Positive solutions of nonresonant singular boundary value problem of second order differential equations (Q2731034)

The authors consider the singular boundary value problems for the second-order ordinary differential equation NEWLINE\[NEWLINE-x^{\prime \prime }+\rho p(t)x=f(t,x),\;t\in (0,1),NEWLINE\]NEWLINE subject to the boundary conditions \(ax(0)-bx'(0)=0\), \(cx( 1) +dx'(1)=0\), where \(\rho >0\) is such that the boundary value problem NEWLINE\[NEWLINE-x^{\prime \prime }+\rho p(t)x=0,\;t\in (0,1),\;ax(0)-bx'(0)=0,\;cx( 1) +dx'(1)=0,NEWLINE\]NEWLINE has only the trivial solution, and with \(p(t)\in C(0,1),\) \(p(t)\geq 0,\) \(t\in (0,1),\) \(a\geq 0,\) \(b\geq 0,\) \(c\geq 0,\) \(d\geq 0,\) \(a+b>0,\) \(c+d>0,\) \(ac+ad+bc>0.\) Under some hypothesis the authors give a necessary and sufficient condition for the existence of \(C[0,1]\) positive solutions as well as \(C^{1}[0,1]\) positive solutions by using the method of lower and upper solutions with fixed-point theorems.NEWLINENEWLINENEWLINEIt should be pointed out that several particular results for special data were known (see for instance [\textit{S. D. Taliaferro}, Nonlinear Anal., Theory Methods Appl. 3, 897-904 (1979; Zbl 0421.34021); \textit{Y. Zhang}, J. Math. Anal. Appl. 185, No. 1, 215-222 (1994; Zbl 0823.34030); \textit{Z. Wei}, Positive solutions to singular boundary value problems for negative exponent Emden-Fowler equations (Chinese), Acta Math. Sinica 41, No. 3, 655-662 (1998); and Syst. Sci. Math. Sci. 11, No. 1, 82-88 (1998; Zbl 0902.34020)]) but the present article may be considered more than unifying them. (See Remark 2 on page 135).