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Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. (English) Zbl 1221.34071

From the introduction: For $J=\left[0,1\right]$, let $0={t}_{0}<{t}_{1}<\cdots <{t}_{m}<{t}_{m+1}=1$. Put ${J}^{\text{'}}=\left(0,1\right)\setminus \left\{{t}_{1},{t}_{2},\cdots ,{t}_{m}\right\}$. Put ${ℝ}_{+}=\left[0,\infty \right)$ and ${J}_{k}=\left({t}_{k},{t}_{k+1}\right]$, $k=0,1,\cdots ,m-1$, ${J}_{m}=\left({t}_{m},{t}_{m+1}\right)$.

Let us consider second-order impulsive differential equations of the type

${x}^{\text{'}\text{'}}\left(t\right)+\alpha \left(t\right)f\left(t,x\left(t\right)\right)=0,\phantom{\rule{1.em}{0ex}}t\in {J}^{\text{'}},$
${\Delta }{x}^{\text{'}}\left({t}_{k}\right)={Q}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{1.em}{0ex}}k=1,2,\cdots ,m,$
$x\left(0\right)=0,\phantom{\rule{1.em}{0ex}}x\left(1\right)=\lambda \left[x\right],$

where as usual ${\Delta }{x}^{\text{'}}\left({t}_{k}\right)={x}^{\text{'}}\left({t}_{k}^{+}\right)-{x}^{\text{'}}\left({t}_{k}^{-}\right)$; ${x}^{\text{'}}\left({t}_{k}^{+}\right)$ and ${x}^{\text{'}}\left({t}_{k}^{-}\right)$ denote the right and left limits of ${x}^{\text{'}}$ at ${t}_{k}$, respectively. Here $\lambda \left[u\right]$ denotes a linear functional of $C\left(J\right)$ given by

$\lambda \left[u\right]={\int }_{0}^{1}u\left(t\right)\phantom{\rule{0.166667em}{0ex}}d{\Lambda }\left(t\right)$

involving a Stieltjes integral with a suitable function ${\Lambda }$ of bounded variation.

The existence of at least three positive solutions to impulsive second-order differential equations as above is investigated. Sufficient conditions which guarantee the existence of positive solutions are obtained, by using the Avery-Peterson theorem. An example is added to illustrate the results.

MSC:
 34B37 Boundary value problems for ODE with impulses 34B10 Nonlocal and multipoint boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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