Positive solutions for a class of fractional differential equations with integral boundary conditions

Applied Mathematics Letters 34 (2014) 17–21

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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

Positive solutions for a class of fractional differential equations with integral boundary conditions Yongping Sun ∗ , Min Zhao College of Electron and Information, Zhejiang University of Media and Communications, Hangzhou 310018, Zhejiang, People’s Republic of China

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abstract

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Article history: Received 14 January 2014 Received in revised form 9 March 2014 Accepted 11 March 2014 Available online 19 March 2014

The existence of positive solutions for a class of fractional equations involving the Riemann–Liouville fractional derivative with integral boundary conditions is investigated. By means of the monotone iteration method and some inequalities associated with the Green function, we obtain the existence of a positive solution and establish the iterative sequence for approximating the solution. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Fractional differential equations Boundary value problems Positive solution Monotone iterative method

1. Introduction The aim of this paper is to investigate the existence of positive solutions for the fractional differential equation with integral boundary conditions

 α D0+ u(t ) + q(t )f (t , u(t )) = 0,  0 < t < 1, 1 u(1) = g (s)u(s)ds, u(0) = u′ (0) = 0,

(1.1)

0

where 2 < α 6 3, Dα0+ is the standard Riemann–Liouville fractional derivative of order α which is defined as follows: α

D0+ h(t ) =

1

Γ (n − α)



d dt

n 

t

(t − s)n−α−1 h(s)ds,

n = [α] + 1,

0

where Γ denotes the Euler gamma function and [α] denotes the integer part of number α , provided that the right side is pointwise defined on (0, ∞), see [1]. Here, by a positive solution to the problem (1.1), we mean a function u ∈ C [0, 1], which is positive on (0, 1), and satisfies (1.1). Fractional differential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, economics and biology. In recent years, there has been a great deal of research on the questions of existence and/or uniqueness of solutions (or positive solutions) to boundary value problems for fractional-order differential equations. Among them some work is devoted to the solvability of nonlinear fractional differential equations with integral boundary conditions. For details, see,

∗

Corresponding author. Tel.: +86 0571 86832139. E-mail addresses: [email protected], [email protected] (Y. Sun).

http://dx.doi.org/10.1016/j.aml.2014.03.008 0893-9659/© 2014 Elsevier Ltd. All rights reserved.

18

Y. Sun, M. Zhao / Applied Mathematics Letters 34 (2014) 17–21

[2–20] and the references therein. In particular, Cabada and Hamdi in [2] investigated the existence of positive solutions for the fractional boundary value problem

 α D0+ u(t ) + f (t , u(t )) = 0, u(0) = u (0) = 0,

0 < t < 1,

u(1) = λ

′

1



(1.2)

u(s)ds,

0

where 2 < α 6 3, λ ̸= α , Dα0+ is the standard Riemann–Liouville fractional derivative and f is a continuous function. The authors obtained some existence results by Guo–Krasnoselskii’s fixed point theorem. Karakostas [3] provided sufficient conditions for the non-existence of solutions of the boundary-value problems (1.2) in the Caputo sense. Zhao et al. [4] studied the fractional boundary value problem (1.1) replaced q(t )f (t , u(t )) by λh(t )f (u(t )). The authors obtained some existence results of positive solutions when the nonlinear term satisfies different requirements of superlinearity, sublinearity and the parameter lies in some intervals. The non-existence of positive solutions was also studied, but the question of the computational methods of approximating solutions was not treated. Motivated by the above mentioned work, our purpose in this paper is to show the existence and iteration of a positive solution to the problem (1.1) by using the monotone iterative method. We not only obtain the existence of a positive solution, but also establish an iterative sequence for approximating the solution. The first term of the iterative sequence may be taken to be a constant function or a simple function. The monotone iterative method has been successfully applied to boundary value problems of integer order ordinary differential equations, see, for example, [21–25] and the references cited therein. 2. Several lemmas In this section, we present several lemmas that are useful to the proof of our main results. For the forthcoming analysis, we need the following assumptions:

(H1) f : [0, 1] × [0, ∞) → [0, ∞) is continuous andf (t , 0) ̸≡ 0 on [0, 1];  1 1 (H2) g : [0, 1] → [0, ∞) with g ∈ L1 [0, 1] and σ = 0 sα−1 g (s)ds < 1, θ = 0 sα g (s)ds;  1 (H3) q : [0, 1] → [0, ∞) with q ∈ L1 [0, 1] and 0 < 0 (1 − s)α−1 q(s)ds < ∞. In [4], the authors obtained the Green function associated with the problem (1.1). More precisely, the authors proved the following lemma. Lemma 2.1 ([4]). For any h ∈ C [0, 1], the unique solution of the boundary value problem

 α D0+ u(t ) + h(t ) = 0,

0 6 t 6 1,

u(0) = u (0) = 0,

u(1) =

′

1



g (s)u(s)ds, 0

is given by u(t ) =

1



G(t , s)h(s)ds,

t ∈ [0, 1],

0

where G(t , s) = G1 (t , s) + G2 (t , s), G1 (t , s) =

α−1

(t , s) ∈ [0, 1] × [0, 1], α−1

(2.1)

α−1

1 t ( 1 − s) − (t − s) , 0 6 s 6 t 6 1, α−1 (1 − s)α−1 , 0 6 t 6 s 6 1, Γ (α) t



(2.2)

and G2 (t , s) =

t α−1 1−σ



1

G1 (τ , s)g (τ )dτ .

(2.3)

0

Obviously, G(t , s) is continuous on the unit square [0, 1] × [0, 1]. Lemma 2.2 ([5]). The function G1 (t , s) defined by (2.2) has the following properties: t α−1 (1 − t )s(1 − s)α−1

Γ (α)

6 G1 (t , s) 6

s(1 − s)α−1

Γ (α − 1)

,

∀ t , s ∈ [0, 1].

(2.4)

The following properties of the Green function play an important role in this paper. Lemma 2.3. The Green function G(t , s) defined by (2.1) satisfies the inequalities p(t )G(s) 6 G(t , s) 6 G(s),

∀ t , s ∈ [0, 1],

(2.5)

Y. Sun, M. Zhao / Applied Mathematics Letters 34 (2014) 17–21

19

here G(s) = and δ =

1 0

(1 − σ + δ)s(1 − s)α−1 , (1 − σ )Γ (α − 1)

s ∈ [0, 1],

p(t ) =

[1 − σ + (α − 1)(σ − θ )](1 − t )t α−1 , (α − 1)(1 − σ + δ)

t ∈ [0, 1], (2.6)

g (s)ds, σ , θ are given in (H2).

Proof. For any t , s ∈ [0, 1], by (2.1), (2.3) and the right inequality of (2.4), we get G(t , s) = G1 (t , s) + G2 (t , s) = G1 (t , s) + s(1 − s)α−1

6

Γ (α − 1)

t α−1

+

1−σ

Γ (α − 1)

0

1



1−σ

s(1 − s)α−1

1



t α−1

G1 (τ , s)g (τ )dτ 0

g (τ )dτ 6

s(1 − s)α−1

Γ (α − 1)

+

δ s(1 − s)α−1 = G(s). (1 − σ )Γ (α − 1)

On the other hand, by (2.1), (2.3) and the left inequality of (2.4), we have G(t , s) = G1 (t , s) + G2 (t , s) = G1 (t , s) + t α−1 (1 − t )s(1 − s)α−1

>

Γ (α) t α−1 (1 − t )s(1 − s)α−1

=

Γ (α) t α−1 (1 − t )s(1 − s)α−1

>

Γ (α − 1)

The lemma is proved.

t α−1 1−σ

t α−1

+



1

1



G1 (τ , s)g (τ )dτ 0

s(1 − s)α−1 α−1 τ (1 − τ )g (τ )dτ Γ (α − 1)

1−σ 0 (σ − θ )t α−1 s(1 − s)α−1 + (1 − σ )Γ (α − 1)   1 σ −θ + = p(t )G(s). α−1 1−σ



Remark 2.1. It is obvious that p(t ) > 0 for t ∈ (0, 1). Lemma 2.4. The Green function G(t , s) defined by (2.1) satisfies the inequality G(t , s) 6

t α−1 (1 − s)α−1

(1 − σ )Γ (α)

,

∀ t , s ∈ [0, 1].

(2.7)

Proof. It is evident by (2.2) that t α−1 (1 − s)α−1

G1 (t , s) 6

Γ (α)

,

∀ t , s ∈ [0, 1].

(2.8)

Thus, by (2.1), (2.3) and (2.8), we have G(t , s) = G1 (t , s) + G2 (t , s) = G1 (t , s) + 6

t α−1 (1 − s)α−1

Γ (α)

Then the proof is completed.

+

t α−1 1−σ

1

 0

t α−1

1



1−σ

G1 (τ , s)g (τ )dτ 0

t α−1 (1 − s)α−1 τ α−1 (1 − s)α−1 g (τ )dτ = , Γ (α) (1 − σ )Γ (α)

∀ t , s ∈ [0, 1].



3. Main result Consider the Banach space E = C [0, 1] with the usual supremum norm ∥u∥ = sup06t 61 |u(t )| and define the cone

K ⊂ E , by K = {u ∈ C [0, 1] : u(t ) > 0, u(t ) > p(t )∥u∥, t ∈ [0, 1]} , where p(t ) is defined in (2.6). We also define the operator T : K → E by

(T u)(t ) =

1



G(t , s)q(s)f (s, u(s))ds,

t ∈ [0, 1].

(3.1)

0

Lemma 3.1. Assume that (H1)–(H3) hold. Then T : K → K is completely continuous. Proof. In view of (2.5) we conclude that T (K ) ⊆ K . Applying the Arzelà–Ascoli theorem and standard arguments, we conclude that T is a completely continuous operator. The proof is completed. 

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Y. Sun, M. Zhao / Applied Mathematics Letters 34 (2014) 17–21

For convenience, we denote,





1

1

(1 − s)α−1 q(s)ds

−1

. (1 − σ )Γ (α) 0 By the condition (H3) we deduce that Λ > 0 is well defined. Λ=

(3.2)

Theorem 3.1. Suppose (H1)–(H3) hold. In addition, we assume that there exists a > 0, such that f (t , x) 6 f (t , y) 6 Λa,

for 0 6 x 6 y 6 a, t ∈ [0, 1],

(3.3)

where Λ is defined by (3.2). Then the problem (1.1) has at least one positive solution u with 0 < ∥u ∥ 6 a. Moreover, there exists ∗ α−1 a monotone non-increasing sequence {un }∞ , t ∈ [0, 1], un+1 = T un , n = n=1 such that limn→∞ un = u , where u0 (t ) = at 0, 1, 2, . . . . ∗

∗

Proof. Set Ka = {u ∈ K : ∥u∥ 6 a}. If u ∈ Ka , then 0 6 u(s) 6 ∥u∥ 6 a, s ∈ [0, 1]. Thus by (2.7) and (3.3), we have G(t , s)q(s)f (s, u(s))ds 6 0

1

6

t α−1

1



(T u)(t ) =

(1 − σ )Γ (α)

Λa



1

1



(1 − σ )Γ (α)

(1 − s)α−1 q(s)f (s, a)ds

0

(1 − s)α−1 q(s)ds = a,

t ∈ [0, 1].

(3.4)

0

Then (3.4) shows that ∥T u∥ 6 a, consequently T (Ka ) ⊆ Ka . Obviously, u0 ∈ Ka . In view of the fact that T : Ka → Ka , it follows that un ∈ T (Ka ) ⊆ Ka , n = 1, 2, . . . . Since T ∞ is completely continuous, therefore, we assert that the sequence {un }∞ n=1 has a convergent subsequence {unk }k=1 such that ∗ limk→∞ unk = u ∈ Ka . Since u1 = T u0 ∈ Ka , it follows from (2.7) to (3.3) that

(T u0 )(t ) =

1



G(t , s)q(s)f (s, u0 (s))ds 6 0

6

t α−1

(1 − σ )Γ (α)

Λa

1



t α−1

1



(1 − σ )Γ (α)

(1 − s)α−1 q(s)f (s, a)ds

0

(1 − s)α−1 q(s)ds = at α−1 = u0 (t ),

t ∈ [0, 1],

(3.5)

0

which implies u1 6 u0 . Therefore, by (3.3), u2 (t ) = (T u1 )(t ) =



1

G(t , s)q(s)f (s, u1 (s))ds

0 1



G(t , s)q(s)f (s, u0 (s))ds = (T u0 )(t ) = u1 (t ),

6

t ∈ [0, 1].

0

By the induction, one has un+1 6 un , n = 0, 1, 2, . . . . Hence, limn→∞ un = u∗ . By the continuity of T and taking the limit n → ∞ in un+1 = T un yields T u∗ = u∗ . Moreover, because the zero function is not a solution of the problem (1.1), thus, ∥u∗ ∥ > 0. It follows from the definition of the cone K , that u∗ (t ) > p(t )∥u∗ ∥ > 0, t ∈ (0, 1), i.e., u∗ is a positive solution of problem (1.1). The proof is completed.  Remark 3.1. The iterative sequence in Theorem 3.1 begins with a simple function which is helpful for computational purpose. Remark 3.2. Choose the first item u0 (t ) = 0, t ∈ [0, 1], in a similar way, we can prove that {un }∞ n=1 is non-decreasing and there exists v ∗ such that limn→∞ ∥un − v ∗ ∥ = 0. Moreover, v ∗ is also a positive solution of problem (1.1) with 0 < ∥v ∗ ∥ 6 ∥u∗ ∥ 6 a. Certainly, u∗ = v ∗ may happen. For example, in case the Lipschitz condition is satisfied by the functions involved, the solutions u∗ and v ∗ coincide, and then problem (1.1) will have a unique solution in Ka . Acknowledgments The authors are very grateful to anonymous referees for their valuable and detailed suggestions and comments to improve the original manuscript. This work was supported by Zhejiang Provincial Natural Science Foundation of China (No: Y12A01012). References [1] I. Podlubny, Fractional Differential Equations, in: Math. Sci. Eng., vol. 198, Academic Press, San Diego, 1999. [2] A. Cabada, Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput. 228 (2014) 251–257. [3] G.L. Karakostas, Non-existence of solutions for two-point fractional and third-order boundary-value problems, Electron. J. Differential Equations 152 (2013) 1–19.

Y. Sun, M. Zhao / Applied Mathematics Letters 34 (2014) 17–21

21

[4] X. Zhao, C. Chai, W. Ge, Existence and nonexistence results for a class of fractional boundary value problems, J. Appl. Math. Comput. 41 (2013) 17–31. [5] C. Yuan, Multiple positive solutions for (n − 1, 1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual. Theory Differ. Equ. 36 (2010) 1–12. [6] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012) 403–411. [7] J. Caballero, I. Cabrera, K. Sadarangani, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Abstr. Appl. Anal. 2012 (2012) 1–11. Article ID 303545. [8] B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009 (2009) 1–9. Article ID 494720. [9] C. Yuan, Two positive solutions for (n − 1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations, Commun. Nonlinear. Sci. Numer. Simul. 17 (2012) 930–942. [10] X. Zhang, Y. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett. 25 (2012) 555–560. [11] S. Zhang, L. Hu, A. Shi, Existence result for a nonlinear fractional differential equation with integral boundary conditions at resonance, Adv. Differential Equations 353 (2013) 1–12. [12] W. Jiang, Solvability for a coupled system of fractional differential equations with integral boundary conditions at resonance, Adv. Differential Equations 324 (2013) 1–13. [13] X. Zhang, L. Liu, Y. Wu, Y. Lu, The iterative solutions of nonlinear fractional differential equations, Appl. Math. Comput. 219 (2013) 4680–4691. [14] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a singular fractional differential system involving derivatives, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1400–1409. [15] J.J. Nieto, J. Pimentel, Positive solutions of a fractional thermostat model, Bound. Value Probl. 5 (2013) 1–11. [16] X. Zhang, L. Liu, Y. Wu, The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives, Appl. Math. Comput. 218 (2012) 8526–8536. [17] W. Wu, X. Zhou, Eigenvalue of fractional differential equations with p-Laplacian operator, Discrete Dyn. Nat. Soc. 2013 (2013) 1–8. Article ID 137890. [18] X. Zhang, L. Liu, Y. Wu, Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives, Appl. Math. Comput. 219 (2012) 1420–1433. [19] X. Zhang, L. Liu, B. Wiwatanapataphee, Y. Wu, Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives, Abstr. Appl. Anal. 2012 (2012) 1–16. Article ID 512127. [20] X. Zhang, L. Liu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modelling 55 (2012) 1263–1274. [21] D. Ma, X. Yang, Existence and iteration of positive solutions to third order three-point BVP with increasing homeomorphism and positive homomorphism, Rocky Mountain J. Math. 43 (2013) 539–550. [22] B. Sun, W. Ge, Successive iteration and positive pseudo-symmetric solutions for a three-point second-order p-Laplacian boundary value problems, Appl. Math. Comput. 188 (2007) 1772–1779. [23] J. Sun, X. Wang, Existence and iteration of monotone positive solution of BVP for an elastic beam equation, Math. Probl. Eng. 2011 (2011) 1–10. Article ID 705740. [24] Y. Sun, X. Zhang, M. Zhao, Successive iteration and positive solutions for fourth-order two-point boundary value problems, Abstr. Appl. Anal. 2013 (2013) 1–9. Article ID 621315. [25] Q. Yao, Successively iterative technique of a classical elastic beam equation with carathéodory nonlinearity, Acta Appl. Math. 108 (2009) 385–394.