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.     

Radial solutions for a prescribed mean curvature equation with exponential nonlinearity

We establish an existence result for radial solutions for a prescribed mean curvature equation with exponential nonlinearity. Our methods are based on degree theory combined with a time map analysis. We also obtain two nonexistence results for positive solutions for more general f; one of them is not limited to radial solutions.     

On nontrivial solutions of critical polyharmonic elliptic systems

In the present work, we consider elliptic systems involving polyharmonic operators and critical exponents. We discuss the existence and nonexistence of nontrivial solutions to these systems. Our theorems improve and/or extend the ones established by Bartsch and Guo [T. Bartsch, Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differential Equations 220 (2006) 531-543] in both aspects of spectral interaction and regularity of lower order perturbations.     

Nontrivial solutions for critical potential elliptic systems

We consider potential elliptic systems involving p-Laplace operators, critical nonlinearities and lower-order perturbations. Suitable necessary and sufficient conditions for existence of nontrivial solutions are presented. In particular, a number of results on Brezis-Nirenberg type problems are extended in a unified framework.     

Approximate controllability of a class of semilinear degenerate systems with boundary control

This paper concerns a class of control systems governed by semilinear degenerate equations with boundary control in one-dimensional space. The control is proposed on the ‘degenerate’ part of the boundary. The control systems are shown to be approximately controllable by Kakutani's fixed point theorem.     

Uniqueness and decay estimates for a class of parabolic partial differential equations with hysteresis and convection

We deal in detail with the question of existence, uniqueness and asymptotic behaviour of solutions to a parabolic equation with hysteresis and convection. This equation is part of a model system which describes the magnetohydrodynamic (MHD) flow of a conducting fluid between two ferromagnetic plates. The result of this paper complements the content of a previous paper of the first author, where existence of the solution has been proved under fairly general assumptions on the hysteresis operator and the uniqueness was only obtained for a restricted class of hysteresis operators.     

Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations

We investigate various properties of time maps for one-dimensional prescribed mean curvature equations. Using these properties, we obtain some exact multiplicity results of positive solutions and sign-changing solutions. As it turned out, these quasilinear problems show many different phenomena from semilinear problems. Our methods are based on a detailed analysis of time maps.     

Stability analysis of neutral type systems in Hilbert space

The asymtoptic stability properties of neutral type systems are studied mainly in the critical case when the exponential stability is not possible. We consider an operator model of the system in Hilbert space and use recent results on the existence of a Riesz basis of invariant finite-dimensional subspaces in order to verify its dissipativity. The main results concern the conditions of asymptotic non-exponential stability. We show that the property of asymptotic stability is not determinated only by the spectrum of the system but essentially depends on the geometric spectral characteristic of its main neutral term. Moreover, we present an example of two systems of neutral type which have both the same spectrum in the open left-half plane and the main neutral term but one of them is asymptotically stable while the other is unstable.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Approximate controllability of evolution systems with nonlocal conditions

We study the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to show the application of our result.     

A new criterion for global robust stability of interval delayed neural networks

A novel criterion for the global robust stability of Hopfield-type interval neural networks with delay is presented. An example showing the effectiveness of the present criterion is given.     

Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems

In this paper, based on homotopy perturbation method (HPM) and reproducing kernel method (RKM), a new method is presented for solving nonlinear systems of second order boundary value problems (BVPs). HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKM is also an analytical technique, which can solve powerfully linear BVPs. Homotopy perturbation-reproducing kernel method (HP-RKM) combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear BVPs. Three numerical examples are presented to illustrate the strength of the method.     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.     

Solutions to the seepage face model for dual porosity flows with hysteresis

This paper is concerned with theoretical analysis of the initial-boundary value problem for the dual porosity model of Darcian flows with the Preisach hysteresis operator. Due to the presence of the so-called unilateral boundary condition, the problem comes as an evolution variational inequality. We apply the penalty method to prove the global-in-time existence of the variational solution by constructing approximate solutions satisfying suitable energy type estimates.     

Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.