Wellposedness of impulsive functional abstract second-order differential equations with state-dependent delay. (English) Zbl 07715034
Summary: This study investigates the functional abstract second order impulsive differential equation with state-dependent delay. The major result of this study is that the abstract second-order impulsive differential equation with state-dependent delay system has at least one solution and is unique. After that, the wellposed condition is defined. Following that, we look at whether the proposed problem is wellposed. Finally, some illustrations of our findings are provided.
References:
| [1] | H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Amsterdam, North-Holland Mathematics Studies, 1985. · Zbl 0564.34063 |
| [2] | M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, New York, Hindawi Publishing Corporation, 2006. · Zbl 1130.34003 |
| [3] | V. Lakshmikantham, D. D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, Teaneck, NJ, World Scientific Publishing Co. Inc., 1989. · Zbl 0719.34002 |
| [4] | D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Harlow, Longman Scientific & Technical, 1993. · Zbl 0815.34001 |
| [5] | J. Diestel and J. J. Uhl, Vector Measures, Providence, American Mathematical Society, 1972. |
| [6] | M. Buga and M. Martin, “The escaping disaster: a problem related to state-dependent delays,” Z. Angew. Math. Phys., vol. 55, pp. 547-574, 2004. · Zbl 1062.34080 |
| [7] | W. G. Aiello, H. I. Freedman, and J. Wu, “Analysis of a model representing stage-structured population growth with state-dependent time delay,” SIAM J. Appl. Math., vol. 52, pp. 855-869, 1992. doi:10.1137/0152048. · Zbl 0760.92018 · doi:10.1137/0152048 |
| [8] | G. Arthi, J. H. Park, and H. Y. Jung, “Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay,” Appl. Math. Comput., vol. 248, pp. 328-341, 2014. doi:10.1016/j.amc.2014.09.084. · Zbl 1338.34153 · doi:10.1016/j.amc.2014.09.084 |
| [9] | I. Chueshov and A. Rezounenko, “Dynamics of second order in time evolution equations with state-dependent delay,” Nonlinear Anal., vol. 123, pp. 126-149, 2015. doi:10.1016/j.na.2015.04.013. · Zbl 1322.35155 · doi:10.1016/j.na.2015.04.013 |
| [10] | S. Das, D. N. Pandey, and N. Sukavanam, “Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay,” Acta Math. Sci. B Engl. Ed., vol. 36, pp. 1509-1523, 2016. doi:10.1016/s0252-9602(16)30086-8. · Zbl 1374.35433 · doi:10.1016/s0252-9602(16)30086-8 |
| [11] | R. D. Driver, “A neutral system with state-dependent delay,” J. Differ. Equ., vol. 54, pp. 73-86, 1984. doi:10.1016/0022-0396(84)90143-8. · Zbl 1428.34119 · doi:10.1016/0022-0396(84)90143-8 |
| [12] | E. Hernández, K. Azevedo, and D. O’Regan, “On second order differential equations with state-dependent delay,” Hist. Anthropol., vol. 97, pp. 2610-2617, 2018. doi:10.1080/00036811.2017.1382685. · Zbl 06959358 · doi:10.1080/00036811.2017.1382685 |
| [13] | Y. K. Chang, M. M. Arjunan, and V. Kavitha, “Existence results for a second order impulsive functional differential equation with state-dependent delay,” Differ. Equ. Appl., vol. 1, pp. 325-339, 2009. doi:10.7153/dea-01-17. · Zbl 1192.34091 · doi:10.7153/dea-01-17 |
| [14] | F. Hartung and J. Turi, “On differentiability of solutions with respect to parameters in state-dependent delay equations,” J. Differ. Equ., vol. 135, pp. 192-237, 1997. doi:10.1006/jdeq.1996.3238. · Zbl 0877.34045 · doi:10.1006/jdeq.1996.3238 |
| [15] | F. Hartung, “Differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays,” J. Math. Anal. Appl., vol. 324, pp. 504-524, 2006. doi:10.1016/j.jmaa.2005.12.025. · Zbl 1115.34077 · doi:10.1016/j.jmaa.2005.12.025 |
| [16] | F. Hartung, T. Krisztin, H. O. Walther, and J. Wu, “Functional differential equations with state-dependent delays: theory and applications,” Handbook Differ. Equ., vol. 3, pp. 435-545, 2006. |
| [17] | H. R. Henríquez and C. H. Vásquez, “Differentiabilty of solutions of the second order abstract Cauchy problem,” Semigroup Forum, vol. 64, pp. 472-488, 2002. doi:10.1007/s002330010092. · Zbl 1032.47026 · doi:10.1007/s002330010092 |
| [18] | E. Hernández, A. Prokopczyk, and L. Ladeira, “A note on partial functional differential equations with state-dependent delay,” Nonlinear Anal. R. World Appl., vol. 7, pp. 510-519, 2006. doi:10.1016/j.nonrwa.2005.03.014. · Zbl 1109.34060 · doi:10.1016/j.nonrwa.2005.03.014 |
| [19] | Y. K. Chang, A. Anguraj, and K. Karthikeyan, “Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators,” Nonlinear Anal., vol. 71, pp. 4377-4386, 2009. doi:10.1016/j.na.2009.02.121. · Zbl 1178.34071 · doi:10.1016/j.na.2009.02.121 |
| [20] | E. Hernández, “A second order impulsive Cauchy problem,” Int. J. Math. Math. Sci., vol. 31, pp. 451-461, 2002. · Zbl 1013.34061 |
| [21] | E. Hernández and H. R. Henríquez, “Impulsive partial neutral differential equations,” Appl. Math. Lett., vol. 19, pp. 215-222, 2006. doi:10.1016/j.aml.2005.04.005. · Zbl 1103.34068 · doi:10.1016/j.aml.2005.04.005 |
| [22] | E. Hernández, M. Pierri, and J. Wu, “C1+α-strict solutions and wellposedness of abstract differential equations with state dependent delay,” J. Differ. Equ., vol. 261, pp. 6856-6882, 2016. doi:10.1016/j.jde.2016.09.008. · Zbl 1353.34093 · doi:10.1016/j.jde.2016.09.008 |
| [23] | E. Hernández, “Existence of solutions for a second order abstract functional differential equation with state-dependent delay,” Electron. J. Differ. Equ., vol. 2007, pp. 1-10, 2007. · Zbl 1113.47061 |
| [24] | J. Kisyński, “On cosine operator functions and one parameter group of operators,” Stud. Math., vol. 49, pp. 93-105, 1972. · Zbl 0216.42104 |
| [25] | N. Kosovalic, Y. Chen, and J. Wu, “Algebraic-delay differential systems:C0-extendable submanifolds and linearization,” Trans. Am. Math. Soc., vol. 369, pp. 3387-3419, 2017. · Zbl 1362.34098 |
| [26] | N. Kosovalic, F. M. G. Magpantay, Y. Chen, and J. Wu, “Abstract algebraic-delay differential systems and age structured population dynamics,” J. Differ. Equ., vol. 255, pp. 593-609, 2013. · Zbl 1291.34124 |
| [27] | T. Krisztin and A. Rezounenko, “Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold,” J. Differ. Equ., vol. 260, pp. 4454-4472, 2016. doi:10.1016/j.jde.2015.11.018. · Zbl 1334.35374 · doi:10.1016/j.jde.2015.11.018 |
| [28] | Y. Lv, R. Yuan, and Y. Pei, “Smoothness of semiflows for parabolic partial differential equations with state-dependent delay,” J. Differ. Equ., vol. 260, pp. 6201-6231, 2016. doi:10.1016/j.jde.2015.12.037. · Zbl 1334.35376 · doi:10.1016/j.jde.2015.12.037 |
| [29] | J. H. Liu, “Nonlinear impulsive evolution equations,” Dyn. Contin. Discrete Impuls. Sys., vol. 6, pp. 77-85, 1999. · Zbl 0932.34067 |
| [30] | B. Radhakrishnan and K. Balachandran, “Controllability of neutral evolution integrodifferential systems with state dependent delay,” J. Optim. Theor. Appl., vol. 153, pp. 85-97, 2012. doi:10.1007/s10957-011-9934-z. · Zbl 1237.93029 · doi:10.1007/s10957-011-9934-z |
| [31] | A. V. Rezounenko, “A condition on delay for differential equations with discrete state-dependent delay,” J. Math. Anal. Appl., vol. 385, pp. 506-516, 2012. doi:10.1016/j.jmaa.2011.06.070. · Zbl 1242.34136 · doi:10.1016/j.jmaa.2011.06.070 |
| [32] | A. V. Rezounenko, “Partial differential equations with discrete and distributed state-dependent delays,” J. Math. Anal. Appl., vol. 326, pp. 1031-1045, 2007. doi:10.1016/j.jmaa.2006.03.049. · Zbl 1178.35370 · doi:10.1016/j.jmaa.2006.03.049 |
| [33] | A. V. Rezounenko and J. Wu, “A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors,” J. Comput. Appl. Math., vol. 190, pp. 99-113, 2006. doi:10.1016/j.cam.2005.01.047. · Zbl 1082.92039 · doi:10.1016/j.cam.2005.01.047 |
| [34] | Y. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dyn. Contin. Discrete Impuls. Sys., vol. 3, pp. 57-88, 1997. · Zbl 0879.34014 |
| [35] | Y. Rogovchenko, “Nonlinear impulse evolution systems and applications to population models,” J. Math. Anal. Appl., vol. 207, pp. 300-315, 1997. doi:10.1006/jmaa.1997.5245. · Zbl 0876.34011 · doi:10.1006/jmaa.1997.5245 |
| [36] | R. Sakthivel, E. R. Anandhi, and N. I. Mahmudov, “Approximate controllability of second-order systems with state-dependent delay,” Numer. Funct. Anal. Optim., vol. 29, pp. 1347-1362, 2008. doi:10.1080/01630560802580901. · Zbl 1153.93006 · doi:10.1080/01630560802580901 |
| [37] | C. C. Travis and G. F. Webb, “Compactness, regularity and uniform continuity properties of strongly continuous cosine families,” Houst. J. Math., vol. 4, pp. 555-567, 1977. · Zbl 0386.47024 |
| [38] | C. C. Travis and G. F. Webb, “Cosine families and abstract nonlinear second order differential equations,” Acta Math. Acad. Sci. Hungar., vol. 32, pp. 76-96, 1978. doi:10.1007/bf01902205. · Zbl 0388.34039 · doi:10.1007/bf01902205 |
| [39] | V. V. Vasil’ev and S. I. Piskarev, “Differential equations in Banach spaces II. Theory of cosine operator functions,” J. Math. Sci., vol. 122, pp. 3055-3174, 2004. · Zbl 1115.34058 |
| [40] | J. G. Si and X. P. Wang, “Analytic solutions of a second-order functional-differential equation with a state derivative dependent delay,” Colloq. Math., vol. 79, pp. 273-281, 1999. doi:10.4064/cm-79-2-273-281. · Zbl 0927.34077 · doi:10.4064/cm-79-2-273-281 |
| [41] | V. Vijayakumar and H. R. Henríquez, “Existence of global solutions for a class of abstract second-order nonlocal cauchy problem with impulsive conditions in Banach spaces,” Numer. Funct. Anal. Optim., vol. 39, pp. 704-736, 2018. doi:10.1080/01630563.2017.1414060. · Zbl 1392.34071 · doi:10.1080/01630563.2017.1414060 |
| [42] | V. Vijayakumar, S. K. Panda, K. S. Nisar, and H. M. Baskonus, “Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay,” Numer. Methods Part. Differ. Equ., vol. 37, no. 2, pp. 1200-1221, 2021. doi:10.1002/num.22573. · Zbl 07776009 · doi:10.1002/num.22573 |
| [43] | V. Vijayakumar, R. Udhayakumar, and C. Dineshkumar, “Approximate controllability of second order nonlocal neutral differential evolution inclusions,” IMA J. Mat. Con. Inf., vol. 38, no. 1, pp. 192-210, 2021. doi:10.1093/imamci/dnaa001. · Zbl 1475.93017 · doi:10.1093/imamci/dnaa001 |
| [44] | N. I. Mahmudov, R. Udhayakumar, and V. Vijayakumar, “On the approximate controllability of second-order evolution hemivariational inequalities,” Results Math., vol. 75, 2020. doi:10.1007/s00025-020-01293-2. · Zbl 1453.93023 · doi:10.1007/s00025-020-01293-2 |
| [45] | V. Vijayakumar and R. Murugesu, “Controllability for a class of second-order evolution differential inclusions without compactness,” Hist. Anthropol., vol. 98, pp. 1367-1385, 2019. doi:10.1080/00036811.2017.1422727. · Zbl 1414.93038 · doi:10.1080/00036811.2017.1422727 |
| [46] | W. Kavitha, V. Vijayakumar, R. Udhayakumar, and K. S. Nisar, “A new study on existence and uniqueness of nonlocal fractional delay differential systems of order 1 < r < 2 in Banach spaces,” Numer. Methods Part. Differ. Equ., vol. 37, pp. 1-13, 2020. doi:10.1002/num.22560. · Zbl 07775996 · doi:10.1002/num.22560 |
| [47] | V. Vijayakumar, R. Udhayakumar, Y. Zhou, and N. Sakthivel, “Approximate controllability results for soblolev-type delay differential system of fractional order without uniqueness,” Numer. Methods Part. Differ. Equ., pp. 1-20, 2020. doi:10.1002/num.22642. · Zbl 1535.35189 · doi:10.1002/num.22642 |
| [48] | M. Mohan Raja, V. Vijayakumar, and R. Udhayakumar, “Results on the existence and controllability of fractional integro-differential system of order 1 < r < 2 via measure of noncompactness,” Chaos, Solit. Fractals, vol. 139, p. 110299, 2020. · Zbl 1490.34093 |
| [49] | M. Mohan Raja, V. Vijayakumar, R. Udhayakumar, and Y. Zhou, “A new approach on the approximate controllability of fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces,” Chaos, Solit. Fractals, vol. 141, p. 110310, 2020. doi:10.1016/j.chaos.2020.110310. · Zbl 1496.34021 · doi:10.1016/j.chaos.2020.110310 |
| [50] | M. Mohan Raja, V. Vijayakumar, and R. Udhayakumar, “A new exploration on existence of Sobolev-type Hilfer fractional neutral integro-differential equations with infinite delay,” Numer. Methods Part. Differ. Equ., vol. 37, pp. 750-766, 2020. · Zbl 1534.35410 |
| [51] | C. Dineshkumar and R. Udhayakumar, “New results concerning to approximate controllability of Hilfer fractional neutral stochastic delay integro-differential systems,” Numer. Methods Part. Differ. Equ., vol. 37, no. 2, pp. 1072-1090, 2021. doi:10.1002/num.22567. · Zbl 07776003 · doi:10.1002/num.22567 |
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