Blow-up behavior of Hammerstein-type delay Volterra integral equations

Abstract

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations (DVIEs). Two types of delays, i.e., vanishing delay (pantograph delay) and non-vanishing delay (constant delay), are considered. With the same assumptions of Volterra integral equations (VIEs), in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. he blow-up behaviors of DVIEs with non-vanishing delay vary with different nitial functions and the length of the lag, while DVIEs with pantograph delay wn the same blow-up behavior of VIEs. Some examples and applications to elay differential equations illustrate this influence.

Keywords

Delay Volterra integral equation (DVIE) / non-vanishing delay / vanishing delay / blow-up of solution

References

[1] Banaś J. An existence theorem for nonlinear Volterra integral equation with deviating argument. Rend Circ Mat Palermo, 1986, 35: 82-89

[2] Bélair J. Population models with state-dependent delays. In: Arino O, Axelrod D E, Kimmel M, eds. Mathematical Population Dynamics. Lecture Notes in Pure and Appl Math, vol 131. New York: Marcel Dekker, 1991, 165-176

[3] Bownds J M, Cushing J M, Schutte R. Existence, uniqueness, and extendibility of solutions of Volterra integral systems with multiple, variable lags. Funkcial Ekvac, 1976, 19: 101-111

[4] Brauer F, van den Driessche P. Some directions for mathematical epidemiology. In: Ruan S, Wolkowicz G S K, Wu J, eds. Dynamical Systems and Their Applications in Biology (Cape Breton, 2001). Fields Institute Communications, Vol 36. Providence: Amer Math Soc, 2003, 95-112

[5] Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge Monographs on Applied and Computational Mathematics, Vol 15. Cambridge: Cambridge University Press, 2004

[6] Brunner H, Yang Z W. Blow-up behavior of Hammerstein-type Volterra integra equations. J Integral Equations Appl (to appear)

[7] Busenberg S, Cooke K L. The effect of integral conditions in certain equations modelling epidemics and population growth. J Math Biol, 1980, 10: 13-32

[8] Cañada A, Zertiti A. Method of upper and lower solutions for nonlinear delay integral equations modelling epidemics and population growth. Math Models Methods Appl Sci, 1994, 4: 107-119

[9] Cañada A, Zertiti A. Systems of nonlinear delay integral equations modelling population growth in a periodic environment. Comment Math Univ Carolin, 1994, 35: 633-644

[10] Cardone A, Del Prete I, Nitsch C. Gaussian direct quadrature methods for double delay Volterra integral equations. Electron Trans Numer Anal, 2009, 35: 201-216

[11] Chambers Ll G. Some properties of the functional equation ϕ(x) = f(x) + ∫0λxg(x, y, ϕ(y))dy.Internat J Math Sci, 1990, 14: 27-44

[12] Cooke K L. An epidemic equation with immigration. Math Biosci, 1976, 29: 135-158

[13] Cooke K L, Yorke J A. Some equations modelling growth processes and epidemics. Math Biosci, 1973, 16: 75-101

[14] Ezzinbi K, Jazar M. Blow-up Results for some nonlinear delay differential equations. Positivity, 2006, 10: 329-341

[15] Golaszewska A, Turo J. On nonlinear Volterra integral equations with state dependent delays in several variables. Z Anal Anwend, 2010, 29: 91-106

[16] Hethcote H W, van den Driessche P. Two SIS epidemiologic models with delays. J Math Biol, 2000, 40: 3-26

[17] Linz P. Analytical and Numerical Methods for Volterra Equations. Philadelphia: SIAM, 1985

[18] Malolepszy T. Nonlinear Volterra integral equations and the Schrŏder functional equation. Nonlinear Anal, 2011, 74: 424-432

[19] Metz J A J, Diekmann O. The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomath, Vol 68. Berlin-Heidelberg: Springer-Verlag, 1986

[20] Miller R K. Nonlinear Volterra Integral Equations. Menlo Park: W A Benjamin, 1971

[21] Mydlarczyk W. The blow-up solutions of integral equations. Colloq Math, 1999, 79: 147-156

[22] Olmstead W E. Ignition of a combustible half space. SIAMJ Appl Math, 1983, 43: 1-15

[23] Smith H L. On periodic solutions of a delay integral equation modelling epidemics. J Math Biol, 1977, 4: 69-80

[24] Waltham P. Deterministic Threshold Models in the Theory of Epidemics. Lecture Notes in Biomath, Vol 1. Berlin-Heidelberg: Springer-Verlag, 1974

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg