The iterative solution of the equation

$f\in x+Tx$ for a monotone operator T in

${L}^{p}$ spaces.

*(English)* Zbl 0606.47067
Suppose T is a multivalued monotone operator (not necessarily continuous) with open domain D(T) in ${L}_{p}$ $(2\le p<\infty )$, $f\in R(I+T)$ and the equation $f\in x+Tx$ has a solution $q\in D\left(T\right)$. Then there exists a neighbourhood $B\subset D\left(T\right)$ of q and a real number ${r}_{1}>0$ such that for any $r\ge {r}_{1}$, for any initial guess ${x}_{1}\in B$, and any single-valued section ${T}_{0}$ of T, the sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ generated from ${x}_{1}$ by ${x}_{n+1}=(1-{C}_{n}){x}_{n}+{C}_{n}(f-{T}_{0}{x}_{n})$ remains in D(T) and converges strongly to q with $\parallel {x}_{n}-q\parallel =O\left({n}^{-}\right)$. Furthermore, for $X={L}_{p}\left(E\right)$, $\mu \left(E\right)<\infty $, $\mu =Lebesgue$ measure and $1<p<2$, suppose T is a single-valued locally Lipschitzian monotone operator with open domain D(T) in X. For $f\in R(I+T)$, a solution of the equation $x+Tx=f$ is obtained as the limit of an iteratively constructed sequence with an explicit error estimate.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H06 | Accretive operators, dissipative operators, etc. (nonlinear) |

65J15 | Equations with nonlinear operators (numerical methods) |