Coincidence principle and applications. (Q2761570)

This monograph is dedicated to the presentation of some basic results concerning the existence of solution (in various function spaces) of the equations which can be written in the form NEWLINE\[NEWLINELu= Nu,\tag{E}NEWLINE\]NEWLINE where \(L\) and \(N\) are operators acting on various function spaces, usually chosen as dictated by the applications.NEWLINENEWLINENEWLINEThe first chapter contains existence of coincidence points under conditions stated by several authors (Goebel, Kirk, Anderson). Results due to Peetre and I. A. Rus are also included. The continuous dependence of the coincidence points with respect to perturbation of the operators is also investigated. Connection with fixed point theorems (particularly Banach contraction mapping) are also discussed.NEWLINENEWLINENEWLINEThe second chapter is dedicated to the problem of existence of coincidence points for (E) in partially ordered spaces. An immediate connection with integral inequalities is exploited (Gronwall's type, linear and nonlinear). Applications are presented for elliptic or parabolic equations, but also for evolution equations.NEWLINENEWLINENEWLINEThe third chapter treats the concept of ``universal operators''. The equivalence of these operators with the so-called zero-epi operators is established. The coincidence degree is studied in the fourth chapter. Applications are given to the problem of periodic solutions to functional differential equations.NEWLINENEWLINENEWLINEThe fifth chapter is concerned with the concept of weak closedness between two operators acting on a Banach space. This concept, introduced by the author and her associate (A. Domokos) is a generalization of the concept of closedness defined by S. Campanato. Properties which are preserved by weak close perturbations are investigated. The application are mainly concerned with elliptic equations which are ``totally'' nonlinear. More precisely, equations of the form NEWLINE\[NEWLINEa(x,u, Du, D^2u)= f,NEWLINE\]NEWLINE under adequate conditions on the function \(a\). Results related to those of Gilbarg and Trudinger, S. Campanato, A. Tarsia are generalized.NEWLINENEWLINENEWLINEThe list of references contains 195 items and covers a good deal of the existing literature related to the subject. The author is a member of the group at the University of Cluj-Napoca dealing with nonlinear functional analysis and its applications to functional equations (ordinary or partial differential equations and related types) including researchers like I. A. Rus, R. Precup, A. Petrusel a.o. The group is editing the publication Seminar on Fixed Point Theory, which has about 20 years of existence.