Singular initial value problem for a system of integro-differential equations. (English) Zbl 1258.45005

Šmarda, Zdeněk; Khan, Yasir

Abstr. Appl. Anal. 2012, Article ID 918281, 18 p. (2012).

Summary: Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: \[ g_i(t)y'_i(t) = a_iy_i(t)\left(1 + f_i\left(t, \mathbf{y}(t), \int^t_{0^+} K_i(t, s, \text\textbf{y}(t), \text\textbf{y}(s))ds\right)\right), \quad y_i(0^+) = 0, t \in (0, t_0], \] where \(\mathbf{y} = (y_i, \dots, y_n), a_i > 0, i = 1, \dots, n\) are constants and \(t_0 > 0\). An approach which combines topological method of T. Ważewski and Schauder’s fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.

MSC:

45J05	Integro-ordinary differential equations
45G15	Systems of nonlinear integral equations
45G05	Singular nonlinear integral equations

Keywords:

system of integro-differential equations; singular initial value problem

References:

[1]	R. P. Agarwal, D. O’Regan, and O. E. Zernov, “A singular initial value problem for some functional differential equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2004, no. 3, pp. 261-270, 2004. · Zbl 1065.34053
[2]	V. A. \vCe\vcik, “Investigation of systems of ordinary differential equations with a singularity,” Trudy Moskovskogo Matemati\vceskogo Ob\vs\vcestva, vol. 8, pp. 155-198, 1959 (Russian).
[3]	I. Diblík, “Asymptotic behavior of solutions of a differential equation partially solved with respect to the derivative,” Siberian Mathematical Journal, vol. 23, no. 5, pp. 654-662, 1982 (Russian). · Zbl 0521.34003
[4]	J. Diblík, “Existence of solutions of a real system of ordinary differential equations entering into a singular point,” Ukrainian Mathematical Journal, vol. 38, no. 6, pp. 588-592, 1986 (Russian). · Zbl 0654.34001
[5]	J. Ba\vstinec and J. Diblík, “On existence of solutions of a singular Cauchy-Nicoletti problem for a system of integro-differential equations,” Demonstratio Mathematica, vol. 30, no. 4, pp. 747-760, 1997. · Zbl 0910.45002
[6]	J. Diblík, “On the existence of \sum k=1n(ak1t+ak2x)(x’)k=b1t+b2x+f(t,x,x’),x(0)=0-curves of a singular system of differential equations,” Mathematische Nachrichten, vol. 122, pp. 247-258, 1985 (Russian). · Zbl 0584.34022
[7]	J. Diblík and C. Nowak, “A nonuniqueness criterion for a singular system of two ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 64, no. 4, pp. 637-656, 2006. · Zbl 1111.34007
[8]	J. Diblík and M. Rů\vzi, “Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 3, pp. 923-944, 2004. · Zbl 1080.34001
[9]	J. Diblík and M. R. Rů\vzi, “Inequalities for solutions of singular initial problems for Caratheodory systems via Wa\Dzewski’s principle,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 12, pp. 657-656, 2008.
[10]	Z. \vSmarda, “On the uniqueness of solutions of the singular problem for certain class of integro-differential equations,” Demonstratio Mathematica, vol. 25, no. 4, pp. 835-841, 1992. · Zbl 0781.45007
[11]	Z. \vSmarda, “On a singular initial value problem for a system of integro-differential equations depending on a parameter,” Fasciculi Mathematici, no. 25, pp. 123-126, 1995. · Zbl 0832.45004
[12]	Z. \vSmarda, “On an initial value problem for singular integro-differential equations,” Demonstratio Mathematica, vol. 35, no. 4, pp. 803-811, 2002. · Zbl 1161.45308
[13]	Z. \vSmarda, “Implicit singular integrodifferential equations of Fredholm type,” Tatra Mountains Mathematical Publications, vol. 38, pp. 255-263, 2007. · Zbl 1164.45005
[14]	A. E. Zernov and Yu. V. Kuzina, “Qualitative investigation of the singular Cauchy problem \sum k=1n(ak1t+ak2x)(x\(^{\prime}\))k=b1t+b2x+f(t,x,x\(^{\prime}\)),x(0)=0,” Ukrainian Mathematical Journal, vol. 55, no. 10, pp. 1419-1424, 2003 (Russian). · Zbl 1080.34503
[15]	A. E. Zernov and Yu. V. Kuzina, “Geometric analysis of a singular Cauchy problem,” Nonlinear Oscillations, vol. 7, no. 1, pp. 67-80, 2004 (Russian). · Zbl 1100.34008
[16]	A. E. Zernov and O. R. Chaĭchuk, “Asymptotic behavior of solutions of a singular Cauchy problem for a functional-differential equation,” Journal of Mathematical Sciences, vol. 160, no. 1, pp. 123-135, 2009. · Zbl 1186.34108
[17]	R. Srzednicki, “Wa\Dzewski method and Conley index,” in Handbook of Differential Equations: Ordinary Differential Equations, A. Canada, P. Drabek, and A. Fonda, Eds., vol. 1, pp. 591-684, Elsevier, Amsterdam, The Netherlands, 2004. · Zbl 1091.37006
[18]	P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0125.32102
[19]	E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, vol. 108 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1999. · Zbl 0834.46002