On nonlinear scalar Volterra integral equations. I
HTML articles powered by AMS MathViewer
- by Hans Engler PDF
- Trans. Amer. Math. Soc. 291 (1985), 319-336 Request permission
Abstract:
The scalar nonlinear Volterra integral equation \[ u(t) + \int _0^t {g(t,s,u(s)) ds = f(t)\qquad (0 \leqslant t)} \] is studied. Conditions are given under which the difference of two solutions can be estimated by the variation of the difference of the corresponding right-hand sides. Criteria for the existence of $\lim u(t)$ (as $t \to \infty$) are given, and existence and uniqueness questions are also studied.References
- Hans Engler, Bounds and asymptotics for a scalar Volterra integral equation, J. Integral Equations 7 (1984), no. 3, 209–227. MR 770148
- Gustaf Gripenberg, An abstract nonlinear Volterra equation, Israel J. Math. 34 (1979), no. 3, 198–212 (1980). MR 570881, DOI 10.1007/BF02760883
- Gustaf Gripenberg, Bounded solutions of a Volterra equation, J. Differential Equations 28 (1978), no. 1, 18–22. MR 463827, DOI 10.1016/0022-0396(78)90076-1
- 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, DOI 10.1016/0022-0396(79)90012-3
- T. Kiffe and M. Stecher, $L^{2}$ solutions of Volterra integral equations, SIAM J. Math. Anal. 10 (1979), no. 2, 274–280. MR 523843, DOI 10.1137/0510026 V. Lakshmikantham and S. Leela, Differential and integral inequalities. I, II, Academic Press, New York, 1969.
- J. J. Levin, A bound on the solutions of a Volterra equation, Arch. Rational Mech. Anal. 52 (1973), 339–349. MR 336269, DOI 10.1007/BF00247468
- J. J. Levin, Some a priori bounds for nonlinear Volterra equations, SIAM J. Math. Anal. 7 (1976), no. 6, 872–897. MR 420170, DOI 10.1137/0507070
- Stig-Olof Londen, On the solutions of a nonlinear Volterra equation, J. Math. Anal. Appl. 39 (1972), 564–573. MR 313743, DOI 10.1016/0022-247X(72)90180-1
- Richard K. Miller, Nonlinear Volterra integral equations, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. MR 0511193
- J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, 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. MR 0433481
- 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, DOI 10.1016/0022-247X(81)90251-1
- Olof J. Staffans, A priori bounds for a discontinuous Volterra equation, J. Integral Equations 3 (1981), no. 3, 231–243. MR 628353
- 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 44009
- 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
- Olof J. Staffans, A note on a Volterra equation with several nonlinearities, J. Integral Equations 7 (1984), no. 3, 249–252. MR 770151
- Hans Engler, A note on scalar Volterra integral equations. II, J. Math. Anal. Appl. 115 (1986), no. 2, 363–395. MR 836233, DOI 10.1016/0022-247X(86)90002-8
Additional Information
- © Copyright 1985 American Mathematical Society
- 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