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New results for the periodic boundary value problem for impulsive integro-differential equations. (English) Zbl 1166.45002

Consider J=[0,T], T>0, the continuous function f:J× 3 , the continuous functions I k :, 1km, 0=t 0 <t 1 <<t m <t m+1 =T, the set D={(t,s)J×J; ts}, the functions KC(D,[0,+)), HC(J×J,[0,+)) and the functions

[𝒯u](t)= 0 t K(t,s)u(s)ds,tJ,[𝒮u](t)= 0 T H(t,s)u(s)ds,tJ,

where u:J.

Suppose that there exist the limits

u(t k + )=lim tt k t<t k u(t),u(t k - )=lim tt k t>t k u(t),1km,

and denote Δu(t k )=u(t k + )-u(t k - ), 1km.

The authors consider the first-order impulsive integrodifferential equation

u ' (t)=f(t,u(t),[𝒯u](t),[𝒮u](t)),tJ{t 1 ,,t m }(1)

with periodic boundary value conditions

Δu(t k )=I k (u(t k )),1km,u(0)=u(T)(2)

and prove some comparison principles and establish existence results for extremal solutions u of the problem (1)(2) using these principles and the monotone iterative technique.

For example, they consider the Banach spaces (PC(J),· PC ) and (PC 1 (J),· PC 1 ), where

PC(J)={u:J;u| (t k ,t k+1 ] C((t k ,t k+1 [,),0km,u(t k + ),u(t k - )=u(t k ),1km}, PC 1 (J)={uPC(J);u| (t k ,t k+1 ) C 1 ((t k ,t k+1 ],),0km,u ' (0 + ),u ' (T - ),u ' (t k + ),u ' (t k - ),1km}

with the norms u PC =sup{|u(t)|;tJ}, respectively, u PC 1 =u PC +u ' PC and if there exist the functions α and β in PC 1 (J), αβ, satisfying some hypotheses, then there exist monotone sequences (α n ) n , (β n ) n of functions with

α=α 0 α n β n β 0 =β,n,

which converge uniformly on J to the extremal solutions u of the problem (1)(2) in


45J05Integro-ordinary differential equations
45L05Theoretical approximation of solutions of integral equations