zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Nagumo type existence results for second-order nonlinear dynamic BVPS. (English) Zbl 1072.34015
A time scale ${\Bbb T}$ is a nonempty closed subset of ${\Bbb R}$. This paper studies the second-order nonlinear dynamic equation $y^{\Delta\Delta}(t)=f(t,y^\sigma(t),y^\Delta(t))$ on time scales with certain functional boundary conditions. The authors introduce a Nagumo-type condition (restricting the growth of $f$ in $y^\Delta$) and establish the existence of at least one solution of the given boundary value problem lying between the associated lower and upper solutions. An example is presented illustrating the main result.

34B15Nonlinear boundary value problems for ODE
39A12Discrete version of topics in analysis
Full Text: DOI
[1] A. Adje. Sur et Sous-Solutions dans les Equations Differentielles Discontinues avec des Conditions aux Limites non Linéaires, Ph. D., Université Catholique de Louvain-la-Neuve, 1987.
[2] Bohner, M.; Peterson, A.: Dynamic equations on time scales, an introduction with applications. (2001) · Zbl 0978.39001
[3] Cabada, A.; Espinar, V. Otero --; Pouso, R. L.: Existence and approximation of solutions for discontinuous first order difference problems with nonlinear functional boundary conditions in the presence of lower and upper solutions. Comput. math. Appl. 39, 21-33 (2000) · Zbl 0972.39002
[4] A. Cabada, D.R. Vivero, Expression of the Lebesgue \Delta  -- integral on time scales as a usual Lebesgue integral, Application to the calculus of \Delta  -- antiderivatives, preprint. · Zbl 1092.39017
[5] Eloe, P. W.: A boundary value problem for a system of difference equations. Nonlinear anal. 7, 813-820 (1983) · Zbl 0518.39002
[6] D. Franco, D. O’Regan, Existence of solutions to second order problems with nonlinear boundary conditions, Dynamical Systems and Differential Equations (Wilmington, NC, 2002); Discrete Contininuous Dyn. Syst. Suppl. (2003) 273 -- 280.
[7] Habets, P.; Pouso, R. L.: Examples on the nonexistence of a solution in the presence of upper and lower solutions. Anziam J. 44, 591-594 (2003) · Zbl 1048.34036
[8] Hilger, S.: Analysis on measure chains --- a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) · Zbl 0722.39001
[9] Kaymakçalan, B.; Laksmikantham, V.; Sivasundaram, S.: Dynamical systems on measure chains. (1996)
[10] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations. (1985) · Zbl 0658.35003
[11] Nagumo, M.: Ueber die differentialgleichung y″=f(x,y,$y^{\prime}$). Proc. phys.-math. Soc. Japan 19, No. 3, 861-866 (1935) · Zbl 63.1021.04
[12] Nagumo, M.: On principally linear elliptic differential equations of the second order. Osaka math. J. 6, 207-229 (1954) · Zbl 0057.08201
[13] Picard, E.: Sur lapplication des methodes d’approximations successives a l’étude de certaines équations différentielles ordinaires. J. de math. 9, 217-271 (1893) · Zbl 25.0507.02
[14] Dragoni, G. Scorza: Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenzile del secondo ordine. Giornale mat. (Battaglini) 69, 77-112 (1931) · Zbl 0002.25702
[15] Zhuang, W.; Chen, Y.; Cheng, S. S.: Monotone methods for a discrete boundary problem. Comput. math. Appl. 32, 41-49 (1996) · Zbl 0872.39005