American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 283-290. doi: 10.3934/proc.2013.2013.283

Positive solutions of nonlocal fractional boundary value problems

John R. Graef 1, , Lingju Kong 1, , Qingkai Kong 2, and  Min Wang 1,

1. 

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States, United States
2. 

Department of Mathematics, Northern Illinois University, DeKalb, Il 60115

Received  August 2012 Published  November 2013

The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.
Keywords: fractional calculus., positive solution, Green's function.
Mathematics Subject Classification: Primary: 34B15, 34B1.
Citation: John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283
References:
[1]

R. Agarwal, D. O'Regan, and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,, J. Math. Anal. Appl. 371 (2010), 371 (2010), 57. Google Scholar

[2]

B. Ahmad and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,, Comput. Math. Appl. 58 (2009), 58 (2009), 1838. Google Scholar

[3]

Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation,, J. Math. Anal. Appl. 311 (2005), 311 (2005), 495. Google Scholar

[4]

K. Deimling, "Nonlinear Functional Analysis'',, Springer-Verlag, (1985). Google Scholar

[5]

M. Feng, X. Zhang, and W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions,, Bound. Value Probl. (2011), (2011). Google Scholar

[6]

C. Goodrich, Existence of a positive solution to a class of fractional differential equations,, Appl. Math. Lett. 23 (2010), 23 (2010), 1050. Google Scholar

[7]

J. R. Graef, L. Kong, Q. Kong, and M. Wang, Fractional boundary value problems with integral boundary conditions,, Appl. Anal. 92 (2013), 92 (2013), 2008. Google Scholar

[8]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones'',, Academic Press, (1988). Google Scholar

[9]

R. Hilfer, "Applications of Fractional Calculus in Physics'',, World Scientific, (2000). Google Scholar

[10]

D. Jiang and C. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,, Nonlinear Anal. 72 (2010), 72 (2010), 710. Google Scholar

[11]

Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions,, Electron. J. Qual. Theory Differ. Equ., (2012), 1. Google Scholar

[12]

V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media'',, Springer-Verlag, (2011). Google Scholar

[13]

L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems,, Appl. Math. Lett. 23 (2010), 23 (2010), 1095. Google Scholar

[14]

W. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions,, Comput. Math. Appl. 63 (2012), 63 (2012), 288. Google Scholar

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems'',, Springer-Verlag, (1986). Google Scholar

[16]

C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems,, Nonlinear Anal., 75 (2012), 2542. Google Scholar

[17]

S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation,, Comput. Math. Appl. 59 (2010), 59 (2010), 1300. Google Scholar

show all references

References:
[1]

R. Agarwal, D. O'Regan, and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,, J. Math. Anal. Appl. 371 (2010), 371 (2010), 57. Google Scholar

[2]

B. Ahmad and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,, Comput. Math. Appl. 58 (2009), 58 (2009), 1838. Google Scholar

[3]

Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation,, J. Math. Anal. Appl. 311 (2005), 311 (2005), 495. Google Scholar

[4]

K. Deimling, "Nonlinear Functional Analysis'',, Springer-Verlag, (1985). Google Scholar

[5]

M. Feng, X. Zhang, and W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions,, Bound. Value Probl. (2011), (2011). Google Scholar

[6]

C. Goodrich, Existence of a positive solution to a class of fractional differential equations,, Appl. Math. Lett. 23 (2010), 23 (2010), 1050. Google Scholar

[7]

J. R. Graef, L. Kong, Q. Kong, and M. Wang, Fractional boundary value problems with integral boundary conditions,, Appl. Anal. 92 (2013), 92 (2013), 2008. Google Scholar

[8]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones'',, Academic Press, (1988). Google Scholar

[9]

R. Hilfer, "Applications of Fractional Calculus in Physics'',, World Scientific, (2000). Google Scholar

[10]

D. Jiang and C. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,, Nonlinear Anal. 72 (2010), 72 (2010), 710. Google Scholar

[11]

Q. Kong and M. Wang, Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions,, Electron. J. Qual. Theory Differ. Equ., (2012), 1. Google Scholar

[12]

V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media'',, Springer-Verlag, (2011). Google Scholar

[13]

L. Yang and H. Chen, Unique positive solutions for fractional differential equation boundary value problems,, Appl. Math. Lett. 23 (2010), 23 (2010), 1095. Google Scholar

[14]

W. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions,, Comput. Math. Appl. 63 (2012), 63 (2012), 288. Google Scholar

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems'',, Springer-Verlag, (1986). Google Scholar

[16]

C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems,, Nonlinear Anal., 75 (2012), 2542. Google Scholar

[17]

S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation,, Comput. Math. Appl. 59 (2010), 59 (2010), 1300. Google Scholar

[1]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[2]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[3]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307
[4]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657
[5]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[6]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[7]

Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110
[8]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[9]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[10]

Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77
[11]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
[12]

Olga Kharlampovich and Alexei Myasnikov. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements, 1998, 4: 101-108.

[13]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[14]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[15]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[16]

Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 539-560. doi: 10.3934/dcdss.2020030
[17]

Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321
[18]

Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055
[19]

Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407
[20]

Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61

 Impact Factor: 

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences