Abstract
In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter λ. The arguments are based upon the fixed point index theory and the upper and lower solutions method.
Highlights
Consider the following second-order singular boundary value problem (BVP) 1 p(t) (p(t)x (t))+ λg(t)f (x(t)) = 0, ax(0) − b lim p(t)x (t) = 0, t→0+cx(1) + d lim p(t)x (t) = 0, t→1− 0 < t < 1,(1.1)λ where a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, ac + bc + ad > 0; λ > 0
Nonlinear singular boundary value problems has become an important area of investigation in previous years; see[1,2,3,4,5, 7,8,9,10,11,12,13,14,15] and references therein
Motivated by the results mentioned above, in this paper we study the existence, multiplicity, and nonexistence of positive solutions for the BVP (1.1) by new technique(different from [3,5,11,12,13,14]) to overcome difficulties arising from the appearances of p(t) and p(t) is singular at t = 0 and t = 1
Summary
In the special case iii)f (t, x) = q(t)g(x), q(t) is singular only at t = 0 and g(x) ≥ ex, the existence of multiple positive solutions for the BVP (1.2) have been studied by Ha and Lee in [3] with the sub-super solutions method. Motivated by the results mentioned above, in this paper we study the existence, multiplicity, and nonexistence of positive solutions for the BVP (1.1) by new technique(different from [3,5,11,12,13,14]) to overcome difficulties arising from the appearances of p(t) and p(t) is singular at t = 0 and t = 1. Fixed point index theorems have been applied to various boundary value problems to show the existence of multiple positive solutions. Lemma 1.2.[16,17,18,19,20] Suppose A : P ∩ Ω → P is a completely continuous operator, and satisfies
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