Abstract
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.
Highlights
The investigation of impulsive differential equations has a long history; see, e.g., the monographs [ – ]
Most papers dealing with impulsive differential equations subject to boundary conditions focus their attention on impulses at fixed moments
In our paper, we present an approach leading to a new existence principle for impulsive boundary value problems
Summary
The investigation of impulsive differential equations has a long history; see, e.g., the monographs [ – ]. Most papers dealing with impulsive differential equations subject to boundary conditions focus their attention on impulses at fixed moments. This is a very particular case of a more complicated case with state-dependent impulses. In our paper, we present an approach leading to a new existence principle for impulsive boundary value problems. This approach is applicable to each linear boundary condition which is considered with some first-order differential equation subject to statedependent impulses. Since C[a, b] ⊂ GL[a, b] ⊂ L∞[a, b], we equip the sets C[a, b] and GL[a, b] with the norm · ∞ and get Banach spaces (cf [ ]). ΧA is the characteristic function of a set A, where A ⊂ R
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