The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory     

Differential equations with state-dependent piecewise constant argument

A new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations—the existence and uniqueness of solutions, the existence of periodic solutions, and the stability of the zero solution—are obtained. Appropriate examples are constructed.     

Entire solutions in periodic lattice dynamical systems

This paper deals with entire solutions of periodic lattice dynamical systems. Unlike homogeneous problems, the periodic equation studied here lacks symmetry between increasing and decreasing pulsating traveling fronts, which affects the construction of entire solutions. In the bistable case, the existence, uniqueness and Liapunov stability of entire solutions are proved by constructing different sub- and supersolutions. In the monostable case, the existence and asymptotic behavior of spatially periodic solutions connecting two steady states are first established. Some new types of entire solutions are then constructed by combining leftward and rightward pulsating traveling fronts with different speeds and a spatially periodic solution. Various qualitative features of the entire solutions are also investigated.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

Iterated Hamiltonian type systems and applications

We discuss, in arbitrary dimension, certain Hamiltonian type systems and prove existence, uniqueness and regularity properties, under the independence condition. We also investigate the critical case, define a class of generalized solutions and prove existence and basic properties. Relevant examples and counterexamples are also indicated. The applications concern representations of implicitly defined manifolds and their perturbations, motivated by differential systems involving unknown geometries.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Existence theorems for solutions to random fuzzy differential equations

We examine a random fuzzy initial value problem with two kinds of fuzzy derivatives. For both cases we establish the results of existence and uniqueness of local solutions to random fuzzy differential equations. The existence of global solutions is also obtained.     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

A system of ODEs with a hysteresis effect

The paper deals with a class of ordinary differential systems which contains a differential inclusion describing input–output relations of hysteresis type. Existence and uniqueness of local and global solutions of the systems under consideration are proved.     

Nonlinear differential problems with first- or second-order differences in Hilbert spaces

In a real Hilbert space, we investigate the existence, uniqueness and asymptotic properties of the generalized solutions of some nonlinear differential systems with first- or second-order differences, subject to extreme conditions and initial data. Some generalizations of Minty’s theorem for the characterization of maximal monotone operators are also presented.     

A composition theorem for weighted pseudo-almost automorphic functions and applications

In this paper, we establish a composition theorem for weighted pseudo-almost automorphic functions under a weaker Lipschitz condition. Our composition theorem generalizes some known results. Moreover, the existence and uniqueness of pseudo-almost automorphic solutions for abstract semilinear evolution equations are studied.     

Almost periodic solutions for a class of non-instantaneous impulsive differential equations

Abstract

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order

This paper studies the existence and uniqueness of solutions of second-order three-point boundary value problems with lower and upper solutions in the reversed order, obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method, and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution.     

Existence results for fractional order semilinear functional differential equations with nondense domain

In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann-Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray-Schauder type.     

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.     

Existence of periodic solutions for a system of delay differential equations

In this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two-neuron network model of Hopfield type with time-delayed connections between the neurons. We first examine the local stability of the trivial solution, propose some sufficient conditions for the uniqueness of equilibria and then apply the Poincaré-Bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions. In addition, a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given. Numerical examples are provided to support the theoretical analysis.     

Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks

We study some properties of piecewise linear differential systems describing gene regulatory networks, where the dynamics are governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing discontinuous differential equations with differential inclusions. The global existence and some other basic properties of the Filippov solutions such as continuous dependence on parameters are studied. We also study the uniqueness and non-uniqueness of the Filippov solutions in singular domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two ways of defining the singular stationary points: one is based on the Filippov theory and the other consists in replacing step functions with steep sigmoids and investigating the smooth systems thus obtained. The results are illustrated by a number of examples.     

Existence of solutions to an initial-boundary value problem of multidimensional radiation hydrodynamics

In this paper, we investigate the well-posedness of an initial-boundary value problem for the equations of multidimensional radiation hydrodynamics which are a hyperbolic-Boltzmann coupled system. We obtain the local existence and uniqueness of smooth solutions to this problem by using the energy method.     

Solvability of a fractional boundary value problem with fractional integral condition

Using Banach contraction principle and Leray-Schauder nonlinear alternative we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems for fractional differential equations with fractional integral condition, involving the Caputo fractional derivative. Some examples are given to illustrate our results.