## Abstract

The aim of this paper is to discuss some basic problems (existence and uniqueness, data dependence) of the Cauchy problem for a hybrid differential equation with maxima using weakly Picard operators technique.

## Authors

D. **Otrocol**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy,

## Keywords

## Cite this paper as:

D. Otrocol, *Hybrid differential equations with maxima via Picard operators theory*, Stud. Univ. Babes-Bolyai Math., 61 (2016) no. 4, pp. 421-428.

## About this paper

##### Journal

Studia Universitatis Babes-Bolyai Mathematica

##### Publisher Name

Univ. Babes-Bolyai, Cluj-Napoca, Romania

##### DOI

##### Print ISSN

0252-1938

##### Online ISSN

2065-961X

##### MR

MR3583207

##### ZBL

1397.34108

## Google Scholar

[1] Bainov, D.D., Hristova, S., *Differential equations with maxima*, Chapman & Hall/CRC Pure and Applied Mathematics, 2011.

[2] Bainov, D.D., Petrov, V.A., Proyteheva, V.S., *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.

[3] Dhage, B.C., Lakshmikantham, V., *Basic results on hybrid differential equations*, Nonlinear Anal. Hybrid Syst., 4(2010), 414-424.

[4] Dobritoiu, M., Rus, I.A., Serban, M.A., *An integral equation arising from infections diseases, via Picard operators*, Stud. Univ. Babe¸s-Bolyai Math., 52(2007), no. 3, 81-94.

[5] Georgiev, L., Angelov, V.G., *On the existence and uniqueness of solutions for maximum equations*, Glas. Mat., 37(2002), no. 2, 275-281.

[6] Hale, J*., Theory of functional differential equations*, Springer, 1977.

[7] Otrocol, D., *Systems of functional differential equations with maxima, of mixed type*, Electron. J. Qual. Theory Differ. Eq., 2014(2014), no. 5, 1-9.

[8] Otrocol, D., Ilea, V.A., *Qualitative properties of a functional differential equation*, Electron. J. Qual. Theory Differ. Eq., 47(2014), 1-8.

[9] Otrocol, D., Rus, I.A., *Functional-differential equations with “maxima” via weakly Picard operators theory*, Bull. Math. Soc. Sci. Math. Roumanie, 51(99)(2008), no. 3, 253-261.

[10] Otrocol, D., Rus, I.A., *Functional-differential equations with maxima of mixed type argument*, Fixed Point Theory, 9(2008), no. 1, 207-220.

[11] Precup, R., *Monotone techniques to the initial values problem for a delay integral equation from Biomathematics*, Stud. Univ. Babes-Bolyai Math., 40(1995), no. 2, 63-73.

[12] Rus, I.A., *Picard operators and applications*, Sci. Math. Jpn., 58(2003), no. 1, 191-219.

[13] Rus, I.A., *Functional differential equations of mixed type, via weakly Picard operators*, Seminar on Fixed Point Theory Cluj-Napoca, 3(2002), 335-346.

[14] Rus, I.A., *Generalized contractions and applications*, Cluj University Press, 2001.

[15] Rus, I.A., *Weakly Picard operators and applications*, Seminar on Fixed Point Theory, Cluj-Napoca, 2(2001), 41-58.

[16] Stepanov, E., *On solvability of same boundary value problems for differential equations with “maxima“*, Topol. Methods Nonlinear Anal., 8(1996), 315-326.

[17] Zhang, B.G., Zhang, G., *Qualitative properties of functional equations with “maxima”*, Rocky Mountain J. Math., 29(1999), no. 1, 357-367.