Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle,fixed point theorems and parametric optimization problems

In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder’s fixed point theorem are also given.     

The strong Ekeland variational principle

In this paper, we consider the strong Ekeland variational principle due to Georgiev [P.G. Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988) 1–21]. We discuss it for functions defined on Banach spaces and on compact metric spaces. We also prove the τ-distance version of it.     

A variational principle for vector equilibrium problems

A variational principle is described and analysed for the solutions of vector equilibrium problems.     

Large time behavior for nonlinear higher order convection–diffusion equations

We study the large time asymptotic behavior, in L^p (1p∞), of higher derivatives D^γu(t) of solutions of the nonlinear equation

(1)

where the integers n and θ are bigger than or equal to 1, a is a constant vector in with . The function ψ is a nonlinearity such that and ψ(0)=0, and is a higher order elliptic operator with nonsmooth bounded measurable coefficients on . We also establish faster decay when .     

Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G′/G)-expansion method

In this paper, we demonstrate the effectiveness of the so-called (G′/G)-expansion method by examining some nonlinear evolution equations with physical interest. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.     

On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

A countable Fréchet–Urysohn space of uncountable character

We shall construct a countable Fréchet–Urysohn α₄ not α₃ space X such that all finite powers of X are Fréchet–Urysohn.     

Quartets in maximal weakly compatible split systems

Weakly compatible split systems are a generalization of unrooted evolutionary trees and are commonly used to display reticulate evolution or ambiguity in biological data. They are collections of bipartitions of a finite set X of taxa (e.g. species) with the property that, for every four taxa, at least one of the three bipartitions into two pairs (quartets) is not induced by any of the X-splits. We characterize all split systems where exactly two quartets from every quadruple are induced by some split. On the other hand, we construct maximal weakly compatible split systems where the number of induced quartets per quadruple tends to 0 with the number of taxa going to infinity.     

Existence and multiplicity results for some nonlinear problems with singular -Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems

where l(u,u^′)=0 denotes the Dirichlet, periodic or Neumann boundary conditions on [0,T], is an increasing homeomorphism, (0)=0. The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.     

Finite element analysis of an A– method for time-dependent Maxwell’s equations based on explicit-magnetic-field scheme

In this paper, error analysis of a finite element A– method for the time-dependent Maxwell’s equations is presented. An explicit-magnetic-field scheme is applied. Provided that the time-stepsize τ is sufficiently small, the proposed algorithm yields for finite time T an error of in the L²-norm for the electric field E and the magnetic field H, where h is the mesh size.     

Reduction of the Gibbs phenomenon for smooth functions with jumps by the -algorithm

Recently, Brezinski has proposed to use Wynn's ε-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Padé–Fourier and Padé–Chebyshev approximants, including those recently studied by Kaber and Maday.     

Existence and global attractiveness of a square‐mean ‐pseudo almost automorphic solution for some stochastic evolution equation driven by Lévy noise

In this work, we introduce the concept of μ‐pseudo almost automorphic processes in distribution. We use the μ‐ergodic process to define the spaces of μ‐pseudo almost automorphic processes in the square mean sense. We establish many interesting results on the functional space of such processes like a composition theorem. Under some appropriate assumptions, we establish the existence, the uniqueness and the stability of the square‐mean μ‐pseudo almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise. We provide an example to illustrate our results.     

A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.     

Existence and subharmonicity of solutions for nonlinear non-smooth periodic systems with a -Laplacian

In the present paper, we consider the nonlinear periodic systems driven by a vectorial p-Laplacian with a locally Lipschitz nonlinearity. Some existence, multiplicity and subharmonic results are obtained by using non-smooth critical point theory.     

On scatteredly continuous maps between topological spaces

A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset AX the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G_δ-measurability (the preimage of each open set is of type G_δ). Also under Martin Axiom, we construct a G_δ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.