Skip to main content

Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions

Abstract

This paper investigates the existence and uniqueness of nontrivial solutions to a class of fractional nonlocal multi-point boundary value problems of higher order fractional differential equation, this kind of problems arise from viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system. Some sufficient conditions for the existence and uniqueness of nontrivial solutions are established under certain suitable growth conditions, our proof is based on Leray-Schauder nonlinear alternative and Schauder fixed point theorem.

MSC:34B15, 34B25.

1 Introduction

The purpose of this paper is to establish the existence and uniqueness of nontrivial solutions to the following higher fractional differential equation:

{ D α x ( t ) = f ( t , x ( t ) , D μ 1 x ( t ) , D μ 2 x ( t ) , , D μ n 1 x ( t ) ) , 0 < t < 1 , x ( 0 ) = 0 , D μ i x ( 0 ) = 0 , D μ x ( 1 ) = j = 1 p 2 a j D μ x ( ξ j ) , 1 i n 1 ,
(1.1)

where n3, nN, n1<αn, nl1<α μ l <nl, for l=1,2,,n2, and μ μ n 1 >0, α μ n 1 2, αμ>1, a j [0,+), 0< ξ 1 < ξ 2 << ξ p 2 <1, j = 1 p 2 a j ξ j α μ 1 1, D α is the standard Riemann-Liouville derivative, and f:[0,1]× R n R is continuous.

Differential equations of fractional order occur more frequently in different research areas such as engineering, physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system, etc. [16].

For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas et al. [7], Miller and Ross [8], Podlubny [9], the papers [1024] and the references therein.

Recently, Salem [10] has investigated the existence of Pseudo solutions for the nonlinear m-point boundary value problem of a fractional type. In particular, he considered the following boundary value problem:

{ D α x ( t ) + q ( t ) f ( t , x ( t ) ) = 0 , 0 < t < 1 , α ( n 1 , n ] , n 2 , x ( 0 ) = x ( 0 ) = x ( 0 ) = = x ( n 2 ) ( 0 ) = 0 , x ( 1 ) = i = 1 m 2 ξ i x ( η i ) ,
(1.2)

where x takes values in a reflexive Banach space E0< η 1 < η 2 << η m 2 <1 and ξ i >0 with j = 1 m 2 ξ j η j α 1 <1. x ( k ) denotes the k th Pseudo-derivative of x and D α denotes the Pseudo fractional differential operator of order α. By means of the fixed point theorem attributed to O’Regan, a criterion was established for the existence of at least one Pseudo solution for the problem (1.2).

More recently, Zhang [11] has considered the following problem whose nonlinear term and boundary condition contain integer order derivatives of unknown functions:

{ D α x ( t ) + q ( t ) f ( x , x , , x ( n 2 ) ) = 0 , 0 < t < 1 , n 1 < α n , x ( 0 ) = x ( 0 ) = = x ( n 2 ) ( 0 ) = x ( n 2 ) ( 1 ) = 0 ,
(1.3)

where D α is the standard Riemann-Liouville fractional derivative of order α q may be singular at t=0 and f may be singular at x=0 x =0,, x ( n 2 ) =0. By using the fixed point theorem of a mixed monotone operator, a unique existence result of positive solution to the problem (1.3) was established. And then, Goodrich [12] was concerned with a partial extension of the problem (1.3) by extending boundary conditions

{ D α x ( t ) = f ( t , x ( t ) ) , 0 < t < 1 , n 1 < α n , n > 3 , x ( i ) ( 0 ) = 0 , 0 i n 2 , D α x ( 1 ) = 0 , 1 α n 2 .
(1.4)

The author derived the Green’s function for the problem (1.4) and showed that it satisfies certain properties. Then, by using cone theoretic techniques, a general existence theorem for (1.4) was obtained when f(t,x) satisfies some growth conditions.

In recent work [13], Rehman and Khan have investigated the multi-point boundary value problems for fractional differential equations of the form

{ D α y ( t ) = f ( t , y ( t ) , D β y ( t ) ) , t ( 0 , 1 ) , y ( 0 ) = 0 , D β y ( 1 ) i = 1 m 2 ζ i D β y ( ξ i ) = y 0 ,
(1.5)

where 1<α20<β<10< ξ i <1 ζ i [0,+) with i = 1 m 2 ζ i ξ i α β 1 <1. By using the Schauder fixed point theorem and the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for BVP (1.5) provided that the nonlinear function f:[0,1]×R×R is continuous and satisfies certain growth conditions. However, Rehman and Khan only considered the case 1<α2 and the case of the nonlinear term f was not considered comprehensively.

Notice that the results dealing with the existence and uniqueness of solution for multi-point boundary value problems of fractional order differential equations are relatively scarce when the nonlinear term f and the boundary conditions all involve fractional derivatives of unknown functions. Thus, the aim of this paper is to establish the existence and uniqueness of nontrivial solutions for the higher nonlocal fractional differential equations (1.1) where nonlinear term f and the boundary conditions all involve fractional derivatives of unknown functions. In our study, the proof is based on the reduced order method as in [11] and the main tool is the Leray-Schauder nonlinear alternative and the Schauder fixed point theorem.

2 Basic definitions and preliminaries

Definition 2.1 A function x is said to be a solution of BVP (1.1) if xC[0,1] and satisfies BVP (1.1). In addition, x is said to be a nontrivial solution if x0 for t(0,1) and x is solution of BVP (1.1).

For the convenience of the reader, we present some definitions, lemmas, and basic results that will be used later. These and other related results and their proofs can be found, for example, in [69].

Definition 2.2 (see [8])

Let α>0 with αR. Suppose that x:[a,)R then the α th Riemann-Liouville fractional integral is defined by

I α x(t)= 1 Γ ( α ) a t ( t s ) α 1 x(s)ds

whenever the right-hand side is defined. Similarly, with α>0 with αR, we define the α th Riemann-Liouville fractional derivative to be

D α x(t)= 1 Γ ( n α ) ( d d t ) ( n ) a t ( t s ) n α 1 x(s)ds,

where nN is the unique positive integer satisfying n1α<n and t>a.

Remark 2.1 If x,y:(0,+)R with order α>0, then

D α ( x ( t ) + y ( t ) ) = D α x(t)+ D α y(t).

Lemma 2.1 (see [7])

  1. (1)

    If x L 1 (0,1), ρ>σ>0, then

    I ρ I σ x(t)= I ρ + σ x(t), D σ I ρ x(t)= I ρ σ x(t), D σ I σ x(t)=x(t).
  2. (2)

    If ρ>0, ν>0, then

    D ρ t ν 1 = Γ ( ν ) Γ ( ν ρ ) t ν ρ 1 .

Lemma 2.2 (see [8])

Assume thatxC(0,1) L 1 (0,1)with a fractional derivative of orderα>0, then I α D α x(t)=x(t)+ c 1 t α 1 + c 2 t α 2 ++ c n t α n , where c i R, i=1,2,,n (n=[α]+1). Here I α stands for the standard Riemann-Liouville fractional integral of orderα>0and D α denotes the Riemann-Liouville fractional derivative as Definition 2.1.

Lemma 2.3 If1<α μ n 1 2, αμ>1andh L 1 [0,1], then the boundary value problem

{ D α μ n 1 w ( t ) = h ( t ) , w ( 0 ) = 0 , D μ μ n 1 w ( 1 ) = j = 1 p 2 a j D μ μ n 1 w ( ξ j ) ,
(2.1)

has the unique solution

w(t)= 0 1 K(t,s)h(s)ds,

whereK(t,s)is the Green function of BVP (2.1), and

K(t,s)= k 1 (t,s)+ t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 j = 1 p 2 a j k 2 ( ξ j ,s),
(2.2)
(2.3)

Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation

w(t)= I α μ n 1 h(t)+ c 1 t α μ n 1 1 + c 2 t α μ n 1 2 , c 1 , c 2 R.
(2.4)

Note that w(0)=0 and (2.4), we have c 2 =0. Consequently, a general solution of (2.3) is

w(t)= I α μ n 1 h(t)+ c 1 t α μ n 1 1 .
(2.5)

By (2.5) and Lemma 2.1, we have

D μ μ n 1 w ( t ) = D μ μ n 1 I α μ n 1 h ( t ) + c 1 D μ μ n 1 t α μ n 1 1 = I α μ h ( t ) + c 1 Γ ( α μ n 1 ) Γ ( α μ ) t α μ 1 = 0 t ( t s ) α μ 1 Γ ( α μ ) h ( s ) d s + c 1 Γ ( α μ n 1 ) Γ ( α μ ) t α μ 1 .
(2.6)

So, from (2.6), we have

D μ μ n 1 w ( 1 ) = 0 1 ( 1 s ) α μ 1 Γ ( α μ ) h ( s ) d s + c 1 Γ ( α μ n 1 ) Γ ( α μ ) , D μ μ n 1 w ( ξ j ) = 0 ξ j ( ξ j s ) α μ 1 Γ ( α μ ) h ( s ) d s + c 1 Γ ( α μ n 1 ) Γ ( α μ ) ξ j α μ 1 , for j = 1 , 2 , , p 2 .
(2.7)

By D μ μ n 1 w(1)= j = 1 p 2 a j D μ μ n 1 w( ξ j ), combining with (2.7), we obtain

c 1 = 0 1 ( 1 s ) α μ 1 h ( s ) d s j = 1 p 2 a j 0 ξ j ( ξ j s ) α μ 1 h ( s ) d s Γ ( α μ n 1 ) ( 1 j = 1 p 2 a j ξ j α μ 1 ) .

So, substituting c 1 into (2.5), the unique solution of the problem (2.1) is

w ( t ) = 0 t ( t s ) α μ n 1 1 Γ ( α μ n 1 ) h ( s ) d s + t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 × { 0 1 ( 1 s ) α μ 1 Γ ( α μ n 1 ) h ( s ) d s j = 1 p 2 a j 0 ξ j ( ξ j s ) α μ 1 Γ ( α μ n 1 ) h ( s ) d s } = 0 t ( t s ) α μ n 1 1 Γ ( α μ n 1 ) h ( s ) d s + ( 1 j = 1 p 2 a j ξ j α μ 1 + j = 1 p 2 a j ξ j α μ 1 ) t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 × 0 1 ( 1 s ) α μ 1 Γ ( α μ n 1 ) h ( s ) d s t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 j = 1 p 2 a j 0 ξ j ( ξ j s ) α μ 1 Γ ( α μ n 1 ) h ( s ) d s = 0 t ( t s ) α μ n 1 1 Γ ( α μ n 1 ) h ( s ) d s + 0 1 ( 1 s ) α μ 1 t α μ n 1 1 Γ ( α μ n 1 ) h ( s ) d s + t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 j = 1 p 2 a j 0 1 ( 1 s ) α μ 1 ξ j α μ 1 Γ ( α μ n 1 ) h ( s ) d s t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 j = 1 p 2 a j 0 ξ j ( ξ j s ) α μ 1 Γ ( α μ n 1 ) h ( s ) d s = 0 1 ( k 1 ( t , s ) + t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 j = 1 p 2 a j k 2 ( ξ j , s ) ) h ( s ) d s = 0 1 K ( t , s ) h ( s ) d s .

The proof is completed. □

Lemma 2.4|K(t,s)|M ( 1 s ) α μ 1 , fort,s[0,1], where

M= 1 + j = 1 p 2 a j | 1 j = 1 p 2 a j ξ j α μ 1 | Γ ( α μ n 1 ) .
(2.8)

Proof Obviously, for t,s[0,1], we have k i (t,s) ( 1 s ) α μ 1 Γ ( α μ n 1 ) , i=1,2. Thus

| K ( t , s ) | = | k 1 ( t , s ) + t α μ n 1 1 1 j = 1 p 2 a j ξ j α μ 1 j = 1 p 2 a j k 2 ( ξ j , s ) | ( 1 s ) α μ 1 Γ ( α μ n 1 ) + j = 1 p 2 a j ( 1 s ) α μ 1 Γ ( α μ n 1 ) | 1 j = 1 p 2 a j ξ j α μ 1 | ( 1 + j = 1 p 2 a j | 1 j = 1 p 2 a j ξ j α μ 1 | ) ( 1 s ) α μ 1 Γ ( α μ n 1 ) .

This completes the proof. □

Now let us consider the following modified problem of BVP (1.1)

{ D α μ n 1 v ( t ) = f ( t , I μ n 1 v ( t ) , I μ n 1 μ 1 v ( t ) , , I μ n 1 μ n 2 v ( t ) , v ( t ) ) , v ( 0 ) = 0 , D μ μ n 1 v ( 1 ) = j = 1 p 2 a j D μ μ n 1 v ( ξ j ) .
(2.9)

Lemma 2.5 Letx(t)= I μ n 1 v(t), v(t)C[0,1]. Then (2.9) can be transformed into (1.1). Moreover, ifvC([0,1],R)is a solution of the problem (2.9), then the functionx(t)= I μ n 1 v(t)is a solution of the problem (1.1).

Proof Substituting x(t)= I μ n 1 v(t) into (1.1), by Lemmas 2.1 and 2.2, we can obtain that

(2.10)

and also D μ n 1 x(0)=v(0)=0. It follows from D μ x(t)= D μ I μ n 1 v(t)= d n d t n I n μ I μ n 1 v(t)= D μ μ n 1 v(t) that D μ μ n 1 v(1)= j = 1 p 2 a j D μ μ n 1 v( ξ j ). Using x(t)= I μ n 1 v(t), vC[0,1], (2.9) is transformed into (1.1).

Now, let vC([0,1],R) be a solution for the problem (2.9). Then, from Lemma 2.1, (2.9) and (2.10), one has

D α x ( t ) = d n d t n I n α x ( t ) = d n d t n I n α I μ n 1 v ( t ) = d n d t n I n α + μ n 1 v ( t ) = D α μ n 1 v ( t ) = f ( t , I μ n 1 v ( t ) , I μ n 1 μ 1 v ( t ) , , I μ n 1 μ n 2 v ( t ) , v ( t ) ) = f ( t , x ( t ) , D μ 1 x ( t ) , D μ 2 x ( t ) , , D μ n 1 x ( t ) ) , 0 < t < 1 .

Notice

I α v(t)= 1 Γ ( α ) 0 t ( t s ) α 1 v(s)ds,

which implies that I α v(0)=0. Thus from (2.10), for i=1,2,,n1, we have

x(0)=0, D μ i x(0)=0, D μ x(1)= j = 1 p 2 a j D μ x( ξ j ).

Moreover, it follows from the monotonicity and property of I μ n 1 that

I μ n 1 vC ( [ 0 , 1 ] , [ 0 , + ) ).

Consequently, x(t)= I μ n 1 v(t) is a solution of the problem (1.1). □

Now let us define an operator T:C[0,1]C[0,1] by

(Tv)(t)= 0 1 K(t,s)f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) ds.
(2.11)

Clearly, the fixed point of the operator T is a solution of BVP (2.9); and consequently is also a solution of BVP (1.1) from Lemma 2.5.

Lemma 2.6T:C[0,1]C[0,1]is a completely continuous operator.

Proof Noticing that f:[0,1]× R n R is continuous, by using the Ascoli-Arzela theorem and standard arguments, the result can easily be shown. □

Lemma 2.7 (see [25])

Let X be a real Banach space, Ω be a bounded open subset of X, whereθΩ, T: Ω ¯ Xis a completely continuous operator. Then, either there existsxΩ, λ>1such thatT(x)=λx, or there exists a fixed point x Ω ¯ .

3 Main results

For the convenience of expression in rest of the paper, we let μ 0 =0.

Theorem 3.1 Supposef(t,0,,0)0for anyt[0,1]. Moreover, there exist nonnegative functions p 1 , p 2 ,, p n ,q L 1 [0,1]such that

| f ( t , u 1 , u 2 , , u n ) | i = 1 n p i (t)| u i |+q(t),a.e.(t, u 1 , u 2 ,, u n )[0,1]× R n ,
(3.1)

and

M 0 1 ( 1 s ) α μ 1 i = 1 n p i (s)ds< ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 ,
(3.2)

where M is defined by (2.8). Then BVP (1.1) has at least one nontrivial solution.

Proof Since f(t,0,,0)0, there exists [σ,τ][0,1] such that

min t [ σ , τ ] |f(t,0,,0)|>0.

By condition (3.1), we have q(t)|f(t,0,,0)|, a.e. t[0,1], thus

0 1 ( 1 s ) α μ 1 q(s)ds>0.

On the other hand, from (3.2), we know

( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i (s)ds<1.

Take

r= M 0 1 ( 1 s ) α μ 1 q ( s ) d s 1 ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s

then r>0.

Now let Ω r ={vC[0,1]:x<r}, suppose v Ω r , λ>1 such that Tv=λv. Then

λ r = λ v = T v = max t [ 0 , 1 ] | T v ( t ) | M 0 1 ( 1 s ) α μ 1 f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) d s .
(3.3)

Moreover, for i=0,1,2,,n2,

| I μ n 1 μ i v ( t ) | =| 0 t ( t s ) μ n 1 μ i 1 v ( s ) Γ ( μ n 1 μ i ) ds| v Γ ( μ n 1 μ i ) ,

thus we have, by hypothesis (3.1),

| f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) | p 1 ( s ) | I μ n 1 v ( s ) | + p 2 ( s ) | I μ n 1 μ 1 v ( s ) | + + p n 1 ( s ) | I μ n 1 μ n 2 v ( s ) | + p n ( s ) | v ( s ) | + q ( s ) v Γ ( μ n 1 ) p 1 ( s ) + v Γ ( μ n 1 μ 1 ) p 2 ( s ) + + v Γ ( μ n 1 μ n 2 ) p n 1 ( s ) + v p n ( s ) + q ( s ) ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) v [ p 1 ( s ) + p 2 ( s ) + + p n 1 ( s ) + p n ( s ) ] + q ( s ) .

Consequently, from (3.3), we have

λ r ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s v + M 0 1 ( 1 s ) α μ 1 q ( s ) d s = r ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s + M 0 1 ( 1 s ) α μ 1 q ( s ) d s .

Therefore,

λ ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s + M 0 1 ( 1 s ) α μ 1 q ( s ) d s r = 1 .

This contradicts λ>1. By Lemma 2.7, T has a fixed point v Ω ¯ , since f(t,0,,0)0; so then, by Lemma 2.5, BVP (1.1) has a nontrivial solution v . This completes the proof. □

Theorem 3.2 Supposef(t,0,,0)0for anyt[0,1]. Moreover, there exist nonnegative functions p 1 , p 2 ,, p n ,q L 1 [0,1]such that

| f ( t , u 1 , u 2 , , u n ) | i = 1 n p i (t) | u i | σ i +q(t),a.e.(t, u 1 , u 2 ,, u n )[0,1]× R n ,
(3.4)

where0< σ 1 , σ 2 ,, σ n <1are nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.

Proof By Lemma 2.6, we know T:C[0,1]C[0,1] is a completely continuous operator.

Let

a = ( 1 + i = 0 n 2 1 Γ σ i + 1 ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s , b = M 0 1 ( 1 s ) α μ 1 q ( s ) d s .

Choose

R { ( n + 1 ) b , [ ( n + 1 ) a ] 1 1 σ 1 , [ ( n + 1 ) a ] 1 1 σ 2 , , [ ( n + 1 ) a ] 1 1 σ n }

and define a ball M={vC[0,1]:vR,t[0,1]}. For every vM, we have

| T v ( t ) | 0 1 K ( t , s ) | f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) | d s M 0 1 ( 1 s ) α μ 1 | f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) | d s .

On the other hand, it follows from (3.4) that

| f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) | p 1 ( s ) | I μ n 1 v ( s ) | σ 1 + p 2 ( s ) | I μ n 1 μ 1 v ( s ) | σ 2 + + p n 1 ( s ) | I μ n 1 μ n 2 v ( s ) | σ n 1 + p n ( s ) | v ( s ) | σ n + q ( s ) v σ 1 Γ σ 1 ( μ n 1 ) p 1 ( s ) + v σ 2 Γ σ 2 ( μ n 1 μ 1 ) p 2 ( s ) + + v σ n 1 Γ σ n 1 ( μ n 1 μ n 2 ) p n 1 ( s ) + v σ n p n ( s ) + q ( s ) ( v σ n + i = 0 n 2 v σ i + 1 Γ σ i + 1 ( μ n 1 μ i ) ) [ p 1 ( s ) + p 2 ( s ) + + p n 1 ( s ) + p n ( s ) ] + q ( s ) ( 1 + i = 0 n 2 1 Γ σ i + 1 ( μ n 1 μ i ) ) i = 1 n v σ i i = 1 n p i ( s ) + q ( s ) .
(3.5)

In view of (3.5), we have the following estimate:

| T v ( t ) | ( 1 + i = 0 n 2 1 Γ σ i + 1 ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s i = 1 n v σ i + M 0 1 ( 1 s ) α μ 1 q ( s ) d s = a i = 1 n v σ i + b n R n + 1 + R n + 1 = R .

Therefore, TvR. Thus we have T:MM. Hence the Schauder fixed point theorem implies the existence of a solution in M for BVP (2.9). Since f(t,0,,0)0, then by Lemma 2.5, BVP (1.1) has a nontrivial solution v . This completes the proof. □

Theorem 3.3 Supposef(t,0,,0)0for anyt[0,1]. Moreover, there exist nonnegative functions p 1 , p 2 ,, p n ,q L 1 [0,1]such that

| f ( t , u 1 , u 2 , , u n ) | i = 1 n p i (t) | u i | σ i +q(t),a.e.(t, u 1 , u 2 ,, u n )[0,1]× R n ,
(3.6)

where σ 1 , σ 2 ,, σ n >1are nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.

Proof The proof is similar to that of Theorem 3.2, so it is omitted. □

Remark 3.1 In [13], the authors studied the cases 1<α2 μ 1 = μ 2 == μ n 1 =β0<β<1, but the case of σ i =1i=1,2,,n was not considered. Here we extend the results of [13] and fill the case σ i =1i=1,2,,n.

Theorem 3.4 Supposef(t,0,,0)0for anyt[0,1]. Moreover, there exist nonnegative functions p 1 , p 2 ,, p n L 1 [0,1]such that

| f ( t , u 1 , u 2 , , u n ) f ( t , v 1 , v 2 , , v n ) | i = 1 n p i ( t ) | u i v i | , a.e. ( t , u 1 , u 2 , , u n ) , ( t , v 1 , v 2 , , v n ) [ 0 , 1 ] × R n ,
(3.7)

and (3.2) holds. Then BVP (1.1) has a unique nontrivial solution.

Proof In fact, if v 1 = v 2 == v n 0, then we have

| f ( t , u 1 , u 2 , , u n ) | i = 1 n p i (t)| u i |+ | f ( t , 0 , 0 , , 0 ) | .

From Theorem 3.1, we know BVP (1.1) has a nontrivial solution.

But in this case, we prefer to concentrate on the uniqueness of a nontrivial solution for BVP (1.1). Let T be given in (2.11), we shall show that T is a contraction. In fact, by (3.7), a similar method to Theorem 3.1, we have

| f ( s , I μ n 1 u ( s ) , I μ n 1 μ 1 u ( s ) , , I μ n 1 μ n 2 u ( s ) , u ( s ) ) f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) | ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) u v i = 1 n p i ( s ) .

And then

T u T v M 0 1 ( 1 s ) α μ 1 | f ( s , I μ n 1 u ( s ) , I μ n 1 μ 1 u ( s ) , , I μ n 1 μ n 2 u ( s ) , u ( s ) ) f ( s , I μ n 1 v ( s ) , I μ n 1 μ 1 v ( s ) , , I μ n 1 μ n 2 v ( s ) , v ( s ) ) | d s ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s u v .

Then (3.2) implies that T is indeed a contraction. Finally, we use the Banach fixed point theorem to deduce the existence of a unique nontrivial solution to BVP (1.1). □

Corollary 3.1 Supposef(t,0,,0)0for anyt[0,1], and (3.1) holds. Then BVP (1.1) has at least one nontrivial solution if one of the following conditions holds

  1. (1)

    There exists a constant p>1 such that

    0 1 [ i = 1 n p i ( s ) ] p ds< ( p ( α μ 1 ) p 1 + 1 ) p 1 ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) p .
    (3.8)
  2. (2)

    There exists a constant λ>1 such that

    i = 1 n p i (s)< Γ ( α + λ μ 1 ) Γ ( α μ ) Γ ( λ + 1 ) ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) 1 s λ .
    (3.9)
  3. (3)

    There exists a constant λ>1 such that

    i = 1 n p i (s)<(α+λμ) ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) 1 ( 1 s ) λ .
    (3.10)
  4. (4)

    p i (s) (i=1,2,,n) satisfy

    i = 1 n p i (s)<(αμ) ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) 1 .
    (3.11)

Proof Let

R=M 0 1 ( 1 s ) α μ 1 i = 1 n p i (s)ds.

From the proof of Theorem 3.1, we only need to prove

R< ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .
  1. (1)

    If (3.8) holds, let 1 p + 1 q =1, and by using H o ¨ lder inequality,

    R M ( 0 1 [ i = 1 n p i ( s ) ] p d s ) 1 p ( 0 1 ( 1 s ) q ( α μ 1 ) d s ) 1 q = M [ q ( α μ 1 ) + 1 ] 1 q ( 0 1 [ i = 1 n p i ( s ) ] p d s ) 1 p = M [ p ( α μ 1 ) p 1 + 1 ] p 1 p ( 0 1 [ i = 1 n p i ( s ) ] p d s ) 1 p < ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .
  2. (2)

    In this case, it follows from (3.9) that

    R < M Γ ( α + λ μ 1 ) Γ ( α μ ) Γ ( λ + 1 ) ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) 1 0 1 ( 1 s ) α μ 1 s λ d s = ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .
  3. (3)

    In this case, it follows from (3.10) that

    R < M ( α + λ μ ) ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) 1 0 1 ( 1 s ) α μ 1 ( 1 s ) λ d s = ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .
  4. (4)

    If (3.11) is satisfied, we have

    R < M ( α μ ) ( M + i = 0 n 2 M Γ ( μ n 1 μ i ) ) 1 0 1 ( 1 s ) α μ 1 d s = ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .

This completes the proof of Corollary 3.1. □

Corollary 3.2 Supposef(t,0,,0)0for anyt[0,1]. Moreover,

0 lim sup i = 1 n | u i | + max t [ 0 , 1 ] | f ( t , u 1 , u 2 , , u n ) | i = 1 n | u i | < α μ M ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .
(3.12)

Then BVP (1.1) has at least one nontrivial solution.

Proof Take ϵ>0 such that

α μ M ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 ϵ>0,

by (3.12), there exists a large enough constant R 0 >0 such that for any t[0,1], i = 1 n | u i | R 0 , one has

|f(t, u 1 , u 2 ,, u n )| ( α μ M ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 ϵ ) i = 1 n | u i |.

Let

ħ= max t [ 0 , 1 ] , i = 1 n | u i | R 0 | f ( t , u 1 , u 2 , , u n ) | .

Then for any (t, u 1 , u 2 ,, u n )[0,1]× R n , we have

| f ( t , u 1 , u 2 , , u n ) | ħ+ ( α μ M ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 ϵ ) i = 1 n | u i |.

Let

i = 1 n p i (s)= ( α μ M ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 ϵ ) ,q(s)=ħ,

we prove

R< ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .

In fact,

R = M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s M ( α μ M ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 ϵ ) 0 1 ( 1 s ) α μ 1 d s < ( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 .

Then it follows from Theorem 3.1 that BVP (1.1) has at least one nontrivial solution. □

4 Examples

Example 4.1 Consider the boundary value problem

{ D 5 2 x ( t ) = t sin x ( t ) 100 π + | x ( t ) | D 9 8 x ( t ) 10 2 + | D 5 4 x ( t ) | D 5 2 x ( t ) = + ( 1 + t 2 ) D 5 4 x ( t ) 100 + t 3 2 + cos t , t ( 0 , 1 ) , x ( 0 ) = D 9 8 x ( 0 ) = D 5 4 x ( 0 ) = 0 , D 11 8 x ( 1 ) = 2 2 D 11 8 ( 1 4 ) + 1 5 D 11 8 ( 1 2 ) .
(4.1)

Proof Let α= 5 2 , μ 1 = 9 8 , μ 2 = 5 4 , μ= 11 8 , and set

f ( t , u 1 , u 2 , u 3 ) = t sin u 1 100 π + | u 1 | u 2 10 2 + | u 3 | + ( 1 + t 2 ) u 3 100 + t 3 2 + cos t , p 1 ( t ) = t 100 π , p 2 ( t ) = 1 10 2 , p 3 ( t ) = 1 + t 2 100 , q ( t ) = t 3 2 + cos t .

Then

| f ( t , u 1 , u 2 , u 3 ) | p 1 (t)| u 1 |+ p 2 (t)| u 2 |+ p 3 (t)| u 3 |+q(t),

and

( 1 + i = 0 n 2 1 Γ ( μ n 1 μ i ) ) 1 = ( 1 Γ ( 5 4 ) + 1 Γ ( 1 8 ) + 1 ) 1 0.4472 , M = 1 + j = 1 p 2 a j | 1 j = 1 p 2 a j ξ j α μ 1 | Γ ( α μ n 1 ) 2.3934 .

Thus we have

M 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s = 2.3934 0 1 ( 1 s ) α μ 1 i = 1 n p i ( s ) d s 0.07679 × 2.3934 0.1838 < 0.4472 .

Thus the condition (3.2) in Theorem 3.1 is satisfied, and from Theorem 3.1, BVP (4.1) has a nontrivial solution. □

Example 4.2 Consider the boundary value problem

{ D 8 3 x ( t ) = 1 2 ( t t 2 ) x 5 ( t ) ( sin t + e t ) [ D 7 6 x ( t ) ] 9 8 D 8 3 x ( t ) = + 2 t 3 [ D 4 3 x ( t ) ] 3 + e t + t , t ( 0 , 1 ) , x ( 0 ) = D 7 6 x ( 0 ) = D 4 3 x ( 0 ) = 0 , D 3 2 x ( 1 ) = 1 π D 3 2 ( 1 3 ) 2 D 3 2 ( 2 3 ) + 1 2 D 3 2 ( 3 4 ) .
(4.2)

Proof Let

f ( t , u 1 , u 2 , u 3 ) = 1 2 ( t t 2 ) | u 1 | 5 + ( sin t + e t ) | u 2 | 9 8 + 2 t 3 | u 3 | 3 + e t + t , p 1 ( t ) = 1 2 ( t t 2 ) , p 2 ( t ) = sin t + e t , p 3 ( t ) = 2 t 3 , q ( t ) = e t + t .

Then

| f ( t , u 1 , u 2 , u 3 ) | p 1 (t) | u 1 | 5 + p 2 (t) | u 2 | 9 8 + p 3 (t) | u 3 | 3 +q(t),t[0,1].

Thus Theorem 3.4 guarantees a nontrivial solution for BVP (4.2). □

References

  1. Diethelm K, Freed AD: On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity. In Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Mackens W, Voss H, Werthers J. Springer, Heidelberg; 1999.

    Google Scholar 

  2. Gaul L, Klein P, Kempffe S: Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5: 81-88. 10.1016/0888-3270(91)90016-X

    Article  Google Scholar 

  3. Glockle WG, Nonnenmacher TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 1995, 68: 46-53. 10.1016/S0006-3495(95)80157-8

    Article  Google Scholar 

  4. Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanics. In Fractal and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri CA, Mainardi F. Springer, Vienna; 1997.

    Google Scholar 

  5. Metzler F, Schick W, Kilian HG, Nonnenmache TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 1995, 103: 7180-7186. 10.1063/1.470346

    Article  Google Scholar 

  6. Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.

    Google Scholar 

  7. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

    Google Scholar 

  8. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.

    Google Scholar 

  9. Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.

    Google Scholar 

  10. Salem AH: On the fractional m -point boundary value problem in reflexive Banach space and the weak topologies. J. Comput. Appl. Math. 2009, 224: 565-572. 10.1016/j.cam.2008.05.033

    Article  MathSciNet  Google Scholar 

  11. Zhang S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 2010, 59: 1300-1309. 10.1016/j.camwa.2009.06.034

    Article  MathSciNet  Google Scholar 

  12. Goodrich CS: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 2010, 23: 1050-1055. 10.1016/j.aml.2010.04.035

    Article  MathSciNet  Google Scholar 

  13. Rehman M, Khan R: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 2010, 23: 1038-1044. 10.1016/j.aml.2010.04.033

    Article  MathSciNet  Google Scholar 

  14. Zhang X, Liu L, Wu Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 2012, 55: 1263-1274. 10.1016/j.mcm.2011.10.006

    Article  MathSciNet  Google Scholar 

  15. Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033

    Article  MathSciNet  Google Scholar 

  16. Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011:

    Google Scholar 

  17. Goodrich CS: Existence of a positive solution to systems of differential equations of fractional order. Comput. Math. Appl. 2011, 62: 1251-1268. 10.1016/j.camwa.2011.02.039

    Article  MathSciNet  Google Scholar 

  18. Goodrich CS: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61: 191-202. 10.1016/j.camwa.2010.10.041

    Article  MathSciNet  Google Scholar 

  19. Goodrich CS: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2012, 75: 417-432. 10.1016/j.na.2011.08.044

    Article  MathSciNet  Google Scholar 

  20. Zhang X, Han Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett. 2012, 25: 555-560. 10.1016/j.aml.2011.09.058

    Article  MathSciNet  Google Scholar 

  21. Wu J, Zhang X, Liu L, Wu Y: Positive solutions of higher order nonlinear fractional differential equations with changing-sign measure. Adv. Differ. Equ. 2012., 2012:

    Google Scholar 

  22. Zhang X, Liu L, Wiwatanapataphee B, Wu Y: Solutions of eigenvalue problems for a class of fractional differential equations with derivatives. Abstr. Appl. Anal. 2012., 2012:

    Google Scholar 

  23. Wu T, Zhang X: Solutions of sign-changing fractional differential equation with the fractional. Abstr. Appl. Anal. 2012., 2012:

    Google Scholar 

  24. Zhang X, Liu L, Wu Y: The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives. Appl. Math. Comput. 2012, 218: 8526-8536. 10.1016/j.amc.2012.02.014

    Article  MathSciNet  Google Scholar 

  25. Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.

    Book  Google Scholar 

Download references

Acknowledgement

The authors thank the referee for helpful comments and suggestions which led to an improvement of the paper. The authors were supported financially by the National Natural Science Foundation of China (11071141) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2010091), the National Science Foundation for Post-doctoral Scientists of China (Grant No. 2012M510956).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Min Jia or Xinguang Zhang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between all authors. Each of the authors contributed to every part of this study equally and read and approved the final version of the manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Jia, M., Zhang, X. & Gu, X. Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions. Bound Value Probl 2012, 70 (2012). https://doi.org/10.1186/1687-2770-2012-70

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2012-70

Keywords