A nonlinear Volterra equation with rapidly decaying solutions
Author:
Olof J. Staffans
Journal:
Trans. Amer. Math. Soc. 258 (1980), 523530
MSC:
Primary 45J05; Secondary 45D05
MathSciNet review:
558188
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study the asymptotic behavior of the solutions of a nonlinear integrodifferential Volterra equation with a convolution kernel. More specifically, we give conditions which imply that a solution x satisfies , where p is an arbitrary, positive real number.
 [1]
C.
M. Dafermos and J.
A. Nohel, Energy methods for nonlinear hyperbolic Volterra
integrodifferential equations, Comm. Partial Differential Equations
4 (1979), no. 3, 219–278. MR 522712
(80b:45018), http://dx.doi.org/10.1080/03605307908820094
 [2]
R.
C. MacCamy, An integrodifferential equation with application in
heat flow, Quart. Appl. Math. 35 (1977/78),
no. 1, 1–19. MR 0452184
(56 #10465)
 [3]
R.
C. MacCamy, A model for onedimensional, nonlinear
viscoelasticity, Quart. Appl. Math. 35 (1977/78),
no. 1, 21–33. MR 0478939
(57 #18395)
 [4]
R.
C. MacCamy and J.
S. W. Wong, Exponential stability for a nonlinear functional
differential equation, J. Math. Anal. Appl. 39
(1972), 699–705. MR 0313614
(47 #2168)
 [5]
J.
A. Nohel and D.
F. Shea, Frequency domain methods for Volterra equations,
Advances in Math. 22 (1976), no. 3, 278–304. MR 0500024
(58 #17748)
 [6]
Daniel
F. Shea and Stephen
Wainger, Variants of the WienerLévy theorem, with
applications to stability problems for some Volterra integral
equations, Amer. J. Math. 97 (1975), 312–343.
MR
0372521 (51 #8728)
 [7]
Olof
J. Staffans, Positive definite measures with
applications to a Volterra equation, Trans.
Amer. Math. Soc. 218 (1976), 219–237. MR 0458086
(56 #16289), http://dx.doi.org/10.1090/S00029947197604580865
 [8]
Olof
J. Staffans, An inequality for positive definite
Volterra kernels, Proc. Amer. Math. Soc. 58 (1976), 205–210.
MR
0500049 (58 #17761), http://dx.doi.org/10.1090/S00029939197605000490
 [9]
Olof
J. Staffans, Boundedness and asymptotic behavior of solutions of a
Volterra equation, Michigan Math. J. 24 (1977),
no. 1, 77–95. MR 0463861
(57 #3800)
 [10]
Olof
J. Staffans, A nonlinear Volterra integral equation with square
integrable solutions, Volterra equations (Proc. Helsinki Sympos.
Integral Equations, Otaniemi, 1978) Lecture Notes in Math.,
vol. 737, Springer, Berlin, 1979, pp. 281–286. MR 551047
(81d:45005)
 [1]
 C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4 (1979), 219278. MR 522712 (80b:45018)
 [2]
 R. C. MacCamy, An integrodifferential equation with applications in heat flow, Quart. Appl. Math. 35 (1977), 119. MR 0452184 (56:10465)
 [3]
 , A model for onedimensional, nonlinear viscoelasticity, Quart. Appl. Math. 35 (1977), 2133. MR 0478939 (57:18395)
 [4]
 R. C. MacCamy and J. S. W. Wong, Exponential stability for a nonlinear functional differential equation, J. Math. Anal. Appl. 39 (1972), 699705. MR 0313614 (47:2168)
 [5]
 J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Advances in Math. 22 (1976), 278304. MR 0500024 (58:17748)
 [6]
 D. F. Shea and S. Wainger, Variants of the WienerLévy theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97 (1975), 312343. MR 0372521 (51:8728)
 [7]
 O. J. Staffans, Positive definite measures with applications to a Volterra equation, Trans. Amer. Math. Soc. 218 (1976), 219237. MR 0458086 (56:16289)
 [8]
 , An inequality for positive definite Volterra kernels, Proc. Amer. Math. Soc. 58 (1976), 205210. MR 0500049 (58:17761)
 [9]
 , Boundedness and asymptotic behavior of solutions of a Volterra equation, Michigan Math. J. 24 (1977), 7795. MR 0463861 (57:3800)
 [10]
 , A nonlinear Volterra integral equation with square integrable solutions, Volterra Equations, Lecture Notes in Math., vol. 737, SpringerVerlag, Berlin and New York, 1979, pp. 281286. MR 551047 (81d:45005)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
45J05,
45D05
Retrieve articles in all journals
with MSC:
45J05,
45D05
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719800558188X
PII:
S 00029947(1980)0558188X
Article copyright:
© Copyright 1980
American Mathematical Society
