Existence of positive solutions for second order $m$-point boundary value problems

Ruyun Ma¹

¹ Department of Mathematics Northwest Normal University Lanzhou 730070, Gansu, People's Republic of China

Annales Polonici Mathematici, Tome 79 (2002) no. 3, pp. 265-276.

Résumé

Let $\alpha, \beta, \gamma, \delta \geq 0$ and $\varrho:=\gamma\beta+\alpha\gamma+\alpha\delta>0$. Let $\psi(t)=\beta+\alpha t$, $\phi(t)=\gamma+\delta -\gamma t$, $t\in [0,1]$. We study the existence of positive solutions for the $m$-point boundary value problem $$\cases{ u' ' + h(t) f(u)=0, \quad\ 0 t 1 ,\cr \alpha u(0)- \beta u'(0)=\sum^{m-2}_{i=1}a_i u(\xi_i),\cr \gamma u(1)+\delta u'(1)=\sum^{m-2}_{i=1}b_i u(\xi_i),\cr } $$ where $\xi_i\in (0,1)$, $a_i, b_i\in (0,\infty)$ (for $i\in \{1,\ldots, m-2\}$) are given constants satisfying $\varrho -\sum^{m-2}_{i=1} a_i \phi(\xi_i)>0$, $\varrho -\sum^{m-2}_{i=1} b_i \psi(\xi_i)>0$ and $${\mit\Delta}:= \left| \matrix{ -\sum^{m-2}_{i=1}a_i\psi(\xi_i) \varrho -\sum^{m-2}_{i=1}a_i\phi(\xi_i)\cr \varrho -\sum^{m-2}_{i=1}b_i\psi(\xi_i) -\sum^{m-2}_{i=1}b_i\phi(\xi_i)\cr} \right| 0. $$ We show the existence of positive solutions if $f$ is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and Wang for two-point BVPs and a result established by the author for three-point BVPs.

Détail

DOI : 10.4064/ap79-3-4

Keywords: alpha beta gamma delta geq varrho gamma beta alpha gamma alpha delta psi beta alpha phi gamma delta gamma study existence positive solutions m point boundary value problem cases quad alpha beta sum m gamma delta sum m where infty ldots m given constants satisfying varrho sum m phi varrho sum m psi mit delta matrix sum m psi varrho sum m phi varrho sum m psi sum m phi right existence positive solutions either superlinear sublinear simple application fixed point theorem cones result extends result established erbe wang two point bvps result established author three point bvps