Abstract
In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x0, where A is an accretive operator in a general Banach space X and f ε L1(0,T;X). Their method involves proving convergence in the L∞-norm of a sequence of step function approximations αn(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show a.e. existence of mild solution by proving convergence of the step functions αn(σ, τ) in the L1-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane.
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References
M.G. Crandall,An introduction to evolution governed by accretive operators, Dynamical Systems, Academic Press, New York, 1976, 131–165.
M.G. Crandall and L.C. Evans,On the relation of the operator ∂/∂s+∂/∂τto evolution governed by accretive operators, Israel J. Math. 21 (1975), 261–278.
M.G. Crandall and T.M. Liggett,Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298.
M.A. Freedman,Further investigation of the relation of the operator ∂/∂σ+∂/∂τto evolution governed by accretive operators, Houston J. Math, to appear.
K. Kobayasi, Y. Kobayashi and S. Oharu,Nonlinear evolution operators in Banach spaces, Osaka J. Math. 21 (1984), 281–310.
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Communicated by Jerome A. Goldstein
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Freedman, M.A. A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators. Semigroup Forum 36, 117–126 (1987). https://doi.org/10.1007/BF02575009
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DOI: https://doi.org/10.1007/BF02575009