Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:423

Article

Kohlenbach, Ulrich¹  Koutsoukou-Argyraki, Angeliki¹ 

[1] Tech Univ Darmstadt, Dept Math, D-64289 Darmstadt, Germany

关键词: Proof mining;    Accretive operator;    Cauchy problems;    Rate of convergence;    Rate of metastability;    Partial differential equations;   

DOI  :  10.1016/j.jmaa.2014.10.035

来源: Elsevier

【 摘 要 】

We extract rates of convergence and rates of metastability (in the sense of Tao) for convergence results regarding abstract Cauchy problems generated by phi-accretive at zero operators A : D(A)(subset of X) -> 2(X) where X is a real Banach space, proved in [8], by proof-theoretic analysis of the proofs in [8] and having assumed a new, stronger notion of uniform accretivity at zero, yielding a new notion of modulus of accretivity. We compute the rate of metastability for the convergence of the solution of the abstract Cauchy problem generated by a uniformly accretive at zero operator to the unique zero of A via proof mining based on a result by the first author. Finally, we apply our results to a special class of Cauchy problems considered in [8]. This work is the first application of proof mining to the theory of partial differential equations. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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