A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=Nu and the nonlinear time dependent problem u^′=(L+N)u. Under appropriate assumptions on L and N we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.     

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1,?. . .?, pn)-dominated multilinear operators and dominated (p1,?. . .?, pn; σ)-continuous multilinear operators.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

On fixed point theorems of mixed monotone operators and applications

In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators.     

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator

The paper deals with second order nonlinear evolution inclusions and their applications. We study evolution inclusions involving a Volterra-type integral operator, which are considered within the framework of an evolution triple of spaces. First, we deliver a result on the unique solvability of the Cauchy problem for the inclusion by combining a surjectivity result for multivalued pseudomonotone operators and the Banach contraction principle. Next, we provide a theorem on the continuous dependence of the solution to the inclusion with respect to the operators involved in the problem. Finally, we consider a dynamic frictional contact problem of viscoelasticity for materials with long memory and indicate how the result on evolution inclusion is applicable to the model of the contact problem.     

Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators

In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

Perturbation semi-linéaire des résolvantes et des semi-groupes

Résumé La perturbation semi-linéaire des résolvantes et des semi-groupes linéaires, nous donne des résolvantes et des semi-groupes non linéaires. Nous étudions alors les propriétés de ces opérateurs non linéaires et en particulier les fonctions surmédianes et excessives associées.

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We are concerned with nonlinear resolvents and semi-groups. They are obtained by perturbing linear ones. Properties of these nonlinear operators are investigated, particularly supermedian and excessive functions.

Ce travail est soutenu par la fondation nationale pour la recherche scientifique. Projet MA4-89-FST.     

A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems

We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators.     

Semilinear evolution equations with nonlinear constraints and applications

This paper deals with a general class of evolution problems for semilinear equations coupled with nonlinear constraints. Those constraints may contain compositions of nonlinear operators and unbounded linear operators, and hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. Accordingly, a family of equivalent norms is introduced to discuss 'quasidissipativity' in a local sense of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators on bounded sets. It is a feature of our treatment that the resultant solution operators are obtained as nonlinear semigroups on the whole space which are not 'quasicontractive' but locally equi-Lipschitz continuous.     

On mappings covering at a nonlinear rate and their perturbation stability

The present paper contains a study of covering (alias, openness) properties at a nonlinear rate for set-valued mappings between metric spaces. Such study is focussed on the stability of these properties in the presence of perturbations. A crucial result valid for linear openness, known as Milyutin’s theorem, is extended to set-valued mappings covering at a nonlinear rate under possibly non-Lipschitz perturbations. Consequently, a Lyusternik type theorem is derived from such extension and a general penalization principle for constrained optimization problems, which exploits nonlinear covering properties, is presented.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Hierarchies of Huygens' Operators and Hadamard's Conjecture

We develop a new unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard. The underlying method is essentially algebraic and based on a certain nonlinear extension of similarity (gauge) transformations in the ring of analytic differential operators.The paper provides a systematic and self-consistent review of classical and recent results on Huygens' principle in Minkowski spaces. Most of these results are carried over to more general pseudo-Riemannian spaces with the metric of a plane gravitational wave.A particular attention is given to various connections of Huygens' principle with integrable systems and the soliton theory. We discuss the link to nonlinear KdV-type evolution equations, Darboux–Bäcklund transformations and the bispectral problem in the sense of Duistermaat, Grünbaum and Wilson.     

Essential spectra of spectral operators

The various essential spectra of a linear operator have been surveyed byB. Gramsch andD. Lay [4]. In this paper we characterize the essential spectra and the related quantities nullity, defect, ascent and descent of bounded spectral operators. It is shown that a number of these spectra coincide in the case of a spectral or a scalar type operator. Some results known for normal operators in Hilbert space are extended to spectral operators in Banach space.     

Approximation of regularized evolution operators

This paper is devoted to the approximation of a regularized evolution operator by a sequence of regularized discrete parameter evolution operators, and the results obtained here are applied to some degenerate partial differential equation.Received: 4 December 2001     

Lur’e feedback systems with both unbounded control and observation: Well-posedness and stability using nonlinear semigroups

This paper is a complement of information to Grabowski and Callier (2006) [1]. A SISO Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static incremental sector type controller is considered. Well-posedness and a criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a novel nonlinear semigroup approach. A quadratic form Lyapunov functional is considered via a Lur’e type linear operator inequality. A sufficient strict circle criterion of solvability of the latter is found, using the solution of an operator Riccati equation by a novel self contained exposition, via reciprocal systems with bounded generating operators as recently studied and used by R.F. Curtain. The noncoercive case is finally considered using, in a novel way, LaSalle’s invariance principle.     

A contraction proximal point algorithm with two monotone operators

It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature.     

On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators

In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

Energy-critical Hartree equation with harmonic potential for radial data

In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions (n≥5) and show the global well-posedness and scattering theory in the space Σ=H¹∩FH¹. We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao’s approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory.