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Description of Individual Course Units| Course Unit Code | Course Unit Title | Type of Course Unit | Year of Study | Semester | Number of ECTS Credits | | MAT-23-121 | NONLINEAR FUNCTIONAL ANALYSYS AND APPLICATIONS I | Elective | 1 | 1 | 6 |
| | Level of Course Unit | | Third Cycle | | Objectives of the Course | | To explain the properties of nonlinear transformations (operators, functions) in Banach spaces. To explain the application to the problems of the basic results obtained for examining the problems with nonlinear weak compact operators. Explaining the application of the abstractly described approaches to the differential, integral and other nonlinear transformations in the Sobolev (v. s.) spaces with fixed or variable exponents. The students can understand the works about these themes. | | Name of Lecturer(s) | | Prof. Dr. Kamal SOLTANOV | | Learning Outcomes | | 1 | To explain the dual spaces of the Sobolev spaces with fixed and variable exponents | | 2 | To explain to students the properties of nonlinear weak compact transformations (operators, functions) in Banach and topological spaces in the abstract form and examples | | 3 | To explain to students the methods of the investigations of the differential-operator equations |
| | Mode of Delivery | | Daytime Class | | Prerequisites and co-requisities | | It is need to know the basic facts of Mathematical Analysis, Linear Algebra, Real analysis, Functional Analysis, Ordinary and Partial Differential Equations, Algebra and Topology in the level of graduate or, at least, undergraguate. | | Recommended Optional Programme Components | | None | | Course Contents | | The space of the generalized functions and derivative of the generalized functions
Vector Sobolev spaces and the dual spaces. Examples.
The embedding theorems and compactly embedding theorems for vector Sobolev spaces, Examples.
The embedding theorems and compactly embedding theorems for vector Sobolev spaces, Examples.
Nonlinear operators and functionals, Examples..
The Gateaux and Frechet derivative of of the nonlinear operators and functionals, Examples.
On the methods of the investigation of the nonlinear diferential-operator equations. Examples
On the methods of the investigation of the nonlinear diferential-operator equations (Compactness method) .
On the methods of the investigation of the nonlinear diferential-operator equations (Compactness method) .
On the methods of the investigation of the nonlinear diferential-operator equations (Method of the weakly compactness).
On the methods of the investigation of the nonlinear diferential-operator equations (Method of the weakly compactness).
Method of monotonness and duality operators. Examples
Method of monotonnes
| | Weekly Detailed Course Contents | |
| 1 | The space of the generalized functions and derivative of the generalized functions | | | | 2 | Vector Sobolev spaces and the dual spaces. Examples. | | | | 3 | The embedding theorems and compactly embedding theorems for vector Sobolev spaces, Examples. | | | | 4 | The embedding theorems and compactly embedding theorems for vector Sobolev spaces, Examples. | | | | 5 | Nonlinear operators and functionals, Examples.. | | | | 6 | The Gateaux and Frechet derivative of of the nonlinear operators and functionals, Examples. | | | | 7 | MID TERM EXAM | | | | 8 | On the methods of the investigation of the nonlinear diferential-operator equations. Examples | | | | 9 | On the methods of the investigation of the nonlinear diferential-operator equations (Compactness method) . Examples | | | | 10 | On the methods of the investigation of the nonlinear diferential-operator equations (Compactness method) . Examples | | | | 11 | On the methods of the investigation of the nonlinear diferential-operator equations (Method of the weakly compactness). Examples | | | | 12 | On the methods of the investigation of the nonlinear diferential-operator equations (Method of the weakly compactness). Examples | | | | 13 | Method of monotonness and duality operators. Examples | | | | 14 | Method of monotonness and preparation to the final exam | | | | 15 | Final Exam | | |
| | Recommended or Required Reading | | Aubin, J.P., Ekeland, I. - Applied Nonlinear Analysis, New York ,Wiley, 1984
Deimling, K. - Nonlinear Functional Analysis. Springer Verlag, 1985
Nirenberg, L. - Topics in Nonlinear Functional Analysis. Courant Institute, 1974
Zeidler, E. - Nonlinear Functional Analysis and its Applications 2/A, 2/B, 4 - Application to mathematical physics , Springer Verlag, 1990, 1988
Lions J.-L. - Quelques Methodes de Resolution des Problemes aux Limites Non Linearies. Dunod, Parıs, 1969.
Lions, J.-L. - Magenes E. – Non-homogeneus boundary value problems and Applications. Springer – Verlag, 1972.
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011, Springer, N.Y.
Soltanov, K.N. – Yayınlar
| | Planned Learning Activities and Teaching Methods | | | Assessment Methods and Criteria | |
| Midterm Examination | 1 | 100 | | SUM | 100 | |
| Final Examination | 1 | 100 | | SUM | 100 | | Term (or Year) Learning Activities | 50 | | End Of Term (or Year) Learning Activities | 50 | | SUM | 100 |
| | Language of Instruction | | Turkish | | Work Placement(s) | | None |
| | Workload Calculation | |
| Midterm Examination | 1 | 1 | 1 | | Final Examination | 1 | 2 | 2 | | Attending Lectures | 14 | 3 | 42 | | Question-Answer | 14 | 3 | 42 | | Individual Study for Mid term Examination | 1 | 50 | 50 | | Individual Study for Final Examination | 1 | 50 | 50 | |
| Contribution of Learning Outcomes to Programme Outcomes | | LO1 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | | LO2 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | | LO3 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| | * Contribution Level : 1 Very low 2 Low 3 Medium 4 High 5 Very High |
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