Investigation of solutions of state-dependent multi-impulsive boundary value problems Andr´ as Ront´ o, Irena Rach˚ unkov´ a, Mikl´ os Ront´ o, Luk´ aˇ s Rach˚ unek Abstract We discuss a reduction technique allowing one to combine an analysis of the existence of solutions with an efficient construction of approximate solutions for a state-dependent multi-impulsive boundary value problem which consists of the nonlinear system of differential equations du(t) dt = f (t, u(t)) for a.e. t ∈ [a, b] , subject to the state-dependent impulse condition u(t+) - u(t-)= γt (u (t-)) for t ∈ (a, b) such that g (t, u (t-)) = 0, and the nonlinear two-point boundary condition V (u(a),u(b)) = 0. Mathematics Subject Classification 2010: 34B37, 34B15 Key words: State-dependent multi-impulsive systems, nonlinear boundary value problem, parametrization tech- nique, successive approximations 1 Problem setting We consider the nonlinear system of differential equations du(t) dt = f (t, u(t)) for a.e. t ∈ [a, b] , (1.1) with −∞ <a<b< ∞ and a continuous vector-function f :[a, b] × R n → R n . Eq. (1.1) is subject to the state-dependent impulse condition u(t+) − u(t−)= γ t (u (t−)) for t ∈ (a, b) such that g (t, u (t−)) = 0. (1.2) Here the impulse vector-functions γ t : R n → R n and the barrier function g :[a, b] × R n → R are continuous, and the impulse instants t in (1.2) are unknown since they depend on a solution u through the equation g(t, u(t−)) = 0. The impulsive problem (1.1), (1.2) is investigated together with the nonlinear two-point boundary condition V (u(a),u(b)) = 0, (1.3) where the vector-function V : R n × R n → R n is continuous. The set G = {(t, x) ∈ [a, b] × R n : g(t, x)=0} (1.4) determined by the function g in (1.2) is called a barrier. Studies of real life problems with state-dependent impulses can be found in [1]- [4], [13], [14]. But the majority of existence results on impulsive boundary value problems concern fixed-time impulses. This is due to the fact that state-dependent impulses significantly change properties of boundary value problems. This is explained in detail in [6] or [7], where new existence results for boundary value problems with state-dependent impulses as well as with fixed-time impulses are proven. The results about state-dependent impulses concern linear boundary conditions and barriers in the form t = g(x) which are special cases of (1.3) and (1.4), respectively. Only solutions having exactly one intersection point with a barrier are discussed in the citied papers. Moreover, at present, according to the authors’ knowledge, no numerical results for boundary value problems with state- dependent impulses are available in the literature, except of [5]. In particular, [5] contains the existence result for state-dependent impulsive boundary value problems with linear boundary conditions where a solution can have just one intersection point with a barrier. Here we study solutions of the fully nonlinear problem (1.1)-(1.3) which are allowed to meet a barrier having the general form (1.4) finitely many times. Consequently, we specify solutions of problem (1.1)-(1.3) by the next definition. 1