ε-solutions in vector minimization problems

This paper presents some properties of -solutions for vector minimization problems where the function to be optimized takes its values in the Euclidean space ^p . The results obtained generalize the classical ones for exact Pareto solutions.     

L ∞-solutions for some nonlinear degenerate elliptic and parabolic equations

In this paper we prove the existence of bounded solutions for equations whose prototype is:

C~∞-solutions for the Second Type of Generalized Feigenbaum's Functional Equations

This work focuses on the second type of generalized Feigenbaum's equation(f(x)) = f(f((x))),f(0) = 1, 0 ≤ f(x) ≤ 1, x ∈ [0, 1],where (x) is C∞-increasing function on [0, 1] and satisfies that (0) = 0, 0 (x) 1(x ∈ [0, 1]).Using constructive method, we discuss the existence of C∞-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation.     

On nonlinear Schrödinger equations,II.H S -solutions and unconditional well-posedness

We consider the nonlinear Schrödinger equation (NLS) (see below) with a general “potential”F(u), for which there are in general no conservation laws. The main assumption onF(u) is a growth rateO(|u| ^k ) for large |u|, in addition to some smoothness depending on the problem considered. A uniqueness theorem is proved with minimal smoothness assumption onF andu, which is useful in eliminating the “auxiliary conditions” in many cases. A new local existence theorem forH ^S -solutions is proved using an auxiliary space of Lebesgue type (rather than Besov type); here the main assumption is thatk≤1+4/(m?2s) ifs/2,k<∞ ifs=m/2 (no assumption ifs>m/2). Moreover, a general existence theorem is proved for globalH ^S -solutions with small initial data, under the main additional condition thatF(u)=O(|u|^1+4/m) for small |u|; in particularF(u) need not be (quasi-) homogeneous or in the critical case. The results are valid for alls≥0 ifm≤6; there are some restrictions ifm≥7 and ifF(u) isnot a polynomial inu and $\bar u$ .     

On variational principles,level sets,well-posedness,and ∈-solutions in vector optimization

Using the concept for -efficient solutions introduced in Refs. 1 and 2, we extend the Ekeland variational principle and another variational principle given in Ref. 3 to vector-valued objective functions. This enables us to establish a kind of well-posedness for the resulting perturbed vector optimization problems. Based on this definition of -efficiency, we also formulate a concept for level sets and prove some results about the Kuratowski limits of level sets.This research was supported by the Bulgarian Ministry of Education and Science and by the Deutsche Forschungsgemeinschaft.     

On smoothness of <Emphasis Type="Italic">L</Emphasis><Subscript>3,∞</Subscript>-solutions to the Navier–Stokes equations up to boundary

We show that L_3,-solutions to the three-dimensional Navier-Stokes equations near a flat part of the boundary are smooth.Mathematics Subject Classification (1991): 35K, 76D