Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.     

Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.     

Center manifolds theorem for parameterized delay differential equations with applications to zero singularities

The goal of this paper is to develop a center manifold theory for delay differential equations with parameters. As applications, we use the center manifold theorem to establish fold and Bogdanov-Takens bifurcations. In particular, we obtain the versal unfoldings of delayed predator-prey systems with predator harvesting at the Bogdanov-Takens singularity.     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Nonautonomous bifurcation patterns for one-dimensional differential equations

Although, bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this article, two different approaches are discussed which are based on special notions of attractivity and repulsivity. Generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation are obtained.     

Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces

In this paper we deal with the existence of impulsive mild solutions for semilinear differential inclusions with nonlocal conditions, where the linear part generates an evolution system and the nonlinearity satisfies the lower Scorza-Dragoni property. Our theorems extend the existence propositions proved by Fan in 2010. An example is presented.     

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

The existence of multiple positive solutions of systems of singular Hammerstein integral equations is studied, where the nonlinearities involved are allowed to have singularities in their second variables and satisfy weaker conditions involving the first eigenvalues of the corresponding linear Hammerstein integral operators. Such systems contain some mathematical models arising in science and engineering. Applications are given to the existence of multiple positive radial solutions of systems of semilinear singular elliptic equations in annuli on which, to the best of our knowledge, there has been little study.     

On the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearities

In this paper, we study the existence of positive solutions for the semilinear elliptic problem

     

Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations

In this paper, we study a linear Sturm-Liouville impulsive problem and a nonlinear Sturm-Liouville impulsive problem. By applying a new approach via critical point theory and variational methods, the existence results of positive solutions are obtained.     

Attractors for differential equations with unbounded delays

We prove the existence of attractors for some types of differential problems containing infinite delays. Applications and examples are provided to illustrate the theory, which is valid for both cases with and without explicit dependence of time, and with or without uniqueness of solutions, as well.     

Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations

This paper uses critical point theory and variational methods to investigate the subharmonic solutions with prescribed minimal period for a class of second-order impulsive differential equations. The conditions for the existence of subharmonic solutions are established. Our results rectify some known results in the literature and will allow us, in the future, to deal with the subharmonic solutions for more extensive impulsive problems.     

On semilinear elliptic equations with indefinite nonlinearities

This paper concerns semilinear elliptic equations whose nonlinear term has the formW(x)f(u) whereW changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part ofW is contained in a condition which is shown to be necessary for homogeneousf. More general existence questions are also discussed.Supported in part by NSF grant DMS9003149.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

Modulating pulse solutions for quasilinear wave equations

This paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave train; they are also referred to as ‘moving breathers’ since they are time periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with two stable, two unstable and infinitely many neutral directions. Using a partial normal form and a generalisation of local invariant-manifold theory to the quasilinear setting we prove the existence of modulating pulses on arbitrarily large, but finite domains in space and time.     

Large-time behavior of solutions to the equations of a viscous polytropic ideal gas

We study Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field. After Routh's reduction, such systems become nonautonomous with Lagrangian quadratic in time. We prove the existence of solutions tending to an orbit of the symmetry group as t± . As an example, we study doubly asymptotic solutions for the Kirchhoff problem of a heavy rigid body in an infinite volume of incompressible ideal fluid performing a potential motion.Supported by GNFM and by MURST (40%: «Equazioni di evoluzione...»).Supported by Russian Foundation of Basic Research and by INTAS.     

Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions

The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.     

Multiple periodic solutions of ordinary differential equations with double resonance

In this paper we study by Morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin.     

Autonomous and non-autonomous attractors for differential equations with delays

The asymptotic behaviour of some types of retarded differential equations, with both variable and distributed delays, is analysed. In fact, the existence of global attractors is established for different situations: with and without uniqueness, and for both autonomous and non-autonomous cases, using the classical notion of attractor and the recently new concept of pullback one, respectively.