A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces

In this paper, we consider the following boundary-value problems for second-order three-point nonlinear impulsive integro-differential equation of mixed type in a real Banach space EE: x″(t)+f(t,x(t),x′(t),(Ax)(t),(Bx)(t))=θ,t∈J,t≠tk,Δx|t=tk=Ik(x(tk)),Δx′|t=tk=Īk(x(tk),x′(tk)),k=1,2,…,m,x(0)=θ,x(1)=ρx(η), where θθ is the zero element of EE, (Ax)(t)=∫0tg(t,s)x(s)ds,(Bx)(t)=∫01h(t,s)x(s)ds,g∈C[D,R+],D={(t,s)∈J×J:t≥s},h∈C[J×J,R], and Δx|t=tkΔx|t=tk denotes the jump of x(t)x(t) at t=tkt=tk, Δx′|t=tkΔx′|t=tk denotes the jump of x′(t)x′(t) at t=tkt=tk. Some new results are obtained for the existence and multiplicity of positive solutions of the above problems by using the fixed-point index theory and fixed-point theorem in the cone of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results.