## 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.

### 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
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### References:

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