Second order differential operators with Feller–Wentzell type boundary conditions

Abstract

In this paper, we study a basic generation problem concerning the second order differential operator a d 2 d x 2 + b d d x + c in the space C [ 0 , 1 ] of complex continuous functions equipped with Feller–Wentzell type boundary conditions, which originates from the work of Feller [W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952) 468–519]. We prove successfully that the operator, under suitable assumptions, generates a strongly continuous cosine function on C [ 0 , 1 ] (or on a subspace of C [ 0 , 1 ] ), by means of an operator matrix analysis combined with perturbation, approximation, and similarity techniques.