Theory Of Monotone Operators Research Articles

Global Attracting Behavior of Non-autonomous Stage-structured Population Dynamical System with Diffusion

A non-autonomous single species dispersal model is considered, in which individual member of the population has a life history that goes through two stages, immature and mature. By applying the theory of monotone and concave operators to functional differential equations, we establish conditions under which the system admits a positive periodic solution which attracts all other positive solutions.

On a class of non-smooth oscillators

The aim of this paper is to provide a unifying framework for several models of non-smooth oscillators with hysteresis which have attracted considerable interest in the engineering literature over the past three decades. By using the theory of maximal monotone operators in conjunction with a relatively recent result of Haraux and Ôtani (1990, Journal of the Mathematical Society of Japan , 42 : 277-294) we are able to characterize the attractors of these systems; in particular, it is shown that the study of their bounded orbits lies at the heart of understanding their possible dynamics. The interplay between material nonlinearity and viscous damping is analysed for a special case of the original model and some new features of this problem are revealed.

An Error Estimate for the Finite Element Solution of an Elliptic Problem with a Nonlinear Newton Boundary Condition in Nonpolygonal Domains

The article is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional domain with a curved boundary. The existence and uniqueness of the weak solution of the continuous problem is a consequence of the monotone operator theory. The problem is discretized with the use of the finite element method. The main attention is paid to the effect of the approximation of the curved boundary by a piecewise linear boundary and of the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlámal's ideal triangulation and interpolation, the error estimate for the solution of the discrete problem is derived.

The initial boundary value problem for thin-plate flow of incompressible non-Newtonian viscous fluids

In this paper, the particular model that describes the thin-plate flow of incompressible non-Newtonian viscous fluids is investigated. This model, which arises from polymer processing, concerns the initial boundary problem of nonstationary thin-plate flow of non-Newtonian viscous incompressible fluids. By using monotone operator theory and Schauder's fixed-point theorem, an existence and uniqueness theorem for a generalized solution is obtained, and the regularity for this solution is tested.

The composition of projections onto closed convex sets in Hilbert space is asymptotically regular

The composition of finitely many projections onto closed convex sets in Hilbert space arises naturally in the area of projection algorithms. We show that this composition is asymptotically regular, thus proving the so-called “zero displacement conjecture” of Bauschke, Borwein and Lewis. The proof relies on a rich mix of results from monotone operator theory, fixed point theory, and convex analysis.

Cycle Stability for Hopf Bifurcation Generated by Sublinear Terms

The paper is concerned with the study of small stable cycles of autonomous quasilinear systems depending on a parameter. Sufficient conditions are presented for the existence of such cycles for control theory equations with scalar nonlinearities. The principal distinction of the case considered from usual results on Hopf bifurcations is that the linear part of the problem is degenerate for all the parameter values (not only at a bifurcation point). Small sublinear nonlinearities play the main role in our results. The proofs are based on the theory of monotone operators.

On the Finite Element Analysis of Problems with Nonlinear Newton Boundary Conditions in Nonpolygonal Domains

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional nonpolygonal domain with a curved boundary. The existence and uniqueness of the solution of the continuous problem is a consequence of the monotone operator theory. The main attention is paid to the effect of the basic finite element variational crimes: approximation of the curved boundary by a polygonal one and the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlamal's ideal triangulation and interpolation, the convergence of the method is analyzed.

Nonlinear elliptic problems of Neumann-type

In this paper we study a nonlinear elliptic differential equation driven by thep-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem of the range of the sum of monotone operators, we prove the existence of a (strong) solution.

Analysis of a quasistatic displacement-traction problem for viscoelastic materials

We consider an initial and boundary value problem which describes the evolution of a viscoelastic body submitted to body forces and surface tractions. The viscoelastic constitutive law is assumed to be nonlinear and the process is quasistatic. We prove the existence and the uniqueness of the solution using arguments of monotone operators theory and a version of Cauchy-Lypchitz theorem. We establish the continuous dependence of the solution on the elasticity operator. Finally, we study the behavior of the solution when the viscosity operator converges to zero.

The existence of periodic solution and behavior of the generalized solution when t→+∞ of boundary problem of non-Newtonian fluids

In this paper, we discuss the existence of periodic solution and the behavior of the generalized solution when t→+∞ of the first boundary problem of incompressible non-Newtonian fluids. This problem arises from polymer processing which concerns with the first initial-boundary value problem of non-stationary thin-plate flow of the non-Newtonian viscous incompressible fluids. By using the monotone operator theory and the Schauder's fixed point theorem, we have obtained the existence theorem of periodic solution and the characteristics of generalized solution when t→+∞.

Nonlinear boundary value problems with maximal monotone terms

In this paper we consider a second order nonlinear differential equation with maximal monotone terms and nonlinear, possibly multivalued boundary conditions. Using the theory of maximal monotone operators and the Leray—Schauder alternative theorem we establish the existence of solutions. Our formulation is very general and includes as special cases the Dirichlet, Neumann and periodic problems.

Nonlinear Elliptic Problems of Neumann-Type

In this paper we study a nonlinear elliptic differential equation driven by the p-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem on the range of the sum of monotone operators, we prove the existence of a (strong) solution.

Parameter identification in nonlocal nonlinear evolution equations

We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed.

Nonlinear boundary value problems

In this paper we consider two quasilinear boundary value problems. The first is vector valued and has periodic boundary conditions. The second is scalar valued with nonlinear boundary conditions determined by multivalued maximal monotone maps. Using the theory of maximal monotone operators for reflexive Banach spaces and the Leray-Schauder principle we establish the existence of solutions for both problems.

Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems

In this paper we consider a general nonlinear boundary value problem for second-order differential inclusions. We prove two existence theorems, one for the ``convex'' problem and the other for the ``nonconvex'' problem. Then we show that the solution set of the latter is dense in the C 1 (T,R N ) -norm to the solution set of the former (relaxation theorem). Subsequently for a Dirichlet boundary value problem we prove the existence of extremal solutions and we show that they are dense in the solutions of the convexified problem for the C 1 (T,R N ) -norm . Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray—Schauder principle.

Error estimates for the finite element solution of elliptic problems with nonlinear newton boundary conditions

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous pioblem is a consequence of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numeriral integration for the evaluation of boundary integrals. The error estimates for the solution of the discrete finite element problem are derived

Farthest points and monotone operators

We apply the theory of monotone operators to study farthest points in closed bounded subsets of real Banach spaces. This new approach reveals the intimate connection between the farthest point mapping and the subdifferential of the farthest distance function. Moreover, we prove that a typical exception set in the Baire category sense is pathwise connected. Stronger results are obtained in Hilbert spaces.

Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as $x \to \pm\infty$, to periodic stationary states with different wave-numbers $\eta_\pm$. These solutions are stable with respect to small perturbations, and approach as $t \to +\infty$ a universal diffusive profile depending only on the values of $\eta_\pm$. This extends a previous result of Bricmont and Kupiainen by removing the assumption that $\eta_\pm$ should be close to zero. The existence of the diffusive profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed.

Weak solutions of second-order quasilinear parabolic equations with double non-linearity

The first boundary-value problem for the equation is considered in a bounded subdomain of . The function is assumed to be continuous and satisfy the following growth conditions: The other coefficients satisfy the standard conditions of the theory of monotone operators. An existence theorem for a global weak solution is proved.