Abstract
In this paper we study the following second order functional-differential equations
with maxima, of mixed type,%
\[
-x^{n}\left( t\right) -f(t,x\left( t\right) ,\max_{t-h_{1}\leq \xi \leq
t}x\left( \xi \right) ,\max_{t\leq \xi \leq t+h_{2}}x\left( \xi)\right)
,\ \ t\in \left[ a,b\right]
\]
with \textquotedblright boundary\textquotedblright \ conditions%
\[
\left \{
\begin{array}
[c]{c}%
x\left( t\right) -\varphi \left( t\right) ,\ t\in \left[ a-h_{1},a\right]
,\\
x\left( t\right) =\psi \left( t\right) ,\ t\in \left[ b,b+h_{2}\right] .
\end{array}
\right.\]
The plan of the paper is the following: 1. Introduction 2. Picard and weakly Picard operator 3. The operator max 4. Existence and uniqueness 5. Inequalities of Caplygin type \ 6. Data dependence: monotony 7. Data dependence: continuity 8. Examples.
Authors
Diana Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy
Ioan A. Rus
Department of Applied Mathematics Babes-Bolyai University Cluj-Napoca, Romania
Keywords
Picard operator; weakly Picard operators; equation of mixed type; equations with maxima; fixed points, data dependence.
Paper coordinates
D. Otrocol, I.A. Rus, Functional-differential equations with maxima of mixed type, Fixed Point Theory, Volume 9, No. 1, 2008, 207-220.
About this paper
Journal
Publisher Name
Print ISSN
1583-5022
Online ISSN
2066 – 9208
google scholar link
[1] N.V. Azbelev (ed.), Functional-Differential Equations, Perm. Politekh. Inst. Perm, 1985 (in Russian).
[2] D. Bainov, D. Mishev, Oscillation Theory of Operator-Differential Equations, World Scientific, Singapore, 1995.
[3] D.D. Bainov, V. A. Petrov, V.S. Proyteheva, Existence and asymptotic behavior of nonoscillatory solutions of second order neutral differential equations with ”maxima”, J. Comput. Appl. Math., 83(1997), no. 2, 237-249.
[4] L. Georgiev, V.G. Angelov, On the existence and uniqueness of solutions for maximum equations, Glas. Mat., 37(2002), 275-281.
[5] J. Hale, Theory of Functional Differential Equations, Springer, 1977.
[6] T. Jankovskyi, System of differential equations with maxima, Dopov, Akad. Nauk. Ukr., 8(1997), 57-60.
[7] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Acad. Publ., Dordrecht, 1992.
[8] E. Liz, Monotone iterative technique in ordered Banach spaces, Nonlinear Analysis, 30(1997), 5179-5190.
[9] A.R. Magomedov, Existence and uniqueness theorems for solutions of differential equations with maxima containing a functional parameter, Arch. Math., Brne, 28(1992), 139-154 (in Russian).
[10] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae, 58(2003), 191-219.
[11] I.A. Rus, Functional differential equations of mixed type, via weakly Picard operators, Seminar on Fixed Point Theory Cluj-Napoca, 3(2002), 335-346.
[12] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, 2001.
[13] I.A. Rus, Weakly Picard operators and applications, Seminar on Fixed Point Theory Cluj-Napoca, 2(2001), 41-58.
[14] I.A. Rus, A. Petrusel, M. S¸erban, Weakly Picard operators: equivalent definitions, applications and open problems, Fixed Point Theory, 7(2006), 3-22.
[15] E. Stepanov, On solvability of same boundary value problems for differential equations with ”maxima”, Topological Methods in Nonlinear Analysis, 8(1996), 315-326.
[16] B.G. Zhang, G. Zhang, Qualitative properties of functional equations with ”maxima”, Rocky Mount. J. Math., 29(1999), 357-367.
[17] M. Zima, Applications of the spectral radius to same integral equations, Comment. Math. Univ. Caroline, 36(1997), 695-703.