Approximate method for boundary value problems of anti-periodic type for differential equations with ‘maxima’

  • Snezhana Hristova1Email author,

    Affiliated with

    • Angel Golev1 and

      Affiliated with

      • Kremena Stefanova1

        Affiliated with

        Boundary Value Problems20132013:12

        DOI: 10.1186/1687-2770-2013-12

        Received: 16 October 2012

        Accepted: 9 January 2013

        Published: 25 January 2013

        Abstract

        An algorithm for constructing two sequences of successive approximations of a solution of the nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’ is given. The case of a boundary condition of anti-periodic type is investigated. This algorithm is based on the monotone iterative technique. Two sequences of successive approximations are constructed. It is proved both sequences are monotonically convergent. Each term of the constructed sequences is a solution of an initial value problem for a linear differential equation with ‘maxima’ and it is a lower/upper solution of the given problem. A computer realization of the algorithm is suggested and it is illustrated on a particular example.

        MSC:34K10, 34K25, 34B15.

        Keywords

        differential equations with ‘maxima’ nonlinear boundary value problem approximate solution computer realization

        1 Introduction

        Differential equations with ‘maxima’ are adequate models of real world problems, in which the present state depends significantly on its maximum value on a past time interval (see [14], monograph [5]).

        Note that usually differential equations with ‘maxima’ are not possible to be solved in an explicit form and that requires the application of approximate methods. In the current paper, the monotone iterative technique [6, 7], based on the method of lower and upper solutions, is theoretically proved to a boundary value problem for a nonlinear differential equation with ‘maxima’. The case when the nonlinear boundary function is a nondecreasing one with respect to its second argument is studied. This type of the boundary function covers the case of an anti-periodic boundary condition. An improved algorithm of monotone-iterative techniques is suggested. The main advantage of this scheme is connected with the construction of the initial conditions.

        2 Preliminary notes and definitions

        Let 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq1_HTML.gif be a given fixed point and h be a positive constant. Consider the set
        P ( h , T ) = { u : [ h , T ] R : u C ( [ h ; 0 ] , R ) , u C 1 ( [ 0 , T ] , R ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equa_HTML.gif
        Consider the following nonlinear differential equation with ‘maxima’:
        x ( t ) = f ( t , x ( t ) , max s [ t h , t ] x ( s ) ) for  t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ1_HTML.gif
        (1)
        with a boundary condition
        g ( x ( 0 ) , x ( T ) ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ2_HTML.gif
        (2)
        and an initial condition
        x ( t ) = x ( 0 ) for  t [ h , 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ3_HTML.gif
        (3)

        where x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq2_HTML.gif, f : [ 0 , T ] × R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq3_HTML.gif, g : R × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq4_HTML.gif.

        In this paper, we study boundary condition (2) in the case when the function g ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq5_HTML.gif is nondecreasing with respect to its second argument y. So, the anti-periodic boundary value problem is a partial case of boundary condition (2). Note that similar problems are investigated for ordinary differential equations [8], delay differential equations [9] and impulsive differential equations [10], and some approximate methods are suggested. The presence of the maximum of the unknown function requires additionally some new comparison results, existence results as well as a new algorithm for constructing successive approximations to the exact unknown solution.

        Let α , β C ( [ h , T ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq6_HTML.gif be such that α ( t ) β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq7_HTML.gif on [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq8_HTML.gif. Define the following sets:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equb_HTML.gif

        Definition 1 The function g : W ( α , β ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq9_HTML.gif is said to be from the class L ( γ , α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq10_HTML.gif if for any v [ α ( T ) , β ( T ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq11_HTML.gif and for any u 1 , u 2 [ α ( 0 ) , β ( 0 ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq12_HTML.gif such that u 1 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq13_HTML.gif, the inequality g ( u 1 , v ) g ( u 2 , v ) γ ( u 1 u 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq14_HTML.gif holds.

        Definition 2 The function g : W ( α , β ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq9_HTML.gif is said to be quasi-nondecreasing in W ( α , β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq15_HTML.gif if for any x [ α ( 0 ) , β ( 0 ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq16_HTML.gif and for any y 1 , y 2 [ α ( T ) , β ( T ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq17_HTML.gif such that y 1 y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq18_HTML.gif, the inequality g ( x , y 1 ) g ( x , y 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq19_HTML.gif holds.

        In connection with the construction of successive approximations, we will introduce a couple of quasi-solutions of boundary value problem (1)-(3).

        Definition 3 We will say that the functions α , β P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq20_HTML.gif form a couple of quasi-solutions of boundary value problem (1)-(3), if they satisfy the equations g ( α ( 0 ) , β ( T ) ) = g ( β ( 0 ) , α ( T ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq21_HTML.gif, (1) and (3).

        Definition 4 We will say that the functions α , β P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq20_HTML.gif form a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3), if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ4_HTML.gif
        (4)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ5_HTML.gif
        (5)

        In the proof of our main results, we will use the following lemma.

        Lemma 1 (Comparison result)

        Let the following conditions be fulfilled:
        1. 1.
          The functions M , L C ( [ 0 , T ] , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq22_HTML.gif satisfy the inequality
          max t [ 0 , T ] [ M ( t ) + L ( t ) ] T 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ6_HTML.gif
          (6)
           
        2. 2.
          The function u P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq23_HTML.gif satisfies the inequalities
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ7_HTML.gif
          (7)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ8_HTML.gif
          (8)
           

        Then u ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq24_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif.

        Proof Assume the statement of Lemma 1 is not true. Consider the following two cases.

        Case 1: Let u ( 0 ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq26_HTML.gif. According to the assumption, it follows that there exists η ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq27_HTML.gif such that u ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq28_HTML.gif for t [ h , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq29_HTML.gif, u ( η ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq30_HTML.gif and u ( η 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq31_HTML.gif.

        Denote min t [ h , η ] u ( t ) = λ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq32_HTML.gif, where λ is a positive constant. Let the point ξ [ 0 , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq33_HTML.gif be such that u ( ξ ) = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq34_HTML.gif.

        According to the mean value theorem, it follows that there exists ζ ( ξ , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq35_HTML.gif such that
        u ( η ) u ( ξ ) = u ( ζ ) ( η ξ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ9_HTML.gif
        (9)
        From inequalities λ min s [ ζ h , ζ ] u ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq36_HTML.gif, λ u ( ζ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq37_HTML.gif and (7), we obtain
        λ = u ( η ) u ( ξ ) = u ( ζ ) ( η ξ ) [ M ( ζ ) u ( ζ ) L ( ζ ) min s [ ζ h , ζ ] u ( s ) ] ( η ξ ) [ M ( ζ ) + L ( ζ ) ] λ ( η ξ ) < ( M ( ζ ) + L ( ζ ) ) λ T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ10_HTML.gif
        (10)

        Inequality (10) contradicts (6).

        Case 2: Let u ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq38_HTML.gif. Define a function u ˜ P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq39_HTML.gif by the equality u ˜ ( t ) = u ( t ) δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq40_HTML.gif, where δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq41_HTML.gif is a small enough constant.

        Therefore, u ˜ ( 0 ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq42_HTML.gif and u ˜ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq43_HTML.gif satisfies inequality (7). From case 1 it follows u ˜ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq44_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq45_HTML.gif. Take a limit as δ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq46_HTML.gif and obtain u ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq24_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif. □

        In our further investigations, we will use the following result for differential equations with ‘maxima’ which is a partial case of Theorem 3.1.1 [5].

        Lemma 2 (Existence and uniqueness)

        Let the following conditions be fulfilled:
        1. 1.

          The function Q C ( [ 0 , T ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq47_HTML.gif.

           
        2. 2.

          The functions M , L C ( [ 0 , T ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq48_HTML.gif and satisfy inequality (6).

           
        Then the initial value problem for a linear differential equation with ‘maxima’
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equc_HTML.gif

        has a unique solution u ( t ) P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq49_HTML.gif.

        3 Monotone-iterative method

        We will give an algorithm for obtaining an approximate solution of the boundary value problem for a nonlinear differential equation with ‘maxima’ (1)-(3).

        Theorem 1 Let the following conditions be fulfilled:
        1. 1.

          The functions α 0 , β 0 P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq50_HTML.gif form a couple of quasi-lower and quasi-upper solutions of (1)-(3) such that α 0 ( t ) β 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq51_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif.

           
        2. 2.

          The function g C ( W ( α 0 , β 0 ) , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq52_HTML.gif is quasi-nondecreasing in W ( α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq53_HTML.gif and g L ( γ , α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq54_HTML.gif.

           
        3. 3.
          The function f C ( Ω ( α 0 , β 0 ) , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq55_HTML.gif and for ( t , x 1 , y 1 ) , ( t , x 2 , y 2 ) Ω ( α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq56_HTML.gif such that x 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq57_HTML.gif, y 1 y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq18_HTML.gif the inequality
          f ( t , x 1 , y 1 ) f ( t , x 2 , y 2 ) M ( t ) [ x 1 x 2 ] L ( t ) [ y 1 y 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equd_HTML.gif
           

        holds, where the functions M , L C ( [ 0 , T ] , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq22_HTML.gif satisfy inequality (6).

        Then there exist two sequences { α n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq58_HTML.gif and { β n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq59_HTML.gif such that
        1. (a)

          The functions α n , β n P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq60_HTML.gif ( n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq61_HTML.gif) and ( α n , β n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq62_HTML.gif is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).

           
        2. (b)

          The sequence { α n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq63_HTML.gif is nondecreasing.

           
        3. (c)

          The sequence { β n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq64_HTML.gif is nonincreasing.

           
        4. (d)
          For t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif the inequalities
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ11_HTML.gif
          (11)
           
        hold.
        1. (e)

          Both sequences are uniformly convergent on [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq8_HTML.gif, and ( V , W ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq65_HTML.gif is a couple of quasi-solutions of boundary value problem (1)-(3) in S ( α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq66_HTML.gif.

           
        2. (f)

          If additionally the function f ( t , x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq67_HTML.gif is Lipschitz in Ω ( α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq68_HTML.gif, then there exists a unique solution u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq69_HTML.gif of boundary value problem (1)-(3) and lim n α n ( t ) = lim n β n ( t ) = V ( t ) = W ( t ) = u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq70_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif.

           

        Proof We will give an algorithm for construction of successive approximations to the unknown exact solution of nonlinear boundary value problem (1)-(3).

        Assume the functions α j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq71_HTML.gif and β j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq72_HTML.gif, j = 0 , 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq73_HTML.gif, are constructed. Then consider both initial value problems for the linear differential equations with ‘maxima’
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ12_HTML.gif
        (12)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ13_HTML.gif
        (13)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ14_HTML.gif
        (14)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ15_HTML.gif
        (15)
        where
        Q n + 1 ( t ) = f ( t , α n ( t ) , max s [ t h , t ] α n ( s ) ) + M ( t ) α n ( t ) + L ( t ) max s [ t h , t ] α n ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Eque_HTML.gif
        and
        P n + 1 ( t ) = f ( t , β n ( t ) , max s [ t h , t ] β n ( s ) ) + M ( t ) β n ( t ) + L ( t ) max s [ t h , t ] β n ( s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equf_HTML.gif

        According to Lemma 2, initial value problems (12), (13) and (14), (15) have unique solutions α n + 1 , β n + 1 P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq74_HTML.gif.

        So, step by step we can construct two sequences of functions { α n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq75_HTML.gif and { β n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq76_HTML.gif.

        Now, we will prove by induction that for j = 0 , 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq77_HTML.gif ,

        (H1) α j + 1 ( t ) α j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq78_HTML.gif and β j + 1 ( t ) β j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq79_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif;

        (H2) α j + 1 ( t ) β j + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq80_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif;

        (H3) ( α j + 1 , β j + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq81_HTML.gif is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).

        Assume the claims (H1)-(H3) are satisfied for j = 0 , 1 , , n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq82_HTML.gif.

        We will prove (H1) for j = n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq83_HTML.gif.

        Define the function p 1 P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq84_HTML.gif by the equality p 1 ( t ) = α n ( t ) α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq85_HTML.gif.

        Let t [ h , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq86_HTML.gif. Then according to condition 2 of Theorem 1, the inductive assumption and the definition of the functions α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq87_HTML.gif, α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq88_HTML.gif, we have
        p 1 ( t ) = α n 1 ( 0 ) α n ( 0 ) + 1 γ [ g ( α n ( 0 ) , β n ( T ) ) g ( α n 1 ( 0 ) , β n 1 ( T ) ) ] = α n 1 ( 0 ) α n ( 0 ) + 1 γ [ g ( α n ( 0 ) , β n ( T ) ) g ( α n 1 ( 0 ) , β n ( T ) ) ] + 1 γ [ g ( α n 1 ( 0 ) , β n ( T ) ) g ( α n 1 ( 0 ) , β n 1 ( T ) ) ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ16_HTML.gif
        (16)
        Let t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif. From (H1) for j = n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq90_HTML.gif, condition 3 of Theorem 1, the definition of the functions α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq87_HTML.gif, α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq88_HTML.gif and (12), we get
        p 1 ( t ) M ( t ) [ α n ( t ) α n + 1 ( t ) ] L ( t ) [ max s [ t h , t ] α n ( s ) max s [ t h , t ] α n + 1 ( s ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ17_HTML.gif
        (17)
        Note that for any t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif the following inequality holds:
        max s [ t h , t ] α n ( s ) max s [ t h , t ] α n + 1 ( s ) min s [ t h , t ] [ α n ( s ) α n + 1 ( s ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ18_HTML.gif
        (18)
        From inequalities (17) and (18) it follows
        p 1 ( t ) M ( t ) p 1 ( t ) L ( t ) min s [ t h , t ] p 1 ( s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equg_HTML.gif

        According to Lemma 1, we get p 1 ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq91_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq45_HTML.gif. Thus, α n ( t ) α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq92_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif.

        Define the function p 2 P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq93_HTML.gif by the equality p 2 ( t ) = β n + 1 ( t ) β n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq94_HTML.gif. Then for t [ h , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq86_HTML.gif we have
        p 2 ( t ) = β n ( 0 ) β n 1 ( 0 ) + 1 γ [ g ( β n 1 ( 0 ) , α n 1 ( T ) ) g ( β n ( 0 ) , α n ( T ) ) ] = β n ( 0 ) β n 1 ( 0 ) + 1 γ [ g ( β n 1 ( 0 ) , α n 1 ( T ) ) g ( β n ( 0 ) , α n 1 ( T ) ) ] + 1 γ [ g ( β n ( 0 ) , α n 1 ( T ) ) g ( β n ( 0 ) , α n ( T ) ) ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ19_HTML.gif
        (19)
        From equation (14), the inductive assumption, the definition of the functions β n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq95_HTML.gif, β n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq96_HTML.gif and condition 3 of Theorem 1, it follows the validity of the inequality
        p 2 ( t ) M ( t ) p 2 ( t ) L ( t ) min s [ t h , t ] p 2 ( s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equh_HTML.gif

        According to Lemma 1, we get p 2 ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq97_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq45_HTML.gif, i.e., the claim (H1) is true for j = n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq83_HTML.gif.

        Define the function p 3 P ( h , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq98_HTML.gif by the equality p 3 ( t ) = α n + 1 ( t ) β n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq99_HTML.gif.

        Let t [ h , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq86_HTML.gif. From condition 2 of Theorem 1, the inductive assumption and the definition of the functions α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq88_HTML.gif, β n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq96_HTML.gif, we obtain
        p 3 ( t ) = α n ( 0 ) β n ( 0 ) + 1 γ [ g ( β n ( 0 ) , α n ( T ) ) g ( α n ( 0 ) , α n ( T ) ) ] + 1 γ [ g ( α n ( 0 ) , α n ( T ) ) g ( α n ( 0 ) , β n ( T ) ) ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equi_HTML.gif
        Let t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif. According to the choice of the functions α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq88_HTML.gif, β n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq96_HTML.gif, condition 3 of Theorem 1 and inequality max s [ t h , t ] α n + 1 ( s ) max s [ t h , t ] β n + 1 ( s ) min s [ t h , t ] [ α n + 1 ( s ) β n + 1 ( s ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq100_HTML.gif, we get
        p 3 ( t ) M ( t ) p 3 ( t ) L ( t ) min s [ t h , t ] p 3 ( s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equj_HTML.gif

        According to Lemma 1, it follows p 3 ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq101_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq45_HTML.gif. Therefore, the claim (H2) is satisfied for j = n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq83_HTML.gif.

        Now, we will prove the claim (H3) for j = n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq83_HTML.gif.

        Let t [ h , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq86_HTML.gif. Then from (13) we get
        α n + 1 ( t ) = α n ( 0 ) 1 γ g ( α n ( 0 ) , β n ( T ) ) = α n + 1 ( 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ20_HTML.gif
        (20)
        From (H1) for j = n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq83_HTML.gif, condition 2 of Theorem 1 and the choice of the function α n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq88_HTML.gif, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ21_HTML.gif
        (21)
        Let t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif. From condition 3 of Theorem 1, inequalities (18) and (H1), we get
        α n + 1 ( t ) = M ( t ) [ α n + 1 ( t ) α n ( t ) ] L ( t ) [ max s [ t h , t ] α n + 1 ( s ) max s [ t h , t ] α n ( s ) ] + f ( t , α n + 1 ( t ) , max s [ t h , t ] α n + 1 ( s ) ) + [ f ( t , α n ( t ) , max s [ t h , t ] α n ( s ) ) f ( t , α n + 1 ( t ) , max s [ t h , t ] α n + 1 ( s ) ) ] f ( t , α n + 1 ( t ) , max s [ t h , t ] α n + 1 ( s ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ22_HTML.gif
        (22)

        Similarly, we prove the function β n + 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq96_HTML.gif satisfies inequalities (5). Therefore, the claim (H3) is true for j = n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq83_HTML.gif. Furthermore, the functions α n + 1 ( t ) , β n + 1 ( t ) S ( α n , β n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq102_HTML.gif.

        For any fixed t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif, the sequences { α n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq103_HTML.gif and { β n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq104_HTML.gif are nondecreasing and nonincreasing, respectively, and they are bounded by α 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq105_HTML.gif and β 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq106_HTML.gif.

        Therefore, both sequences converge pointwisely and monotonically. Let lim n α n ( t ) = V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq107_HTML.gif and lim n β n ( t ) = W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq108_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif. According to Dini’s theorem, both sequences converge uniformly and the functions V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq109_HTML.gif, W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq110_HTML.gif are continuous. Additionally, the claims (H1), (H2) prove V , W S ( α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq111_HTML.gif.

        Now, we will prove that for any t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif the following equality holds:
        lim n [ max ξ [ t h , t ] α n ( ξ ) ] = max ξ [ t h , t ] [ lim n α n ( ξ ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ23_HTML.gif
        (23)

        For any t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif, we introduce the notation max ξ t [ t h , t ] α n ( ξ t ) = A n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq112_HTML.gif. From condition (H1) it follows that for any ξ t [ t h , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq113_HTML.gif the inequalities α n 1 ( ξ t ) α n ( ξ t ) A n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq114_HTML.gif hold and thus, A n 1 ( t ) A n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq115_HTML.gif, n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq61_HTML.gif , i.e., the sequence { A n ( t ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq116_HTML.gif is monotone nondecreasing and bounded from above by β 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq117_HTML.gif for any t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif. Therefore, there exists the limit A ( t ) = lim n A n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq118_HTML.gif.

        From the monotonicity of the sequence of the quasi-lower solutions α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq87_HTML.gif, we get that for ξ t [ t h , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq113_HTML.gif the inequality α n ( ξ t ) V ( ξ t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq119_HTML.gif holds. Let η t [ t h , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq120_HTML.gif be such that max ξ t [ t h , t ] V ( ξ t ) = V ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq121_HTML.gif.

        Assume V ( η t ) < A ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq122_HTML.gif. Then there exists a natural number N such that the inequalities V ( η t ) < A N ( η t ) A ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq123_HTML.gif hold. Therefore, there exists ξ t [ η t h , η t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq124_HTML.gif such that α N ( ξ t ) = max ξ t [ η t h , η t ] α N ( ξ t ) = A N ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq125_HTML.gif or V ( η t ) < α N ( ξ t ) V ( ξ t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq126_HTML.gif. The obtained contradiction proves the assumption is not valid.

        Assume V ( η t ) > A ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq127_HTML.gif. According to the definition of the function V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq109_HTML.gif, it follows that for the fixed number η t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq128_HTML.gif, we have lim n α n ( η t ) = V ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq129_HTML.gif. Then there exists a natural number N such that A ( η t ) < α N ( η t ) V ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq130_HTML.gif and max η t [ t h , t ] α N ( η t ) = A N ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq131_HTML.gif. Therefore, α N ( η t ) max η t [ t h , t ] α N ( η t ) A ( η t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq132_HTML.gif. The obtained contradiction proves the assumption is not valid.

        Therefore, the required equality (23) is fulfilled.

        In a similar way, we can prove that for any t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif the equality
        lim n [ max ξ [ t h , t ] β n ( ξ ) ] = max ξ [ t h , t ] [ lim n β n ( ξ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ24_HTML.gif
        (24)

        holds.

        Take a limit as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq133_HTML.gif in (13) and get
        V ( t ) = V ( 0 ) 1 γ g ( V ( 0 ) , W ( T ) ) for  t [ h , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ25_HTML.gif
        (25)

        From (25) for t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq134_HTML.gif, we get g ( V ( 0 ) , W ( T ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq135_HTML.gif.

        Taking a limit in the integral equation equivalent to (12), we obtain the function V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq109_HTML.gif satisfies equation (1) for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif.

        In a similar way, we can prove that W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq110_HTML.gif satisfies equation (1) for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq89_HTML.gif and g ( W ( 0 ) , V ( T ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq136_HTML.gif. Therefore, the couple ( V , W ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq65_HTML.gif is a couple of quasi-solutions of (1)-(3) in S ( α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq66_HTML.gif such that V ( t ) W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq137_HTML.gif for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif.

        Let the function f ( t , x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq67_HTML.gif be Lipschitz. Then if (1) has a solution u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq69_HTML.gif, it is unique (see [11]). In this case, V ( t ) W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq138_HTML.gif and for t [ h , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq25_HTML.gif,
        lim n α n ( t ) = lim n β n ( t ) = V ( t ) = W ( t ) = u ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equk_HTML.gif

         □

        4 Applications

        We will apply the given above algorithm for approximate solving of a nonlinear boundary value problem.

        Example

        Consider the following nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ26_HTML.gif
        (26)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ27_HTML.gif
        (27)

        Boundary value problem (26), (27) is of type (1)-(3), where h = 0.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq139_HTML.gif, T = 0.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq140_HTML.gif, f ( t , x , y ) = 1 2 x 2 y 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq141_HTML.gif and g ( x , y ) = 3 x + x 2 + e y 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq142_HTML.gif.

        Let α 0 ( t ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq143_HTML.gif and β 0 ( t ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq144_HTML.gif. The couple ( α 0 ( t ) , β 0 ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq145_HTML.gif is a couple of quasi-lower and quasi-upper solutions of boundary value problem (26), (27).

        Let ( t , x 1 , x 2 ) , ( t , y 1 , y 2 ) { ( t , u , v ) [ 0 , 0.3 ] × [ 1 , 1 ] × [ 1 , 1 ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq146_HTML.gif and x i y i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq147_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq148_HTML.gif. Therefore,
        f ( t , x 1 , x 2 ) f ( t , y 1 , y 2 ) = x 1 y 1 ( 2 x 1 ) ( 2 y 1 ) 2 [ x 2 y 2 ] M ( t ) ( x 1 y 1 ) L ( t ) ( x 2 y 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equl_HTML.gif

        where M ( t ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq149_HTML.gif, L ( t ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq150_HTML.gif for t [ 0 , 0.3 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq151_HTML.gif. Thus, condition 3 of Theorem 1 holds.

        The function g ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq152_HTML.gif is quasi-nondecreasing with respect to y and g L ( γ , α 0 , β 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq54_HTML.gif, γ = 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq153_HTML.gif.

        The above given problem has a zero solution. We will apply the procedure given in Theorem 1 to obtain two sequences, which are monotonically convergent to 0.

        The function α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq87_HTML.gif, n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq154_HTML.gif, is a solution of problem (12), (13), which is reduced to the following linear initial value problem:
        α n ( t ) = α n ( t ) 2 max s [ t 0.1 , t ] α n ( s ) 1 2 α n ( t ) = + 1 2 α n 1 ( t ) + α n 1 ( t ) , t [ 0 , 0.3 ] , α n ( t ) = 0.4 α n 1 ( 0 ) 0.2 ( α n 1 ( 0 ) ) 2 α n ( t ) = 0.2 e β n 1 ( 0.3 ) + 0.2 , t [ 0.1 , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ28_HTML.gif
        (28)
        The function β n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq95_HTML.gif, n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq154_HTML.gif, is a solution of problem (14), (15), which is reduced to the following linear initial value problem:
        β n ( t ) = β n ( t ) 2 max s [ t 0.1 , t ] β n ( s ) 1 2 β n ( t ) = + 1 2 β n 1 ( t ) + β n 1 ( t ) , t [ 0 , 0.3 ] , β n ( t ) = 0.4 β n 1 ( 0 ) 0.2 ( β n 1 ( 0 ) ) 2 β n ( t ) = 0.2 e α n 1 ( 0.3 ) + 0.2 , t [ 0.1 , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Equ29_HTML.gif
        (29)

        According to Lemma 2, initial value problems (28) and (29) have unique solutions α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq87_HTML.gif and β n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq95_HTML.gif, respectively. Because of the presence of the maximum of the unknown function over a past time interval, there is no explicit formula for the exact solutions of (28) and (29). We use a computer program based on a modified numerical method to solve these problems (see [12]).

        Also, by a computer realization of the scheme given in Theorem 1 and applied to problems (28) and (29), we obtain the values in Table 1.
        Table 1

        Values of the successive approximations α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq87_HTML.gif and β n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq95_HTML.gif , n = 1 , 2 , 3 , 4 , 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq155_HTML.gif

        t

        0.001

        0.002

        0.299

        0.3

        β 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq156_HTML.gif

        0.32694

        0.32746

        0.42923

        0.42944

        β 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq157_HTML.gif

        0.20089

        0.20071

        0.18124

        0.18127

        β 3 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq158_HTML.gif

        0.14342

        0.14324

        0.09975

        0.09966

        β 4 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq159_HTML.gif

        0.10273

        0.10261

        0.06662

        0.06652

        β 5 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq160_HTML.gif

        0.0692

        0.06912

        0.04563

        0.04556

        α 5 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq161_HTML.gif

        −0.10599

        −0.10593

        −0.08765

        −0.08759

        α 4 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq162_HTML.gif

        −0.20876

        −0.20861

        −0.16411

        −0.16397

        α 3 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq163_HTML.gif

        −0.39236

        −0.39197

        −0.28498

        −0.28466

        α 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq164_HTML.gif

        −0.66195

        −0.66107

        −0.44103

        −0.44044

        α 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq165_HTML.gif

        −0.94199

        −0.94034

        −0.61509

        −0.61441

        From Table 1 and Figure 1, it is obvious that the sequence { α n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq166_HTML.gif is increasing and the sequence { β n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq167_HTML.gif is decreasing and both monotonically converge to the unique solution 0 of nonlinear boundary value problem (26), (27).
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_Fig1_HTML.jpg
        Figure 1

        Graphic of the successive approximations α n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq168_HTML.gif and β n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq169_HTML.gif , n = 1 , 2 , 3 , 4 , 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-12/MediaObjects/13661_2012_Article_266_IEq170_HTML.gif .

        Declarations

        Authors’ Affiliations

        (1)
        Faculty of Mathematics and Informatics, Plovdiv University

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        Copyright

        © Hristova 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.