Open Access

Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter

Boundary Value Problems20132013:7

DOI: 10.1186/1687-2770-2013-7

Received: 26 July 2012

Accepted: 29 December 2012

Published: 16 January 2013

Abstract

In this paper, we study the multiplicity of positive doubly periodic solutions for a singular semipositone telegraph equation. The proof is based on a well-known fixed point theorem in a cone.

MSC:34B15, 34B18.

Keywords

semipositone telegraph equation doubly periodic solution singular cone fixed point theorem

1 Introduction

Recently, the existence and multiplicity of positive periodic solutions for a scalar singular equation or singular systems have been studied by using some fixed point theorems; see [19]. In [10], the authors show that the method of lower and upper solutions is also one of common techniques to study the singular problem. In addition, the authors [11] use the continuation type existence principle to investigate the following singular periodic problem:
( | u | p 2 u ) + h ( u ) u = g ( u ) + c ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equa_HTML.gif
More recently, using a weak force condition, Wang [12] has built some existence results for the following periodic boundary value problem:
{ u t t u x x + c 1 u t + a 11 ( t , x ) u + a 12 ( t , x ) v = f 1 ( t , x , u , v ) + χ 1 ( t , x ) , v t t v x x + c 2 v t + a 21 ( t , x ) u + a 22 ( t , x ) v = f 2 ( t , x , u , v ) + χ 2 ( t , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equb_HTML.gif

The proof is based on Schauder’s fixed point theorem. For other results concerning the existence and multiplicity of positive doubly periodic solutions for a single regular telegraph equation or regular telegraph system, see, for example, the papers [1317] and the references therein. In these references, the nonlinearities are nonnegative.

On the other hand, the authors [18] study the semipositone telegraph system
{ u t t u x x + c 1 u t + a 1 ( t , x ) u = b 1 ( t , x ) f ( t , x , u , v ) , v t t v x x + c 2 v t + a 2 ( t , x ) v = b 2 ( t , x ) g ( t , x , u , v ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equc_HTML.gif

where the nonlinearities f, g may change sign. In addition, there are many authors who have studied the semipositone equations; see [19, 20].

Inspired by the above references, we are concerned with the multiplicity of positive doubly periodic solutions for a general singular semipositone telegraph equation
{ u t t u x x + c u t + a ( t , x ) u = λ f ( t , x , u ) , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equ1_HTML.gif
(1)
where c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq1_HTML.gif is a constant, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq2_HTML.gif is a positive parameter, a ( t , x ) C ( R × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq3_HTML.gif, f ( t , x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq4_HTML.gif may change sign and is singular at u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq5_HTML.gif, namely,
lim u 0 + f ( t , x , u ) = + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equd_HTML.gif

The main method used here is the following fixed-point theorem of a cone mapping.

Lemma 1.1 [21]

Let E be a Banach space, and K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq6_HTML.gif be a cone in E. Assume Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq7_HTML.gif, Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq8_HTML.gif are open subsets of E with 0 Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq9_HTML.gif, Ω ¯ 1 Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq10_HTML.gif, and let T : K ( Ω ¯ 2 Ω 1 ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq11_HTML.gif be a completely continuous operator such that either
  1. (i)

    T u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq12_HTML.gif, u K Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq13_HTML.gif and T u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq14_HTML.gif, u K Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq15_HTML.gif; or

     
  2. (ii)

    T u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq14_HTML.gif, u K Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq13_HTML.gif and T u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq12_HTML.gif, u K Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq15_HTML.gif.

     

Then T has a fixed point in K ( Ω ¯ 2 Ω 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq16_HTML.gif.

The paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, we give the main result.

2 Preliminaries

Let 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq17_HTML.gif be the torus defined as
2 = ( R / 2 π Z ) × ( R / 2 π Z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Eque_HTML.gif
Doubly 2π-periodic functions will be identified to be functions defined on 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq17_HTML.gif. We use the notations
L p ( 2 ) , C ( 2 ) , C α ( 2 ) , D ( 2 ) = C ( 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equf_HTML.gif

to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space D ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq18_HTML.gif denotes the space of distributions on 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq17_HTML.gif.

By a doubly periodic solution of Eq. (1) we mean that a u L 1 ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq19_HTML.gif satisfies Eq. (1) in the distribution sense, i.e.,
2 u ( φ t t φ x x c φ t + a ( t , x ) φ ) d t d x = λ 2 f ( t , x , u ) φ d t d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equg_HTML.gif
First, we consider the linear equation
u t t u x x + c u t ξ u = h ( t , x ) , in  D ( 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equ2_HTML.gif
(2)

where c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq1_HTML.gif, μ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq20_HTML.gif, and h ( t , x ) L 1 ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq21_HTML.gif.

Let £ ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq22_HTML.gif be the differential operator
£ ξ u = u t t u x x + c u t ξ u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equh_HTML.gif
acting on functions on 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq17_HTML.gif. Following the discussion in [14], we know that if ξ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq23_HTML.gif, £ ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq22_HTML.gif has the resolvent R ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq24_HTML.gif,
R ξ : L 1 ( 2 ) C ( 2 ) , h i ( t , x ) u i ( t , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equi_HTML.gif

where u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq25_HTML.gif is the unique solution of Eq. (2), and the restriction of R ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq24_HTML.gif on L p ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq26_HTML.gif ( 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq27_HTML.gif) or C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq28_HTML.gif is compact. In particular, R ξ : C ( 2 ) C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq29_HTML.gif is a completely continuous operator.

For ξ = c 2 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq30_HTML.gif, the Green function G ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq31_HTML.gif of the differential operator £ ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq22_HTML.gif is explicitly expressed; see Lemma 5.2 in [14]. From the definition of G ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq31_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equj_HTML.gif

For convenience, we assume the following condition holds throughout this paper:

(H1) a ( t , x ) C ( 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq32_HTML.gif, 0 a ( t , x ) c 2 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq33_HTML.gif on 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq17_HTML.gif, and 2 a ( t , x ) d t d x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq34_HTML.gif.

Finally, if −ξ is replaced by a ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq35_HTML.gif in Eq. (2), the author [13] has proved the following unique existence and positive estimate result.

Lemma 2.1 Let h ( t , x ) L 1 ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq21_HTML.gif. Then Eq. (2) has a unique solution u ( t , x ) = P [ h ( t , x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq36_HTML.gif, P : L 1 ( 2 ) C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq37_HTML.gif is a linear bounded operator with the following properties:
  1. (i)

    P : C ( 2 ) C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq38_HTML.gif is a completely continuous operator;

     
  2. (ii)
    If h ( t , x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq39_HTML.gif, a.e ( t , x ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq40_HTML.gif, P [ h ( t , x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq41_HTML.gif has the positive estimate
    G ̲ h L 1 P [ h ( t , x ) ] G ¯ G ̲ a L 1 h L 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equ3_HTML.gif
    (3)
     

3 Main result

Theorem 3.1 Assume (H1) holds. In addition, if f ( t , x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq4_HTML.gif satisfies

(H2) lim u 0 + f ( t , x , u ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq42_HTML.gif, uniformly ( t , x ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq43_HTML.gif,

(H3) f : 2 × ( 0 , + ) ( , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq44_HTML.gif is continuous,

(H4) there exists a nonnegative function h ( t , x ) C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq45_HTML.gif such that
f ( t , x , u ) + h ( t , x ) 0 , ( t , x ) 2 , u > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equk_HTML.gif

(H5) 2 F ( t , x ) d t d x = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq46_HTML.gif, where the limit function F ( t , x ) = lim inf u + f ( t , x , u ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq47_HTML.gif,

then Eq. (1) has at least two positive doubly periodic solutions for sufficiently small λ.

C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq28_HTML.gif is a Banach space with the norm u = max ( t , x ) 2 | u ( t , x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq48_HTML.gif. Define a cone K C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq49_HTML.gif by
K = { u C ( 2 ) : u 0 , u ( t , x ) δ u } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equl_HTML.gif

where δ = G ̲ 2 a L 1 G ¯ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq50_HTML.gif. Let K r = { u K : u = r } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq51_HTML.gif, [ u ] + = max { u , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq52_HTML.gif. By Lemma 2.1, it is easy to obtain the following lemmas.

Lemma 3.2 If h ( t , x ) C ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq45_HTML.gif is a nonnegative function, the linear boundary value problem
{ u t t u x x + c u t + a ( t , x ) u = λ h ( t , x ) , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equm_HTML.gif
has a unique solution ω ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq53_HTML.gif. The function ω ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq53_HTML.gif satisfies the estimates
λ G ̲ h L 1 ω ( t , x ) = λ P ( h ( t , x ) ) λ G ¯ G ̲ a L 1 h L 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equn_HTML.gif
Lemma 3.3 If the boundary value problem
{ u t t u x x + c u t + a ( t , x ) u = λ [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equo_HTML.gif

has a solution u ˜ ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq54_HTML.gif with u ˜ > λ G ¯ 2 G ̲ 3 a L 1 2 h L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq55_HTML.gif, then u ( t , x ) = u ˜ ( t , x ) ω ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq56_HTML.gif is a positive doubly periodic solution of Eq. (1).

Proof of Theorem 3.1 Step 1. Define the operator T as follows:
( T u ) ( t , x ) = λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equp_HTML.gif

We obtain the conclusion that T ( K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq57_HTML.gif, and T : K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq58_HTML.gif is completely continuous.

For any u K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq59_HTML.gif, then [ u ( t , x ) ω ( t , x ) ] + > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq60_HTML.gif, and T is defined. On the other hand, for u K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq59_HTML.gif, the complete continuity is obvious by Lemma 2.1. And we can have
( T u ) ( t , x ) = λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] λ G ̲ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) L 1 G ̲ G ̲ a L 1 G ¯ T ( u ) δ T u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equq_HTML.gif

Thus, T ( K { u K : u ( t , x ) ω ( t , x ) } ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq61_HTML.gif.

Now we prove that the operator T has one fixed point u ˜ K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq62_HTML.gif and u ˜ > λ G ¯ 2 G ̲ 3 a L 1 2 h L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq55_HTML.gif for all sufficiently small λ.

Since 2 F ( t , x ) d t d x = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq46_HTML.gif, there exists r 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq63_HTML.gif such that
2 f ( t , x , u ) u d t d x 1 δ , u δ r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equr_HTML.gif
Furthermore, we have 2 f ( t , x , δ r 1 ) d t d x r 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq64_HTML.gif. It follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equs_HTML.gif
Let Φ ( t , x ) = max { f ( t , x , u ) : δ 2 r 1 u r 1 } + h ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq65_HTML.gif. Then Φ L 1 ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq66_HTML.gif and 2 Φ ( t , x ) d t d x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq67_HTML.gif. Set
λ = min { δ 2 2 G ̲ h L 1 , 2 G ̲ a L 1 G ¯ Φ L 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equt_HTML.gif
For any u K r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq68_HTML.gif and 0 < λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq69_HTML.gif, we can verify that
u ( t , x ) ω ( t , x ) δ u ω ( t , x ) = δ r 1 ω ( t , x ) δ r 1 λ G ¯ G ̲ a L 1 h L 1 δ r 1 δ r 1 2 = δ r 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equu_HTML.gif
Then we have
T u = λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] λ G ¯ G ̲ a L 1 f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) L 1 λ G ¯ G ̲ a L 1 Φ ( t , x ) L 1 < 2 r 1 = u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equv_HTML.gif
On the other hand,
lim inf u + f ( t , x , u ω ( t , x ) ) u = lim inf u + f ( t , x , u ) u = F ( t , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equw_HTML.gif
By the Fatou lemma, one has
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equx_HTML.gif
Hence, there exists a positive number r 2 > δ r 2 > r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq70_HTML.gif such that
2 f ( t , x , u ω ( t , x ) ) + h ( t , x ) u d t d x λ 1 δ 1 G ̲ 1 ( 4 π 2 ) 1 , u δ r 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equy_HTML.gif
Hence, we have
2 f ( t , x , u ω ( t , x ) ) + h ( t , x ) d t d x λ 1 G ̲ 1 ( 4 π 2 ) 1 r 2 , u δ r 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equz_HTML.gif
For any u K r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq71_HTML.gif, we have δ r 2 = δ u u ( t , x ) u = r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq72_HTML.gif. On the other hand, since 0 < λ < λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq69_HTML.gif, we can get
u ( t , x ) ω ( t , x ) δ r 2 ω ( t , x ) δ r 2 δ λ G ¯ G ̲ a L 1 δ r 2 δ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equaa_HTML.gif
From above, we can have
T u λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] λ G ̲ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) L 1 λ G ̲ 4 π 2 λ 1 G ̲ 1 ( 4 π 2 ) 1 r 2 = r 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equab_HTML.gif
Therefore, by Lemma 1.1, the operator T has a fixed point u ˜ ( t , x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq73_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equac_HTML.gif

So, Eq. (1) has a positive solution u ˆ ( t , x ) = u ˜ ( t , x ) ω ( t , x ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq74_HTML.gif.

Step 2. By conditions (H2) and (H3), it is clear to obtain that
u 0 = inf { u K : f ( t , x , u ) 0 , ( t , x ) 2 } > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equad_HTML.gif
Let r 4 = min { δ 2 , δ u 0 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq75_HTML.gif. For any u ( 0 , r 4 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq76_HTML.gif, we have f ( t , x , u ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq77_HTML.gif. Then define the operator A as follows:
( A u ) ( t , x ) = λ P ˆ [ f ( t , x , u ( t , x ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equae_HTML.gif

It is easy to prove that A ( K { u C ( 2 ) : 0 < u < r 4 } ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq78_HTML.gif, and A : K { u C ( 2 ) : 0 < u < r 4 } K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq79_HTML.gif is completely continuous.

And for any ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq80_HTML.gif, define
M ( ρ ) = max { f ( t , x , u ) : u R + , δ ρ u ρ , ( t , x ) 2 } > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equaf_HTML.gif
Furthermore, for any u K r 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq81_HTML.gif, we have
A u = λ P ˆ [ f ( t , x , u ( t , x ) ) ] λ G ¯ G ̲ a L 1 f ( t , x , u ( t , x ) ) L 1 λ G ¯ G ̲ a L 1 M ( r 4 ) 4 π 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equag_HTML.gif
Thus, from the above inequality, there exists λ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq82_HTML.gif such that
A u < u , for  u K r 4 , 0 < λ < λ ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equah_HTML.gif
Since lim u 0 + f ( t , x , u ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq83_HTML.gif, then there is 0 < r 3 < r 4 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq84_HTML.gif such that
f ( t , x , u ) μ u , for  u R +  with  0 < u r 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equai_HTML.gif
where μ satisfies λ G ̲ μ δ > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq85_HTML.gif. For any u K r 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq86_HTML.gif, then we have
f ( t , x , u ) μ u ( t , x ) , for  ( t , x ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equaj_HTML.gif
By Lemma 2.1, it is clear to obtain that
A u = λ P ˆ [ f ( t , x , u ( t , x ) ) ] λ G ̲ f ( t , x , u ( t , x ) ) L 1 λ G ̲ μ δ r 3 > r 3 = u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equak_HTML.gif

Therefore, by Lemma 1.1, A has a fixed point in u ¯ ( t , x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq87_HTML.gif and u ¯ r 4 δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq88_HTML.gif, which is another positive periodic solution of Eq. (1).

Finally, from Step 1 and Step 2, Eq. (1) has two positive doubly periodic solutions u ˆ ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq89_HTML.gif and u ¯ ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq90_HTML.gif for sufficiently small λ. □

Example

Consider the following problem:
{ u t t u x x + 2 u t + sin 2 ( t + x ) u = λ [ 1 u + min { u 2 , u | 1 t π | | 1 x π | } 10 ] , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_Equal_HTML.gif

It is clear that f ( t , x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-7/MediaObjects/13661_2012_Article_267_IEq4_HTML.gif satisfies the conditions (H1)-(H5).

Declarations

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions for improving this paper.

Authors’ Affiliations

(1)
College of Science, Hohai University
(2)
Department of Mathematics, Nanjing University of Aeronautics and Astronautics

References

  1. Chu J, Torres PJ, Zhang M: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 2007, 239: 196-212. 10.1016/j.jde.2007.05.007MATHMathSciNetView Article
  2. Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665-675. 10.1016/j.jmaa.2011.09.061MATHMathSciNetView Article
  3. Chu J, Zhang Z: Periodic solutions of second order superlinear singular dynamical systems. Acta Appl. Math. 2010, 111: 179-187. 10.1007/s10440-009-9539-9MATHMathSciNetView Article
  4. Chu J, Li M: Positive periodic solutions of Hill’s equations with singular nonlinear perturbations. Nonlinear Anal. 2008, 69: 276-286. 10.1016/j.na.2007.05.016MATHMathSciNetView Article
  5. Chu J, Torres PJ: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. Lond. Math. Soc. 2007, 39: 653-660. 10.1112/blms/bdm040MATHMathSciNetView Article
  6. Jiang D, Chu J, Zhang M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 2005, 211: 282-302. 10.1016/j.jde.2004.10.031MATHMathSciNetView Article
  7. Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3MATHView Article
  8. Torres PJ: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 2007, 232: 277-284. 10.1016/j.jde.2006.08.006MATHView Article
  9. Wang H: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 2010, 249: 2986-3002. 10.1016/j.jde.2010.08.027MATHView Article
  10. DeCoster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. 371. In Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations CISM-ICMS. Edited by: Zanolin F. Springer, New York; 1996:1-78.View Article
  11. Jebelean P, Mawhin J: Periodic solutions of forced dissipative p -Liénard equations with singularities. Vietnam J. Math. 2004, 32: 97-103.MATHMathSciNet
  12. Wang F: Doubly periodic solutions of a coupled nonlinear telegraph system with weak singularities. Nonlinear Anal., Real World Appl. 2011, 12: 254-261. 10.1016/j.nonrwa.2010.06.012MATHMathSciNetView Article
  13. Li Y: Positive doubly periodic solutions of nonlinear telegraph equations. Nonlinear Anal. 2003, 55: 245-254. 10.1016/S0362-546X(03)00227-XMATHMathSciNetView Article
  14. Ortega R, Robles-Perez AM: A maximum principle for periodic solutions of the telegraph equations. J. Math. Anal. Appl. 1998, 221: 625-651. 10.1006/jmaa.1998.5921MATHMathSciNetView Article
  15. Wang F, An Y: Nonnegative doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 2008, 338: 91-100. 10.1016/j.jmaa.2007.05.008MATHMathSciNetView Article
  16. Wang F, An Y: Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 2009, 349: 30-42. 10.1016/j.jmaa.2008.08.003MATHMathSciNetView Article
  17. Wang F, An Y: Nonnegative doubly periodic solutions for nonlinear telegraph system with twin-parameters. Appl. Math. Comput. 2009, 214: 310-317. 10.1016/j.amc.2009.03.069MATHMathSciNetView Article
  18. Wang F, An Y: On positive solutions of nonlinear telegraph semipositone system. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2009, 16: 209-219.MATHMathSciNet
  19. Xu X: Positive solutions for singular semi-positone three-point systems. Nonlinear Anal. 2007, 66: 791-805. 10.1016/j.na.2005.12.019MATHMathSciNetView Article
  20. Yao Q: An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem. Appl. Math. Lett. 2010, 23: 1401-1406. 10.1016/j.aml.2010.06.025MATHMathSciNetView Article
  21. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.MATH

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© Wang and An; licensee Springer. 2013

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