doi: 10.3934/dcdss.2020342

Higher order convergence for a class of set differential equations with initial conditions

College of Mathematics and Information Science, Hebei University, Baoding, Hebei 071002, China

* Corresponding author: Peiguang Wang

Received  July 2019 Revised  August 2019 Published  April 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (grant number 11771115, 11271106)

In this paper, we obtain some rapid convergence results for a class of set differential equations with initial conditions. By introducing the partial derivative of set valued function and the $ m $-hyperconvex/hyperconcave functions ($ m\ge 1 $), and using the comparison principle and quasilinearization, we derive two monotone iterative sequences of approximate solutions of such equations that involve the sum of two functions, and these approximate solutions converge uniformly to the unique solution with higher order.

Citation: Peiguang Wang, Xiran Wu, Huina Liu. Higher order convergence for a class of set differential equations with initial conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020342
References:
[1]

U. Abbas and V. Lupulescu, Set functional differential equations, Commun. Appl. Nonlinear Anal., 18 (2011), 97-110.   Google Scholar

[2]

B. Ahmad, Stability of impulsive hybrid set valued differential equations with delay by perturbing Lyapunov functions, J. Appl. Anal., 14 (2008), 209-218.  doi: 10.1515/JAA.2008.209.  Google Scholar

[3]

T. G. Bhaskar and V. Lakshmikantham, Set differential equations and flow invariance, Applicable Analysis, 82 (2003), 357-368.  doi: 10.1080/0003681031000101529.  Google Scholar

[4]

T. G. Bhaskar and V. Lakshmikantham, Lyapunov stability for set differential equations, Dynamic Systems and Applications, 13 (2004), 1-10.   Google Scholar

[5]

T. G. Bhaskar and J. V. Devi, Nonuniform stability and boundedness criteria for set differential equations, Applicable Analysis, 84 (2005), 131-142.  doi: 10.1080/00036810410001724346.  Google Scholar

[6]

T. G. BhaskarV. Lakshmikantham and J. V. Devi, Nonlinear variation of parameters formula for set differential equations in a metric space, Nonlinear Analysis, 63 (2005), 735-744.  doi: 10.1016/j.na.2005.02.036.  Google Scholar

[7]

J. V. Devi, Basic results in impulsive set differential equations, Nonlinear Studies, 10 (2003), 259-272.   Google Scholar

[8]

J. V. Devi, Extremal solutions and continuous dependences for set differential equations involving causal operators with memory, Communications in Applied Analysis, 15 (2011), 113-124.   Google Scholar

[9]

J. V. Devi and A. S. Vatsala, Monotone iterative technique for impulsive set differential equations, Nonlinear Studies, 11 (2004), 639-658.   Google Scholar

[10]

J. V. Devi, Generalized monotone iterative technique for set differential equations involving causal operators with memory, Int. J. Adv. Eng. Sci. Appl. Math., 3 (2011), 74-83.  doi: 10.1007/s12572-011-0031-1.  Google Scholar

[11]

J. V. Devi and A. S. Vatsala, A study of set differential equations with delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 11 (2004), 287-300.   Google Scholar

[12]

J. V. Devi, Comparison theorems and existence results for set causal operators with memory, Nonlinear Studies, 18 (2011), 603-610.   Google Scholar

[13]

D. B. Dhaigudel and C. A. Naidu, Monotone iterative technique for periodic boundary value problem of set differential equations involving causal operators, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24 (2017), 133-146.  doi: 10.1007/s10732-017-9360-y.  Google Scholar

[14]

G. N. GalanisaT. G. BhaskarV. Lakshmikantham and P. K. Palamides, Set value functions in Fréchet spaces: Continuity, Hukuhara differentiability and applications to set differential equations, Nonlinear Analysis, 61 (2005), 559-575.  doi: 10.1016/j.na.2005.01.004.  Google Scholar

[15]

S. H. Hong, Stability criteria for set dynamic equations on time scales, Comput. Math. Appl., 59 (2010), 3444-3457.  doi: 10.1016/j.camwa.2010.03.033.  Google Scholar

[16]

J. F JiangC. F. Li and H. T. Chen, Existence of solutions for set differential equations involving causal operator with memory in Banach space, J. Appl. Math. Comput., 41 (2013), 183-196.  doi: 10.1007/s12190-012-0604-6.  Google Scholar

[17]

R. N. MohapatraK. Vajravelu and Y. Yin, Extension of the method of quasilinearization and rapid convergence, Journal of Optimization Theory and Applications, 96 (1998), 667-682.  doi: 10.1023/A:1022620813436.  Google Scholar

[18]

G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Advanced Texts and Surveys in Pure and Applied Mathematics, 27. Pitman, Boston, MA, distributed by John Wiley & Sons, Inc., New York, 1985.  Google Scholar

[19]

V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Nonlinear Problems, Mathematics and its Applications, 440. Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-1-4757-2874-3.  Google Scholar

[20]

V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006.  Google Scholar

[21]

A. J. B. Lopes PintoF. S. De Blasi and F. Iervolino, Uniqueness and existence theorems for differential equations with compact convex-valued solutions, Boll. Un. Mat. Ital., 3 (1970), 47-54.   Google Scholar

[22]

V. Lupulescu, Successive approximations to solutions of set differential equations in Banach spaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 391-401.   Google Scholar

[23]

M. T. Malinowski, On set differential equations in Banach spaces - a second type Hukuhara differentiability approach, Appl. Math. Comput., 219 (2012), 289-305.  doi: 10.1016/j.amc.2012.06.019.  Google Scholar

[24]

M. T. Malinowski, Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Appl. Math. Comput., 218 (2012), 9427-9437.  doi: 10.1016/j.amc.2012.03.027.  Google Scholar

[25]

F. A. McRae and J. V. Devi, Impulsive set differential equations with delay, Applicable Analysis, 8 (2005), 329-341.  doi: 10.1080/00036810410001731483.  Google Scholar

[26]

F. A. McRaeJ. V. Devi and Z. Drici, Existence result for periodic boundary value problem of set differential equations using monotone iterative technique, Communications in Applied Analysis, 19 (2015), 245-256.   Google Scholar

[27]

T. G. Melton and A. S. Vatsala, Generalized quasilinearization and higher order of convergence for first order initial value problems, Dynamic Systems and Applications, 15 (2006), 375-393.   Google Scholar

[28]

N. D. PhuL. T. Quang and N. V. Hoa, On the existence of extremal solutions for set differential equations, Journal of Advanced Research in Dynamical and Control Systems, 4 (2012), 18-28.   Google Scholar

[29]

L. T. QuangN. D. PhuN. V. Hoa and H. Vu, On maximal and minimal solutions for set integro-differential equations with feedback control, Nonlinear Studies, 20 (2013), 39-56.   Google Scholar

[30]

N. N. Tu and T. T. Tung, Stability of set differential equations and applications, Applicable Analysis, 71 (2009), 1526-1533.  doi: 10.1016/j.na.2008.12.045.  Google Scholar

[31]

P. G. Wang and W. Gao, Quasilinearization of an initial value problem for a set valued integro-differential equation, Comput. Math. Appl., 61 (2011), 2111-2115.  doi: 10.1016/j.camwa.2010.08.084.  Google Scholar

show all references

References:
[1]

U. Abbas and V. Lupulescu, Set functional differential equations, Commun. Appl. Nonlinear Anal., 18 (2011), 97-110.   Google Scholar

[2]

B. Ahmad, Stability of impulsive hybrid set valued differential equations with delay by perturbing Lyapunov functions, J. Appl. Anal., 14 (2008), 209-218.  doi: 10.1515/JAA.2008.209.  Google Scholar

[3]

T. G. Bhaskar and V. Lakshmikantham, Set differential equations and flow invariance, Applicable Analysis, 82 (2003), 357-368.  doi: 10.1080/0003681031000101529.  Google Scholar

[4]

T. G. Bhaskar and V. Lakshmikantham, Lyapunov stability for set differential equations, Dynamic Systems and Applications, 13 (2004), 1-10.   Google Scholar

[5]

T. G. Bhaskar and J. V. Devi, Nonuniform stability and boundedness criteria for set differential equations, Applicable Analysis, 84 (2005), 131-142.  doi: 10.1080/00036810410001724346.  Google Scholar

[6]

T. G. BhaskarV. Lakshmikantham and J. V. Devi, Nonlinear variation of parameters formula for set differential equations in a metric space, Nonlinear Analysis, 63 (2005), 735-744.  doi: 10.1016/j.na.2005.02.036.  Google Scholar

[7]

J. V. Devi, Basic results in impulsive set differential equations, Nonlinear Studies, 10 (2003), 259-272.   Google Scholar

[8]

J. V. Devi, Extremal solutions and continuous dependences for set differential equations involving causal operators with memory, Communications in Applied Analysis, 15 (2011), 113-124.   Google Scholar

[9]

J. V. Devi and A. S. Vatsala, Monotone iterative technique for impulsive set differential equations, Nonlinear Studies, 11 (2004), 639-658.   Google Scholar

[10]

J. V. Devi, Generalized monotone iterative technique for set differential equations involving causal operators with memory, Int. J. Adv. Eng. Sci. Appl. Math., 3 (2011), 74-83.  doi: 10.1007/s12572-011-0031-1.  Google Scholar

[11]

J. V. Devi and A. S. Vatsala, A study of set differential equations with delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 11 (2004), 287-300.   Google Scholar

[12]

J. V. Devi, Comparison theorems and existence results for set causal operators with memory, Nonlinear Studies, 18 (2011), 603-610.   Google Scholar

[13]

D. B. Dhaigudel and C. A. Naidu, Monotone iterative technique for periodic boundary value problem of set differential equations involving causal operators, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24 (2017), 133-146.  doi: 10.1007/s10732-017-9360-y.  Google Scholar

[14]

G. N. GalanisaT. G. BhaskarV. Lakshmikantham and P. K. Palamides, Set value functions in Fréchet spaces: Continuity, Hukuhara differentiability and applications to set differential equations, Nonlinear Analysis, 61 (2005), 559-575.  doi: 10.1016/j.na.2005.01.004.  Google Scholar

[15]

S. H. Hong, Stability criteria for set dynamic equations on time scales, Comput. Math. Appl., 59 (2010), 3444-3457.  doi: 10.1016/j.camwa.2010.03.033.  Google Scholar

[16]

J. F JiangC. F. Li and H. T. Chen, Existence of solutions for set differential equations involving causal operator with memory in Banach space, J. Appl. Math. Comput., 41 (2013), 183-196.  doi: 10.1007/s12190-012-0604-6.  Google Scholar

[17]

R. N. MohapatraK. Vajravelu and Y. Yin, Extension of the method of quasilinearization and rapid convergence, Journal of Optimization Theory and Applications, 96 (1998), 667-682.  doi: 10.1023/A:1022620813436.  Google Scholar

[18]

G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Advanced Texts and Surveys in Pure and Applied Mathematics, 27. Pitman, Boston, MA, distributed by John Wiley & Sons, Inc., New York, 1985.  Google Scholar

[19]

V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Nonlinear Problems, Mathematics and its Applications, 440. Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-1-4757-2874-3.  Google Scholar

[20]

V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006.  Google Scholar

[21]

A. J. B. Lopes PintoF. S. De Blasi and F. Iervolino, Uniqueness and existence theorems for differential equations with compact convex-valued solutions, Boll. Un. Mat. Ital., 3 (1970), 47-54.   Google Scholar

[22]

V. Lupulescu, Successive approximations to solutions of set differential equations in Banach spaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 391-401.   Google Scholar

[23]

M. T. Malinowski, On set differential equations in Banach spaces - a second type Hukuhara differentiability approach, Appl. Math. Comput., 219 (2012), 289-305.  doi: 10.1016/j.amc.2012.06.019.  Google Scholar

[24]

M. T. Malinowski, Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Appl. Math. Comput., 218 (2012), 9427-9437.  doi: 10.1016/j.amc.2012.03.027.  Google Scholar

[25]

F. A. McRae and J. V. Devi, Impulsive set differential equations with delay, Applicable Analysis, 8 (2005), 329-341.  doi: 10.1080/00036810410001731483.  Google Scholar

[26]

F. A. McRaeJ. V. Devi and Z. Drici, Existence result for periodic boundary value problem of set differential equations using monotone iterative technique, Communications in Applied Analysis, 19 (2015), 245-256.   Google Scholar

[27]

T. G. Melton and A. S. Vatsala, Generalized quasilinearization and higher order of convergence for first order initial value problems, Dynamic Systems and Applications, 15 (2006), 375-393.   Google Scholar

[28]

N. D. PhuL. T. Quang and N. V. Hoa, On the existence of extremal solutions for set differential equations, Journal of Advanced Research in Dynamical and Control Systems, 4 (2012), 18-28.   Google Scholar

[29]

L. T. QuangN. D. PhuN. V. Hoa and H. Vu, On maximal and minimal solutions for set integro-differential equations with feedback control, Nonlinear Studies, 20 (2013), 39-56.   Google Scholar

[30]

N. N. Tu and T. T. Tung, Stability of set differential equations and applications, Applicable Analysis, 71 (2009), 1526-1533.  doi: 10.1016/j.na.2008.12.045.  Google Scholar

[31]

P. G. Wang and W. Gao, Quasilinearization of an initial value problem for a set valued integro-differential equation, Comput. Math. Appl., 61 (2011), 2111-2115.  doi: 10.1016/j.camwa.2010.08.084.  Google Scholar

[1]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[2]

Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020296

[3]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[4]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[5]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[6]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[7]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[8]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[9]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051

[10]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[11]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[12]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[13]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[14]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[15]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[16]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[17]

Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025

[18]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[19]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[20]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

2019 Impact Factor: 1.233

Article outline

[Back to Top]