First-order nonlinear differential equations with state-dependent impulses

• Published in: Boundary value problems Vol. 2013; no. 1; p. 195
• Main Authors: Rachůnek, Lukáš, Rachůnková, Irena
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 28.08.2013; BioMed Central Ltd
• Subjects: Analysis; Approximations and Expansions; Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Kurzweil-Stieltjes integral; existence; first-order ODE; state-dependent impulses; transversality conditions; general linear boundary conditions
• ISSN: 1687-2770; 1687-2770
• DOI: 10.1186/1687-2770-2013-195

The paper deals with the state-dependent impulsive problem z ′ ( t ) = f ( t , z ( t ) ) for a.e.  t ∈ [ a , b ] , z ( τ + ) − z ( τ ) = J ( τ , z ( τ ) ) , γ ( z ( τ ) ) = τ , ℓ ( z ) = c 0 , where [ a , b ] ⊂ R , c 0 ∈ R , f fulfils the Carathéodory conditions on [ a , b ] × R , the impulse function is continuous on [ a , b ] × R , the barrier function γ has a continuous first derivative on some subset of ℝ and ℓ is a linear bounded functional which is defined on the Banach space of left-continuous regulated functions on [ a , b ] equipped with the sup-norm. The functional ℓ is represented by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of first-order differential equations subject to state-dependent impulse conditions. Here, sufficient and effective conditions guaranteeing the solvability of the above problem are presented for the first time. MSC: 34B37, 34B15.