Accretive Operators In Banach Spaces Research Articles

On the Ranges of Nonlinear Set-Valued Accretive Operators in Banach Spaces

LetEbe a real Banach space andA:D(A)⊆E→2E,B:D(B)⊆E→2Ebe two set valued accretive operators. We study the range problems of (A+λI) and (A+B+λI) for λ>0. The results obtained in this paper generalize some other results.

Iterative solution of nonlinear equations with strongly accretive operators in Banach spaces

Let X be a real Banach space with a uniformly convex dual X*. Let T: X→X be a Lipschitzian and strongly accretive mapping with a Lipschitzian constant L≥1 and a strongly accretive constant k∈(0,1). Let {±n} and {²n} be two real sequences in [0,1] statisfying: $$\alpha _n \to 0 as n \to \infty ;$$ ( i ) $$\beta _n< \frac{{k(1 - k)}}{{L(1 + L)}},for all n \geqslant 0;$$ ( ii ) $$\prod\limits_{n = 0}^\infty {\alpha _n = \infty .} $$ ( iii ) Set Sx = f-Tx+x, ∀x∈X. Assume that {un}n=0∞ and {vn}n=0∞ be two sequences in X satisfying ∥un∥ For arbitrary x0∈X, the iteration sequence {xn} is defined by $$(IS)_1 \left\{ \begin{gathered} x_{n + 1} = (1 - a_n )x_n + a_n Sy_n + u_n , \hfill \\ y_n = (1 - \beta _n )x_n + \beta _n Sx_n + v_n (n \geqslant 0), \hfill \\ \end{gathered} \right.$$ then {xn} converges strongly to the unique solution of the equation Tx=f.

Iterative approximation of solutions to nonlinear equations of strongly accretive operators in Banach spaces

Iterative approximation of solutions to nonlinear equations of strongly accretive operators in Banach spaces

On conditions of weak convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

On conditions of weak convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces

Let X be a real Banach space and G a bounded, open and convex subset of X. The solvability of the fixed point problem (*) Tx + Cx 3 x in D(T) n G is considered, where T . X D D(T) -* 2X is a possibly discontinuous m-dissipative operator and C: G -X is completely continuous. It is assumed that X is uniformly convex, D(T) n G $& 0 and (T + C)(D(T) n OG) C G. A result of Browder, concerning single-valued operators T that are either uniformly continuous or continuous with X* uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let IF {= 7Z+ -+ R-; /3(r) -O 0 as r -* oo}. The effect of a weak boundary condition of the type (u + Cx, x) > -/3(11xfl)flxl12 on the range of operators T + C is studied for m-accretive and maximal monotone operators T. Here, ,B E F, x C D(T) with sufficiently large norm and u C Tx. Various new eigenvalue results are given involving the solvability of Tx + ACx 3 0 with respect to (A, x) E (0, oo) x D(T). Several results do not require the continuity of the operator C. Four open problems are also given, the solution of which would improve upon certain results of the paper.

On the perturbation theory of $m$-accretive operators in Banach spaces

On the perturbation theory of $m$-accretive operators in Banach spaces

Nonzero solutions of nonlinear equations involving compact perturbations of accretive operators in Banach spaces

Nonzero solutions of nonlinear equations involving compact perturbations of accretive operators in Banach spaces

$p$-regular mappings and alternative results for perturbations of $m$-accretive operators in Banach spaces

$p$-regular mappings and alternative results for perturbations of $m$-accretive operators in Banach spaces

A further necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

Sets in the ranges of nonlinear accretive operators in Banach spaces

Let X be a real Banach space and G ⊂ X open and bounded. Assume that one of the following conditions is satisfied: (i) X* is uniformly convex and T:Ḡ→ X is demicontinuous and accretive; (ii) T:Ḡ→ X is continuous and accretive; (iii) T:X ⊃ D(T)→ X is m-acc

Iterative Solutions to Nonlinear Equations of Strongly Accretive Operators in Banach Spaces

Iterative Solutions to Nonlinear Equations of Strongly Accretive Operators in Banach Spaces

Strong convergence theorems for accretive operators in Banach spaces

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm and let A ⊂ E × E be m-accretive. Assume that S: cl( D( A)) → E is bounded, strongly accretive, and continuous. For x ϵ E and t > 0, let x t be the unique solution of the inclusion x ϵ Sx t + tAx t . It is proved that if there exists a nonexpansive retraction P of E onto cl( D( A)), then the strong lim t → 0 x t exists. Moreover, it is shown that if every weakly compact convex subset of E has the fixed point property for nonexpansive mappings and 0 ϵ R( A), then the strong lim t → ∞ x t exists and belongs to A −10. An interesting application to a convergence result for an implicit iterative scheme is also included.

On a fully nonlinear parabolic equation and the asymptotic behaviour of its solutions

Abstract : This paper deals with the first boundary problem associated with the fully nonlinear equation u sub t = Min(thi, delta u) on the set omega x (0, infinity), where omega is a domain of R to the N power and thi(x) is a given obstacle such that thi > or = 0 on omega. Formulating the problem (occurring in heat control) as an Evolution Variational Inequality, H. Brezis obtained the existence and uniqueness of weak solutions in the space H sub 0 to the 1st (omega) as well as weak convergence to an unknown equilibrium point of the equation (when t goes to infinity). The strong convergence of the solution to the zero equilibrium point is shown here, provided the obstacle is positive and subharmonic. If in addition thi(x) > or = gamma > 0 then the asymptotic behaviour is completely described in the sense that the solution satisfies the linear heat equation u sub t = delta u on (T sub O, infinity) x omega, T sub 0 being a finite time. To do this the results are first presented for strong solutions (that is, those which satisfy the equation a.e.). The fact that under more regularity on the initial datum the weak solution is also a strong one and certain useful comparison principles are proved by using the theory of accretive operators in Banach spaces. (Author)

Some mapping theorems for accretive operators in Banach spaces

Some mapping theorems for accretive operators in Banach spaces

Strong convergence theorems for resolvents of accretive operators in Banach spaces

Strong convergence theorems for resolvents of accretive operators in Banach spaces

Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces

LetA be anm-accretive operator in a Banach spaceE. Suppose thatA −10 is not empty and that bothE andE * are uniformly convex. We study a general condition onA that guarantees the strong convergence of the semigroup generated by—A and of related implicit and explicit iterative schemes to a zero ofA. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also establish strong convergence under the assumption that the interior ofA −10 is not empty. In Hilbert space this result is due to H. Brezis.

Zeros of accretive operators

In the investigation of accretive operators in Banach spaces X, the existence of zeros plays an important role, since it yields surjectivity results as well as fixed point theorems for operators S such that I-S is accretive. Let D⊂X and T: D→X an operator such that the initial value problems

Nonlinear perturbations of linear accretive operators in Banach spaces

It is shown that a strongly continuous semi-group of nonlinear nonexpansive operators can be constructed as lim n→∞ ((I+t/nB)−1 (I+t/nB)−1) n whereA is a linearm-accretive operator,B is a nonlinearm-accretive operator, andB satisfies a boundedness condition relative toA.

Continuous nonlinear perturbations of linear accretive operators in banach spaces

Let A be a linear, closed, densely defined m-accretive operator from a Banach space X to itself, and let T( t), t ⩾ 0, be the semigroup of operators which has − A as its infinitesimal generator. Let B be a nonlinear, continuous, everywhere defined accretive operator from X to itself, and let S( t), t ⩾ 0, be the semigroup of nonlinear operators which has − B as its infinitesimal generator. It is shown that for all x ϵ X, t ⩾ 0, lim n → ∞(T( t n ) S( t n )) nx = U(t)x exists, U(t)x = T(t)x − ∝ 0 t T(t − s) BU(s) x ds, and U(t), t ⩾ 0 , is a strongly continuous semigroup of nonlinear contractions on X. It is shown also that −( A + B) is the infinitesimal generator of U( t), t ⩾ 0, and A + B is m-accretive on X.

Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces

Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces