I. Background in Functional Analysis.- 1. Topology and measure.- Banach spaces.- Elements of the geometry of Banach spaces.- Continuous mappings.- Vitali’s theorem.- 2. Convex analysis.- Continuity of convex functions.- Separation of convex sets.- Lower semi-continuous functions.- The subdifferential.- Smooth norms.- 3. Related topics and exercises.- Bibliographical comments.- II. Methods of Compactness.- 1. Compact operators.- Linear integral operators.- Integral operators of potential type.- 2. Sobolev spaces.- Mollifiers.- Generalized derivatives.- Imbedding theorems.- Generalities about the trace.- Imbeddings in the case of unbounded domains.- 3. Theory of topological degree.- Degree of a mapping in N-dimensional spaces.- Properties of the degree.- Fixed point theorems.- Topological degree in Banach spaces.- 4. Related topics and exercises.- Bibliographical comments.- III. Nonlinear Mappings of Monotone Type.- 1. Pseudo-monotone mappings.- Mappings of variational calculus type.- The class of pseudomonotone mappings.- Mappings with generalized pseudomonotone property.- 2. Monotone mappings.- Definitions and examples.- Local boundedness and continuity.- Maximal monotone mappings.- Normalized duality map.- Linear maximal monotone mappings.- Surjectivity of coercive mappings.- A maximality criterion.- Subdifferentials of convex functions.- 3. Perturbation of maximal monotone mappings.- The Yosida approximant.- Surjectivity of perturbed mappings.- Maximality of sums.- 4. Noncoercive operators.- Operators of type (S).- Surjectivity of noncoercive mappings.- 5. Mappings of type (M).- Extent of the class.- Surjectivity of mappings of type (M).- 6. Related topics and exercises.- Bibliographical comments.- IV. Hammerstein Equations.- 1. The Nemitskyi operator.- The case Lp(?) with 1 < p < ?.- The case L1(?).- 2. Abstract Hammerstein equations.- Equations with linear compact mappings.- Pseudo-monotonicity methods.- Hilbertean case.- 3. Angle-bounded mappings.- Angle-bounded linear and nonlinear operators.- The splitting of linear maps.- Equations with angle-bounded operators.- An approximation method.- A class of Urysohn equations.- Hammerstein integral equations.- 4. Variational methods.- Banach space case.- Hilbert space case.- 5. Related topics and exercises.- Bibliographical comments.- V. Homotopy Arguments.- 1. Odd mappings.- Infinite-dimensional variant of Borsuk’s theorem.- Perturbations homotopic to odd operators.- 2. Eigenvalue problems for maximal monotone operators.- Mappings with a compact resolvent.- The subdifferential case.- A Fredholm alternative.- 3. Range of sums of monotone mappings.- Condition (*) for nonlinear mappings.- The case (Tu, S?u) ? 0.- 4. Related topics and exercises.- Bibliographical comments.- VI. Variational Problems and Inequalities.- 1. Variational inequalities.- Equivalent formulations of minimum problems.- The penalty method.- Convergence of solutions of variational inequalities.- 2. Variational boundary-value problems.- Generalized divergence equations.- The existence of variational solutions.- Other assumptions.- 3. Strongly nonlinear problems.- Mappings of type (M) with respect to two Banach spaces.- Strongly nonlinear variational problems.- Strongly nonlinear inequalities.- 4. Other methods.- Odd perturbations.- A problem with upper and lower solutions.- Landesman-Lazer theorem.- 5. Boundary-value problems in unbounded domains.- An approach with approximate domains.- Elliptic super-regularization.- 6. Related topics and exercises.- Bibliographical comments.- Suggestions for further study.- References.
