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Existence of positive solutions for singular second-order m-point boundary value problems. (English) Zbl 1080.34517

The singular second-order m-point boundary value problem

(p(t)x ' (t)) ' +q(t)x(t)+h(t)f(x(t))=0,0<t<1,x(0)=0,x(1)= j=1 m-2 a j x(ξ j ),(1)

is considered, with pC 1 [0,1], p>0, qC 0 [0,1], q0, hC 0 (0,1)L 1 [0,1], h0, fC 0 (0,), f0, 0<ξ 1 <<ξ m-2 <1 and a j >0. The existence of positive solutions of problem (1) is obtained by means of fixed-point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operators. The existence theorems generalize the results by R. Ma and H. Wang [J. Math. Anal. Appl. 279, 216–227 (2003; Zbl 1028.34014)] and R. Ma [Acta Math. Sin. 46, No. 4, 785–794 (2003; Zbl 1056.34015)].

34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
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