Steepest Descent Method for Equilibrium Points of Nonlinear Systems with Accretive Operators

AbstractLet E be a real normed linear space and let A: E↦2E be a bounded uniformly continuous φ-strongly accretive multi-valued map with nonempty closed values such that the inclusion 0∈Ax has a solution x*∈E. It is proved that both the Ishikawa and the Mann iteration processes are bounded and converge strongly to x*. Related results deal with the convergence of the iteration processes to fixed points of φ-strongly pseudocontractive maps and the iterative solution of the equation 0∈x+Fx for an accretive map F. Furthermore, the Ishikawa and Mann iteration methods with errors are discussed and proofs are sketched on how our theorems extend to these methods. Our method of proof is of independent interest