zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations. (English) Zbl 1080.34001
This work considers initial value problems (IVPs) defined by a diagonally implicit ODE of the form $g(x)y'=A(x)\alpha(y) - \omega(x)$, together with $y(0^+)=0;$ here $y \in \Bbb{R}^n$, $y'=dy/dx$, $g(x)=\text{diag}(g_1(x), \ldots, g_n(x))$, $A(x)=(a_{ij}(x))$ is a non-singular matrix with non-vanishing diagonal entries, $\alpha(y)=(\alpha_1(y_1), \ldots, \alpha_n(y_n)),$ and $\omega(x)=(\omega_1(x), \ldots, \omega_n(x))$; $g$ is continuous, and $A$, $\alpha$, $\omega$ are $C^1$. The condition $g_i(0^+)=0$ for some $i$ defines a singular Cauchy problem. Under componentwise positivity hypotheses on $g(x)$, $\alpha(y)$, $\alpha'(y)$, $\Omega(x) \equiv A^{-1}(x)\omega(x)$ and $\Omega'(x)$, and assuming additionally $\alpha(0^+)=\Omega(0^+)=0$, $\alpha_i(y) \leq M\alpha'_i(y)$, the authors first show that there exists a well-defined, positive, $C^1$ mapping $z=\varphi(x)=\alpha^{-1}(\Omega(x))$ with $\varphi(0^+)=0$. This allows them to construct a “funnel”, that is, a domain which will contain the graph of a solution to the singular IVP; specifically, under additional conditions on $a_{ij},$ $\omega_i,$ $g_i$ and $\Omega_i$, they prove using a topological retract method that there exists a family of positive solutions to the IVP. Some generalizations, simplification for linear cases (in which $\alpha_i = \text{id}$), and examples are discussed.

MSC:
 34A09 Implicit equations, differential-algebraic equations 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions
Full Text:
References:
 [1] K. Balla, Solution of singular boundary value problems for non-linear systems of ordinary differential equations , USSR Comput. Math. Math. Phys. 20 (1980), 100-115. · Zbl 0465.34013 · doi:10.1016/0041-5553(80)90275-X [2] Ch. Conley, Isolated invariant sets and the Morse index , Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, 1978. · Zbl 0397.34056 [3] J. Diblík, Some asymptotic properties of solutions of certain classes of ordinary differential equations , J. Math. Anal. Appl. 165 (1992), 288-304. · Zbl 0754.34043 · doi:10.1016/0022-247X(92)90080-W [4] --------, On asymptotic behaviour of solutions of certain classes of ordinary differential equations , J. Differential Equations 95 (1992), 203-217. · Zbl 0754.34042 · doi:10.1016/0022-0396(92)90029-M · eudml:34606 [5] Ph. Hartman, Ordinary differential equations , Second edition, Birkhäuser, Berlin, 1982. · Zbl 0476.34002 [6] V.A. Chechyk, Investigations of systems ordinary differential equations with singularities , Trudy Moskovsk. Mat. Obsc. 8 (1959), 155-198 (in Russian). [7] A.F. Izé, Asymptotic behavior of solutions of a class of differential equations in infinite-dimensional spaces , Nonlinear Anal., Theory, Methods Appl. 39 (2000), 289-303. · Zbl 0944.34051 · doi:10.1016/S0362-546X(98)00167-9 [8] I.T. Kiguradze, Some singular boundary value problems for ordinary differential equations , Izd. Tbilisskovo Univ., Tbilisi, 1975 (in Russian). · Zbl 0307.34003 [9] N.B. Konyukhova, Singular Cauchy problems for systems of ordinary differential equations , USSR Comput. Math. Math. Phys. 23 (1983), 72-82. · Zbl 0555.34002 · doi:10.1016/S0041-5553(83)80104-9 [10] Chr. Nowak, Some remarks on a paper by Samimi on nonuniqueness criteria for ordinary differential equations , Appl. Anal. 47 (1992), 39-44. · Zbl 0792.34002 · doi:10.1080/00036819208840130 [11] D. O’Regan, Nonresonant nonlinear singular problems in the limit circle case , J. Math. Anal. Appl. 197 (1996), 708-725. · Zbl 0855.34023 · doi:10.1006/jmaa.1996.0047 [12] J. Smoller, Shock waves and reaction-diffusion equations , Springer-Verlag, Berlin, 1983. · Zbl 0508.35002 [13] R. Srzednicki, On the structure of the Conley index , in Equadiff $99$, Internat. Conf. on Differential Equations (Berlin 1999), Vol. 1 (2000), 231-240. · Zbl 0974.37009 [14] --------, On periodic solutions inside isolating chains , J. Differential Equations 165 (2000), 42-60. · Zbl 0970.34038 · doi:10.1006/jdeq.2000.3764 [15] --------, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations , Nonlinear. Anal., Theory, Methods Appl. 22 (1994), 707-737. · Zbl 0801.34041 · doi:10.1016/0362-546X(94)90223-2 [16] T. Wa $\dotz$ewski, Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires , Ann. De La Soc. Polon. De Mathèmatique 20 (1947), Krakow, (1948), 279-313. · Zbl 0032.35001 [17] F.W. Wilson and J.A. Yorke, Liapunov functions and isolating blocks , J. Differential Equations 13 (1973), 106-123. · Zbl 0249.34037 · doi:10.1016/0022-0396(73)90034-X