Abstract
We consider the nonlinear Cauchy problem for second order differential-functional equations of parabolic type, and present two existence theorems: in the class of bounded and in the class of unbounded viscosity solutions. These are based on differential inequalities and on the contraction method. The functional dependence in the equation is of Hale type. Our results cover equations with retarded and deviated argument, and differential-integral problems.
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Topolski, K.A. On the existence of viscosity solutions for the parabolic differential-functional Cauchy problem. Acta Math Hung 129, 277–296 (2010). https://doi.org/10.1007/s10474-010-0029-3
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DOI: https://doi.org/10.1007/s10474-010-0029-3