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

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.


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
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