# American Institute of Mathematical Sciences

September  2021, 14(9): 3233-3248. 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  September 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (9) : 3233-3248. doi: 10.3934/dcdss.2020342
##### References:
 [1] U. Abbas and V. Lupulescu, Set functional differential equations, Commun. Appl. Nonlinear Anal., 18 (2011), 97-110. [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. [3] T. G. Bhaskar and V. Lakshmikantham, Set differential equations and flow invariance, Applicable Analysis, 82 (2003), 357-368.  doi: 10.1080/0003681031000101529. [4] T. G. Bhaskar and V. Lakshmikantham, Lyapunov stability for set differential equations, Dynamic Systems and Applications, 13 (2004), 1-10. [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. [6] T. G. Bhaskar, V. 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. [7] J. V. Devi, Basic results in impulsive set differential equations, Nonlinear Studies, 10 (2003), 259-272. [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. [9] J. V. Devi and A. S. Vatsala, Monotone iterative technique for impulsive set differential equations, Nonlinear Studies, 11 (2004), 639-658. [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. [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. [12] J. V. Devi, Comparison theorems and existence results for set causal operators with memory, Nonlinear Studies, 18 (2011), 603-610. [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. [14] G. N. Galanisa, T. G. Bhaskar, V. 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. [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. [16] J. F Jiang, C. 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. [17] R. N. Mohapatra, K. 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. [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. [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. [20] V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006. [21] A. J. B. Lopes Pinto, F. 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. [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. [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. [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. [25] F. A. McRae and J. V. Devi, Impulsive set differential equations with delay, Applicable Analysis, 8 (2005), 329-341.  doi: 10.1080/00036810410001731483. [26] F. A. McRae, J. 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. [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. [28] N. D. Phu, L. 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. [29] L. T. Quang, N. D. Phu, N. V. Hoa and H. Vu, On maximal and minimal solutions for set integro-differential equations with feedback control, Nonlinear Studies, 20 (2013), 39-56. [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. [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.

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##### References:
 [1] U. Abbas and V. Lupulescu, Set functional differential equations, Commun. Appl. Nonlinear Anal., 18 (2011), 97-110. [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. [3] T. G. Bhaskar and V. Lakshmikantham, Set differential equations and flow invariance, Applicable Analysis, 82 (2003), 357-368.  doi: 10.1080/0003681031000101529. [4] T. G. Bhaskar and V. Lakshmikantham, Lyapunov stability for set differential equations, Dynamic Systems and Applications, 13 (2004), 1-10. [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. [6] T. G. Bhaskar, V. 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. [7] J. V. Devi, Basic results in impulsive set differential equations, Nonlinear Studies, 10 (2003), 259-272. [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. [9] J. V. Devi and A. S. Vatsala, Monotone iterative technique for impulsive set differential equations, Nonlinear Studies, 11 (2004), 639-658. [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. [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. [12] J. V. Devi, Comparison theorems and existence results for set causal operators with memory, Nonlinear Studies, 18 (2011), 603-610. [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. [14] G. N. Galanisa, T. G. Bhaskar, V. 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. [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. [16] J. F Jiang, C. 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. [17] R. N. Mohapatra, K. 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. [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. [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. [20] V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006. [21] A. J. B. Lopes Pinto, F. 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. [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. [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. [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. [25] F. A. McRae and J. V. Devi, Impulsive set differential equations with delay, Applicable Analysis, 8 (2005), 329-341.  doi: 10.1080/00036810410001731483. [26] F. A. McRae, J. 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. [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. [28] N. D. Phu, L. 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. [29] L. T. Quang, N. D. Phu, N. V. Hoa and H. Vu, On maximal and minimal solutions for set integro-differential equations with feedback control, Nonlinear Studies, 20 (2013), 39-56. [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. [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.
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