Nonlocal problems in the theory of hyperbolic differential equations

Abstract:

There are relatively few local problems for general hyperbolic differential equations in a bounded domain on the plane, and all these problems are well studied, and, in simple cases, are included in almost any textbook on partial differential equations. On the contrary, nonlocal problems (even more general than boundary problems) remain practically not studied, although a number of problems of this type were successfully studied in connection with elliptic or parabolic equations. In the present paper, we consider two nonlocal quasiboundary problems of sufficiently general type in the characteristic rectangle for equations of the above type. In both cases we find conditions for unique solvability and (for the first time in the theory of hyperbolic equations) the conditions for problems to be Fredholm. Examples show that these conditions are sharp: if they are violated, the resulting problems may fail to have the required solvability properties. The proofs (in their nonanalytic part) are given in the framework of perturbation theory of operators in Banach spaces.