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Periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in Banach spaces. (English) Zbl 0849.45006

The author considers the existence of minimal and maximal solutions for the periodic boundary value problems of second order impulsive integro-differential equations of mixed type in Banach spaces

-u '' =f(t,u,Tu,Su)fortt k ,Δu| t=t k =I k u ( t k ),Δu ' | t=t k =I ¯ k u ( t k )(k=1,2,,m)

u(0)=u(2π), u ' (0)=u ' (2π), where fC[J×E 3 ;E], J=[0,2π], E is a real Banach space, 0<t 1 <<t m <2π, I k C[E;E], I ¯ k C[E;E], Δu| t=t k =u(t k + )-u(t k - ), Δu ' | t=t k =u ' (t k + )-u ' (t k - ) (k=1,2,,m). The operators T,S are given by

Tu(t)= 0 t k(t,s)u(s)ds,Su(t)= 0 2π k 1 (t,s)u(s)ds

with kC[D,R], D={(t,s) 2 :0st2π}, k 1 C[J×J;]. The method of proof is based on the monotone iterative technique and cone theory.

45N05Abstract integral equations, integral equations in abstract spaces
45J05Integro-ordinary differential equations
45M15Periodic solutions of integral equations
45G15Systems of nonlinear integral equations
45L05Theoretical approximation of solutions of integral equations