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

Boundary Value Problems20122012:47

DOI: 10.1186/1687-2770-2012-47

Received: 14 October 2011

Accepted: 23 April 2012

Published: 23 April 2012

Abstract

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

- u + q u λ F ( , u ) , u ( 0 ) = 0 , u ( 1 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equa_HTML.gif

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.

Keywords

climate model differential inclusion eigenvalue positive solutions

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 [819].

In this article, we are concerned with the following differential inclusion problem which raises from a Budyko-North type energy balance climate models:
- u ( x ) + q ( x ) u ( x ) λ F ( x , u ( x ) ) , a .e . x ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ1_HTML.gif
(1.1)

(see [2025] 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) qC([0, 1],(0,+∞));

(H2) f+C ([0, 1] × [0,+∞), (0,+∞)), inf x [ 0 , 1 ] f + ( x , 0 ) > 0 , lim s + f + ( x , s ) s = b ( x ) C ( [ 0 , 1 ] , ( 0 , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq1_HTML.gif.

Let the set-valued function F in (1.1) is given by
F ( x , y ) = { f + ( x , y ) } , x [ 0 , 1 ] , y > 0 , [ 0 , f + ( x , 0 ) ] , x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equb_HTML.gif
Notice that if f + (x, 0) ≡ 0, x ∈ [0, 1], then the differential inclusion problem (1.1) reduces to the BVP of differential equation
- u ( x ) + q ( x ) u ( x ) = λ f + ( x , u ( x ) ) , x ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ2_HTML.gif
(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
lim sup l Z l : = x X : lim inf l dist ( x , Z l ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equc_HTML.gif
is called the upper limit of the sequence {Z l }, whereas
lim inf l Z l : = x X : lim l dist ( x , Z l ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equd_HTML.gif

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 , AY A , BY 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∥ | xA n } = ∞;

(iii) for every R > 0, n = 1 A n B R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq2_HTML.gif is a relatively compact set of Z, where B R = {xZ | ∥x∥ ≤ R}. Then there exists an unbounded component http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq3_HTML.gif in lim sup l A l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq4_HTML.gif and z * C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq5_HTML.gif.

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 lim inf l A l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq6_HTML.gif and ∪l∈ℕ A l is relatively compact. Then lim sup l A l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq4_HTML.gif is nonempty, compact and connected.

Next, we introduce the result of global solution behavior of the bifurcation branches of the equation
x = μ ( L x + N x ) , μ R , x X , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ3_HTML.gif
(2.1)

to wit the following lemma.

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

(C1) The operators L, N: XX are compact on the Banach space X over R. Furthermore, L is linear andNx∥/∥x∥ → 0 asx∥ → 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 xX can be represented as x = x1 - x2, where x1, x2K. 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 + : = { ( μ , x ) R × X : ( μ , x ) i s a s o l u t i o n o f ( 2 . 1 ) w i t h μ > 0 , x > 0 } ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Eque_HTML.gif

contains an unbounded solution component C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq7_HTML.gif which passes through (μ0, 0).

If additionally

(C3+) The linear operator L is strongly positive, then ( μ , x ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq8_HTML.gif and μμ0 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
- u ( x ) + q ( x ) u ( x ) = 0 , x ( 0 , 1 ) , u ( 0 ) = 0 , u ( 0 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equf_HTML.gif
and
- u ( x ) + q ( x ) u ( x ) = 0 , x ( 0 , 1 ) , u ( 1 ) = 0 , u ( 1 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equg_HTML.gif
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
- u ( x ) + q ( x ) u ( x ) = 0 , x ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equh_HTML.gif
is explicitly given by
G ( x , s ) = - 1 ψ ( 0 ) ψ ( x ) φ ( s ) , 0 s x 1 , φ ( x ) ψ ( s ) , 0 x s 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ4_HTML.gif
(2.2)
Moreover, we have that
0 < G ( x , s ) G ( s , s ) , ( x , s ) [ 0 , 1 ] × [ 0 , 1 ] ; σ G ( s , s ) G ( x , s ) , ( x , s ) [ 0 , 1 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ5_HTML.gif
(2.3)

with σ : = min 1 ψ ( 0 ) , 1 φ ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq9_HTML.gif.

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 ξ j 1 N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq10_HTML.gif and a small enough constant δ such that ξ1 < σ(ξ2 - δ) and
Φ ( ξ 2 j - 1 ) < 1 2 0 1 G ( s , s ) d s - 1 ( ξ 2 j - 1 - δ ) , j * ; Ψ ( ξ 2 j ) > 2 0 1 G 1 2 , s d s - 1 ( ξ 2 j + δ ) , j * , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equi_HTML.gif
where
Φ ( l ) : = max { f + ( t , c ) : 0 t 1 , 0 c l + δ } , Ψ ( l ) : = min { f + ( t , c ) : 0 t 1 , σ ( l - δ ) c l + δ } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equj_HTML.gif

then there exits an unbounded component C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq7_HTML.gif in Σ with ( 0 , 0 ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq11_HTML.gif. Moreover,

(i) ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif withu = ξ2j- 1 for some j ∈ ℕ* implies that λ ≥ 2;

(ii) ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif withu = ξ2j for some j ∈ ℕ* implies that λ 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq13_HTML.gif.

Actually, such continua C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq7_HTML.gif 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 : = inf { f + ( x , 0 ) : x [ 0 , 1 ] } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ6_HTML.gif
(3.1)

fixes l0 ∈ ℕ such that d g l 0 < ξ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq14_HTML.gif, 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 0 , d g 2 l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq15_HTML.gif;

(A2) d g 2 f l ( x , y ) f + ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq16_HTML.gif for x ∈ [0, 1] and y d g 2 l , d g l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq17_HTML.gif;

(A3) f l (x,y) = f + (x, y) for x ∈ [0, 1] and y d g l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq18_HTML.gif;

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

Next, we show that the continuous problem
- v ( x ) + q ( x ) v ( x ) = λ f l ( x , v ( x ) ) , x ( 0 , 1 ) , v ( 0 ) = 0 , v ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ7_HTML.gif
(3.2l)
has an unbounded closed subsets C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq19_HTML.gif of the positive solutions set of (3.2 l ) with
  1. (a)

    λ 1 l , 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq20_HTML.gif is the bifurcation point contained in C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq19_HTML.gif;

     
  2. (b)

    If ( μ , ϑ ) C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq21_HTML.gif and ϑ ≢ 0, then ϑ is positive on (0,1).

     
It is easy to see that (3.2 l ) equivalent to
v ( x ) = λ 0 1 G ( x , s ) f l ( s , v ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ8_HTML.gif
(3.3)
Let
( L v ) ( x ) : = l 0 1 G ( x , s ) v ( s ) d s , ( N v ) ( x ) : = 0 1 G ( x , s ) ( f l ( s , v ( s ) ) - l v ( s ) ) d s , v C [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equk_HTML.gif
Then according to (3.3), (3.2 l ) can be written as the following operator equation
v = λ ( L v + N v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equl_HTML.gif

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
N v v = 0 1 G ( x , s ) f l ( s , v ( s ) ) - l v ( s ) v d s 0 1 G ( s , s ) f l ( s , v ( s ) ) - l v ( s ) v d s 0 , as v 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equm_HTML.gif

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

Denote the principal eigenvalue of
- ω ( x ) + q ( x ) ω ( x ) = λ ω ( x ) , x ( 0 , 1 ) , ω ( 0 ) = 0 , ω ( 1 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ9_HTML.gif
(3.4)
by λ1, then we know that λ1 > 0 (see [33]). Since (3.4) is equivalent to operator equation
ω = λ l L ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equn_HTML.gif

we have that ( r ( L ) ) - 1 = λ 1 l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq22_HTML.gif. Therefore, the conditions (C2)(C2+) of Lemma 2.3 are satisfied.

Let the cone K in C[0, 1] is given by
K = u C [ 0 , 1 ] u ( x ) 0 , 0 x 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equo_HTML.gif

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 λ 1 l , 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq20_HTML.gif 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 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq19_HTML.gif of the positive solutions set of (3.2 l ) with (a) and (b).

Combining the above with the fact
lim l λ 1 l , 0 = ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equp_HTML.gif
and utilizing Lemma 2.2, it concludes that there exits an unbounded component C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq7_HTML.gif with
( 0 , 0 ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ10_HTML.gif
(3.5)
and
C 1 + lim sup l C 1 , l + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ11_HTML.gif
(3.6)
Denote the cone P in C[0, 1] by
P = u C [ 0 , 1 ] min 0 x 1 u ( x ) σ u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equq_HTML.gif
Define an operator Tλ : PC[0, 1] by
T λ u ( x ) = λ 0 1 G ( x , s ) f l ( s , u ( s ) ) d s , x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equr_HTML.gif

It is easy to get the following lemma.

Lemma 3.1. Assume that (H1), (H2) and (A1)-(A4) hold. Then T λ : PP 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 λ u λ M r 0 1 G ( s , s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equs_HTML.gif

where M r = max 0 x 1 , 0 s r { f l ( x , s ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq23_HTML.gif.

Proof. Since f l (x, u(x)) ≤ M r for x ∈ [0, 1], it follows from (2.3) that
T λ u = λ 0 1 G ( x , s ) f l ( s , u ( s ) ) d s λ 0 1 G ( s , s ) f l ( s , u ( s ) ) d s λ M r 0 1 G ( s , s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equt_HTML.gif
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 λ u λ m r 0 1 G 1 2 , s d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equu_HTML.gif

where m r = min 0 x 1 , σ ( r - δ ) s r + δ { f l ( x , s ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq24_HTML.gif.

Proof. Since f l (x, u(x)) ≥ m r for x ∈ [0, 1], it follows that
T λ u λ 0 1 G 1 2 , s f l ( s , u ( s ) ) d s λ m r 0 1 G 1 2 , s d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equv_HTML.gif

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

(i) ( λ , u ) C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq25_HTML.gif withu ∈ (ξ2j-1 - δ2j-1 + δ) for some j ∈ ℕ* implies that λ > 2;

(ii) ( λ , u ) C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq25_HTML.gif, withu ∈ ( ξ2j - δ,ξ2j + δ) for some j ∈ ℕ* implies that λ < 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq26_HTML.gif.

Proof. (i) Assume that ( λ , u ) C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq25_HTML.gif with ∥u ∈ (ξ2j-1 - δ, ξ2j-1 + δ) for some j ∈ ℕ*, then u = Tλ u and
0 u ( x ) ξ 2 j - 1 + δ for x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equw_HTML.gif
By Lemma 3.2 and (H3), it follows that
u = T λ u λ M ξ 2 j - 1 + δ 0 1 G ( s , s ) d s = λ max 0 x 1 , 0 s ξ 2 j - 1 + δ { f l ( x , s ) } 0 1 G ( s , s ) d s λ Φ ( ξ 2 j - 1 ) 0 1 G ( s , s ) d s < λ 1 2 0 1 G ( s , s ) d s - 1 ( ξ 2 j - 1 - δ ) 0 1 G ( s , s ) d s . = 1 2 λ ( ξ 2 j - 1 - δ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equx_HTML.gif
Thus λ > 2.
  1. (ii)
    Assume that ( λ , u ) C 1 , l + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq25_HTML.gif with ∥u ∈ (ξ2j - δ, ξ2j + δ) for some j ∈ ℕ*, then u = Tλu and
    σ ( ξ 2 j - δ ) u ( x ) ξ 2 j + δ for x [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equy_HTML.gif
     
By Lemma 3.3 and the assumption (H3), it follows that
u = T λ u λ m ξ 2 j 0 1 G 1 2 , s d s = λ min 0 x 1 , σ ( ξ 2 j - δ ) s ξ 2 j + δ { f l ( x , s ) } 0 1 G 1 2 , s d s = λ Ψ ( ξ 2 j ) 0 1 G 1 2 , s d s > λ 2 0 1 G 1 2 , s d s - 1 ( ξ 2 j + δ ) 0 1 G 1 2 , s d s . = 2 λ ( ξ 2 j + δ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equz_HTML.gif

Thus λ < 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq26_HTML.gif.

Lemma 3.5. If ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif, then (λ, u) is a solution of (1.1) and uW2,∞(0, 1).

Proof. Let ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif. By the definition of C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq7_HTML.gif there exists a sequence {l k } ∈ ℕ0 strictly increasing, and ( λ l k , v l k ) [ 0 , ) × C 1 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq27_HTML.gif with ( λ l k , v l k ) C 1 , l k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq28_HTML.gif for k ∈ ℕ and
( λ l k , v l k ) ( λ , u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equaa_HTML.gif
Since { f l k ( , v l k ( ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq29_HTML.gif is uniformly bounded, i.e.
f l k L 2 M , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ12_HTML.gif
(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 ( x ) > ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq30_HTML.gif for all k > k0 and x ∈ (x0 - τ, x0 + τ). Choose k1 > k0 with d g l k 1 < ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq31_HTML.gif. Then f l k ( x , v l k ( x ) ) = f + ( x , v l k ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq32_HTML.gif for all kk1 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 meas ( K ) f + ( x , y ) - f + ( x , 0 ) ε 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq33_HTML.gif for x ∈ [0, 1] and y ∈ [0, η]. Choosing k2 ∈ ℕ with v l k - u < η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq34_HTML.gif for kk2. One obtains for kk2:
K [ ϕ ( x ) - f l k ( x , v l k ( x ) ) ] d x = K [ ϕ ( x ) - f + ( x , 0 ) ] d x + K [ f + ( x , 0 ) - f l k ( x , v l k ( x ) ) ] d x = ε + K [ f + ( x , 0 ) - f + ( x , v l k ( x ) ) ] d x ε 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equab_HTML.gif

which contradicts f l k ( , v l k ( ) ) ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq35_HTML.gif. Thus, meas(K) = 0.

Now, let A be the closed linear operator in L2(0, 1) defined by
dom ( A ) : = { φ W 2 , 2 [ 0 , 1 ] : φ ( 0 ) = 0 = φ ( 1 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equac_HTML.gif
and := -φ" + . Clearly,
f l k ( x , v l k ( x ) ) ϕ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ13_HTML.gif
(3.8)
hence v l k u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq36_HTML.gif and the fact that A is weakly closed yields
A u = λ ϕ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ14_HTML.gif
(3.9)
i.e.
A u λ F ( , u ( ) ) a .e . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equad_HTML.gif
Finally, we show that uW2,∞(0,1). In fact, from (3.9) we have
u ( x ) = q ( x ) u ( x ) - λ ϕ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ15_HTML.gif
(3.10)
According to (H1) and the boundedness of u we have
q u L ( 0 , 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ16_HTML.gif
(3.11)
We claim that ϕL(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
ϕ ( s ) > M w L 2 + 1 - [ 0 , 1 ] \ E ϕ ( x ) w ( x ) d x E w ( x ) d x , s E , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ17_HTML.gif
(3.12)
where M is given by (3.7) and wL2(0,1). On the one hand, for k larger enough from (3.7), (3.8) and (H2) we have
0 1 ϕ ( x ) w ( x ) d x 0 1 f l k ( x , v l k ( x ) ) w ( x ) d x + 1 M w L 2 + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ18_HTML.gif
(3.13)
On the other hand, from (3.12) we have
0 1 ϕ ( x ) w ( x ) d x = E ϕ ( x ) w ( x ) d x + [ 0 , 1 ] \ E ϕ ( x ) w ( x ) d x > M v L 2 + 1 - [ 0 , 1 ] \ E ϕ ( x ) w ( x ) d x E w ( x ) d x E w ( x ) d x + [ 0 , 1 ] \ E ϕ ( x ) w ( x ) d x = M w L 2 + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equae_HTML.gif
which contradicts (3.13). Thus,
ϕ L ( 0 , 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ19_HTML.gif
(3.14)

Therefore, from (3.10), (3.11) and (3.14) we obtain uW2,∞(0,1).

Now we are in the position to prove Theorem 3.1.

Proof of Theorem 3.1.

Assume that ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif. We divide the proof into two cases.

Case l. If ∥u = ξ2j- 1 for some j ∈ ℕ*, then λ ≥ 2.

Since ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif, there exists a sequence ( λ k i , z k i ) C 1 , k i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq37_HTML.gif, such that
lim i λ k i = λ , lim i z k i = u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equaf_HTML.gif
Hence, for δ > 0 there exists i0 ∈ ℕ, such that
z k i - u < δ , i > i 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equag_HTML.gif
i.e.
ξ 2 j - 1 - δ < z k i < ξ 2 j - 1 + δ , i > i 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equah_HTML.gif
By using Lemma 3.4, we obtain that
λ k i > 2 , i > i 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equai_HTML.gif
Hence, we get
λ = lim i λ k i 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equaj_HTML.gif

Case 2. If ∥u = ξ2j for some j ∈ ℕ*, then λ 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq13_HTML.gif.

Since ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif, there exists a sequence ( λ k i , z k i ) C 1 , k i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq37_HTML.gif, such that
lim i λ k i = λ , lim i z k i = u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equak_HTML.gif
Hence, for δ > 0 there exists i0 ∈ ℕ, such that
z k i - u < δ , i > i 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equag_HTML.gif
i.e.
ξ 2 j - δ < z k i < ξ 2 j + δ , i > i 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equal_HTML.gif
By using lemma 3.4, we obtain that
λ k i < 1 2 , i > i 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equam_HTML.gif
Hence, we get
λ = lim i λ k i 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equan_HTML.gif

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

(i) for each λ ( 0 , 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq38_HTML.gif, (1.1) has at least one positive solution: u 0 C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq39_HTML.gif;

(ii) for each λ [ 1 2 , 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq40_HTML.gif, (1.1) has N positive solutions:
u k , k = 1 , 2 , , N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equao_HTML.gif

which satisfy that u k C 1 + , k = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq41_HTML.gif.

Proof. According to Theorem 3.1, the boundary value problem (1.1) has an unbounded component C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq42_HTML.gif in Σ with ( 0 , 0 ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ10_HTML.gif. Moreover,

( ĩ ) ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq43_HTML.gif with ∥u = ξ2j-1 for some j ∈ ℕ* implies that λ ≥ 2;

( i i ̃ ) ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq44_HTML.gif with ∥u = ξ2j for some j ∈ ℕ* implies that λ 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq13_HTML.gif.

From the facts ( 0 , 0 ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ10_HTML.gif, and ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif with ∥u = ξ1 implies that λ ≥ 2 and the connectivity of C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq42_HTML.gif, we obtain
C 1 + λ × C 1 [ 0 , 1 ] , λ 0 , 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equap_HTML.gif

which implies for each λ ( 0 , 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq38_HTML.gif, (1.1) has at least one positive solution: u 0 C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq39_HTML.gif.

Let
C 1 + , k : = { ( λ , u ) C 1 + | ξ k - 1 u < ξ k } , k = 1 , 2 , , N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equaq_HTML.gif
where ξ0 = 0, ξ k (k = 1, 2,..., N) is given by (H3). Then according to ( i ̃ ) ( i i ̃ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq45_HTML.gif and the connectivity of C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq42_HTML.gif, we obtain
C 1 + , k λ × C 1 [ 0 , 1 ] , λ 1 2 , 2 , k = 1 , 2 , , N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equar_HTML.gif
which implies for each λ ( 1 2 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq46_HTML.gif, (1.1) has N positive solutions:
u k , k = 1 , 2 , , N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equao_HTML.gif

and u k C 1 + , k C 1 + , k = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq47_HTML.gif.

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
- u ( x ) + u ( x ) λ F ( u ( x ) ) , a .e . x ( 0 , 1 ) , u ( 0 ) = 0 , u ( 1 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ20_HTML.gif
(4.1)
where the set-valued function F in (4.1) is given by
F ( y ) = 49 2 y - 195 2 , y 4 , 1 16 y + 1 4 , 0 < y < 4 , [ 0 , 1 4 ] , y = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equas_HTML.gif
Obviously, (H 1), (H 2) conditions of Theorem 3.1 are satisfied. Moreover, Green's function of the associated linear problem
- u + u = 0 , u ( 0 ) = 0 , u ( 1 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equat_HTML.gif
can be explicitly expressed by
G ( x , s ) = 1 2 ( e - e - 1 ) ( e x - 1 + e 1 - x ) ( e s + e - s ) , 0 s x 1 , ( e x + e - x ) ( e s - 1 + e 1 - s ) , 0 x s 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equau_HTML.gif

By calculation we can get 0 1 G ( s , s ) d s = e e - e - 1 , 0 1 G 1 2 , s d s = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq48_HTML.gif and σ = 2 e + e - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq49_HTML.gif.

Let ξ1 = 3, ξ2 = 11, δ = 1, then we can check that ξ1 = 3 < 5 < σ(ξ2 - δ), and
Φ ( ξ 1 ) = 1 2 < 3 4 < 1 - e - 2 = 1 2 0 1 G ( s , s ) d s - 1 ( ξ 1 - δ ) , Ψ ( ξ 2 ) 25 > 24 = 2 0 1 G 1 2 , s d s - 1 ( ξ 2 + δ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equav_HTML.gif
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 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq42_HTML.gif in Σ with ( 0 , 0 ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_Equ10_HTML.gif. Moreover,
  1. (i)

    ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif with ∥u = 3 implies that λ ≥ 2;

     
  2. (ii)

    ( λ , u ) C 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq12_HTML.gif with ∥u = 11 implies that λ 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-47/MediaObjects/13661_2011_Article_157_IEq13_HTML.gif.

     

Declarations

Acknowledgements

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.

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University

References

  1. Aubin JP, Cellina A: Differential Inclusion, vol. 264. In Grundlehren Math Wiss. Springer-Verlag, Berlin; 1984.
  2. Kunze M: Nonsmooth dynamical systems. Lecture Notes in Mathematics, vol. 1744. Springer-Verlag, Berlin; 2000.
  3. Clarke FH, Ldeyaev YS, Stern RJ, Wolenski PR: Nonsmooth Analysis and Control Theory. Springer-Verlag, New York; 1998.
  4. Clarke FH: Optimization and Nonsmooth Analysis. SIAM, Philadelphia; 1990.View Article
  5. Leine RI, Nijjmeijer H: Dynamics and bifurcation of nonsmooth mechanical systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer-Verlag, Berlin (2004);
  6. Gasiésdi L, Papageorgion NS: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and Hall/CRC, Boca Raton; 2005.
  7. Deimling K: Multivalued Differential Equation. Springer-Verlag, Berlin; 1985.
  8. Kowalczyk P, Piiroinen PT: Two-parameter sliding bifurcation of periodic solutions in a dry-friction oscillator. Physica D 2008, 237: 1053-1073. 10.1016/j.physd.2007.12.007MATHMathSciNetView Article
  9. Deimling K: Resonance and clulomb friction. Diff Integr Equ 1994, 7(3):759-765.MATHMathSciNet
  10. Ma R: Existence of periodic solutions of a generalized friction oscillator. Nonlinear Anal Real World Appl 2010, 11: 3316-3322. 10.1016/j.nonrwa.2009.11.024MATHMathSciNetView Article
  11. Chang KC: Variational methods for nondifferentable functionals and their applications to partial differential equations. J Math Anal Appl 1981, 80: 102-112. 10.1016/0022-247X(81)90095-0MATHMathSciNetView Article
  12. Kourogenis NC, Papageorgiou NS: Nonsmooth critical point theory and nonlinear elliptic equation at resonance. Kodai Math J 2000, 23: 128-135.MathSciNetView Article
  13. Zykov PS: On two-point boundary value problems for second-order differential inclusions on manifolds. Appl Anal 2009, 88(6):895-902. 10.1080/00036810903042232MATHMathSciNetView Article
  14. Hannelore L, Csaba V: Multiple solutions for a differential inclusion problem with nonhomogeneous boundary conditions. Numer Funct Anal Optim 2009, 30(5-6):566-581. 10.1080/01630560902987857MATHMathSciNetView Article
  15. Ntouyas SK, O' Regan D: Existence results for semilinear neutral differential inclusions with nonlocal conditions. Diff Equ Appl 2009, 1(1):41-65.MATHMathSciNet
  16. Papageorgion NS, Staicu V: The method of upper-lower solutions for nonlinear second order differential inclusions. Nonlinear Anal 2007, 67(3):708-726. 10.1016/j.na.2006.06.023MathSciNetView Article
  17. Kyritsi S, Matzakos N, Papageorgion NS: Periodic problems for strongly nonlinear second-order differential inclusions. J Diff Equ 1982, 183: 279-302.View Article
  18. Dhage BC, Ntouyas SK, Cho YJ: On the second order discontinuous differential inclusions. J Appl Funt Anal 2006, 1(4):469-476.MATHMathSciNet
  19. Benchchra M, Graef JR, Ouahab A: Oscillatory and nonoscillatory solutions of multivalued differential inclusions. Comput Math Appl 2005, 49(9-10):1347-1354. 10.1016/j.camwa.2004.12.007MathSciNetView Article
  20. Budyko MI: The effect of solar radiation variations on the climate of the earth. Tellus 1969, 21: 611-619. 10.1111/j.2153-3490.1969.tb00466.xView Article
  21. Diaz JI: Mathematical Analysis of Some Diffusive Energy Balance Models. In Math Climate Environ. Mason, Paris; 1993.
  22. Diaz JI: The Mathematics of Models for Climatology and Environment, NATO ASI Series I: Global Environmental Changes, vol. 48. Springer-Verlag, New York; 1997.View Article
  23. Diaz JI, Hernandez J, Tello L: On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology. J Math Anal Appl 1997, 216: 593-613. 10.1006/jmaa.1997.5691MATHMathSciNetView Article
  24. Henderson-Sellers A, McGuffie KA: A Climate Modeling Primer. Wiley, Chich-ester; 1987.
  25. Hetzer G: A bifurcation result for Sturm-Liouville problem with a set-valued term. Mississippi State University; 1997.
  26. Jiang D, Liu H: Existence of positive solutions to second order Neumann boundary value problem. J Math Res Expo 2000, 20: 360-364.MATH
  27. Chu JF, Sun YG, Chen H: Positive solutions of Neumann problems with singularities. J Math Anal Appl 2008, 337: 1267-1272. 10.1016/j.jmaa.2007.04.070MATHMathSciNetView Article
  28. Sun Y, Cho YJ, O' Regan D: Positive solution for singular second order Neumann boundary value problems via a cone fixed point theorem. Appl Math Comput 2009, 210: 80-86. 10.1016/j.amc.2008.11.025MATHMathSciNetView Article
  29. Whyburn GT: Topological Analysis. Princeton University Press, Princeton, NJ; 1964.
  30. Ma R, An Y: Global structure of positive solutions for nonlocal boundary value problems involving integral conditions. Nonlinear Anal TMA 2009, 71(10):4364-4376. 10.1016/j.na.2009.02.113MATHMathSciNetView Article
  31. Kuratowski C: Topologie II. Warszawa; 1950.
  32. Zeidler E: Nonlinear Functional Analysis and its Applications I (Fixed-point theorems). Springer-Verlag, New York; 1986.View Article
  33. Mavinga N, Nkashama MN: Steklov-Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions. J Diff Equ 2010, 248: 1212-1229. 10.1016/j.jde.2009.10.005MATHMathSciNetView Article

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