一类二阶时滞泛函微分方程周期解的存在与唯一性_英文_.docxVIP

Existence an d Un iqueness of Per iodic Sol ut ionsf or a Kin d of Secon d Order Funct ional D if f erent ialEqua t ions with a Devia t ing ArgumentWAN G Na1 ,L U Shi2pi ng2 ,ZHAN G J ia2shun1 ,SH EN Zuo2mi n1(1 . Depart ment of Mat hematics , Chizho u College , Chizho u 247000 , China ; 2 . College of Mat hematics and Co mp uter Science , Anhui No r malU niversit y , Wuhu 241000 , China)Abstract : In t his paper , t he aut ho rs employ t he t heo ry of Maw hin’s co ntinuatio n t heo rem to st udy t heexistence and uniqueness of perio dic solutio ns fo r a kind of seco nd o rder f unctio nal differential equatio ns wit h a deviating argument :x″( t ) + f ( t , x ( t ) , x ( t - τ) ) = p ( t ) .So me new result s o n t he existence and uniqueness of perio dic solutio ns are o btained.Key words : perio dic solutio n ; deviating argument ; f unctio nal differential equatio ns ; Maw hin ’s co ntinuatio n t heo remPa per No :1001 - 2443 (2008) 05 - 0409 - 06Cla ssif icat ion No :O175 . 6Document code :A1Introduct ion an d Ma in Lemma sThe p ro blems of perio dic solutio ns fo r seco nd o rder o rdinary differential equatio n have been extensivelyst udied1 - 4 ,9 . But t he wo r k to st udy t he existence and uniqueness of perio dic solutio ns fo r seco nd o rder f unctio nal differential equatio ns by using Maw hin’s co ntinuatio n t heo rem rarely appeared5 ,7 . The reaso n fo r itlies in t hat a p rio ri bo und of solutio ns is not easy to estimate .In t his paper , we co nsider t he seco nd o rder f unctio nal differential equatio nsx″( t ) + f ( t , x ( t ) , x ( t - τ) )(1)= p ( t ) ,w here f ∈ C ( R 3 , R ) wit h f ( t +T , x 0 , x 1 ) ≡ f ( t , x 0 , x 1 ) fo r Π ( x 0 , x 1 ) ∈ R 2 , p ∈ C ( R , R ) wit h p ( t )= p ( t + T ) , andτis a co nstant . Our aim is to establish so me criteria to guarantee t he existence and uniqueness of perio dic solutio ns fo r Eq. (1) by using Maw hin’s co ntinuat