# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${L}_{p}$ $\left(2\le p<\infty \right)$, $f\in R\left(I+T\right)$ 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 ${x}_{n+1}=\left(1-{C}_{n}\right){x}_{n}+{C}_{n}\left(f-{T}_{0}{x}_{n}\right)$ 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, suppose T is a single-valued locally Lipschitzian monotone operator with open domain D(T) in X. For $f\in R\left(I+T\right)$, 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)