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...

The aim of this paper is to propose weak assumptions to prove maximal L q regularity for Cauchy problem : du dt (t) − Lu(t) = f (t). Mainly we only require " off-diagonal " estimates on the real semigroup (e tL) t>0 to obtain maximal L q regularity. The main idea is to use a one kind of Hardy space H 1 adapted to this problem and then use interpolation results. These techniques permit us to pro...

We give elementary proofs of the theorems mentioned in the title. Our methods rely on a simple version of Ramsey theory and a martingale difference lemma. They also provide quantitative results: if a Banach space contains l only with a bad constant then every bounded sequence admits a subsequence which is “nearly” a weak Cauchy sequence.

We prove that the Cauchy problem for Muskat equation is well-posed locally in time any initial data critical space of Lipschitz functions with three-half derivative $$L^2$$ . Moreover, we solution exists globally under a smallness assumption.

This paper is concerned with the Cauchy problem of the modified Kawahara equation. By using the Fourier restriction norm method introduced by Bourgain, and using the I-method as well as the L 2 conservation law, we prove that the modified Kawahara equation is globally well-posed for the initial data in the Sobolev space H s (R) with s > − 3 22 .

Let (x n) be a bounded sequence in a Banach space X. Rosenthal's l 1-theorem states that there is essentially only one exceptional situation where it is not possible to extract a subsequence which is a weak Cauchy sequence: This happpens if (x n) is the sequence of unit vectors in l 1. The aim of these lectures is twofold: On the one hand results from the last few years centering around this th...

The quasilinear degenerate evolution equation of parabolic type d(Mv) dt + L(Mv)v = F (Mv), 0 < t ≤ T considered in a Banach space X is written, putting Mv = u, in the form du dt + A(u)u 3 F (u), 0 < t ≤ T , where A(u) = L(u)M−1 are multivalued linear operators in X for u ∈ K, K being a bounded ball ‖u‖Z < R in another Banach space Z continuously embedded in X. Existence and uniqueness of the l...

In this paper, we consider the Cauchy problem in \(\mathbb {R}^N\), \(N\ge 1\), for semi-linear Schrödinger equations with space–time fractional derivatives. We discuss nonexistence of global \(L^1\) or \(L^2\) weak solutions subcritical and critical cases under some conditions on initial data nonlinear term. Furthermore, local supercritical case are studied.

Let X be a Banach space and (fn)n be a bounded sequence in L (X). We prove a complemented version of the celebrated Talagrand’s dichotomy i.e we show that if (en)n denotes the unit vector basis of c0, there exists a sequence gn ∈ conv(fn, fn+1, . . . ) such that for almost every ω, either the sequence (gn(ω)⊗en) is weakly Cauchy in X⊗̂πc0 or it is equivalent to the unit vector basis of l. We the...

We deene a residue current of a holomorphic mapping , or more generally a holomorphic section to a holomorphic vector bundle, by means of Cauchy-Fantappie-Leray type formulas , and prove that a holomorphic function that annihilates this current belongs to the corresponding ideal locally. We also prove that the residue current coincides with the Colee-Herrera current in the case of a complete in...