We prove the existence of positive solutions of the Sturm-Liouville boundary value problem −(r(t)φ(u′))′ = λg(t)f(t, u), t ∈ (0, 1), au(0)− bφ−1(r(0))u′(0) = 0, cu(1) + dφ−1(r(1))u′(1) = 0, where φ(u′) = |u′|p−2u′, p > 1, f : (0, 1) × (0,∞) → R satisfies a p-sublinear condition and is allowed to be singular at u = 0 with semipositone structure. Our results extend previously known results in the...

We establish the results on the existence of three positive solutions to Sturm-Liouville boundary value problems of the singular fractional differential equation with the nonlinearity depending on D 0+u ⎪⎨ ⎪⎩ D0+u(t)+ f (t,u(t),D μ 0+ u(t)) = 0, t ∈ (0,1),1 < α < 2, a limt→0 I2−α 0+ u(t)−b limt→0 [ I2−α 0+ u(t) ]′ = ∫ 1 0 g(s,u(s),D μ 0+u(s))ds, c D 0+u(1)+du(1) = ∫ 1 0 h(s,u(s),D μ 0+u(s))ds. ...

wherem≥ 3 and 1≤ q≤m – 2. We note that the nonlinear term F involves derivatives. This makes the problem challenging, and such cases are seldom investigated in the literature. In this paper we develop a new technique to obtain existence criteria for one or multiple positive solutions of the boundary value problem. Several examples with known positive solutions are presented to dwell upon the us...

We consider the existence of positive solutions for a class of discrete second-order four-point boundary value problem with p-Laplacian. Using the well known Krasnosel’skii’s fixed point theorem, some new existence criteria for positive solutions of the boundary value problem are presented.

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression