Generalized linear differential equations in a Banach space: Continuous dependence on a parameter

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January 2013, 33(1): 283-303. doi: 10.3934/dcds.2013.33.283

Generalized linear differential equations in a Banach space: Continuous dependence on a parameter

Giselle A. Monteiro ^1, and Milan Tvrdý ^2,

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

Received July 2011 Revised November 2011 Published September 2012

Abstract
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This paper deals with integral equations of the form \begin{eqnarray*} x(t)=\tilde{x}+∫_a^td[A]x+f(t)-f(a), t∈[a,b], \end{eqnarray*} in a Banach space $X,$ where $-\infty\ < a < b < \infty$, $\tilde{x}∈ X,$ $f:[a,b]→X$ is regulated on [a,b] and $A(t)$ is for each $t∈[a,b], $ a linear bounded operator on $X,$ while the mapping $A:[a,b]→L(X)$ has a bounded variation on [a,b] Such equations are called generalized linear differential equations. Our aim is to present new results on the continuous dependence of solutions of such equations on a parameter. Furthermore, an application of these results to dynamic equations on time scales is given.

Keywords: time scale dynamics, continuous dependence, Generalized differential equations in Banach space, Kurzweil-Stieljes integral..

Mathematics Subject Classification: Primary: 45A05, 34A30; Secondary: 34N0.

Citation: Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283

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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