On nonlinear scalar Volterra integral equations. I

Author:
Hans Engler

Journal:
Trans. Amer. Math. Soc. **291** (1985), 319-336

MSC:
Primary 45D05

DOI:
https://doi.org/10.1090/S0002-9947-1985-0797063-7

MathSciNet review:
797063

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Abstract | References | Similar Articles | Additional Information

Abstract: The scalar nonlinear Volterra integral equation

**[1]**H. Engler,*Bounds and asymptotics for a scalar Volterra integral equation*, J. Integral Equations**7**(1984), 209-227. MR**770148 (86j:45005)****[2]**G. Gripenberg,*An abstract nonlinear Volterra equation*, Israel J. Math.**34**(1979), 198-212. MR**570881 (81g:45011)****[3]**-,*Bounded solutions of a Volterra equation*, J. Differential Equations**28**(1978), 18-22. MR**0463827 (57:3766)****[4]**T. Kiffe and M. Stecher,*An abstract Volterra integral equation in a reflexive Banach space*, J. Differential Equations**34**(1979), 303-325. MR**550048 (81d:45020)****[5]**-, -*solutions of Volterra integral equations*, SIAM J. Math. Anal.**10**(1979), 274-280. MR**523843 (80c:45005)****[6]**V. Lakshmikantham and S. Leela,*Differential and integral inequalities*. I, II, Academic Press, New York, 1969.**[7]**J. J. Levin,*A bound on the solutions of a Volterra equation*, Arch. Rational Mech. Anal.**52**(1973), 339-349. MR**0336269 (49:1045)****[8]**-,*Some a priori bounds for nonlinear Volterra equations*, SIAM J. Math. Anal.**7**(1976), 872-896. MR**0420170 (54:8185)****[9]**S. O. London,*On the solutions of a nonlinear Volterra equation*, J. Math. Anal. Appl.**39**(1972), 564-573. MR**0313743 (47:2297)****[10]**R. K. Miller,*Nonlinear Volterra integral equations*, Benjamin, Menlo Park, 1971. MR**0511193 (58:23394)****[11]**J. T. Schwartz,*Nonlinear functional analysis*, Gordon & Breach, New York, 1969. MR**0433481 (55:6457)****[12]**O. J. Staffans,*A bound on the solutions of a nonlinear Volterra equation*, J. Math. Anal. Appl.**83**(1981), 127-134. MR**632331 (82j:45004)****[13]**-,*A priori bounds for a discontinuous Volterra equation*, J. Integral Equations**3**(1981), 231-243. MR**628353 (82j:45003)****[14]**J. H. Roberts and W. R. Mann,*On a certain nonlinear integral equation of the Volterra type*, Pacific J. Math.**1**(1951), 431-445. 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. 233-255. MR**0402437 (53:6257)****[16]**O. J. Staffans,*A note on a Volterra equation with several nonlinearities*, J. Integral Equations**7**(1984), 249-252. 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:
https://doi.org/10.1090/S0002-9947-1985-0797063-7

Keywords:
Nonlinear scalar Volterra equation,
uniqueness,
existence,
continuous dependence on data,
a priori estimates,
asymptotic behavior,
log-convex kernels

Article copyright:
© Copyright 1985
American Mathematical Society