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Solutions of abstract integro‐differential equations via Poisson transformation
Author(s) -
Lizama Carlos,
Ponce Rodrigo
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6042
Subject(s) - mathematics , resolvent , banach space , discretization , kernel (algebra) , linear map , integrable system , norm (philosophy) , initial value problem , semigroup , domain (mathematical analysis) , poisson kernel , mathematical analysis , poisson distribution , pure mathematics , political science , law , statistics
We study the initial value problem( * )u ( n + 1 ) − u ( n ) = A u ( n + 1 ) + ∑ k = 0 n + 1a ( n + 1 − k ) A u ( k ) , n ∈ N 0u ( 0 ) = x ,where A is closed linear operator defined on a Banach space X , x belongs to the domain of A , and the kernel a is a particular discretization of an integrable kernel a ∈ L 1 ( R + ) . Assuming that A generates a resolvent family, we find an explicit representation of the solution to the initial value problem (*) as well as for its inhomogeneous version, and then we study the stability of such solutions. We also prove that for a special class of kernels a , it suffices to assume that A generates an immediately norm continuous C 0 ‐semigroup. We employ a new computational method based on the Poisson transformation.