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Symmetric solutions of a multi-point boundary value problem. (English) Zbl 1085.34011

The author considers the second-order nonlinear multipoint boundary value problem \[ -u''(t)=a(t)f(t,u(t),| u'(t)| ), \quad t\in (0,1),\tag{1} \]
\[ u(0)=\sum_{i=1}^{n} \mu_{i}u(\xi_i),\tag{2} \]
\[ u(t)=u(1-t), \quad t\in [0,1],\tag{3} \]
with \(0<\xi_1<\xi_2<\cdots<\xi_n \leq \frac{1}{2},\) \(\mu_i>0, i=1,2,\cdots,n\) and \(\sum_{i=1}^{n} \mu_{i}<1, n\geq 2.\) It is assumed that the inhomogeneous term of (1) satisfies the following conditions: \(f\) is continuous and \(a\) belongs to \(L^p\) for some \(1\leq p \leq \infty.\) Moreover, \(a(t)\geq 0\) and, in fact, \(a(t)\geq m\) a.e. on \([0,1]\) for some positive constant \(m\). Adding other suitable growth conditions on \(f\), the author proves the existence of at least three symmetric positive solutions of the boundary value problem. In order to reach that, he transforms this question into an equivalent problem of showing the existence of fixed-points for the completely continuous operator \[ Su=\int_{0}^{1}G(t,s)a(s)f(s,u(s),| u'(t)| )ds \] whose kernel \(G(t,s)\) is precisely the Green function of the equation \(u''(t)=-g(t),\) \(g\in C[0,1],\) subject to conditions \((2)-(3),\) and then he applies a fixed-point result due to R.I. Avery and A.C. Peterson [Comput. Math. Appl. 42, 313-322 (2001; Zbl 1005.47051)].

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems

Citations:

Zbl 1005.47051
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References:

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