Positive solutions for nonlinear eigenvalue problems. (English) Zbl 0876.34023

The authors are concerned with determining values of \(\lambda\) (eigenvalues), for which there exist positive solutions of the boundary value problem \[ (1_\lambda)\quad u''+\lambda a(t)f(u)=0,\;0<t<1, \qquad (2) \quad u(0)= u(1)=0, \] where \(f:[0,\infty)\to [0,\infty)\) is continuous; \(a:[0,1]\to [0,\infty)\) is continuous and does not vanish identically on any subinterval, and \[ f_0= \lim_{x\to 0} (f(x)/x), \qquad f_\infty= \lim_{x\to\infty} (f(x)/x) \] exist. To research the problem \((1_\lambda)\), (2) the Krasnosel’skij methods of solutions of nonlinear operator equations in a space with a cone are applicable.


34B15 Nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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