Class Of Nonlinear Operators Research Articles

Existence of fixed points of large MR-Kannan contractions in Banach Spaces

The purpose of this paper is to introduce the class of large MR-Kannan contractions on Banach space that contains the classes of Kannan, enriched Kannan, large Kannan, MR-Kannan contractions and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. We prove the existence of a unique fixed point for such a class of operators in Banach spaces.

On joint parameterizations of linear and nonlinear functionals in neural networks

The paper proposes a new class of nonlinear operators and a dual learning paradigm where optimization jointly concerns both linear convolutional weights and the parameters of these nonlinear operators. The nonlinear class proposed to perform a rich functional representation is composed by functions called rectified parametric sigmoid units. This class is constructed to benefit from the advantages of both sigmoid and rectified linear unit functions, while rejecting their respective drawbacks. Moreover, the analytic form of this new neural class involves scale, shift and shape parameters to obtain a wide range of activation shapes, including the standard rectified linear unit as a limit case. Parameters of this neural transfer class are considered as learnable for the sake of discovering the complex shapes that can contribute to solving machine learning issues. Performance achieved by the joint learning of convolutional and rectified parametric sigmoid learnable parameters are shown to be outstanding in both shallow and deep learning frameworks. This class opens new prospects with respect to machine learning in the sense that main learnable parameters are attached not only to linear transformations, but also to a wide range of nonlinear operators.

Fixed point results of enriched interpolative Kannan type operators with applications

The purpose of this paper is to introduce the class of enriched interpolative Kannan type operators on Banach space that contains theclasses of enriched Kannan operators, interpolative Kannan type contraction operators and some other classes of nonlinear operators. Some examples are presented to support the concepts introduced herein. A convergence theorem for the Krasnoselskij iteration method to approximate fixed point of the enriched interpolative Kannan type operators is proved. We study well-posedness, Ulam-Hyers stability and periodic point property of operators introduced herein. As an application of the main result, variational inequality problems is solved.

Reachability results for perturbed heat equations

This work studies the reachable space of infinite dimensional control systems which are null controllable in any positive time, the typical example being the heat equation controlled from the boundary or from an arbitrary open set. The focus is on the robustness of the reachable space with respect to linear or nonlinear perturbations of the generator. More precisely, our first main result asserts that this space is invariant under perturbations which are small (in an appropriate sense). A second main result asserts the invariance of the reachable space with respect to perturbations which are compact (again in an appropriate sense), provided that a Hautus type condition is satisfied. Moreover, our methods give precise information on the behavior of the reachable space when the generator is perturbed by a class of nonlinear operators. When applied to the classical heat equation, our results provide detailed information on the reachable space when the generator is perturbed by a small potential or by a class of non local operators, and in particular in one space dimension, we deduce from our analysis that the reachable space for perturbations of the 1-d heat equation is a space of holomorphic functions. We also show how our approach leads to reachability results for a class of semi-linear parabolic equations.

A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and infty -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

Spaces generated by the cone of sublinear operators

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

On extensions of some nonlinear maps in vector lattices

We show that an Urysohn lattice pre-homomorphism defined on a normal sublattice D of a vector lattice E can be extended to the whole space E and the extended operator is an Urysohn lattice homomorphism. We introduce a new class of nonlinear operators which called φ-operators and describe some of their properties. Finally we investigate a structure of positive orthogonally additive operators dominated by an Urysohn lattice homomorphism defined on a vector lattice E and taking value in Dedekind complete vector lattice F.

Determination of the potential form of operators

The author extends ideas of duality [see, for example, B. Noble and M. J. Sewell, J. Inst. Math. Appl. 9 (1972), 123–193; MR0307012] to a class of nonlinear operators on Banach spaces. Let U, V be Banach spaces and a(u,v) a bilinear form on U×V. Let N be a (nonlinear) operator N:U→V. GN(u)h denotes the Gâteaux derivative of N in the direction of h, computed at the point u∈U. Let us assume that a separates points in U×V (as defined by Marshall Stone). If there is v∈V such that a(h,v)=⟨h,Gf(u)⟩ for a functional f:U→R then v is called the gradient of f(u). The operator N is called potential if a suitable functional f satisfying this condition exists. The problem of symmetrizing N involves a suitable choice of the bilinear form a. For example, the operator N(u(t))=[(du/dt)2−g(t)] is not potential with respect to the usual L2 product. The author formulates a number of variational principles and discusses specific examples. This is an interesting article, supplementing the ideas of E. Tonti and of R. W. Atherton and G. M. Homsy [Studies in Appl. Math. 54 (1975), no. 1, 31–60; MR0458271].

Approximating the minimum-norm fixed point of pseudocontractive mappings

We introduce an iterative process which converges strongly to the minimum-norm fixed point of Lipschitzian pseudocontractive mapping. As a consequence, convergence result to the minimum-norm zero of monotone mappings is proved. In addition, applications to convexly constrained linear inverse problems and convex minimization problems are included. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

Algorithms for Solutions of Variational Inequalities in the Set of Common Fixed Points of Finite Family of λ-Strictly Pseudocontractive Mappings

In this article, we introduce an algorithm which has strong convergence for solving the variational inequality problem for η-inverse strongly accretive mappings in the set of common fixed points of finite family of λ-strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

Strong convergence theorems for a common fixed point of a finite family of Bregman weak relativity nonexpansive mappings in reflexive Banach spaces.

We introduce an iterative process for finding an element of a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

The split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators

The main purpose in this paper is to introduce an original technique, namely, a specific way of choosing the indexes of the involved operators, by which a new iterative algorithm is developed for solving the split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators and some strong and weak convergence theorems are established for this class of nonlinear operators in the framework of infinite-dimensional Hilbert spaces. The result is more applicable than those of other authors with related interest.

A new class of non-linear stability preserving operators

We extend Brändén's recent proof of a conjecture of Fisk, Stanley, McNamara and Sagan to a new class of non-linear operators that preserve weak Hurwitz stability and the Laguerre–Pólya class.

Minimum-norm solution of variational inequality and fixed point problem in banach spaces

We introduce an iterative process which converges strongly to a common minimum-norm solution of a variational inequality problem for an-inverse strongly monotone mapping and a fixed point of relatively non-expansive mapping in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of non-linear operators.

Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptoticallyk-Strict Pseudocontractive Mappings

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptoticallyk-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

Fixed-Point Methods for a Certain Class of Operators

We introduce in this paper a new class of nonlinear operators, which contains, among others, the class of operators with semimonotone additive inverse and also the class of nonexpansive mappings. We study this class and discuss some of its properties. Then we present iterative procedures for computing fixed points of operators in this class, which allow for inexact solutions of the subproblems and relative error criteria. We prove weak convergence of the generated sequences in the context of Hilbert spaces. Strong convergence is also discussed.

Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems

Abstract We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for continuous monotone mapping, solution of equilibrium problem and a common fixed point of finite family of asymptotically regular uniformly continuous relatively asymptotically nonexpansive mappings in Banach spaces. Our scheme does not involve computation of C n+1from C n for each n ≥ 1. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. Mathematics Subject Classification (2000): 47H05, 47H09, 47H10, 47J05, 47J25

Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems

We introduce an iterative process for finding an element of common solutions of finite family of equilibrium problems and finite family of variational inequality problems for continuous monotone mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of non-linear operators.

Interpolation of nonlinear operators in weighted L p -spaces

We show that Peetre’s classical interpolation theorem in weighted L p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.