The exact solution for solving a class nonlinear operator equations in the reproducing kernel space. (English) Zbl 1034.47030

Li, Chunli; Cui, Minggen

Appl. Math. Comput. 143, No. 2-3, 393-399 (2003).

The authors consider the Sobolev space \(W^1_2(\Omega^n)\) and give the reproducing properties of this space. As a results, the class of nonlinear operator equations \(\sum^n_{i=1} \prod^{n_i}_{j=1} (A_{ij}u)=f\) is transformed to the \(n\)-dimensional linear operator equation \(Au=f\). Finally, they give the exact solution of this nonlinear operator equation.

Reviewer: Ulrich Kosel (Freiberg)

MSC:

47J05	Equations involving nonlinear operators (general)
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

nonlinear operator equation; reproducing kernel

References:

[1]	Zhang, M.; Cui, M.G.; Deng, Z.X., A new uniformly convergent iterative method by interpolation where error decreases monotonically, Journal of computational mathematics (China), 3, 4, 365-372, (1985) · Zbl 0595.65013
[2]	Li, C.-L.; Cui, M.-G., How to solve the equation aubu+cu=f, Applied mathematics and computation, 133, 643-653, (2002) · Zbl 1051.47009
[3]	Han, G.; Ken, H.; Kokichi, S.; Wang, J., Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations, Applied mathematics and computation, 112, 49-61, (2000) · Zbl 1023.65140
[4]	Haidar; Nassar, H.S., Nonlinearly weighted L1–solvability of the first kind Fredholm integral equation, Journal of mathematical analysis and applications, 245, 69-86, (2000) · Zbl 0971.45001