# On some equations y'(x) = f(x,y(h(x)+g(y(x))))

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2011)

- Volume: 31, Issue: 2, page 173-182
- ISSN: 1509-9407

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topZbigniew Grande. "On some equations y'(x) = f(x,y(h(x)+g(y(x))))." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.2 (2011): 173-182. <http://eudml.org/doc/271189>.

@article{ZbigniewGrande2011,

abstract = {In [4] W. Li and S.S. Cheng prove a Picard type existence and uniqueness theorem for iterative differential equations of the form y'(x) = f(x,y(h(x)+g(y(x)))). In this article I show some analogue of this result for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy's problem is obtained.},

author = {Zbigniew Grande},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {iterative differential equation; existence and uniqueness theorem; Picard approximation; derivative; (S)-continuity; (S)-path continuity},

language = {eng},

number = {2},

pages = {173-182},

title = {On some equations y'(x) = f(x,y(h(x)+g(y(x))))},

url = {http://eudml.org/doc/271189},

volume = {31},

year = {2011},

}

TY - JOUR

AU - Zbigniew Grande

TI - On some equations y'(x) = f(x,y(h(x)+g(y(x))))

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2011

VL - 31

IS - 2

SP - 173

EP - 182

AB - In [4] W. Li and S.S. Cheng prove a Picard type existence and uniqueness theorem for iterative differential equations of the form y'(x) = f(x,y(h(x)+g(y(x)))). In this article I show some analogue of this result for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy's problem is obtained.

LA - eng

KW - iterative differential equation; existence and uniqueness theorem; Picard approximation; derivative; (S)-continuity; (S)-path continuity

UR - http://eudml.org/doc/271189

ER -

## References

top- [1] A.M. Bruckner, Differentiation of real functions, Lectures Notes in Math. 659, Springer-Verlag, Berlin, 1978.
- [2] Z. Grande, A theorem about Carathéodory's superposition, Math. Slovaca 42 (1992), 443-449. Zbl0766.34036
- [3] Z. Grande, When derivatives of solutions of Cauchy's problem are (S)-continuous?, Tatra Mt. Math. Publ. 34 (2006), 173-177. Zbl1135.26004
- [4] W. Li and S.S. Cheng, A Picard theorem for iterative differential equations, Demonstratio Math. 42 (2) 2009, 371-380. Zbl1180.34007
- [5] B.S. Thomson, Real Functions, Lectures Notes in Math., Vol. 1170, Springer-Verlag, Berlin, 1980.