Abstract
The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse Proofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space . Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions. MSC:34B37, 34B15.
Highlights
1 Introduction Differential equations involving impulse effects appear as a natural description of observed evolution phenomena of several real world problems
Papers dealing with state-dependent impulses, called impulses at variable times, focus their attention on initial value problems or periodic problems
In this paper we provide a new approach to boundary value problems with statedependent impulses based on a construction of proper sets and operators and the topological degree arguments
Summary
Differential equations involving impulse effects appear as a natural description of observed evolution phenomena of several real world problems. For each compact set K ⊂ A, there exists a function mK ∈ L ([a, b]) such that |f (t, x)| ≤ mK (t) for a.e. t ∈ [a, b] and each x ∈ K . We say that z : [ , T] → R is a solution of problem ( )-( ), if z is continuous on [ , T], there exists unique τ ∈ ( , T) such that γ (z(τ )) = τ , z|[ ,τ] and z|[τ,T] have absolutely continuous first derivatives, z satisfies equation ( ) for a.e. t ∈ [ , T] and fulfills conditions ( ), ( ). Let us consider K of ( ) and define the set
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