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Multiple positive solutions for some four-point boundary value problems with \(p\)-Laplacian. (English) Zbl 1157.34015
The paper discusses the four-point boundary value problem
\[ \begin{cases}(\varphi_p(u'))'+a(t)f(u(t),u'(t))=0,\quad & 0<t<1\\ \alpha\varphi_p(u(0))-\beta\varphi_p(u'(\xi))=0,\quad \gamma\varphi_p(u(1))+\delta\varphi_p(u'(\eta))=0 \end{cases} \]
where \(\varphi_p(s)=| s|^{p-1}s\), \(p>1\), is the \(p\)-Laplacian, \(\alpha,\gamma>0\), \(\beta, \delta\geq0\) and \(0<\xi<\eta<1.\) The functions \(f\) and \(a\) are positive and continuous, \(a\) may exhibit a singularity at \(t=0\) or \(t=1\) but is integrably bounded, that is, \(0<\int_0^1a(t)dt<\infty.\) Under additional assumptions involving local growth conditions on the nonlinearity \(f\), the author proves existence of at least three concave, positive solutions (at least two of them are nontrivial). When \(f\) does not depend on the first derivative, the Legget-Williams fixed point theorem is used [R. W. Leggett, L. R. Williams, Indiana Univ. Math. J. 28, 673–688 (1979; Zbl 0421.47033)]. In case, it does depend, the author appeals to the Avery-Peterson fixed point theorem [R. I. Avery, A. C. Peterson, Comput. Math. Appl. 42, 313–322 (2001; Zbl 1005.47051)]. Note the same problem is also considered in [H. Su, Z. Wei and F. Xu [Appl. Math. Comput. 181, No. 2, 826–836 (2006; Zbl 1111.34020)] of the paper, where the existence of one or two positive solutions is proved using fixed point theory. In [H. Su, Z. Wei, B. Wang, Nonlinear Anal., Theory Methods Appl. 66, 2204–2217 (2007; Zbl 1126.34017)], the proof of the existence of a solution involves a discussion on the limits \(\lim_{u\to0}\frac{f(u)}{u^{p-1}}\) and \(\lim_{u\to+\infty}\frac{f(u)}{u^{p-1}}\).
Finally, notice, for further readings, that the existence of positive symmetric solutions is also considered in the following papers: [D. Ji, W. Ge, Y. Yang, Appl. Math. Comput. 189, No. 2, 1087–1098 (2007; Zbl 1126.34322)] and [B. Sun, Ch. Miao, W. Ge, Appl. Math. Comput. 201, No. 1–2, 481–488 (2008; Zbl 1160.34019)].

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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[2] Bai, C.Z.; Fang, J.X., Existence of multiple positive solutions for nonlinear m-point boundary-value problems, Appl. math. comput., 140, 297-305, (2003) · Zbl 1033.34019
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