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A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces. (English) Zbl 1161.34357

Summary: We consider the following boundary-value problems for a nonlinear impulsive integro-differential equation of mixed type in a real Banach space \(E\):
\[ \begin{aligned} & x''(t)+f(t,x(t),x'(t),(Ax)(t),(Bx)(t))=\theta,\quad t\in J,\;t\neq t_k,\\ & \Delta x|_{t=t_k}=I_k(x(t_k)),\quad \Delta x'|_{t=t_k}=\overline I_k(x(t_k),x'(t_k)),\quad k=1,2,\dots,m,\\ & x(0)=\theta,\quad x(1)=\rho x(\eta),\end{aligned} \]
where \(\theta\) is the zero element of \(E\),
\[ (Ax)(t)=\int^t_0 g(t,s)x(s)\,ds,(Bx)(t)=\int^1_0 h(t,s)x(s)\,ds, \]
\(g\in C[D,R^+]\), \(D=\{(t,s)\in J\times J:t\geq s\}\), \(h\in C[J\times J,R]\), and \(\Delta x|_{t=t_k}\) denotes the jump of \(x(t)\) at \(t=t_k\), \(\Delta x'|_{t=t_k}\) denotes the jump of \(x'(t)\) at \(t=t_k\). Some results are obtained for the existence and multiplicity of positive solutions of the above problem by using the a fixed-point index theory and fixed-point theorem in a cone of a strict set contraction operators. An example is presented to demonstrate the main results.

MSC:

34K45 Functional-differential equations with impulses
34K30 Functional-differential equations in abstract spaces
34K10 Boundary value problems for functional-differential equations
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