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RESEARCH PRODUCT

On the construction of Ljusternik-Schnirelmann critical values in banach spaces

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w h e r e f a n d g are functionals on a Banach space X, are considered in many papers. The existence theorems are based on the existence of a critical vector with respect to the manifold M,={xEX: f(x)=r}. Morse theory can often be used to obtain precise information about the behaviour of the functional close to the critical level. However, this would limit the study to Hilbert spaces and functions with nondegenerate critical points. These assumptions are not always satisfied in applications and are not rleeded when applying the Ljusternik--Schnirelmann theory. Therefore, Ljusternik--Schnirelmann theory has been widely used to study various nonlinear eigenvalue problems. Very general results for Banach spaces can be found in H. Amann [1] and in E. Zeidler [12], which also contains an extensive bibliography on critical point theories. In the case of a Hilbert space iterative methods for the construction of all Ljusternik--Schnirelmann critical values and critical vectors has been presented by J. Ne~as [7], by A. Kratochvll and J. Ne6as [4], [5] and by J. Ne6as, A. Lehtonen and P. Neittaanmfiki [8]. In [8J we also present numerical examples of the method. In this paper we shall give an extension of the method used in [5] and [8] to study the eigenvalue problem for the constraint function f(x)=[]xlt in uniformly convex Banach spaces.