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Generalized linear differential equations in a Banach space: Continuous dependence on a parameter
1. | Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil |
2. | Institute of Mathematics, Academy of Sciences of Czech Republic, Žitná 25, CZ 115 67 Praha 1, Czech Republic |
References:
[1] |
S. Afonso, E. M. Bonotto, M. Federson and Š. Schwabik, Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invarianceprinciple for non-autonomous systems with impulses, J. Differential Equations, 250 (2011), 2969-3001.
doi: 10.1016/j.jde.2011.01.019. |
[2] |
Z. Artstein, Continuous dependence on parameters: On the best possible results, J. Differential Equations, 19 (1975), 214-225.
doi: 10.1016/0022-0396(75)90002-9. |
[3] |
M. Ashordia, On the correctness of linear boundary value problems for systems of generalized ordinarydifferential equations, Proc. Georgian Acad. Sci. Math., 1 (1993), 385-394. |
[4] |
M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications," Birkhäuser, Boston, 2001. |
[5] |
M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser, Boston, 2003. |
[6] |
M. Brokate and P. Krejčí,, Duality in the space of regulated functions and the play operator, Math. Z., 245 (2003), 667-688.
doi: 10.1007/s00209-003-0563-6. |
[7] |
M. Federson and Š. Schwabik, Generalized ordinary differential equations approach to impulsive retarded functionaldifferential equations, Differential and Integral Equations, 19 (2006), 1201-1234. |
[8] |
D. Fraňková, Continuous dependence on a parameter of solutions of generalized differential equations, časopis pěst. mat., 114 (1989), 230-261. |
[9] |
Z. Halas, Continuous dependence of solutions of generalized linear ordinary differential equationson a parameter, Mathematica Bohemica, 132 (2007), 205-218. |
[10] |
Z. Halas, G. Monteiro and M. Tvrdý, Emphatic convergence and sequential solutions of generalized linear differentialequations, Mem.Differential Equations Math. Phys., 54 (2011), 27-49. |
[11] |
Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on aparameter, Funct. Differ. Equ., 16 (2009), 299-313. |
[12] |
T. H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations, Illinois J. Math., 3 (1959), 352-373. |
[13] |
Ch. S. Hönig, "Volterra Stieltjes-integral Equations," North Holland and American Elsevier, Mathematics Studies 16. Amsterdam and New York, 1975. |
[14] |
C. Imaz and Z. Vorel, Generalized ordinary differential equations in Banach spaces and applicationsto functional equations, Bol. Soc. Mat. Mexicana, 11 (1966), 47-59. |
[15] |
I. Kiguradze, Boundary value problems for systems of ordinary differential equations, (in Russian), Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh, 30 (1987), 3-103; English transl.: J. Sov. Math., 43 (1988), 2259-2339. |
[16] |
M. A. Krasnoselskij and S. G. Krein, On the averaging principle in nonlinear mechanics, (in Russian), Uspekhi mat. nauk, 10 (1955), 147-152. |
[17] |
P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. |
[18] |
J. Kurzweil and Z. Vorel, Continuous dependence of solutions of differential equations on a parameter, Czechoslovak Math. J., 7 (1957), 568-583. |
[19] |
J. Kurzweil, Generalized ordinary differential equation and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449. |
[20] |
J. Kurzweil, Generalized ordinary differential equations, Czechoslovak Math. J., 8 (1958), 360-388. |
[21] |
G. Meng and M. Zhang, Continuity in weak topology: First order linear system of ODE, Acta Math. Sinica, 26 (2010), 1287-1298.
doi: 10.1007/s10114-010-8103-x. |
[22] |
G. Meng and M. Zhang, Measure differential equations I. Continuity of solutions in measures with weak topology, Tsinghua University, preprint (2009). Available from http://faculty.math.tsinghua.edu.cn/~mzhang/publs/mde1.pdf. |
[23] |
G. Meng and M. Zhang, Measure differential equations II. Continuity of eigenvalues in measures with weak topology, Tsinghua University, preprint (2009). Available from http://faculty.math.tsinghua.edu.cn/~mzhang/publs/mde2.pdf. |
[24] |
G. A. Monteiro and M. Tvrdý, On Kurzweil-Stieltjes integral in Banach space, Math. Bohem., 137 (2013), 365-381. |
[25] |
F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equations, Bol. Soc. Mat. Mexicana, 11 (1966), 40-46. |
[26] |
Z. Opial, Continuous parameter dependence in linear systems of differential equations, J. Differential Equations, 3 (1967), 571-579.
doi: 10.1016/0022-0396(67)90017-4. |
[27] |
Š. Schwabik, "Generalized Ordinary Differential Equations," World Scientific. Singapore, 1992. |
[28] |
Š. Schwabik, Abstract Perron-Stieltjes integral, Math. Bohem., 121 (1996), 425-447. |
[29] |
Š. Schwabik, Linear Stieltjes integral equations in Banach spaces, Math. Bohem., 124 (1999), 433-457. |
[30] |
Š. Schwabik, Linear Stieltjes integral equations in Banach spaces II; Operator valued solutions, Math. Bohem., 125 (2000), 431-454. |
[31] |
Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations: Boundary Value Problems and Adjoint," Academia and Reidel. Praha and Dordrecht, 1979. |
[32] |
A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl., 385 (2012), 534-550.
doi: 10.1016/j.jmaa.2011.06.068. |
[33] |
A. Taylor, "Introduction to Functional Analysis," Wiley, 1958. |
[34] |
M. Tvrdý, On the continuous dependence on a parameter of solutions of initial value problems for linear generalized differential equations, Funct. Differ. Equ., 5 (1999), 483-498. |
[35] |
M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1-104. |
show all references
References:
[1] |
S. Afonso, E. M. Bonotto, M. Federson and Š. Schwabik, Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invarianceprinciple for non-autonomous systems with impulses, J. Differential Equations, 250 (2011), 2969-3001.
doi: 10.1016/j.jde.2011.01.019. |
[2] |
Z. Artstein, Continuous dependence on parameters: On the best possible results, J. Differential Equations, 19 (1975), 214-225.
doi: 10.1016/0022-0396(75)90002-9. |
[3] |
M. Ashordia, On the correctness of linear boundary value problems for systems of generalized ordinarydifferential equations, Proc. Georgian Acad. Sci. Math., 1 (1993), 385-394. |
[4] |
M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications," Birkhäuser, Boston, 2001. |
[5] |
M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser, Boston, 2003. |
[6] |
M. Brokate and P. Krejčí,, Duality in the space of regulated functions and the play operator, Math. Z., 245 (2003), 667-688.
doi: 10.1007/s00209-003-0563-6. |
[7] |
M. Federson and Š. Schwabik, Generalized ordinary differential equations approach to impulsive retarded functionaldifferential equations, Differential and Integral Equations, 19 (2006), 1201-1234. |
[8] |
D. Fraňková, Continuous dependence on a parameter of solutions of generalized differential equations, časopis pěst. mat., 114 (1989), 230-261. |
[9] |
Z. Halas, Continuous dependence of solutions of generalized linear ordinary differential equationson a parameter, Mathematica Bohemica, 132 (2007), 205-218. |
[10] |
Z. Halas, G. Monteiro and M. Tvrdý, Emphatic convergence and sequential solutions of generalized linear differentialequations, Mem.Differential Equations Math. Phys., 54 (2011), 27-49. |
[11] |
Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on aparameter, Funct. Differ. Equ., 16 (2009), 299-313. |
[12] |
T. H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations, Illinois J. Math., 3 (1959), 352-373. |
[13] |
Ch. S. Hönig, "Volterra Stieltjes-integral Equations," North Holland and American Elsevier, Mathematics Studies 16. Amsterdam and New York, 1975. |
[14] |
C. Imaz and Z. Vorel, Generalized ordinary differential equations in Banach spaces and applicationsto functional equations, Bol. Soc. Mat. Mexicana, 11 (1966), 47-59. |
[15] |
I. Kiguradze, Boundary value problems for systems of ordinary differential equations, (in Russian), Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh, 30 (1987), 3-103; English transl.: J. Sov. Math., 43 (1988), 2259-2339. |
[16] |
M. A. Krasnoselskij and S. G. Krein, On the averaging principle in nonlinear mechanics, (in Russian), Uspekhi mat. nauk, 10 (1955), 147-152. |
[17] |
P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. |
[18] |
J. Kurzweil and Z. Vorel, Continuous dependence of solutions of differential equations on a parameter, Czechoslovak Math. J., 7 (1957), 568-583. |
[19] |
J. Kurzweil, Generalized ordinary differential equation and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449. |
[20] |
J. Kurzweil, Generalized ordinary differential equations, Czechoslovak Math. J., 8 (1958), 360-388. |
[21] |
G. Meng and M. Zhang, Continuity in weak topology: First order linear system of ODE, Acta Math. Sinica, 26 (2010), 1287-1298.
doi: 10.1007/s10114-010-8103-x. |
[22] |
G. Meng and M. Zhang, Measure differential equations I. Continuity of solutions in measures with weak topology, Tsinghua University, preprint (2009). Available from http://faculty.math.tsinghua.edu.cn/~mzhang/publs/mde1.pdf. |
[23] |
G. Meng and M. Zhang, Measure differential equations II. Continuity of eigenvalues in measures with weak topology, Tsinghua University, preprint (2009). Available from http://faculty.math.tsinghua.edu.cn/~mzhang/publs/mde2.pdf. |
[24] |
G. A. Monteiro and M. Tvrdý, On Kurzweil-Stieltjes integral in Banach space, Math. Bohem., 137 (2013), 365-381. |
[25] |
F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equations, Bol. Soc. Mat. Mexicana, 11 (1966), 40-46. |
[26] |
Z. Opial, Continuous parameter dependence in linear systems of differential equations, J. Differential Equations, 3 (1967), 571-579.
doi: 10.1016/0022-0396(67)90017-4. |
[27] |
Š. Schwabik, "Generalized Ordinary Differential Equations," World Scientific. Singapore, 1992. |
[28] |
Š. Schwabik, Abstract Perron-Stieltjes integral, Math. Bohem., 121 (1996), 425-447. |
[29] |
Š. Schwabik, Linear Stieltjes integral equations in Banach spaces, Math. Bohem., 124 (1999), 433-457. |
[30] |
Š. Schwabik, Linear Stieltjes integral equations in Banach spaces II; Operator valued solutions, Math. Bohem., 125 (2000), 431-454. |
[31] |
Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations: Boundary Value Problems and Adjoint," Academia and Reidel. Praha and Dordrecht, 1979. |
[32] |
A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl., 385 (2012), 534-550.
doi: 10.1016/j.jmaa.2011.06.068. |
[33] |
A. Taylor, "Introduction to Functional Analysis," Wiley, 1958. |
[34] |
M. Tvrdý, On the continuous dependence on a parameter of solutions of initial value problems for linear generalized differential equations, Funct. Differ. Equ., 5 (1999), 483-498. |
[35] |
M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1-104. |
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