Convergence analysis of iterative scheme and error estimation of positive solution for a fractional differential equation
Abstract
In this paper, we focus on the iterative scheme and error estimation of positive solutions for a class of p-Laplacian fractional order differential equation subject to Riemann-Stieltjes integral boundary condition. Under a weaker growth condition of nonlinearity, by using a monotone iterative technique, we first establish a new result on the sufficient condition for the existence of a unique positive solution to the above problem, then construct an iterative scheme which converges to the unique positive solution, and then present an error estimation and the exact convergence rate of the approximate solution.
Keyword : uniqueness, fractional p-Laplacian equation, monotone iterative technique, error estimation
How to Cite
Wu, J., Zhang, X., Liu, L., Wu, Y., & Cui, Y. (2018). Convergence analysis of iterative scheme and error estimation of positive solution for a fractional differential equation. Mathematical Modelling and Analysis, 23(4), 611-626. https://doi.org/10.3846/mma.2018.037
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Liu and A. Qian. Ground state solution for a Schrodinger–Poisson equation with critical growth.Nonlinear Analysis: Real World Applications, 40:428–443, 2018. https://doi.org/10.1016/j.nonrwa.2017.09.008
A. Mao and H. Chang. Kirchhoff type problems in rn with radial potentialsand locally lipschitz functional. Applied Mathematics Letters, 62:49–54, 2016. https://doi.org/10.1016/j.aml.2016.06.014
A. Mao and W. Wang. Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in R3. Journal of Mathematical Analysis and Applications, 459(1):556–563, 2018.https://doi.org/10.1016/j.jmaa.2017.10.020
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T. Ren, S. Li, X. Zhang and L. Liu. Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes. Boundary Value Problems, 2017(1):118, 2017.https://doi.org/10.1186/s13661-017-0849-y
M. Shao and A. Mao. Multiplicity of solutions to Schrodinger–Poisson system with concave–convex nonlinearities. Applied Mathematics Letters, 83:212–218, 2018. https://doi.org/10.1016/j.aml.2018.04.005
Y. Wang, L. Liu and Y. Wu. Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Analysis: Theory, Methods & Applications, 74(17):6434–6441, 2011. https://doi.org/10.1016/j.na.2011.06.026
Y. Wang, L. Liu, X. Zhang and Y. Wu. Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection. Applied Mathematics and Computation, 258:312–324, 2015. https://doi.org/10.1016/j.amc.2015.01.080
J.R.L. Webb and M. Zima. Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems. Nonlinear Analysis: Theory, Methods & Applications, 71(3-4):1369–1378, 2009. https://doi.org/10.1016/j.na.2008.12.010
J. Wu, X. Zhang, L. Liu and Y. Wu. Positive solution of singular fractional differential system with nonlocal boundary conditions. Advances in Difference Equations, 2014(1):323, 2014.https://doi.org/10.1186/1687-1847-2014-323
J. Wu, X. Zhang, L. Liu and Y. Wu. Twin iterative solutions for a fractional differential turbulent flow model. Boundary Value Problems, 2016(1):98, 2016. https://doi.org/10.1186/s13661-016-0604-9
J. Wu, X. Zhang, L. Liu, Y. Wu and Y. Cui. The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Boundary Value Problems, 2018(82), 2018. https://doi.org/10.1186/s13661-018-1003-1
J. Wu, X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee. Iterative algorithm and estimation of solution for a fractional order differential equation. Boundary Value Problems, 2016(1):116, 2016.https://doi.org/10.1186/s13661-016-0608-5
X. Zhang and L. Liu. A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions. Applied Mathematics and Computation, 215(10):3501–3508, 2010.https://doi.org/10.1016/j.amc.2009.10.044
X. Zhang, L. Liu, B. Wiwatanapataphee and Y. Wu. The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Applied Mathematics and Computation, 235:412–422, 2014.https://doi.org/10.1016/j.amc.2014.02.062
X. Zhang, L. Liu and Y. Wu. The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives. Applied Mathematics and Computation, 218:8526–8536, 2012.https://doi.org/10.1016/j.amc.2012.02.014
X. Zhang, L. Liu and Y. Wu. Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives.Applied Mathematics and Computation, 219:1420–1433, 2012. https://doi.org/10.1016/j.amc.2012.07.046
X. Zhang, L. Liu and Y. Wu. Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Mathematical and Computer Modelling, 55(3-4):1263–1274, 2012.https://doi.org/10.1016/j.mcm.2011.10.006
X. Zhang, L. Liu and Y. Wu. The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium.Applied Mathematics Letters, 37:26–33, 2014. https://doi.org/10.1016/j.aml.2014.05.002
X. Zhang, L. Liu and Y. Wu. Variational structure and multiple solutions for a fractional advection–dispersion equation. Computers & Mathematics with Applications, 68(12, Part A):1794–1805, 2014. https://doi.org/10.1016/j.camwa.2014.10.011
X. Zhang, L. Liu and Y. Wu. The entire large solutions for a quasilinear Schrodinger elliptic equation by the dual approach. Applied Mathematics Letters, 55:1–9, 2016.https://doi.org/10.1016/j.aml.2015.11.005
X. Zhang, L. Liu, Y. Wu and L. Caccetta. Entire large solutions for a class of Schrodinger systems with a nonlinear random operator. Journal of Mathematical Analysis and Applications, 423(2):1650–1659, 2015. https://doi.org/10.1016/j.jmaa.2014.10.068
X. Zhang, L. Liu, Y. Wu and Y. Cui. Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a non-square diffusion term. Applied Mathematics Letters,74:85–93, 2017. https://doi.org/10.1016/j.aml.2017.05.010
X. Zhang, L. Liu, Y. Wu and Y. Cui. New result on the critical exponent for solution of an ordinary fractional differential problem. Journal of Function Spaces, 1:3976469, 2017. https://doi.org/10.1155/2017/3976469
X. Zhang, L. Liu, Y. Wu and Y. Cui. The existence and nonexistence of entire large solutions for a quasilinear Schrodinger elliptic system by dual approach. Journal of Mathematical Analysis and Applications, 464(2):1089–1106, 2018.https://doi.org/10.1016/j.jmaa.2018.04.040
X. Zhang, L. Liu, Y. Wu and Y. Cui. Existence of infinitely solutions for a modified nonlinear schrodinger equation via dual approach.Electronic Journal of Differential Equations, 2018(147):1–15, 2018.
X. Zhang, L. Liu, Y. Wu and Y. Lu. The iterative solutions of nonlinear fractional differential equations. Applied Mathematics and Computation, 219(9):4680–4691, 2013. https://doi.org/10.1016/j.amc.2012.10.082
X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee. The spectral analysis for a singular fractional differential equation with a signed measure. Applied Mathematics and Computation, 257:252–263, 2015. https://doi.org/10.1016/j.amc.2014.12.068
X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee. Nontrivial solutions for afractional advection dispersion equation in anomalous diffusion. Applied Mathematics Letters,66:1–8, 2017. https://doi.org/10.1016/j.aml.2016.10.015
X. Zhang, C. Mao, L. Liu and Y. Wu. Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qualitative Theory of Dynamical Systems,16(1):205–222, 2017. https://doi.org/10.1007/s12346-015-0162-z
X. Zhang, Y. Wu and Y. Cui. Existence and nonexistence of blow-up solutions for a Schrodinger equation involving a nonlinear operator. Applied Mathematics Letters, 82:85–91, 2018. https://doi.org/10.1016/j.aml.2018.02.019
Z. Zhao. Existence and uniqueness of fixed points for some mixed monotone operators. Nonlinear Analysis: Theory, Methods & Applications, 73(6):1481–1490, 2010. https://doi.org/10.1016/j.na.2010.04.008
B. Zhu, L. Liu and Y. Wu. Local and global existence of mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay. Applied Mathematics Letters, 61:73–79, 2016.https://doi.org/10.1016/j.aml.2016.05.010
B. Zhu, L. Liu and Y. Wu. Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations. Fractional Calculus and Applied Analysis, 20(6):1338–1355, 2017.https://doi.org/10.1515/fca-2017-0071
M. Zuo, X. Hao, L. Liu and Y. Cui. Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Boundary Value Problems, 2017(1):161, 2017.https://doi.org/10.1186/s13661-017-0892-8
Y. Cui and Y. Zou. Monotone iterative method for differential systems with coupled integral boundary value problems. Boundary Value Problems, 2013:245, 2013.https://doi.org/10.1186/1687-2770-2013-245
H.A.A. El-Saka. The fractional-order sis epidemic model with variable population size. Journal of the Egyptian Mathematical Society, 22(1):50–54, 2014. ISSN1110-256X. https://doi.org/10.1016/j.joems.2013.06.006
L. Guo, L. Liu and Y. Wu.Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions. Nonlinear Analysis: Modelling and Control, 21(5):635–650, 2016. https://doi.org/10.15388/NA.2016.5.5
X. Hao, H. Wang, L. Liu and Y. Cui.Positive solutions for a systemof nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Boundary Value Problems, 2017(1):182, 2017. https://doi.org/10.1186/s13661-017-0915-5
J. Jiang, L. Liu and Y. Wu. Positive solutions to singular fractional differential system with coupled boundary conditions. Communications in Nonlinear Science and Numerical Simulation, 18(11):3061–3074, 2013. https://doi.org/10.1016/j.cnsns.2013.04.009
J. Liu and A. Qian. Ground state solution for a Schrodinger–Poisson equation with critical growth.Nonlinear Analysis: Real World Applications, 40:428–443, 2018. https://doi.org/10.1016/j.nonrwa.2017.09.008
A. Mao and H. Chang. Kirchhoff type problems in rn with radial potentialsand locally lipschitz functional. Applied Mathematics Letters, 62:49–54, 2016. https://doi.org/10.1016/j.aml.2016.06.014
A. Mao and W. Wang. Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in R3. Journal of Mathematical Analysis and Applications, 459(1):556–563, 2018.https://doi.org/10.1016/j.jmaa.2017.10.020
J. Mao, Z. Zhao and N. Xu. On existence and uniqueness of positive solutions for integral boundary boundary value problems. Electronic Journal of Qualitative Theory of Differential Equations, 16:1–8, 2010.https://doi.org/10.14232/ejqtde.2010.1.16
K. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
T. Ren, S. Li, X. Zhang and L. Liu. Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes. Boundary Value Problems, 2017(1):118, 2017.https://doi.org/10.1186/s13661-017-0849-y
M. Shao and A. Mao. Multiplicity of solutions to Schrodinger–Poisson system with concave–convex nonlinearities. Applied Mathematics Letters, 83:212–218, 2018. https://doi.org/10.1016/j.aml.2018.04.005
Y. Wang, L. Liu and Y. Wu. Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Analysis: Theory, Methods & Applications, 74(17):6434–6441, 2011. https://doi.org/10.1016/j.na.2011.06.026
Y. Wang, L. Liu, X. Zhang and Y. Wu. Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection. Applied Mathematics and Computation, 258:312–324, 2015. https://doi.org/10.1016/j.amc.2015.01.080
J.R.L. Webb and M. Zima. Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems. Nonlinear Analysis: Theory, Methods & Applications, 71(3-4):1369–1378, 2009. https://doi.org/10.1016/j.na.2008.12.010
J. Wu, X. Zhang, L. Liu and Y. Wu. Positive solution of singular fractional differential system with nonlocal boundary conditions. Advances in Difference Equations, 2014(1):323, 2014.https://doi.org/10.1186/1687-1847-2014-323
J. Wu, X. Zhang, L. Liu and Y. Wu. Twin iterative solutions for a fractional differential turbulent flow model. Boundary Value Problems, 2016(1):98, 2016. https://doi.org/10.1186/s13661-016-0604-9
J. Wu, X. Zhang, L. Liu, Y. Wu and Y. Cui. The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Boundary Value Problems, 2018(82), 2018. https://doi.org/10.1186/s13661-018-1003-1
J. Wu, X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee. Iterative algorithm and estimation of solution for a fractional order differential equation. Boundary Value Problems, 2016(1):116, 2016.https://doi.org/10.1186/s13661-016-0608-5
X. Zhang and L. Liu. A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions. Applied Mathematics and Computation, 215(10):3501–3508, 2010.https://doi.org/10.1016/j.amc.2009.10.044
X. Zhang, L. Liu, B. Wiwatanapataphee and Y. Wu. The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Applied Mathematics and Computation, 235:412–422, 2014.https://doi.org/10.1016/j.amc.2014.02.062
X. Zhang, L. Liu and Y. Wu. The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives. Applied Mathematics and Computation, 218:8526–8536, 2012.https://doi.org/10.1016/j.amc.2012.02.014
X. Zhang, L. Liu and Y. Wu. Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives.Applied Mathematics and Computation, 219:1420–1433, 2012. https://doi.org/10.1016/j.amc.2012.07.046
X. Zhang, L. Liu and Y. Wu. Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Mathematical and Computer Modelling, 55(3-4):1263–1274, 2012.https://doi.org/10.1016/j.mcm.2011.10.006
X. Zhang, L. Liu and Y. Wu. The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium.Applied Mathematics Letters, 37:26–33, 2014. https://doi.org/10.1016/j.aml.2014.05.002
X. Zhang, L. Liu and Y. Wu. Variational structure and multiple solutions for a fractional advection–dispersion equation. Computers & Mathematics with Applications, 68(12, Part A):1794–1805, 2014. https://doi.org/10.1016/j.camwa.2014.10.011
X. Zhang, L. Liu and Y. Wu. The entire large solutions for a quasilinear Schrodinger elliptic equation by the dual approach. Applied Mathematics Letters, 55:1–9, 2016.https://doi.org/10.1016/j.aml.2015.11.005
X. Zhang, L. Liu, Y. Wu and L. Caccetta. Entire large solutions for a class of Schrodinger systems with a nonlinear random operator. Journal of Mathematical Analysis and Applications, 423(2):1650–1659, 2015. https://doi.org/10.1016/j.jmaa.2014.10.068
X. Zhang, L. Liu, Y. Wu and Y. Cui. Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a non-square diffusion term. Applied Mathematics Letters,74:85–93, 2017. https://doi.org/10.1016/j.aml.2017.05.010
X. Zhang, L. Liu, Y. Wu and Y. Cui. New result on the critical exponent for solution of an ordinary fractional differential problem. Journal of Function Spaces, 1:3976469, 2017. https://doi.org/10.1155/2017/3976469
X. Zhang, L. Liu, Y. Wu and Y. Cui. The existence and nonexistence of entire large solutions for a quasilinear Schrodinger elliptic system by dual approach. Journal of Mathematical Analysis and Applications, 464(2):1089–1106, 2018.https://doi.org/10.1016/j.jmaa.2018.04.040
X. Zhang, L. Liu, Y. Wu and Y. Cui. Existence of infinitely solutions for a modified nonlinear schrodinger equation via dual approach.Electronic Journal of Differential Equations, 2018(147):1–15, 2018.
X. Zhang, L. Liu, Y. Wu and Y. Lu. The iterative solutions of nonlinear fractional differential equations. Applied Mathematics and Computation, 219(9):4680–4691, 2013. https://doi.org/10.1016/j.amc.2012.10.082
X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee. The spectral analysis for a singular fractional differential equation with a signed measure. Applied Mathematics and Computation, 257:252–263, 2015. https://doi.org/10.1016/j.amc.2014.12.068
X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee. Nontrivial solutions for afractional advection dispersion equation in anomalous diffusion. Applied Mathematics Letters,66:1–8, 2017. https://doi.org/10.1016/j.aml.2016.10.015
X. Zhang, C. Mao, L. Liu and Y. Wu. Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qualitative Theory of Dynamical Systems,16(1):205–222, 2017. https://doi.org/10.1007/s12346-015-0162-z
X. Zhang, Y. Wu and Y. Cui. Existence and nonexistence of blow-up solutions for a Schrodinger equation involving a nonlinear operator. Applied Mathematics Letters, 82:85–91, 2018. https://doi.org/10.1016/j.aml.2018.02.019
Z. Zhao. Existence and uniqueness of fixed points for some mixed monotone operators. Nonlinear Analysis: Theory, Methods & Applications, 73(6):1481–1490, 2010. https://doi.org/10.1016/j.na.2010.04.008
B. Zhu, L. Liu and Y. Wu. Local and global existence of mild solutions for a class of nonlinear fractional reaction–diffusion equations with delay. Applied Mathematics Letters, 61:73–79, 2016.https://doi.org/10.1016/j.aml.2016.05.010
B. Zhu, L. Liu and Y. Wu. Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations. Fractional Calculus and Applied Analysis, 20(6):1338–1355, 2017.https://doi.org/10.1515/fca-2017-0071
M. Zuo, X. Hao, L. Liu and Y. Cui. Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Boundary Value Problems, 2017(1):161, 2017.https://doi.org/10.1186/s13661-017-0892-8