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Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space. (English) Zbl 0939.45004

The initial value problem is investigated for the second-order nonlinear impulsive integro-differential equations of Volterra type

x '' =f(t,x,Tx),tJ,tt i ,
Δx| t=t i =- j=1 i α ij x(t j )+ j=1 i β ij x ' (t j ),
Δx ' | t=t i =- j=1 i γ ij x(t j ),
x(0)=x 0 ,x ' (0)=x 1 ,

where fC[J×E×E,E], J=[0,a](a>0), 0<t 1 <<t i <<t m <a, α ij , β ij , γ ij (ij,i=1,2,,m) are nonnegative constants x 0 ,x 1 E, and

(Tx)(t)= 0 t k(t,s)x(s)ds,foralltJ,

kC[D,R + ], D={(t,s)J×J:ts}, R + is the set of all nonnegative numbers, in a real Banach space by means of upper and lower solutions. Conditions for the existence of maximal and minimal solutions are established.

45J05Integro-ordinary differential equations
45N05Abstract integral equations, integral equations in abstract spaces
45G10Nonsingular nonlinear integral equations
45L05Theoretical approximation of solutions of integral equations