Positive solutions for 2nth-order singular sub-linear m-point boundary value problems

Abstract

This paper investigates the existence of positive solutions for 2nth-order singular sub-linear m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²ⁿ⁻² [a, b] ∩ C²ⁿ(a, b) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²ⁿ⁻²[0, 1] as well as C²ⁿ⁻¹[0, 1] positive solutions. Our nonlinearity f(t, x₁, x₂, … , x_n) may be singular at x_i = 0, i = 1, 2, … , n, t = 0 and/or t = 1.

Introduction

The singular ordinary differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer and so on. The theory of singular boundary value problems has become an important area of investigation in recent years (see [1], [2], [3], [4], [5], [6] and the references therein).

The existence of solutions of multi-point boundary value problems has been studied by many authors using nonlinear alternative of Leray–Schauder, Coincidence degree theory and fixed point theorem in cones (see [7], [8], [9], [10], [11] and references therein).

Very recently, the existence of positive solutions and multiple positive solutions of singular second order multi-point boundary value problem have been studied by papers [12], [13], [14] using the fixed point index and approximate process; Zhang and Wang in [15] gave some sufficient conditions for the existence results of a class of singular nonlinear second order three-point boundary value problems by the upper and lower solution method and the monotone iterative technique. But they only considered the case: nonlinear function f(t, x) cannot be singular at x = 0.

On the boundary value problems of 2nth-order ordinary differential equation

$$\left\{\begin{array}{c}(-1{)}^n{x}^{(2n)}(t)=f(t\text{,}x(t)\text{,}-x^{″}(t)\text{,}\dots \text{,}(-1{)}^i{x}^{(2i)}(t)\text{,}\dots \text{,}(-1{)}^{n-1}{x}^{(2n-2)}(t))\text{,}t\in (0\text{,}1)\text{,}\hfill \\ x^{(2i)}(0)=x^{(2i)}(1)=0\text{,}i=0\text{,}1\text{,}2\text{,}\dots \text{,}n-1.\hfill \end{array}\right.$$

For the special case n = 2, function f ∈ C([0, 1] × R × R, R) in (*), i.e. f is continuous, problem (*) is nonsingular, the existence and uniqueness of positive solutions of (*) have been studied by papers [16], [17], [18], [19].

A sufficient condition for the existence of solutions of the singular problem (*) was given by O’Regan in [20] with a topological transversal theorem.

For the general case n ⩾ 2: the function f ∈ C([0, 1] × Rⁿ, R) in (*), i.e. f is continuous, problem (*) is nonsingular, the existence of one or more solutions of (*) a have been studied by papers [21], [22], [23]. The function $f\in C((0\text{,}1)\times R^{{+}^n}\text{,}{R}^{+})$ in (1.1), i.e. f is singular at t = 0 or t = 1, but f is continuous at x_i, i = 1, 2, … , n, paper [24] has investigated super-linear singular boundary value problem (*) and obtained some necessary and sufficient conditions for the existence of C²ⁿ⁻²[0, 1] as well as C²ⁿ⁻¹[0, 1] positive solutions by means of the fixed point theorems on cones.

In this paper, we shall consider the existence of positive solutions for 2nth-order singular m-point boundary value problems of the following differential equation:

$$(-1{)}^n{x}^{(2n)}(t)=f(t\text{,}x(t)\text{,}-x^{‴}(t)\text{,}\dots \text{,}(-1{)}^i{x}^{(2i)}(t)\text{,}\dots \text{,}(-1{)}^{n-1}{x}^{(2n-2)}(t))\text{,}t\in (0\text{,}1)\text{,}$$

$$x^{(2i)}(0)=0\text{,}{x}^{(2i)}(1)=\sum _{j=1}^{m-2}{k}_{\mathit{ij}}{x}^{(2i)}({\eta }_j)\text{,}i=0\text{,}1\text{,}2\text{,}\dots \text{,}n-1\text{,}$$

where 0 < k_ij < 1, j = 1, 2, … , m − 2, ${\sum }_{j=1}^{m-2}{k}_{\mathit{ij}}<1\text{,}i=1\text{,}2\text{,}\dots \text{,}n-1$, 0 < η₁ < η₂ < ⋯ < η_m−2 < 1, are constants, m ⩾ 3 and n > 1 is an integer, and f satisfies the following hypothesis.

(H): f ∈ C((0, 1) × (0, ∞)ⁿ, [0, ∞)), and there exist constants λ_i, μ_i, (−∞ < λ_i ⩽ 0 ⩽ μ_i, i = 1, 2, … , n, μ_n < 1, ${\sum }_{i=1}^n{\mu }_i<1$) such that for t ∈ (0, 1), x_i ∈ (0, ∞)

$$c^{{\mu }_i}f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{x}_i\text{,}\dots \text{,}{x}_n)⩽f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{\mathit{cx}}_i\text{,}\dots \text{,}{x}_n)⩽c^{{\lambda }_i}f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{x}_i\text{,}\dots \text{,}{x}_n)\text{if}0<c⩽1\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{.}$$

Remark 1

Inequality (1.3) implies

$$c^{{\lambda }_i}f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{x}_i\text{,}\dots \text{,}{x}_n)⩽f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{\mathit{cx}}_i\text{,}\dots \text{,}{x}_n)⩽c^{{\mu }_i}f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{x}_i\text{,}\dots \text{,}{x}_n)\text{if}c⩾1\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{.}$$

Typical functions that satisfy the above sub-linear hypothesis are those taking the form $f(t\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n)={\sum }_{i=1}^m{p}_i(t)x_1^{{\ell }_{i1}}{x}_2^{{\ell }_{i2}}\cdots x_n^{{\ell }_{\mathit{in}}}$; here p_j(t) ∈ C(0, 1), p_j(t) > 0 on (0, 1), ℓ_jk ∈ R, ℓ_jn < 1, j = 1, 2, … , m, k = 1, 2, … , n, ${\sum }_{i=1}^m{\sum }_{k=1}^n\max \{0\text{,}{\ell }_{\mathit{ik}}\}<1$.

By singularity we mean that the function f(t, x₁, x₂, … , x_i, … , x_n) in (1.1) is allowed to be unbounded at x_i, i = 1, 2, … , n, t = 0 and/or t = 1. A function x(t) ∈ C²ⁿ⁻² [0, 1] ∩ C²ⁿ(0, 1) is called a C²ⁿ⁻²[0, 1] (positive) solution of (1.1), (1.2) if it satisfies (1.1), (1.2) ((−1)ⁱx⁽²ⁱ⁾(t) > 0, i = 0, 1, 2, … , n − 1 for t ∈ (0, 1)). A C²ⁿ⁻²[0, 1] (positive) solution of (1.1), (1.2) is called a C²ⁿ⁻¹[0, 1] (positive) solution if x⁽²ⁿ⁻¹⁾(0⁺) and x⁽²ⁿ⁻¹⁾(1⁻) both exist ((−1)ⁱx⁽²ⁱ⁾(t) > 0, i = 0, 1, 2, … , n − 1 for t ∈ (0, 1)).

For the general case n ⩾ 2: the function f ∈ C([0, 1] × Rⁿ, R) in (1.1), i.e. f is continuous, problem (1.1), (1.2) is nonsingular, the existence of one or more solutions of (1.1), (1.2) have been studied by papers [25], [26] with the fixed point theorem.

In this paper, we shall study the existence of positive solutions for 2nth-order singular sub-linear m-point boundary value problem (1.1), (1.2). Firstly, we establish a comparison theorem, then we define a partial ordering in C²ⁿ⁻²[a, b] ∩ C²ⁿ(a, b) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²ⁿ⁻² [0, 1] as well as C²ⁿ⁻¹[0, 1] positive solutions. Our nonlinearity f(t, x₁, x₂, … , x_n) may be singular at x_i = 0, i = 1, 2, … , n, t = 0 and/or t = 1.