Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2000"

On the vectoral Ekeland's variational principle and minimal points in product spaces

Abstract: Phelps [13] noticed that the (scalar) Ekeland's variational principle (EVP) is equivalent to the existence of a minimal point of the epigraph of the corresponding function with respect to an appropriate order. Attouch and Riahi [1] showed that EVP is equivalent to the existence of maximal points with respect to cones satisfying some additional conditions. Taking these into account, Gopfert and Tammer ([6], [7]) established a maximal point theorem in a product space. The aim of this paper is to obtain several minimal point theorems in product spaces and the corresponding variants of the vectorial EVP.

Homogenizing the acoustic properties of the seabed: part I

Abstract: One of the pressing problems in underwater acoustics today is formulating and then solving a model for interaction of acoustic waves in a shallow ocean with the seabed. Shallow-water/seabed waveguide, direct and inverse wave propagation problems are ubiquitous in applied science and technology. One such application is for inverse imaging of objects submerged in the ocean or the seabed. As much of the acoustic energy passes into the seabed, this imagery is possible only if the sea environment (water, sediment, interfaces), in the absence of the object, is properly characterized beforehand. This means that a suitable model of the sediment and of propagation of sound therein must be developed and a method be proposed for solving the inverse problem of the identification of the mechanical parameters involved in this model. This model, as well as the sediment parameter and object identification scheme, must be able to take into account sound speed and density variations in the water as well as the behavior of sound in the seabed. In general, either an acoustic pulse, or a monochromatic signal with frequency o is used. Consequently, not only acoustic signals with acoustic frequencies spread about a central frequency, but time-harmonic solutions are of interest. There have been several acoustic models of the seabed [9,10,16]; however, the primary one in usage goes back

First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions

Abstract: The monotone iterative technique is a powerful method that have been used to approximate the extremal solutions of several problems [1, 2] On the other hand, impulsive differential equations are a basic tool to study some problems of biology, medicine, engineering, and physics [3] In this paper we deal with the following nonlinear boundary value problem for first order differential equation with impulses at fixed points

Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems

Abstract: In this paper we study existence of positive solutions to singular elliptic boundary value problems involving divergence terms in general domains. By constructing suitable upper and lower solutions and making comparison, we obtain suucient conditions for existence and nonexistence of solutions. We also study a concrete example to show that the conditions imposed on parameters appearing in the structure conditions of nonlinear terms are optimal and our results can be used to get boundary regularity of solutions.

Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

Abstract: Here a; b?0 and p?1, m?1. In case of IBVP, in a bounded domain ⊂Rn with Dirichlet boundary conditions, the following results are known: 1. When a=0, it is proved (see [1, 3, 8, 14, 16]) that the solution blows up in nite time for su ciently large initial data. 2. When b=0; Haraux and Zuazua [5] and Kopackova [7] prove the global existence result for large initial data. The behavior of the solution of Eq. (1.1) with nonlinear source and linear damping (case m=1) in an abstract setting was considered by Levine in [9]. More precisely, he showed that the solutions with negative initial energy are nonglobal.

Bean's critical-state model as the p → limit of an evolutionary p -Laplacian equation

Abstract: We consider magnetization of type II superconductors characterized by a multi valued current voltage relation the Bean model and show that for the longitudinal and thin lm con gurations the problems are equivalent to similar evolutionary variational inequalities with a gradient constraint It is proved that the unique solutions to these inequalities are the p limits of the solutions to the evo lutionary p Laplacian equations that appear in the corresponding magnetization models obeying the power current voltage law with exponent p Introduction The Bean critical state model provides a description for the magnetization of type II su perconductors in a nonstationary external magnetic eld The model was rst formulated for the simplest con guration of a cylindrical superconductor in a parallel eld see Since then more complicated cases have also been considered in particular a very thin superconducting lm in a perpendicular external eld see and the references therein Phenomenologically the problem can be understood as a nonlinear eddy current problem In accordance with the Faraday law of electromagnetic induction the eddy currents in a conductor are driven by the electric elds induced by time variations of the magnetic ux In an ordinary conductor the vectors of the electric eld and the current density are usually related by the linear Ohm law Type II superconductors are instead characterized in the Bean model by a highly nonlinear current voltage relation This non linearity gives rise to an interesting free boundary problem which is considered here for the two speci c geometrical con gurations mentioned above a long cylinder in a parallel magnetic eld and a thin lm in a perpendicular eld Acknowledges travel support from Marks and Spencer Ltd Present address CEEP Blaustein Inst for Desert Research Ben Gurion University of the Negev Sede Boqer Campus Israel In these cases the electric eld e inside the isotropic superconductor has the same direction as the current density j and the superconducting material may be characterized by a scalar current voltage law This nonlinear constitutive relation is given in the Bean model by a multivalued monotone graph jej if jjj if jjj if jjj Here we have adopted units in which the critical current density jc The magne tization model with this current voltage law is equivalent to an evolutionary variational inequality see and such a formulation is convenient for both the numerical approxi mation and theoretical study of these magnetization problems In simple cases the solution to the Bean model can be found analytically see e g the two examples in the next section Physicists however usually approximate by a smooth function in order to sim plify the numerical discretization or to account for the thermally activated creep of the magnetic ux see The power law approximation jej jjj for a xed large p R is the most often adopted This approximation leads to evolu tionary equations involving the p Laplacian operator and it was assumed in the physical literature that their solutions converge to the Bean model solution as p In this paper we study the behaviour of the solutions to these evolutionary equations for two geometrical con gurations and prove rigorously that the convergence does indeed take place in each case to the unique solution of the corresponding evolutionary variational inequality equivalent to the Bean critical state model for that con guration For long cylinders in a parallel eld the variational inequality problem can be written in terms of the magnetic eld and involves a gradient constraint This problem is similar to that arising in another critical state model the sandpile growth model see Recently Aronsson Evans and Wu have shown that the sandpile growth model can be obtained as the p limit of the Cauchy problem for an evolutionary p Laplacian equation We partially adopt their techniques in our consideration of the corresponding limits of the similar boundary value problems in superconductivity It should be noted that the similarity between the magnetization of type II superconductors and the growth of sandpiles is well known see In the case of a thin superconducting lm placed into a perpendicular magnetic eld a variational inequality with the same gradient constraint as in the cylindrical case can be derived for the stream function of a divergence free two dimensional d sheet current density This evolutionary variational inequality is implicit with respect to the time derivative and we prove that it is the p limit of an implicit evolutionary equation involving the p Laplacian operator In the next section we derive variational formulations for the power law and Bean magnetization problems for these two speci c geometrical con gurations Although the two con gurations lead to di erent mathematical problems these can be regarded as two special cases of a more general evolutionary problem involving the p Laplacian Therefore in section we analyse the well posedness of this more general problem and study its limit as p Variational formulation of the models Let R be a bounded connected domain with a Lipschitz boundary If is not simply connected we allow it to have a nite number of holes i i I with i being a bounded domain with a connected boundary i We set

The Knaster-Kuratowski and Mazurkiewicz theroy in hyperconvex metric spaces and some of its applications

Abstract: The Knaster-Kuratowski and Mazurkiewicz principle is characterized in hyperconvex metric spaces, leading to a characterization theorem for a family of subsets with the finite intersection property in such setting. The theorem is illustrated by giving hyperconvex versions of Fan's celebrated minimax principle and Fan's best approximation theorem for set-valued mappings. These are applied to obtain formulations of the Browder-Fan fixed point theorem and the Schauder-Tychonoff fixed point theorem in hyperconvex metric spaces for set-valued mappings. In addition, existence theorems for saddle points, intersection theorems and Nash equilibria are obtained.

Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications

Abstract: where I ⊂R is an open interval, f(x) is a given function, is a nonlinear di erential operator, and W ( ) ∈ L(I) is a piecewise Gâteaux di erentiable function of = (u); Ua is a closed convex subspace of a re exive Banach space U. This general nonconvex, nonsmooth variational problem appears in many nonlinear systems. For example, in the nonlinear equilibrium problem of Ericksen’s bar subjected to axial extension [17], or the post-buckling analysis of extended nonlinear beam subjected to a compressed load [26], the nite strain = (u)= 2u 2 ; x − is a quadratic operator,