Abstract
Suppose X is a real q‐uniformly smooth Banach space and F, K : X → X with D(K) = F(X) = X are accretive maps. Under various continuity assumptions on F and K such that 0 = u + KFu has a solution, iterative methods which converge strongly to such a solution are constructed. No invertibility assumption is imposed on K and the operators K and F need not be defined on compact subsets of X. Our method of proof is of independent interest.
Highlights
Let X be a real normed linear space with dual X∗
For 1 < q < ∞, we denote by Jq, the generalized duality mapping from X to 2X∗ defined by
It is known that many physically significant problems can be modelled by initial-value problems of the form x (t) + Ax(t) = 0, x(0) = x0, (1.3)
Summary
Let X be a real normed linear space with dual X∗. For 1 < q < ∞, we denote by Jq, the generalized duality mapping from X to 2X∗ defined by. In the special case in which the operators are defined on subsets D of X which are compact (or more generally, angle-bounded), Brezis and Browder [2] have proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.8) (see [4]). It is our purpose in this paper to introduce a new method that contains an auxiliary operator, defined in an appropriate real Banach space in terms of K and F, which under certain conditions, is accretive whenever K and F are, and whose zeros are solutions of (1.8). Our method which does not involve K−1 provides an explicit algorithm for the computation of solutions of (1.8)
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