Nonlinear Operator Theory Research Articles

Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials

We draw a fundamental compendium of the most valuable results of the theory of summing linear operators and detail those that are not shared by known multilinear and polynomial extensions of absolutely summing linear operators. The lack of such results in the theory of non-linear summing operators justifies the introduction of a class of polynomials and multilinear operators that satisfies at once all related non-linear results. Surprisingly enough, this class, defined by means of a summing inequality, happens to be the well known ideal of composition with a summing operator.

Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances

This paper is concerned with the boundary stabilisation of a Timoshenko beam with a tip payload under the boundary external disturbances. Nonlinear feedback control laws are designed to reduce the effects of the external disturbances. Since the controlled system is nonlinear, the well posedness of the nonlinear closed-loop system is investigated using the theory of nonlinear maximal monotone operators and the variational principle. In addition, we show the exponential stability of the nonlinear controlled system using the Lyapunov direct method. In the end, the numerical simulations are displayed to illustrate the effectiveness of the proposed boundary controls.

An algorithm for finding common solutions of various problems in nonlinear operator theory

Abstract In this paper, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets. MSC:47H06, 47H09, 47J05, 47J25.

Geometry of Banach Spaces, Operator Theory, and Their Applications

In recent years, with developments of computer, high technique, and life science, more and more requirements were proposed to functional analysis. There were a large number of research articles published in the relative area and many major discoveries have been obtained. At the present time the ideas, terminology, and methods of functional analysis have penetrated deeply, not only into natural sciences, but also into applied disciplines. Many real world problems were analyzed and solved by using the ideas and results of function analysis. To fit the requirements, we edit this special issue. The special issue represented the recent development in geometric structure of Banach spaces, Bergman spaces, and Minkowski spaces, random normed modules on function space, linear and nonlinear operator theory, and the applications of modern analysis in related areas of mathematics, as well as other disciplines, such as economics, finance and risk management, dynamic systems, the natural and life sciences, medicine, physics, and other real world problems. The special issue consisted of twenty research articles contributed by participants of the 4th International Conference on Modern Analysis and Its Applications which was held at the Tianjin University, China, from August 1 to 4, 2013. With over 240 participants fromdifferent countries, the conference proved highly successful in bringing together experts and researchers and provided an opportunity to evaluate and disseminate new ideas and methods in modern analysis. Each paper has been peer-reviewed and the editors thank the referees for their time and efforts to help ensure a high standard for this special issue. We hope that readers interested in function spaces and operator theory will find in this special issue not only new information about development of functional analysis, but also important questions be resolved and directions for further study. At same time, we hope that in the near future, we can see new results to be published based on this special issue.

Method of equivalent differences for the neutron transport equation

For boundary value problems in neutron transport theory, numerical methods are described that are based on the reduction of these problems to systems of nonlinear algebraic equations. Techniques based on the theory of nonlinear positive operators in partially ordered spaces are used to establish major results concerning the existence and uniqueness of solutions and solution-finding methods.

Well-posed problem of nonlinear singular distributed parameter systems and nonlinear GE-semigroup

According to the well-posed problem of nonlinear singular distributed parameter systems, first of all, the nonlinear GE-semigroup induced by a continuous (possibly nonlinear) operator is introduced in Banach space, which is a generalization of GE-semigroup (i.e., generalized operator semigroup), and the properties of nonlinear GE-semigroup are discussed; and then the existence, uniqueness and constructive expression for the strong solution of nonlinear singular distributed parameter system are discussed by nonlinear GE-semigroup; at last, the exponential stability of nonlinear singular distributed parameter system is studied by using nonlinear GE-semigroup, functional analysis and operator theory in Banach space.

Classification of ξ(s)-Quadratic Stochastic Operators on 2D simplex

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some QSO has been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for the quadratic stochastic operators. To study this problem it was investigated several classes of such QSO. In this paper we study ξ(s)-QSO class of operators. We study such kind of operators on 2D simplex. We first classify these ξ(s)-QSO into 20 classes. Further, we investigate the dynamics of one class of such operators.

Onξ(s)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. We studyξ(s)-QSO defined on 2D simplex. We first classifyξ(s)-QSO into 20 nonconjugate classes. Further, we investigate the dynamics of three classes of such operators.

Lagrange and practical stability criteria for dynamical systems with nonlinear perturbations

Lagrange and practical stability criteria for dynamical systems with nonlinear perturbationsIn the paper two classes of nonlinear dynamical system with perturbations are considered. The sufficient conditions for robust Lagrange and practical stability are proven with theorems, applying the theory of nonlinear operators of the functional analysis. The presented criteria give also the bounds of the analyzed dynamical processes. Three examples comparing the numerical computer solutions and the analytical investigation of the stability of the systems are given. The method can be applied to analytical and computer modeling of nonlinear dynamical systems, synthesis of computer control and optimization.

Regularity theory for parabolic nonlinear integral operators

This article is dedicated to the regularity theory for solutions to a class of nonlinear integral variational problems. Those problems are involved in nonlocal image and signal processing.

Existence of heteroclinic orbits in simplified longitudinal flight dynamics

The existence of an infinite chain of heteroclinic orbits connecting saddle-nodes is proved, in the case of a simplified system of differential equations governing the longitudinal flight of an aircraft, using fixed point theory for nonlinear Volterra operators with convolution kernels. Numerical simulations are given and the importance of the obtained results is discussed from the point of view of the appearance of oscillatory behavior in longitudinal flight.

Weak convergence of a projection algorithm for variational inequalities in a Banach space

Let C be a nonempty, closed convex subset of a Banach space E. In this paper, motivated by Alber [Ya.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lecture Notes Pure Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15–50], we introduce the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly-monotone operator A in a Banach space: x 1 = x ∈ C and x n + 1 = Π C J −1 ( J x n − λ n A x n ) for every n = 1 , 2 , … , where Π C is the generalized projection from E onto C, J is the duality mapping from E into E ∗ and { λ n } is a sequence of positive real numbers. Then we show a weak convergence theorem (Theorem 3.1). Finally, using this result, we consider the convex minimization problem, the complementarity problem, and the problem of finding a point u ∈ E satisfying 0 = A u .

A fixed point theorem for a class of differentiable stable operators in banach spaces

We study Frechet differentiable stable operators in real Banach spaces. We present the theory of linear and nonlinear stable operators in a systematic way and prove solvability theorems for operator equations with differentiable expanding operators. In addition, some relations to the theory of monotone operators in Hilbert spaces are discussed. Using the obtained solvability results, we formulate the corresponding fixed point theorem for a class of nonlinear expanding operators.

Extremal Solutions and Strong Relaxation for Second Order Multivalued Boundary Value Problems

In this paper we study semilinear second order differential inclusions involving a multivalued maximal monotone operator. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain “extremal” solutions and we prove a strong relaxation theorem.

Quasivariational evolution inequalities for a class of reaction–diffusion systems

This paper is concerned with a class of doubly nonlinear quasivariational evolution inequalities for a class of reaction–diffusion models with time-derivative constraint. We prove an existence result by using the fixed point argument with the monotonicity technique in the nonlinear operator theory.

Algorithms for Countable State Markov Decision Models with an Absorbing Set

We consider a countable state Markov decision process with bounded reward function and an absorbing set. At first we generalize known properties and derive new properties of the critical discount factor, which is roughly defined as the maximal discount factor under which for V, the maximal expected infinite-stage discounted reward, there is guaranteed existence, boundedness, and computability by the successive approximation method. The emphasis of the paper is on algorithms for computing V exactly (recursion in state space and policy iteration) or approximately (value iteration combined with an extrapolation and finite state approximation). Our extrapolation method is motivated by and based on the Perron--Frobenius theory for nonlinear operators. As a by-product we obtain an efficient algorithm for determining the distribution of the entrance time of a Markov chain into an absorbing set. Further results concern asymptotically $\epsilon$-optimal policies and a new turnpike theorem. Some of the results need tightness of the transition law, which turns out to be equivalent to compactness of a nonlinear operator, which is crucial for our study.

Existence of solutions for strongly nonlinear elliptic differential inclusions with unilateral constraints

We study a nonlinear elliptic differential inclusion involving monotone and nonmonotone terms. Our formulation incorporates variational-hemivariational inequalities. The problem is strongly nonlinear in the sense that on the nonmonotone multivalued term we do not impose any growth condition and instead we use a generalized sign condition. Our approach is based on methods and techniques from multivalued analysis and from the theory of nonlinear operators of monotone type.

Mini-Workshop: Nonlinear Spectral and Eigenvalue Theory

What is the state-of-the-art of abstract spectral and eigenvalue theory for nonlinear operators, and how may this theory be applied to nonlinear equations involving the p-Laplace operator? These two questions have provided the main focus of the Mini-Workshop. Accordingly, the main topics covered by the talks on this Mini-Workshop have been Of course, these three topics are not mutually independent, but there are various interconnections between them which are of particular interest. For example, sets of eigenvalues (point spectra) may be regarded, as in the linear case, as an important part of the spectrum; conversely, nonlinear eigenvalue theory is one of the historical roots of nonlinear spectral theory. Moreover, the p-Laplace operator is one of the most interesting homogeneous (though nonlinear) operators which may not only serve as a “model operator” in nonlinear eigenvalue problems, but also occurs quite frequently in various applications to physics, mechanics, and elasticity. The aim and scope of the Mini-Workshop was to bring together experts on nonlinear spectral analysis and operator theory, on the one hand, and more application-oriented specialists in eigenvalue problems for nonlinear partial differential equations (like the p-Laplace equation), on the other. As a result, 15 leading experts in the field from 10 different countries discussed recent progress and open problems in the theory, methods, and applications of spectra and eigenvalues of nonlinear operators.

ON NONLINEAR ELLIPTIC HEMIVARIATIONAL INEQUALITIES OF SECOND ORDER

In this paper, a nonlinear hemivariational inequality of second order with a forcing term of subcritical growth is studied. Using techniques from multivalued analysis and the theory of nonlinear operators of monotone type, an existence theorem for the Dirichlet boundary value problem is proved.

TOWARD A THEORY OF CHAOS

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.