Oscillating global continua of positive solutions of second order Neumann problem with a set-valued term

Abstract

In this note, we study the oscillating global continua of the differential inclusion of the form

$$\left\{\begin{array}{c}\hfill -u^{″}+qu\in \lambda F\left(\cdot ,u\right),\hfill \\ \hfill u^{\prime }\left(0\right)=0,u^{\prime }\left(1\right)=0,\hfill \end{array}\right.$$

where F is a "set-valued representation" of a function with jump discontinuities along the line segment [0, 1] × {0}, and λ ∈ [0, ∞) is a parameter. The proof of our main result relies on an approximation procedure.

Mathematics Subject Classification 2000: 34B16; 34B18.

1 Introduction

In recent years, nonsmooth analysis has come to play an important role in functional analysis [1], dynamical systems [2], control theory [3], optimization [4], mechanical systems [5], differential equation [6, 7] etc. Since many mathematical and physical problems may be reduced to ODES or PDES with discontinuous nonlinearities, the existence of multiple solutions for differential inclusion problems has been widely investigated [8–19].

In this article, we are concerned with the following differential inclusion problem which raises from a Budyko-North type energy balance climate models:

$$\left\{\begin{array}{c}-u^{″}\left(x\right)+q\left(x\right)u\left(x\right)\in \lambda F\left(x,u\left(x\right)\right),\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}x\in \left(0,1\right),\hfill \\ u^{\prime }\left(0\right)=0,u^{\prime }\left(1\right)=0\hfill \end{array}\right.$$

(1.1)

(see [20–25] and the references therein). In particular, the set-valued right hand side arise from a jump discontinuity of the albedo at the ice-edge in these models. By filling such a gap, one arrives at the set-valued problem (1.1). As in [25], we are here interested in a considerably simplify version as compared to the situation from climate modeling, e.g. a one-dimensional regular Sturm-Liouville differential operator substitutes for a two-dimensional Laplace-Beltrami operator or a singular Legendre-type operator, and the jump discontinuity is transformed to u = 0 in a way, which resembles only locally the climatological problem.

We are concerned with the set-valued problem (1.1) under the following assumptions

(H1) q∈C([0, 1],(0,+∞));

(H2) f⁺∈ C ([0, 1] × [0,+∞), (0,+∞)), $\underset{x\in \left[0,1\right]}{\mathsf{\text{inf}}}{f}^{+}\left(x,0\right)>0,\underset{s\to +\infty }{\mathsf{\text{lim}}}\frac{f^{+}\left(x,s\right)}{s}=b\left(x\right)\in C\left(\left[0,1\right],\left(0,\infty \right)\right)$.

Let the set-valued function F in (1.1) is given by

$$F\left(x,y\right)=\left\{\begin{array}{c}\left\{f^{+}\left(x,y\right)\right\},x\in \left[0,1\right],y>0,\hfill \\ \left[0,f^{+}\left(x,0\right)\right],x\in \left[0,1\right].\hfill \end{array}\right.$$

Notice that if f⁺(x, 0) ≡ 0, x ∈ [0, 1], then the differential inclusion problem (1.1) reduces to the BVP of differential equation

$$\left\{\begin{array}{c}-u^{″}\left(x\right)+q\left(x\right)u\left(x\right)=\lambda f^{+}\left(x,u\left(x\right)\right),x\in \left(0,1\right),\hfill \\ u^{\prime }\left(0\right)=0,u^{\prime }\left(1\right)=0.\hfill \end{array}\right.$$

(1.2)

In the last 20 years, the positive solutions of (1.2) have been studied by several authors, see Jiang and Liu [26], Chu et al. [27] and Sun et al. [28].

The purpose of this article is to investigate the oscillating global continua of positive solutions of the differential inclusion problem (1.1). The proof of our main result relies on an approximation procedure. The rest of the article is organized as follows. In Section 2, we state some notations and prove some preliminary results. In Section 3, we state and prove our main result. In Section 4, an example is given to illustrate the application of our main result.

2 Notations and preliminaries

Recall Kuratowski's notion of lower and upper limits of sequence of sets.

Definition 2.1. [29] Let X be a metric space and {Z _l }_l∈ℕ be a sequence of subsets of X. The set

$$\underset{l\to \infty }{\mathsf{\text{lim sup}}}{Z}_l:=\left\{x\in X:\underset{l\to \infty }{\mathsf{\text{lim inf}}}\mathsf{\text{dist}}\left(x,Z_l\right)=0\right\}$$

is called the upper limit of the sequence {Z _l }, whereas

$$\underset{l\to \infty }{\mathsf{\text{lim inf}}}{Z}_l:=\left\{x\in X:\underset{l\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{dist}}\left(x,Z_l\right)=0\right\}$$

is called the lower limit of the sequence {Z _l }.

Definition 2.2. [29] A component of a set M is meant a maximal connected subset of M.

Lemma 2.1. [29] Suppose that Y is a compact metric space, A and B are non-intersecting closed subsets of Y, and no component of Y intersects both A and B. Then there exist two disjoint compact subsets Y _A and Y _B , such that Y = Y _A ∪Y _B , A ⊂ Y _A , B ⊂ Y _B .

Using the above Whyburn Lemma, Ma and An [30] proved the following

Lemma 2.2. [30, Lemma 2.1] Let Z be a Banach space and let {A _n } be a family of closed connected subsets of Z. Assume that

(i) there exist z _n ∈ A _n , n = 1, 2, ..., and z* ∈ Z, such that z _n → z*;

(ii) r _n = sup {∥x∥ | x ∈ A _n } = ∞;

(iii) for every R > 0, $\left({\bigcup }_{n=1}^{\infty }{A}_n\right)\cap B_R$ is a relatively compact set of Z, where B _R = {x ∈ Z | ∥x∥ ≤ R}. Then there exists an unbounded component in $\underset{l\to \infty }{\mathsf{\text{lim sup}}}{A}_l$ and $z^{*}\in \mathcal{C}$.

Remark 2.1. The limiting processes for sets go back at least to the work of Kuratowski [31]. Lemma 2.2 is a slight generalization of the following well-know result due to Whyburn [29]:

Proposition 2.1. (Whyburn [29, p. 12]) Let Z be a Banach space and {A _n } be a family of closed connected subsets of Z. Let $\underset{l\to \infty }{\mathsf{\text{lim inf}}}{A}_l\ne \mathrm{\varnothing }$ and ∪_l∈ℕ A _l is relatively compact. Then $\underset{l\to \infty }{\mathsf{\text{lim sup}}}{A}_l$ is nonempty, compact and connected.

Next, we introduce the result of global solution behavior of the bifurcation branches of the equation

$$x=\mu \left(Lx+Nx\right),\mu \in R,x\in X,$$

(2.1)

to wit the following lemma.

Lemma 2.3. [32] (Dancer (1974)) Assume that

(C1) The operators L, N: X → X are compact on the Banach space X over R. Furthermore, L is linear and ∥Nx∥/∥x∥ → 0 as ∥x∥ → 0;

(C2) The real number μ ₀ is a characteristic number of L of odd algebraic multiplicity;

(C1+) The real Banach space X has an order cone K with X = K-K, i.e., every x ∈ X can be represented as x = x₁ - x₂, where x₁, x₂ ∈ K. Furthermore, L + N is positive, i.e., L + N maps K into K;

(C2+) The spectral radius r(L) of L is positive. We set μ₀ = (r(L))^-1.

Then (μ₀, 0) is a bifurcation point of equation (2.1) and

$$S_{+}:=\overline{\left\{\left(\mu ,x\right)\in R\times X:\left(\mu ,x\right)isasolutionof\left(2.1\right)with\mu >0,x>0\right\}}$$

contains an unbounded solution component $C_1^{+}$ which passes through (μ₀, 0).

If additionally

(C3+) The linear operator L is strongly positive, then $\left(\mu ,x\right)\in C_1^{+}$ and μ ≠ μ₀ always implies x > 0 and μ > 0.

Remark 2.2. This result is often called the nonlinear Krein-Rutman theorem. It will play an important role in the proof of our main result.

Let φ and ψ be the unique solution of the problems

$$\left\{\begin{array}{c}-u^{″}\left(x\right)+q\left(x\right)u\left(x\right)=0,x\in \left(0,1\right),\hfill \\ u^{\prime }\left(0\right)=0,u\left(0\right)=1\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{c}-u^{″}\left(x\right)+q\left(x\right)u\left(x\right)=0,x\in \left(0,1\right),\hfill \\ u^{\prime }\left(1\right)=0,u\left(1\right)=1\hfill \end{array}\right.$$

respectively. Then it is easy to check φ(·) is nondecreasing on (0,1), ψ(·) is nonincreasing on (0,1), and the Green's function G(x, s) of

$$\left\{\begin{array}{c}-u^{″}\left(x\right)+q\left(x\right)u\left(x\right)=0,x\in \left(0,1\right),\hfill \\ u^{\prime }\left(0\right)=0,u^{\prime }\left(1\right)=0\hfill \end{array}\right.$$

is explicitly given by

$$G\left(x,s\right)=-\frac{1}{{\psi }^{\prime }\left(0\right)}\left\{\begin{array}{cc}\psi \left(x\right)\phi \left(s\right),\hfill & 0\le s\le x\le 1,\hfill \\ \phi \left(x\right)\psi \left(s\right),\hfill & 0\le x\le s\le 1.\hfill \end{array}\right.$$

(2.2)

Moreover, we have that

$$\begin{array}{cc}0<G\left(x,s\right)\le G\left(s,s\right),\hfill & \left(x,s\right)\in \left[0,1\right]\times \left[0,1\right];\hfill \\ \sigma G\left(s,s\right)\le G\left(x,s\right),\hfill & \left(x,s\right)\in \left[0,1\right]\times \left[0,1\right]\hfill \end{array}$$

(2.3)

with $\sigma :=\mathsf{\text{min}}\left\{\frac{1}{\psi \left(0\right)},\frac{1}{\phi \left(1\right)}\right\}.$.

3 The main result

Let Σ be the closure of the set of positive solutions of (1.1) in [0, ∞) × C¹[0, 1], and ℕ* := {1, 2,..., N}. The main result of this article is the following theorem.

Theorem 3.1. Assume that (H1)-(H2) hold. If

(H3) there is an increasing sequence of positive numbers ${\left\{{\xi }_j\right\}}_1^N$ and a small enough constant δ such that ξ₁ < σ(ξ₂ - δ) and

$$\begin{array}{c}\Phi \left({\xi }_{2j-1}\right)<\frac{1}{2}{\left(\underset{0}{\overset{1}{\int }}G\left(s,s\right)ds\right)}^{-1}\left({\xi }_{2j-1}-\delta \right),j\in {\mathbb{N}}^{*};\\ \Psi \left({\xi }_{2j}\right)>2{\left(\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)ds\right)}^{-1}\left({\xi }_{2j}+\delta \right),j\in {\mathbb{N}}^{*},\end{array}$$

where

$$\begin{array}{c}\Phi \left(l\right):=\mathsf{\text{max}}\left\{f^{+}\left(t,c\right):0\le t\le 1,0\le c\le l+\delta \right\},\\ \Psi \left(l\right):=\mathsf{\text{min}}\left\{f^{+}\left(t,c\right):0\le t\le 1,\sigma \left(l-\delta \right)\le c\le l+\delta \right\},\end{array}$$

then there exits an unbounded component $C_1^{+}$ in Σ with $\left(0,0\right)\in C_1^{+}$. Moreover,

(i) $\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = ξ_{2j- 1} for some j ∈ ℕ* implies that λ ≥ 2;

(ii) $\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = ξ_2j for some j ∈ ℕ* implies that $\lambda \le \frac{1}{2}$.

Actually, such continua $C_1^{+}$ can be obtained as upper limits in the sense of Kura-towski of sequence of solution continua from associated continuous problems. To this end one sets

$$d_g:=\mathsf{\text{inf}}\left\{f^{+}\left(x,0\right):x\in \left[0,1\right]\right\},$$

(3.1)

fixes l₀ ∈ ℕ such that $\frac{d_g}{l_0}<{\xi }_1$, and selects an approximation sequence {f _l } ⊂ C ([0, 1] × ℝ, ℝ) (l > l₀) of F satisfying:

(A1) f _l (x, y) = ly for x ∈ [0, 1] and $y\in \left[0,\frac{d_g}{2l}\right]$;

(A2) $\frac{d_g}{2}\le f_l\left(x,y\right)\le f^{+}\left(x,y\right)$ for x ∈ [0, 1] and $y\in \left[\frac{d_g}{2l},\frac{d_g}{l}\right]$;

(A3) f _l (x,y) = f⁺(x, y) for x ∈ [0, 1] and $y\ge \frac{d_g}{l}$;

(A4) {f _l (x, y)}_l∈ℕ is nondecreasing in l for (x, y) ∈ [0, 1] × (0,∞).

Next, we show that the continuous problem

$$\left\{\begin{array}{c}-v^{″}\left(x\right)+q\left(x\right)v\left(x\right)=\lambda f_l\left(x,v\left(x\right)\right),x\in \left(0,1\right),\hfill \\ v^{\prime }\left(0\right)=0,v^{\prime }\left(1\right)=0\hfill \end{array}\right.$$

(3.2l)

has an unbounded closed subsets $C_{1,l}^{+}$ of the positive solutions set of (3.2 _l ) with

• (a) $\left(\frac{{\lambda }_1}{l},0\right)$ is the bifurcation point contained in $C_{1,l}^{+}$;

• (b) If $\left(\mu ,\vartheta \right)\in C_{1,l}^{+}$ and ϑ ≢ 0, then ϑ is positive on (0,1).

It is easy to see that (3.2 _l ) equivalent to

$$v\left(x\right)=\lambda \underset{0}{\overset{1}{\int }}G\left(x,s\right)f_l\left(s,v\left(s\right)\right)\mathsf{\text{d}}s.$$

(3.3)

Let

$$\begin{array}{c}\left(Lv\right)\left(x\right):=l\underset{0}{\overset{1}{\int }}G\left(x,s\right)v\left(s\right)\mathsf{\text{d}}s,\hfill \\ \left(Nv\right)\left(x\right):=\underset{0}{\overset{1}{\int }}G\left(x,s\right)\left(f_l\left(s,v\left(s\right)\right)-lv\left(s\right)\right)\mathsf{\text{d}}s,v\in C\left[0,1\right].\hfill \end{array}$$

Then according to (3.3), (3.2 _l ) can be written as the following operator equation

$$v=\lambda \left(Lv+Nv\right).$$

Clearly, the operators L, N : C[0, 1] → C[0, 1] are compact on the Banach space

C[0, 1]. Furthermore, L is linear and thanks to (2.3)(A1) that

$$\begin{array}{c}\frac{{∥Nv∥}_{\infty }}{{∥v∥}_{\infty }}={∥\underset{0}{\overset{1}{\int }}G\left(x,s\right)\frac{f_l\left(s,v\left(s\right)\right)-lv\left(s\right)}{{∥v∥}_{\infty }}\mathsf{\text{d}}s∥}_{\infty }\hfill \\ \le \underset{0}{\overset{1}{\int }}G\left(s,s\right)\frac{\left|f_l\left(s,v\left(s\right)\right)-lv\left(s\right)\right|}{{∥v∥}_{\infty }}\mathsf{\text{d}}s\to 0,\mathsf{\text{as}}{∥v∥}_{\infty }\to 0,\hfill \end{array}$$

which implies that the condition (C1) of Lemma 2.3 is satisfied.

Denote the principal eigenvalue of

$$\left\{\begin{array}{c}-{\omega }^{″}\left(x\right)+q\left(x\right)\omega \left(x\right)=\lambda \omega \left(x\right),x\in \left(0,1\right),\hfill \\ {\omega }^{\prime }\left(0\right)=0,{\omega }^{\prime }\left(1\right)=0,\hfill \end{array}\right.$$

(3.4)

by λ₁, then we know that λ₁ > 0 (see [33]). Since (3.4) is equivalent to operator equation

$$\omega =\frac{\lambda }{l}L\omega ,$$

we have that ${\left(r\left(L\right)\right)}^{-1}=\frac{{\lambda }_1}{l}$. Therefore, the conditions (C2)(C2+) of Lemma 2.3 are satisfied.

Let the cone K in C[0, 1] is given by

$$K=\left\{u\in C\left[0,1\right]\left|u\left(x\right)\ge 0,0\le x\le 1\right.\right\}.$$

It is easy to see thanks to (A1)-(A4) and (2.3) that the (C1+)(C3+) conditions of Lemma 2.3 are satisfied.

According to Lemma 2.3, we obtain that $\left(\frac{{\lambda }_1}{l},0\right)$ is a bifurcation point of the positive solutions set of (3.2 _l ) for every l ∈ {l₀ + 1, l₀ + 2, ...} =: ℕ₀, and for each l ∈ ℕ₀ there exits an unbounded closed subsets $C_{1,l}^{+}$ of the positive solutions set of (3.2 _l ) with (a) and (b).

Combining the above with the fact

$$\underset{l\to \infty }{\mathsf{\text{lim}}}\left(\frac{{\lambda }_1}{l},0\right)=\left(0,0\right)$$

and utilizing Lemma 2.2, it concludes that there exits an unbounded component $C_1^{+}$ with

$$\left(0,0\right)\in C_1^{+}$$

(3.5)

and

$$C_1^{+}\subset \underset{l\to \infty }{\mathsf{\text{lim sup}}}{C}_{1,l}^{+}.$$

(3.6)

Denote the cone P in C[0, 1] by

$$P=\left\{\left.u\in C\left[0,1\right]\right|\underset{0\le x\le 1}{\mathsf{\text{min}}}u\left(x\right)\ge \sigma {∥u∥}_{\infty }\right\}.$$

Define an operator T_λ : P → C[0, 1] by

$$\left(T_{\lambda }u\right)\left(x\right)=\lambda \underset{0}{\overset{1}{\int }}G\left(x,s\right)f_l\left(s,u\left(s\right)\right)\mathsf{\text{d}}s,x\in \left[0,1\right].$$

It is easy to get the following lemma.

Lemma 3.1. Assume that (H1), (H2) and (A1)-(A4) hold. Then T _λ : P → P is completely continuous.

Lemma 3.2. Assume that (H1), (H2) and (A1)-(A4) hold. If 0 ≤ u(x) ≤ r, r > 0, for x ∈ [0, 1], then

$${∥T_{\lambda }u∥}_{\infty }\le \lambda M_r\underset{0}{\overset{1}{\int }}G\left(s,s\right)ds,$$

where $M_r=\underset{0\le x\le 1,0\le s\le r}{\mathsf{\text{max}}}\left\{f_l\left(x,s\right)\right\}$.

Proof. Since f _l (x, u(x)) ≤ M _r for x ∈ [0, 1], it follows from (2.3) that

$$\begin{array}{ll}\hfill {∥T_{\lambda }u∥}_{\infty }& ={∥\lambda \underset{0}{\overset{1}{\int }}G\left(x,s\right)f_l\left(s,u\left(s\right)\right)\mathsf{\text{d}}s∥}_{\infty }\\ \le \lambda \underset{0}{\overset{1}{\int }}G\left(s,s\right)f_l\left(s,u\left(s\right)\right)\mathsf{\text{d}}s\\ \le \lambda M_r\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s.\end{array}$$

Lemma 3.3. Assume that (H1), (H2) and (A1)-(A4) hold. If σ(r - δ) ≤ u(x) ≤ r + δ, r > δ, for x ∈ [0, 1], then

$${∥T_{\lambda }u∥}_{\infty }\ge \lambda m_r\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)ds,$$

where $m_r=\underset{0\le x\le 1,\sigma \left(r-\delta \right)\le s\le r+\delta }{\mathsf{\text{min}}}\left\{f_l\left(x,s\right)\right\}$.

Proof. Since f _l (x, u(x)) ≥ m _r for x ∈ [0, 1], it follows that

$$\begin{array}{c}{∥T_{\lambda }u∥}_{\infty }\ge \lambda \underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)f_l\left(s,u\left(s\right)\right)\mathsf{\text{d}}s\\ \ge \lambda m_r\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s.\end{array}$$

Lemma 3.4. Assume that (H1), (H2), (H3) and (A1)-(A4) hold. then

(i) $\left(\lambda ,u\right)\in C_{1,l}^{+}$ with ∥u∥_∞ ∈ (ξ_2j-1 - δ,ξ_2j-1 + δ) for some j ∈ ℕ* implies that λ > 2;

(ii) $\left(\lambda ,u\right)\in C_{1,l}^{+}$, with ∥u∥_∞ ∈ ( ξ_2j - δ,ξ_2j + δ) for some j ∈ ℕ* implies that $\lambda <\frac{1}{2}$.

Proof. (i) Assume that $\left(\lambda ,u\right)\in C_{1,l}^{+}$ with ∥u∥_∞ ∈ (ξ_2j-1 - δ, ξ_2j-1 + δ) for some j ∈ ℕ*, then u = T_λ u and

$$0\le u\left(x\right)\le {\xi }_{2j-1}+\delta \mathsf{\text{for}}x\in \left[0,1\right].$$

By Lemma 3.2 and (H3), it follows that

$$\begin{array}{ll}\hfill {∥u∥}_{\infty }& ={∥T_{\lambda }u∥}_{\infty }\\ \le \lambda M_{{\xi }_{2j-1}+\delta }\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s\\ =\lambda \underset{0\le x\le 1,0\le s\le {\xi }_{2j-1}+\delta }{\mathsf{\text{max}}}\left\{f_l\left(x,s\right)\right\}\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s\\ \le \lambda \Phi \left({\xi }_{2j-1}\right)\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s\\ <\lambda \left[\frac{1}{2}{\left(\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s\right)}^{-1}\left({\xi }_{2j-1}-\delta \right)\right]\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s.\\ =\frac{1}{2}\lambda \left({\xi }_{2j-1}-\delta \right).\end{array}$$

Thus λ > 2.

• (ii) Assume that $\left(\lambda ,u\right)\in C_{1,l}^{+}$ with ∥u∥_∞ ∈ (ξ_2j - δ, ξ_2j + δ) for some j ∈ ℕ*, then u = T_λu and

  $$\sigma \left({\xi }_{2j}-\delta \right)\le u\left(x\right)\le {\xi }_{2j}+\delta \mathsf{\text{for}}x\in \left[0,1\right].$$

By Lemma 3.3 and the assumption (H3), it follows that

$$\begin{array}{ll}\hfill {∥u∥}_{\infty }& ={∥T_{\lambda }u∥}_{\infty }\\ \ge \lambda m_{{\xi }_{2j}}\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s\\ =\lambda \underset{0\le x\le 1,\sigma \left({\xi }_{2j}-\delta \right)\le s\le {\xi }_{2j}+\delta }{\mathsf{\text{min}}}\left\{f_l\left(x,s\right)\right\}\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s\\ =\lambda \Psi \left({\xi }_{2j}\right)\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s\\ >\lambda \left[2{\left(\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s\right)}^{-1}\left({\xi }_{2j}+\delta \right)\right]\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s.\\ =2\lambda \left({\xi }_{2j}+\delta \right).\end{array}$$

Thus $\lambda <\frac{1}{2}$.

Lemma 3.5. If $\left(\lambda ,u\right)\in C_1^{+}$, then (λ, u) is a solution of (1.1) and u ∈ W^2,∞(0, 1).

Proof. Let $\left(\lambda ,u\right)\in C_1^{+}$. By the definition of $C_1^{+}$ there exists a sequence {l _k } ∈ ℕ₀ strictly increasing, and $\left({\lambda }_{l_k},v_{l_k}\right)\in \left[0,\infty \right)\times C^1\left[0,1\right]$ with $\left({\lambda }_{l_k},v_{l_k}\right)\in C_{1,l_k}^{+}$ for k ∈ ℕ and

$$\left({\lambda }_{l_k},v_{l_k}\right)\to \left(\lambda ,u\right).$$

Since $\left\{f_{l_k}\left(\cdot ,v_{l_k}\left(\cdot \right)\right)\right\}$ is uniformly bounded, i.e.

$${∥f_{l_k}∥}_{L^2}\le M,$$

(3.7)

we can assume after passing to a subsequence, if necessary, that it converges weekly in L²(0, 1) to some ϕ. We claim that ϕ(x) ∈ F(x, u(x)) a.e. on (0, 1).

Let x₀ ∈ (0, 1) with u(x₀) > 0. Then there exist ρ > 0 and τ ∈ (0, min{x₀, 1-x₀}) with u(x) > ρ for all x ∈ (x₀ - τ, x₀ + τ), hence there is a k₀ ∈ ℕ with $v_{l_k}\left(x\right)>\frac{\rho }{2}$ for all k > k₀ and x ∈ (x₀ - τ, x₀ + τ). Choose k₁ > k₀ with $\frac{d_g}{l_{k_1}}<\frac{\rho }{2}$. Then $f_{l_k}\left(x,v_{l_k}\left(x\right)\right)=f^{+}\left(x,v_{l_k}\left(x\right)\right)$ for all k ≥ k₁ and x ∈ (x₀ - τ, x₀ + τ), which yields ϕ(x) = f⁺(x, u(x)) for x ∈ (x₀ - τ, x₀ + τ) a.e.

Next, if u ≡ 0, let K: = {x ∈ (0, 1) : ϕ(x) > f⁺(x, 0)}. We claim that meas(K) = 0. Suppose that meas(K) > 0. Then ε := ∫ _K [ϕ(x) - f⁺(x, 0)] dx > 0, and one finds η ∈ (0, ∞) with $\mathsf{\text{meas}}\left(K\right)\left|f^{+}\left(x,y\right)-f^{+}\left(x,0\right)\right|\le \frac{\epsilon }{2}$ for x ∈ [0, 1] and y ∈ [0, η]. Choosing k₂ ∈ ℕ with ${∥v_{l_k}-u∥}_{\infty }<\eta$ for k ≥ k₂. One obtains for k ≥ k₂:

$$\begin{array}{c}\underset{K}{\int }\left[\varphi \left(x\right)-f_{l_k}\left(x,v_{l_k}\left(x\right)\right)\right]\mathsf{\text{d}}x\hfill \\ =\underset{K}{\int }\left[\varphi \left(x\right)-f^{+}\left(x,0\right)\right]\mathsf{\text{d}}x+\underset{K}{\int }\left[f^{+}\left(x,0\right)-f_{l_k}\left(x,v_{l_k}\left(x\right)\right)\right]\mathsf{\text{d}}x\hfill \\ =\epsilon +\underset{K}{\int }\left[f^{+}\left(x,0\right)-f^{+}\left(x,v_{l_k}\left(x\right)\right)\right]\mathsf{\text{d}}x\hfill \\ \ge \frac{\epsilon }{2},\hfill \end{array}$$

which contradicts $f_{l_k}\left(\cdot ,v_{l_k}\left(\cdot \right)\right)\rightharpoonup \varphi$. Thus, meas(K) = 0.

Now, let A be the closed linear operator in L²(0, 1) defined by

$$\mathsf{\text{dom}}\left(A\right):=\left\{\phi \in W^{2,2}\left[0,1\right]:{\phi }^{\prime }\left(0\right)=0={\phi }^{\prime }\left(1\right)\right\}$$

and Aφ := -φ" + qφ. Clearly,

$$f_{l_k}\left(x,v_{l_k}\left(x\right)\right)\rightharpoonup \varphi \left(x\right),$$

(3.8)

hence $v_{l_k}\to u$ and the fact that A is weakly closed yields

$$Au=\lambda \varphi ,$$

(3.9)

i.e.

$$Au\in \lambda F\left(\cdot ,u\left(\cdot \right)\right)\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}$$

Finally, we show that u ∈ W^2,∞(0,1). In fact, from (3.9) we have

$$u^{″}\left(x\right)=q\left(x\right)u\left(x\right)-\lambda \varphi \left(x\right).$$

(3.10)

According to (H1) and the boundedness of u we have

$$qu\in L^{\infty }\left(0,1\right).$$

(3.11)

We claim that ϕ ∈ L^∞(0,1). Suppose on the contrary that there exists a set E ⊂ [0, 1], meas(E) > 0 such that |ϕ| is unbounded on E. Without loss of generality, we assume that

$$\left|\varphi \left(s\right)\right|>\frac{M{∥w∥}_{L^2}+1-{\int }_{\left[0,1\right]\backslash E}\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x}{{\int }_E\left|w\left(x\right)\right|\mathsf{\text{d}}x},s\in E,$$

(3.12)

where M is given by (3.7) and w ∈ L²(0,1). On the one hand, for k larger enough from (3.7), (3.8) and (H2) we have

$$\underset{0}{\overset{1}{\int }}\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x\le \underset{0}{\overset{1}{\int }}\left|f_{l_k}\left(x,v_{l_k}\left(x\right)\right)w\left(x\right)\right|\mathsf{\text{d}}x+1\le M{∥w∥}_{L^2}+1.$$

(3.13)

On the other hand, from (3.12) we have

$$\begin{array}{c}\underset{0}{\overset{1}{\int }}\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x\hfill \\ =\underset{E}{\int }\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x+\underset{\left[0,1\right]\backslash E}{\int }\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x\hfill \\ >\frac{M{∥v∥}_{L^2}+1-{\int }_{\left[0,1\right]\backslash E}\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x}{{\int }_E\left|w\left(x\right)\right|\mathsf{\text{d}}x}\underset{E}{\int }\left|w\left(x\right)\right|\mathsf{\text{d}}x+\underset{\left[0,1\right]\backslash E}{\int }\left|\varphi \left(x\right)\right|\left|w\left(x\right)\right|\mathsf{\text{d}}x\hfill \\ =M{∥w∥}_{L^2}+1,\hfill \end{array}$$

which contradicts (3.13). Thus,

$$\varphi \in L^{\infty }\left(0,1\right).$$

(3.14)

Therefore, from (3.10), (3.11) and (3.14) we obtain u ∈ W^2,∞(0,1).

Now we are in the position to prove Theorem 3.1.

Proof of Theorem 3.1.

Assume that $\left(\lambda ,u\right)\in C_1^{+}$. We divide the proof into two cases.

Case l. If ∥u∥_∞ = ξ_{2j- 1} for some j ∈ ℕ*, then λ ≥ 2.

Since $\left(\lambda ,u\right)\in C_1^{+}$, there exists a sequence $\left({\lambda }_{k_i},z_{k_i}\right)\in C_{1,k_i}^{+}$, such that

$$\underset{i\to \infty }{\mathsf{\text{lim}}}{\lambda }_{k_i}=\lambda ,\underset{i\to \infty }{\mathsf{\text{lim}}}{z}_{k_i}=u.$$

Hence, for δ > 0 there exists i₀ ∈ ℕ, such that

$${∥z_{k_i}-u∥}_{\infty }<\delta ,i>i_0,$$

i.e.

$${\xi }_{2j-1}-\delta <{∥z_{k_i}∥}_{\infty }<{\xi }_{2j-1}+\delta ,i>i_0.$$

By using Lemma 3.4, we obtain that

$${\lambda }_{k_i}>2,i>i_0.$$

Hence, we get

$$\lambda =\underset{i\to \infty }{\mathsf{\text{lim}}}{\lambda }_{k_i}\ge 2.$$

Case 2. If ∥u∥_∞ = ξ_2j for some j ∈ ℕ*, then $\lambda \le \frac{1}{2}$.

Since $\left(\lambda ,u\right)\in C_1^{+}$, there exists a sequence $\left({\lambda }_{k_i},z_{k_i}\right)\in C_{1,k_i}^{+}$, such that

$$\underset{i\to \infty }{\mathsf{\text{lim}}}{\lambda }_{k_i}=\lambda ,\underset{i\to \infty }{\mathsf{\text{lim}}}{z}_{k_i}=u.$$

Hence, for δ > 0 there exists i₀ ∈ ℕ, such that

$${∥z_{k_i}-u∥}_{\infty }<\delta ,i>i_0,$$

i.e.

$${\xi }_{2j}-\delta <{∥z_{k_i}∥}_{\infty }<{\xi }_{2j}+\delta ,i>i_0.$$

By using lemma 3.4, we obtain that

$${\lambda }_{k_i}<\frac{1}{2},i>i_0.$$

Hence, we get

$$\lambda =\underset{i\to \infty }{\mathsf{\text{lim}}}{\lambda }_{k_i}\le \frac{1}{2}.$$

Corollary 3.1. Assume that (H1)-(H3) hold. Then

(i) for each $\lambda \in \left(0,\frac{1}{2}\right)$, (1.1) has at least one positive solution: $u_0\in C_1^{+}$;

(ii) for each $\lambda \in \left[\frac{1}{2},2\right]$, (1.1) has N positive solutions:

$$u_k,k=1,2,\dots ,N,$$

which satisfy that $u_k\in C_1^{+},k=1,2,\dots ,N$.

Proof. According to Theorem 3.1, the boundary value problem (1.1) has an unbounded component $C_1^{+}$ in Σ with $\left(0,0\right)\in C_1^{+}$. Moreover,

$\left(ĩ\right)\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = ξ_2j-1 for some j ∈ ℕ* implies that λ ≥ 2;

$\left(\widetilde{ii}\right)\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = ξ_2j for some j ∈ ℕ* implies that $\lambda \le \frac{1}{2}$.

From the facts $\left(0,0\right)\in C_1^{+}$, and $\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = ξ₁ implies that λ ≥ 2 and the connectivity of $C_1^{+}$, we obtain

$$C_1^{+}\cap \lambda \times C^1\left[0,1\right]\ne \mathrm{\varnothing },\forall \lambda \in \left(0,\frac{1}{2}\right),$$

which implies for each $\lambda \in \left(0,\frac{1}{2}\right)$, (1.1) has at least one positive solution: $u_0\in C_1^{+}$.

Let

$$C_1^{+,k}:=\left\{\left(\lambda ,u\right)\in C_1^{+}|{\xi }_{k-1}\le ∥u∥<{\xi }_k\right\},k=1,2,\dots ,N,$$

where ξ₀ = 0, ξ _k (k = 1, 2,..., N) is given by (H3). Then according to $\left(\tilde{i}\right)\left(\widetilde{ii}\right)$ and the connectivity of $C_1^{+}$, we obtain

$$C_1^{+,k}\cap \lambda \times C^1\left[0,1\right]\ne \mathrm{\varnothing },\forall \lambda \in \left(\frac{1}{2},2\right),k=1,2,\dots ,N,$$

which implies for each $\lambda \in \left(\frac{1}{2},2\right)$, (1.1) has N positive solutions:

$$u_k,k=1,2,\dots ,N,$$

and $u_k\in C_1^{+,k}\subset C_1^{+},k=1,2,\dots ,N$.

4 Example

In this section, an example is given to illustrate the application of our main result (Theorem 3.1). Consider second order Neumann differential inclusion problem

$$\left\{\begin{array}{c}-u^{″}\left(x\right)+u\left(x\right)\in \lambda F\left(u\left(x\right)\right),\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}x\in \left(0,1\right),\hfill \\ u^{\prime }\left(0\right)=0,u^{\prime }\left(1\right)=0,\hfill \end{array}\right.$$

(4.1)

where the set-valued function F in (4.1) is given by

$$F\left(y\right)=\left\{\begin{array}{c}\frac{49}{2}y-\frac{195}{2},y\ge 4,\hfill \\ \frac{1}{16}y+\frac{1}{4},0<y<4,\hfill \\ \left[0,\frac{1}{4}\right],y=0.\hfill \end{array}\right.$$

Obviously, (H 1), (H 2) conditions of Theorem 3.1 are satisfied. Moreover, Green's function of the associated linear problem

$$\left\{\begin{array}{c}-u^{″}+u=0,\hfill \\ u^{\prime }\left(0\right)=0,u^{\prime }\left(1\right)=0,\hfill \end{array}\right.$$

can be explicitly expressed by

$$G\left(x,s\right)=\frac{1}{2\left(e-e^{-1}\right)}\left\{\begin{array}{cc}\left(e^{x-1}+e^{1-x}\right)\left(e^s+e^{-s}\right),\hfill & 0\le s\le x\le 1,\hfill \\ \left(e^x+e^{-x}\right)\left(e^{s-1}+e^{1-s}\right),\hfill & 0\le x\le s\le 1.\hfill \end{array}\right.$$

By calculation we can get ${\int }_0^{1}G\left(s,s\right)\mathsf{\text{d}}s=\frac{e}{e-e^{-1}},{\int }_0^{1}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s=1$ and $\sigma =\frac{2}{e+e^{-1}}$.

Let ξ₁ = 3, ξ₂ = 11, δ = 1, then we can check that ξ₁ = 3 < 5 < σ(ξ₂ - δ), and

$$\begin{array}{c}\Phi \left({\xi }_1\right)=\frac{1}{2}<\frac{3}{4}<1-e^{-2}=\frac{1}{2}{\left(\underset{0}{\overset{1}{\int }}G\left(s,s\right)\mathsf{\text{d}}s\right)}^{-1}\left({\xi }_1-\delta \right),\\ \Psi \left({\xi }_2\right)\ge 25>24=2{\left(\underset{0}{\overset{1}{\int }}G\left(\frac{1}{2},s\right)\mathsf{\text{d}}s\right)}^{-1}\left({\xi }_2+\delta \right).\end{array}$$

So that (H 3) condition of Theorem 3.1 is satisfied. Therefore, according to Theorem 3.1 the differential inclusion problem (4.1) has an unbounded component $C_1^{+}$ in Σ with $\left(0,0\right)\in C_1^{+}$. Moreover,

• (i) $\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = 3 implies that λ ≥ 2;

• (ii) $\left(\lambda ,u\right)\in C_1^{+}$ with ∥u∥_∞ = 11 implies that $\lambda \le \frac{1}{2}$.