Perturbations In Banach Spaces Research Articles

Regularity for evolution equations with non-autonomous perturbations in Banach spaces

We provide regularity of solutions to a large class of evolution equations on Banach spaces where the generator is composed of a static principal part plus a non-autonomous perturbation. Regularity is examined with respect to the graph norm of the iterations of the principal part. The results are applied to the Schrödinger equation and conditions on a time-dependent scalar potential for the regularity of the solution in higher Sobolev spaces are derived.

A new perturbation theorem for Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is “the generalized Neumann lemma” which is quite different from the method in [12] where “the generalized Banach lemma” was used. By the method of the perturbation analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

Convex Sweeping Processes with Noncompact Perturbations and with Delay in Banach Spaces

We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state.

Existence and asymptotic properties of solutions of nonlinear multivalued differential inclusions with nonlocal conditions

This paper deals with a nonlocal problem governed by the nonlinear differential inclusions with multivalued perturbations in Banach spaces. First using the approach of geometry of Banach space, Hausdorff metric, the measure of noncompactness and fixed point, existence results are obtained. We have removed from previous papers the crucial restriction on the semigroup. Then we establish the asymptotic properties of integral solutions when t→+∞. Finally, for illustration, a partial differential equation is worked out.

Well-posedness by perturbations of variational-hemivariational inequalities with perturbations

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a class of variational-hemivariational inequalities with perturbations in Banach spaces, which includes as a special case the class of mixed variational inequalities. Under very mild conditions, we establish some metric characterizations for the well-posed variational-hemivariational inequality, and show that the well-posedness by perturbations of a variational-hemivariational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Furthermore, in the setting of finite-dimensional spaces we also derive some conditions under which the variational-hemivariational inequality is strongly generalized well-posed-like by perturbations.

Regularization of ill-posed mixed variational inequalities with non-monotone perturbations

In this paper, we study a regularization method for ill-posed mixed variational inequalities with non-monotone perturbations in Banach spaces. The convergence and convergence rates of regularized solutions are established by using a priori and a posteriori regularization parameter choice that is based upon the generalized discrepancy principle.

THE PERSISTENCE OF SNAP-BACK REPELLER UNDER SMALL C1 PERTURBATIONS IN BANACH SPACES

In this paper, we consider the persistence of snap-back repellers under small C1 perturbations in Banach spaces. Let X be a Banach space and f be a C1-map from X into itself. We show that if f has a snap-back repeller, then any small C1 perturbations of f has a snap-back repeller and exhibits chaos in the sense of Devaney. The obtained results further extend the existing relevant results in finite-dimensional Euclidean spaces. As applications, we will discuss the chaotic behavior of two nonlocal population models.

Some new perturbation theorems for generalized inverses of linear operators in Banach spaces

et X , Y be Banach spaces and let T ∈ B ( X , Y ) be a linear operator with closed range R ( T ) . We consider the perturbation problem for the bounded outer inverse, the bounded (2,3)-inverse, and the bounded generalized inverse of T with respect to the projectors. Some new perturbation theorems are given by the generalized Neumman lemma and the concept of stable perturbations in Banach spaces.

Nonlocal differential equations with multivalued perturbations in Banach spaces

We discuss the existence of mild solutions for nonlocal differential inclusions with multivalued perturbations in Banach spaces and establish new existence theorems for related Cauchy problems, which extend some existing results in this area. Using the established results, we investigate a special nonlocal problem. Finally, we also consider a partial functional differential equation.

Ordinary differential equations involving perturbations in Banach spaces

Ordinary differential equations involving perturbations in Banach spaces

Discretely uniform approximation of continuous functions

This paper studies discretely uniform approximation of continuous functions and the associated discrete limit spaces. In particular, those conditions are established which must be satisfied in order that discretely uniform convergence exist and be surjective and that the corresponding limit space be a metric discrete limit space. Further, discretely uniform convergence is characterized by discretely continuous convergence. These results are applied to spaces of continuous functions defined on subsets of R n and of continuous functionals defined on subspaces of a reflexive Banach space. This theory is of interest in connection with Galerkin approximations, finite element methods and singular perturbations in Banach spaces.