First-order nonlinear differential equations with state-dependent impulses

Boundary value problems

Irena Rachnkov¹  Luk Rachnek² 

[1] University, Olomouc, Czech Republic;Department of Mathematics, Faculty of Science, Palacký

关键词: first-order ODE;    state-dependent impulses;    transversality conditions;    general linear boundary conditions;    existence;    Kurzweil-Stieltjes integral;   

DOI  :  10.1186/1687-2770-2013-195

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

The paper deals with the state-dependent impulsive problemz′(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)=c0,where[a,b]⊂R,c0∈R, f fulfils the Carathéodory conditions on[a,b]×R, the impulse functionis 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.

【 授权许可】

CC BY