u , v , w ) Ω 2 , we have ( u , v , w ) = ( m 1 , m 2 , m 3 ) , m 1 , m 2 , m 3 R . Then from N ( u , v , w ) Im L ,

0 1 ( 1 s ) α 2 f 1 ( s , m 2 , 0 ) d s = 0 ,

0 1 ( 1 s ) β 2 f 2 ( s , m 3 , 0 ) d s = 0

and

0 1 ( 1 s ) γ 2 f 3 ( s , m 1 , 0 ) d s = 0.

From ( H 3 ) imply that | m 1 | , | m 2 | , | m 3 | M * . Thus, we get

( u , v , w ) X ¯ M * .

Therefore Ω 2 is bounded.

Lemma 8. Assume that the first part of ( H 3 ) holds, then the set

Ω 3 = { ( u , v , w ) Ker L | λ ( u , v , w ) + ( 1 λ ) Q N ( u , v , w ) = ( 0 , 0 , 0 ) , λ [ 0 , 1 ] }

is bounded.

Proof. For ( u , v , w ) Ω 3 , we have ( u , v , w ) = ( m 1 , m 2 , m 3 ) , m 1 , m 2 , m 3 R and

λ m 1 + ( 1 λ ) ( α 1 ) 0 1 ( 1 s ) α 2 f 1 ( s , m 2 ,0 ) d s = 0 , (15)

λ m 2 + ( 1 λ ) ( β 1 ) 0 1 ( 1 s ) β 2 f 2 ( s , m 3 , 0 ) d s = 0 (16)

and

λ m 3 + ( 1 λ ) ( γ 1 ) 0 1 ( 1 s ) γ 2 f 3 ( s , m 1 ,0 ) d s = 0. (17)

If λ = 0 , then by ( H 3 ) , we get | m 1 | , | m 2 | , | m 3 | M * . If λ = 1 , then m 1 = m 2 = m 3 = 0 . For λ ( 0,1 ] , we obtain | m 1 | , | m 2 | , | m 3 | M * . Otherwise, if | m 1 | or | m 2 | or | m 3 | > M * , from ( H 3 ) , one has

λ m 1 2 + ( 1 λ ) ( α 1 ) 0 1 ( 1 s ) α 2 m 1 f 1 ( s , m 2 , 0 ) d s > 0

or

λ m 2 2 + ( 1 λ ) ( β 1 ) 0 1 ( 1 s ) β 2 m 2 f 2 ( s , m 3 , 0 ) d s > 0

or

λ m 3 2 + ( 1 λ ) ( γ 1 ) 0 1 ( 1 s ) γ 2 m 3 f 3 ( s , m 1 ,0 ) d s > 0,

which contradict to (15) or (16) or (17). Hence, Ω 3 is bounded.

Remark 1 Suppose the second part of ( H 3 ) holds, then the set

Ω 3 = { ( u , v , w ) Ker L | λ ( u , v , w ) + ( 1 λ ) Q N ( u , v , w ) = ( 0 , 0 , 0 ) , λ [ 0 , 1 ] }

is bounded.

Proof of the Theorem 1: Set Ω = { ( u , v , w ) X ¯ | ( u , v , w ) X ¯ < max { K , M * } + 1 } .

From the Lemma 4 and Lemma 5 we can get L is a Fredholm operator of index zero and N is L-compact on Ω ¯ . By Lemma 6 and Lemma 7, we obtain

(1) L ( u , v , w ) λ N ( u , v , w ) for every ( ( u , v , w ) , λ ) [ ( dom L \ Ker L ) Ω ] × ( 0 , 1 ) ;

(2) N x Im L for every ( u , v , w ) Ker L Ω .

Choose

H ( ( u , v , w ) , λ ) = ± λ ( u , v , w ) + ( 1 λ ) Q N ( u , v , w ) .

By Lemma 8 (or Remark 1), we get H ( ( u , v , w ) , λ ) 0 for ( u , v , w ) Ker L Ω . Therefore

deg ( Q N | Ker L , Ker L Ω , 0 ) = deg ( H ( . , 0 ) , Ker L Ω , 0 ) = deg ( H ( . , 1 ) , Ker L Ω , 0 ) = deg ( ± I , Ker L Ω , 0 ) 0.

Thus, the condition (3) of Lemma 1 is satisfied. By Lemma 1, we obtain L ( u , v , w ) = N ( u , v , w ) has at least one solution in dom L Ω ¯ . Hence Neumann boundary value problem (1) has at least one solution. This completes the proof.

4. Examples

In this section, we give two examples to illustrate our main results.

Example 1. Consider the following Neumann boundary value problem of fractional differential equation of the form

D 0 + 5 4 u ( t ) = 1 8 ( v ( t ) 6 ) + t 2 8 ( 1 + v ( t ) ) 1 2 , t ( 0 , 1 ) , D 0 + 3 2 v ( t ) = 1 6 ( w ( t ) 4 ) + t 3 6 cos 2 w ( t ) , t ( 0 , 1 ) , D 0 + 7 4 w ( t ) = 1 10 ( u ( t ) 8 ) + t 3 10 sin 2 u ( t ) , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , w ( 0 ) = w ( 1 ) = 0. (18)

Here α = 5 4 , β = 3 2 , γ = 7 4 . Moreover,

f 1 ( t , v ( t ) , v ( t ) ) = 1 8 ( v ( t ) 6 ) + t 2 8 ( 1 + v ( t ) ) 1 2 ,

f 2 ( t , w ( t ) , w ( t ) ) = 1 6 ( w ( t ) 4 ) + t 3 6 cos 2 w ( t ) ,

f 3 ( t , u ( t ) , u ( t ) ) = 1 10 ( u ( t ) 8 ) + t 3 10 sin 2 u ( t ) .

Now let us compute a 1 ( t ) , b 1 ( t ) , c 1 ( t ) from f 1 ( t , v ( t ) , v ( t ) ) .

f 1 ( t , v ( t ) , v ( t ) ) = 1 8 ( v ( t ) 6 ) + t 2 8 ( 1 + v ( t ) ) 1 2 = 1 8 ( v ( t ) 6 ) + t 2 8 ( 1 + 1 2 v ( t ) + ) 1 8 ( v ( t ) 6 ) + t 2 8 .

| f 1 ( t , v ( t ) , v ( t ) ) | 1 8 | v ( t ) | + 7 8 .

From the above inequality, we get a 1 ( t ) = 7 8 , b 1 ( t ) = 1 8 , c 1 ( t ) = 0. Also,

f 2 ( t , w ( t ) , w ( t ) ) = 1 6 ( w ( t ) 4 ) + t 3 6 cos 2 w ( t )

| f 2 ( t , w ( t ) , w ( t ) ) | 1 6 | w ( t ) | + 5 6 .

Here, a 2 ( t ) = 5 6 , b 2 ( t ) = 1 6 , c 2 ( t ) = 0 . Finally,

f 3 ( t , u ( t ) , u ( t ) ) = 1 10 ( u ( t ) 8 ) + t 3 10 sin 2 u ( t ) .

| f 3 ( t , u ( t ) , u ( t ) ) | 1 10 | u ( t ) | + 8 10 .

We get, a 3 ( t ) = 4 5 , b 3 ( t ) = 1 10 , c 3 ( t ) = 0 . And we get, B 1 ( t ) = 1 8 , B 2 ( t ) = 1 6 , B 3 ( t ) = 1 10 , C 1 ( t ) = C 2 ( t ) = C 3 ( t ) = 0 . Choose M = M * = 8 . Also,

Γ ( α ) Γ ( β ) Γ ( γ ) ( B 1 + C 1 ) ( B 2 + C 2 ) ( B 3 + C 3 ) Γ ( α ) Γ ( β ) Γ ( γ ) = Γ ( 5 4 ) Γ ( 3 2 ) Γ ( 7 4 ) ( B 1 B 2 B 3 ) Γ ( 5 4 ) Γ ( 3 2 ) Γ ( 7 4 ) 0.73605543 > 0 ,

where Γ ( 1 4 ) 3.625 , Γ ( 3 4 ) 1.225416702 and Γ ( 1 2 ) = π . All the condi-

tions of Theorem 1 are satisfied. Hence, boundary value problem (18) has at least one solution.

Example 2. Consider the Neumann boundary value problem of fractional differential equation of the following form

D 0 + 4 3 u ( t ) = 1 17 ( v ( t ) 15 ) + t 4 20 log ( 1 + v ( t ) ) , t ( 0 , 1 ) , D 0 + 5 4 v ( t ) = 1 9 ( w ( t ) 7 ) + t 6 25 ( 1 + w ( t ) ) 1 3 , t ( 0 , 1 ) , D 0 + 3 2 w ( t ) = 1 13 ( u ( t ) 11 ) + t 7 15 arctan u ( t ) , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , w ( 0 ) = w ( 1 ) = 0. (19)

Here α = 4 3 , β = 5 4 , γ = 3 2 . Moreover,

f 1 ( t , v ( t ) , v ( t ) ) = 1 17 ( v ( t ) 15 ) + t 4 20 log ( 1 + v ( t ) ) ,

f 2 ( t , w ( t ) , w ( t ) ) = 1 9 ( w ( t ) 7 ) + t 6 25 ( 1 + w ( t ) ) 1 3 ,

f 3 ( t , u ( t ) , u ( t ) ) = 1 13 ( u ( t ) 11 ) + t 7 15 arctan u ( t ) .

Now let us compute a 1 ( t ) , b 1 ( t ) , c 1 ( t ) from f 1 ( t , v ( t ) , v ( t ) ) .

f 1 ( t , v ( t ) , v ( t ) ) = 1 17 ( v ( t ) 15 ) + t 4 20 log ( 1 + v ( t ) ) = 1 17 ( v ( t ) 15 ) + t 4 20 ( v ( t ) ( v ( t ) ) 2 2 ! + )

| f 1 ( t , v ( t ) , v ( t ) ) | 1 17 | v ( t ) | + 15 17 + 1 20 | v ( t ) |

From the above inequality, we get a 1 ( t ) = 15 17 , b 1 ( t ) = 1 17 , c 1 ( t ) = 1 20 . Also,

f 2 ( t , w ( t ) , w ( t ) ) = 1 9 ( w ( t ) 7 ) + t 6 25 ( 1 + w ( t ) ) 1 3 = 1 9 ( w ( t ) 7 ) + t 6 25 ( 1 + 1 3 w ( t ) + )

| f 2 ( t , w ( t ) , w ( t ) ) | 1 9 | w ( t ) | + 7 9 + 1 25 ( 1 + 1 3 | w ( t ) | + ) 1 9 | w ( t ) | + 184 225 + 1 75 | w ( t ) | .

Here, a 2 ( t ) = 184 225 , b 2 ( t ) = 1 9 , c 2 ( t ) = 1 75 . Similarly,

f 3 ( t , u ( t ) , u ( t ) ) = 1 13 ( u ( t ) 11 ) + t 7 10 arctan u ( t ) . = 1 13 ( u ( t ) 11 ) + t 7 15 ( u ( t ) ( u ( t ) 3 ) 3 + ) .

| f 3 ( t , u ( t ) , u ( t ) ) | 1 13 | u ( t ) | + 11 13 + 1 15 u ( t ) .

Here, a 3 ( t ) = 11 13 , b 3 ( t ) = 1 13 , c 3 ( t ) = 1 15 . We get, B 1 ( t ) = 1 17 , B 2 ( t ) = 1 9 , B 3 ( t ) = 1 13 , C 1 ( t ) = 1 20 , C 2 ( t ) = 1 75 , C 3 ( t ) = 1 15 . Choose M = M * = 15 .

Also,

Γ ( α ) Γ ( β ) Γ ( γ ) ( B 1 + C 1 ) ( B 2 + C 2 ) ( B 3 + C 3 ) Γ ( α ) Γ ( β ) Γ ( γ ) = Γ ( 4 3 ) Γ ( 5 4 ) Γ ( 3 2 ) ( ( 1 17 + 1 20 ) ( 1 9 + 1 75 ) ( 1 13 + 1 15 ) ) Γ ( 4 3 ) Γ ( 5 4 ) Γ ( 3 2 ) 0.96872 > 0 ,

where Γ ( 1 4 ) 3.625 , Γ ( 1 3 ) 0.1924 and Γ ( 1 2 ) = π . Hence all the condi-

tions of Theorem 1 are satisfied. Therefore, boundary value problem (19) has at least one solution.

5. Conclusion

We have investigated some existence results for three-dimensional fractional differential system with Neumann boundary condition. By using Mawhin’s coin- cidence degree theory, we established that the given boundary value problem admits at least one solution. We also presented examples to illustrate the main results.

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the manuscript.

Cite this paper
Sadhasivam, V. , Kavitha, J. and Deepa, M. (2017) Existence of Solutions of Three-Dimensional Fractional Differential Systems. Applied Mathematics, 8, 193-208. doi: 10.4236/am.2017.82016.
References

[1]   Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.

[2]   Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon.

[3]   Hilfer, Z. (2000) Appliations of Fractional Calculus in Physics. World Scientific, Singapore.
https://doi.org/10.1142/3779

[4]   Metzler, R. and Klafter, J. (2000) Boundary Value Problems for Fractional Diffusion Equations. Physics A, 278, 107-125.
https://doi.org/10.1016/S0378-4371(99)00503-8

[5]   Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.

[6]   Lakshmikantham, V., Leela, S. and Vasundhara Devi, J. (2009) Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge.

[7]   Mainardi, A. (2010) Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London.
https://doi.org/10.1142/p614

[8]   Abbas, S., Benchora, M. and N’Guerekata, G.M. (2012) Topics in Fractional Differential Equations. Springer, New York.
https://doi.org/10.1007/978-1-4614-4036-9

[9]   Zhou, Y. (2014) Basic Theory of Fractional Differential Equations. World Scientific, Singapore.
https://doi.org/10.1142/9069

[10]   Ahmad, B. and Nieto, J.J. (2009) Existence Results for a Coupled System of Nonlinear Fractional Differential Equations with Three-Point Boundary Conditions. Computers and Mathematics with Applications, 58, 1838-1843.
https://doi.org/10.1016/j.camwa.2009.07.091

[11]   Liu, Y., Ahmad, B. and Agarwal, R.P. (2013) Existence of Solutions for a Coupled System of Nonlinear Fractional Differential Equations with Fractional Boundary Conditions on the Half-Line. Advances in Difference Equations, 2013, 46.
https://doi.org/10.1186/1687-1847-2013-46

[12]   Aphithana, A., Ntouyas, S.K. and Tariboon, J. (2015) Existence and Uniqueness of Symmetric Solutions for Fractional Differential Equations with Multi-Point Fractional Integral Conditions. Boundary Value Problems, 2015, 68.
https://doi.org/10.1186/s13661-015-0329-1

[13]   Wang, Y. (2016) Positive Solutions for Fractional Differential Equation Involving the Riemann-Stieltjes Integral Conditions with Two Parameters. Journal of Nonlinear Science and Applications, 9, 5733-5740.

[14]   Mawhin, J. (1993) Topological Degree and Boundary Value Problems for Nonlinear Differential Equations in Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, 1537, 74-142.
https://doi.org/10.1007/BFb0085076

[15]   Kosmatov, N. (2010) A Boundary Value Problem of Fractional Order at Resonance. Electronic Journal of Differential Equations, 135, 1-10.

[16]   Bai, Z. and Zhang, Y. (2010) The Existence of Solutions for a Fractional Multi-Point Boundary Value Problem. Computers and Mathematics with Applications, 60, 2364-2372.
https://doi.org/10.1016/j.camwa.2010.08.030

[17]   Wang, G., Liu, W., Zhu, S. and Zheng, T. (2011) Existence Results for a Coupled System of Nonlinear Fractional 2m-Point Boundary Value Problems at Resonance. Advances in Difference Equations, 44, 1-17.
https://doi.org/10.1155/2011/783726

[18]   Zhang, Y., Bai, Z. and Feng, T. (2011) Existence Results for a Coupled System of Nonlinear Fractional Three-Point Boundary Value Problems at Resonance. Computers and Mathematics with Applications, 61, 1032-1047.
https://doi.org/10.1016/j.camwa.2010.12.053

[19]   Jiang, W. (2012) Solvability for a Coupled System of Fractional Differential Equations at Resonance. Nonlinear Analysis, 13, 2285-2292.
https://doi.org/10.1016/j.nonrwa.2012.01.023

[20]   Hu, Z., Liu, W. and Rui, W. (2012) Existence of Solutions for a Coupled System of Fractional Differential Equations. Springer, Berlin, 1-15.
https://doi.org/10.1186/1687-2770-2012-98

[21]   Hu, L. (2016) On the Existence of Positive Solutions for Fractional Differential Inclusions at Resonance. SpringerPlus, 5, 957.
https://doi.org/10.1186/s40064-016-2665-8

[22]   Hu, Z., Liu, W. and Chen, T. (2011) Two-Point Boundary Value Problems for Fractional Differential Equations at Resonance. Bulletin of the Malaysian Mathematical Society Series, 3, 747-755.

[23]   Hu, Z., Liu, W. and Chen, T. (2012) Existence of Solutions for a Coupled System of Fractional Differential Equations at Resonance. Boundary Value Problems, 2012, 98.
https://doi.org/10.1186/1687-2770-2012-98

 
 
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