Fixed point of a non-contracting mapping. (Point fixe d’une application non contractante.) (French) Zbl 1111.47053

Lemarié-Rieusset, Pierre Gilles

Rev. Mat. Iberoam. 22, No. 1, 339-356 (2006).

This article deals with the equation \(x = x_0 + Tx^k\) in a Banach space \(E\), where \(T\) is a continuous \(k\)-linear operator in \(E\) satisfying the inequalities \(\| T(x_1,\dots,x_k)\| \leq c_0\| x_1\| \dots \| x_k\| \) for all \(x, \dots , x_k\). The main result is the unique solvability of this equation for all \(x_0\), \(\| x_0\| \leq (k - 1)k^{-1}(c_0k)^{\frac1{k-1}}\), and the convergence of Picard’s approximations to this solution. The convergence of Newton’s approximations is also studied.
Some refinements of these statements are given in the case when the operator \(T\) satisfies the inequalities \(\| T(x_1,\dots,x_k)\| _F \leq c_1 \min_{1 \leq i \leq k} \left(\| x_i\| _F \prod_{j \neq i} \| x_j\| \right)\); here, \(F\) is a Banach space continuously embedded in \(E\).

Reviewer: Peter Zabreiko (Minsk)

MSC:

47H60	Multilinear and polynomial operators
47J05	Equations involving nonlinear operators (general)
35Q30	Navier-Stokes equations
47H10	Fixed-point theorems
47J25	Iterative procedures involving nonlinear operators
47N20	Applications of operator theory to differential and integral equations
65J99	Numerical analysis in abstract spaces
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

fixed point theorem; generalized contraction; \(k\)-linear operators; \(k\)-forms

References:

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