András Rontó

UDC 517.9 We show how an appropriate parametrization technique and special successive approximations can help to control unknown jumps in the case of nonlinear boundary-value problems with state-dependent impulses. The practical application of the technique is shown on a numerical example.

We consider a weighted initial value problem for first‐order functional differential equations with monotone non‐linearities which may have a non‐integrable singularity with respect to the independent variable. Conditions for the existence of solutions in a suitable weighted space of locally absolutely continuous functions are given.

We show the use of parametrization techniques and successive approximations for the effective construction of solutions of linear boundary value problems for differential systems with multiple argument deviations. The approach is illustrated with a numerical example.

It is shown that a class of symmetric solutions of scalar non-linear functional differential equations can be investigated by using the theory of boundary value problems. We reduce the question to a two-point boundary value problem on a bounded interval and present several conditions ensuring the existence of a unique symmetric solution.

We describe 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 non-linear system of differential equations u ′ ⁢ ( t ) = f ⁢ ( t , u ⁢ ( t ) ) for a.e. ⁢ t ∈ [ a , b ] , u^{\prime}(t)=f(t,u(t))\quad\text{for a.e. }t\in[a,b], 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 , u(t+)-u(t-)=\gamma_{t}(u(t-))\quad\text{for }t\in(a,b)\text{ such that }g(t,u(% t-))=0, and the non-linear two-point boundary condition V ⁢ ( u ⁢ ( a ) , u ⁢ ( b ) ) = 0 . V(u(a),u(b))=0.