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Existence results for systems with coupled nonlocal initial conditions. (English) Zbl 1288.34019
In this interesting paper the authors discuss the system of the first order differential equations \[ u_j'=f_j(t,u_1,\dots,u_n), \, j=1, 2,\dots,n, \] a.e. on the interval \([0,1]\) associated with the conditions of the form \[ u_j(0)=\sum_{i=1}^n\alpha_{ji}[u_i], \] where the symbol \(\alpha_{ji}[u_i]\) stands for the known Riemann-Stieltjes integral \(\int_0^1u_{ji}(s)dA_{ji}(s)\) and the functions \(A_{ji}\) have bounded variations. The function \(f\) satisfies Lipschitz type conditions vectorially and on intervals \([0,t_0]\), \([t_0,1]\) separately. This condition permits to write the corresponding integral vector operator as a sum of a Fredholm and of a Volterra operator. Then the authors choose a norm which makes the “slope” of the second operator small enough, thus the original operator stays contracting, if the Fredholm one is such. The contraction fixed point theorem completes the proof. The Schauder’s fixed point theorem is applied in section 2.2, when \(f_i(t,u)\) is bounded by expressions like \[ \sum_{i=1}^na_{ji}|u_{j}|+\bar{a}_i(t) \] and the Schaefer’s fixed point applies in the general case in Section 2.3, when \[ |f_i(t,u)|\leq \omega(t,\|u\|), \text{ if } t\in[0,t_0] \text{ and }\leq\beta(|u|)\gamma(t), \text{ if } t\in[t_0,1]. \] Some examples are given to illustrate the results.

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
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