An Alternative Way of Utilizing Fixed Point Theory

Abstract

The author proposes an alternative way of using fixed point theory to get the existence for semilinear equations. As an example, a nonlocal ordinary differential equation is considered. The idea is to solve homogeneous equations in the linearization. One feature of this method is that it does not need the equation to have special structures, for instance, variational structures, maximum principle, etc. Roughly speaking, the existence comes from good properties of the suitably linearized equation. The idea may have wider application.

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Acknowledgements

The author would like to thank Robert Brown, Hongjie Dong, Jiaxing Hong, Vladimir Sverak and Yijun Yao for wonderful discussions. The author also thanks the anonymous referee for the careful reading of the manuscript and many suggestions for its improvement.

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Correspondence to Dapeng Du.

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Du, D. An Alternative Way of Utilizing Fixed Point Theory. Chin. Ann. Math. Ser. B 41, 861–872 (2020). https://doi.org/10.1007/s11401-020-0237-2

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Keywords

  • Existence
  • Semi-linear equations
  • Fixed point theory
  • Homogeneous linearization

2000 MR Subject Classification

  • 35A99