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On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. (English) Zbl 0795.45007
Lith. Math. J. 32, No. 2, 238-264 (1992) and Liet. Mat. Rink. 32, No. 2, 299-331 (1992).
We consider the Cauchy problem in Sobolev and Hölder spaces for the operators $$L^{(\alpha)} = A^{(\alpha)} + B^{(\alpha)}$$, $$0<\alpha \leq 2$$, where the principal part of the operator $$L^{(\alpha)}$$ is the pseudo-differential operator $$A^{(\alpha)}$$ of the order $$\alpha$$, and $$B^{(\alpha)}$$ is the integro-differential perturbing operator. We prove several existence and uniqueness theorems as well as error estimates for the solution of the Cauchy problem $$({\partial \over \partial t} + L^{(\alpha)} - \lambda) u=f$$.

##### MSC:
 45K05 Integro-partial differential equations 35S05 Pseudodifferential operators as generalizations of partial differential operators
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