions, Global Existence, Stability Criteria, Nonuniform Stability Criteria
Week 10: Criteria for Boundedness, Set Differential Systems, The Method of Vector Lyapunov Functions
Week 11: The method of Variation of Parameters.
Week 12: Non-smooth Analysis, Lyapunov Stability Criteria. Midterm Exam II.
Week 13: Connection to Fuzzy Differential Equations(FDEs); Lyapunov-like functions, Connection with SDEs, Upper Semicontinuous Case Continued, Impulsive FDEs, Hybrid FDEs, Another Formulation.
Week 14: Miscellaneous Topics; Impulsive Set Differential Equations (SDEs), Monotone Iterative Technique, Set Differential Equations with Delay. Impulsive Set Differential Equations with Delay, Set Difference Equations, Set Differential Equations with Causal Operators, Lyapunov-like Functions in K_c (R_+^d ), Set Differential Equations in (K_c(E),D).
Week 15*: -
Week 16*: Final Exam.
Textbooks and materials: Lakshmikantham, V. Bhaskar Gnana T. And Devi Vasundhara J., Theory of Set Differential Equations in Metric Spaces. Cambridge Scientific Publishers 2006.
Recommended readings: 1: Lakshmikantham, V. Leela, S. , Dirici, Z. and McRae, F.A., Theory of Causal Differential Equations. Atlantis Press/ World Scientific Publishers 2009.
2 : Brauer, F. and Nohel, J., The Qualitative Theory of Ordinary Differential Equations, W.A. Benjamin, Inc., New York 1969.
3 : Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities, Vol. 1, Academic Press, New York 1969.
4 : Lakshmikantham, V., Leela, S. and Martynyuk, A.A., Stability Analysis of Nonlinear Systems, Marcel Dekker, New York 1989.
5 : Shaw, M.D. and Yakar, C., Generalized variation of parameters with initial time difference and a comparison result in term Lyapunov-like functions, International Journal of Non-linear Differential Equations-Theory Methods and Applications 5, (1999) 86-108.
6 : Shaw, M.D. and Yakar, C., Stability criteria and slowly growing motions with initial time difference, Problems of Nonlinear Analysis in Engineering Systems 1, (2000) 50-66.
7 : Yakar, C. Boundedness Criteria in Terms of Two Measures with Initial Time Difference. Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Mathematical Analysis. Watam Press. Waterloo. Page: 270-275. DCDIS 14 (S2) 1-305 (2007).
8. Yakar C., Strict Stability Criteria of Perturbed Systems with respect to Unperturbed Systems in term of Initial Time Difference. Proceedings of the Conference on Complex Analysis and Potential Theory. World Scientific Publishing. Page: 239-248 (2007).
9: Yakar C. and Shaw, M.D., A Comparison Result and Lyapunov Stability Criteria with Initial Time Difference. Dynamics of Continuous, Discrete and Impulsive Systems. A: Mathematical Analysis. Volume 12, Number 6 (2005) (731-741).
10 : Yakar C. and Shaw, M.D., Initial Time Difference Stability in Terms of Two Measures and Variational Comparison Result. Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Mathematical Analysis 15 (2008) 417-425.
11 : Yakar C. and Shaw, M.D., Practical stability in terms of two measures with initial time difference. Nonlinear Analysis: Theory, Methods & Applications. Vol. 71 (2009) e781-e785.
12 : Yakar C., Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference. Abstract and Applied Analysis. (Accepted) Vol 3. Volume 2010, Article ID 762857, 16 pages doi:10.1155/2010/762857. (2010)
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 6, 12 40
Other in-term studies: 7, 13 5
Project: 8, 14 5
Homework: 2,3,4,9,10,13 5
Quiz: 5,11 5
Final exam: 16 40
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 2 14
Practice, Recitation: 0 0
Homework: 6 10
Term project: 6 2
Term project presentation: 1 1
Quiz: 1 2
Own study for mid-term exam: 10 2
Mid-term: 2 2
Personal studies for final exam: 10 1
Final exam: 3 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
-->