Journal of Integral Equations and Applications

A multiple nonlinear Abel type integral equation

W. Mydlarczyk

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We discuss a multiple nonlinear Abel type integral equation. The basic results provide criteria for the existence of nontrivial everywhere positive solutions. They are expressed in terms of the generalized Osgood condition. The global behavior of the solution, especially the conditions when it experiences blow-up, is also considered.

Article information

J. Integral Equations Applications, Volume 27, Number 4 (2015), 555-572.

First available in Project Euclid: 8 February 2016

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Zentralblatt MATH identifier

Primary: 45D05: Volterra integral equations [See also 34A12]
Secondary: 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) [See also 47B35] 45G05: Singular nonlinear integral equations 45G10: Other nonlinear integral equations

Nonlinear Abel type integral equations the existence and uniqueness of solutions to integral Volterra equations nontrivial solutions the maximal solution blow-up solutions


Mydlarczyk, W. A multiple nonlinear Abel type integral equation. J. Integral Equations Applications 27 (2015), no. 4, 555--572. doi:10.1216/JIE-2015-27-4-555.

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  • P.J. Bushell and W. Okrasiński, Uniqueness of solutions for a class of nonlinear Volterra integral equations with convolution kernel, Math. Proc. Camb. Phil. Soc. 106 (1989), 547–552.
  • C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, 1991.
  • G. Gripenberg, Unique solutions of some Volterra integral equations, Math. Scand. 48 (1981), 59–67.
  • G. Gripenberg, S.O. Londen and O. Staffans, Volterra integral and functional equations, Encycl. Math. Appl. 34, Cambridge University Press, New York, 1990.
  • D.G. Lasseigne and W.E. Olmstead, Ignition or nonignition of a combustible solid with marginal heating, Quart. Appl. Math. 49 (1991), 303–312.
  • T. Małolepszy and W. Okrasiński, Conditions for blow-up of solutions of some nonlinear Volterra integral equations, J. Comp. Appl. Math. 205 (2007), 744–-750.
  • W. Mydlarczyk, The existence of nontrivial solutions of Volterra equations, Math. Scand. 68 (1991), 83–88.
  • ––––, A condition for finite blow-up time for a Volterra equations, J. Math. Anal. Appl. 181 (1994), 248–253.
  • W.E. Olmstead, Ignition of a combustible half space, SIAM J. Appl. Math. 43 (1983), 1–15.
  • C.A. Roberts, Analysis of explosion for nonlinear Volterra equations with blow-up solutions, J. Comp. Appl. Math. 97 (1998), 153–166.
  • C.A. Roberts, D.G. Lassigne and W.E. Olmstead, Volterra equations which models explosion in a diffusive medium, J. Int. Equat. Appl. 5 (1993), 531–546.