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

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• 作者：Irena Rach?nková ; Jan Tome?ek
• 关键词：34B37 ; 34B10 ; 34B15 ; system of ODEs of the first order ; state ; dependent impulses ; general linear boundary conditions ; transversality conditions ; fixed point problem
• 刊名：Boundary Value Problems
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：2014
• 期：1
• 全文大小：1,258 KB
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• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770

文摘

The paper investigates a fixed point problem in the space \(({\mathbb{W}}^{1,\infty} ([a,b];\mathbb{R}^{n}) )^{p+1}\) which is connected to boundary value problems with state-dependent impulses of the form \(z'(t) = f(t,z(t))\) , a.e. \(t\in[a,b]\subset\mathbb{R}\) , \(z(\tau_{i}+) - z(\tau_{i}) = J_{i}(\tau_{i},z(\tau_{i}))\) , \(\ell(z) = c_{0}\) . Here, the impulse instants \(\tau_{i}\) are determined as solutions of the equations \(\tau_{i} = \gamma_{i}(z(\tau_{i}))\) , \(i = 1,\ldots,p\) . We assume that \(n,p\in\mathbb{N}\) , \(c_{0} \in\mathbb{R}^{n}\) , the vector function f satisfies the Carathéodory conditions on \([a,b]\times\mathbb{R}^{n}\) , the impulse functions \(J_{i}\) , \(i=1,\ldots,p\) , are continuous on \([a,b]\times\mathbb {R}^{n}\) , and the barrier functions \(\gamma_{i}\) , \(i = 1,\ldots,p\) , are continuous on \(\mathbb{R}^{n}\) . The operator ?/em> is an arbitrary linear and bounded operator on the space of left-continuous regulated on \([a,b]\) vector valued functions and is represented by the Kurzweil-Stieltjes integral. Provided the data functions f and \(J_{i}\) 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.