On Hyperbolic Variational Inequalities of First Order and Some Applications

This paper presents further developments in the study of strong solutions and their coincidence sets to the obstacle problem for linear transport operators. Under natural mild assumptions, the strong convergence of the solutions and the characteristic functions of their coincidence sets is obtained in the passage of second order to first order problems. An application to a steady-state chemotaxis problem is given. The extension to the two obstacles problem of first order is also presented, having in view an application to a porous media model in presence of saturation arising in petroleum engineering.     

ON THE HYPERBOLIC OBSTACLE PROBLEM OF FIRST ORDER

InroductlonThe classical obstacle problem canbe formulated as the     

Strong unique continuation for higher order elliptic equations with Gevrey coefficients

We address the strong unique continuation problem for higher order elliptic partial differential equations in 2D with Gevrey coefficients. We provide a quantitative estimate of unique continuation (observability estimate) and prove that the solutions satisfy the strong unique continuation property for ranges of the Gevrey exponent strictly including non-analytic Gevrey classes. As an application, we obtain a new upper bound on the Hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients.     

The nonlinear <Emphasis Type="Italic">N</Emphasis>-membranes evolution problem

The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution are studied in what concerns the approximation, regularity, and stability of variational solutions. To the evolutionary case is extended the characterization of the Lagrange multipliers associated with the ordering constraint in terms of characteristic functions of coincidence sets. Continuous dependence results are given, and the asymptotic behavior as t → ∞ of the solution and the coincidence sets, showing that they converge to their stationary counterparts, is studied. Bibliography: 22 titles. Dedication to V. A. Solonnikov on the occasion of his 75th birthday with admiration and friendship Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 303–324.     

二阶常微分方程周期边值问题的三解定理

在有两对严格上下解的假定下,通过一些构造和迭合度的计算,证明了 一般的非线性二阶常微分方程周期边值问题至少有三个不同的解的结果.作为应用, 讨论了Li'enard方程的多解问题.     

Bifurcation of fixed points from a manifold of trivial fixed points in the infinite-dimensional case

We obtain a necessary as well as a sufficient condition for the existence of bifurcation points of a coincidence equation, and, in particular, of a parametrized fixed point problem. In both cases the trivial solutions are assumed to form a finite-dimensional submanifold of a Banach manifold. An application is given to a delay differential equation on a manifold: we detect periodic solutions that rotate close to an equilibrium point. To Albrecht Dold and Edward Fadell, superb mathematicians and first rate friends     

Differential Constraints and Exact Solution to Riemann Problems for a Traffic Flow Model

In view of solving in a closed form initial and/or boundary value problems of interest in nonlinear hyperbolic and dissipative wave processes it is considered a reduction approach based upon appending differential constraints to quasilinear nonhomogeneous hyperbolic systems of first order PDEs. In this context a governing model of traffic flow is analyzed thoroughly and a classification of possible constraints along with sets of consistent response functions involved therein is worked out whereupon the classes of corresponding exact solutions are determined. To some extent these solutions generalize the classical simple wave solutions and may also incorporate dissipative effects. Furthermore, in order to solve a Riemann Problem, an exact rarefaction wave-like solution is built. Finally an application of the results to the so-called ??green traffic light problem?? is also illustrated.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Single machine scheduling with symmetric earliness and tardiness penalties

This research focuses on scheduling jobs with varying processing times and distinct due dates on a single machine subject to earliness and tardiness penalties. Hence, this work will find application in a just-in-time (JIT) production environment. The scheduling problem is formulated as a 0–1 linear integer program with three sets of constraints, where the objective is to minimize the sum of the absolute deviations between job completion times and their respective due dates. The first two sets of constraints are equivalent to the supply and demand constraints of an assignment problem. The third set, which represents the process time non-overlap constraints, is relaxed to form the Lagrangian dual problem. The dual problem is then solved using the subgradient algorithm. Efficient heuristics have also been developed in this work to yield initial primal feasible solutions and to convert primal infeasible solutions to feasibility. The computational results show that the relative deviation from optimality obtained by the subgradient algorithm is less than 3% for problem sizes varying from 10 to 100 jobs.     

Fuzzy programming and linear programming with several objective functions

In the recent past numerous models and methods have been suggested to solve the vectormaximum problem. Most of these approaches center their attention on linear programming problems with several objective functions. Apart from these approaches the theory of fuzzy sets has been employed to formulate and solve fuzzy linear programming problems. This paper presents the application of fuzzy linear programming approaches to the linear vectormaximum problem. It shows that solutions obtained by fuzzy linear programming are always efficient solutions. It also shows the consequences of using different ways of combining individual objective functions in order to determine an “optimal” compromise solution.     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture   for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N$\mathbb{N}$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–?ech compactification βN$\beta \mathbb{N}$. This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture  . The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–?ech compactification of N$\mathbb{N}$.     

Oscillating solutions of a nonlinear fourth order ordinary differential equation

We study the existence of periodic solutions for a nonlinear fourth order ordinary differential equation. Under suitable conditions we prove the existence of at least one solution of the problem applying coincidence degree theory and the method of upper and lower solutions.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

扰动变分不等式和扰动相补问题研究的重合指数和不动点指数方法

本文首次利用重合指数和不动点指数方法研究具单值和多值扰动的变分不等式和相补问题解的存在性问题及其解指数理论．     

Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory

We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type with a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally smooth. Afterward, we deal with discontinuous entropy solutions, and obtain the global well posedness of the Cauchy problem in this class. Eventually, we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data. As a byproduct, we establish the existence of an utility process which is a solution to a backward-forward stochastic differential equation, for a given class of final utilities, which is relevant for financial applications.     

Four positive periodic solutions for the first order differential system

In this paper, we establish the existence of four positive periodic solutions for the first order differential system by using the continuation theorem of coincidence degree theory. When our result is applied to a competition Lotka-Volterra population model, we obtain the existence of four positive periodic solutions for this model.     

Scheduling coupled-operation jobs with exact time-lags

Scheduling coupled-operation jobs with exact time-lags on a single machine has a wide range of applications. In that problem, each job consists of two operations with given processing times, which should be scheduled on a single machine observing a given time-lag. The general case of the problem with arbitrary processing times of operations and arbitrary time lags is known to be NP-hard in the strong sense and the problem remains NP-hard for many special cases. In order to develop a local search algorithm for the problem, we first explore two possible approaches for representing feasible solutions and their neighborhoods based on maintaining a permutation of first operations of the jobs or maintaining a full permutation of all operations. The first representation appears to be unpromising since, as we prove, the problem of finding an optimal sequence of second operations for a fixed sequence of first operations is NP-hard in the strong sense even in the case of unit processing times. We elaborate the second approach by developing a tabu search heuristic based on efficient job re-insertion. Empirical evaluation demonstrates superiority of the developed algorithm in comparison with the earlier published algorithms.     

A third order boundary value problem with nonlinear growth at resonance on the half‐axis

This paper is concerned with the solvability of a fully nonlinear third‐order m‐point boundary value problem at resonance posed on the half line. The nonlinearity which depends on the first and the second derivatives satisfies a sublinear‐like growth condition. Our main existence result is based on Mawhin's coincidence degree theory. An illustrative example of application is included. Copyright © 2016 John Wiley & Sons, Ltd.     

一阶混合型积分微分方程的周期边值问题

在有一般化的上下解的条件下,研究了一阶混合型积分微分方程的周期边值问题的可解性和最小最大解的存在性。讨论基于迭合度方法,单调迭代方法和新的比较定理。