Abstract
The paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions z (t) = A(t)z(t) + f(t, z(t)) for a.e. t ∈ [a, b] ⊂ R, z(ti+)− z(ti) = Ji(z(ti)), i = 1, . . . , p, l(z) = c0, c0 ∈ R n . Here p, n ∈ N, a < t1 < . . . < tp < b, A ∈ L ([a, b];R), f ∈ Car([a, b] × R;R), Ji ∈ C(R ;R), i = 1, . . . , p, and l is a linear bounded operator on the space of left-continuous regulated functions on interval [a, b]. The operator l is expressed by means of the KurzweilStieltjes integral and covers all linear boundary conditions for solutions of the above system subject to impulse conditions. The representation, which is based on the Green matrix to a corresponding linear homogeneous problem, leads to an existence principle for the original problem. A special case of the n-th order scalar differential equation is discussed. This approach can be also used for analogical problems with state-dependent impulses. Mathematics Subject Classification 2010: 34B37, 34B15, 34B27
Highlights
In the literature there is a large amount of papers investigating the solvability of impulsive boundary value problems with impulses at fixed times
Where all possible linear boundary conditions are covered by condition (3)
The approach presented here can be applied to problems with state-dependent impulses, which will be shown in our papers
Summary
In the literature there is a large amount of papers investigating the solvability of impulsive boundary value problems with impulses at fixed times. The reason is to obtain a general tool, which can be applied to problems with state-dependent impulsive conditions Solutions of such problems are left-continuous and can have discontinuities anywhere in the interval (a, b). Remark 9 Let us note that the Green matrix of problem (10), (11) is not determined uniquely. The linear impulsive boundary value problem (12), (13), (3) has a unique solution z which has the form b p z(t) = G(t, s)q(s) ds + G(t, ti)Ii + Y (t) [l(Y )]−1 c0, t ∈ [a, b],. From Lemma 8 and (c) of Definition 7, it follows that the function b x(t) = G(t, s)q(s) ds, t ∈ [a, b], a is a unique solution of problem (12),(11).
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