Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations
Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations
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DOI：
10.1619/fesi.55.479
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发表时间：
2012
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影响因子：
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通讯作者：
H. Matsunaga;S. Murakami;V. M. Nguyen
中科院分区：
文献类型：
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作者：
H. Matsunaga;S. Murakami;V. M. Nguyen
． 本文讨论了x ð t ޼Рt (cid:1) y K ð t (cid:1) s Þ x ð s Þ ds；其中K ð s Þ是s中可测量的矩阵，满足有界性条件。我们提出了一种动态系统的方法来研究这些方程的行为，将它们视为具有无限延迟的泛函方程。结果得到了与方程相关的解半群的生成子的一组正规特征值所对应的相空间的分解。此外，我们建立了非齐次积分方程解的表示公式（在相空间中称为“常数变分公式”），并在相空间分解的基础上给出了分解公式。最后，我们应用该公式研究了这些方程的函数空间的可容许性。
. The present paper is concerned with linear integral equations of the form x ð t Þ ¼ Ð t (cid:1) y K ð t (cid:1) s Þ x ð s Þ ds ; where K ð s Þ is a matrix measurable in s , and satisfies some boundedness condition. We propose a dynamical-systems approach to studying the behavior of the equations by treating them as functional equations with infinite delay. As a result we obtain a decomposition of the phase space corresponding to a set of normal eigenvalues of the generator of solution semigroups associated with the equations. Moreover, we establish a representation formula (which is called ‘‘a variation-of-constants formula’’ in the phase space) for solutions of nonhomogeneous integral equations, together with a decomposed formula based on the decomposition of the phase space. Finally, we apply the formula to investigate the admissibility of function spaces with respect to these equations.