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Fixed point problem associated with state-dependent impulsive boundary value problems

Research paper by Irena Rachůnková, Jan Tomeček

Indexed on: 24 Sep '14Published on: 24 Sep '14Published in: Boundary Value Problems



Abstract

The paper investigates a fixed point problem in the space (W1,∞([a,b];Rn))p+1Open image in new window which is connected to boundary value problems with state-dependent impulses of the form z′(t)=f(t,z(t))Open image in new window, a.e. t∈[a,b]⊂ROpen image in new window, z(τi+)−z(τi)=Ji(τi,z(τi))Open image in new window, ℓ(z)=c0Open image in new window. Here, the impulse instants τiOpen image in new window are determined as solutions of the equations τi=γi(z(τi))Open image in new window, i=1,…,pOpen image in new window. We assume that n,p∈NOpen image in new window, c0∈RnOpen image in new window, the vector function f satisfies the Carathéodory conditions on [a,b]×RnOpen image in new window, the impulse functions JiOpen image in new window, i=1,…,pOpen image in new window, are continuous on [a,b]×RnOpen image in new window, and the barrier functions γiOpen image in new window, i=1,…,pOpen image in new window, are continuous on RnOpen image in new window. The operator ℓ is an arbitrary linear and bounded operator on the space of left-continuous regulated on [a,b]Open image in new window vector valued functions and is represented by the Kurzweil-Stieltjes integral. Provided the data functions f and JiOpen image in new window are bounded, transversality conditions which guarantee that this fixed point problem is solvable are presented. As a result it is possible to realize the construction of a solution of the above impulsive problem.MSC: 34B37, 34B10, 34B15.