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Positive solutions for impulsive equations of third order in Banach space. (English) Zbl 1210.34080

Summary: The purpose of this paper is to establish the existence of positive solutions for the following third-order three-point boundary value problem in a Banach space \(E\)
\[ -x'''(t)=\lambda f_1(t,x(t),y(t)),\quad t\in[0,1]\setminus [t_1,t_2,\dots,t_m], \]
\[ -y'''(t)=\mu f_2(t,x(t),y(t)),\quad t\in[0,1]\setminus [t_1,t_2,\dots,t_m], \]
\[ \Delta x''(t_k)=-I_{1,k}(x(t_k)),\quad \Delta y''(t_k)=-I_{2,k}(y(t_k)),\quad k=1,2,\dots,m, \]
\[ x(0)=x'(0)=\theta,\quad x'(1)-\alpha x'(\eta)=\theta,\quad y(0)=y'(0)=\theta,\quad y'(1)-\alpha y'(\eta)=\theta, \]
where \(f_i\in C([0,1]\times P\times P,P)\), \(I_{i,k}\in C(P,P)\), \(i=1,2\), \(k=1,2,\dots,m\). \(\Delta x''(t_k)=x''(t^+_k)-x''(t^-_k)\), \(\Delta y''(t_k)=y''(t^+_k)-y''(t_k^-)\), \(\mu>0\), \(\lambda>0\). \(\theta\) is the zero element of \(E\).
Using a fixed-point theorem, we prove multiple and single positive solutions. An example is given to illustrate the main results.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34B37 Boundary value problems with impulses for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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