Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators. (English) Zbl 1221.47111
Summary: Suppose that \(X\) is a real \(q\)-uniformly smooth Banach space and \(F,K:X\rightarrow X\) are bounded strongly accretive maps with \(D(K)=F(X)=X\). Let \(u^{*}\) denote the unique solution of the Hammerstein equation \(u+KFu=0\). A new explicit coupled iteration process is shown to converge strongly to \(u^{*}\). No invertibility assumption is imposed on \(K\) and the operators \(K\) and \(F\) need not be defined on compact subsets of \(X\). Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H04 | Set-valued operators |
| 47H06 | Nonlinear accretive operators, dissipative operators, etc. |
Keywords:
accretive operators; \(q\)-uniformly smooth spaces; duality maps; strong convergence; coupled iteration processReferences:
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