On equations with perturbed accretive mappings
This article deals with the equation \(Ax+ Cx=f\), where \(A: X\to 2^X\) is an accretive operator (\(X\) is a reflexive Banach space), \(C: X\to X\) is a bounded single-valued pseudoaccretive operator. The author calls an operator \(C: X\to X\) pseudoaccretive if the relations \(x\rightharpoonup x\), \(\limsup\langle U(x_n- x),Ax_n\rangle\leq 0\) imply that \(\liminf\langle U(x_n- x)_, Ax_n\rangle\geq \langle U(x- z), Ax\rangle\) for each \(z\). The basic result is the existence theorem for such equations under the assumptions that \(X\) and \(X^*\) are uniformly convex spaces, the dual mapping \(U: X\to X^*\) is weakly continuous, and \[ \langle Ux,y+ Cx- f\rangle\geq 0\qquad (y\in Ax,\;\| x\|= r) \] for some \(r>0\). The corresponding solution of the original equation is not unique; the result on the strong convergence of special approximate solutions to the solution \(\overline x\), defined by the inequalities \[ \langle U(\overline x- x),\overline x\rangle\leq 0\qquad (x: Ax+ Cx= f), \] is presented.
scientific article; zbMATH DE number 1175186