Non-local boundary value problems for impulsive fractional integro-differential equations in Banach spaces

  • Hilmi Ergören1 and

    Affiliated with

    • Adem Kılıçman2Email author

      Affiliated with

      Boundary Value Problems20122012:145

      DOI: 10.1186/1687-2770-2012-145

      Received: 23 July 2012

      Accepted: 21 November 2012

      Published: 11 December 2012

      Abstract

      In this study, we establish some conditions for existence and uniqueness of the solutions to semilinear fractional impulsive integro-differential evolution equations with non-local conditions by using Schauder’s fixed point theorem and the contraction mapping principle.

      MSC:26A33, 34A37.

      Keywords

      boundary value problem Caputo type fractional derivative existence and uniqueness fixed point theorem impulsive integro-differential equation nonlocal condition

      1 Introduction

      The topic of fractional differential equations has received a great deal of attention from many scientists and researchers during the past decades; see, for instance, [17]. This is mostly due to the fact that fractional calculus provides an efficient and excellent instrument to describe many practical dynamical phenomena which arise in engineering and science such as physics, chemistry, biology, economy, viscoelasticity, electrochemistry, electromagnetic, control, porous media; see [813]. Moreover, many researchers study the existence of solutions for fractional differential equations; see [1416] and the references therein.

      In particular, several authors have considered a nonlocal Cauchy problem for abstract evolution differential equations having fractional order. Indeed, the nonlocal Cauchy problem for abstract evolution differential equations was studied by Byszewski [17, 18] initially. Afterwards, many authors [1921] discussed the problem for different kinds of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces. Balachandran et al. [22, 23] established the existence of solutions of quasilinear integrodifferential equations with nonlocal conditions. N’Guérékata [24] and Balachandran and Park [25] researched the existence of solutions of fractional abstract differential equations with a nonlocal initial condition. Ahmad [26] obtained some existence results for boundary value problems of fractional semilinear evolution equations. Recently, Balachandran and Trujillo [27] have investigated the nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces.

      On the other hand, the theory of impulsive differential equations for integer order has emerged in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years. One can see a significant development in impulsive theory. We refer the readers to [2831] for the general theory and applications of impulsive differential equations. Besides, some researchers (see [3235] and the references therein) have addressed the theory of boundary value problems for impulsive fractional differential equations.

      However, only a few studies were concerned with the Cauchy problem for impulsive evolution differential equations of fractional order; see [3638]. Further, in [38], Balachandran et al. studied the existence of solutions for fractional impulsive integrodifferential equations of the following type:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equa_HTML.gif

      where 0 t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq1_HTML.gif and 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq2_HTML.gif, by using the contraction mapping principle.

      Motivated by the aforementioned works, in this paper, we deal with the existence and uniqueness of solutions for a boundary value problem (BVP), for the following impulsive fractional semilinear integro-differential equation with nonlocal conditions:
      { D q C u ( t ) = A ( t ) u ( t ) + f ( t , u ( t ) , 0 t k ( t , s , u ( s ) ) d s ) , t J : = [ 0 , 1 ] , t t k , Δ u ( t k ) = I k ( u ( t k ) ) , Δ u ( t k ) = I k ( u ( t k ) ) , k = 1 , 2 , , p , α u ( 0 ) + β u ( 0 ) = g 1 ( u ) , α u ( 1 ) + β u ( 1 ) = g 2 ( u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ1_HTML.gif
      (1.1)
      where 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq3_HTML.gif, D α C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq4_HTML.gif is the Caputo fractional derivative, A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq5_HTML.gif is a bounded linear operator on a Banach space X, f C ( J × X × X , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq6_HTML.gif, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq7_HTML.gif , I k , I k C ( X , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq8_HTML.gif, g 1 , g 2 : P C ( J , X ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq9_HTML.gif ( P C ( J , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq10_HTML.gif will be defined in the next section),
      Δ u ( t k ) = u ( t k + ) u ( t k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equb_HTML.gif
      with
      u ( t k + ) = lim h 0 + u ( t k + h ) , u ( t k ) = lim h 0 u ( t k + h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equc_HTML.gif

      and Δ u ( t k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq11_HTML.gif has a similar meaning for u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq12_HTML.gif, 0 = t 0 < t 1 < t 2 < < t p < t p + 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq13_HTML.gif, and α , β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq14_HTML.gif. Here http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq15_HTML.gif . For brevity, let us take K u ( t ) = 0 t k ( t , s , u ( s ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq16_HTML.gif.

      Meanwhile, nonlinear functions f of this type with the integral term k occur in mathematical problems that are concerned with the heat flow in materials having memory and viscoelastic problems; see [39]. Also, as indicated in [40, 41], nonlocal conditions can be more useful than standard conditions to describe physical phenomena. For example, in [41], the author described the diffusion phenomenon of a small amount of gas in a transparent tube by using the formula
      g ( u ) = i = 1 m η i u ( ξ i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equd_HTML.gif

      where η i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq17_HTML.gif, i = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq18_HTML.gif are given constants and 0 < ξ 1 < ξ 2 < < ξ m < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq19_HTML.gif.

      Note that in this work, to the best of our knowledge, it is the first time that a general boundary value problem for impulsive semilinear evolution integrodifferential equations of fractional order 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq3_HTML.gif with nonlocal conditions has been considered.

      The rest of this paper is organized as follows. In Section 2, we present some notations and preliminary results about fractional calculus and differential equations to be used in the following sections. In Section 3, we discuss some existence and uniqueness results for solutions of BVP (1.1). Namely, the first result is based on Schauder’s fixed point theorem and the second one is based on Banach’s fixed point theorem. Finally, we shall give an illustrative example for our results.

      2 Preliminaries

      In order to model the real world application, the fractional differential equations are considered by using the fractional derivatives. There are many different starting points for the discussion of classical fractional calculus; see, for example, [42]. One can begin with a generalization of repeated integration. If f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq20_HTML.gif is absolutely integrable on [ 0 , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq21_HTML.gif, it can be found [42, 43]
      0 t d t n 0 t n d t n 1 0 t 3 d t 2 0 t 2 f ( t 1 ) d t 1 = 1 ( n + 1 ) ! 0 t ( t t 1 ) n 1 f ( t 1 ) d t 1 = 1 ( n + 1 ) ! t n 1 f ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Eque_HTML.gif
      where n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq22_HTML.gif and 0 t b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq23_HTML.gif. On writing Γ ( n ) = ( n 1 ) ! http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq24_HTML.gif, an immediate generalization in the form of the operation I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq25_HTML.gif defined for α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq26_HTML.gif is
      ( I α f ) ( t ) = 1 Γ ( α ) 0 t ( t t 1 ) α 1 f ( t 1 ) d t 1 = 1 Γ ( α ) t α 1 f ( t ) , 0 t < b , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ2_HTML.gif
      (2.1)

      where Γ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq27_HTML.gif is the gamma function and t α 1 f ( t ) = 0 t f ( t t 1 ) α 1 ( t 1 ) d t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq28_HTML.gif is called the convolution product of t α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq29_HTML.gif and f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq20_HTML.gif. Now Eq. (2.1) is known as a fractional integral of order α for the function f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq20_HTML.gif.

      Next, we give some basic definitions and properties of fractional calculus theory used in this paper; see [1, 4, 28, 31, 32].

      Let J 0 = [ 0 , t 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq30_HTML.gif, J 1 = ( t 1 , t 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq31_HTML.gif, …,  J k 1 = ( t k 1 , t k ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq32_HTML.gif, J k = ( t k , t k + 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq33_HTML.gif, and J : = [ 0 , T ] { t 1 , t 2 , , t p } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq34_HTML.gif, then we define the set of functions as follows:

      P C ( J , X ) = { u : J X : u C ( ( t k , t k + 1 ] , X ) , k = 0 , 1 , 2 , , p  and there exist  u ( t k + )  and u ( t k ) , k = 1 , 2 , , p  with  u ( t k ) = u ( t k ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq35_HTML.gif and

      P C 1 ( J , X ) = { u P C ( J , X ) , u C ( ( t k , t k + 1 ] , X ) , k = 0 , 1 , 2 , , p  and there exist  u ( t k + )  and u ( t k ) , k = 1 , 2 , , p  with  u ( t k ) = u ( t k ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq36_HTML.gif which is a Banach space with the norm
      u = sup t J { u P C , u P C } where  u P C : = sup { | u ( t ) | : t J } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equf_HTML.gif

      Now, B ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq37_HTML.gif denotes the Banach space of bounded linear operators from X into X with the norm A B ( X ) = sup { A ( u ) : u = 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq38_HTML.gif.

      Definition 1 [1, 4]

      The fractional (arbitrary) order integral of the function h L 1 ( J , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq39_HTML.gif of order q R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq40_HTML.gif is defined by
      I 0 + q h ( t ) = 1 Γ ( q ) 0 t ( t s ) q 1 h ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equg_HTML.gif

      where Γ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq41_HTML.gif is the Euler gamma function.

      Definition 2 [1, 4]

      For a function h given on the interval J, the Caputo-type fractional derivative of order q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq42_HTML.gif is defined by
      D 0 + q C h ( t ) = 1 Γ ( n q ) 0 t ( t s ) n q 1 h ( n ) ( s ) d s , n = [ q ] + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equh_HTML.gif

      where the function h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq43_HTML.gif has absolutely continuous derivatives up to order ( n 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq44_HTML.gif.

      Lemma 1 [1]

      Let q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq42_HTML.gif, then the differential equation
      D q C h ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equi_HTML.gif
      has the following solution:
      h ( t ) = c 0 + c 1 t + c 2 t 2 + + c n 1 t n 1 , c i R , i = 0 , 1 , 2 , , n 1 , n = [ q ] + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equj_HTML.gif

      Lemma 2 [14]

      Let q > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq42_HTML.gif, then
      I q C D q h ( t ) = h ( t ) + c 0 + c 1 t + c 2 t 2 + + c n 1 t n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equk_HTML.gif

      for some c i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq45_HTML.gif, i = 0 , 1 , 2 , , n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq46_HTML.gif, n = [ q ] + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq47_HTML.gif.

      Now, by using the Kronecker convolution product, see [7], the fractional integral becomes
      ( I α f ) ( x ) = 1 Γ ( α ) x α 1 f ( x ) ξ T 1 Γ ( α ) { x α 1 ϕ m ( x ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ3_HTML.gif
      (2.2)
      Thus, if x α 1 ϕ m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq48_HTML.gif can be integrated, then expanded in block pulse functions, the fractional integral is solved via the block pulse functions operational matrix as follows:
      1 Γ ( α ) 0 t ( t t 1 ) α 1 ϕ m ( t 1 ) d t 1 F α ϕ m ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equl_HTML.gif
      where
      ψ m ( t ) = { 1 ( m 1 i ) b t < ( m i ) b , 0 elsewhere http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equm_HTML.gif
      for m = 1 , 2 , , i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq49_HTML.gif and
      F α = ( b m ) α 1 Γ ( α + 2 ) [ 1 ξ 2 ξ 3 ξ m 0 1 ξ 2 ξ m 1 0 0 1 ξ m 2 0 0 0 0 0 0 0 1 ] ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equn_HTML.gif

      see [7].

      Now, we need the following lemma for our study.

      Lemma 3 Let 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq3_HTML.gif and h : J X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq50_HTML.gif be continuous. A function u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq51_HTML.gif is a solution of the fractional integral equation
      u ( t ) = { 0 t ( t s ) q 1 Γ ( q ) h ( s ) d s + ( β α t ) [ t k 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + i = 1 k ( t i 1 t i ( t i s ) q 1 Γ ( q ) h ( s ) d s + I i ( u ( t i ) ) ) + i = 1 k ( β α + 1 t k ) ( t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + I i ( u ( t i ) ) ) + 1 α ( g 1 ( u ) g 2 ( u ) ) ] + g 1 ( u ) α , t J 0 , t k t ( t s ) q 1 Γ ( q ) h ( s ) d s + i = 1 k ( t i 1 t i ( t i s ) q 1 Γ ( q ) h ( s ) d s + I i ( u ( t i ) ) ) + i = 1 k ( t t i ) t i 1 t i ( ( t i s ) q 2 Γ ( q 1 ) h ( s ) d s + I i ( u ( t i ) ) ) + ( β α t ) [ t k 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + i = 1 k ( t i 1 t i ( t i s ) q 1 Γ ( q ) h ( s ) d s + I i ( u ( t i ) ) ) + i = 1 k ( β α + 1 t k ) ( t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + I i ( u ( t i ) ) ) + 1 α ( g 1 ( u ) g 2 ( u ) ) ] + g 1 ( u ) α , t J k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ4_HTML.gif
      (2.3)
      if and only if u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq51_HTML.gif is a solution of the fractional BVP
      D q C u ( t ) = h ( t ) , t J Δ u ( t k ) = I k ( u ( t k ) ) , Δ u ( t k ) = I k ( u ( t k ) ) , α u ( 0 ) + β u ( 0 ) = g 1 ( u ) , α u ( 1 ) + β u ( 1 ) = g 2 ( u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ5_HTML.gif
      (2.4)

      where k = 1 , 2 , , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq52_HTML.gif.

      Proof Let u be the solution of (2.4). If t J 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq53_HTML.gif, then Lemma 2 implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equo_HTML.gif

      for some c 0 , c 1 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq54_HTML.gif.

      Applying the boundary condition α u ( 0 ) + β u ( 0 ) = g 1 ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq55_HTML.gif for t J 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq56_HTML.gif, we find that
      u ( t ) = 0 t ( t s ) q 1 Γ ( q ) h ( s ) d s + c 1 ( β α t ) + g 1 ( u ) α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ6_HTML.gif
      (2.5)
      If t J 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq57_HTML.gif, then Lemma 2 implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equp_HTML.gif
      for some d 0 , d 1 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq58_HTML.gif. Thus, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equq_HTML.gif
      In the view of
      Δ u ( t 1 ) = u ( t 1 + ) u ( t 1 ) = I 1 ( u ( t 1 ) ) and Δ u ( t 1 ) = u ( t 1 + ) u ( t 1 ) = I 1 ( u ( t 1 ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equr_HTML.gif
      we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equs_HTML.gif
      Hence,
      u ( t ) = t 1 t ( t s ) q 1 Γ ( q ) h ( s ) d s + 0 t 1 ( t 1 s ) q 1 Γ ( q ) h ( s ) d s + ( t t 1 ) [ 0 t 1 ( t 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + I 1 ( u ( t 1 ) ) ] + I 1 ( u ( t 1 ) ) + c 1 ( β α t ) + g 1 ( u ) α , for  t J 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equt_HTML.gif
      By repeating the process, for t J k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq59_HTML.gif, we have
      u ( t ) = t k t ( t s ) q 1 Γ ( q ) h ( s ) d s + i = 1 k [ t i 1 t i ( t i s ) q 1 Γ ( q ) h ( s ) d s + I i ( u ( t i ) ) ] + i = 1 k ( t t i ) [ t i 1 t i ( t i s ) q 2 Γ ( q 1 ) h ( s ) d s + I i ( u ( t i ) ) ] + c 1 ( β α t ) + g 1 ( u ) α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ7_HTML.gif
      (2.6)
      Now, applying the boundary condition
      α u ( 1 ) + β u ( 1 ) = g 2 ( u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equu_HTML.gif
      we find that
      c 1 = t k 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + i = 1 k [ t i 1 t i ( t i s ) q 1 Γ ( q ) h ( s ) d s + I i ( u ( t i ) ) ] + i = 1 k ( β α + 1 t k ) [ t i 1 t i ( t i s ) q 2 Γ ( q 1 ) h ( s ) d s + I i ( u ( t i ) ) ] + 1 α [ g 1 ( u ) g 2 ( u ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equv_HTML.gif

      Substituting the value of c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq60_HTML.gif in (2.5) and (2.6), we obtain Eq. 2.3.

      Conversely, if we assume that u satisfies the impulsive fractional integral equation (2.3), then by direct computation, we can easily see that the solution given by (2.3) satisfies (2.4). Thus, the proof of Lemma 3 is complete. □

      3 Main results

      Definition 3 A function u P C 1 ( J , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq61_HTML.gif with its q-derivative existing on J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq62_HTML.gif is said to be a solution of (1.1) if u satisfies the equation
      D q C u ( t ) = A ( t ) u ( t ) + f ( t , u ( t ) , K u ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equw_HTML.gif
      on J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq63_HTML.gif and satisfies the conditions
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equx_HTML.gif
      Now, we define the operator T : P C 1 ( J , X ) P C 1 ( J , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq64_HTML.gif by
      T u ( t ) = { t k t ( t s ) q 1 Γ ( q ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + i = 1 k ( t i 1 t i ( t i s ) q 1 Γ ( q ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + I i ( u ( t i ) ) ) + i = 1 k ( t t i ) × ( t i 1 t i ( t i s ) q 2 Γ ( q 1 ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + I i ( u ( t i ) ) ) + ( β α t ) [ t k 1 ( 1 s ) q 1 Γ ( q ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + i = 1 k ( t i 1 t i ( t i s ) q 1 Γ ( q ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + I i ( u ( t i ) ) ) + i = 1 k ( β α + 1 t k ) × ( t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) ( A ( s ) u ( s ) + f ( s , u ( s ) , K u ( s ) ) ) d s + I i ( u ( t i ) ) ) + 1 α ( g 1 ( u ) g 2 ( u ) ) ] + g 1 ( u ) α , t J k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ8_HTML.gif
      (3.1)

      Clearly, the fixed points of the operator T are the solutions of problem (1.1). To begin with, we need the following assumptions to prove the existence and uniqueness of a solution of the integral equation (2.3) which satisfies BVP (1.1):

      (A1) A : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq65_HTML.gif is a continuous bounded linear operator and there exists a constant A 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq66_HTML.gif such that A ( u ) B ( X ) A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq67_HTML.gif for all u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq68_HTML.gif;

      (A2) The function f : J × X × X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq69_HTML.gif is continuous and there exists a constant M 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq70_HTML.gif such that M 1 = max s J { f ( s , u ( s ) , K u ( s ) ) , u X } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq71_HTML.gif;

      (A3) I k , I k : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq72_HTML.gif are continuous and there exist constants M 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq73_HTML.gif and M 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq74_HTML.gif such that I k ( u ) M 2 , I k ( u ) M 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq75_HTML.gif for each u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq68_HTML.gif and k = 1 , 2 , , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq52_HTML.gif;

      (A4) There exist constants G i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq76_HTML.gif and g i : P C 1 ( J , X ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq77_HTML.gif are continuous functions such that g i ( u ) G i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq78_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq79_HTML.gif;

      (A5) There exists a constant L 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq80_HTML.gif such that
      f ( t , u 1 , v 1 ) f ( t , u 2 , v 2 ) L 1 ( u 1 u 2 + v 1 v 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equy_HTML.gif

      t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq81_HTML.gif, and u 1 , u 2 , v 1 , v 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq82_HTML.gif;

      (A6) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq83_HTML.gif is continuous and there exists a constant L 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq84_HTML.gif such that
      k ( t , s , u ) k ( t , s , v ) L 2 u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equz_HTML.gif

      for all u , v X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq85_HTML.gif;

      (A7) There exist constants L 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq86_HTML.gif, L 4 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq87_HTML.gif such that I k ( u ) I k ( v ) L 3 u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq88_HTML.gif, I k ( u ) I k ( v ) L 4 u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq89_HTML.gif for each u , v X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq85_HTML.gif and k = 1 , 2 , , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq52_HTML.gif;

      (A8) There exist constants b i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq90_HTML.gif such that g i ( u ) g i ( v ) b i u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq91_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq79_HTML.gif.

      The following are the main results of this paper. Our first result relies on Schauder’s fixed point theorem which gives an existence result for solutions of BVP (1.1).

      Theorem 1 Assume that the assumptions (A1)-(A4) hold. Then BVP (1.1) has at least one solution on J.

      Proof In order to show the existence of a solution of BVP (1.1), we need to transform BVP (1.1) to a fixed point problem by using the operator T in (3.1). Now, we shall use Schauder’s fixed point theorem to prove T has a fixed point which is then a solution of BVP (1.1). First, let us define B r = { u P C 1 ( J ) : u r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq92_HTML.gif for any r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq93_HTML.gif. Then it is clear that the set B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq94_HTML.gif is a closed, bounded and convex. The proof will be given in several steps.

      Step 1: T is continuous.

      Let { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq95_HTML.gif be a sequence such that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq96_HTML.gif in P C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq97_HTML.gif. Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equaa_HTML.gif

      Since A is a continuous operator and f, g, I, I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq98_HTML.gif are continuous functions, we have T u n T u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq99_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq100_HTML.gif.

      Step 2: T maps bounded sets into bounded sets.

      Now, it is enough to show that there exists a positive constant l such that T u l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq101_HTML.gif for each u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq102_HTML.gif. Then we have, for each t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq103_HTML.gif,
      | ( T u ) ( t ) | t k t ( t s ) q 1 Γ ( q ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | ) d s + i = 1 k t i 1 t i ( t i s ) q 1 Γ ( q ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + i = 1 k ( t t i ) t i 1 t i ( t i s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + | β α t | [ t k 1 ( 1 s ) q 1 Γ ( q ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | ) d s + i = 1 k t i 1 t i ( t i s ) q 1 Γ ( q ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + i = 1 k ( β α + 1 t k ) t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + 1 α ( | g 1 ( u ) | + | g 2 ( u ) | ) ] + 1 α | g 1 ( u ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equab_HTML.gif
      Thus,
      | ( T u ) ( t ) | ( A 1 r + M 1 ) t k t ( t s ) q 1 Γ ( q ) d s + ( A 1 r + M 1 + M 2 ) i = 1 k t i 1 t i ( t i s ) q 1 Γ ( q ) d s + ( A 1 r + M 1 + M 3 ) i = 1 k | t t i | t i 1 t i ( t i s ) q 2 Γ ( q 1 ) d s + ( β α + 1 ) [ ( A 1 r + M 1 ) ( t k 1 ( 1 s ) q 1 Γ ( q ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) d s ) + ( A 1 r + M 1 + M 2 ) i = 1 k t i 1 t i ( t i s ) q 1 Γ ( q ) d s + ( A 1 r + M 1 + M 3 ) i = 1 k ( β α + 1 ) t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) d s + 1 α ( G 1 + G 2 ) ] + 1 α G 1 1 Γ ( q + 1 ) ( β α + 2 ) ( ( A 1 r + M 1 ) ( 1 + p ) + p M 2 ) + 1 Γ ( q ) [ p ( A 1 r + M 1 + M 3 ) ( 1 + ( β α + 1 ) 2 ) + ( A 1 r + M 1 ) β α ( β α + 1 ) ] + 1 α ( β α + 1 ) ( G 1 + G 2 ) + G 1 = l . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equac_HTML.gif

      Then it follows that T u l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq101_HTML.gif.

      Step 3: T maps bounded sets into equicontinuous sets.

      Let B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq94_HTML.gif be a bounded set of P C 1 ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq104_HTML.gif as in Step 2, and let u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq102_HTML.gif. Then, letting τ 1 , τ 2 J k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq105_HTML.gif with τ 1 < τ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq106_HTML.gif, 0 k p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq107_HTML.gif, we have
      | ( T u ) ( τ 2 ) ( T u ) ( τ 1 ) | τ 1 τ 2 | ( T u ) ( s ) | d s l ˜ ( τ 2 τ 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equad_HTML.gif
      where
      | ( T u ) ( t ) | t k t ( t s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | ) d s + i = 1 k t i 1 t i ( t i s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + t k 1 ( 1 s ) q 1 Γ ( q ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | ) d s + i = 1 k t i 1 t i ( t i s ) q 1 Γ ( q ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + i = 1 k ( β α + 1 t k ) t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) ( | A ( s ) | | u ( s ) | + | f ( s , u ( s ) , K u ( s ) ) | + | I i ( u ( t i ) ) | ) d s + 1 α ( | g 1 ( u ) | + | g 2 ( u ) | ) | ( T u ) ( t ) | ( A 1 r + M 1 ) [ t k t ( t s ) q 2 Γ ( q 1 ) d s + t k 1 ( 1 s ) q 1 Γ ( q ) d s + β α t k 1 ( 1 s ) q 2 Γ ( q 1 ) d s ] + ( A 1 r + M 1 + M 2 ) i = 1 k t i 1 t i ( t i s ) q 1 Γ ( q ) d s + ( A 1 r + M 1 + M 3 ) × [ i = 1 k t i 1 t i ( t i s ) q 2 Γ ( q 1 ) d s + i = 1 k ( β α + 1 ) t i 1 t i ( t 1 s ) q 2 Γ ( q 1 ) d s ] + 1 α ( G 1 + G 2 ) 1 Γ ( q + 1 ) [ 2 ( A 1 r + M 1 ) + M 2 ] + 1 α ( G 1 + G 2 ) + 1 Γ ( q ) [ ( A 1 r + M 1 ) ( β α + 1 ) + p ( A 1 r + M 1 + M 3 ) ( β α + 2 ) ] : = l ˜ for any  t J k , 0 k p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equae_HTML.gif

      Hence, T ( B r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq108_HTML.gif is equicontinuous on all the subintervals J k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq109_HTML.gif, k = 0 , 1 , 2 , , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq110_HTML.gif. Then we can deduce that T : P C 1 ( J , X ) P C 1 ( J , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq64_HTML.gif is completely continuous as a result of the Arzela-Ascoli theorem together with Steps 1 to 3.

      As a consequence of Schauder’s fixed point theorem, we conclude that T has a fixed point. That is, BVP (1.1) has at least one solution. The proof is complete. □

      Our second result is about the uniqueness of the solution of BVP (1.1). And it depends on Banach’s fixed point theorem.

      Theorem 2 Assume that (A1)-(A8) hold with
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ9_HTML.gif
      (3.2)
      Proof First, we show that T B r B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq111_HTML.gif. Indeed, in order to do this, it is adequate to replace l with r in Step 2 in Theorem 1. Thus, T maps B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq94_HTML.gif into itself. Now, define the mapping T : C ( J , B r ) C ( J , B r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq112_HTML.gif. Then, for each t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq103_HTML.gif, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equaf_HTML.gif
      Observing the inequality
      | f ( s , u ( s ) , K u ( s ) ) f ( s , v ( s ) , K v ( s ) ) | L 1 ( | u ( s ) v ( s ) | + | K u ( s ) K v ( s ) | ) L 1 ( 1 + L 2 ) | u ( s ) v ( s ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equag_HTML.gif
      we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equah_HTML.gif
      Thus,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equai_HTML.gif
      which implies that
      ( T u ) ( t ) ( T v ) ( t ) Ω A 1 , L 1 , L 2 , L 3 , L 4 , b 1 , b 2 , q , α , β u v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equaj_HTML.gif

      Therefore, by (3.2), the operator T is a contraction. As a consequence of Banach’s fixed point theorem, we deduce that T has a fixed point which is a unique solution of BVP (1.1). □

      Example 1 Consider the following boundary value problem for impulsive integrodifferential evolution equation of fractional order:
      D 3 2 C u ( t ) = 1 20 ( cos 2 t ) u ( t ) + ( sin 7 t ) | u ( t ) | ( t + 5 ) 4 ( 1 + | u ( t ) | ) D 3 2 C u ( t ) = + 0 t e 1 25 u ( s ) d s , t [ 0 , 1 ] , t 1 2 , Δ u ( 1 2 ) = | u ( 1 2 ) | 15 + | u ( 1 2 ) | , Δ u ( 1 2 ) = | u ( 1 2 ) | 10 + | u ( 1 2 ) | 3 u ( 0 ) + u ( 0 ) = i = 1 m η i u ( ξ i ) , 3 u ( 1 ) + u ( 1 ) = j = 1 m η ˜ j u ˜ ( ξ i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equ10_HTML.gif
      (3.3)

      where 0 < η 1 < η 2 < < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq113_HTML.gif, 0 < η ˜ 1 < η ˜ 2 < < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq114_HTML.gif, and η i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq17_HTML.gif, η ˜ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq115_HTML.gif are given positive constants with i = 1 m η i < 2 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq116_HTML.gif and j = 1 m η ˜ j < 3 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq117_HTML.gif.

      Here, α = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq118_HTML.gif, β = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq119_HTML.gif, q = 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq120_HTML.gif, p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq121_HTML.gif. Obviously, A 1 = 1 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq122_HTML.gif, L 1 = 1 125 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq123_HTML.gif, L 2 = 1 25 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq124_HTML.gif, L 3 = 1 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq125_HTML.gif, L 4 = 1 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq126_HTML.gif, b 1 = 2 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq127_HTML.gif, b 2 = 3 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq128_HTML.gif and by (2.5), it can be found that
      Ω A 1 , L 1 , L 2 , L 3 , L 4 , b 1 , b 2 , q , α , β = 1 , 203 , 709 843 , 750 π + 38 135 = 0.63361 < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_Equak_HTML.gif

      Therefore, due to the fact that all the assumptions of Theorem 2 hold, BVP (3.3) has a unique solution. Besides, one can easily check the result of Theorem (1) for BVP (3.3).

      Conclusion

      In the literature, the authors consider impulsive fractional semilinear evolution integro-differential equations of order 0 < q < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq2_HTML.gif in different aspects as mentioned above. Besides, either impulsive fractional semilinear equations of order 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq3_HTML.gif or impulsive fractional integro-differential equations of order 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq3_HTML.gif are studied by different authors (see, for instance, [44, 45]). But, to the best of our knowledge, no study considering both cases has been carried out. Thus, in this article, we consider a general boundary value problem for impulsive fractional semilinear evolution integro-differential equations of order 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-145/MediaObjects/13661_2012_Article_239_IEq3_HTML.gif with nonlocal conditions. Therefore, the present results are new and complementary to previously known literature.

      Declarations

      Acknowledgements

      The authors express their sincere thanks to the referees for the careful and noteworthy reading of the manuscript and very helpful suggestions that improved the manuscript substantially. The second author gratefully acknowledges that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme (project No. 5527068).

      Authors’ Affiliations

      (1)
      Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University
      (2)
      Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia

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