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

2008, Computers & Mathematics with Applications

https://doi.org/10.1016/J.CAMWA.2008.01.033

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Abstract

The singular boundary value problem u + q(t)g(t, u, u) = 0, t ∈ (0, 1), η ∈ (0, 1), γ ∈ (0, 1] u(0) = 0, u(1) = γ u(η), is studied in this paper.The singularity may appear at t = 0 and the function g may be superlinear at u = ∞ and change sign. The existence of solutions is obtained via an upper and lower solutions method.

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...

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR nth-ORDER THREE-POINT NONHOMOGENEOUS BOUNDARY VALUE PROBLEM

2017

This paper is concerned with four-point boundary value problem of second order singular differential equations on whole line. Green's function G(t, s) for problem −(ρ(t)x (t)) = 0, lim t→−∞ x(t) − kx(ξ) = 0, lim t→+∞ x(t) − lx(η) = 0 is obtained. We proved that G(t, s) ≥ 0 under some assumptions which actually generalize a corresponding result in Liu (Appl Math Comput 155(1):179-203, 2004) (see Remark 2.1). Sufficient conditions to guarantee the existence and non-existence of positive solutions are established. We give two examples and remarks to illustrate main theorems.

Multiple Solutions for Nonlinear Doubly Singular Three-Point Boundary Value Problems with Derivative Dependence

International Journal of Differential Equations, 2012

We study the existence of multiple nonnegative solutions for the doubly singular threepoint boundary value problem with derivative dependent data function − p t y t q t f t, y t , p t y t , 0 < t < 1, y 0 0, y 1 α 1 y η. Here, p ∈ C 0, 1 ∩ C 1 0, 1 with p t > 0 on 0, 1 and q t is allowed to be discontinuous at t 0. The fixed point theory in a cone is applied to achieve new and more general results for existence of multiple nonnegative solutions of the problem. The results are illustrated through examples.

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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.

A Sharper Condition for the Solvability of a Three-Point Second Order Boundary Value Problem

Journal of Mathematical Analysis and Applications, 1997

This problem was studied earlier by Gupta, Ntouyas, and Tsamatos when ␣ F 1 and when ␣) 1 with ␣-1. In the general case this problem was studied by Gupta as a multi-point boundary value problem. In this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case.

Two-Point Boundary Value Problems for a Class of Second-Order Ordinary Differential Equations

International Journal of Mathematics and Mathematical Sciences, 2012

We study the general semilinear second-order ODE u g t, u, u 0 under different twopoint boundary conditions. Using the method of upper and lower solutions, we obtain an existence result. Moreover, under a growth condition on g, we prove that the set of solutions of u g t, u, u 0 is homeomorphic to the two-dimensional real space.

Upper and lower solutions for higher order boundary value problems

This paper is concerned with the study of the higher order boundary value problem 8 > > < > > : u (n) (t) = f (t; u(t); u 0 (t); :::; u (n 1) (t)); n 2; u (i) (0) = 0; i = 0; :::; n 3; a u (n 2) (0) b u (n 1) (0) = A; c u (n 2) (1) + d u (n 1) (1) = B; with f : [0; 1] R n ! R a continuous function, a; c; A; B 2 R and b; d 0 such that a 2 + b > 0 and c 2 + d > 0: It is assumed that lower and upper solutions exist and that f satis…es Nagumotype conditions. An existence result is obtained

On some third order nonlinear boundary value problems

Journal of Mathematical Analysis and Applications, 2008

We prove an Ambrosetti-Prodi type result for the third order fully nonlinear equationu'''(t)+f(t,u(t),u'(t),u''(t))=sp(t) with and continuous functions, , under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori estimate on u'' is obtained. An existence and location result will be proved, by degree theory, for such that there exist lower and upper solutions. The location part can be used to prove the existence of positive solutions if a non-negative lower solution is considered. The existence, nonexistence and multiplicity of solutions will be discussed as s varies.