Existence and uniqueness of solutions for fourth-order periodic boundary value problems under two-parameter nonresonance conditions

  • He Yang1Email author,

    Affiliated with

    • Yue Liang2 and

      Affiliated with

      • Pengyu Chen1

        Affiliated with

        Boundary Value Problems20132013:14

        DOI: 10.1186/1687-2770-2013-14

        Received: 16 December 2012

        Accepted: 18 January 2013

        Published: 4 February 2013

        Abstract

        This paper deals with the existence and uniqueness of solutions of the fourth-order periodic boundary value problem

        { u ( 4 ) ( t ) = f ( t , u ( t ) , u ( t ) ) , t [ 0 , 1 ] , u ( i ) ( 0 ) = u ( i ) ( 1 ) , i = 0 , 1 , 2 , 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equa_HTML.gif

        where f : [ 0 , 1 ] × R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq1_HTML.gif is continuous. Under two-parameter nonresonance conditions described by rectangle and ellipse, some existence and uniqueness results are obtained by using fixed point theorems. These results improve and extend some existing results.

        MSC:34B15.

        Keywords

        existence uniqueness two-parameter nonresonance condition equivalent norm

        1 Introduction and main results

        In mathematics, the equilibrium state of an elastic beam is described by fourth-order boundary value problems. According to the difference of supported condition on both ends, it brings out various fourth-order boundary value problems; see [1]. In this paper, we deal with the periodic boundary value problem (PBVP) of the fourth-order ordinary differential equation
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ1_HTML.gif
        (1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ2_HTML.gif
        (2)

        where f : [ 0 , 1 ] × R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq1_HTML.gif is continuous. PBVP (1)-(2) models the deformations of an elastic beam in equilibrium state with a periodic boundary condition. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors; see [26].

        Throughout this paper, we denote that I = [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq2_HTML.gif, R = ( , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq3_HTML.gif, Z = { , 2 , 1 , 0 , 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq4_HTML.gif, N = { 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq5_HTML.gif, N = N { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq6_HTML.gif. In [710], authors showed the existence of solutions to Eq. (1) under the boundary condition
        u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ3_HTML.gif
        (3)
        At first, the existence of a solution to two-point boundary value problem (BVP) (1)-(3) was studied by Aftabizadeh in [7] under the restriction that f is a bounded function. Then, under the following growth condition:
        | f ( t , u , v ) | a | u | + b | v | + c , a , b , c > 0 , a π 4 + b π 2 < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equb_HTML.gif

        Yang in [[8], Theorem 1] extended Aftabizadeh’s result and showed the existence to BVP (1)-(3). Later, Del Pino and Manasevich in [9] further extended the result of Aftabizadeh and Yang in [7, 8] and obtained the following existence theorem.

        Theorem A Assume that the pair ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif satisfies
        α ( k π ) 4 + β ( k π ) 2 1 , k N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ4_HTML.gif
        (4)
        and that there are positive constants a, b, and c such that
        a max k N 1 | ( k π ) 4 α β ( k π ) 2 | + b max k N ( k π ) 2 | ( k π ) 4 α β ( k π ) 2 | < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ5_HTML.gif
        (5)
        and f satisfies the growth condition
        | f ( t , u , v ) ( α u β v ) | a | u | + b | v | + c , t I , u , v , R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equc_HTML.gif

        Then BVP (1)-(3) possesses at least one solution.

        Condition (4)-(5) trivially implies that
        a + b ( k π ) 2 | ( k π ) 4 α β ( k π ) 2 | < 1 , k N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ6_HTML.gif
        (6)
        It is easy to prove that condition (6) is equivalent to the fact that the rectangle
        R ( α , β ; a , b ) = [ α a , α + a ] × [ β b , β + b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equd_HTML.gif

        does not intersect any of the eigenlines of the two-parameter linear eigenvalue problem corresponding to BVP (1)-(3).

        In [2], Ma applied Theorem A to PBVP (1)-(2) successfully and obtained the following existence theorem.

        Theorem B Assume that the pair ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif satisfies
        α + β ( 2 k π ) 2 ( 2 k π ) 4 , k N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ7_HTML.gif
        (7)
        and that there are positive constants a, b, and c such that
        a max k N 1 | ( 2 k π ) 4 α β ( 2 k π ) 2 | + b max k N ( 2 k π ) 2 | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ8_HTML.gif
        (8)
        and f satisfies the growth condition
        | f ( t , u , v ) ( α u β v ) | a | u | + b | v | + c , t I , u , v , R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ9_HTML.gif
        (9)

        Then PBVP (1)-(2) has at least one solution.

        Condition (7)-(9) concerns a nonresonance condition involving the two-parameter linear eigenvalue problem (LEVP)
        { u ( 4 ) ( t ) + β u ( t ) α u ( t ) = 0 , t I , u ( i ) ( 0 ) = u ( i ) ( 1 ) , i = 0 , 1 , 2 , 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ10_HTML.gif
        (10)
        In [2], it has been proved that ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif is an eigenvalue pair of LEVP (10) if and only if α + β ( 2 k π ) 2 = ( 2 k π ) 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq8_HTML.gif, k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq9_HTML.gif. Hence, for each k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq9_HTML.gif, the straight line
        k = { ( α , β ) | α + β ( 2 k π ) 2 = ( 2 k π ) 4 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Eque_HTML.gif
        is called an eigenline of LEVP (10). Condition (7)-(8) trivially implies that
        a + b ( 2 k π ) 2 | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 , k N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ11_HTML.gif
        (11)

        It is easy to prove that condition (11) is equivalent to the fact that the rectangle R ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq10_HTML.gif does not intersect any of the eigenline k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq11_HTML.gif of LEVP (10). Hence, we call (11) and (9) the two-parameter nonresonance condition described by rectangle, which is a direct extension from a single-parameter nonresonance condition to a two-parameter one.

        The purpose of this paper is to improve and extend the above-mentioned results. Different from the two-parameter nonresonance condition described by rectangle, we will present new two-parameter nonresonance conditions described by ellipse and circle. Under these nonresonance conditions, we obtain several existence and uniqueness theorems.

        The main results are as follows.

        Theorem 1 Assume that the pair ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif satisfies (7). If there exist positive constants a, b, and c such that (11) and
        | f ( t , u , v ) ( α u β v ) | a 2 u 2 + b 2 v 2 + c , t I , u , v R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ12_HTML.gif
        (12)

        hold, then PBVP (1)-(2) has at least one solution.

        When the partial derivatives f u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq12_HTML.gif and f v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq13_HTML.gif exist, if u 2 + v 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq14_HTML.gif is large enough such that
        ( f u ( t , u , v ) , f v ( t , u , v ) ) E ( α , β ; a , b ) , t I , u 2 + v 2 R 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ13_HTML.gif
        (13)
        where E ( α , β ; a , b ) = { ( x , y ) | ( x α ) 2 a 2 + ( y β ) 2 b 2 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq15_HTML.gif is a certain ellipse, and the corresponding close rectangle R ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq10_HTML.gif satisfies
        R ( α , β ; a , b ) k = , k N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ14_HTML.gif
        (14)

        by the theorem of differential mean value, we easily see that (7), (11), and (12) hold. Hence, by Theorem 1, we have the following corollary.

        Corollary 1 Assume that the partial derivatives f u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq12_HTML.gif and f v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq13_HTML.gif exist in I × R × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq16_HTML.gif. If there exists an ellipse E ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq17_HTML.gif such that (13) holds for a positive real number R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq18_HTML.gif large enough, and the corresponding close rectangle R ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq10_HTML.gif satisfies (14), then PBVP (1)-(2) has at least one solution.

        Condition (11) is weaker than condition (8), but condition (12) is stronger than condition (9). Hence, Theorem 1 and Corollary 1 partly improve Theorem B.

        In the nonresonance condition of Theorem 1, condition (11) can be weakened as
        a 2 + b 2 ( 2 k π ) 4 | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 , k N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ15_HTML.gif
        (15)

        In this case, we have the following results.

        Theorem 2 Assume that the pair ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif satisfies (7). If there exist positive constants a, b, and c such that (12) and (15) hold, then PBVP (1)-(2) has at least one solution.

        Condition (15) is equivalent to the fact that
        E ( α , β ; a , b ) k = , k N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ16_HTML.gif
        (16)

        Condition (16) indicates that the ellipse E ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq17_HTML.gif does not intersect any of the eigenline k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq11_HTML.gif of LEVP (10). Hence, we call (15) and (12) the two-parameter nonresonance condition described by ellipse, which is another extension of a single-parameter nonresonance condition. Similar to Corollary 1, we have the following corollary.

        Corollary 2 Assume that the partial derivatives f u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq12_HTML.gif and f v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq13_HTML.gif exist in I × R × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq16_HTML.gif. If there exists an ellipse E ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq17_HTML.gif such that (13) and (16) hold for a positive real number R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq18_HTML.gif large enough, then PBVP (1)-(2) has at least one solution.

        Theorem 3 Assume that the partial derivatives f u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq12_HTML.gif and f v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq13_HTML.gif exist in I × R × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq16_HTML.gif. If there exists an ellipse E ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq17_HTML.gif such that (16) and
        ( f u ( t , u , v ) , f v ( t , u , v ) ) E ( α , β ; a , b ) , t I , u , v R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ17_HTML.gif
        (17)

        hold, then PBVP (1)-(2) has a unique solution.

        In Theorem 2, Theorem 3, and Corollary 2, we present a new two-parameter nonresonance condition described by ellipse, which is another extension of a single-parameter nonresonance condition. As a special case, we replace the ellipse E ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq17_HTML.gif by a circle
        B ¯ ( α , β ; r ) = { ( x , y ) | ( x α ) 2 + ( y β ) 2 r 2 } , r > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equf_HTML.gif

        and obtain the following results.

        Corollary 3 Assume that there exist a circle B ¯ ( α , β ; r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq19_HTML.gif and a positive constant c such that
        B ¯ ( α , β ; r ) k = , k N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ18_HTML.gif
        (18)
        and f satisfies the growth condition
        | f ( t , u , v ) ( α u β v ) | r u 2 + v 2 + c , t I , u , v R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ19_HTML.gif
        (19)

        Then PBVP (1)-(2) has at least one solution.

        Condition (18) indicates that the circle B ¯ ( α , β ; r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq19_HTML.gif does not intersect any of the eigenline k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq11_HTML.gif of LEVP (10). Hence, we call condition (18)-(19) the two-parameter nonresonance condition described by circle, which is also an extension of a single-parameter nonresonance condition. Similarly to Corollary 2 and Theorem 3, we have the following corollaries.

        Corollary 4 Assume that the partial derivatives f u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq12_HTML.gif and f v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq13_HTML.gif exist in I × R × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq16_HTML.gif. If there exists a circle B ¯ ( α , β ; r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq19_HTML.gif such that (18) and
        ( f u ( t , u , v ) , f v ( t , u , v ) ) B ¯ ( α , β ; r ) , t I , u 2 + v 2 R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ20_HTML.gif
        (20)

        hold for a positive real number R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq18_HTML.gif large enough, then PBVP (1)-(2) has at least one solution.

        Corollary 5 Assume that the partial derivatives f u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq12_HTML.gif and f v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq13_HTML.gif exist in I × R × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq16_HTML.gif. If there exists a circle B ¯ ( α , β ; r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq19_HTML.gif such that (18) and
        ( f u ( t , u , v ) , f v ( t , u , v ) ) B ¯ ( α , β ; r ) , t I , u , v R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ21_HTML.gif
        (21)

        hold, then PBVP (1)-(2) has a unique solution.

        2 Preliminaries

        Let ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif be not eigenvalue pair of LEVP (10), i.e., ( α , β ) L : = k = 0 + k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq20_HTML.gif. For any h L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq21_HTML.gif, we consider the linear periodic boundary value problem (LPBVP)
        { u ( 4 ) ( t ) + β u ( t ) α u ( t ) = h ( t ) , t I , u ( i ) ( 0 ) = u ( i ) ( 1 ) , i = 0 , 1 , 2 , 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ22_HTML.gif
        (22)
        By the Fredholm alternative, LPBVP (22) has a unique solution u H 4 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq22_HTML.gif. If h C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq23_HTML.gif, then the solution u C 4 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq24_HTML.gif. We define an operator T by
        T h = u , h L 2 ( I ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equg_HTML.gif

        Then T : L 2 ( I ) H 4 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq25_HTML.gif is a bounded linear operator, and we call it the solution operator of LPBVP (22). By compactness of the embedding H 4 ( I ) H 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq26_HTML.gif, T : L 2 ( I ) H 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq27_HTML.gif is a compact linear operator.

        Let a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq28_HTML.gif. We choose an equivalent norm in the Sobolev space H 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq29_HTML.gif by
        u E a , b = a 2 u 2 2 + b 2 u 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equh_HTML.gif

        and denote the Banach space H 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq29_HTML.gif reendowed norm E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq30_HTML.gif by E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq31_HTML.gif.

        Lemma 1 Let ( α , β ) L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq32_HTML.gif. Then the solution operator of LPBVP (22) T : L 2 ( I ) E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq33_HTML.gif is a compact linear operator and its norm satisfies
        T B ( L 2 ( I ) , E a , b ) max k N a 2 + b 2 ( 2 k π ) 4 | ( 2 k π ) 4 α β ( 2 k π ) 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ23_HTML.gif
        (23)

        Proof We only need to prove that (23) holds.

        Since { e 2 k π i t | k Z } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq34_HTML.gif is a complete orthogonal system of L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq35_HTML.gif, every h L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq21_HTML.gif can be expressed by the Fourier series expansion
        h ( t ) = k = h k e 2 k π i t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equi_HTML.gif
        where h k = 0 1 h ( s ) e 2 k π i s d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq36_HTML.gif, k Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq37_HTML.gif. By the Parseval equality, we have
        h 2 2 = k = | h k | 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equj_HTML.gif
        where 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq38_HTML.gif is the norm in L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq35_HTML.gif. Now, by uniqueness of the Fourier series expansion, the solution u = T h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq39_HTML.gif of LPBVP (22) has the Fourier series expansion
        u ( t ) = k = h k ( 2 k π ) 4 α β ( 2 k π ) 2 e 2 k π i t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equk_HTML.gif
        and u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq40_HTML.gif can be expressed by the Fourier series expansion
        u ( t ) = k = ( 2 k π ) 2 h k ( 2 k π ) 4 α β ( 2 k π ) 2 e 2 k π i t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equl_HTML.gif
        Hence, by the Parseval equality, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ24_HTML.gif
        (24)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ25_HTML.gif
        (25)
        From (24) and (25), we have
        T h E a , b 2 = u E a , b 2 = a 2 u 2 2 + b 2 u 2 2 = k = ( a 2 + b 2 ( 2 k π ) 4 ) | h k | 2 | ( 2 k π ) 4 α β ( 2 k π ) 2 | 2 ( max k N a 2 + b 2 ( 2 k π ) 4 | ( 2 k π ) 4 α β ( 2 k π ) 2 | ) 2 k = | h k | 2 = ( max k N a 2 + b 2 ( 2 k π ) 4 | ( 2 k π ) 4 α β ( 2 k π ) 2 | ) 2 h 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equm_HTML.gif

        This implies that (23) holds. The proof of Lemma 1 is completed. □

        Lemma 2 Let α , β L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq41_HTML.gif and a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq28_HTML.gif. Then the rectangle R ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq10_HTML.gif satisfies condition (14) if and only if condition (11) holds.

        Proof Condition (14) holds

        ( α a , β b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq42_HTML.gif and ( α + a , β + b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq43_HTML.gif on the same side of every eigenline k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq11_HTML.gif,

        ( 2 k π ) 4 ( α a ) ( β b ) ( 2 k π ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq44_HTML.gif and ( 2 k π ) 4 ( α + a ) ( β + b ) ( 2 k π ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq45_HTML.gif have the same sign,

        ( ( 2 k π ) 4 α β ( 2 k π ) 2 ) 2 ( a + b ( 2 k π ) 2 ) 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq46_HTML.gif,

        a + b ( 2 k π ) 2 | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq47_HTML.gif.

        The proof of Lemma 2 is completed. □

        Lemma 3 Let α , β L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq41_HTML.gif and a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq28_HTML.gif. Then the ellipse E ( α , β ; a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq17_HTML.gif satisfies condition (16) if and only if condition (15) holds.

        Proof Condition (16) holds

        ⇔ for θ [ 0 , 2 π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq48_HTML.gif, ( α a cos θ , β b sin θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq49_HTML.gif and ( α + a cos θ , β + b sin θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq50_HTML.gif on the same side of every eigenline k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq11_HTML.gif,

        ( 2 k π ) 4 ( α a cos θ ) ( β b sin θ ) ( 2 k π ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq51_HTML.gif and ( 2 k π ) 4 ( α + a cos θ ) ( β + b sin θ ) ( 2 k π ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq52_HTML.gif have the same sign,

        ( ( 2 k π ) 4 α β ( 2 k π ) 2 ) 2 ( a cos θ + b sin θ ( 2 k π ) 2 ) 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq53_HTML.gif,

        | a cos θ + b sin θ ( 2 k π ) 2 | | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq54_HTML.gif,

        max θ [ 0 , 2 π ] | a cos θ + b sin θ ( 2 k π ) 2 | | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq55_HTML.gif,

        a 2 + b 2 ( 2 k π ) 4 | ( 2 k π ) 4 α β ( 2 k π ) 2 | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq56_HTML.gif.

        The proof of Lemma 3 is completed. □

        3 Proof of the main results

        Proof of Theorem 1 We define a mapping F : E a , b L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq57_HTML.gif by
        F ( u ) ( t ) = f ( t , u ( t ) , u ( t ) ) α u ( t ) + β u ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ26_HTML.gif
        (26)
        It follows from (12) that F : E a , b L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq58_HTML.gif is continuous and satisfies
        F ( u ) 2 u E a , b + c , u E a , b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ27_HTML.gif
        (27)
        Therefore, the mapping defined by
        Q = T F : E a , b E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equ28_HTML.gif
        (28)

        is a completely continuous mapping. By the definition of the operator T, the solution of PBVP (1)-(2) is equivalent to the fixed point of the operator Q.

        From (7), (11), and Lemma 1, it follows that T B ( L 2 ( I ) , E a , b ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq59_HTML.gif. We choose R c T B ( L 2 ( I ) , E a , b ) 1 T B ( L 2 ( I ) , E a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq60_HTML.gif. Let B ¯ ( θ , R ) = { u E a , b | u E a , b R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq61_HTML.gif. Then for any u B ¯ ( θ , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq62_HTML.gif, from (27) and (28), we have
        Q u E a , b = T ( F ( u ) ) E a , b T B ( L 2 ( I ) , E a , b ) F ( u ) 2 T B ( L 2 ( I ) , E a , b ) ( u E a , b + c ) T B ( L 2 ( I ) , E a , b ) ( R + c ) R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equn_HTML.gif

        Therefore, Q ( B ¯ ( θ , R ) ) B ¯ ( θ , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq63_HTML.gif. By the Schauder’s fixed point theorem, Q has at least one fixed point in B ¯ ( θ , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq64_HTML.gif, which is a solution of PBVP (1)-(2). □

        By Lemma 2, we can obtain the following existence result:

        Corollary 6 Assume that the pair ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq7_HTML.gif satisfies (7). If there exist positive constants a, b, and c such that (12) and (14) hold, then PBVP (1)-(2) has at least one solution.

        Proof of Theorem 2 Let F : E a , b L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq58_HTML.gif be a mapping defined by (26). Then it follows from (12) that F : E a , b L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq65_HTML.gif is continuous and satisfies
        F ( u ) 2 u E a , b + c , u E a , b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equo_HTML.gif

        Thus, the mapping Q = T F : E a , b E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq66_HTML.gif is completely continuous. By using (7), (15), and Lemma 1, a similar argument as in the proof of Theorem 1 shows that Q has at least one fixed point in B ¯ ( θ , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq64_HTML.gif, which is the solution of PBVP (1)-(2). □

        Proof of Theorem 3 Let F : E a , b L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq58_HTML.gif be defined by (26). Then F : E a , b L 2 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq58_HTML.gif is continuous. For any u 1 , u 2 E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq67_HTML.gif, from (17), we have
        | F ( u 2 ) F ( u 1 ) | = | f ( t , u 2 , u 2 ) α u 2 + β u 2 [ f ( t , u 1 , u 1 ) α u 1 + β u 1 ] | = | ( f u α ) ( u 2 u 1 ) + ( f v + β ) ( u 2 u 1 ) | = | f u α a a ( u 2 u 1 ) + f v + β b b ( u 2 u 1 ) | ( f u α ) 2 a 2 + ( f v + β ) 2 b 2 a 2 ( u 2 u 1 ) 2 + b 2 ( u 2 u 1 ) 2 a 2 ( u 2 u 1 ) 2 + b 2 ( u 2 u 1 ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equp_HTML.gif
        It follows from the above that F ( u 2 ) F ( u 1 ) 2 u 2 u 1 E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq68_HTML.gif. Thus, Q = T F : E a , b E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq66_HTML.gif is a continuous mapping and it satisfies
        Q ( u 2 ) Q ( u 1 ) a , b = T ( F ( u 2 ) F ( u 1 ) ) E a , b T B ( L 2 ( I ) , E a , b ) F ( u 2 ) F ( u 1 ) 2 T B ( L 2 ( I ) , E a , b ) u 2 u 1 E a , b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_Equq_HTML.gif

        It follows from (16) and Lemma 3 that (15) holds. By (15) and Lemma 1, it is easy to see that T B ( L 2 ( I ) , E a , b ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq59_HTML.gif. Hence, Q : E a , b E a , b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-14/MediaObjects/13661_2012_Article_271_IEq69_HTML.gif is a contraction mapping. By the Banach contraction mapping principle, Q has a unique fixed point, which is the unique solution of PBVP (1)-(2). □

        As in Corollary 6, in Theorem 2 we can use condition (16) to replace condition (15), and in Theorem 3, we use condition (15) to replace condition (16).

        Declarations

        Acknowledgements

        Research supported by the NNSF of China (Grant No. 11261053), the Fundamental Research Funds for the Gansu Universities and the Project of NWNU-LKQN-11-3.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Northwest Normal University
        (2)
        Science College, Gansu Agricultural University

        References

        1. Gupta C: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 1988, 26: 289-304. 10.1080/00036818808839715MATHMathSciNetView Article
        2. Ma R: The existence of solutions of a fourth-order periodic boundary value problem. Acta Sci. Math. 1995, 15: 315-318. (in Chinese)MATH
        3. Kong L, Jiang D: Multiple solutions of a nonlinear fourth order periodic boundary value problem. Ann. Pol. Math. 1998, LXIV: 265-270.MathSciNet
        4. Li Y: Positive solutions of fourth-order periodic boundary value problems. Nonlinear Anal. 2003, 54: 1069-1078. 10.1016/S0362-546X(03)00127-5MATHMathSciNetView Article
        5. Yao Q: Existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem. Nonlinear Anal. 2005, 63: 237-246. 10.1016/j.na.2005.05.009MATHMathSciNetView Article
        6. Jiang D, Liu H, Zhang L: Optimal existence theory for single and multiple positive solutions to fourth-order periodic boundary value problems. Nonlinear Anal., Real World Appl. 2006, 7: 841-852. 10.1016/j.nonrwa.2005.05.003MATHMathSciNetView Article
        7. Aftabizadeh A: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 1986, 116: 415-426. 10.1016/S0022-247X(86)80006-3MATHMathSciNetView Article
        8. Yang Y: Fourth-order two-point boundary value problems. Proc. Am. Math. Soc. 1988, 104: 175-180. 10.1090/S0002-9939-1988-0958062-3MATHView Article
        9. Del Pino MA, Manasevich RF: Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition. Proc. Am. Math. Soc. 1991, 112: 81-86.MATHMathSciNetView Article
        10. Li Y: Two-parameter nonresonance condition for the existence of fourth-order boundary value problems. J. Math. Anal. Appl. 2005, 308(1):121-128. 10.1016/j.jmaa.2004.11.021MATHMathSciNetView Article

        Copyright

        © Yang et al.; licensee Springer. 2013

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.