Showing papers in "Nonlinear Analysis-theory Methods & Applications in 1990"

Nonexpansive iterations in hyperbolic spaces

Abstract: ONE OF THE most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings. Most of the results, however, have been obtained in normed linear spaces. It is natural, therefore, to try to develop a theory of nonexpansive iterations in more general infinite-dimensional manifolds. This is the purpose of the present paper. More specifically, we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general, and of iterative processes for nonexpansive mappings in particular. This class of metric spaces, which is defined in Section 2, includes all normed linear spaces and Hadamard manifolds, as well as the Hilbert ball and the Cartesian product of Hilbert balls. In Section 3 we introduce co-accretive operators and their resolvents, and present some of their properties. In the fourth section we discuss the concept of uniform convexity for hyperbolic spaces. Section 5 is devoted to two new geometric properties of (infinite-dimensional) Banach spaces. Theorem 5.6 provides a characterization of Banach spaces having these properties in terms of nonlinear accretive operators. In Sections 6, 7 and 8 we study explicit, implict and continuous iterations, repectively, using the same approach in all three sections. We illustrate this common approach with the following special case. Let C be a closed convex subset of a hyperbolic space (X, p), let T: C --f C be a nonexpansive mapping, and let x be a point in C. In order to study the iteration (T”x: n = 0, 1,2, . . .), we set z,, = (1 (l/n))x 0 (l/n)T”x, K = clco(zj;j I l), and d = inf(p(y, Ty): y E C). The first step is to show that p(x, K) = lim p(x, T”x)/n = d. This leads to the convergence “+m of lz,) when X is uniformly convex and to the weak convergence of (z,,] when X is a Banach space which is reflexive and strictly convex. When T is an averaged mapping we are also able to establish the following triple equality. For all k 2 1,

Existence results for the flow of viscoelastic fluids with a differential constitutive law

Abstract: where 2 is the (symmetric) extra-stress tensor (the total stress is given by g = -pL + I, where p is the hydrodynamic pressure); 4 is the rate of deformation tensor, g[u] = *(Vu + Vu’), u being the velocity field; A, is the relaxation time, A2 the retardation time, 0 I A, < A1 ; and Q is the fluid viscosity. The symbol a),/d)t denotes an objective (frame indifferent) derivative [I, 181. More precisely,

Positivity properties of viscosity solutions of a degenerate parabolic equation

Abstract: We consider the problem: (I) {u t =uΔu−γ|⊇u| 2 in R N ×R + , u(x, 0)=u 0 (x) in R N , where γ is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on R N

Existence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation

Abstract: A mathematical model for the transverse deflection of an extensible beam of length L whose ends are held at fixed distance apart is equation ∂ 2 u/∂t 2 +α∂ 4 u/∂x 4 +(β+∫ 0 L uξ 2 (ξ,t)dξ)(−∂ 2 u/∂x 2 )=0, which has been proposed by Woinowsky and Krieger [18], where α is a positive constant, β is a constant, not necessarily positive, and the nonlinear term represents the change in the tension of the beam due to its extensibility

On a quasilinear wave equation with memory

Abstract: Equation (1. l), with a’ = 0 and n = 1, has its roots in a model for the small amplitude vibrations of a string in which the dependence of the tension on the deformation cannot be ignored (see for example, Bernstein [2], Carrier [3, 41, Dickey [S] and Nishida [13]). When a’ # 0, (1.1) may be used to describe the dynamics of an extensible string with fading memory. This equation states that the dynamic equilibrium of a body depends not only on the present state of deformation, but also on the previous history of the deformation gradient. Since the publication of the aforementioned papers, equation (1.1) (with a’ = 0) has been the subject of intense scrutiny (see for example Arosio and Spagnolo [ 11, Lions [9], Medeiros [lo], Menzala [ll], Nishihara [14, 151, Pohoiaev [16], and Yamada [17, 181, to name but a few). Among the facts thus far proved, Pohoiaev’s results (and their extension to the degenerate case of Arosio and Spagnolo) are of particular significance; they show that if the data satisfy an analytic-type condition, then solutions are global in time. In spite of the abundant literature on the memoryless form of equation (l.l), to our best knowledge, equation (1.1) (with a’ f 0) appears to be untouched. Our main results (theorem 5.1, theorem 5.2 and theorem 5.4) will show that under appropriate assumptions on the kernel

Global existence of solutions for a strongly coupled quasilinear parabolic system

Abstract: We are interested in existence, uniqueness and a priori estimates of non-negative solutions to the following problem: $$ \begin{gathered} {u_{t}} = \Delta \left[ {\left( {1 + \alpha v} \right)u} \right] + uf(u,v), \hfill \\ {v_{t}} = \Delta v + vg(u,v)\quad in\;(0,\infty ) \times \Omega , \hfill \\ u = v = 0\quad in\;(0,\infty ) \times \partial \Omega , \hfill \\ u(0) = {u_{{0,}}}v(0) = {v_{0}}\quad in\;\Omega \hfill \\ \end{gathered} $$ (1.0) .

An inverse hyperbolic integrodifferential problem arising in geophysics II

Abstract: eG3, t) = g(t), t E K4 Tl, (1.4) x0 being a fixed point in IF’. The problem of determining h arises in Geophysics (see [ 1, 21) and was studied by Yanno [ 1 l] for the one-dimensional equation, using a supplementary information different from (1.4). In [5] we proved local (in time) existence, global uniqueness and stability for the inverse problem (1 . l)-(1.4) by means of a technique very different from that used in [ 111. Here we establish analogous results holding also for a general (bounded) geometry.