Narrow Results

It is well known that a finite-dimensional linear system cannot be chaotic. In this article, by introducing a weak topology into a two-dimensional Euclidean space, it shows that Li–Yorke chaos can be generated by a linear map, where the weak topology is induced by a linear functional. Some examples of linear systems are presented, some are chaotic while some others regular. Consequently, several open problems are posted.

This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator.

We will study the dependence of λ(a, b), half-eigenvalues of the one-dimensional p-Laplacian, on potentials , 1 ≤ γ ≤ ∞, where . Two results are obtained. One is the continuity of half-eigenvalues in , where w_γ is the weak topology in space. The other is the continuous differentiability of half-eigenvalues in , where ‖ ⋅ ‖_γ is the L^γ norm of . These results will be used to study extremal problems of half-eigenvalues in future work.

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of $\mathrm{\Delta }$-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc.60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and ${\ell }^p$-spaces, but not in $L^p({ℝ}^N)$, $p\ne 2$. $\mathrm{\Delta }$-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of $\mathrm{\Delta }$-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.

We give a counterexample to Theorem 3.2 in Toumi and Toumi [The range of an orthomorphism, J. Algebra Appl.8 (2009) 863–868]. In the process we add a new equivalent statement to those in Theorem 7.48 of Aliprantis and Burkinshaw [Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, Vol. 105 (American Mathematical Society, 2003)].

The relation between the differential equations in a special abstract function space and the integral equations in the space is discussed by using the continuous wavelet transform; the result that the differential equations can be transformed into the integral equations is obtained; they are equivalent not only in the weak topology but also in the strong topology.