Remarks on the nonlinear eigenvalue problem \(Tx=\lambda Sx\) (Q1107810)

The authors prove two existence theorems for the eigenvalue problem \(Tx=\lambda Sx\). The first theorem refers to continuous operators T and S in a finite-dimensional space X, the second one to P-compact operators T and S in a Banach space X. The crucial hypothesis on T and S is a Leray- Schauder type boundary condition, e.g. the relation \(Tx=\alpha x+\lambda Sx\) for \(\| x\| =r\) implies that \(\alpha\leq 0\). A parallel ``random'' analogue to these theorems is also given, generalizing previous results by \textit{M. Joshi} [Indian J. Pure Appl. Math. 11, 791- 799 (1980; Zbl 0441.60067)].