Renewal theorems for a system of integral equations
Journal Article
·
· Sbornik. Mathematics
- Centre of Mathematical Physics Byurakan Astrophysical Observatory National Academy of Sciences of Armenia (Armenia)
The system of renewal integral equations {phi}{sub i}(x) = g{sub i}(x) + {sigma}{sub j=1}{sup m}{integral}{sub 0}{sup x}u{sub ij}(x-t){phi}{sub j}(t)dt, i=1,...,m is considered, where the matrix-valued function u=(u{sub ij}) satisfies the condition of conservativeness 0{<=}u{sub ij} element of L{sub 1}{sup +}{identical_to}L{sub 1}(0;{infinity}), and the matrix A={integral}{sub 0}{sup {infinity}}u(x) dx is irreducible and of spectral radius. The existence of a limit at +{infinity} of the solution {phi}=({phi}{sub 1},...,{phi}{sub m}){sup T} is established in the case when the vector-valued function g=(g{sub 1},...,g{sub m}){sup T} element of L{sub 1}{sup m} is bounded and g(+{infinity})=0. This limit is evaluated. The structure of {phi} for g element of L{sub 1}{sup m} is determined; namely, {phi}(x)={mu}+{rho}{sub 0}(x)+{psi}(x), where {rho}{sub 0} element of C{sub 0}{sup m} and {psi} element of L{sub 1}{sup m}. A similar formula for the resolvent matrix-valued function is obtained.
- OSTI ID:
- 21202825
- Journal Information:
- Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 12 Vol. 189; ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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