Regularization of nonlinear Ill-posed equations with accretive operators
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We study the regularization methods for solving equations with arbitrary accretive operators. We establish the strong convergence of these methods and their stability with respect to perturbations of operators and constraint sets in Banach spaces. Our research is motivated by the fact that the fixed point problems with nonexpansive mappings are namely reduced to such equations. Other important examples of applications are evolution equations and co-variational inequalities in Banach spaces. 1. Introduction Let E be a real normed linear space with dual E∗ . The normalized duality mapping j : E → ∗ 2E is defined by
j(x) := x∗ ∈ E∗ : x,x∗ = x2 , x∗ ∗ = x ,
(1.1)
where x,φ denotes the dual product (pairing) between vectors x ∈ E and φ ∈ E∗ . It is well known that if E∗ is strictly convex, then j is single valued. We denote the single valued normalized duality mapping by J. A map A : D(A) ⊆ E → 2E is called accretive if for all x, y ∈ D(A) there exists J(x − y) ∈ j(x − y) such that
u − v,J(x − y) ≥ 0,
∀u ∈ Ax, ∀v ∈ Ay.
(1.2)
If A is single valued, then (1.2) is replaced by
Ax − Ay,J(x − y) ≥ 0.
(1.3)
A is called uniformly accretive if for all x, y ∈ D(A) there exist J(x − y) ∈ j(x − y) and a strictly increasing function ψ : R+ := [0, ∞) → R+ , ψ(0) = 0 such that
Ax − Ay,J(x − y) ≥ ψ x − y .
Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 11–33 DOI: 10.1155/FPTA.2005.11
(1.4)
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Nonlinear Ill-posed problems with accretive operators
It is called strongly accretive if there exists a constant k > 0 such that in (1.4) ψ(t) = kt2 . If E is a Hilbert space, accretive operators are also called monotone. An accretive operator A is said to be hemicontinuous at a point x0 ∈ D(A) if the sequence {A(x0 + tn x)} converges weakly to Ax0 for any element x such that x0 + tn x ∈ D(A), 0 ≤ tn ≤ t(x0 ) and tn → 0, n → ∞. An accretive operator A is said to be maximal accretive if it is accretive and the inclusion G(A) ⊆ G(B), with B accretive, where G(A) and G(B) denote graphs of A and B, respectively, implies that A = B. It is known (see, e.g., [14]) that an accretive hemicontinuous operator A : E → E with a domain D(A) = E is maximal accretive. In a smooth Banach space, a maximal accretive operator is strongly-weakly demiclosed on D(A). An accretive operator A is said to be m-accretive if R(A + αI) = E for all α > 0, where I is the identity operator in E. Interest in accretive maps stems mainly from their firm connection with fixed point problems, evolution equations and co-variational inequalites in a Banach space (see, e.g. [6, 7, 8, 9, 10, 11, 12, 26]). Recall that each nonexpansive mapping is a continuous accretive operator [7, 19]. It is known that many physically significant problems can be modeled by initial-value problems of the form (see, e.g., [10, 12, 26]) x (t) + Ax(t) = 0,
x(0) = x0 ,
(1.5)
where A is an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be
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