Abstract
This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument-x′′t-atx′t+btxt=ωtft, xαt, t∈0, 1; x′0=0, x1-∫01htxtdt=0, wherea∈0, 1, 0, +∞, b∈C0, 1, 0, +∞and,ωmay be singular att=0or/and att=1. Furthermore, several new and more general results are obtained for the existence of positive solutions for the above problem by using Krasnosel’skii’s fixed point theorem. We discuss our problems with a delayed argument, which may concern optimization issues of some technical problems. Moreover, the approach to express the integral equation of the above problem is different from earlier approaches. Our results cover a second-order boundary value problem without deviating arguments and are compared with some recent results.
Highlights
Boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3], and so on
This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument −x (t) − a (t) x (t) + b (t) x (t) = ω (t) f (t, x (α (t))), t ∈ (0, 1) ; x (0) = 0, x (1) − ∫01 h (t) x (t) dt = 0, where a ∈ ([0, 1], [0, +∞)), b ∈ C ([0, 1], (0, +∞)) and, ω may be singular at t = 0 or/and at t = 1
Few papers have reported the same problems where the solution is without concavity; for example, see some recent excellent results and applications of the case of ordinary differential equations with deviating arguments to a variety of problems from Jankowski [21,22,23], Jiang and Wei [24], Wang [25], Wang et al [26], and Hu et al [27]
Summary
Boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3], and so on. The initial set reduces to one point t = 0, and we cannot apply the step method To our knowledge, it is the first paper in which positive solution has been investigated for a secondorder singular differential equation with a delayed argument under the case that Lx := −x − ax + bx. Being directly inspired by [5, 12, 20, 21], the authors will prove several new and more general results for the existence of positive solutions for problem (3) by using fixed point theories in a cone Another contribution of this paper is to study the expression and properties of Green’s function associated with problem (3).
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