We present a solution of Exercise 1.2.1 in [1] which yields a short new proof of a key step in one of proofs of Brouwer’s fixed point theorem (1910). A few people asked the author about the details of the solution, and this note might be interesting to a broader audience. Our approach is absolutely different from the ones using algebraic or differential topology or differential calculus and is based on a simple observation which somehow escaped many authors treating this theorem in the past.
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N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Am. Math. Soc., Providence, RI (2008).
N. Dunford and J.T. Schwartz, Linear Operators. I. General Theory, Interscience, New York etc. (1958).
W. Kulpa, “An integral theorem and its applications to coincidence theorems,” Acta Univ. Carol., Math. Phys. 30, No 2, 83–90 (1989).
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Translated from Problemy Matematicheskogo Analiza 126, 2024, pp. 17-20.
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Krylov, N.V. On a New Proof of the Key Step in the Proof of Brouwer’s Fixed Point Theorem. J Math Sci 279, 468–471 (2024). https://doi.org/10.1007/s10958-024-07025-z
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DOI: https://doi.org/10.1007/s10958-024-07025-z