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On positive solutions for fourth-order boundary value problem with impulse. (English) Zbl 1169.34318

The author considers the impulsive boundary value problem for the ordinary differential equation of fourth order
\[ \begin{aligned} &y^{(4)} - q(x)y'' = f(x,y),\quad x \in [a,c)\cup(c,b], \\ &\alpha_1y(a) - \beta_1y'(a) = 0,\quad \gamma_1y(b) + \delta_1y'(b) = 0, \\ &y''(c - 0) = d_1y''(c + 0), \quad y'''(c - 0) = d_2y'''(c + 0),\\ &\alpha_2y''(a) - \beta_2y'''(a) = 0, \quad \gamma_2y''(b) + \delta_2y'''(b) = 0, \end{aligned} \]
where \(f\) is continuous on \(([a,c)\cup(c,b])\times {\mathbb R}\), \(q\) is nonnegative and integrable on \([a,b]\). Sufficient conditions for the existence of multiple positive solutions are found. The eigenvalue problem \(y^{(4)} - q(x)y'' = \lambda f(x,y)\) with the same impulsive and boundary conditions is investigated for \(\lambda > 0\), also. In the proofs there is used the Krasnoselskii fixed point theorem.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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