Positive solutions to one-dimensional quasilinear impulsive indefinite boundary value problems

Abstract

Consider the one-dimensional quasilinear impulsive boundary value problem involving the p-Laplace operator ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩-(φ p (u)) = λω(t)f (u), 0 < t < 1,-u| t=t k = μI k (u(t k)), k = 1, 2,. .. , n, u | t=t k = 0, k = 1, 2,. .. , n, u (0) = 0, u(1) = 1 0 g(t)u(t) dt, where λ, μ > 0 are two positive parameters, φ p (s) is the p-Laplace operator, i.e., φ p (s) = |s| p-2 s, p > 1, ω(t) changes sign on [0, 1]. Several new results are obtained for the above quasilinear indefinite problem.

Multiple positive solutions to singular fractional differential equations with integral boundary conditions involving $p-q$-order derivatives

“…The existence of positive solutions for FDEs integral boundary problems (BVPs) with one or two parameters in boundary conditions is obtained by means of fixed point theory [5,14,22,28]. The existence of positive solutions for integer-order differential equations with integral boundary conditions can be found in the literature (see [8,13,15,18,24] for instance). As is well known, semipositone problems arise in bulking of mechanical systems, chemical reactions, astrophysics, combustion, management of natural resources, etc.…”

“…As is well known, semipositone problems arise in bulking of mechanical systems, chemical reactions, astrophysics, combustion, management of natural resources, etc. Details are available in the works [2,6,15,21,23,24,26,29,32,34,35,37,39]. Studying positive solutions for semipositone problems is more difficult than that for positive problems.…”

Positive solutions to second-order singular nonlocal problems: existence and sharp conditions

“…Problems with integral boundary conditions come naturally from thermal conduction problems [29] and hydrodynamic problems [30]. In recent years there has been a lot of investigation of boundary value problems with integral boundary conditions (see for instance [31][32][33][34][35][36][37][38][39][40][41][42][43]). In particular, Boucherif [44] used the fixed point theorem in cones to consider the following problem: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u (t) = f (t, u(t)), 0 < t < 1, u(0)cu (0) = 1 0 g 0 (t)u(t) dt, u(1)du (1) = 1 0 g 1 (t)u(t) dt.…”