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Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. (English) Zbl 1181.34035

Summary: This paper is concerned with the second-order singular Sturm-Liouville integral boundary value problems

-u '' (t)=λh(t)f(t,u(t)),0<t<1,αu(0)=βu ' (0)= 0 1 a(s)u(s)dsγu(0)=δu ' (1)= 0 1 b(s)u(s)ds,

where λ>0, h is allowed to be singular at t=0,1 and f(t,x) may be singular at x=0. By using the fixed point theory in cones, an explicit interval for λ is derived such that for any λ in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and non-singular cases.

MSC:
34B24Sturm-Liouville theory
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
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