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

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
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