Upper and lower solutions method and a second order three-point singular boundary value problem (original) (raw)

Upper and lower solutions for a second-order three-point singular boundary-value problem

Electronic Journal of Differential Equations, 2009

We study the singular boundary-value problem u′′ + q(t)g(t, u) = 0, t ∈ (0, 1), η ∈ (0, 1), γ > 0 u(0) = 0, u(1) = γu(η) . The singularity may appear at t = 0 and the function g may be superlinear at infinity and may change sign. The existence of solutions is obtained via an upper and lower solutions method.

On a singular multi-point third-order boundary value problem on the half-line

Mathematica Bohemica, 2019

We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel’skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity f = f(t, x, y) which satisfies upper and lower-homogeneity conditions in the space variables x, y may be also singular at time t = 0. Two examples of applications are included to illustrate the existence theorems.

Triple Solutions for Second-Order Three-Point Boundary Value Problems

Journal of Mathematical Analysis and Applications, 2002

We establish the existence of at least three positive solutions to the second-order three-point boundary value problem, u + f t u = 0 u 0 = 0 αu η = u 1 , where η 0 < η < 1 0 < α < 1/η, and f 0 1 × 0 ∞ → 0 ∞ is continuous. We accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works.

The upper and lower solution method for nonlinear fourth-order boundary value problem

Journal of Physics: Conference Series, 2011

This paper is concerned with the following fourth-order three-point boundary value problem u (4) (t) + f (t, u(t), u ′ (t), u ′′ (t)) = 0, t ∈ [0, 1], u ′ (0) = 0, u(0) = λu(1), u ′′ (0) = 0, u ′′ (1) = αu ′′ (η), where 0 < η < 1, 0 ≤ α < 1, 0 ≤ λ < 1 and f ∈ C([0, 1] × R 3 , R). Some existence results are established for this problem via upper and lower solution method and fixed point.

A Study on Solutions for a Class of Higher-Order System of Singular Boundary Value Problem

Symmetry

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation 1rβrβ1rβ(rβΘ′)′′′=12rβK11μ′Θ′2+2μΘ′Θ″+K12μ′φ′2+2μφ′φ″+λ1G1(r),1rβrβ1rβ(rβφ′)′′′=12rβK21μ′Θ′2+2μΘ′Θ″+K22μ′φ′2+2μφ′φ″+λ2G2(r), where λ1≥0 and λ2≥0 are two parameters, μ=pr2β−2,p∈R+, G1,G2∈L1[0,1] such that M1*≥G1(r)≥M1>0,M2*≥G2(r)≥M2>0 and K12>0, K11≥0, and K21>0, K22≥0 are constants that are connected by the relation (K12+K22)≥(K11+K21) and β>1. To study the governing equation, we consider three different types of homogeneous boundary conditions. We use the transformation t=r1+β1+β to deduce the second-order singular boundary value problem. Also, for β=p=G1(r)=G2(r)=1, it admits dual solutions. We show the existence of at least one solution in continuous space. We derive a sign of solutions. Furthermore, we compute the approximate bound of the parameters to point out the region of nonex...