Given a Radon measure μ on R, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space BMO(μ) when μ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming μ doubling. For instance, Calderón-Zygmund operators which are bounded on L(μ) are also bou...

Consider the anisotropic porous medium equation, ut = n ∑ i=1 (um)xixi , where mi > 0, (i = 1, 2, · · · , n) satisfying min 1≤i≤n {mi} ≤ 1, n ∑ i=1 mi > n−2, and max 1≤i≤n {mi} ≤ 1 n (2+ n ∑ i=1 mi). Assuming that the initial data belong only to L(R), we establish the existence and uniqueness of the solution for the Cauchy problem in the space, C([0,∞), L(R))∩C(R× (0,∞)) ∩ L(R × [ε,∞)), where ε...

A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exists non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a ra...

In this paper, we study the Schrödinger-Robertson uncertainty relations as corollaries of equalities in a scalar product space. Moreover, we give a number of characterizations in the case where the associated inequalities are in fact equalities. Our presentation is based exclusively on an algebraic observation on the standard Cauchy-Schwarz inequality and could presumably provide a clear and ex...

We prove a representation formula for solutions of Schrödinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L or in the energy space of model equations arising in Bose Einstein condensation [1] or in fiber optics [2]. Our results also give a...

The numerical solution of the Cauchy problem for elliptic equations is considered. We reformulate the original problem as a variational inequality problem, which we solve using the finite element method. Moreover, we prove the convergence of the approximate solution. Let <$> be a bounded open set in the space R" and d6^ be the boundary of fy. Then ßr = ty X (0, T) is a bounded open set in Rn+I....

in the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. in particular, it isshown that the cauchy-schwarz inequality holds. moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy hilbert space has a complementary subspace.finally, the notions of fuzzy bo...

We consider the Cauchy problem in nonlinear thermoelasticity with second sound in one space dimension. Due to Cattaneo’s law, replacing Fourier’s law for heat conduction, the system is hyperbolic. We first investigate the local well-posedness as a strictly hyperbolic system, and then discuss the relation between energy estimates for non-symmetric hyperbolic systems and well-posedness. For the g...

The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L∞. The solutions attained are unique and exclusive. Performance is established by utilizing the maximum principle for certain generalized time and space-fractional diffusion equations. The fractional differenti...

In the present paper we study lightlike submanifolds of almost paracontact metric manifolds. We define invariant lightlike submanifolds. We study radical transversal lightlike submanifolds of para-Sasakian manifolds and investigate the geometry of distributions. Also we introduce a general notion of paracontact Cauchy-Riemann (CR) lightlike submanifolds and we derive some necessary and sufficie...