Theory of fractional hybrid differential equations with linear perturbations of second type

  • Hongling Lu1,

    Affiliated with

    • Shurong Sun1Email author,

      Affiliated with

      • Dianwu Yang1 and

        Affiliated with

        • Houshan Teng1

          Affiliated with

          Boundary Value Problems20132013:23

          DOI: 10.1186/1687-2770-2013-23

          Received: 30 November 2012

          Accepted: 22 January 2013

          Published: 11 February 2013

          Abstract

          In this paper, we develop the theory of fractional hybrid differential equations with linear perturbations of second type involving Riemann-Liouville differential operators of order 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif. An existence theorem for fractional hybrid differential equations is proved under the φ-Lipschitz condition. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle which will be useful for further study of qualitative behavior of solutions is proved.

          MSC:34A40, 34A12, 34A99.

          Keywords

          fractional differential inequalities existence theorem comparison principle

          1 Introduction

          Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [114]. Although the tools of fractional calculus have been available and applicable to various fields of study, there are few papers on the investigation of the theory of fractional differential equations; see [1519]. The differential equations involving Riemann-Liouville differential operators of fractional order 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif are very important in modeling several physical phenomena [2022] and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations.

          In recent years, quadratic perturbations of nonlinear differential equations have attracted much attention. The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases. This class of hybrid differential equations includes the perturbations of original differential equations in different ways. There have been many works on the theory of hybrid differential equations, and we refer the readers to the articles [2329]. Dhage and Lakshmikantham [24] discussed the following first-order hybrid differential equation with linear perturbations of first type:
          { d d t [ x ( t ) f ( t , x ( t ) ) ] = g ( t , x ( t ) ) , a.e.  t J , x ( t 0 ) = x 0 R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equa_HTML.gif
          where f C ( J × R , R { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq2_HTML.gif and g C ( J × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq3_HTML.gif. Dhage and Jadhav [25] discussed the following first-order hybrid differential equation with linear perturbations of second type:
          { d d t [ x ( t ) f ( t , x ( t ) ) ] = g ( t , x ( t ) ) , t J , x ( t 0 ) = x 0 R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equb_HTML.gif

          where f C ( J × R , R { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq2_HTML.gif and g C ( J × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq3_HTML.gif. They established the existence and uniqueness results and some fundamental differential inequalities for hybrid differential equations initiating the study of theory of such systems and proved utilizing the theory of inequalities, its existence of extremal solutions and a comparison result.

          From the above works, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif. In this paper, we initiate the basic theory of fractional hybrid differential equations of mixed perturbations of second type involving three nonlinearities and prove the basic result such as the strict and nonstrict fractional differential inequalities, an existence theorem and maximal and minimal solutions etc. We claim that the results of this paper are a basic and important contribution to the theory of nonlinear fractional differential equations.

          2 Fractional hybrid differential equation

          Let ℝ be a real line and J = [ t 0 , t 0 + a ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq4_HTML.gif be a bounded interval in ℝ for some t 0 , a R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq5_HTML.gif with a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq6_HTML.gif. Let C ( J × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq7_HTML.gif denote the class of continuous functions f : J × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq8_HTML.gif.

          Definition 2.1 [19]

          The Riemann-Liouville fractional derivative of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq9_HTML.gif of a continuous function f : ( t 0 , + ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq10_HTML.gif is given by
          D α f ( t ) = 1 Γ ( n α ) ( d d t ) ( n ) t 0 t f ( s ) ( t s ) α n + 1 d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equc_HTML.gif

          where n = [ α ] + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq11_HTML.gif, [ α ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq12_HTML.gif denotes the integer part of number α, provided that the right-hand side is pointwise defined on ( t 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq13_HTML.gif.

          Definition 2.2 [19]

          The Riemann-Liouville fractional integral of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq9_HTML.gif of a function f : ( 0 , + ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq14_HTML.gif is given by
          I α f ( t ) = 1 Γ ( α ) t 0 t ( t s ) α 1 f ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equd_HTML.gif

          provided that the right-hand side is pointwise defined on ( t 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq13_HTML.gif.

          We consider fractional hybrid differential equations (in short FHDE) involving Riemann-Liouville differential operators of order 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif,
          { D q [ x ( t ) f ( t , x ( t ) ) ] = g ( t , x ( t ) ) , t J , x ( t 0 ) = x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ1_HTML.gif
          (2.1)

          where f , g C ( J × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq15_HTML.gif.

          By a solution of FHDE (2.1), we mean a function x C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq16_HTML.gif such that
          1. (i)

            the function t x f ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq17_HTML.gif is continuous for each x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq18_HTML.gif, and

             
          2. (ii)

            x satisfies the equations in (2.1).

             

          The theory of strict and nonstrict differential inequalities related to ODEs and hybrid differential equations is available in the literature (see [24, 25, 28, 29]). It is known that differential inequalities are useful for proving the existence of extremal solutions of ODEs and hybrid differential equations defined on J.

          3 Existence result

          In this section, we prove the existence results for FHDE (2.1) on the closed and bounded interval J = [ t 0 , t 0 + a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq19_HTML.gif under mixed Lipschitz and compactness conditions on the nonlinearities involved in it.

          We place FHDE (2.1) in the space C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq20_HTML.gif of continuous real-valued functions defined on J. Define a supremum norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq21_HTML.gif in C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq20_HTML.gif by x = sup t J | x ( t ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq22_HTML.gif. Clearly, C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq23_HTML.gif is a Banach algebra with respect to the above norm.

          We prove the existence of a solution for FHDE (2.1) by a fixed point theorem in the Banach algebra due to Dhage [30].

          Definition 3.1 Let X be a Banach space. A mapping T : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq24_HTML.gif is called φ-Lipschitzian if there exists a continuous and nondecreasing function φ : R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq25_HTML.gif such that
          T x T y φ ( x y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Eque_HTML.gif

          for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq26_HTML.gif, where φ ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq27_HTML.gif.

          Further, if φ satisfies the condition φ ( r ) < r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq28_HTML.gif, r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq29_HTML.gif, then T is called a nonlinear contraction with a control function φ.

          Lemma 3.1 [30]

          Let S be a nonempty, closed convex and bounded subset of the Banach algebra X and let A : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq30_HTML.gif and B : S X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq31_HTML.gif be two operators such that
          1. (a)

            A is nonlinear contraction,

             
          2. (b)

            B is completely continuous,

             
          3. (c)

            A x + B x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq32_HTML.gif for all x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq33_HTML.gif.

             

          Then the operator equation A x + B x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq34_HTML.gif has a solution in S.

          We consider the following hypotheses in what follows.

          (A0) The function x x f ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq35_HTML.gif is increasing in ℝ for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif.

          (A1) There exist constants M L > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq37_HTML.gif such that
          | f ( t , x ) f ( t , y ) | L | x y | M + | x y | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equf_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif and x , y R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq38_HTML.gif.

          (A3) There exists a continuous function h C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq39_HTML.gif such that
          | g ( t , x ) | h ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equg_HTML.gif

          for all x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq18_HTML.gif.

          Lemma 3.2 [19]

          Let 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif and u L 1 ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq40_HTML.gif.

          (H1) The equality D q I q u ( t ) = u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq41_HTML.gif holds.

          (H2) The equality
          I q D q u ( t ) = u ( t ) I 1 q u ( t 0 ) Γ ( q ) ( t t 0 ) q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equh_HTML.gif

          holds almost everywhere on J.

          The following lemma is useful in what follows.

          Lemma 3.3 Assume that hypothesis (A0) holds. Then, for any h C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq39_HTML.gif and 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif, the function x C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq42_HTML.gif is a solution of the FHDE
          D q [ x ( t ) f ( t , x ( t ) ) ] = h ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ2_HTML.gif
          (3.1)
          and
          x ( t 0 ) = x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ3_HTML.gif
          (3.2)
          if and only if x satisfies the hybrid integral equation (HIE)
          x ( t ) = x 0 f ( t 0 , x 0 ) + f ( t , x ( t ) ) + 1 Γ ( q ) t 0 t ( t s ) q 1 h ( s ) d s , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ4_HTML.gif
          (3.3)
          Proof Let x be a solution of the Cauchy problem (3.1) and (3.2). Since the Riemann-Liouville fractional integral I q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq43_HTML.gif is a monotone operator, thus we apply the fractional integral I q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq43_HTML.gif on both sides of (3.1). By Lemma 3.2, we have
          I q D q [ x ( t ) f ( t , x ( t ) ) ] = x ( t ) f ( t , x ( t ) ) I 1 q [ x ( t ) f ( t , x ( t ) ) ] | t = t 0 Γ ( q ) ( t t 0 ) q 1 = I q h ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equi_HTML.gif
          then by (3.2), we get
          x ( t ) f ( t , x ( t ) ) = I q h ( t ) + I 1 q [ x ( t ) f ( t , x ( t ) ) ] | t = t 0 Γ ( q ) ( t t 0 ) q 1 = x 0 f ( t 0 , x 0 ) + I q h ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equj_HTML.gif
          i.e.,
          x ( t ) = x 0 f ( t 0 , x 0 ) + f ( t , x ( t ) ) + I q h ( t ) = x 0 f ( t 0 , x 0 ) + f ( t , x ( t ) ) + 1 Γ ( q ) t 0 t ( t s ) q 1 h ( s ) d s , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equk_HTML.gif

          Thus, (3.3) holds.

          Conversely, assume that x satisfies HIE (3.3). Then applying D q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq44_HTML.gif on both sides of (3.3), (3.1) is satisfied. Again, substituting t = t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq45_HTML.gif in (3.3) yields
          x ( t 0 ) f ( t 0 , x ( t 0 ) ) = x 0 f ( t 0 , x 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equl_HTML.gif

          The map x x f ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq35_HTML.gif is increasing in ℝ for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif, the map x x f ( t 0 , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq46_HTML.gif is injective in ℝ, hence x ( t 0 ) = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq47_HTML.gif. The proof is completed. □

          Now, we are in a position to prove the following existence theorem for FHDE (2.1).

          Theorem 3.1 Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a solution defined on J.

          Proof Set X = C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq48_HTML.gif and define a subset S of X defined by
          S = { x X x N } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ5_HTML.gif
          (3.4)

          where N = | x 0 f ( t 0 , x 0 ) | + L + F 0 + a q Γ ( q + 1 ) h L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq49_HTML.gif and F 0 = sup t J | f ( t , 0 ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq50_HTML.gif.

          Clearly, S is a closed, convex and bounded subset of the Banach algebra X. Now, using the hypotheses (A0)-(A2), it can be shown by an application of Lemma 3.3, FHDE (2.1) is equivalent to the nonlinear HIE
          x ( t ) = x 0 f ( t 0 , x 0 ) + f ( t , x ( t ) ) + 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ6_HTML.gif
          (3.5)
          Define two operators A : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq30_HTML.gif and B : S X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq31_HTML.gif by
          A x ( t ) = f ( t , x ( t ) ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ7_HTML.gif
          (3.6)
          and
          B x ( t ) = x 0 f ( t 0 , x 0 ) + 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ8_HTML.gif
          (3.7)
          Then HIE (3.5) is transformed into the operator equation as
          A x ( t ) + B x ( t ) = x ( t ) , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ9_HTML.gif
          (3.8)

          We will show that the operators A and B satisfy all the conditions of Lemma 3.1.

          First, we show that A is a Lipschitz operator on X with the Lipschitz constant L. Let x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq51_HTML.gif. Then by hypothesis (A1),
          | A x ( t ) A y ( t ) | = | f ( t , x ( t ) ) f ( t , y ( t ) ) | L | x ( t ) y ( t ) | M + | x ( t ) y ( t ) | L x y M + x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equm_HTML.gif
          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Taking supremum over t, we obtain
          A x A y L x y M + x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equn_HTML.gif

          for all x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq51_HTML.gif. This shows that A is a nonlinear contraction on X with a control function φ defined by φ = L r M + r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq52_HTML.gif.

          Next, we show that B is a compact and continuous operator on S into X. First, we show that B is continuous on S. Let { x n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq53_HTML.gif be a sequence in S converging to a point x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq54_HTML.gif. Then, by the Lebesgue dominated convergence theorem,
          lim n B x n ( t ) = x 0 f ( t 0 , x 0 ) + lim n 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , x n ( s ) ) d s = x 0 f ( t 0 , x 0 ) + 1 Γ ( q ) t 0 t ( t s ) q 1 lim n g ( s , x n ( s ) ) d s = x 0 f ( t 0 , x 0 ) + 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s = B x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equo_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. This shows that B is a continuous operator on S.

          Now, we show that B is a compact operator on S. It is enough to show that B ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq55_HTML.gif is a uniformly bounded and equicontinuous set in X. On the one hand, let x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq54_HTML.gif be arbitrary. Then by hypothesis (A2),
          | B x ( t ) | = | x 0 f ( t 0 , x 0 ) | + | 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s | | x 0 f ( t 0 , x 0 ) | + 1 Γ ( q ) t 0 t ( t s ) q 1 | g ( s , x ( s ) ) | d s | x 0 f ( t 0 , x 0 ) | + 1 Γ ( q ) t 0 t ( t s ) q 1 h ( s ) d s | x 0 f ( t 0 , x 0 ) | + a q Γ ( q + 1 ) h L 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equp_HTML.gif
          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Taking supremum over t,
          B x | x 0 f ( t 0 , x 0 ) | + a q Γ ( q + 1 ) h L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equq_HTML.gif

          for all x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq54_HTML.gif. This shows that B is uniformly bounded on S.

          On the other hand, let t 1 , t 2 J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq56_HTML.gif with t 1 < t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq57_HTML.gif. Then, for any x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq54_HTML.gif, one has
          | B x ( t 1 ) B x ( t 2 ) | = | 1 Γ ( q ) t 0 t 1 ( t 1 s ) q 1 g ( s , x ( s ) ) d s 1 Γ ( q ) t 0 t 2 ( t 2 s ) q 1 g ( s , x ( s ) ) d s | | 1 Γ ( q ) t 0 t 1 ( t 1 s ) q 1 g ( s , x ( s ) ) d s 1 Γ ( q ) t 0 t 1 ( t 2 s ) q 1 g ( s , x ( s ) ) d s | + | 1 Γ ( q ) t 0 t 1 ( t 2 s ) q 1 g ( s , x ( s ) ) d s 1 Γ ( q ) t 0 t 2 ( t 2 s ) q 1 g ( s , x ( s ) ) d s | h L 1 Γ ( q + 1 ) [ | ( t 2 t 0 ) q ( t 1 t 0 ) q ( t 2 t 1 ) q | + ( t 2 t 1 ) q ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equr_HTML.gif
          Hence, for ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq58_HTML.gif, there exists a δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq59_HTML.gif such that
          | t 1 t 2 | < δ | B x ( t 1 ) B x ( t 2 ) | < ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equs_HTML.gif

          for all t 1 , t 2 J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq56_HTML.gif and for all x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq54_HTML.gif. This shows that B ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq55_HTML.gif is an equicontinuous set in X. Now, the set B ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq55_HTML.gif is a uniformly bounded and equicontinuous set in X, so it is compact by the Arzela-Ascoli theorem. As a result, B is a complete continuous operator on S.

          Next, we show that hypothesis (c) of Lemma 3.1 is satisfied. Let x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq54_HTML.gif. Then, by assumption (A1), we have
          | A x ( t ) + B x ( t ) | | A x ( t ) | + | B x ( t ) | | x 0 f ( t 0 , x 0 ) | + | f ( t , x ( t ) ) | + | 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , y ( s ) ) d s | | x 0 f ( t 0 , x 0 ) | + [ | f ( t , x ( t ) ) f ( t , 0 ) | + | f ( t , 0 ) | ] + ( 1 Γ ( q ) t 0 t ( t s ) q 1 | g ( s , x ( s ) ) | d s ) | x 0 f ( t 0 , x 0 ) | + L + F 0 + ( 1 Γ ( q ) t 0 t ( t s ) q 1 h ( s ) d s ) | x 0 f ( t 0 , x 0 ) | + L + F 0 + T q Γ ( q + 1 ) h L 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equt_HTML.gif
          Taking supremum over t,
          x | x 0 f ( t 0 , x 0 ) | + L + F 0 + T q Γ ( q + 1 ) h L 1 = N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equu_HTML.gif

          Thus, all the conditions of Lemma 3.1 are satisfied and hence the operator equation A x + B x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq34_HTML.gif has a solution in S. As a result, FHDE (2.1) has a solution defined on J. This completes the proof. □

          4 Fractional hybrid differential inequalities

          We discuss a fundamental result relative to strict inequalities for FHDE (2.1).

          Lemma 4.1 [17]

          Let m : R + R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq60_HTML.gif be locally Hölder continuous such that for any t 1 ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq61_HTML.gif, we have
          m ( t 1 ) = 0 and m ( t ) 0 for t 0 t t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ10_HTML.gif
          (4.1)
          Then it follows that
          D q m ( t 1 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ11_HTML.gif
          (4.2)
          Theorem 4.1 Assume that hypothesis (A0) holds. Suppose that there exist functions y , z : J R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq62_HTML.gif that are locally Hölder continuous such that
          D q [ y ( t ) f ( t , y ( t ) ) ] g ( t , y ( t ) ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ12_HTML.gif
          (4.3)
          and
          D q [ z ( t ) f ( t , z ( t ) ) ] g ( t , z ( t ) ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ13_HTML.gif
          (4.4)
          one of the inequalities being strict. Then
          y ( t 0 ) < z ( t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ14_HTML.gif
          (4.5)
          implies
          y ( t ) < z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ15_HTML.gif
          (4.6)

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif.

          Proof Suppose that inequality (4.4) is strict. Assume that the claim is false. Then there exists a t 1 J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq63_HTML.gif, t 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq64_HTML.gif, such that y ( t 1 ) = z ( t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq65_HTML.gif and y ( t ) < z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq66_HTML.gif for t 0 t < t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq67_HTML.gif.

          Define
          Y ( t ) = y ( t ) f ( t , y ( t ) ) and Z ( t ) = z ( t ) f ( t , z ( t ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equv_HTML.gif
          Then we have Y ( t 1 ) = Z ( t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq68_HTML.gif and by virtue of hypothesis (A0), we get Y ( t ) < Z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq69_HTML.gif for all t 0 t < t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq67_HTML.gif. Setting m ( t ) = Y ( t ) Z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq70_HTML.gif, t 0 t t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq71_HTML.gif, we find that m ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq72_HTML.gif, t 0 t t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq71_HTML.gif and m ( t 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq73_HTML.gif. Then by Lemma 4.1, we have D q m ( t 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq74_HTML.gif. By (4.3) and (4.4), we obtain
          g ( t 1 , y ( t 1 ) ) D q Y ( t 1 ) D q Z ( t 1 ) > g ( t 1 , z ( t 1 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equw_HTML.gif

          This is a contradiction to y ( t 1 ) = z ( t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq65_HTML.gif. Hence, the conclusion (4.6) is valid and the proof is complete. □

          The next result is concerned with nonstrict fractional differential inequalities which require a kind of one-sided φ-Lipshitz condition.

          Theorem 4.2 Assume that the conditions of Theorem 4.1 hold. Suppose that there exists a real number M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq75_HTML.gif such that
          g ( t , x 1 ) g ( t , x 2 ) M 1 + t q ( x 1 f ( t , x 1 ) ) ( x 2 f ( t , x 2 ) ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ16_HTML.gif
          (4.7)
          for all x 1 , x 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq76_HTML.gif with x 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq77_HTML.gif. Then y ( 0 ) z ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq78_HTML.gif implies, provided M a q 1 Γ ( 1 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq79_HTML.gif,
          y ( t ) z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ17_HTML.gif
          (4.8)

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif.

          Proof

          We set
          z ε ( t ) f ( t , z ε ( t ) ) = z ( t ) f ( t , z ( t ) ) + ε ( 1 + t q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equx_HTML.gif
          for small ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq58_HTML.gif, so that we have
          z ε ( t ) f ( t , z ε ( t ) ) > z ( t ) f ( t , z ( t ) ) z ε ( t ) > z ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ18_HTML.gif
          (4.9)
          Define Z ε ( t ) = z ε ( t ) f ( t , z ε ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq80_HTML.gif and Z ( t ) = z ( t ) f ( t , z ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq81_HTML.gif for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Since
          g ( t , z ε ( t ) ) g ( t , z ( t ) ) M 1 + t q ( ( z ε ( t ) f ( t , z ε ( t ) ) ) ( z ( t ) f ( t , z ( t ) ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equy_HTML.gif
          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif and M a q 1 Γ ( 1 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq79_HTML.gif, one has
          D q Z ε ( t ) = D q Z ( t ) + ε D q ( 1 + t q ) g ( t , z ( t ) ) + ε ( 1 t q Γ ( 1 q ) + Γ ( 1 + q ) ) g ( t , z ε ( t ) ) M ε + ε 1 t q Γ ( 1 q ) + ε Γ ( 1 + q ) > g ( t , z ε ( t ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equz_HTML.gif

          Also, we have z ε ( 0 ) > z ( 0 ) y ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq82_HTML.gif. Hence, by an application of Theorem 4.1 with z = z ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq83_HTML.gif yields that y ( t ) < z ε ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq84_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. By the arbitrariness of ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq58_HTML.gif, taking the limits as ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq85_HTML.gif, we have y ( t ) z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq86_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. This completes the proof. □

          Remark 4.1 Let f ( t , x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq87_HTML.gif and g ( t , x ) = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq88_HTML.gif. We can easily verify that f and g satisfy the condition (4.7).

          5 Existence of maximal and minimal solutions

          In this section, we prove the existence of maximal and minimal solutions for FHDE (2.1) on J = [ t 0 , t 0 + a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq89_HTML.gif. We need the following definition in what follows.

          Definition 5.1 A solution r of FHDE (2.1) is said to be maximal if for any other solution x to FHDE (2.1), one has x ( t ) r ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq90_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Similarly, a solution ρ of FHDE (2.1) is said to be minimal if ρ ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq91_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif, where x is any solution of FHDE (2.1) on J.

          We discuss the case of a maximal solution only, as the case of a minimal solution is similar and can be obtained with the same arguments with appropriate modifications. Given an arbitrary small real number ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq58_HTML.gif, consider the following initial value problem of FHDE of order 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq1_HTML.gif,
          { D q [ x ( t ) f ( t , x ( t ) ) ] = g ( t , x ( t ) ) + ε , a.e.  t J , x ( t 0 ) = x 0 + ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ19_HTML.gif
          (5.1)

          where f , g C ( J × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq92_HTML.gif.

          An existence theorem for FHDE (5.1) can be stated as follows.

          Theorem 5.1 Assume that hypotheses (A0)-(A2) hold. Then, for every small number ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq58_HTML.gif, FHDE (5.1) has a solution defined on J.

          Proof The proof is similar to Theorem 3.1 and we omit the details. □

          Our main existence theorem for a maximal solution for FHDE (2.1) is as follows.

          Theorem 5.2 Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a maximal solution defined on J.

          Proof Let { ε n } 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq93_HTML.gif be a decreasing sequence of positive real numbers such that lim n ε n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq94_HTML.gif. By Theorem 5.1, then there exists a solution r ( t , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq95_HTML.gif of the FHDE defined on J
          { D q [ x ( t ) f ( t , x ( t ) ) ] = g ( t , x ( t ) ) + ε n , t J , x ( t 0 ) = x 0 + ε n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ20_HTML.gif
          (5.2)
          Then, for any solution u of FHDE (2.1), any solution of auxiliary problem (5.2) satisfies
          D q [ r ( t , ε n ) f ( t , r ( t , ε n ) ) ] = g ( t , r ( t , ε n ) ) + ε n > g ( t , r ( t , ε n ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equaa_HTML.gif
          where u ( t 0 ) = x 0 x 0 + ε n = r ( t 0 , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq96_HTML.gif. By Theorem 4.2, we infer that
          u ( t ) r ( t , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ21_HTML.gif
          (5.3)

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif and n N { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq97_HTML.gif.

          Since x 0 + ε 2 = r ( t 0 , ε 2 ) r ( t 0 , ε 1 ) = x 0 + ε 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq98_HTML.gif, then by Theorem 4.2, we infer that r ( t , ε 2 ) r ( t , ε 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq99_HTML.gif. Therefore, r ( t , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq95_HTML.gif is a decreasing sequence of positive real numbers, the limit
          r ( t ) = lim n r ( t , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ22_HTML.gif
          (5.4)
          exists. We show that the convergence in (5.4) is uniform on J. To finish, it is enough to prove that the sequence r ( t , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq95_HTML.gif is equicontinuous in C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq100_HTML.gif. Let t 1 , t 2 J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq101_HTML.gif with t 1 < t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq57_HTML.gif be arbitrary. Then
          | r ( t 1 , ε n ) r ( t 2 , ε n ) | = | ( f ( t 1 , r ( t 1 , ε n ) ) + 1 Γ ( q ) t 0 t 1 ( t 1 s ) q 1 ( g ( s , r ( s , ε n ) ) + ε n ) d s ) ( f ( t 2 , r ( t 2 , ε n ) ) + 1 Γ ( q ) t 0 t 2 ( t 2 s ) q 1 ( g ( s , r ( s , ε n ) ) + ε n ) d s ) | | f ( t 1 , r ( t 1 , ε n ) ) f ( t 2 , r ( t 2 , ε n ) ) | + 1 Γ ( q ) t 1 t 2 ε n d s + 1 Γ ( q ) | t 0 t 1 ( ( t 1 s ) q 1 ( t 2 s ) q 1 ) g ( s , r ( s , ε n ) ) d s | + 1 Γ ( q ) | t 1 t 2 ( t 2 s ) q 1 g ( s , r ( s , ε n ) ) d s | | f ( t 1 , r ( t 1 , ε n ) ) f ( t 2 , r ( t 2 , ε n ) ) | + ε n Γ ( q ) ( t 2 t 1 ) + h L 1 Γ ( q + 1 ) [ ( t 2 t 0 ) q ( t 1 t 0 ) q ] + ( t 2 t 1 ) q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equab_HTML.gif
          Since f is continuous on a compact set J × [ N , N ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq102_HTML.gif, it is uniformly continuous there. Hence,
          | f ( t 1 , r ( t 1 , ε n ) ) f ( t 2 , r ( t 2 , ε n ) ) | 0 as  t 1 t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equac_HTML.gif

          uniformly for all n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq103_HTML.gif.

          Therefore, from the above inequality, it follows that
          | r ( t 1 , ε n ) r ( t 2 , ε n ) | 0 as  t 1 t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equad_HTML.gif
          uniformly for all n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq103_HTML.gif. Therefore,
          r ( t , ε n ) r ( t ) as  n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equae_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif.

          Next, we show that the function r ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq104_HTML.gif is a solution of FHDE (2.1) defined on J. Now, since r ( t , ε n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq95_HTML.gif is a solution of FHDE (5.2), we have
          r ( t , ε n ) = x 0 + ε n + f ( t , r ( t , ε n ) ) + 1 Γ ( q ) t 0 t ( t s ) q 1 ( g ( s , r ( s , ε n ) ) + ε n ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ23_HTML.gif
          (5.5)
          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Taking the limit as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq105_HTML.gif in above Eq. (5.5) yields
          r ( t ) = f ( t , r ( t ) ) + 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , r ( s ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equaf_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Thus, the function r is a solution of FHDE (2.1) on J. Finally, from inequality (5.3), it follows that u ( t ) r ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq106_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Hence, FHDE (2.1) has a maximal solution on J. This completes the proof. □

          6 Comparison theorems

          The main problem of differential inequalities is to estimate a bound for the solution set for the differential inequality related to FHDE (2.1). In this section, we prove that the maximal and minimal solutions serve as bounds for the solutions of the related differential inequality to FHDE (2.1) on J = [ t 0 , t 0 + a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq107_HTML.gif.

          Theorem 6.1 Assume that hypotheses (A0)-(A2) hold. Suppose that there exists a real number M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq75_HTML.gif such that
          g ( t , x 1 ) g ( t , x 2 ) M 1 + t q [ ( x 1 f ( t , x 1 ) ) ( x 2 f ( t , x 2 ) ) ] , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equag_HTML.gif
          for all x 1 , x 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq76_HTML.gif with x 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq77_HTML.gif, where M a q 1 Γ ( 1 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq79_HTML.gif. Furthermore, if there exists a function u C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq108_HTML.gif such that
          { D q [ u ( t ) f ( t , u ( t ) ) ] g ( t , u ( t ) ) , a.e. t J , u ( t 0 ) x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ24_HTML.gif
          (6.1)
          Then
          u ( t ) r ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ25_HTML.gif
          (6.2)

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif, where r is a maximal solution of FHDE (2.1) on J.

          Proof Let ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq58_HTML.gif be arbitrary small. By Theorem 5.2, r ( t , ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq109_HTML.gif is a maximal solution of FHDE (5.1) and the limit
          r ( t ) = lim ε 0 r ( t , ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ26_HTML.gif
          (6.3)
          is uniform on J and the function r is a maximal solution of FHDE (2.1) on J. Hence, we obtain
          { D q [ r ( t , ε ) f ( t , r ( t , ε ) ) ] = g ( t , r ( t , ε ) ) + ε , t J , r ( t 0 , ε ) = x 0 + ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equah_HTML.gif
          From the above inequality, it follows that
          { D q [ r ( t , ε ) f ( t , r ( t , ε ) ) ] > g ( t , r ( t , ε ) ) , a.e.  t J , r ( t 0 , ε ) > x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ27_HTML.gif
          (6.4)

          Now, we apply Theorem 4.2 to inequalities (6.1) and (6.4) and conclude that u ( t ) < r ( t , ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq110_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. This further, in view of limit (6.3), implies that inequality (6.2) holds on J. This completes the proof. □

          Theorem 6.2 Assume that hypotheses (A0)-(A2) hold. Suppose that there exists a real number M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq75_HTML.gif such that
          g ( t , x 1 ) g ( t , x 2 ) M 1 + t q [ ( x 1 f ( t , x 1 ) ) ( x 2 f ( t , x 2 ) ) ] , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equai_HTML.gif
          for all x 1 , x 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq76_HTML.gif with x 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq77_HTML.gif, where M T q 1 Γ ( 1 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq111_HTML.gif. Furthermore, if there exists a function u C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq108_HTML.gif such that
          { D q [ v ( t ) f ( t , v ( t ) ) ] g ( t , v ( t ) ) , a.e. t J , v ( t 0 ) > x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equaj_HTML.gif
          Then
          ρ ( t ) v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equak_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif, where ρ is a minimal solution of FHDE (2.1) on J.

          Note that Theorem 6.1 is useful to prove the boundedness and uniqueness of the solutions for FHDE (2.1) on J. A result in this direction is as follows.

          Theorem 6.3 Assume that hypotheses (A0)-(A2) hold. Suppose that there exists a function G : J × R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq112_HTML.gif such that
          | g ( t , x 1 ) g ( t , x 2 ) | G ( t , | ( x 1 ( s ) f ( t , x 1 ( s ) ) ) ( x 2 ( s ) f ( t , x 2 ( s ) ) ) | ) , a.e. t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equal_HTML.gif
          for all x 1 , x 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq76_HTML.gif. If an identically zero function is the only solution of the differential equation
          D q m ( t ) = G ( t , m ( t ) ) a.e. t J , m ( t 0 ) = x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ28_HTML.gif
          (6.5)

          then FHDE (2.1) has a unique solution on J.

          Proof By Theorem 3.1, FHDE (2.1) has a solution defined on J. Suppose that there are two solutions u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq113_HTML.gif and u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq114_HTML.gif of FHDE (2.1) existing on J. Define a function m : J R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq115_HTML.gif by
          m ( t ) = | ( u 1 ( t ) f ( t , u 1 ( t ) ) ) ( u 2 ( t ) f ( t , u 2 ( t ) ) ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equam_HTML.gif
          As D α ( | x ( t ) | ) | D α x ( t ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq116_HTML.gif for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif, we have
          D q m ( t ) = | D q ( u 1 ( t ) f ( t , u 1 ( t ) ) ) D q ( u 2 ( t ) f ( t , u 2 ( t ) ) ) | = | g ( t , u 1 ( t ) ) g ( t , u 2 ( t ) ) | G ( t , ( u 1 f ( t , u 1 ) ) ( u 2 f ( t , u 2 ) ) ) = G ( t , m ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equan_HTML.gif

          for almost everywhere t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif, and m ( t 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq117_HTML.gif.

          Now, we apply Theorem 6.1 with f ( t , x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq87_HTML.gif to get that m ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq118_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. This gives
          u 1 f ( t , u 1 ) = ( u 2 f ( t , u 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equao_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Then we can get u 1 = u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq119_HTML.gif in view of hypothesis (A0). This completes the proof. □

          7 Existence of extremal solutions in a vector segment

          Sometimes it is desirable to have knowledge of the existence of extremal positive solutions for FHDE (2.1) on J. In this section, we prove the existence of maximal and minimal positive solutions for FHDE (2.1) between the given upper and lower solutions on J = [ t 0 , t 0 + a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq19_HTML.gif. We use a hybrid fixed point theorem of Dhage [26] in ordered Banach spaces for establishing our results. We need the following preliminaries in what follows.

          A nonempty closed set K in a Banach algebra X is called a cone with vertex 0 if
          1. (i)

            K + K K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq120_HTML.gif,

             
          2. (ii)

            λ K K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq121_HTML.gif for λ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq122_HTML.gif, λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq123_HTML.gif,

             
          3. (iii)

            ( K ) K = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq124_HTML.gif, where 0 is the zero element of X,

             
          4. (iv)

            A cone K is called positive if K K K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq125_HTML.gif, where ∘ is a multiplication composition in X.

             

          We introduce an order relation ‘≤’ in X as follows. Let x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq51_HTML.gif. Then x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq126_HTML.gif if and only if y x K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq127_HTML.gif. A cone K is called normal if the norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq21_HTML.gif is semi-monotone increasing on K, that is, there is a constant N > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq128_HTML.gif such that x N y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq129_HTML.gif for all x , y K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq130_HTML.gif with x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq126_HTML.gif. It is known that if the cone K is normal in X, then every order-bounded set in X is norm-bounded. The details of cones and their properties appear in Heikkilä and Lakshmikantham [31].

          Lemma 7.1 [26]

          Let K be a positive cone in a real Banach algebra X and let u 1 , u 2 , v 1 , v 2 K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq131_HTML.gif be such that u 1 v 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq132_HTML.gif and u 2 v 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq133_HTML.gif. Then u 1 u 2 v 1 v 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq134_HTML.gif.

          For any a , b X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq135_HTML.gif, the order interval [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif is a set in X given by
          [ a , b ] = { x X : a x b } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equap_HTML.gif

          Definition 7.1 A mapping T : [ a , b ] X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq137_HTML.gif is said to be nondecreasing or monotone increasing if x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq126_HTML.gif implies T x T y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq138_HTML.gif for all x , y [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq139_HTML.gif.

          We use the following fixed point theorems of Dhage [27] for proving the existence of extremal solutions for IVP (2.1) under certain monotonicity conditions.

          Lemma 7.2 [27]

          Let K be a cone in a Banach algebra X and let a , b X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq135_HTML.gif be such that a b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq140_HTML.gif. Suppose that A , B : [ a , b ] K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq141_HTML.gif are two nondecreasing operators such that
          1. (a)

            A is a nonlinear contraction,

             
          2. (b)

            B is completely continuous,

             
          3. (c)

            A x + B x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq142_HTML.gif for each x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq143_HTML.gif.

             

          Further, if the cone K is positive and normal, then the operator equation A x + B x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq34_HTML.gif has a least and a greatest positive solution in [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif.

          We equip the space C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq20_HTML.gif with the order relation ≤ with the help of a cone K defined by
          K = { x C ( J , R ) : x ( t ) 0 , t J } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equ29_HTML.gif
          (7.1)

          It is well known that the cone K is positive and normal in C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq144_HTML.gif. We need the following definitions in what follows.

          Definition 7.2 A function a C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq145_HTML.gif is called a lower solution of FHDE (2.1) defined on J if it satisfies (4.3). Similarly, a function a C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq146_HTML.gif is called an upper solution of FHDE (2.1) defined on J if it satisfies (4.4). A solution to FHDE (2.1) is a lower as well as an upper solution for FHDE (2.1) defined on J and vice versa.

          We consider the following set of assumptions:

          (B0) f : J × R R + { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq147_HTML.gif, g : J × R R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq148_HTML.gif.

          (B1) FHDE (2.1) has a lower solution a and an upper solution b defined on J with a b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq140_HTML.gif.

          (B2) The function x x f ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq149_HTML.gif is increasing in the interval [ min t J a ( t ) , max t J b ( t ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq150_HTML.gif almost everywhere for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif.

          (B3) The functions f ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq151_HTML.gif and g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq152_HTML.gif are nondecreasing in x almost everywhere for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif.

          (B4) There exists a function k L 1 ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq153_HTML.gif such that g ( t , b ( t ) ) k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq154_HTML.gif.

          We remark that hypothesis (B4) holds in particular if f is continuous and g is L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq155_HTML.gif-Carathéodory on J × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq156_HTML.gif.

          Theorem 7.1 Suppose that assumptions (A1) and (B0)-(B4) hold. Then FHDE (2.1) has a minimal and a maximal positive solution defined on J.

          Proof Now, FHDE (2.1) is equivalent to integral equation (3.5) defined on J. Let X = C ( J , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq157_HTML.gif. Define two operators A and B on X by (3.6) and (3.7) respectively. Then the integral equation (3.5) is transformed into an operator equation A x ( t ) + B x ( t ) = x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq158_HTML.gif in the Banach algebra X. Notice that hypothesis (B0) implies A , B : [ a , b ] K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq159_HTML.gif. Since the cone K in X is normal, [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif is a norm bounded set in X. Now it is shown, as in the proof of Theorem 3.1, that A is a Lipschitzian with the Lipschitz constant L and B is a completely continuous operator on [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif. Again, hypothesis (B3) implies that A and B are nondecreasing on [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif. To see this, let x , y [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq139_HTML.gif be such that x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq126_HTML.gif. Then, by hypothesis (B3),
          A x ( t ) = f ( t , x ( t ) ) f ( t , y ( t ) ) = A y ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equaq_HTML.gif
          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. Similarly, we have
          B x ( t ) = x 0 f ( t 0 , x 0 ) + 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s x 0 f ( t 0 , x 0 ) + 1 Γ ( q ) t 0 t ( t s ) q 1 g ( s , y ( s ) ) d s = B y ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equar_HTML.gif
          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif. So, A and B are nondecreasing operators on [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif. Lemma 7.1 and hypothesis (B3) together imply that
          a ( t ) x 0 f ( t 0 , x 0 ) + f ( a , a ( t ) ) f ( t , a ( t ) ) Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s x 0 f ( t 0 , x 0 ) + f ( t , x ( t ) ) + f ( t , x ( t ) ) Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s x 0 f ( t 0 , x 0 ) + f ( t , b ( t ) ) + f ( t , b ( t ) ) Γ ( q ) t 0 t ( t s ) q 1 g ( s , x ( s ) ) d s b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_Equas_HTML.gif

          for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif and x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq143_HTML.gif. As a result, a ( t ) A x ( t ) + B x ( t ) b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq160_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq36_HTML.gif and x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq143_HTML.gif. Hence, A x + B x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq142_HTML.gif for all x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq143_HTML.gif.

          Now, we apply Lemma 7.2 to the operator equation A x B x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq161_HTML.gif to yield that FHDE (2.1) has a minimal and a maximal positive solution in [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-23/MediaObjects/13661_2012_Article_273_IEq136_HTML.gif defined on J. This completes the proof. □

          Declarations

          Acknowledgements

          Dedicated to Professor Hari M Srivastava.

          This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

          Authors’ Affiliations

          (1)
          School of Mathematical Sciences, University of Jinan

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          © Lu et al.; licensee Springer. 2013

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