Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions

  • Bashir Ahmad1Email author and

    Affiliated with

    • Sotiris K Ntouyas2

      Affiliated with

      Boundary Value Problems20122012:55

      DOI: 10.1186/1687-2770-2012-55

      Received: 30 December 2011

      Accepted: 9 May 2012

      Published: 9 May 2012

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      Abstract

      This article studies a new class of nonlocal boundary value problems of nonlinear differential equations and inclusions of fractional order with strip conditions. We extend the idea of four-point nonlocal boundary conditions x 0 = σ x μ , x 1 = η x v , σ , η , 0 < μ , v < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq1_HTML.gif to nonlocal strip conditions of the form: x ( 0 ) = σ α β x ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq2_HTML.gif, x ( 1 ) = η γ δ x ( s ) d s , 0 < α < β < γ < δ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq3_HTML.gif. These strip conditions may be regarded as six-point boundary conditions. Some new existence and uniqueness results are obtained for this class of nonlocal problems by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed.

      MSC 2000: 26A33; 34A12; 34A40.

      Keywords

      fractional differential equations fractional differential inclusions nonlocal boundary conditions fixed point theorems Leray-Schauder degree

      1 Introduction

      The subject of fractional calculus has recently evolved as an interesting and popular field of research. A variety of results on initial and boundary value problems of fractional order can easily be found in the recent literature on the topic. These results involve the theoretical development as well as the methods of solution for the fractional-order problems. It is mainly due to the extensive application of fractional calculus in the mathematical modeling of physical, engineering, and biological phenomena. For some recent results on the topic, see [119] and the references therein.

      In this article, we discuss the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations and inclusions of order q ∈ (1, 2] with nonlocal strip conditions. As a first problem, we consider the following boundary value problem of fractional differential equations
      c D q x ( t ) = f ( t , x ( t ) , 0 < t < 1 , 1 < q 2 , x ( 0 ) = σ α β x ( s ) d s , x ( 1 ) = η γ δ x ( s ) d s , 0 < α < β < γ < δ < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ1_HTML.gif
      (1.1)

      where c D q denotes the Caputo fractional derivative of order q, f : 0 , 1 × http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq4_HTML.gif is a given continuous function and σ, η are appropriately chosen real numbers.

      The boundary conditions in the problem (1.1) can be regarded as six-point nonlocal boundary conditions, which reduces to the typical integral boundary conditions in the limit α, γ → 0, β, δ → 1. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. For a detailed description of the integral boundary conditions, we refer the reader to the articles [20, 21] and references therein. Regarding the application of the strip conditions of fixed size, we know that such conditions appear in the mathematical modeling of real world problems, for example, see [22, 23].

      As a second problem, we study a two-strip boundary value problem of fractional differential inclusions given by
      c D q x ( t ) F ( t , x ( t ) ) , 0 < t < 1 , 1 < q 2 , x ( 0 ) = σ α β x ( s ) d s , x ( 1 ) = η γ δ x ( s ) d s , 0 < α < β < γ < δ < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ2_HTML.gif
      (1.2)

      where F : 0 , 1 × P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq5_HTML.gif is a multivalued map, P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq6_HTML.gif is the family of all subsets of ℝ.

      We establish existence results for the problem (1.2), when the right-hand side is convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.

      The methods used are standard, however their exposition in the framework of problems (1.1) and (1.2) is new.

      2 Linear problem

      Let us recall some basic definitions of fractional calculus [2426].

      Definition 2.1 For at least n-times continuously differentiable function g : [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq7_HTML.gif, the Caputo derivative of fractional order q is defined as
      c D q g ( t ) = 1 Γ ( n - q ) 0 t ( t - s ) n - q - 1 g ( n ) ( s ) d s , n - 1 < q < n , n = [ q ] + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equa_HTML.gif

      where [q] denotes the integer part of the real number q.

      Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as
      I q g ( t ) = 1 Γ ( q ) 0 t g ( s ) ( t - s ) 1 - q d s , q > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equb_HTML.gif

      provided the integral exists.

      By a solution of (1.1), we mean a continuous function x(t) which satisfies the equation c D q x(t) = f (t, x(t)), 0 < t < 1, together with the boundary conditions of (1.1).

      To define a fixed point problem associated with (1.1), we need the following lemma, which deals with the linear variant of problem (1.1).

      Lemma 2.3 For a given g C 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq8_HTML.gif, the solution of the fractional differential equation
      c D q x ( t ) = g ( t ) , 1 < q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ3_HTML.gif
      (2.1)
      subject to the boundary conditions in (1.1) is given by
      x ( t ) = 1 Γ ( q ) 0 t ( t - s ) q - 1 g ( s ) d s + σ Δ - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) α β 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s + 1 Δ σ 2 β 2 - α 2 - ( σ ( β - α ) - 1 ) t η γ δ 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s - 0 1 ( 1 - s ) q - 1 Γ ( q ) g ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ4_HTML.gif
      (2.2)
      where
      Δ = η 2 ( δ 2 - γ 2 ) - 1 σ ( β - α ) - 1 - σ 2 ( β 2 - α 2 ) η ( δ - γ ) - 1 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equc_HTML.gif
      Proof. It is well known that the solution of (2.1) can be written as [24]
      x ( t ) = I q g ( t ) - c 0 - c 1 t = 0 t ( t - s ) q - 1 Γ ( q ) g ( s ) d s - c 0 - c 1 t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ5_HTML.gif
      (2.3)
      where c 0 , c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq9_HTML.gif are constants. Applying the boundary conditions given in (1.1), we find that
      ( σ ( β - α ) - 1 ) c 0 + σ 2 ( β 2 - α 2 ) c 1 = σ α β 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s , ( η ( δ - γ ) - 1 ) c 0 + ( η 2 ( δ 2 - γ 2 ) - 1 ) c 1 = η γ δ 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s - 0 1 ( 1 - s ) q - 1 Γ ( q ) g ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equd_HTML.gif
      Solving these equations simultaneously, we find that
      c 0 = 1 Δ η 2 ( δ 2 - γ 2 ) - 1 σ α β 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s - σ 2 ( β 2 - α 2 ) η γ δ 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s - 0 1 ( 1 - s ) q - 1 Γ ( q ) g ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Eque_HTML.gif
      c 1 = 1 Δ - ( η ( δ - γ ) - 1 ) σ α β 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s + ( σ ( β - α ) - 1 ) η γ δ 0 s ( s - m ) q - 1 Γ ( q ) g ( m ) d m d s - 0 1 ( 1 - s ) q - 1 Γ ( q ) g ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equf_HTML.gif

      Substituting the values of c0 and c1 in (2.3), we obtain the solution (2.2). □

      3 Existence results for single-valued case

      Let C = C 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq10_HTML.gif denotes the Banach space of all continuous functions from [0, 1] → ℝ endowed with the norm defined by x = sup x t , t 0 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq11_HTML.gif.

      In view of Lemma 2.3, we define an operator F : C C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq12_HTML.gif by
      ( F x ) ( t ) = 1 Γ ( q ) 0 t ( t - s ) q - 1 f ( s , x ( s ) ) d s + σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) α β 0 s ( s - m ) q - 1 f ( m , x ( m ) ) d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t γ δ 0 s ( s - m ) q - 1 f ( m , x ( m ) ) d m d s - 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 f ( s , x ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ6_HTML.gif
      (3.1)

      Observe that the problem (1.1) has solutions if and only if the operator equation F x = x has fixed points.

      For the forthcoming analysis, we need the following assumptions:

      (A 1 ) |f (t, x) - f (t, y)| ≤ L|x - y|, ∀t ∈ [0, 1], L > 0, x, y ∈ ℝ;

      (A 2 ) |f (t, x)| ≤ μ(t), ∀(t, x) ∈ [0, 1] × ℝ, and μC([0, 1], ℝ+).

      For convenience, let us set
      Λ = 1 Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ7_HTML.gif
      (3.2)
      where
      σ 2 ( β 2 - α 2 ) + ( σ ( β - α ) - 1 ) : = Δ 1 , η 2 ( δ 2 - γ 2 ) - 1 + ( η ( δ - γ ) - 1 ) : = Δ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equg_HTML.gif

      Theorem 3.1 Assume that f : 0 , 1 × http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq13_HTML.gif is a jointly continuous function and satisfies the assumption (A1) with L < 1/Λ, where Λ is given by (3.2). Then the boundary value problem (1.1) has a unique solution.

      Proof. Setting sup t 0 , 1 f t , 0 = M < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq14_HTML.gif and choosing r Λ M 1 - L Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq15_HTML.gif, we show that F B r B r , where B r = { x C : x r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq16_HTML.gif. For xB r , we have
      | | ( F x ) | | sup t [ 0 , 1 ] 1 Γ ( q ) 0 t ( t - s ) q - 1 | f ( s , x ( s ) ) | d s + σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) × α β 0 s ( s - m ) q - 1 | f ( m , x ( m ) ) | d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t γ δ 0 s ( s - m ) q - 1 | f ( m , x ( m ) ) | d m d s + 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 | f ( s , x ( s ) ) | d s sup t [ 0 , 1 ] 1 Γ ( q ) 0 t ( t - s ) q - 1 ( | f ( s , x ( s ) ) - f ( s , 0 ) | + | f ( s , 0 ) | ) | | d s + | σ | | Δ | Γ ( q ) Δ 2 α β 0 s ( s - m ) q - 1 ( | f ( m , x ( m ) ) - f ( m , 0 ) | + | f ( m , 0 ) | ) d m d s + | η | | Δ | Γ ( q ) Δ 1 γ β 0 s ( s - m ) q - 1 ( | f ( m , x ( m ) ) - f ( m , 0 ) | + | f ( m , 0 ) | ) d m d s + 1 | Δ | Γ ( q ) Δ 1 0 1 ( 1 - s ) q - 1 ( | f ( s , x ( s ) ) - f ( s , 0 ) | + | f ( s , 0 ) | ) d s ( L r + M ) sup t [ 0 , 1 ] 1 Γ ( q ) 0 t ( t - s ) q - 1 d s + | σ | | Δ | Γ ( q ) Δ 2 α β 0 s ( s - m ) q - 1 d m d s + | η | | Δ | Γ ( q ) Δ 1 γ δ 0 s ( s - m ) q - 1 d m d s + 1 | Δ | Γ ( q ) Δ 1 0 1 ( 1 - s ) q - 1 d s ( L r + M ) Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) ( q + 1 ) | Δ | + Δ 1 | η | ( δ q + 1 - γ q + 1 ) ( q + 1 ) | Δ | + Δ 1 | Δ | = ( L r + M ) Λ r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equh_HTML.gif
      Now, for x , y C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq17_HTML.gif we obtain
      | | ( F x ) ( t ) - ( F y ) | | sup t [ 0 , 1 ] 1 Γ ( q ) 0 t ( t - s ) q - 1 | f ( s , x ( s ) ) - f ( s , y ( s ) ) | d s + | σ | | Δ | Γ ( q ) Δ 2 α β 0 s ( s - m ) q - 1 | f ( m , x ( m ) ) - f ( m , y ( m ) ) | d m d s + | η | | Δ | Γ ( q ) Δ 1 γ δ 0 s ( s - m ) q - 1 | f ( m , x ( m ) ) - f ( m , y ( m ) ) | d m d s + 1 | Δ | Γ ( q ) Δ 1 0 1 ( 1 - s ) q - 1 | f ( s , x ( s ) ) - f ( s , y ( s ) ) | d s L Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | | | x - y | | = L Λ | | x - y | | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equi_HTML.gif

      where Λ is given by (3.2). Observe that Λ depends only on the parameters involved in the problem. As L < 1/ Λ, therefore F is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). □

      Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii's fixed point theorem [27].

      Theorem 3.2 (Krasnoselskii's fixed point theorem). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (i) Ax + ByM whenever x, yM; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists zM such that z = Az + Bz.

      Theorem 3.3 Let f : 0 , 1 × http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq18_HTML.gif be a jointly continuous function satisfying the assumptions (A1) and (A2) with
      L Γ ( q + 1 ) Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ8_HTML.gif
      (3.3)

      Then the boundary value problem (1.1) has at least one solution on [0, 1].

      Proof. Letting sup t 0 , 1 μ t = μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq19_HTML.gif, we fix
      r ̄ | | μ | | Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equj_HTML.gif
      and consider B r ̄ = { x C : | | x | | r ̄ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq20_HTML.gif. We define the operators http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq22_HTML.gif on B r ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq23_HTML.gif as
      ( P x ) ( t ) = 0 t ( t - s ) q - 1 Γ ( q ) f ( s , x ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equk_HTML.gif
      ( Q x ) ( t ) = σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) α β 0 s ( s - m ) q - 1 f ( m , x ( m ) ) d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t γ δ 0 s ( s - m ) q - 1 f ( m , x ( m ) ) d m d s - 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 f ( s , x ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equl_HTML.gif
      For x , y B r ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq24_HTML.gif, we find that
      | | P x + Q y | | | | μ | | Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | r ̄ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equm_HTML.gif
      Thus, P x + Q y B r ̄ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq25_HTML.gif It follows from the assumption (A1) together with (3.3) that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq22_HTML.gif is a contraction mapping. Continuity of f implies that the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq26_HTML.gif is continuous. Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq27_HTML.gif is uniformly bounded on B r ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq28_HTML.gif as
      | | P x | | | | μ | | Γ ( q + 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equn_HTML.gif

      Now we prove the compactness of the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq29_HTML.gif .

      In view of (A1), we define sup ( t , x ) [ 0 , 1 ] × B r ̄ | f ( t , x ) = f ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq30_HTML.gif, and consequently we have
      | ( P x ) ( t 1 ) - ( P x ) ( t 2 ) | = 1 Γ ( q ) 0 t 1 [ ( t 2 - s ) q - 1 - ( t 1 - s ) q - 1 ] f ( s , x ( s ) ) d s + t 1 t 2 ( t 2 - s ) q - 1 f ( s , x ( s ) ) d s f ̄ Γ ( q + 1 ) | 2 ( t 2 - t 1 ) q + t 1 q - t 2 q | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equo_HTML.gif

      which is independent of x. Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq31_HTML.gif is equicontinuous. Hence, by the Arzelá-Ascoli Theorem, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq32_HTML.gif is compact on B r ̄ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq33_HTML.gif Thus all the assumptions of Theorem 3.2 are satisfied. So the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on [0, 1]. □

      Our next existence result is based on Leray-Schauder degree theory.

      Theorem 3.4 Let f : 0 , 1 × http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq34_HTML.gif. Assume that there exist constants 0 κ < 1 Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq35_HTML.gif, where Λ is given by (3.2) and M > 0 such that |f(t, x)|κ|x|+M for all t ∈ [0, 1], xC[0, 1]. Then the boundary value problem (1.1) has at least one solution.

      Proof. Consider the fixed point problem
      x = F x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ9_HTML.gif
      (3.4)
      where F is defined by (3.1). In view of the fixed point problem (3.4), we just need to prove the existence of at least one solution xC[0, 1] satisfying (3.4). Define a suitable ball B R C[0, 1] with radius R > 0 as
      B R = { x C [ 0 , 1 ] : max t [ 0 , 1 ] | x ( t ) | < R } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equp_HTML.gif
      where R will be fixed later. Then, it is sufficient to show that F : B ̄ R C 0 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq36_HTML.gif satisfies
      x λ F x , x B R a n d λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ10_HTML.gif
      (3.5)
      Let us set
      H ( λ , x ) = λ F x , x C ( ) λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equq_HTML.gif
      Then, by the Arzelá-Ascoli Theorem, h λ (x) = x - H (λ, x) = x - λF x is completely continuous. If (3.5) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that
      deg ( h λ , B R , 0 ) = deg ( I - λ F , B R , 0 ) = deg ( h 1 , B R , 0 ) = deg ( h 0 , B R , 0 ) = deg ( I , B R , 0 ) = 1 0 , 0 B r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equr_HTML.gif
      where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, h1(t) = x - λ F x = 0 for at least one xB R . In order to prove (3.5), we assume that x = λ F x, λ ∈ [0, 1]. Then for x∂B R and t ∈ [0, 1] we have
      | x ( t ) | = | λ ( F x ) ( t ) | 1 Γ ( q ) 0 t ( t - s ) q - 1 | f ( s , x ( s ) ) | d s + σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 α β 0 s ( s - m ) q - 1 | f ( m , x ( m ) ) | d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t γ δ 0 s ( s - m ) q - 1 | f ( m , x ( m ) ) | d m d s + 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 | f ( s , x ( s ) ) | d s ( κ | | x | | + M ) 1 Γ ( q ) 0 1 ( 1 - s ) q - 1 d s + | σ | | Δ | Γ ( q ) Δ 2 α β 0 s ( s - m ) q - 1 d m d s + | η | | Δ | Γ ( q ) Δ 1 γ δ 0 s ( s - m ) q - 1 d m d s + 1 | Δ | Γ ( q ) Δ 1 0 1 ( 1 - s ) q - 1 d s κ | | x | | + M Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | = ( κ x + M ) Λ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equs_HTML.gif
      which, on taking norm sup t 0 , 1 x t = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq37_HTML.gif and solving for ‖x‖, yields
      | | x | | M Λ 1 - κ Λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equt_HTML.gif

      Letting R = M Λ 1 - κ Λ + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq38_HTML.gif, (3.5) holds. This completes the proof. □

      Example 3.5 Consider the following strip fractional boundary value problem
      c D 3 / 2 x ( t ) = 1 ( t + 2 ) 2 | x | 1 + | x | , t [ 0 , 1 ] , x ( 0 ) = 1 / 3 1 / 2 x ( s ) d s , x ( 1 ) = 2 / 3 3 / 4 x ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ11_HTML.gif
      (3.6)
      Here, q = 3/2, σ = 1, η = 1, α = 1/3, β = 1/2, γ = 2/3, δ = 3/4 and f ( t , x ) = 1 ( t + 2 ) 2 | x | 1 + | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq39_HTML.gif. As | f ( t , x ) - f ( t , y ) | 1 4 | x - y | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq40_HTML.gif, therefore, (A1) is satisfied with L = 1 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq41_HTML.gif. Further, Δ1 = 65/72, Δ2 = 535/288, Δ = 4945/5184, and
      Λ = 1 Γ ( q + 1 ) 1 + Δ 2 | σ | ( β q + 1 - α q + 1 ) + Δ 1 | η | ( δ q + 1 - γ q + 1 ) + ( q + 1 ) Δ 1 ( q + 1 ) | Δ | = 1 . 128765 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equu_HTML.gif

      Clearly, L Λ = 0.282191 < 1. Thus, by the conclusion of Theorem 3.1, the boundary value problem (3.6) has a unique solution on [0, 1].

      Example 3.6 Consider the following boundary value problem
      c D 3 / 2 x ( t ) = 1 ( 4 π ) sin ( 2 π x ) + | x | 1 + | x | , t [ 0 , 1 ] , 1 < q 2 , x ( 0 ) = 1 / 3 1 / 2 x ( s ) d s , x ( 1 ) = 2 / 3 3 / 4 x ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ12_HTML.gif
      (3.7)
      Here,
      | f ( t , x ) | = 1 ( 4 π ) sin ( 2 π x ) + | x | 1 + | x | 1 2 | x | + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equv_HTML.gif
      Clearly M = 1 and
      κ = 1 2 < 1 Λ = 0 . 885924 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equw_HTML.gif

      Thus, all the conditions of Theorem 3.4 are satisfied and consequently the problem (3.7) has at least one solution.

      4 Existence results for multi-valued case

      4.1 Preliminaries

      Let us recall some basic definitions on multi-valued maps [28, 29].

      For a normed space (X, ‖.‖), let P c l X = Y P X : Y is closed http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq42_HTML.gif, P b X = Y P X : Y is bounded http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq43_HTML.gif, P c p X = Y P X : Y is compact http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq44_HTML.gif, and P c p , c X = Y P X : Y is compact and convex http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq45_HTML.gif. A multi-valued map G : X P X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq46_HTML.gifis convex (closed) valued if G(x) is convex (closed) for all xX. The map G is bounded on bounded sets if G ( B ) = x B G ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq47_HTML.gif is bounded in X for all B P b ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq48_HTML.gif (i.e., sup x B { sup { | y | : y G ( x ) } } < ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq49_HTML.gif. G is called upper semi-continuous (u.s.c.) on X if for each x0X, the set G(x0) is a nonempty closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq50_HTML.gif of x0 such that G N 0 N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq51_HTML.gif. G is said to be completely continuous if G ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq52_HTML.gif is relatively compact for every B P b ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq53_HTML.gif. If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., x n x*, y n y*, y n G(x n ) imply y*G(x*). G has a fixed point if there is xX such that xG(x). The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map G : 0 ; 1 P c l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq54_HTML.gif is said to be measurable if for every y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq55_HTML.gif, the function
      t d ( y , G ( t ) ) = inf { | y - z | : z G ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equx_HTML.gif

      is measurable.

      Let C([0, 1]) denotes a Banach space of continuous functions from [0, 1] into ℝ with the norm x = sup t 0 , 1 x t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq56_HTML.gif. Let L1([0, 1], ℝ) be the Banach space of measurable functions x : [0, 1] → ℝ which are Lebesgue integrable and normed by x L 1 = 0 1 | x ( t ) | d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq57_HTML.gif.

      Definition 4.1 A multivalued map F : 0 , 1 × P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq58_HTML.gif is said to be Carathéodory if

      (i) tF (t, x) is measurable for each x ∈ ℝ;

      (ii) xF (t, x) is upper semicontinuous for almost all t ∈ [0, 1];

      Further a Carathéodory function F is called L 1 -Carathéodory if

      (iii) for each α > 0, there exists φ α L 1 0 , 1 , + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq59_HTML.gif such that
      | | F ( t , x ) | | = sup { | v | : v F ( t , x ) } φ α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equy_HTML.gif

      for allxα and for a. e. t ∈ [0, 1].

      For each y C 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq60_HTML.gif, define the set of selections of F by
      S F , y : = { v L 1 ( [ 0 , 1 ] , ) : v ( t ) F ( t , y ( t ) ) for a .e . t [ 0 , 1 ] } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equz_HTML.gif

      Let X be a nonempty closed subset of a Banach space E and G : X P E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq61_HTML.gif be a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set {yX : G(y) ∩ B ≠ ∅} is open for any open set B in E. Let A be a subset of [0, 1] × ℝ. A is L B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq62_HTML.gif measurable if A belongs to the σ-algebra generated by all sets of the form J × D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq63_HTML.gif, where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq64_HTML.gif is Lebesgue measurable in [0, 1] and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq65_HTML.gif is Borel measurable in ℝ. A subset http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq66_HTML.gif of L1([0, 1], ℝ) is decomposable if for all u , v A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq67_HTML.gif and measurable J 0 , 1 = J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq68_HTML.gif, the function u χ J + v χ J - J A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq69_HTML.gif, where χ J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq70_HTML.gif stands for the characteristic function of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq71_HTML.gif .

      Definition 4.2 Let Y be a separable metric space and let N : Y P L 1 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq72_HTML.gif be a multivalued operator. We say N has a property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.

      Let F : 0 , 1 × P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq73_HTML.gif be a multivalued map with nonempty compact values. Define a multivalued operator F : C 0 , 1 × P L 1 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq74_HTML.gif associated with F as
      F ( x ) = { w L 1 ( [ 0 , 1 ] , ) : w ( t ) F ( t , x ( t ) ) for a .e . t [ 0 , 1 ] } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equaa_HTML.gif

      which is called the Nemytskii operator associated with F.

      Definition 4.3 Let F : 0 , 1 × P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq75_HTML.gif be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq76_HTML.gif is lower semi-continuous and has nonempty closed and decomposable values.

      Let (X, d) be a metric space induced from the normed space (X; ‖.‖). Consider H d : P X × P X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq77_HTML.gif given by
      H d ( A , B ) = max { sup a A d ( a , B ) , sup b B d ( A , b ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equab_HTML.gif

      where d(A, b) = infaAd(a; b) and d(a, B) = infbBd(a; b). Then (P b,cl (X), H d ) is a metric space and (P cl (X), H d ) is a generalized metric space (see [30]).

      Definition 4.4 A multivalued operator N : XP cl (X) is called:

      (a) γ-Lipschitz if and only if there exists γ > 0 such that
      H d ( N ( x ) , N ( y ) ) γ d ( x , y ) f o r e a c h x , y X ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equac_HTML.gif

      (b) a contraction if and only if it is γ-Lipschitz with γ < 1.

      The following lemmas will be used in the sequel.

      Lemma 4.5 (Nonlinear alternative for Kakutani maps) [31]. Let E be a Banach space, C is a closed convex subset of E, U is an open subset of C and 0 ∈ U. Suppose that F : U ¯ P c , c v ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq78_HTML.gif is a upper semicontinuous compact map; here P c , c v C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq79_HTML.gif denotes the family of nonempty, compact convex subsets of C. Then either

      (i) F has a fixed point in U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq80_HTML.gif, or

      (ii) there is a u∂U and λ ∈ (0, 1) with uλF(u).

      Lemma 4.6 [32] Let X be a Banach space. Let F : 0 , T × P c p , c X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq81_HTML.gif be an L1-Carathéodory multivalued map and let θ be a linear continuous mapping from L1([0, 1], X) to C([0, 1], X). Then the operator
      Θ S F : C ( [ 0 , 1 ] , X ) P c p , c ( C ( [ 0 , 1 ] , X ) ) , x ( Θ S F ) ( x ) = Θ ( S F , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equad_HTML.gif

      is a closed graph operator in C([0, 1], X) × C([0, 1], X).

      Lemma 4.7 [33] Let Y be a separable metric space and let N : Y P L 1 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq82_HTML.gif be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Y L 1 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq83_HTML.gif such that g(x) ∈ N(x) for every xY .

      Lemma 4.8 [34] Let (X, d) be a complete metric space. If N : XP cl (X) is a contraction, then FixN ≠ ∅.

      Definition 4.9 A function xC2([0, 1], ℝ) is a solution of the problem (1.2) if x 0 = σ α β x ( s ) d s , x ( 1 ) = η γ δ x ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq84_HTML.gif, and there exists a function fL1([0, 1], ℝ) such that

      f(t) ∈ F (t, x(t)) a.e. on [0, 1] and
      x ( t ) = 1 Γ ( q ) 0 t ( t - s ) q - 1 f ( s ) d s + σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) α β 0 s ( s - m ) q - 1 f ( m ) d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t γ δ 0 s ( s - m ) q - 1 f ( m ) d m d s - 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 f ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equ13_HTML.gif
      (4.1)

      4.2 The Carathéodory case

      Theorem 4.10 Assume that:

      (H1) F : 0 , 1 × P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq85_HTML.gif is Carathéodory and has nonempty compact and convex values;

      (H2) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function p L 1 0 , 1 , + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq86_HTML.gif such that
      | | F ( t , x ) | | P : = sup { | y | : y F ( t , x ) } p ( t ) ψ ( | | x | | ) f o r e a c h ( t , x ) [ 0 , 1 ] × . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equae_HTML.gif
      (H3) there exists a constant M > 0 such that
      M ψ ( M ) Γ ( q ) 1 + Δ 1 | Δ | 0 1 p ( s ) d s + | σ | Δ 2 | Δ | α β 0 s ( s - m ) q - 1 p ( m ) d m d s + | η | 1 | | γ δ 0 s ( s - m ) q - 1 p ( m ) d m d s - 1 > 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equaf_HTML.gif

      Then the boundary value problem (1.2) has at least one solution on [0, 1].

      Proof. Define the operator Ω F : C ( [ 0 , 1 ] , ) P ( C ( [ 0 , 1 ] , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq87_HTML.gif by
      Ω F ( x ) = h C ( [ 0 , 1 ] , ) : h ( t ) = 1 Γ ( q ) 0 t ( t - s ) q - 1 f ( s ) d s + σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) × × α β 0 s ( s - m ) q - 1 f ( m ) d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t × × γ δ 0 s ( s - m ) q - 1 f ( m ) d m d s - 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 f ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equag_HTML.gif

      for fS F,x . We will show that Ω F satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that Ω F is convex for each xC([0, 1], ℝ). This step is obvious since S F,x is convex (F has convex values), and therefore we omit the proof.

      In the second step, we show that Ω F maps bounded sets (balls) into bounded sets in C([0, 1], ℝ). For a positive number ρ, let B ρ = {xC([0, 1], ℝ): ‖x‖ ≤ ρ} be a bounded ball in C([0, 1], ℝ). Then, for each h ∈ Ω F (x), xB ρ , there exists fS F,x such that
      h ( t ) = 1 Γ ( q ) 0 t ( t - s ) q - 1 f ( s ) d s + σ Δ Γ ( q ) - η 2 ( δ 2 - γ 2 ) - 1 + t ( η ( δ - γ ) - 1 ) α β 0 s ( s - m ) q - 1 f ( m ) d m d s + η Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t γ δ 0 s ( s - m ) q - 1 f ( m ) d m d s - 1 Δ Γ ( q ) σ 2 ( β 2 - α 2 ) - ( σ ( β - α ) - 1 ) t 0 1 ( 1 - s ) q - 1 f ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equah_HTML.gif
      Then for t∈[0, 1] we have
      h ( t ) 1 Γ ( q ) 0 t ( t - s ) q - 1 f ( s , x ( s ) ) d s + σ Δ Γ t ( q ) - η 2 δ 2 - γ 2 - 1 + t η δ - γ - 1 α β 0 s s - m q - 1 f m d m d s + η Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t γ δ 0 s s - m q - 1 f m d m d s + 1 Δ Γ ( q ) σ 2 β 2 - α 2 - σ β - α - 1 t 0 1 1 - s q - 1 f s d s ψ x 1 Γ q 0 t t - s q - 1 p s d s + σ Δ Γ q Δ 2 α β 0 s s - m q - 1 p m d m d s + | η | Δ Γ q Δ 1 γ δ 0 s s - m q - 1 p m d m d s + 1 Δ Γ q Δ 1 0 1 1 - s q - 1 p s d s ψ x Γ q 1 + Δ 1 Δ 0 1 p s d s + σ Δ 2 Δ α β 0 s s - m q - 1 f m d m d s + η Δ 1 Δ γ δ 0 s s - m q - 1 p m d m d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equai_HTML.gif
      Thus,
      h ψ ρ Γ q 1 + Δ 1 Δ 0 1 p s d s + | σ | Δ 2 Δ α β 0 s s - m q - 1 p m d m d s + η Δ 1 Δ γ δ 0 s s - m q - 1 p m d m d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equaj_HTML.gif

      Now we show that Ω F maps bounded sets into ;equicontinuous sets of C([0, 1], ℝ).

      Let t', t'' ∈ [0, 1] with t' < t'' and xB ρ . For each h ∈ Ω F (x), we obtain
      h t - h t ψ x 0 t t - s q - 1 - t - s q - 1 Γ q p s d s + ψ x t t t - s q - 1 Γ q p s d s + ψ x σ Δ Γ q η δ - γ - 1 t - t α β 0 s s - m q - 1 p m d m d s + η ψ x Δ Γ q σ β - α + 1 t - t γ δ 0 s s - m q - 1 p m d m d s + ψ x Δ Γ q σ β - α + 1 t - t 0 1 1 - s q - 1 p s d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equak_HTML.gif

      Obviously the right-hand side of the above inequality tends to zero independently of xB ρ as t'' - t' → 0. As Ω F satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that Ω F : C 0 , 1 , P C 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq88_HTML.gif is completely continuous.

      In our next step, we show that Ω F has a closed graph. Let x n x * , h n Ω F x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq89_HTML.gif, and h n h * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq90_HTML.gif. Then we need to show that h * Ω F x * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq91_HTML.gif. Associated with h n ∈ Ω F (x n ), there exists f n S F , x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq92_HTML.gif such that for each t ∈ [0, 1],
      h n t = 1 Γ q 0 t t - s q - 1 f n s d s + σ Δ Γ q - η 2 δ 2 - γ 2 - 1 + t η δ - γ - 1 α β 0 s s - m q - 1 f n m d m d s + η Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t γ δ 0 s s - m q - 1 f n m d m d s - 1 Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t 0 1 1 - s q - 1 f n s d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equal_HTML.gif
      Thus it suffices to show that there exists f * S F , x * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq93_HTML.gif such that for each t ∈ [0, 1],
      h * t = 1 Γ q 0 t t - s q - 1 f * s d s + σ Δ Γ q - η 2 δ 2 - γ 2 - 1 + t η δ - γ - 1 α β 0 s s - m q - 1 f * m d m d s + η Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t γ δ 0 s s - m q - 1 f * m d m d s - 1 Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t 0 1 1 - s q - 1 f * s d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equam_HTML.gif
      Let us consider the linear operator θ: L1([0, 1], ℝ) → C([0, 1], ℝ) given by
      f Θ f t = 1 Γ q 0 t t - s q - 1 f s d s + σ Δ Γ q - η 2 δ 2 - γ 2 - 1 + t η δ - γ - 1 α β 0 s s - m q - 1 f m d m d s + η Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t γ δ 0 s s - m q - 1 f m d m d s - 1 Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t 0 1 1 - s q - 1 f s d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equan_HTML.gif
      Observe that
      h n t - h * t = 1 Γ q 0 t t - s q - 1 f n s - f * s d s + σ Δ Γ q - η 2 δ 2 - γ 2 - 1 + t η δ - γ - 1 α β 0 s s - m q - 1 f n m - f * m d m d s + η Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t γ δ 0 s s - m q - 1 f n m - f * m d m d s - 1 Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t 0 1 1 - s q - 1 f n s - f * s d s 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_Equao_HTML.gif

      as n → ∞.

      Thus, it follows by Lemma 4.6 that θ ο S F is a closed graph operator. Further, we have h n t Θ S F , x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq94_HTML.gif. Since x n x * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-55/MediaObjects/13661_2011_Article_145_IEq95_HTML.gif, therefore, we have
      h * t = 1 Γ q 0 t t - s q - 1 f * s d s + σ Δ Γ q - η 2 δ 2 - γ 2 - 1 + t η δ - γ - 1 α β 0 s s - m q - 1 f * m d m d s + η Δ Γ q σ 2 β 2 - α 2 - σ β - α - 1 t γ δ 0 s s - m q - 1 f * m d m d s - 1 Δ Γ q σ 2 β 2 -