Generalized Solutions of Operator Equations and Extreme Elements

Preface1. Fundamental notions, general and auxiliary facts2. Simplest schemes of generalized solution of linear operator equations2.1. Strong generalized solution2.2. Strong almost solution2.3. Weak generalized solution2.4. Weak almost solution2.5. Unique existence of weak generalized solution2.6. Relationship between weak and strong generalized solutions3. A priori estimations for linear continuous operator3.1. A priori inequalities3.2. Generalized solution of operator equation in Banach spaces3.3. Generalized solution of operator equation in locally convex topological spaces3.4. Relationship between generalized solutions in Banach and locally convex topological spaces4. Applications of the theory of generalized solvability of linear equations4.1. Equations with Hilbert-Schmidt operator in Hilbert space L2(-,)4.2. Generalized solution of infinite system of linear algebraic equations4.3. Volterra Integral Equation of the First Kind4.4. Statistics of random processes4.5. Parabolic PDE in a connected domain4.6. Parabolic PDE in a disconnected domain5. Scheme of generalized solutions of linear operator equations5.1. Generalized solution of linear operator equations in locally convex linear topological spaces5.2. Examples of generalized solutions5.3. Properties of generalized solutions in spaces E1, E26. Scheme of generalized solutions of nonlinear operator equations6.1. Generalized solution of nonlinear operator equation6.2. Almost solution of nonlinear operator equations6.3. Unique existence of generalized solution6.4. Correctness of generalized solutions6.5. Pseudo-generalized and essentially generalized solutions6.6. Embedding of space of pseudo-generalized solutions into space of generalized solutions6.7. Examples of operators6.8. Computation of generalized solution7. Generalized extreme elements7.1. Examples of generalized extreme elements7.2. Generalized extreme elements for linear and positively homogeneous convex functional7.3. Generalized extreme elements for general convex functional7.4. Some remarksReference