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On a D. V. Ionescu problem for functional-differential equations. (English) Zbl 1183.47077

The authors study the problem
\[ x'(t)=f(t,x(t), x(\omega(t)), \quad t\in [a,b], \]
\[ x_1(a)=0,\, x_2(t_2)=x_1(t_2), \dots,\;x_{m-1}(t_{m-1})=x_{m-2}(t_{m-1}), \, x_m(b)=0, \]
where \(x=(x_1,x_2,\dots,x_m),\) \(x(\omega)=(x_1(\omega_1), x_2(\omega_2), \dots, x_m(\omega_m)),\) \(a=t_1<t_2<\dots<t_{m-1}<t_m=b,\) \(f\in C([a,b]\times {\mathbb R}^{2m}, {\mathbb R}^{m})\) and \(\omega_i\in C([a,b],[a,b])\), \(i=1,2,\dots, m.\) This problem is a generalization of a problem studied by D.V.Ionescu [“Quelques théorèmes d’existence des intégrales des systèmes d’équations différentielles”, C.R.Acad.Sci.Paris 186, 1262–1264 (1928; JFM 54.0461.04)].
By using weakly Picard operator theory, existence, uniqueness and data dependence results are proved.

MSC:

47N20 Applications of operator theory to differential and integral equations
47H10 Fixed-point theorems
34K07 Theoretical approximation of solutions to functional-differential equations

Citations:

JFM 54.0461.04
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