Solvability of a nonlinear conjugate eigenvalue problem (Q2730911)

The authors consider the nonlinear conjugate eigenvalue problem NEWLINE\[NEWLINE(-1)^{n-k}y^{(n)}(t)=\lambda a(t)f(y), \quad 0\leq t\leq 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEy^{(i)}(0)=0,\quad 0\leq i\leq k-1,\qquad y^{(j)}(1)=0, \quad 0\leq j\leq n-k-1,NEWLINE\]NEWLINE where \(k\) and \(n\) are fixed with \(1\leq k\leq n-1.\) The purpose of the paper is to state QTR{it}{optimal} intervals of eigenvalues \(\lambda \), for which there exists at least one positive solution to the problem mentioned above. The main results require weak conditions on \(a(t)\) as well as on \(f(u).\) In fact, in contrast with several known results [see for instance, \textit{P. W. Eloe} and \textit{J. Henderson}, Positive solutions and nonlinear \(k,n-k\) conjugate eigenvalue problems, Differ. Equ. Dyn. Syst. 6, 309-317 (1998); Nonlinear Anal., Theory Methods Appl. 28, No. 10, 1669-1680 (1997; Zbl 0871.34015); J. Differ. Equations 133, No. 1, 136-151 (1997; Zbl 0870.34031), \textit{L. H. Erbe} and \textit{H. Wang}, Proc. Am. Math. Soc. 120, No. 3, 743-748 (1994; Zbl 0802.34018); \textit{R. Ma} and \textit{H. Wang}, Appl. Anal. 59, No. 1-4, 225-231 (1995; Zbl 0841.34019)], \(a(t)\) is allowed to vanish on some subintervals of [0, 1] and the function \(f(u)\) may be sublinear or superlinear but is not restricted to these two cases only, in contrast with \textit{L. H. Erbe} and \textit{H. Wang} [loc. cit.)] for example. The paper ends with three well commented examples and two interesting remarks.