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.
LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.
Suppose that H is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.
An iteration process studied by Chidume and Zegeye 2002 is proved to convergestronglyto a solution of the equationAu=0whereAis a boundedm-accretive operator on certain real Banach spacesEthat includeLpspaces2≤p<∞.The iteration process does not involve the computation of the resolvent at any step of the process and does not involve the projection of an initial vector onto the intersection of two convex subsets ofE, setbacks associated with the classicalproximal point algorithmof Martinet 1970, Rockafellar 1976 and its modifications by various authors for approximating of a solution of this equation. The ideas of the iteration process are applied to approximate fixed points of uniformly continuous pseudocontractive maps.
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