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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications," Birkhäuser, Boston, 2001.  Google Scholar

[5]

M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser, Boston, 2003.  Google Scholar

[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.  Google Scholar

[7]

M. Federson and Š. Schwabik, Generalized ordinary differential equations approach to impulsive retarded functionaldifferential equations, Differential and Integral Equations, 19 (2006), 1201-1234.  Google Scholar

[8]

D. Fraňková, Continuous dependence on a parameter of solutions of generalized differential equations, časopis pěst. mat., 114 (1989), 230-261.  Google Scholar

[9]

Z. Halas, Continuous dependence of solutions of generalized linear ordinary differential equationson a parameter, Mathematica Bohemica, 132 (2007), 205-218.  Google Scholar

[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. Google Scholar

[11]

Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on aparameter, Funct. Differ. Equ., 16 (2009), 299-313.  Google Scholar

[12]

T. H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations, Illinois J. Math., 3 (1959), 352-373.  Google Scholar

[13]

Ch. S. Hönig, "Volterra Stieltjes-integral Equations," North Holland and American Elsevier, Mathematics Studies 16. Amsterdam and New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. A. Krasnoselskij and S. G. Krein, On the averaging principle in nonlinear mechanics, (in Russian), Uspekhi mat. nauk, 10 (1955), 147-152.  Google Scholar

[17]

P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.  Google Scholar

[18]

J. Kurzweil and Z. Vorel, Continuous dependence of solutions of differential equations on a parameter, Czechoslovak Math. J., 7 (1957), 568-583.  Google Scholar

[19]

J. Kurzweil, Generalized ordinary differential equation and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449.  Google Scholar

[20]

J. Kurzweil, Generalized ordinary differential equations, Czechoslovak Math. J., 8 (1958), 360-388.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

G. A. Monteiro and M. Tvrdý, On Kurzweil-Stieltjes integral in Banach space, Math. Bohem., 137 (2013), 365-381. Google Scholar

[25]

F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equations, Bol. Soc. Mat. Mexicana, 11 (1966), 40-46.  Google Scholar

[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.  Google Scholar

[27]

Š. Schwabik, "Generalized Ordinary Differential Equations," World Scientific. Singapore, 1992.  Google Scholar

[28]

Š. Schwabik, Abstract Perron-Stieltjes integral, Math. Bohem., 121 (1996), 425-447.  Google Scholar

[29]

Š. Schwabik, Linear Stieltjes integral equations in Banach spaces, Math. Bohem., 124 (1999), 433-457.  Google Scholar

[30]

Š. Schwabik, Linear Stieltjes integral equations in Banach spaces II; Operator valued solutions, Math. Bohem., 125 (2000), 431-454.  Google Scholar

[31]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations: Boundary Value Problems and Adjoint," Academia and Reidel. Praha and Dordrecht, 1979.  Google Scholar

[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.  Google Scholar

[33]

A. Taylor, "Introduction to Functional Analysis," Wiley, 1958.  Google Scholar

[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.  Google Scholar

[35]

M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1-104.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications," Birkhäuser, Boston, 2001.  Google Scholar

[5]

M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser, Boston, 2003.  Google Scholar

[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.  Google Scholar

[7]

M. Federson and Š. Schwabik, Generalized ordinary differential equations approach to impulsive retarded functionaldifferential equations, Differential and Integral Equations, 19 (2006), 1201-1234.  Google Scholar

[8]

D. Fraňková, Continuous dependence on a parameter of solutions of generalized differential equations, časopis pěst. mat., 114 (1989), 230-261.  Google Scholar

[9]

Z. Halas, Continuous dependence of solutions of generalized linear ordinary differential equationson a parameter, Mathematica Bohemica, 132 (2007), 205-218.  Google Scholar

[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. Google Scholar

[11]

Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on aparameter, Funct. Differ. Equ., 16 (2009), 299-313.  Google Scholar

[12]

T. H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations, Illinois J. Math., 3 (1959), 352-373.  Google Scholar

[13]

Ch. S. Hönig, "Volterra Stieltjes-integral Equations," North Holland and American Elsevier, Mathematics Studies 16. Amsterdam and New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. A. Krasnoselskij and S. G. Krein, On the averaging principle in nonlinear mechanics, (in Russian), Uspekhi mat. nauk, 10 (1955), 147-152.  Google Scholar

[17]

P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.  Google Scholar

[18]

J. Kurzweil and Z. Vorel, Continuous dependence of solutions of differential equations on a parameter, Czechoslovak Math. J., 7 (1957), 568-583.  Google Scholar

[19]

J. Kurzweil, Generalized ordinary differential equation and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418-449.  Google Scholar

[20]

J. Kurzweil, Generalized ordinary differential equations, Czechoslovak Math. J., 8 (1958), 360-388.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

G. A. Monteiro and M. Tvrdý, On Kurzweil-Stieltjes integral in Banach space, Math. Bohem., 137 (2013), 365-381. Google Scholar

[25]

F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equations, Bol. Soc. Mat. Mexicana, 11 (1966), 40-46.  Google Scholar

[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.  Google Scholar

[27]

Š. Schwabik, "Generalized Ordinary Differential Equations," World Scientific. Singapore, 1992.  Google Scholar

[28]

Š. Schwabik, Abstract Perron-Stieltjes integral, Math. Bohem., 121 (1996), 425-447.  Google Scholar

[29]

Š. Schwabik, Linear Stieltjes integral equations in Banach spaces, Math. Bohem., 124 (1999), 433-457.  Google Scholar

[30]

Š. Schwabik, Linear Stieltjes integral equations in Banach spaces II; Operator valued solutions, Math. Bohem., 125 (2000), 431-454.  Google Scholar

[31]

Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations: Boundary Value Problems and Adjoint," Academia and Reidel. Praha and Dordrecht, 1979.  Google Scholar

[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.  Google Scholar

[33]

A. Taylor, "Introduction to Functional Analysis," Wiley, 1958.  Google Scholar

[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.  Google Scholar

[35]

M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1-104.  Google Scholar

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