Showing papers in "Nonlinear Analysis-theory Methods & Applications in 1993"

Infinitely many solutions of a symmetric Dirichlet problem

Abstract: Here Q is a bounded domain in [R” with smooth boundary and F: I?“’ + R is C’ and satisfies certain growth conditions. If m = 1 and F is an even function, hence, F,: R + R is an odd function, then Ambrosetti and Rabinowitz [l, 21, proved the existence of infinitely many (weak) solutions of (D) using a symmetric version of the mountain pass theorem. A similar result is due to Michalek [3] who considered the case where m = 21, Rm s C’ and F is invariant under a free action of the cyclic group Z/p on C’. Here Z/p is considered as a subgroup of S’ C C acting essentially via scalar multiplication on C’. In this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of (D). Let G be a compact Lie group and p: G -+ O(m) an orthogonal representation of G on lRm. Then G acts on IRm via the map G x IR” -+ IR”, (g, u) y gv := p(g)v. We write V for the space [Rm considered as a G-space. Do there exist infinitely many solutions of (D) if F is invariant under this symmetry, i.e. F(gu) = F(u) for all g E G and u E V? Without further assumptions on p the answer to this question is no, in general. For example, if p(g) = id for all g E G then there is no symmetry at all and one cannot expect infinitely many solutions. More generally, one should exclude the existence of fixed points: VG = (u E VI gu = u for all g E G] = 0. Our first result says that there exist infinitely many solutions of (D) if the following condition is satisfied:

An existence theorem for a nonlinear difference equation

Abstract: where p > 1 and {s,)‘p is a real sequence. We establish conditions under which (1.1) has a positive nondecreasing solution. Here a solution of (1.1) is a real sequence y = (_Y~); which satisfies (1.1). Since (1.1) is a recurrence relation, given real initial values y, and y, , it is clear that we can inductively obtain y, , y, , . . . . Since p is a real number greater than 1, the sequence (yk] may not be real (that is, not a solution). Note, however, that any constant multiple of a solution is again a solution. The existence question originated in [l], where it was shown that Hardy’s inequality for a series [2] can be viewed as a necessary condition for the existence of a positive nondecreasing solution of (1.1). Our existence theorem reduces to a nonoscillation criterion in the case p = 2, since then (1.1) changes to a second order linear difference equation (see, e.g. [3]). A brief outline of the paper is as follows. Section 2 introduces a Riccati-type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution. Section 3 discusses some of the functional analytic background which is needed in order to apply Schauder’s theorem. Section 4 contains the proof of our existence theorem, and Section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in (1.1) are nonnegative.

A limit set trichotomy for self-mappings of normal cones in banach spaces

Abstract: (Here, we use the notation x > y and x 2 y to mean, respectively, x y E k” and x y E K”.) This result extends an earlier theorem of Smith [2] concerning “discrete dynamics of monotone, concave maps”; some interesting applications to differential equations can be found in [l, 21. Another extension of Smith’s theorem has been given by TakaE in [3]. In this paper we shall extend the above trichotomy in two directions. First, we shall allow general normal cones K with nonempty interior in a Banach space. Second, we shall allow a class of maps which is considerably more general than classes allowed in earlier results, even for K, the positive orthant in I?“. The key observation which we shall exploit centers about a metric p, called the part metric or Thompson’s metric (see [l, 4-61 and the references given there; and Section 2 below) which is defined on the interior I? of a cone. The proper class of maps to study seems to be those maps T: I? + k such that T”‘, the mth iterate of T, satisfies

Random fixed points of random multivalued operators on polish spaces

Abstract: RANDOM coincidence point theorems and random fixed point theorems are stochastic generalizations of classical coincidence point theorems and classical fixed point theorems. Random fixed point theorems for contraction mappings in Polish spaces were proved by Spacek [l] and Hans (2, 31. For a complete survey, we refer to Bharucha-Reid [4]. Itoh [5] proved several random fixed point theorems and gave their applications to random differential equations in Banach spaces. Recently, Sehgal and Singh [7], Papageorgiou [8] and Lin [9] have proved different stochastic versions of the well-known Schauder’s fixed point theorem. The aim of this paper is to prove various stochastic versions of Banach type fixed point theorems for multivalued operators. Section 2 is aimed at clarifying the terminology to be used and recalling basic definitions and facts. Section 3 deals with random coincidence point theorems for a pair of compatible random multivalued operators. The structure of common random fixed points of these operators is also studied. In Section 4, the existence of a common random fixed point of two random multivalued operators satisfying the Meir-Keeler type condition in Polish spaces is proved. Section 5 contains a random fixed point theorem for a pair of locally contractive random multivalued operators in .s-chainable Polish spaces. As an application, a theorem on random approximation is also obtained.

A further remark on the regularity of the solutions of the p -Laplacian and its applications to their finite element approximations

Abstract: In this paper, global H 2 regularity is established for the solutions of the p -Laplacian equation, 1 p ≤ 2, and related equations on domains which are either convex or have C 2 boundaries, The results are used to derive some explicit error bounds for the finite element approximation of the p -Laplacian, which were not previously available because of the lack of regularity results.

On minima of a functional of the gradient: necessary conditions

Abstract: In this paper we propose a new method for texture segmentation based on the use of texture feature detectors derived from a decorrelation procedure of a modified version of a Pseudo-Wigner distribution (PWD). The decorrelation procedure is accomplished by a cascade recursive least squared (CRLS) principal component (PC) neural network. The goal is to obtain a more eAcient analysis of images by combining the advantages of using a high-resolution joint representation given by the PWD with an eAective adaptive principal component analysis (PCA) through the use of feedforward neural networks. ” 1999 Elsevier Science B.V. All rights reserved.

On the theory and error estimation of the reduced basis method for multi-parameter problems

Abstract: : In an earlier paper (ZAMM 63, 1983, 21), J.P.Fink and W.C.Rheinboldt developed a priori local error estimates for the scalar-parameter case of the reduced basis method by considering the method in a differential-geometric setting. Here it is shown that an analogous setting can be used for the analysis of the method applied to problems with a multidimensional parameter vector and that this leads to a corresponding local error theory also in this general case.

Finiteness for critical periods of planar analytic vector fields

Abstract: LIMBURGS UNIV CENTRUM,DEPT MATH,B-3610 DIEPENBEEK,BELGIUM.CHICONE, C, UNIV MISSOURI,DEPT MATH,COLUMBIA,MO 65211.