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)
Nonlinear boundary value problems on time scales. (English) Zbl 0995.34016
This paper contains several results on the existence of one or two nonnegative solutions to the nonlinear second-order dynamic equation on a measure chain (time scale) ${\bold T}$, i.e., $$ y^{\Delta\Delta}(t)+f(t,y(\sigma(t)))=0, \quad t\in[a,b]\cap {\bold T}, $$ subject to the boundary conditions $$ y(a)=0, \quad y^\Delta(\sigma(b))=0. $$ The theory of dynamic equations on measure chains unifies and extends the differential (${\bold T}={\bbfR}$) and difference (${\bold T}={\bbfZ}$) equations theories. The results extend the ones by {\it L. Erbe} and {\it A. Peterson} [Math. Comput. Modelling 32, No. 5-6, 571---585 (2000; Zbl 0963.34020)], and are also closely related to results by {\it C. J. Chyan, J. Henderson} and {\it H. C. Lo} [Tamkang J. Math. 30, No. 3, 231-240 (1999; Zbl 0995.34017)]. The proofs are based on an application of a nonlinear alternative of Leray-Schauder-type or Krasnoselskii fixed-point theorems. The paper will be useful for researchers interested in nonnegative solutions to differential or difference equations, and/or in dynamic equations on measure chains (time scales).

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B45Boundary value problems for ODE on graphs and networks
39A99Difference equations
WorldCat.org
Full Text: DOI
References:
[1] Agarwal, R. P.; Bohner, M.: Basic calculus on time scales and some of its applications. Results math. 35, 3-22 (1999) · Zbl 0927.39003
[2] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999.
[3] B. Aulbach, S. Hilgar, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990. · Zbl 0719.34088
[4] Erbe, L. H.; Peterson, A.: Green’s functions and comparison theorems for differential equations on measure chains, dynamics continuous. Discrete impulsive systems 6, 121-138 (1999) · Zbl 0938.34027
[5] L.H. Erbe, A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, Mathematical and Computer Modelling. · Zbl 0963.34020
[6] Hilgar, S.: Analysis on measure chains -- a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990)
[7] B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. · Zbl 0869.34039
[8] J.W. Lee, D. O’Regan, Existence principles for differential equations and systems of equations, in: A. Granas, M. Frigon (Eds.), Topological Methods in Differential Equations and Inclusions, NATO ASI Series C, Vol. 472, Kluwer Academic Publishers, Dordrecht, 1995, pp. 239--289. · Zbl 0834.34074
[9] J. Munkres, Topology, Prentice-Hall, Englewood Cliffs, 1975.