On conditions determining formal automorphisms of integro-differential operators

Lagrangian coordinates in free boundary problems for parabolic equations

Special Properties of Generalized Power Series

Existence of solutions of the abstract Cauchy problem of fractional order

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.

Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem

The car sequencing problem is the ordering of the production of a list of vehicles which are of the same type, but which may have options or variations that require higher work content and longer operation times for at least one assembly workstation. A feasible production sequence is one that does not schedule vehicles with options in such a way that one or more workstations are overloaded. In variations of the problem, other constraints may apply. We describe and compare three approaches to the modeling and solution of this problem. The first uses integer programming to model and solve the problem. The second approaches the question as a constraint satisfaction problem (CSP). The third method proposes an adaptation of the Ant Colony Optimization for the car sequencing problem. Test-problems are drawn from CSPLib, a publicly available set of problems available through the Internet. We quote results drawn both from our own work and from other research. The literature review is not intended to be exhaustive but we have sought to include representative examples and the more recent work. Our conclusions bear on likely research avenues for the solution of problems of practical size and complexity. A new set of larger benchmark problems was generated and solved. These problems are available to other researchers who may wish to solve them using their own methods.

This paper introduces simultaneous generalized hill-climbing (SGHC) algorithms as a framework for simultaneously addressing a set of related discrete optimization problems using heuristics. Many well-known heuristics can be embedded within the SGHC algorithm framework, including simulated annealing, pure local search, and threshold accepting (among others). SGHC algorithms probabilistically move between a set of related discrete optimization problems during their execution according to a problem probability mass function. When an SGHC algorithm moves between discrete optimization problems, information gained while optimizing the current problem is used to set the initial solution in the subsequent problem. The information used is determined by the practitioner for the particular set of problems under study. However, effective strategies are often apparent based on the problem description. SGHC algorithms are motivated by a discrete manufacturing process design optimization problem (that is used throughout the paper to illustrate the concepts needed to implement a SGHC algorithm). This paper discusses effective strategies for three examples of sets of related discrete optimization problems (a set of traveling salesman problems, a set of permutation flow shop problems, and a set of MAX 3-satisfiability problems). Computational results using the SGHC algorithm for randomly generated problems for two of these examples are presented. For comparison purposes, the associated generalized hill-climbing (GHC) algorithms are applied to the individual discrete optimization problems in the sets. These computational results suggest that near-optimal solutions can be reached more effectively and efficiently using SGHC algorithms.

where x ∈ En, S and A are constant matrices, S is in general a singular matrix, f(t, x, λ) is a 2π-periodic function of t, λ ∈ Em, λ is a parameter, t ∈ (−∞,∞), and Ep is the p-dimensional vector space. Further, we assume that f(t, x, λ) = C(t, x, λ) + D(t, x, λ), where C(t, x, λ) is a form of order s > 1 in the variables x and λ, D(t, x, λ) is a finite sum of forms of order > s in the same variables, C(t, 0, λ) ≡ 0, and D(t, 0, λ) ≡ 0. Consequently, x = 0 is a solution of system (1) for any λ. The problem on the existence of a periodic solution of system (1) for S = E (E is the identity matrix) was considered in [1–5]; an important requirement in this connection was that the generating solution is a nonzero solution. If the zero solution of system (1) is chosen as the generating solution, then the zero solution is the unique periodic solution whose existence was guaranteed in the above-mentioned papers. The same problem was considered in [6]. The existence and uniqueness theorem for a periodic solution was proved there by the method of successive approximations without using any properties of the principal solution matrix of the linear approximation system. The contraction mapping principle was used in [7] to obtain conditions for the existence of a periodic solution of system (1) in the form of a sum of a trigonometric polynomial with a prescribed number of harmonics and a remainder term. The existence of a periodic solution of system (1) in the form of a sum of a trigonometric polynomial and a series remainder was proved in [8, p. 375; 9, p. 202 of the Russian translation] by the method of successive approximations; the coefficients of the polynomial were found from an equation that is transcendental in general. Note that the papers [6–9] do not answer the question as to whether the periodic solution whose existence was proved in these papers for the case in which x = 0 is a solution of system (1) for any value of the parameter λ is nonzero. The problem on the existence of solutions and periodic solutions of system (1) was considered in [10–14] in the linear case under the assumption that S can be a singular matrix. In the present paper, we pose the problem of finding conditions for the existence of a nonzero periodic solution of system (1) in a neighborhood of the zero solution. The existence of a periodic solution in the form of a trigonometric series is proved by a fixed point method under a necessary condition for the existence of a solution of some finite-dimensional algebraic system of equations. We introduce the following notation: |u| = maxi {|ui|}, ‖Q‖ = max|u|≤1 |Qu|, u ∈ Ep, Q is a matrix, N is the set of positive integers, and N0 is the set of nonnegative integers. Let Mn be the set of series a0 + ∑∞ k=1 ak cos kt + bk sin kt, where a0 and ak, bk (the coefficients of the series) are n-dimensional vectors for any k ∈ N . The zero element of the set Mn is defined as the series with zero coefficients and is denoted by 0. Let x = a0 + ∑∞ k=1 ak cos kt + bk sin kt ∈ Mn and y = c0 + ∑∞ k=1 ck cos kt + dk sin kt ∈ Mn, let Q be an n× n matrix, and let γ be a real number. Then the operations x+ y, Qx, γx, and ẋ are defined by the formulas

In order to enlarge the set of boundary value problems on time scales, for which we can use the lower and upper solutions technique to get existence of solutions, we extend this method to the case when the pair lacks ordering. We use the degree theory and a priori estimates to obtain the existence of solutions for the second-order Dirichlet boundary value problems. To illustrate a wider application of this result, we conclude with an ex-ample which shows that a combination of well- and nonwell- ordered pairs can yield the existence of multiple solutions. Copyright © 2006 Petr Stehlı́k. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.

The paper is focused on solvability of an inverse boundary problem with an unknown coefficient which depends on time for a second-order parabolic equation with non-classical boundary conditions. The idea of the problem is that together with the solution it is required to determine the unknown coefficient. The problem is considered in the rectangular area. The paper introduces a classical solution of the set problem. At first, an auxiliary inverse boundary problem is examined and the equivalence (in some sense) of the original problem is proved. First, we apply a method of variable separation to analyze the auxiliary inverse boundary problem. Then, we examine a spectral problem for an ordinary second-order differential equation with integral conditions. Having used a formal scheme of the method of variable separation, the solution of a direct boundary problem (in case of specified unknown function) resolves itself into solution of Cauchy problem. After that the solution is limited to the solution of a countable system of integro-differential equations in Fourier coefficients. In its turn, the last system regarding unknown Fourier coefficients is recorded in the form of an integro-differential equation in the desired solution. Using relevant additional conditions of the auxiliary inverse boundary problem, we obtain a system of two nonlinear integral equations for defining unknown functions. Thus, the solution of the auxiliary inverse boundary problem comes down to the system of two nonlinear integro-differential equations in unknown functions. The specific Banach space is designed. Then, in the sphere made of the Banach space we with the help of contracted mapping prove the solvability of the nonlinear integro-differential equations set, which is a unique solution of the additional inverse boundary problem. Using the equivalence of problems, it is concluded about existence and uniqueness of a classical solution of the original problem.

We prove existence and uniqueness of classical solutions for an incomplete second-order abstract Cauchy problem associated with operators which have polynomially bounded resolvent. Some examples of differential operators to which our abstract result appli

A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

For a differential equation with a hysteresis nonlinearity of general type, which can be time-varying, we obtain sufficient conditions for the existence and uniqueness of a solution of the Cauchy problem similar to the well-known Cauchy-Picard, Peano, Perron, and Rosenblatt theorems for ordinary differential equations. We consider examples in which a test for the solution uniqueness similar to the Perron-Rosenblatt theorem is applied to specific differential equations with hysteresis nonlinearities of the form of the Prandtl model of a viscoelastic fiber and a time-varying nonlinearity.

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.