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Multiple positive solutions of a boundary value problem for \(n\)th-order impulsive integro-differential equations in a Banach space. (English) Zbl 1054.45007

The author obtains the existence of multiple positive solutions for a boundary value problem of \(n\)th-order nonlinear impulsive integro-differential equations on an infinite interval in a Banach space by means of fixed point index theory of completely continuous operators. The fixed point index theory of completely continuous operators is used to obtain results on the existence of multiple positive solutions of \(n\)th-order nonlinear impulsive integro-differential equation of mixed type having the form:
(1) \(u^{(n)}(t)=f(t,u(t), \dots,u^{(n-1)}(t)\), \((Su)(t))\), \(t\in J'\);
(2) \(\Delta u^{(i)} |_{t=t_k}= I_{ik} (u (t_k), u'(t_k), \dots,u^{(n-1)}(t_k))\) \((i=0,1,\dots,n-1\;k=1,2,\dots)\);
(3) \(u^{(i)} (0)=0\), \((i=0,1, \dots,n-2)\), \(u^{(n-1)}(\infty) =\beta u^{(n-1)}(0)\).
The problem (1)–(3) is investigated in a real Banach space \(E\) with a normal cone \(P\) generating a partial order in \(E\). Moreover, \(J=\mathbb{R}_+\), \(0<t_1<t_2< \cdots< t_k,\dots\), \(t_k\to\infty\), \(J'=J\setminus \{t_1,t_2,\dots\}\), \(T\) denotes the linear Volterra integral operator while \(S\) is the Fredholm linear operator on \(J\). Apart from this \(\Delta u^{(i)} |_{t=t_k}\) denotes the jump of \(u^{(i)} (t)\) at \(t=t_k\). The assumptions imposed in theorems are too involved to the quoted here.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45M20 Positive solutions of integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
45G10 Other nonlinear integral equations
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References:

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