Periodic solutions of Volterra integral equations

M. N. Islam

International Journal of Mathematics and Mathematical Sciences, 1988, vol. 11, 1-12

Abstract:

Consider the system of equations x ( t ) = f ( t ) + ∫ − ∞ t k ( t , s ) x ( s ) d s , ( 1 ) and x ( t ) = f ( t ) + ∫ − ∞ t k ( t , s ) g ( s , x ( s ) ) d s . ( 2 ) Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k . Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool.

Date: 1988
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:793746

DOI: 10.1155/S016117128800095X

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