Geodesic
On some equations y'(x) = f(x,y(h(x)+g(y(x))))
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 31 (2011) no. 2, pp. 173-182.

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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.
Keywords: iterative differential equation, existence and uniqueness theorem, Picard approximation, derivative, (S)-continuity, (S)-path continuity 
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Grande, Zbigniew. On some equations y'(x) = f(x,y(h(x)+g(y(x)))). Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 31 (2011) no. 2, pp. 173-182. http://geodesic.mathdoc.fr/item/DMDICO_2011_31_2_a2/

[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.

[3] Z. Grande, When derivatives of solutions of Cauchy's problem are (S)-continuous?, Tatra Mt. Math. Publ. 34 (2006), 173-177.

[4] W. Li and S.S. Cheng, A Picard theorem for iterative differential equations, Demonstratio Math. 42 (2) 2009, 371-380.

[5] B.S. Thomson, Real Functions, Lectures Notes in Math., Vol. 1170, Springer-Verlag, Berlin, 1980.