On second order differential equations with state-dependent delay. (English) Zbl 06959358
Summary: We study existence and uniqueness of solutions for a general class of second order abstract differential equations with state-dependent delay. Some examples related to partial differential equations with state dependent delay are presented.
MSC:
| 47D09 | Operator sine and cosine functions and higher-order Cauchy problems |
| 34K30 | Functional-differential equations in abstract spaces |
| 34Kxx | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 34Gxx | Differential equations in abstract spaces |
References:
| [1] | Aiello, WG; Freedman, HI; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J Appl Math, 52, 3, 855-869, (1992) · Zbl 0760.92018 |
| [2] | Hartung, F.; Krisztin, T.; Walther, H-O, Handbook of differential equations: ordinary differential equations, III, Functional differential equations with state-dependent delays: theory and applications, 435-545, (2006), Elsevier/North-Holland, Amsterdam · Zbl 1180.35004 |
| [3] | Driver, RD; LaSalle, J.; Lefschtz, S., International symposium on nonlinear differential equations and nonlinear mechanics, A functional-differential system of neutral type arising in a two-body problem of classical electrodynamics, 474-484, (1963), Academic Press, New York (NY) · Zbl 0129.00102 |
| [4] | Driver, RD, A neutral system with state-dependent delay, J Differ Equ, 54, 73-86, (1984) · Zbl 1428.34119 |
| [5] | Hartung, F., On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, Ann Mat Pura Appl (4), 192, 1, 17-47, (2013) · Zbl 1268.34160 |
| [6] | Hartung, F.; Turi, J., On differentiability of solutions with respect to parameters in state-dependent delay equations, J Differ Equ, 135, 2, 192-237, (1997) · Zbl 0877.34045 |
| [7] | Hartung, F., Differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, J Math Anal Appl, 324, 1, 504-524, (2006) · Zbl 1115.34077 |
| [8] | Walther, H-O, The solution manifold and C1-smoothness for differential equations with state-dependent delay, J Differ Equ, 195, 1, 46-65, (2003) · Zbl 1045.34048 |
| [9] | Henríquez, E.; Prokopczyk, A.; Ladeira, L., A note on partial functional differential equations with state-dependent delay, Nonlinear Anal Real World Appl, 7, 4, 510-519, (2006) · Zbl 1109.34060 |
| [10] | Rezounenko, AV, A condition on delay for differential equations with discrete state-dependent delay, J Math Anal Appl, 385, 1, 506-516, (2012) · Zbl 1242.34136 |
| [11] | Rezounenko, AV, Partial differential equations with discrete and distributed state-dependent delays, J Math Anal Appl, 326, 2, 1031-1045, (2007) · Zbl 1178.35370 |
| [12] | Rezounenko, AV; Wu, J., A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, J Comput Appl Math, 190, 1-2, 99-113, (2006) · Zbl 1082.92039 |
| [13] | Kosovalic, N.; Magpantay, FMG; Chen, Y., Abstract algebraic-delay differential systems and age structured population dynamics, J Differ Equ, 255, 3, 593-609, (2013) · Zbl 1291.34124 |
| [14] | Krisztin, T.; Rezounenko, A., Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold, J Differ Equ, 260, 5, 4454-4472, (2016) · Zbl 1334.35374 |
| [15] | Lv, Y.; Rong, Y.; Yongzhen, P., Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J Differ Equ, 260, 6201-6231, (2016) · Zbl 1334.35376 |
| [16] | Kosovalic, N.; Chen, Y.; Wu, J., Algebraic-delay differential systems: C0-extendable submanifolds and linearization, Trans Amer Math Soc, 369, 5, 3387-3419, (2017) · Zbl 1362.34098 |
| [17] | Hernández, E.; Pierri, M.; Wu, J., C1+α -strict solutions and wellposedness of abstract differential equations with state dependent delay, J Differ Equ, 261, 12, 6856-6882, (2016) · Zbl 1353.34093 |
| [18] | Arthi, G.; Park, JH; Jung, HY, Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay, Appl Math Comput, 248, 328-341, (2014) · Zbl 1338.34153 |
| [19] | Chueshov, I.; Rezounenko, A., Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Anal Theory Methods Appl, 123, 126-149, (2015) · Zbl 1322.35155 |
| [20] | Das, S.; Pandey, DN; Sukavanam, N., Approximate controllability of a second-order neutral stochastic differential equation with state-dependent delay, Nonlinear Anal Model Control, 21, 6, 751-769, (2016) · Zbl 1417.93068 |
| [21] | Das, S.; Pandey, D.; Sukavanam, N., Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay, Acta Math Sci Ser B Engl Ed, 36, 5, 1509-1523, (2016) · Zbl 1374.35433 |
| [22] | Das, S.; Pandey, DN; Sukavanam, N., Approximate controllability of a second order neutral differential equation with state dependent delay, Differ Equ Dyn Syst, 24, 2, 201-214, (2016) · Zbl 1342.34103 |
| [23] | Hernández, E., Existence of solutions for a second order abstract functional differential equation with state-dependent delay, Electron J Differ Equ, 21, (2007) · Zbl 1113.47061 |
| [24] | Radhakrishnan, B.; Balachandran, K., Controllability of neutral evolution integrodifferential systems with state dependent delay, J Optim Theory Appl, 153, 1, 85-97, (2012) · Zbl 1237.93029 |
| [25] | Sakthivel, R.; Anandhi, ER; Mahmudov, NI, Approximate controllability of second-order systems with state-dependent delay, Numer Funct Anal Optim, 29, 11-12, 1347-1362, (2008) · Zbl 1153.93006 |
| [26] | Kisyński, J., On cosine operator functions and one parameter group of operators, Studia Math, 49, (1972) · Zbl 0232.47045 |
| [27] | Fattorini, HO, Second order linear differential equations in Banach spaces, 108, (1985), North-Holland, Amsterdam · Zbl 0564.34063 |
| [28] | Vasil’ev, VV; Piskarev, SI, Differential equations in Banach spaces. II. theory of cosine operator functions. functional analysis, J Math Sci (N Y), 122, 2, 3055-3174, (2004) · Zbl 1115.34058 |
| [29] | Travis, CC; Webb, GF, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J Math, 3, 4, 555-567, (1977) · Zbl 0386.47024 |
| [30] | Travis, CC; Webb, GF, Cosine families and abstract nonlinear second order differential equations, Acta Math Acad Sci Hungaricae, 32, 76-96, (1978) · Zbl 0388.34039 |
| [31] | Diestel, J.; Uhl, JJ, Vector measures, (1972), American Mathematical Society, Providence (RI) |
| [32] | Henríquez, H.; Vásquez, CH, Differentiabilty of solutions of the second order abstract Cauchy problem, Semigroup Forum, 64, 3, 472-488, (2002) · Zbl 1032.47026 |
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