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

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

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.

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:

(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,+∞)), .

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

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

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.

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∈ℕbe a sequence of subsets of X. The set

is called the upper limit of the sequence {Z_l}, whereas

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_Aand 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, is a relatively compact set of Z, where B_R= {x ∈ Z | ∥x∥ ≤ R}. Then there exists an unbounded component in and .

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 and ∪_l∈ℕA_l is relatively compact. Then is nonempty, compact and connected.

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

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 μ₀is 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

contains an unbounded solution component which passes through (μ₀, 0).

If additionally

(C3+) The linear operator L is strongly positive, then and μ ≠ μ₀always 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

and

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

is explicitly given by

Moreover, we have that

with .

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 and a small enough constant δ such that ξ₁ < σ(ξ₂- δ) and

where

then there exits an unbounded component in Σ with . Moreover,

(i) with ∥u∥_∞ = ξ_2j-1for some j ∈ ℕ* implies that λ ≥ 2;

(ii) with ∥u∥_∞ = ξ_2jfor some j ∈ ℕ* implies that .

Actually, such continua 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

fixes l₀ ∈ ℕ such that , 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 ;

(A2) for x ∈ [0, 1] and ;

(A3) f_l(x,y) = f⁺(x, y) for x ∈ [0, 1] and ;

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

Next, we show that the continuous problem

has an unbounded closed subsets of the positive solutions set of (3.2_l) with

(a) is the bifurcation point contained in ;

(b) If and ϑ ≢ 0, then ϑ is positive on (0,1).

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

Let

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

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

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

Denote the principal eigenvalue of

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

we have that . Therefore, the conditions (C2)(C2+) of Lemma 2.3 are satisfied.

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

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 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 of the positive solutions set of (3.2_l) with (a) and (b).

Combining the above with the fact

and utilizing Lemma 2.2, it concludes that there exits an unbounded component with

and

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

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

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

where .

Proof. Since f_l(x, u(x)) ≤ M_r for x ∈ [0, 1], it follows from (2.3) that

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

where .

Proof. Since f_l(x, u(x)) ≥ m_r for x ∈ [0, 1], it follows that

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

(i) with ∥u∥_∞ ∈ (ξ_2j-1 - δ,ξ_2j-1 + δ) for some j ∈ ℕ* implies that λ > 2;

(ii) , with ∥u∥_∞ ∈ ( ξ_2j- δ,ξ_2j + δ) for some j ∈ ℕ* implies that .

Proof. (i) Assume that with ∥u∥_∞ ∈ (ξ_2j-1 - δ, ξ_2j-1 + δ) for some j ∈ ℕ*, then u = T_λu and

By Lemma 3.2 and (H3), it follows that

Thus λ > 2.

(ii) Assume that with ∥u∥_∞ ∈ (ξ_2j - δ, ξ_2j + δ) for some j ∈ ℕ*, then u = T_λu and

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

Thus .

Lemma 3.5. If , then (λ, u) is a solution of (1.1) and u ∈ W^2,∞(0, 1).

Proof. Let . By the definition of there exists a sequence {l_k} ∈ ℕ₀ strictly increasing, and with for k ∈ ℕ and

Since is uniformly bounded, i.e.

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 for all k > k₀ and x ∈ (x₀ - τ, x₀ + τ). Choose k₁ > k₀ with . Then 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 for x ∈ [0, 1] and y ∈ [0, η]. Choosing k₂ ∈ ℕ with for k ≥ k₂. One obtains for k ≥ k₂:

which contradicts . Thus, meas(K) = 0.

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

and Aφ := -φ" + qφ. Clearly,

hence and the fact that A is weakly closed yields

i.e.

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

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

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

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

On the other hand, from (3.12) we have

which contradicts (3.13). Thus,

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 . We divide the proof into two cases.

Case l. If ∥u∥_∞ = ξ_2j-1 for some j ∈ ℕ*, then λ ≥ 2.

Since , there exists a sequence , such that

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

i.e.

By using Lemma 3.4, we obtain that

Hence, we get

Case 2. If ∥u∥_∞ = ξ_2j for some j ∈ ℕ*, then .

Since , there exists a sequence , such that

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

i.e.

By using lemma 3.4, we obtain that

Hence, we get

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

(i) for each , (1.1) has at least one positive solution: ;

(ii) for each , (1.1) has N positive solutions:

which satisfy that .

Proof. According to Theorem 3.1, the boundary value problem (1.1) has an unbounded component in Σ with . Moreover,

with ∥u∥_∞ = ξ_2j-1 for some j ∈ ℕ* implies that λ ≥ 2;

with ∥u∥_∞ = ξ_2j for some j ∈ ℕ* implies that .

From the facts , and with ∥u∥_∞= ξ₁ implies that λ ≥ 2 and the connectivity of , we obtain

which implies for each , (1.1) has at least one positive solution: .

Let

where ξ₀ = 0, ξ_k (k = 1, 2,..., N) is given by (H3). Then according to and the connectivity of , we obtain

which implies for each , (1.1) has N positive solutions:

and .

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

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

Obviously, (H1), (H2) conditions of Theorem 3.1 are satisfied. Moreover, Green's function of the associated linear problem

can be explicitly expressed by

By calculation we can get and .

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

So that (H3) condition of Theorem 3.1 is satisfied. Therefore, according to Theorem 3.1 the differential inclusion problem (4.1) has an unbounded component in Σ with . Moreover,

(i) with ∥u∥_∞ = 3 implies that λ ≥ 2;

(ii) with ∥u∥_∞ = 11 implies that .

The authors declare that they have no competing interests.

The authors express their gratitude to Professors Ma Tian and Ma Ruyun for their guidance and encouragement, also to an anonymous referee for a number of valuable comments and suggestions.

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