# Nonlinear Analysis

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This paper proves that ordinary differential equation systems that are contractive with respect to L-p norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well.

Addresses the disturbance rejection problem in control systems with saturating actuators by using the method of stochastic linearization. The disturbance is assumed to be white noise and the performance measure considered is the output variance. Both analysis and synthesis problems are investigated. In the analysis part, a method is developed to predict disturbance rejection quality of a given control system. In the synthesis part, a systematic controller design method that achieves suboptimal disturbance rejection is presented

The notion of an adjoint operator for a nonlinear mapping has few interpretations in the literature. In this paper a nonlinear Hilbert adjoint operator is proposed. It is shown to unite several existing concepts and provides an essential tool for singular value analysis of nonlinear Hankel operators

A controllability theorem obtained by K. Narukawa and T. Suzuki (1986) for a linear wave equation is generalized to a nonlinear Volterra type. In addition, an inequality condition between the growth rate of a nonlinear function and the regularity of an admissible control function is shown

Deals with the stability analysis of linear functional differential equation (FDEs) of retarded type, with both discrete and distributed delay. The main tool used for solving such problem is based on a comparison principle: the behaviour of the solutions of the initial system is compared, through some regular vector Lyapunov functions (RVLF), to the behaviour of the solution of some simpler FDE

Diagonal type Lyapunov functions are ubiquitous in interconnected systems analysis and design, either by the so called vector or scalar Lyapunov function approaches. Clear justification for this phenomenon has not been given in the extensive literature on the field. This paper explains the reasons for this ubiquity, actually justifying the necessity of diagonal stability in this context

We develop an optimality-based framework to address the problem of nonlinear-nonquadratic control for disturbance rejection of nonlinear systems with bounded exogenous disturbances. Specifically, using nonlinear dissipation theory with appropriate storage functions and supply rates we transform the nonlinear disturbance rejection problem into a optimal control problem by modifying a nonlinear-nonquadratic cost functional to account for the exogenous disturbances. As a consequence, the resulting solution to the modified optimal control problem guarantees disturbance reflection for nonlinear systems with bounded input disturbances. Furthermore, it is shown that the Lyapunov function guaranteeing closed-loop stability is a solution to the steady-state Hamilton-Jacobi-Bellman equation for the controlled system

In this paper we develop an invariance principle for dynamical systems possessing left-continuous flows. As a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems; namely, state-dependent impulsive dynamical systems. These results provide less conservative stability conditions for impulsive systems as compared to classical results in literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems

Geometric properties of V-functions are studied and the application of such functions to nominal equations of a dynamical system is considered. A theorem is proved that provides a framework for quantitative design and investigation of motions. The results allow direct control applications. The application is made to the perturbation equation where classical Liapunov theorems follow as a special case whereby the investigation of locally unstable but Â¿-stable systems is made possible. An example is presented.

We derive optimal MISE kernel radial basis networks in regression estimation problem

In part I of this paper [Nonlinear Anal., Theory Methods Appl. 64, No. 2 (A), 242–252 (2006; Zbl 1085.68150)], focusing on a further development of the well-known algorithms for deriving theorems of the method of Lyapunov vector functions, we suggested the reduction method as a logical technique for formulating hypotheses. This part illustrates its implementation in qualitative analysis of the various properties of dynamical systems represented as systems of motions, differential equations, and automata models with different depths of delays.

In this note, we deal with an iterative scheme of Halpern type for a semigroup of nonexpansive mappings on a compact convex subset of a strictly convex and smooth Banach space with respect to an asymptotically left invariant sequence of means defined on an appropriate space of bounded real valued functions of the semigroup. We improve the corresponding result of [A.T. Lau, H. Miyake, W. Takahashi, Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 1211–1225].

In this paper, we study the blowup of the $N$-dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant and in the sense which $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition% $H_{0}=\int_{0}^{R}rV_{0}dr>0$ blow up on or before the finite time $T=R^{3}/(2H_{0})$ for pressureless fluids or $\gamma>1.$ The main contribution of this article provides the blowup results of the Euler $(\delta=0)$ or Euler-Poisson $(\delta=1)$ equations with repulsive forces, and with pressure $(\gamma>1)$, as the previous blowup papers (\cite{MUK} \cite{MP}, \cite{P} and \cite{CT}) cannot handle the systems with the pressure term, for $C^{1}$ solutions. Comment: Accepted by Nonlinear Analysis Series A: Theory, Methods & Applications Key Words: Euler Equations, Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, $C^{1}$ Solutions, No-Slip Condition

We discuss here a method of proof of the existential part of Hilbert's 16th problem for quadratic vector fields initiated in 1991 [7]. The program consists in proving the finite cyclicity of 121 graphics appearing among quadratic systems. We briefly summarize some of the methods introduced for that purpose and describe some new developments obtained with Świrszcz and Żoldek [20] for graphics surrounding a center.

We consider the system of quasilinear equations describing 1D radiative and reactive viscous flows with arbitrarily large data. The large-time behavior of solutions in the case of first-order kinetics has been recently studied. In this paper, we present new results concerning the case of higher-order kinetics for fairly general kinetics law (unbounded with respect to density and temperature, and dealing with the ignition phenomenon), including L2 and H1-stabilization rate bounds of power type. The power exponents of bounds improve essentially those known for related problems and are partially proved to be sharp. An effect of “faster equaling” of values in space for the concentration of unburned gas is also found. Finally we show how our results are modified in the case of reaction without diffusion.

We consider here the 1D Green-Naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves. We show that the solution of the Green-Naghdi equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition.

We consider a two-dimensional Navier–Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup.Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type bound on the time-averaged energy dissipation rate, independent of viscosity at large Reynolds numbers. Then we establish a version of the Lieb–Thirring inequality for a class of functions defined on the considered non-rectangular flow domain.This research is motivated by a problem from lubrication theory.

Global well-posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2D Boussinesq system in a periodic channel with non-homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved.

We prove global well-posedness of the 2D critical dissipative quasi-geostrophic equation for small initial data in the Triebel-Lizorkin spaces F-p,q(s), s > 2/p, q is an element of (1, infinity). In particular, our result implies the global existence for small initial data in the Sobolev space H-s,H-p, s > 2/p, p is an element of (1, infinity) since F-p,2(s) = H-s,H-p. The key ingredients of our proof are the pointwise exponential decay estimate of the semigroup e(-t kappa(-Delta)alpha) and maximum principle. (c) 2006 Elsevier Ltd. All rights reserved.

In this paper we prove the local-in-time well-posedness for the 2D non-dissipative quasi-geostrophic equation, and study the blow-up criterion in the critical Besov spaces. These results improve the previous one by Constantin et al. [P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994) 1495–1533].

We study the local exponential stabilizability with internally distributed feedback controllers for the incompressible 2D-Navier–Stokes equations with Navier slip boundary conditions. These controllers are localized in a subdomain and take values in a finite-dimensional space.

By using the energy equation method, the existence of the uniform attractor for the 2D non-autonomous Navier–Stokes equations in some unbounded domains with usual settings is obtained without the restriction of the forcing term belonging to some weighted Sobolev space. And for the quasi-periodic forcing term, the estimation of the Hausdorff dimension of such attractors is also obtained.

The 2D quasi-geostrophic equation partial derivative(t)theta + u (.) del theta + kappa(-Delta)(alpha)theta = 0, u = R(theta) is a two-dimensional model of the 3D hydrodynamics equations. When alpha <= 1/2, the issue of existence and uniqueness concerning this equation becomes difficult. It is shown here that this equation with either kappa = 0 or kappa > 0 and 0 <= alpha <= 1/2 has a unique local in time solution corresponding to any initial datum in the space C-r boolean AND L-q for r > 1 and q > 1. (c) 2005 Elsevier Ltd. All rights reserved.

A nonlinear problem for a thermo-viscoelastic Mindlin–Timoshenko plate with hereditary heat conduction is considered here. We prove the existence of a compact global attractor whose fractal dimension is finite. The main aim of the work is to show the upper semicontinuity of the attractor as the relaxation kernels fade in a suitable sense.

In this paper, we study the existence, multiplicity and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, it generalizes and includes some known results.

We study sharp time asymptotics of solutions around the final states of nonlinear Schrödinger equations with quadratic nonlinearities in three space dimensions. In particular, we consider the large time asymptotic estimates of solutions from below.

In a convex domain $\O\subset\R^3$, we consider the minimization of a 3D-Ginzburg-Landau type energy $E_\v(u)=1/2\int_\O|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ with a discontinuous pinning term $a$ among $H^1(\O,\C)$-maps subject to a Dirichlet boundary condition $g\in H^{1/2}(\p\O,\S^1)$. The pinning term $a:\R^3\to\R^*_+$ takes a constant value $b\in(0,1)$ in $\o$, an inner strictly convex subdomain of $\O$, and 1 outside $\o$. We prove energy estimates with various error terms depending on assumptions on $\O,\o$ and $g$. In some special cases, we identify the vorticity defects via the concentration of the energy. Under hypotheses on the singularities of $g$ (the singularities are polarized and quantified by their degrees which are $\pm 1$), vorticity defects are geodesics (computed w.r.t. a geodesic metric $d_{a^2}$ depending only on $a$) joining two paired singularities of $g$ $p_i & n_{\sigma(i)}$ where $\sigma$ is a minimal connection (computed w.r.t. a metric $d_{a^2}$) of the singularities of $g$ and $p_1,...p_k$ are the positive (resp. $n_1,...,n_k$ the negative) singularities.

We prove regularity criteria for the 3D generalized MHD equations. These criteria impose assumptions on the vorticity only. In addition, we also prove a result of global existence for smooth solution under some special conditions.

In this paper, we consider the 3D Boussinesq equations with the incompressibility condition. We obtain a regularity condition for the three-dimensional Boussinesq equations by means of the Littlewood–Paley theory and Bony’s paradifferential calculus.

The growth of a single needle of succinonitrile (SCN) is studied in three dimensional space by using a phase field model. For realistic physical parameters, namely, the large differences in the length scales, i.e., the capillarity length (10^{-8}cm - 10^{-6}cm), the radius of the curvature at the tip of the interface (10^{-3}cm - 10^{-2}cm) and the diffusion length (10^{-3}cm - 10^{-1}cm), resolution of the large differences in length scale necessitates a 500^{3} grid points on the supercomputer. The parameters, initial and boundary conditions used are identical to those of the microgravity experiments of Glicksman et al for SCN. The numerical results for the tip velocity are (i) largely consistent with the Space Shuttle experiments; (ii) compatible with the experimental conclusion that tip velocity does not increase with increased anisotropy; (iii) different for 2D versus 3D by a factor of approximately 1.9; (iv) essentially identical for fully versus rotationally symmetric 3D.

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235–1270], especially, the blow-up of weak solutions in the case of non-negative energy.

We develop Ladyzhenskaya-Prodi-Serrin type spectral regularity criteria for 3D incompressible Navier-Stokes equations in a torus. Concretely, for any $N>0$, let $w_N$ be the sum of all spectral components of the velocity fields whose all three wave numbers are greater than $N$ absolutely. Then, we show that for any $N>0$, the finiteness of the Serrin type norm of $w_N$ implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of $w_N$ and corresponding energy spectrum, in some sense, roughly decays slower than $\kappa^{-2}$

We use Perron method to construct a weak solution to a two-phase free boundary problem with right-hand-side. We thus extend the results of the pioneer work of Caffarelli for the homogeneous case.

In this paper, we study an inhomogeneous variant of the normalized $p$-Laplacian evolution which has been recently treated in \cite{BG1}, \cite{Do}, \cite{MPR} and \cite{Ju}. We show that if the initial datum satisfies the pointwise gradient estimate \eqref{e:main1} a.e., then the unique solution to the Cauchy problem \eqref{main5} satisfies the same gradient estimate a.e. for all later times, see \eqref{e:main} below. A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of equation \eqref{main5} was first obtained in \cite{CGS}. Such estimate generalizes one obtained by L. Modica for the Laplacian, and it has connections to a famous conjecture of De Giorgi.

We consider a mathematical model for the study of the dynamical behavior of suspension bridges. We show that internal resonances, which depend on the bridge structure only, are the origin of torsional instability. We obtain both theoretical and numerical estimates of the thresholds of instability. Our method is based on a finite dimensional projection of the phase space which reduces the stability analysis of the model to the stability of suitable Hill equations. This gives an answer to a long-standing question about the origin of torsional instability in suspension bridges.

In this paper, the stability for the scalar impulsive delay differential equationwhere delay arguments may be bounded or unbounded is investigated. Some new stability theorems are established which improve and extend several known results in the literature.

In this article a class of nonlinear Abel-type integral equations is studied. These equations have at least the trivial solution, but we are interested in a continuous, positive solution. We show that there exists a unique positive solution in an order interval of a cone in a Banach space.

We extend some previous results on the maximum number of isolated periodic solutions of generalized Abel equation and rigid systems. The key hypothesis is a monotonicity assumption on any stability operator (for instance, the divergence) along the solutions of a suitable transversal system. In such a case, at most two isolated periodic solutions exist. Under a simple additional assumption, we also prove a uniqueness result for limit cycles of rigid systems. Our results are easily applicable to special classes of equations, since the hypotheses hold when a suitable convexity property is satisfied.

Consider the vector field $x'= -yG(x, y), y'=xG(x, y),$ where the set of critical points $\{G(x, y) = 0\}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree $n$ and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of $K$ and $n.$ Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for $K\le4$ we recover or improve some results obtained in several previous works.

In this paper we consider an Abelian Gauge Theory in R^4 equipped with the Minkowski metric. This theory leads to a system of equations, the Klein-Gordon-Maxwell equations, which provide models for the interaction between the electromagnetic field and matter. A three dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid. Under suitable assumptions, we prove the existence of vortex-solutions.

In this paper, we establish the existence and uniqueness of the spherically symmetric monopole solutions in SO(5) gauge theory with Higgs scalar fields in the vector representation in six-dimensional Minkowski space–time and obtain sharp asymptotic estimates for the solutions. Our method is based on a dynamical shooting approach that depends on two shooting parameters which provides an effective framework for constructing the generalized monopoles in six-dimensional Minkowski space–time.

The trace identity is generalized to work for the discrete zero-curvature equation associated with the Lie algebra possessing degenerate Killing forms. Then a kind of integrable coupling of the Ablowitz–Ladik (AL) hierarchy is obtained and its Hamiltonian structure is worked out. Moreover, Liouville integrability of the integrable coupling is demonstrated.

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387–409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.

This paper deals with the class of continuous-time singular linear systems with Markovian switching. Sufficient conditions on stochastic stability and robust stochastic stability are developed in the LMI setting. The developed sufficient conditions are used to check if either the nominal or the uncertain systems are regular, impulse-free and stochastically stable or robust stochastically stable.

In this paper we derive the existence and comparison results for second order initial value problems in ordered Banach spaces. The considered problems can be implicit, singular, functional, discontinuous and nonlocal. The main tools are fixed point results in ordered spaces and the theory of locally HL integrable vector-valued functions.

This paper deals with the problem of absolute stability for a class of time-delay singular systems with sector-bounded nonlinearity. Both delay-independent and delay-dependent criteria are presented and formulated in the form of linear matrix inequalities (LMIs). Neither model transformation nor a bounding technique for cross-terms, nor a slack variable method is involved in obtaining the stability criteria. Numerical examples are given to show the effectiveness and improvements over some existing results.

Absolute stability is a basic and important problem in the design of automatic control systems. This paper initiate the study of absolute stability of impulsive control systems with time delay. Several absolute stability criteria are established by constructing suitable Lyapunov functionals. Some examples are also presented to illustrate the main results.

We derive absolute stability results of Popov and circle-criterion types for infinite-dimensional discrete-time systems in an input–output setting. Our results apply to feedback systems in which the linear part is the series interconnection of an l2-stable linear system and an integrator and the nonlinearity satisfies a sector condition which allows for saturation and deadzone effects. The absolute stability theory is then used to prove tracking and disturbance rejection results for integral control schemes in the presence of input and output nonlinearities. Applications of the input–output theory to state-space systems are also provided.

This paper provides finite-dimensional convex conditions to construct homogeneous polynomially parameter-dependent Lur’e functions which ensure the stability of nonlinear systems with state-dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex. The proposed conditions are written as linear matrix inequalities parametrized in terms of the degree g of the parameter-dependent solution and in terms of the relaxation level d of the inequality constraints, based on the algebraic properties of positive matrix polynomials with parameters in the unit simplex. As g and d increase, progressively less conservative solutions are obtained. The results in the paper include as special cases existing conditions for robust stability and for absolute stability analysis. A convex solution suitable for the design of robust nonlinear state feedback stabilizing controllers is also provided. Numerical examples illustrate the efficiency of the proposed conditions.

In this paper we derive existence and comparison results for initial value problems in ordered Banach spaces. The considered problems can be implicit, singular, functional, discontinuous and nonlocal. The main tools are fixed point results in ordered spaces and theory of HL integrable vector-valued functions. Concrete examples are presented and solved.

Top-cited authors
• National Sun Yat-sen University
• University of Santiago de Compostela
• Texas A&M University - Kingsville
• King Abdulaziz University
• Università degli Studi di Palermo