Applications of Alternative Problems
BROWN UNIV PROVIDENCE RI LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
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Many problems in analysis and applied mathematics can be reduced to the solution of equations in a function space or functional equations. The equations often arise from the desire to obtain solutions of ordinary or partial differential equations with subsidiary conditions - periodicity or more general boundary conditions, specified asymptotic behavior, analyticity conditions, etc. Many of the problems involve a linear operator and a nonlinear operator which is small when some parameter is small. If the linear operator has an inverse, conceptually there are no difficulties in obtaining approximate solutions although practically there may be many difficulties. At least one can draw on all existing fixed point techniques. When the linear operator has elements in its null space, then new concepts must be introduced in order to proceed. It is the purpose of these lectures to present a general approach to the solution of functional equations and to indicate how to obtain the appropriate functional equations for a variety of applications.
- Theoretical Mathematics