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Zbl 1221.34071
Jankowski, Tadeusz
Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3775-3785 (2011). ISSN 0362-546X

From the introduction: For $J= [0,1]$, let $0= t_0< t_1<\cdots< t_m< t_{m+1}= 1$. Put $J'= (0,1)\setminus\{t_1, t_2,\dots, t_m\}$. Put $\bbfR_+= [0,\infty)$ and $J_k= (t_k, t_{k+1}]$, $k= 0,1,\dots, m-1$, $J_m= (t_m, t_{m+1})$. Let us consider second-order impulsive differential equations of the type $$x''(t)+ \alpha(t) f(t,x(t))= 0,\quad t\in J',$$ $$\Delta x'(t_k)= Q_k(x(t_k)),\quad k= 1,2,\dots, m,$$ $$x(0)= 0,\quad x(1)=\lambda[x],$$ where as usual $\Delta x'(t_k)= x'(t^+_k)- x'(t^-_k)$; $x'(t^+_k)$ and $x'(t^-_k)$ denote the right and left limits of $x'$ at $t_k$, respectively. Here $\lambda[u]$ denotes a linear functional of $C(J)$ given by $$\lambda[u]= \int^1_0 u(t)\,d\Lambda(t)$$ 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 2000:: *34B37 Boundary value problems with impulses
34B10 Multipoint boundary value problems
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: impulsive second order differential equations; boundary conditions including a Stieltjes integral; positive solutions

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