Abstract
Abstract. The paper is devoted to the study of the following D.V. Ionescu’s problem
\[
\left \{
\begin{array}
[l]{l}%
-x_{1}^{\prime \prime}(t)=f_{1}(t,x_{1}%
(t),x_{2}(t),x_{1}^{\prime}(t),x_{2}^{\prime}(t)),\ t\in \lbrack a,b]\newline \\
-x_{2}^{\prime \prime}(t)=f_{2}(t,x_{1}(t),x_{2}(t),x_{1}^{\prime}%
(t),x_{2}^{\prime}(t))
\end{array}
\right.
\]
with polylocal conditions
\[
\left \{
\begin{array}
[c]{l}%
x_{1}(a)=x_{2}(b)=0\\
x_{1}(c)=x_{2}(c)\\
x_{1}^{\prime}(c)=x_{2}^{\prime}(c).
\end{array}
\right.
\]
Existence, uniqueness and data dependence (monotony, continuity, differentiability with respect to parameter) results of solution for the Cauchy problem are obtained using Perov fixed point theorem and weakly Picard operator theory.
Authors
Veronica Ana IIea
Babes-Bolyai University Kogalniceanu, 1, Cluj-Napoca, Romania
Diana Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis
Keywords
Perov fixed point theorem; weakly Picard operators; polylocal problem; fixed points; data dependence.
Paper coordinates
V.A. Ilea, D. Otrocol, On a D.V. Ionescu’s problem for functional-differential equations of second order, Proceedings of the 10th IC-FPTA, July 9-18, 2012, Cluj-Napoca, Romania, pp. 131-142, http://www.math.ubbcluj.ro/∼fptac
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Proceedings of the 10th IC-FPTA, July 9-18, 2012, Cluj-Napoca, Romania.
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[1] O. Aram˘a, Sur le reste de certaines formules de Runge-Kutta pour l’int´egration numerique des equations differentielles, Acad R.P. Romane Fil. Cluj Stud. Cerc. Mat.,
11(1960), 9-29.
[2] A.M. Bica, Properties of the method of successive approximations for two-point boundary value problems, Journal of Nonlinear Evolution Equations and Applications, 2011(2011), no. 1, 1-22.
[3] H. Chen, P. Li, Three-point boundary value problems for second-order ordinary differential equations in Banach spaces, Computers and Mathematics with Applications, 56(2008), 1852-1860.
[4] Gh. Coman, On D.V. Ionescu practical numerical integration formulas, Mathematical Contributions of D.V. Ionescu, ed. I.A. Rus, Babe¸s-Bolyai University, Departement of Applied Analysis, Cluj-Napoca, 2001, 69-76.
[5] K. Demling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.
[6] D. Guo, V. Lakshmikanthan, X. Lin, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht, 1996.
[7] D. Guo, Boundary value problems for impulsive integro-differential equation on unbounded domains in a Banach space, Appl. Math. Comput., 99(1999), 115.
[8] V.A. Ilea, D. Otrocol, On a D.V. Ionescu’s problem for functional-differential equations, Fixed Point Theory, 10(2009), no. 1, 125-140.
[9] D.V. Ionescu, Quelques th´eorems d’existence des int´egrales des systemes d’equations differentielles, C.R. de l’Acad. Sc. Paris, 186(1929), 1262-1263.
[10] G. Micula, The “D.V. Ionescu method” of constructing approximation formulas, Studia Univ. Babes-Bolyai, 26(1981), 6-13.
[11] A.I. Perov, A.V. Kibenko, On a general method to study boundary value problems, Iz. Akad. Nauk., 30(1966), 249-264.
[12] A. Petru¸sel, I.A. Rus, Mathematical contributions of Professor D.V. Ionescu, Novae Scientiae Mathematicae, 2008, 1-11.
[13] R. Precup, The role of the matrices that are convergent to zero in the study of semilinear operator systems, Math. Computer Modeling, 49(2009), 703-708.
[14] I.A. Rus, Principles ans Applications of the Fixed Point Theory, Romanian, Dacia, Cluj-Napoca, 1979, In Romanian.
[15] I.A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001.
[16] I.A. Rus, Functional differential equations of mixed type, via weakly Picard operators, Seminar on Fixed Point Theory Cluj-Napoca, 3(2002), 335-346.
[17] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae,
58(2003), no. 1, 191-219.