POSITIVE SOLUTIONS FOR A NONLINEAR FRACTIONAL BOUNDARY VALUE PROBLEM
Keywords:
Fractional derivative, Positive solutions, Compact operator, Integral boundary condition, Cone
Abstract
In this paper, we establish the existence of positive solutions to
a nonlinear fractional differential equation with integral boundary conditions.
Our approach is based on the linear operator theory and the application of
Krasnosel’skii fixed-point theorem in a cone. We present two examples to
illustrate the practicability of our main results.
References
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[2] B. Ahmad, J. J. Nieto, A. Alsaed. Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Mathematica Scientia 31 (2011), 2122 - 2130.
[3] M. B. Akorede, P. O. Arawomo. Existence of positive solutions for a coupled system of nonlinear boundary value problems of fractional order with integral boundary conditions. Int. J. Applied Math. 30(4) (2017), 273 - 288. doi:10.12732/ijam.v30i4.1
[4] M. B. Akorede, P. O. Arawomo. Positive solutions to a nonlinear eigenvalue problems of fractional differential equations with integro-differential boundary conditions. Int. J. Pure and Applied Math. 116(2) (2017), 313 - 328. doi:10.12732/ijpam.v116i2.3
[5] A. Anguraj, P. Karthikeyan, J. J. Trujillo. Existence of solutions for fractional mixed integrodifferential equations with nonlocal initial condition. Adv. Difference Equations 2011 (2011), Art.ID 690653, 12pages. doi:10.1155/2011/690653
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[7] Z. Bai. On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. Theory, Methods and Appl. 72(2) (2010), 916 - 924. doi:10.1016/j.na.2009.07.033
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[9] A. Boucherif. Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis 70 (2009), 364 - 371. doi:10.1016/j.na.2007.12.007
[10] A. Cabada, Z. Hamdi. Nonlinear fractional differential equations with integral boundary value conditions. Applied Mathematics and Computation 228 (2014), 251 - 257.
[11] A. Cabada, G. Wang. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389 (2012), 403 - 411. doi:10.1016/j.jmaa.2011.11.065
[12] M. El-Shahed. On the existence of positive solutions for a boundary value problem of fractional order. Thai J. Math. 5(1) (2007), 143 - 150.
[13] M. El-Shahed. Positive solutions for a boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007 (2007), Article ID10368, 8pages. doi:10.1155/2007/10368
[14] M. Feng, X. Zhang, W. Ge. New existence results for higher-order nonlinear fractional differential equations with integral boundary conditions. Boundary Value Problems 2011 (2011), Article ID720702, 20pages. doi:10.1155/2011/720702
[15] M. Feng, X. Liu, H. Feng. The existence of positive solutions for a nonlinear fractional differential equation with integral boundary conditions. Adv. Difference Equations 2011 (2011), Art.ID 546038, 14pages. doi:10.1155/2011/546038
[16] J. Henderson, R. Luca. Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Applied Mathematics and Computation 249 (2014), 182 - 197. doi:10.1016/j.amc.2014.10.028
[17] J. Jin, X. Liu, M. Jia. Existence of positive solutions for singular fractional differential equations with integral boundary conditions. Electron. J. Diff. Equations 2012(63) (2012), 1 - 14.
[18] H. Johnny, L. Rodica, T. Alexandru. Positive solutions for a fractional boundary value problem. Nonlinear Stud. 22(1) (2015), 139 - 151.
[19] P. Kang, Z. Wei, J. Xu. Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Applied Mathematics and Computation 206 (2008), 245 - 256. doi:10.1016/j.amc.2008.09.010
[20] E. R. Kaufmann, E. Mboumi. Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Diff. Equations 2008(3) (2008), 1 - 11.
[21] M. A. Krasnosel’skii. Positive Solutions of Operator Equations. Groningen. Noordhoff (1964).
[22] H. Li, F. Sun. Existence of solutions for integral boundary value problems of second order ordinary differential equations. Boundary Value Problems 2012(147) (2012), 7pages. doi:10.1186/1687-2770-2012-147
[23] L. Liu, X. Hao, Y. Wu. Positive solutions for singular second-order differential equations with integral boundary conditions. Math. Comput. Modelling 57 (2013), 836 - 847. doi:10.1016/j.mcm.2012.09.012
[24] J. Mao, Z. Zhao, N. Xu. On existence and uniqueness of positive solutions for integral boundary value problems. Electron. J. Qual. Theory Diff. Equations 2010(16) (2010), 1 - 8.
[25] K. S. Miller, B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons. New York (1993).
[26] D. R. Smart. Fixed Point Theorems. Cambridge University Press. Cambridge (1980).
[27] Y. Sun, M. Zhao. Positive solutions for a class of fractional differential equations with integral boundary conditions. Applied Mathematics Letters 34 (2014), 17 - 21. doi:10.1016/j.aml.2014.03.008
[28] X. Wang, X. Liu, X. Deng. Existence and nonexistence of positive solutions for fractional integral boundary value problem with two disturbance parameters. Boundary value problems 2015(186) (2015), 13pages. doi:10.1186/s13661-015-0450-1
[29] W. Yang. Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Computers and Mathematics with Appl. 63 (2012), 288 - 297. doi:10.1016/j.camwa.2011.11.021
[30] K. Zhao, P. Gong. Positive Solutions of m-point multi-term fractional integral BVP involving time-delay for fractional differential equation. Boundary value problems 2015(19) (2015), 19pages. doi:10.1186/s13661-014-0280-6
[31] Y. Zhao, X. Zhang. Triple positive solutions of fourth-order impulsive differential equations with integral boundary conditions. Boundary value problems 2015(2) (2015), 14pages. doi:10.1186/s13661-014-0263-7
[32] 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. doi:10.1007/s12190-012-0590-8
[33] L. Zhang, Z. Xuan. Multiple positive solutions for a second-order boundary value problem with integral boundary conditions. Boundary Value Problems 60 (2016), 8pages. doi:10.1186/s13661-016-0568-9
[34] S. Zhang. Positive solutions for boundary value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006(36) (2006), 1 - 12.
[35] X. Zhang, J. Sun. On multiple sign-changing solutions for some second-order integral boundary value problems. Elecron. J. Qual. Theory Diff. Equations 2010(44) (2010), 1 -15.
[36] X. Zhang, L. Wang, Q. Sun. Existence of positive solutions for a class of nonlinear fractional differential equation with integral boundary conditions and a parameter. Applied Mathematics and Computation 226 (2014), 708 - 718.
[37] X. Zhang, M. Feng, W. Ge. Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Analysis 69 (2008), 3310 - 3321.
[38] H. Zhang. Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions. Boundary Value Problems 2014(61) (2014), 13pages. doi:10.1186/1687-2770-2014-61
[2] B. Ahmad, J. J. Nieto, A. Alsaed. Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Mathematica Scientia 31 (2011), 2122 - 2130.
[3] M. B. Akorede, P. O. Arawomo. Existence of positive solutions for a coupled system of nonlinear boundary value problems of fractional order with integral boundary conditions. Int. J. Applied Math. 30(4) (2017), 273 - 288. doi:10.12732/ijam.v30i4.1
[4] M. B. Akorede, P. O. Arawomo. Positive solutions to a nonlinear eigenvalue problems of fractional differential equations with integro-differential boundary conditions. Int. J. Pure and Applied Math. 116(2) (2017), 313 - 328. doi:10.12732/ijpam.v116i2.3
[5] A. Anguraj, P. Karthikeyan, J. J. Trujillo. Existence of solutions for fractional mixed integrodifferential equations with nonlocal initial condition. Adv. Difference Equations 2011 (2011), Art.ID 690653, 12pages. doi:10.1155/2011/690653
[6] Z. Bai, H. Lu. Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311 (2005), 495 - 505. doi:10.1016/j.jmaa.2005.02.052
[7] Z. Bai. On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. Theory, Methods and Appl. 72(2) (2010), 916 - 924. doi:10.1016/j.na.2009.07.033
[8] M. Benchohra, A. Cabada, D. Seba. An existence result for fractional differential equations on Banach spaces. Boundary Value Problems 2009 (2009), Article ID628916, 11pages. doi:10.1155/2009/628916
[9] A. Boucherif. Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis 70 (2009), 364 - 371. doi:10.1016/j.na.2007.12.007
[10] A. Cabada, Z. Hamdi. Nonlinear fractional differential equations with integral boundary value conditions. Applied Mathematics and Computation 228 (2014), 251 - 257.
[11] A. Cabada, G. Wang. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389 (2012), 403 - 411. doi:10.1016/j.jmaa.2011.11.065
[12] M. El-Shahed. On the existence of positive solutions for a boundary value problem of fractional order. Thai J. Math. 5(1) (2007), 143 - 150.
[13] M. El-Shahed. Positive solutions for a boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007 (2007), Article ID10368, 8pages. doi:10.1155/2007/10368
[14] M. Feng, X. Zhang, W. Ge. New existence results for higher-order nonlinear fractional differential equations with integral boundary conditions. Boundary Value Problems 2011 (2011), Article ID720702, 20pages. doi:10.1155/2011/720702
[15] M. Feng, X. Liu, H. Feng. The existence of positive solutions for a nonlinear fractional differential equation with integral boundary conditions. Adv. Difference Equations 2011 (2011), Art.ID 546038, 14pages. doi:10.1155/2011/546038
[16] J. Henderson, R. Luca. Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Applied Mathematics and Computation 249 (2014), 182 - 197. doi:10.1016/j.amc.2014.10.028
[17] J. Jin, X. Liu, M. Jia. Existence of positive solutions for singular fractional differential equations with integral boundary conditions. Electron. J. Diff. Equations 2012(63) (2012), 1 - 14.
[18] H. Johnny, L. Rodica, T. Alexandru. Positive solutions for a fractional boundary value problem. Nonlinear Stud. 22(1) (2015), 139 - 151.
[19] P. Kang, Z. Wei, J. Xu. Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Applied Mathematics and Computation 206 (2008), 245 - 256. doi:10.1016/j.amc.2008.09.010
[20] E. R. Kaufmann, E. Mboumi. Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Diff. Equations 2008(3) (2008), 1 - 11.
[21] M. A. Krasnosel’skii. Positive Solutions of Operator Equations. Groningen. Noordhoff (1964).
[22] H. Li, F. Sun. Existence of solutions for integral boundary value problems of second order ordinary differential equations. Boundary Value Problems 2012(147) (2012), 7pages. doi:10.1186/1687-2770-2012-147
[23] L. Liu, X. Hao, Y. Wu. Positive solutions for singular second-order differential equations with integral boundary conditions. Math. Comput. Modelling 57 (2013), 836 - 847. doi:10.1016/j.mcm.2012.09.012
[24] J. Mao, Z. Zhao, N. Xu. On existence and uniqueness of positive solutions for integral boundary value problems. Electron. J. Qual. Theory Diff. Equations 2010(16) (2010), 1 - 8.
[25] K. S. Miller, B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons. New York (1993).
[26] D. R. Smart. Fixed Point Theorems. Cambridge University Press. Cambridge (1980).
[27] Y. Sun, M. Zhao. Positive solutions for a class of fractional differential equations with integral boundary conditions. Applied Mathematics Letters 34 (2014), 17 - 21. doi:10.1016/j.aml.2014.03.008
[28] X. Wang, X. Liu, X. Deng. Existence and nonexistence of positive solutions for fractional integral boundary value problem with two disturbance parameters. Boundary value problems 2015(186) (2015), 13pages. doi:10.1186/s13661-015-0450-1
[29] W. Yang. Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Computers and Mathematics with Appl. 63 (2012), 288 - 297. doi:10.1016/j.camwa.2011.11.021
[30] K. Zhao, P. Gong. Positive Solutions of m-point multi-term fractional integral BVP involving time-delay for fractional differential equation. Boundary value problems 2015(19) (2015), 19pages. doi:10.1186/s13661-014-0280-6
[31] Y. Zhao, X. Zhang. Triple positive solutions of fourth-order impulsive differential equations with integral boundary conditions. Boundary value problems 2015(2) (2015), 14pages. doi:10.1186/s13661-014-0263-7
[32] 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. doi:10.1007/s12190-012-0590-8
[33] L. Zhang, Z. Xuan. Multiple positive solutions for a second-order boundary value problem with integral boundary conditions. Boundary Value Problems 60 (2016), 8pages. doi:10.1186/s13661-016-0568-9
[34] S. Zhang. Positive solutions for boundary value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006(36) (2006), 1 - 12.
[35] X. Zhang, J. Sun. On multiple sign-changing solutions for some second-order integral boundary value problems. Elecron. J. Qual. Theory Diff. Equations 2010(44) (2010), 1 -15.
[36] X. Zhang, L. Wang, Q. Sun. Existence of positive solutions for a class of nonlinear fractional differential equation with integral boundary conditions and a parameter. Applied Mathematics and Computation 226 (2014), 708 - 718.
[37] X. Zhang, M. Feng, W. Ge. Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Analysis 69 (2008), 3310 - 3321.
[38] H. Zhang. Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions. Boundary Value Problems 2014(61) (2014), 13pages. doi:10.1186/1687-2770-2014-61
Published
2023-04-29
How to Cite
Akorede, M. B., & Arawomo, P. O. (2023). POSITIVE SOLUTIONS FOR A NONLINEAR FRACTIONAL BOUNDARY VALUE PROBLEM. Unilag Journal of Mathematics and Applications, 2(1), 74-89. Retrieved from http://lagjma.unilag.edu.ng/article/view/1722
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Copyright (c) 2022 MOSES B. AKOREDE, PETER O. ARAWOMO
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