TY - JOUR
T1 - On Stability of Linear Barbashin Type Integrodifferential Equations
AU - Gil, Michael
N1 - Publisher Copyright:
© 2015 Michael Gil'.
PY - 2015/1/1
Y1 - 2015/1/1
N2 - We consider the Barbashin type equation ∂u(t,x)/∂t=c(t,x)u(t,x)+∫01k(t,x,s)u(t,s)ds+f(t,x) (t>0; 0≤x≤1), where c(·, ·), k(·, ·, ·), and f(·, ·) are given real functions and u(·, ·) is unknown. Conditions for the boundedness of solutions of this equation are suggested. In addition, a new stability test is established for the corresponding homogeneous equation. These results improve the well-known ones in the case when the coefficients are differentiable in time. Our approach is based on solution estimates for operator equations. It can be considered as the extension of the freezing method for ordinary differential equations.
AB - We consider the Barbashin type equation ∂u(t,x)/∂t=c(t,x)u(t,x)+∫01k(t,x,s)u(t,s)ds+f(t,x) (t>0; 0≤x≤1), where c(·, ·), k(·, ·, ·), and f(·, ·) are given real functions and u(·, ·) is unknown. Conditions for the boundedness of solutions of this equation are suggested. In addition, a new stability test is established for the corresponding homogeneous equation. These results improve the well-known ones in the case when the coefficients are differentiable in time. Our approach is based on solution estimates for operator equations. It can be considered as the extension of the freezing method for ordinary differential equations.
UR - http://www.scopus.com/inward/record.url?scp=84938117636&partnerID=8YFLogxK
U2 - 10.1155/2015/962565
DO - 10.1155/2015/962565
M3 - Article
AN - SCOPUS:84938117636
SN - 1024-123X
VL - 2015
JO - Mathematical Problems in Engineering
JF - Mathematical Problems in Engineering
M1 - 962565
ER -