Operator Inclusions and Applications
Ph.D. Program in Mathematics and Computational Sciences
University of Messina
30 hours
This doctoral course focuses on set-valued analysis and operator inclusions, with particular emphasis on measurable and semicontinuous set-valued mappings, measurable selection theorems, fixed point theorems for multifunctions, Ricceri's existence theorem for operator inclusions, and applications to differential and integral problems.
Course Contents
1. Theory of Set-Valued Mappings
Basic definitions. Graph of a set-valued mapping. Inverse set-valued mapping. Selections of a set-valued mapping. Lower and upper inverse images. Distance from a set and set dilations. Fundamental properties of convex-valued set-valued mappings.
2. Semicontinuous and Measurable Set-Valued Mappings
Lower and upper semicontinuous set-valued mappings. Characterizations of semicontinuity. Properties of the graph of a set-valued mapping. Measurable and weakly measurable set-valued mappings. Measurable graph theorem. Kuratowski–Ryll-Nardzewski measurable selection theorem. Castaing representation theorem.
3. Fixed Point Theorems for Set-Valued Mappings
Fixed points of set-valued mappings. Kakutani's fixed point theorem. Ky Fan's fixed point theorem. Applications of fixed point theorems to the theory of operator inclusions.
4. Ricceri's Existence Theorem for Operator Inclusions
Superposition operators associated with set-valued mappings. Operator inclusions of the form Φ(u)(ω) ∈ F(ω, Ψ(u)(ω)). Statement and proof of Ricceri's Existence Theorem for Operator Inclusions. Role of measurability, lower semicontinuity and convexity assumptions. Main corollaries and applications.
5. Applications to Second-Order Boundary Value Problems
Boundary value problems for second-order ordinary differential inclusions. Operator formulation of the problem. Green's operator. Application of Ricceri's theorem to the existence of solutions. Differential inclusions with homogeneous Dirichlet boundary conditions. Examples and existence results.
6. Applications to Implicit Integral Equations
Integral operators and their fundamental properties. Implicit integral equations with discontinuous right-hand side. Operator formulation of the problem by means of operator inclusions. Application of Ricceri's theorem to the existence of essentially bounded solutions of implicit integral equations.
References
- J. P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
- J. P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
- C. Himmelberg, Measurable Relations, Fundamenta Mathematicae 87 (1), 53–72, 1975.
- K. Ryll-Nardzewski, C. Kuratowski, A General Theorem on Selectors, Sci. Ser. Sci. Math. Astronom. Phys. 13, 397–403, 1965.
- C. Castaing, M. Valadier, Measurable Multifunctions, Springer, 1977.
- K. Fan, Fixed-Point and Minimax Theorems in Locally Convex Topological Linear Spaces, Proceedings of the National Academy of Sciences of the United States of America 38, 121–126, 1952.
- S. Kakutani, A Generalization of Brouwer's Fixed Point Theorem, Duke Mathematical Journal 8, 457–459, 1941.
- O. Naselli Ricceri, B. Ricceri, An Existence Theorem for Inclusions of the Type ψ(u)(t) ∈ F(t, φ(u)(t)) and Application to a Multivalued Boundary Value Problem, Applicable Analysis 38 (4), 259–270, 1990.
- S. A. Marano, Existence Theorems for a Multivalued Boundary Value Problem, Bulletin of the Australian Mathematical Society 45 (2), 249–254, 1992.
- F. Cammaroto, P. Cubiotti, Implicit Integral Equations with Discontinuous Right-Hand Side, Commentationes Mathematicae Universitatis Carolinae 38 (3), 241–246, 1997.