On nonlinear scalar Volterra integral equations. I
Author:
Hans Engler
Journal:
Trans. Amer. Math. Soc. 291 (1985), 319336
MSC:
Primary 45D05
MathSciNet review:
797063
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Abstract: The scalar nonlinear Volterra integral equation is studied. Conditions are given under which the difference of two solutions can be estimated by the variation of the difference of the corresponding righthand sides. Criteria for the existence of (as ) are given, and existence and uniqueness questions are also studied.
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 [2]
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 , Bounded solutions of a Volterra equation, J. Differential Equations 28 (1978), 1822. MR 0463827 (57:3766)
 [4]
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 [5]
 , solutions of Volterra integral equations, SIAM J. Math. Anal. 10 (1979), 274280. MR 523843 (80c:45005)
 [6]
 V. Lakshmikantham and S. Leela, Differential and integral inequalities. I, II, Academic Press, New York, 1969.
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 J. J. Levin, A bound on the solutions of a Volterra equation, Arch. Rational Mech. Anal. 52 (1973), 339349. MR 0336269 (49:1045)
 [8]
 , Some a priori bounds for nonlinear Volterra equations, SIAM J. Math. Anal. 7 (1976), 872896. MR 0420170 (54:8185)
 [9]
 S. O. London, On the solutions of a nonlinear Volterra equation, J. Math. Anal. Appl. 39 (1972), 564573. MR 0313743 (47:2297)
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 R. K. Miller, Nonlinear Volterra integral equations, Benjamin, Menlo Park, 1971. MR 0511193 (58:23394)
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 J. T. Schwartz, Nonlinear functional analysis, Gordon & Breach, New York, 1969. MR 0433481 (55:6457)
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 O. J. Staffans, A bound on the solutions of a nonlinear Volterra equation, J. Math. Anal. Appl. 83 (1981), 127134. MR 632331 (82j:45004)
 [13]
 , A priori bounds for a discontinuous Volterra equation, J. Integral Equations 3 (1981), 231243. MR 628353 (82j:45003)
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 J. H. Roberts and W. R. Mann, On a certain nonlinear integral equation of the Volterra type, Pacific J. Math. 1 (1951), 431445. MR 0044009 (13:354a)
 [15]
 J. J. Levin, Remarks on a Volterra equation, Delay and Functional Differential Equations and their Applications, Academic Press, London, New York, 1972, pp. 233255. MR 0402437 (53:6257)
 [16]
 O. J. Staffans, A note on a Volterra equation with several nonlinearities, J. Integral Equations 7 (1984), 249252. MR 770151 (86e:45019)
 [17]
 H. Engler, A note on scalar Volterra integral equations. II, J. Math. Anal. Appl. (to appear). MR 836233 (87g:45001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507970637
PII:
S 00029947(1985)07970637
Keywords:
Nonlinear scalar Volterra equation,
uniqueness,
existence,
continuous dependence on data,
a priori estimates,
asymptotic behavior,
logconvex kernels
Article copyright:
© Copyright 1985
American Mathematical Society
