Open Access

Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces

  • Dumitru Baleanu1, 2, 3Email author,
  • Ravi P Agarwal4,
  • Hakimeh Mohammadi5 and
  • Shahram Rezapour5
Boundary Value Problems20132013:112

DOI: 10.1186/1687-2770-2013-112

Received: 9 August 2012

Accepted: 16 April 2013

Published: 3 May 2013

Abstract

By using fixed point results on cones, we study the existence and uniqueness of positive solutions for some nonlinear fractional differential equations via given boundary value problems. Examples are presented in order to illustrate the obtained results.

1 Introduction

The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [16] and the references therein). It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. There are some papers dealing with the existence of solutions of nonlinear initial value problems of fractional differential equations by using the techniques of nonlinear analysis such as fixed point results, the Leray-Schauder theorem, stability, etc. (see, for example, [719] and the references therein). In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics (see, for example, [2023] and the references therein). The main advantage of using the fractional nonlinear differential equations is related to the fact that we can describe the dynamics of complex non-local systems with memory. In this line of taught, the equations involving various fractional orders are important from both theoretical and applied view points. We need the following notions.

Definition 1.1 ([1, 4])

For a continuous function f : [ 0 , ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq1_HTML.gif, the Caputo derivative of fractional order α is defined by
D α c f ( t ) = 1 Γ ( n α ) 0 t ( t s ) n α 1 f ( n ) ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equa_HTML.gif

where n 1 < α < n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq2_HTML.gif, n = [ α ] + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq3_HTML.gif and [ α ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq4_HTML.gif denotes the integer part of α.

Definition 1.2 ([1, 4])

The Riemann-Liouville fractional derivative of order α for a continuous function f is defined by
D α f ( t ) = 1 Γ ( n α ) ( d d t ) n 0 t f ( s ) ( t s ) α n 1 d s ( n = [ α ] + 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equb_HTML.gif

where the right-hand side is pointwise defined on ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq5_HTML.gif.

Definition 1.3 ([1, 4])

Let [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq6_HTML.gif be an interval in and α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq7_HTML.gif. The Riemann-Liouville fractional order integral of a function f L 1 ( [ a , b ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq8_HTML.gif is defined by
I a α f ( t ) = 1 γ ( α ) a t f ( s ) ( t s ) 1 α d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equc_HTML.gif

whenever the integral exists.

Suppose that E is a Banach space which is partially ordered by a cone P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq9_HTML.gif, that is, x y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq10_HTML.gif if and only if y x P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq11_HTML.gif. We denote the zero element of E by θ. A cone P is called normal if there exists a constant N > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq12_HTML.gif such that θ x y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq13_HTML.gif implies x N y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq14_HTML.gif (see [24]). Also, we define the order interval [ x 1 , x 2 ] = { x E | x 1 x x 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq15_HTML.gif for all x 1 , x 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq16_HTML.gif [24]. We say that an operator A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq17_HTML.gif is increasing whenever x y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq10_HTML.gif implies A x A y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq18_HTML.gif. Also, x y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq19_HTML.gif means that there exist λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq20_HTML.gif and μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq21_HTML.gif such that λ x y μ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq22_HTML.gif (see [24]). Finally, put P h = { x E | x h } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq23_HTML.gif for all h > θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq24_HTML.gif. It is easy to see that P h P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq25_HTML.gif is convex and λ P h = P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq26_HTML.gif for all λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq27_HTML.gif. We recall the following in our results. Let E be a real Banach space and let P be a cone in E. Let ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq28_HTML.gif be an interval and let τ and φ be two positive-valued functions such that φ ( t ) τ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq29_HTML.gif for all t ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq30_HTML.gif and τ : ( a , b ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq31_HTML.gif is a surjection. We say that an operator A : P P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq32_HTML.gif is τ-φ-concave whenever A ( τ ( t ) x ) φ ( t ) A x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq33_HTML.gif for all t ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq30_HTML.gif and x P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq34_HTML.gif [13]. We say that A is φ-concave whenever τ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq35_HTML.gif for all t [13]. We recall the following result.

Theorem 1.1 ([13])

Let E be a Banach space, let P be a normal cone in E, and let A : P P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq32_HTML.gif be an increasing and τ-φ-concave operator. Suppose that there exists θ h P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq36_HTML.gif such that A h P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq37_HTML.gif. Then there are u 0 , v 0 P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq38_HTML.gif and r ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq39_HTML.gif such that r v 0 u 0 v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq40_HTML.gif and u 0 A u 0 A v 0 v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq41_HTML.gif, the operator A has a unique fixed point x [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq42_HTML.gif, and for x 0 P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq43_HTML.gif and the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq44_HTML.gif with x n = A x n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq45_HTML.gif, we have x n x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq46_HTML.gif.

2 Main results

We study the existence and uniqueness of a solution for the fractional differential equation
D α u ( t ) + f ( t , u ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equd_HTML.gif

on partially ordered Banach spaces with two types of boundary conditions and two types of fractional derivatives, Riemann-Liouville and Caputo.

2.1 Existence results for the fractional differential equation with the Riemann-Liouville fractional derivative

First, we study the existence and uniqueness of a positive solution for the fractional differential equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equ1_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equ2_HTML.gif
(2.2)

where D α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq47_HTML.gif is the Riemann-Liouville fractional derivative of order α. Let E = C [ ε , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq48_HTML.gif. Consider the Banach space of continuous functions on [ ε , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq49_HTML.gif with the sup norm and set P = { y C [ ε , T ] : min t [ ε , T ] y ( t ) 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq50_HTML.gif. Then P is a normal cone.

Lemma 2.1 Let 0 < ε < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq51_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq52_HTML.gif, t [ ε , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq53_HTML.gif, η ( ε , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq54_HTML.gif and 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq55_HTML.gif. Then the problem D α u ( t ) + f ( t , u ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq56_HTML.gif with the boundary value condition u ( η ) = u ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq57_HTML.gif has a solution u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq58_HTML.gif if and only if u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq58_HTML.gif is a solution of the fractional integral equation
u ( t ) = ε T G ( t , s ) f ( s , u ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Eque_HTML.gif
where
G ( t , s ) = { t α 1 ( η s ) α 1 t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε s η t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε η s t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) , ε η t s T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equf_HTML.gif
Proof From D α u ( t ) + f ( t , u ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq56_HTML.gif and the boundary condition, it is easy to see that u ( t ) c 1 t α 1 = I ε α f ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq59_HTML.gif. By the definition of a fractional integral, we get
u ( t ) = c 1 t α 1 ε t ( t s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equg_HTML.gif
Thus, u ( η ) = c 1 η α 1 ε η ( η s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq60_HTML.gif and
u ( T ) = c 1 T α 1 ε T ( T s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equh_HTML.gif
Since u ( η ) = u ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq61_HTML.gif, we obtain
c 1 = 1 η α 1 T α 1 ε η ( η s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s 1 η α 1 T α 1 ε T ( T s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equi_HTML.gif
Hence,
u ( t ) = t α 1 η α 1 T α 1 ε η ( η s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s t α 1 η α 1 T α 1 ε T ( T s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s ε t ( t s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s = ε T G ( t , s ) f ( s , u ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equj_HTML.gif

This completes the proof. □

Now, we are ready to state and prove our first main result.

Theorem 2.2 Let 0 < ε < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq51_HTML.gif be given and let τ and φ be two functions on ( ε , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq62_HTML.gif such that φ ( t ) τ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq63_HTML.gif for all t ( ε , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq64_HTML.gif. Suppose that τ : ( ε , T ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq65_HTML.gif is a surjection and f ( t , u ( t ) ) C ( [ ε , T ] × [ 0 , ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq66_HTML.gif is increasing in u for each fixed t, f ( t , u ( t ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq67_HTML.gif and f ( t , τ ( λ ) u ( t ) ) φ ( λ ) f ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq68_HTML.gif for all t , λ ( ε , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq69_HTML.gif and u P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq70_HTML.gif. Assume that there exist M 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq71_HTML.gif, M 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq72_HTML.gif and θ h P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq73_HTML.gif such that
M 1 h ( t ) ε T G ( t , s ) f ( s , h ( s ) ) d s M 2 h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equk_HTML.gif

for all t [ ε , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq53_HTML.gif, where G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq74_HTML.gif is the green function defined in Lemma 2.1. Then the problem (2.1) with the boundary value condition (2.2) has a unique positive solution u P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq75_HTML.gif. Moreover, for the sequence u n + 1 = ε T G ( t , s ) f ( s , u n ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq76_HTML.gif, we have u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq77_HTML.gif for all u 0 P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq78_HTML.gif.

Proof By using Lemma 2.1, the problem is equivalent to the integral equation
u ( t ) = ε T G ( t , s ) f ( s , u ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equl_HTML.gif
where
G ( t , s ) = { t α 1 ( η s ) α 1 t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε s η t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε η s t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) , ε η t s T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equm_HTML.gif
Define the operator A : P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq79_HTML.gif by A u ( t ) = ε T G ( t , s ) f ( s , u ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq80_HTML.gif. Then u is a solution for the problem if and only if u = A u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq81_HTML.gif. It is easy to check that the operator A is increasing on P. On the other hand,
A ( τ ( λ ) u ) ( t ) = ε T G ( t , s ) f ( s , τ ( λ ) u ( s ) ) d s φ ( λ ) ε T G ( t , s ) f ( s , u ( s ) ) d s = φ ( λ ) A u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equn_HTML.gif
for all λ [ ε , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq82_HTML.gif and u P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq70_HTML.gif. Thus, the operator A is τ-φ-concave. Since
M 1 h ( t ) A h ( t ) = ε T G ( t , s ) f ( s , h ( s ) ) d s M 2 h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equo_HTML.gif

for all t [ ε , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq53_HTML.gif, we get A h P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq83_HTML.gif. Now, by using Theorem 1.1, the operator A has a unique positive solution u P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq75_HTML.gif. This completes the proof. □

Here, we give the following example to illustrate Theorem 2.2.

Example 2.1 Let 0 < ε < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq84_HTML.gif be given. Consider the periodic boundary value problem
D 1 3 u ( t ) + { g ( t ) + [ u ( t ) ] α } = 0 ( t [ ε , 1 ] ) , u ( η ) = u ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equp_HTML.gif
where η ( ε , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq85_HTML.gif, g is continuous on [ ε , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq86_HTML.gif and min t [ ε , 1 ] g ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq87_HTML.gif. Put
G ( t , s ) = { t 2 / 3 ( η s ) 2 / 3 t 2 / 3 ( 1 s ) 2 / 3 ( η 2 / 3 1 2 / 3 ) Γ ( 1 / 3 ) ( t s ) 2 / 3 Γ ( 1 / 3 ) , ε s η t 1 , t 2 / 3 ( 1 s ) 2 / 3 ( η 2 / 3 1 2 / 3 ) Γ ( 1 / 3 ) ( t s ) 2 / 3 Γ ( 1 / 3 ) , ε η s t 1 , t 2 / 3 ( 1 s ) 2 / 3 ( η 2 / 3 1 2 / 3 ) Γ ( 1 / 3 ) , ε η t s 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equq_HTML.gif
Then ε 1 G ( t , s ) d s = t 2 / 3 ( η ε ) 1 / 3 t 2 / 3 ( 1 ε ) 1 / 3 ( t ε ) 1 / 3 ( η 2 / 3 1 ) Γ ( 4 / 3 ) ( η 2 / 3 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq88_HTML.gif. Now, define τ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq35_HTML.gif, φ ( t ) = t 1 / 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq89_HTML.gif, γ 1 = min t [ ε , 1 ] g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq90_HTML.gif, γ 2 = max t [ ε , 1 ] g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq91_HTML.gif and also f ( t , u ) = g ( t ) + u 1 / 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq92_HTML.gif for all t. Then τ : ( 0 , 1 ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq93_HTML.gif is a surjection and φ ( t ) > τ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq94_HTML.gif for all t ( ε , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq95_HTML.gif. For each u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq96_HTML.gif, we have
f ( t , τ ( λ ) u ( t ) ) = f ( t , λ u ( t ) ) = g ( t ) + λ 1 / 3 [ u ( t ) ] 1 / 3 λ 1 / 3 ( g ( t ) + [ u ( t ) ] 1 / 3 ) = φ ( λ ) f ( t , u ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equr_HTML.gif
Now, put h 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq97_HTML.gif, M 1 = ( γ 1 + 1 ) min t [ ε , 1 ] , η [ ε , 1 ] t 2 / 3 ( 1 ε ) 1 / 3 ( t ε ) 1 / 3 ( η 2 / 3 1 ) Γ ( 4 / 3 ) ( η 2 / 3 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq98_HTML.gif and M 2 = ( γ 2 + 1 ) max η [ ε , 1 ] ε 2 / 3 η 1 / 3 Γ ( 4 / 3 ) ( η 2 / 3 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq99_HTML.gif. Then we get
ε 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 1 / 3 } d s ε 1 G ( t , s ) ( γ 2 + 1 ) d s ( γ 2 + 1 ) max t [ ε , 1 ] ε 1 G ( t , s ) d s ( γ 2 + 1 ) ( max η [ ε , 1 ] ε 2 / 3 η 1 / 3 Γ ( 4 / 3 ) ( η 2 / 3 1 ) ) = M 2 h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equs_HTML.gif
and
ε 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 1 / 3 } d s ( γ 1 + 1 ) min t [ ε , 1 ] ε 1 G ( t , s ) d s ( γ 1 + 1 ) min t [ ε , 1 ] , η [ ε , 1 ] t 2 / 3 ( 1 ε ) 1 / 3 ( t ε ) 1 / 3 ( η 2 / 3 1 ) Γ ( 4 / 3 ) ( η 2 / 3 1 ) = M 1 h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equt_HTML.gif

Thus, by using Theorem 2.2, the problem has a unique solution in P h = P 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq100_HTML.gif.

2.2 Existence results for the fractional differential equation with the Caputo fractional derivative

Here, we study the existence and uniqueness of a positive solution for the fractional differential equation
D α c u ( t ) + f ( t , u ( t ) ) = 0 ( t [ 0 , T ] , T 1 , 1 < α < 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equ3_HTML.gif
(2.3)
u ( 0 ) = β 1 u ( η ) , u ( T ) = β 2 u ( η ) ( η ( 0 , t ) , 0 < β 1 < β 2 < 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equ4_HTML.gif
(2.4)
where D α c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq101_HTML.gif is the Caputo fractional derivative of order α. Let E = C [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq102_HTML.gif be the Banach space of continuous functions on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq103_HTML.gif with the sup norm and
P = { y C [ 0 , T ] : min t [ 0 , T ] y ( t ) 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equu_HTML.gif

It is known that P is a normal cone. Similar to the proof of Lemma 2.1, we can prove the following result.

Lemma 2.3 Let 1 < α < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq104_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq52_HTML.gif, t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq105_HTML.gif, η ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq106_HTML.gif and 0 < β 1 < β 2 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq107_HTML.gif. Then the problem D α c u ( t ) + f ( t , u ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq108_HTML.gif with the boundary value conditions u ( 0 ) = β 1 u ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq109_HTML.gif and u ( T ) = β 2 u ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq110_HTML.gif has a solution u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq58_HTML.gif if and only if u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq58_HTML.gif is a solution of the fractional integral equation u ( t ) = 0 T G ( t , s ) f ( s , u ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq111_HTML.gif, where
G ( t , s ) = { [ β 1 T + t ( β 2 β 1 ) ] ( η s ) α 1 + t ( T s ) α 1 T ( t s ) α 1 T Γ ( α ) , 0 s η t T , t ( T s ) α 1 T ( t s ) α 1 T Γ ( α ) , 0 η s t T , t ( T s ) α 1 T Γ ( α ) , 0 η t s T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equv_HTML.gif
Theorem 2.4 Let T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq52_HTML.gif be given and let τ and φ be two positive-valued functions on ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq112_HTML.gif such that φ ( t ) τ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq113_HTML.gif for all t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq114_HTML.gif. Suppose that τ : ( 0 , T ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq115_HTML.gif is a surjection and f ( t , u ( t ) ) C ( [ ε , T ] × [ 0 , ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq66_HTML.gif is increasing in u for each fixed t, f ( t , u ( t ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq116_HTML.gif whenever 0 < η < s < t < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq117_HTML.gif and f ( t , u ( t ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq118_HTML.gif otherwise, and also f ( t , τ ( λ ) u ( t ) ) φ ( λ ) f ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq119_HTML.gif for all t , λ ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq120_HTML.gif and u P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq70_HTML.gif. Assume that there exist M 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq71_HTML.gif, M 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq72_HTML.gif and θ h P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq73_HTML.gif such that
M 1 h ( t ) 0 T G ( t , s ) f ( s , h ( s ) ) d s M 2 h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equw_HTML.gif

for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq105_HTML.gif, where G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq74_HTML.gif is the green function defined in Lemma 2.3. Then the problem (2.3) with the boundary value conditions (2.4) has a unique positive solution u P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq75_HTML.gif. Moreover, for the sequence u n + 1 = ε T G ( t , s ) f ( s , u n ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq76_HTML.gif, we have u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq77_HTML.gif for all u 0 P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq78_HTML.gif.

Proof It is sufficient to define the operator A : P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq79_HTML.gif by
A u ( t ) = 0 T G ( t , s ) f ( s , u ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equx_HTML.gif

Now, by using a similar proof of Theorem 2.2, one can show that A u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq121_HTML.gif for all u P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq122_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq105_HTML.gif, and also the operator A is τ-φ-concave. By using Theorem 1.1, the operator A has a unique positive solution u P h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq75_HTML.gif. This completes the proof by using Lemma 2.3. □

Below we present an example to illustrate Theorem 2.4.

Example 2.2 Let α = 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq123_HTML.gif. Consider the periodic boundary value problem
D α c u ( t ) + g ( t ) + [ u ( t ) ] α = 0 ( t [ 0 , 1 ] ) , u ( 0 ) = 1 3 u ( 1 2 ) u ( 1 ) = 1 2 u ( 1 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equy_HTML.gif
where g is a continuous function on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq124_HTML.gif with min t [ 0 , 1 ] g ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq125_HTML.gif. Put β 2 = η = 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq126_HTML.gif, β 1 = 1 / 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq127_HTML.gif and
G ( t , s ) = { [ 1 3 + 1 6 t ] ( 1 2 s ) 1 / 2 + t ( 1 s ) 1 / 2 ( t s ) 1 / 2 Γ ( 3 / 2 ) , 0 s η t 1 , t ( 1 s ) 1 / 2 ( t s ) 1 / 2 Γ ( 3 / 2 ) , 0 η s t 1 , t ( 1 s ) 1 / 2 Γ ( 3 / 2 ) , 0 η t s 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equz_HTML.gif
Then 0 1 G ( t , s ) d s = [ 1 3 + 1 6 t ] ( 1 2 ) 3 / 2 + t t 3 / 2 Γ ( 5 / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq128_HTML.gif. Now, define τ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq129_HTML.gif, φ ( t ) = t α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq130_HTML.gif, γ 1 = min t [ 0 , 1 ] g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq131_HTML.gif, γ 2 = max t [ 0 , 1 ] g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq132_HTML.gif and f ( t , u ) = g ( t ) + u α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq133_HTML.gif. Then it is easy to see that τ : ( 0 , 1 ) ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq134_HTML.gif is a surjection map and φ ( t ) > τ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq135_HTML.gif for t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq136_HTML.gif. Also, we have
f ( t , τ ( λ ) u ( t ) ) = f ( t , λ u ( t ) ) = g ( t ) + λ α [ u ( t ) ] α λ α ( g ( t ) + [ u ( t ) ] α ) = φ ( λ ) f ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equaa_HTML.gif
for all u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq96_HTML.gif. Now, put h 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq97_HTML.gif, M 1 = ( γ 1 + 1 ) min t [ 0 , 1 ] 1 3 t ( 1 2 ) 3 / 2 t 3 / 2 Γ ( 5 / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq137_HTML.gif and also M 2 = ( γ 2 + 1 ) 5 6 ( 1 2 ) 3 / 2 + 1 Γ ( 5 / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq138_HTML.gif. Then we have
0 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 3 / 2 } d s 0 1 G ( t , s ) ( γ 2 + 1 ) d s ( γ 2 + 1 ) max t [ 0 , 1 ] 0 1 G ( t , s ) d s ( γ 2 + 1 ) 5 6 ( 1 2 ) 3 / 2 + 1 Γ ( 5 / 2 ) = M 2 h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equab_HTML.gif
and
0 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 3 / 2 } d s ( γ 1 + 1 ) min t [ 0 , 1 ] 0 1 G ( t , s ) d s ( γ 1 + 1 ) min t [ 0 , 1 ] 1 3 t ( 1 2 ) 3 / 2 t 3 / 2 Γ ( 5 / 2 ) = M 1 h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_Equac_HTML.gif

Thus, by using Theorem 2.4, the problem has a unique solution in P h = P 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-112/MediaObjects/13661_2012_Article_363_IEq100_HTML.gif.

Declarations

Acknowledgements

This work is partially supported by the Scientific and Technical Research Council of Turkey. Research of the third and forth authors was supported by Azarbaidjan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.

Authors’ Affiliations

(1)
Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University
(2)
Department of Mathematics, Cankaya University
(3)
Institute of Space Sciences
(4)
Department of Mathematics, Texas A&M University
(5)
Department of Mathematics, Azarbaidjan Shahid Madani University

References

  1. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
  2. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.
  3. Oldham KB, Spainer J: The Fractional Calculus. Academic Press, New York; 1974.
  4. Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.
  5. Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivative: Theory and Applications. Gordon & Breach, Switzerland; 1993.
  6. Weitzner H, Zaslavsky GM: Some applications of fractional equations. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 939-945. 10.1016/j.cnsns.2009.05.004MathSciNetView Article
  7. Ahmad B, Nieto JJ: Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations. Abstr. Appl. Anal. 2009., 2009: Article ID 494720
  8. Al-Mdallal M, Syam MI, Anwar MN: A collocation-shooting method for solving fractional boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 3814-3822. 10.1016/j.cnsns.2010.01.020MathSciNetView Article
  9. Belmekki M, Nieto JJ, Rodriguez-Lopez R: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009., 2009: Article ID 324561
  10. Baleanu D, Mohammadi H, Rezapour S: Positive solutions of a boundary value problem for nonlinear fractional differential equations. Abstr. Appl. Anal. 2012., 2012: Article ID 837437
  11. Baleanu D, Mohammadi H, Rezapour S: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2013., 371(1990): Article ID 20120144
  12. Baleanu D, Mustafa OG, Agarwal RP: On the solution set for a class of sequential fractional differential equations. J. Phys. A, Math. Theor. 2010., 43(38): Article ID 385209
  13. Zhai C-B, Cao X-M: Fixed point theorems for τ - φ -concave operators and applications. Comput. Math. Appl. 2010, 59: 532-538. 10.1016/j.camwa.2009.06.016MathSciNetView Article
  14. Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609-625. 10.1006/jmaa.1996.0456MathSciNetView Article
  15. Hashim I, Abdulaziz O, Momani S: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 674-684. 10.1016/j.cnsns.2007.09.014MathSciNetView Article
  16. Jafari H, Daftardar-Gejji V: Positive solution of nonlinear fractional boundary value problems using Adomin decomposition method. J. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007MathSciNetView Article
  17. Zhao Y, Sun SH, Han Z: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 2086-2097. 10.1016/j.cnsns.2010.08.017MathSciNetView Article
  18. Zhang S: The existence of a positive solution for nonlinear fractional differential equation. J. Math. Anal. Appl. 2000, 252: 804-812. 10.1006/jmaa.2000.7123MathSciNetView Article
  19. Zhang S: Existence of positive solutions for some class of nonlinear fractional equation. J. Math. Anal. Appl. 2003, 278: 136-148. 10.1016/S0022-247X(02)00583-8MathSciNetView Article
  20. Agarwal RP, Lakshmikantam V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72: 2859-2862. 10.1016/j.na.2009.11.029MathSciNetView Article
  21. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore; 2012.
  22. Qiu T, Bai Z: Existence of positive solution for singular fractional equations. Electron. J. Differ. Equ. 2008, 146: 1-9.MathSciNet
  23. Sabatier J, Agarwal OP, Machado JAT: Advances in Fractional Calculus. Theorical Developments and Applications in Physics and Engineering. Springer, Berlin; 2007.View Article
  24. Rezapour S, Hamlbarani Haghi R: Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings’. J. Math. Anal. Appl. 2008, 345: 719-724. 10.1016/j.jmaa.2008.04.049MathSciNetView Article

Copyright

© Baleanu et al.; licensee Springer. 2013

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