On nonlinear scalar Volterra integral equations. I

Author:
Hans Engler

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

MSC:
Primary 45D05

MathSciNet review:
797063

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Abstract: The scalar nonlinear Volterra integral equation

**[1]**Hans Engler,*Bounds and asymptotics for a scalar Volterra integral equation*, J. Integral Equations**7**(1984), no. 3, 209–227. MR**770148****[2]**Gustaf Gripenberg,*An abstract nonlinear Volterra equation*, Israel J. Math.**34**(1979), no. 3, 198–212 (1980). MR**570881**, 10.1007/BF02760883**[3]**Gustaf Gripenberg,*Bounded solutions of a Volterra equation*, J. Differential Equations**28**(1978), no. 1, 18–22. MR**0463827****[4]**T. Kiffe and M. Stecher,*An abstract Volterra integral equation in a reflexive Banach space*, J. Differential Equations**34**(1979), no. 2, 303–325. MR**550048**, 10.1016/0022-0396(79)90012-3**[5]**T. Kiffe and M. Stecher,*𝐿² solutions of Volterra integral equations*, SIAM J. Math. Anal.**10**(1979), no. 2, 274–280. MR**523843**, 10.1137/0510026**[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****[8]**J. J. Levin,*Some a priori bounds for nonlinear Volterra equations*, SIAM J. Math. Anal.**7**(1976), no. 6, 872–897. MR**0420170****[9]**Stig-Olof Londen,*On the solutions of a nonlinear Volterra equation*, J. Math. Anal. Appl.**39**(1972), 564–573. MR**0313743****[10]**Richard K. Miller,*Nonlinear Volterra integral equations*, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. MR**0511193****[11]**J. T. Schwartz,*Nonlinear functional analysis*, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR**0433481****[12]**Olof J. Staffans,*A bound on the solutions of a nonlinear Volterra equation*, J. Math. Anal. Appl.**83**(1981), no. 1, 127–134. MR**632331**, 10.1016/0022-247X(81)90251-1**[13]**Olof J. Staffans,*A priori bounds for a discontinuous Volterra equation*, J. Integral Equations**3**(1981), no. 3, 231–243. MR**628353****[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****[15]**J. J. Levin,*Remarks on a Volterra equation*, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 233–255. MR**0402437****[16]**Olof J. Staffans,*A note on a Volterra equation with several nonlinearities*, J. Integral Equations**7**(1984), no. 3, 249–252. MR**770151****[17]**Hans Engler,*A note on scalar Volterra integral equations. II*, J. Math. Anal. Appl.**115**(1986), no. 2, 363–395. MR**836233**, 10.1016/0022-247X(86)90002-8

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DOI:
http://dx.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