We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem: , where n − 1 < α ≤ n, n ≥ 3,0 < β ≤ 1,0 ≤ η ≤ 1, is the standard Riemann-Liouville derivative. Here our nonlinearity f may be singular at u = 0. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.

1. Introduction

Fractional differential equations have been of great interest recently. This is due to the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in selfsimilar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science. For details, see [1–10].

It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. Recently, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional differential equations boundary value problems, see [11–17].

Bai [14] investigated the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation

()

by contraction map principle and fixed-point index theory, where 1 < α ≤ 2, 0 < βη^α−1 < 1, 0 < η < 1,

is the standard Riemann-Liouville derivative. The function f is continuous on [0,1]×[0, ∞).

Li et al. [17] investigated the the existence and multiplicity results of positive solutions for the nonlinear differential equation of fractional order

()

by using some fixed-point theorems, where 1 < α ≤ 2, 0 < β ≤ 1, 0 ≤ a ≤ 1, ξ ∈ (0,1), aξ^α−β−2 ≤ 1 − β, 0 ≤ α − β − 1,

is the standard Riemann-Liouville derivative.

Xu and Fei [18] considered the properties of Green’s function for the nonlinear fractional differential equation boundary value problem

()

where 1 < α ≤ 2, 0 < β ≤ 1, 0 ≤ a ≤ 1, 0 < ξ < 1, α − β − 1 ≥ 0,

is the standard Riemann-Liouville derivative. Here the nonlinearity f may be singular at u = 0. As applications of Green’s function, they give some existence of positive solutions for singular boundary value problems by means of Schauder fixed-point theorem. Here they consider the case: γ_* = 0, γ_* ≥ 0, γ^* ≤ 0.

In this paper, we consider the singular boundary value problem

()

where n − 1 < α ≤ n, n ≥ 3, 0 < β, η < 1 is a real constant,

is the standard Riemann-Liouville fractional derivative. We will deduce a property of Green’s function. The result we establish in Section 2 can be stated as follow.

Theorem 1. The Function G(t, s) defined by (12) is continuous and satisfies

()

where 0 < M = min {1 − βη^α−1, βη^α−2(1 − η), βη^α−1} < 1.

In this paper, we give some existence of positive solutions for singular boundary value problems by means of Schauder fixed-point theorem for the case: γ_* = 0, γ_* ≥ 0, γ^* ≤ 0, γ_* < 0 < γ^*.

The paper is organized as follows. In Section 2, we state some known results and give a property of Green’s function. In Section 3, using Schauder fixed-point theorem, the existence of positive solutions to singular problems are obtained.

2. Background Materials

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory.

Definition 2 (see [7].)The Riemann-Liouville fractional integral of order α > 0 of a function y : (0, ∞) → R is given by

()

provided the right side is pointwise defined on (0, ∞).

Definition 3 (see [7].)The Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : (0, ∞) → R is given by

()

where n = [α] + 1, [α] denotes the integer part of number α, provided that the right side is pointwise defined on (0, ∞).

From the definition of Riemann-Liouville’s derivative, we can obtain the statement.

Lemma 4 (see [7].)Let α > 0, if one assumes that u ∈ C(0,1)∩L(0,1), then the fractional differential equation

()

has u(t) = C₁t^α−1 + C₂t^α−2 + ⋯+C_Nt^α−N, C_i ∈ R, i = 1,2, …, N, as unique solutions, where N is the smallest integer greater than or equal to α.

Lemma 5 (see [7].)Assume that u ∈ C(0,1)∩L(0,1) with a fractional derivative of order α > 0 that belongs to C(0,1)∩L(0,1). Then,

()

for some C_i ∈ R, i = 1,2, …, N, N is the smallest integer greater than or equal to α.

Lemma 6. Given h ∈ C(0,1) the problem

()

is equivalent to

()

where

()

Proof. We can apply Lemma 5 to reduce (10) to an equivalent integral equation

()

for some C₁, C₂, …, C_n ∈ R. Consequently, the general solution of (10) is

()

By u(0) = u^′(0) = ⋯ = u⁽ⁿ⁻²⁾(0) = 0, this is C₂ = C₃ = ⋯ = C_n = 0.

On the other hand, u(1) = βu(η) combining with

()

yields

()

Therefor, the unique solution of problem (10) is

()

For t ≤ η, we have

()

For t ≥ η, we have

()

The proof is complete.

Proof of Theorem 1. It is easy to prove that G(t, s) is continuous on [0,1]×[0,1], here we omit it. In the following, we consider (1 − βη^α−1)Γ(α)G(t, s). When 0 ≤ s < t ≤ 1, s < η, let

()

We have

()

When 0 < η ≤ s < t ≤ 1, let

()

We have

()

When 0 ≤ t ≤ s ≤ η < 1, let g₃(t) = [t(1 − s)] ^α−1 − βt^α−1(η − s) ^α−1, we have

()

When 0 ≤ t ≤ s ≤ 1, η ≤ s, we have

()

It is easy to see that G(t, s) ≤ t^α−1(1 − s) ^α−1/Γ(α)(1 − βη^α−1). Thus, the proof is complete.

Let us fix some notations to be used in the following. “For a.e.” means “for almost every”. Given a ∈ L¹(0,1), we write a≻0 if a ≥ 0 for a.e. t ∈ [0,1], and it is positive in a set of positive measure, we write f ∈ Car((0,1)×(0, +∞), (0, +∞)) if f : (0,1)×(0, +∞)→(0, +∞) is a L¹-caratheodory function, that is, the map x ↦ f(t, x) is continuous for a.e. t ∈ (0,1), and the map t ↦ f(t, x) is measurable for all x ∈ (0, +∞).

Let us define

()

Then,

()

3. Main Results

In this section, we establish the existence of positive solutions for equation

()

where n − 1 < α ≤ n, n ≥ 3, 0 < β ≤ 1, 0 ≤ η ≤ 1, f ∈ Car((0,1)×(0, +∞), (0, +∞)), e(t) ∈ L¹[0,1],

is the standard Riemann-Liouville derivative. The following is the first main result in this section.

Theorem 7. Suppose that the following conditions are satisfied.

(H₁)
For each L > 0, there exists a function ϕ_L≻0 such that f(t, t^α−1x) ≥ ϕ_L(t) for a.e. t ∈ (0,1), all x ∈ (0, L].
(H₂)
There exist g(x), h(x), and k(t)≻0, such that
()
here
()
(H₃)
There exist two positive constants R > r > 0 such that
()
and here
()

Then, (28) has at least one positive solution.

Proof. Let E = (C[0,1], ∥·∥), and Ω is a closed convex set defined as

()

here E = C[0,1] is the Banach space of continuous functions defined on [0,1] with the norm

()

and R > r > 0 are positive constants to be given below.

Now, we define an operator T : Ω → E by

()

Then, (28) is equivalent to the fixed-point problem

()

Let R be the positive constant satisfying (H₃) and

()

Then, we have R > r > 0. Now, we prove T(Ω) ⊂ Ω.

In fact, for each x ∈ Ω and for all t ∈ (0,1), by (H₁) and (H₃)

()

On the other hand, by conditions (H₂) and (H₃), we have

()

In conclusion, T(Ω) ⊂ Ω.

Finally, it is standard that T : Ω → Ω is a continuous and completely continuous operator. By a direct application of Schauder’s fixed-point theorem, (28) has at least one positive solution x(t) ∈ C[0,1], the proof is finished.

Case 1 (γ* = 0). As an application of Theorem 7, we consider the case γ_* = 0. The following corollary is a direct result of Theorem 7 with r = Φ_R1.

Corollary 8. Suppose that f(t, x) satisfies conditions (H₁)-(H₂). Furthermore, assume the following.

There exists a positive constant R > 0 such that
()
and here
()

Then, (28) has at least one positive solution.

From now on, let us define

()

Example 9. Suppose that the nonlinearity in (28) is

()

where k≻0, 0 < λ < 1 and

()

If γ_* = 0, then (28) has at least one positive solution.

Proof. We will apply Corollary 8. To this end, we take

()

then (H₁) and (H₂) are satisfied since ω(λ)<+∞, and the existence condition (

) becomes

()

for some R > 0. Since 0 < λ < 1, we can choose R > 0 large enough such that (46) is satisfied, and the proof is finished.

Example 10. Suppose that the nonlinearity in (28) is

()

where 0 < λ < 1, ν ≥ 0 and μ ≥ 0 is a nonnegative parameter. For each e(t) with γ_* = 0, ω(λ)<+∞,

(i)
if λ + ν < 1 − λ², then (28) has at least one positive solution for each μ ≥ 0.
(ii)
If λ + ν ≥ 1 − λ², then (28) has at least one positive solution for each 0 ≤ μ < μ₁, where μ₁ is some positive constant.

Proof. We will apply Corollary 8. To this end, we take

()

Then, (H₁)-(H₂) are satisfied since ω(λ)<+∞. Now, the existence condition (

) becomes ω(λ)<+∞, and

()

for some R > 0 with R^1+λ > β₁. So, (28) has at least one positive solution for

()

Note that μ₁ = ∞ if λ + ν < 1 − λ² and if λ + ν ≥ 1 − λ², set

()

then, we have

()

Let the function l(R) possess a maximum at R₀, then

()

so we have

()

it is easy to find that

since λ + ν ≥ 1 − λ², and 0 < λ < 1. Finally, it would remain to prove

. This is easily verified through elementary computations since β₁ ≤ β₂. We have the desired results (i) and (ii).

Case 2 (γ* > 0). The next result explores the case when γ_* > 0. In this case r = γ_*.

Corollary 11. Suppose that f(t, x) satisfies (H₂). Furthermore, assume the following.

(H₄)
There exists R > 0, , such that
()

If γ_* > 0, then (28) has at least one positive solution.

Example 12. Suppose that the nonlinearity in (28) be (43) with k≻0, λ > 0. If γ_* > 0, ω(λ)<+∞, then (28) has at least one positive solution.

Proof. We will apply Corollary 11. Take k(t), g(x), and h(x) as the same in the proof of Example 9. Then, (H₂) is satisfied, and the existence condition (H₄) is satisfied if we take R > 0 with , and ω(λ)<+∞.

Example 13. Let the nonlinearity in (28) be (47) with λ > 0 and ν ≥ 0. For each e(t) with γ_* > 0, ω(λ)<+∞,

(i)
if λ + ν < 1, then (28) has at least one positive solution for each μ ≥ 0.
(ii)
If λ + ν ≥ 1, then (28) has at least one positive solution for each 0 ≤ μ < μ₂, where μ₂ is some positive constant.

Proof. We will apply Corollary 11. To this end, we take g(x), h(x), and k(t) as the same in the proof of Example 10, then (H₂) is satisfied, and the existence condition (H₄) becomes ω(λ)<+∞,

()

for some R > 0. So, (28) has at least one positive solution for

()

Note that μ₂ = ∞ if λ + ν < 1 and

if λ + ν = 1, and if λ + ν > 1 set

()

The function l(R) possesses a maximum at

()

then μ₂ = l(R₀). We have the desired results (i) and (ii).

Case 3 (γ* ≤ 0). The next result considers the case γ^* ≤ 0.

Corollary 14. Suppose that f(t, x) satisfies (H₁)-(H₂). Furthermore, assume the following.

(H₅)
There exist two positive constants R > r > 0 such that
()
here
()

Then, (28) has at least one positive solution.

Example 15. Suppose that the nonlinearity in (28) be (43) with k≻0, λ > 0. If γ^* ≤ 0, ω(λ)<+∞,

()

then (28) has at least one positive solution.

Proof. We will apply Corollary 14. Take k(t), g(x) as the same in the proof of Example 9. Then, (H₂) is satisfied, and the existence condition (H₅) is satisfied if we take R > r > 0 with

()

and ω(λ)<+∞. If we fix R = β₂/r^λ, then the first inequality holds if r satisfies

()

or equivalently

()

The function l(r) possesses a minimum at

()

Taking r = r₀, then the first inequality in (63) holds if γ_* ≥ l(r₀), which is just condition (62). The second inequality holds directly from the choice of R, so it remains to prove that R = β₂/r^λ > r₀. This is easily verified through elementary computations.

Example 16. Let the nonlinearity in (28) be (47) with k≻0, λ > 0 and ν ≥ 0. If γ^* ≤ 0, ω(λ)<+∞,

()

here m₀ is the unique solution of the equation

()

Then (28) has at least one positive solution.

Proof. We will apply Corollary 14. To this end, we take g(x), h(x), and k(t) as the same in the proof of Example 10, then (H₂) is satisfied, and the existence condition (H₅) is satisfied if we take R > r > 0 with

()

and ω(λ)<+∞. If we fix R = (1 + μR^λ+ν)β₂r^−λ, then the first inequality holds if R satisfies

()

Let m = 1/R, then

()

Then, we have

()

Let F^′(m) = 0, then we have

()

Now, let us define Φ(m) by

()

It is easy to see that Φ(m) is a nondecreasing function for m ∈ [0, +∞) and Φ(m)→+∞, as m → +∞. Thus, Φ(m) = λ²β₁ has a unique solution m₀ such that

()

and F(m₀) = inf _m>0F(m).

So, it remains to prove that R > r = [(1 + μR^λ+ν)β₂/R] ^1/λ, that is,

()

In fact, by (75), we have

()

that is,

()

Also we have

()

that is,

()

Thus, we have

()

since 0 < λ, ν < 1, and 1 − λ − ν − λ² > 0.

We have the desired results.

Case 4 (γ* < 0 < γ*).

Example 17. Suppose that the nonlinearity in (28) be (43) with k≻0, λ > 0. If γ_* < 0 < γ^*, ω(λ)<+∞,

()

and here m₀ is the unique solution of the equation

()

then (28) has at least one positive solution.

Proof. We will apply Theorem 7. Take k(t), g(x) as the same in the proof of Example 9. Then, (H₂) is satisfied, and the existence condition (H₃) is satisfied if we take R > r > 0 with

()

and ω(λ)<+∞. If we fix R = β₂/r^λ + γ^*, then the first inequality holds if R satisfies

()

Taking m = 1/(R − γ^*), then for m ∈ (0, +∞), we have

()

Then, we have

()

Let F^′(m) = 0, then we have

()

Now, let us define Φ(m) by

()

It is easy to see that Φ(m) is a nondecreasing function for m ∈ [0, +∞), and Φ(m)→+∞, as m → +∞. Thus, Φ(m) = λ²β₁ has a unique solution m₀ such that

()

and F(m₀) = inf _m>0F(m).

So, it remains to prove that R > r = [β₂/(R − γ^*)] ^1/λ, that is,

()

In fact, by (90), we have

()

that is,

()

Thus, we have

()

The proof is complete.

Example 18. Let the nonlinearity in (28) be (47) with k≻0, λ > 0 and ν ≥ 0. If γ_* < 0 < γ^*, ω(λ)<+∞,

()

here m₀ is the unique solution of the equation

()

Then (28) has at least one positive solution.

Proof. We will apply Theorem 7. To this end, we take g(x), h(x), and k(t) as the same in the proof of Example 10, then (H₂) is satisfied, and the existence condition (H₃) is satisfied if we take R > r > 0 with

()

and ω(λ)<+∞. If we fix R = (1 + μR^λ+ν)β₂r^−λ + γ^*, then the first inequality holds if

()

Let m = 1/(R − γ^*), then

()

Then, we have

()

Let F^′(m) = 0, then we have

()

Now, let us define Φ(m) by

()

It is easy to see that Φ(m) is a nondecreasing function for m ∈ [0, +∞), and Φ(m)→+∞, as m → +∞. Thus, Φ(m) = λ²β₁ has a unique solution m₀ such that

()

and F(m₀) = inf _m>0F(m).

So, it remains to prove that R > r = [(1 + μR^λ+ν)β₂/(R − γ^*)] ^1/λ, that is,

()

In fact, by (103), we have

()

that is,

()

Also we have

()

that is,

()

Thus, we have

()

since 0 < λ, ν < 1 and 1 − λ − ν − λ² > 0.

Thus, the proof is complete.

Remark 19. It is easy to find that analogous results to Examples 10, 13, 16, and 18 for the general equation with the nonlinearity in (28) are

()

with b, c≻0, but the notation becomes cumbersome. Here we consider the nonlinearity (47) only for the simplicity.

Acknowledgments

The work was supported by NSFC of China (no. 10971021), the Science and Technology Research Project of Jilin Provincial Department of Education of China (no. 2011176), and the Science and Technology Research Project of Jilin Provincial Department of Education of China (no. 2010391). Supported by Scientific Research Fund of Heilongjiang Provincial Education Department.

References

1 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, 204, Elsevier Science B.V., Amsterdam, The Netherlands, North-Holland Mathematics Studies, MR2218073, https://doi.org/10.1016/S0304-0208(06)80001-0, ZBL1206.26007.
10.1016/S0304-0208(06)80001-0
Google Scholar
2 Oldham K. B. and Spanier J., The Fractional Calculus, 1974, Academic Press, New York, NY, USA, Mathematics in Science and Engineering, vol. 111, MR0361633, ZBL0292.26011.
Google Scholar
3 B. Ross, Fractional Calculus and Its Applications, 1975, Springer, Berlin, Germany, Lecture Notes in Mathematics, vol. 457, MR0367127.
10.1007/BFb0067095
Google Scholar
4 Nonnenmacher T. F. and Metzler R., On the Riemann-Liouville fractional calculus and some recent applications, Fractals. (1995) 3, no. 3, 557–566, https://doi.org/10.1142/S0218348X95000497, MR1410279, ZBL0868.26004.
10.1142/S0218348X95000497
Web of Science® Google Scholar
5 Tatom F. B., The relationship between fractional calculus and fractals, Fractals. (1995) 3, no. 1, 217–229, https://doi.org/10.1142/S0218348X95000175, MR1329030, ZBL0877.28009.
10.1142/S0218348X95000175
Web of Science® Google Scholar
6 Podlubny I., Fractional Differential Equations, 1999, 198, Academic Press, San Diego, Calif, USA, Mathematics in Science and Engineering, MR1658022.
Web of Science® Google Scholar
7 Samko S. G., Kilbas A. A., and Marichev O. I., Fractional Integrals and Derivatives: Theory and Applications, 1993, Gordon and Breach Science, Yverdon, Switzerland, MR1347689.
Google Scholar
8 Magin R., Feng X., and Baleanu D., Solving the fractional order Bloch equation, Concepts in Magnetic Resonance Part A. (2009) 34, no. 1, 16–23, 2-s2.0-59649085150, https://doi.org/10.1002/cmr.a.20129.
10.1002/cmr.a.20129
Web of Science® Google Scholar
9 Baleanu D. and Trujillo J. I., A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives, Communications in Nonlinear Science and Numerical Simulation. (2010) 15, no. 5, 1111–1115, https://doi.org/10.1016/j.cnsns.2009.05.023, MR2563910, ZBL1221.34008.
10.1016/j.cnsns.2009.05.023
Web of Science® Google Scholar
10 Baleanu D., Diethelm K., Scalas E., and Trujillo J. J., Fractional Calculus Models and Numerical Methods, 2012, 3, World Scientific, Hackensack, NJ, USA, Series on Complexity, Nonlinearity and Chaos, https://doi.org/10.1142/9789814355216, MR2894576.
10.1142/8180
Google Scholar
11 Băleanu D., Mustafa O. G., and Agarwal R. P., An existence result for a superlinear fractional differential equation, Applied Mathematics Letters. (2010) 23, no. 9, 1129–1132, https://doi.org/10.1016/j.aml.2010.04.049, MR2659151, ZBL1200.34004.
10.1016/j.aml.2010.04.049
Web of Science® Google Scholar
12 Băleanu D., Agarwal R. P., Mustafa O. G., and Coşulschi M., Asymptotic integration of some nonlinear differential equations with fractional time derivative, Journal of Physics A. (2011) 44, no. 5, 9, 055203, https://doi.org/10.1088/1751-8113/44/5/055203, MR2763455, ZBL1238.26008.
10.1088/1751-8113/44/5/055203
Google Scholar
13 Băleanu D., Mustafa O. G., and Agarwal R. P., On the solution set for a class of sequential fractional differential equations, Journal of Physics A. (2010) 43, no. 38, 7, 385209, https://doi.org/10.1088/1751-8113/43/38/385209, MR2718329, ZBL1216.34004.
10.1088/1751-8113/43/38/385209
Google Scholar
14 Bai Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Analysis: Theory, Methods & Applications. (2010) 72, no. 2, 916–924, https://doi.org/10.1016/j.na.2009.07.033, MR2579357, ZBL1187.34026.
10.1016/j.na.2009.07.033
Web of Science® Google Scholar
15 Wang J., Xiang H., and Liu Z., Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, International Journal of Differential Equations. (2010) 2012, 12, https://doi.org/10.1155/2010/186928, 186928, MR2525726, ZBL1207.34012.
10.1155/2010/186928
Google Scholar
16 Ahmad B. and Nieto J. J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Computers & Mathematics with Applications. (2009) 58, no. 9, 1838–1843, https://doi.org/10.1016/j.camwa.2009.07.091, MR2557562, ZBL1205.34003.
10.1016/j.camwa.2009.07.091
Web of Science® Google Scholar
17 Li C. F., Luo X. N., and Zhou Y., Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Computers & Mathematics with Applications. (2010) 59, no. 3, 1363–1375, https://doi.org/10.1016/j.camwa.2009.06.029, MR2579500, ZBL1189.34014.
10.1016/j.camwa.2009.06.029
Web of Science® Google Scholar
18 Xu X. and Fei X., The positive properties of Green′s function for three point boundary value problems of nonlinear fractional differential equations and its applications, Communications in Nonlinear Science and Numerical Simulation. (2012) 17, no. 4, 1555–1565, https://doi.org/10.1016/j.cnsns.2011.08.032, MR2855448, ZBL1248.35224.
10.1016/j.cnsns.2011.08.032
Web of Science® Google Scholar

Citing Literature

All articles

The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applications

Abstract

1. Introduction

2. Background Materials

3. Main Results

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applications

Abstract

1. Introduction

2. Background Materials

3. Main Results

Acknowledgments

References

Citing Literature

References

Related

Information