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Existence of positive solutions for fourth-order \(m\)-point boundary value problems with a one-dimensional \(p\)-Laplacian operator. (English) Zbl 1175.34026

Summary: This paper discusses the existence of positive solutions for fourth-order \(m\)-point boundary value problems with a one-dimensional \(p\)-Laplacian operator
\[ \begin{cases} (\phi_p(u''(t)(t)))''-g(t)f(u(t))=0,\quad & t\in (0,1),\\ au(0)-bu'(0)=\sum^{m-2}_{i=1}a_iu(\xi_i),\\ cu(1)+du'(1)=\sum^{m-2}_{i=1}b_iu(\xi_i),\end{cases} \]
where \(\phi_p(s)=|s|^{p-2}s\), \(p>1\), \(f\) is a lower semi-continuous function. By using Krasnoselskii’s fixed point theorems in a cone, the existence of one positive solution and multiple positive solutions is established.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations, 23, 803-810 (1987) · Zbl 0668.34025
[2] Gupta, C. P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168, 540-551 (1992) · Zbl 0763.34009
[3] Moshinsky, M., Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. Soc. Mat. Mexicana, 7, 1-25 (1950)
[4] Timoshenko, S., Theory of Elastic Stability (1961), McGraw-Hill: McGraw-Hill New York
[5] Ma, R. Y., Nonlocal Problems for the Nonlinear Ordinary Differential Equation (2004), Science Press: Science Press Beijing, (in Chinese)
[6] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cone (1988), Academic Press: Academic Press Boston
[7] Kaufmann, E. R.; Kosmatov, N., A multiplicity result for boundary value problem with infinitely many singularities, J. Math. Anal., 269, 444-453 (2002) · Zbl 1011.34012
[8] Feng, H. Y.; Ge, W. G.; Jiang, M., Multiple positive solutions for \(m\)-point boundary value problems with a one-dimensional \(p\)-Laplacian, Nonlinear Anal., 68, 2269-2279 (2008) · Zbl 1138.34005
[9] Su, H.; Wei, Z. L.; Wang, B. H., The existence of positive solutions for a nonlinear four-point singular boundary value problem with a \(p\)-Laplacian operator, Nonlinear Anal., 66, 2204-2217 (2007) · Zbl 1126.34017
[10] Wang, Y. Y.; Ge, W. G., Existence of multiple positive solutions for multi-point boundary value problems with a one-dimensional \(p\)-Laplacian, Nonlinear Anal., 67, 476-485 (2007) · Zbl 1130.34009
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