# American Institute of Mathematical Sciences

• Previous Article
On the periodic solutions of a class of Duffing differential equations
• DCDS Home
• This Issue
• Next Article
Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation
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

 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.
Citation: Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. [4] M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications,", Birkhäuser, (2001). [5] M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,", Birkhäuser, (2003). [6] M. Brokate and P. Krejčí，, Duality in the space of regulated functions and the play operator,, Math. Z., 245 (2003), 667. 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. [8] D. Fraňková, Continuous dependence on a parameter of solutions of generalized differential equations,, časopis pěst. mat., 114 (1989), 230. [9] Z. Halas, Continuous dependence of solutions of generalized linear ordinary differential equationson a parameter,, Mathematica Bohemica, 132 (2007), 205. [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. [11] Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on aparameter,, Funct. Differ. Equ., 16 (2009), 299. [12] T. H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations,, Illinois J. Math., 3 (1959), 352. [13] Ch. S. Hönig, "Volterra Stieltjes-integral Equations,", North Holland and American Elsevier, (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. [15] I. Kiguradze, Boundary value problems for systems of ordinary differential equations,, (in Russian), 30 (1987), 3. [16] M. A. Krasnoselskij and S. G. Krein, On the averaging principle in nonlinear mechanics,, (in Russian), 10 (1955), 147. [17] P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159. [18] J. Kurzweil and Z. Vorel, Continuous dependence of solutions of differential equations on a parameter,, Czechoslovak Math. J., 7 (1957), 568. [19] J. Kurzweil, Generalized ordinary differential equation and continuous dependence on a parameter,, Czechoslovak Math. J., 7 (1957), 418. [20] J. Kurzweil, Generalized ordinary differential equations,, Czechoslovak Math. J., 8 (1958), 360. [21] G. Meng and M. Zhang, Continuity in weak topology: First order linear system of ODE,, Acta Math. Sinica, 26 (2010), 1287. 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, (2009). [23] G. Meng and M. Zhang, Measure differential equations II. Continuity of eigenvalues in measures with weak topology,, Tsinghua University, (2009). [24] G. A. Monteiro and M. Tvrdý, On Kurzweil-Stieltjes integral in Banach space,, Math. Bohem., 137 (2013), 365. [25] F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equations,, Bol. Soc. Mat. Mexicana, 11 (1966), 40. [26] Z. Opial, Continuous parameter dependence in linear systems of differential equations,, J. Differential Equations, 3 (1967), 571. 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. [29] Š. Schwabik, Linear Stieltjes integral equations in Banach spaces,, Math. Bohem., 124 (1999), 433. [30] Š. Schwabik, Linear Stieltjes integral equations in Banach spaces II; Operator valued solutions,, Math. Bohem., 125 (2000), 431. [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. 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. [35] M. Tvrdý, Differential and integral equations in the space of regulated functions,, Mem. Differential Equations Math. Phys., 25 (2002), 1.

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. 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. 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. [4] M. Bohner and A. Peterson, "Dynamic Equations on Time Scales: An Introduction with Applications,", Birkhäuser, (2001). [5] M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,", Birkhäuser, (2003). [6] M. Brokate and P. Krejčí，, Duality in the space of regulated functions and the play operator,, Math. Z., 245 (2003), 667. 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. [8] D. Fraňková, Continuous dependence on a parameter of solutions of generalized differential equations,, časopis pěst. mat., 114 (1989), 230. [9] Z. Halas, Continuous dependence of solutions of generalized linear ordinary differential equationson a parameter,, Mathematica Bohemica, 132 (2007), 205. [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. [11] Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on aparameter,, Funct. Differ. Equ., 16 (2009), 299. [12] T. H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations,, Illinois J. Math., 3 (1959), 352. [13] Ch. S. Hönig, "Volterra Stieltjes-integral Equations,", North Holland and American Elsevier, (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. [15] I. Kiguradze, Boundary value problems for systems of ordinary differential equations,, (in Russian), 30 (1987), 3. [16] M. A. Krasnoselskij and S. G. Krein, On the averaging principle in nonlinear mechanics,, (in Russian), 10 (1955), 147. [17] P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159. [18] J. Kurzweil and Z. Vorel, Continuous dependence of solutions of differential equations on a parameter,, Czechoslovak Math. J., 7 (1957), 568. [19] J. Kurzweil, Generalized ordinary differential equation and continuous dependence on a parameter,, Czechoslovak Math. J., 7 (1957), 418. [20] J. Kurzweil, Generalized ordinary differential equations,, Czechoslovak Math. J., 8 (1958), 360. [21] G. Meng and M. Zhang, Continuity in weak topology: First order linear system of ODE,, Acta Math. Sinica, 26 (2010), 1287. 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, (2009). [23] G. Meng and M. Zhang, Measure differential equations II. Continuity of eigenvalues in measures with weak topology,, Tsinghua University, (2009). [24] G. A. Monteiro and M. Tvrdý, On Kurzweil-Stieltjes integral in Banach space,, Math. Bohem., 137 (2013), 365. [25] F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equations,, Bol. Soc. Mat. Mexicana, 11 (1966), 40. [26] Z. Opial, Continuous parameter dependence in linear systems of differential equations,, J. Differential Equations, 3 (1967), 571. 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. [29] Š. Schwabik, Linear Stieltjes integral equations in Banach spaces,, Math. Bohem., 124 (1999), 433. [30] Š. Schwabik, Linear Stieltjes integral equations in Banach spaces II; Operator valued solutions,, Math. Bohem., 125 (2000), 431. [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. 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. [35] M. Tvrdý, Differential and integral equations in the space of regulated functions,, Mem. Differential Equations Math. Phys., 25 (2002), 1.
 [1] Paola Goatin, Philippe G. LeFloch. $L^1$ continuous dependence for the Euler equations of compressible fluids dynamics. Communications on Pure & Applied Analysis, 2003, 2 (1) : 107-137. doi: 10.3934/cpaa.2003.2.107 [2] X. Xiang, Y. Peng, W. Wei. A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces. Conference Publications, 2005, 2005 (Special) : 911-919. doi: 10.3934/proc.2005.2005.911 [3] Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53 [4] Robert Stephen Cantrell, Chris Cosner, William F. Fagan. Edge-linked dynamics and the scale-dependence of competitive. Mathematical Biosciences & Engineering, 2005, 2 (4) : 833-868. doi: 10.3934/mbe.2005.2.833 [5] P.E. Kloeden, Pedro Marín-Rubio. Equi-Attraction and the continuous dependence of attractors on time delays. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 581-593. doi: 10.3934/dcdsb.2008.9.581 [6] Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949 [7] Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11 [8] Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083 [9] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [10] Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837 [11] Mahmut Çalik, Marcel Oliver. Weak solutions for generalized large-scale semigeostrophic equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 939-955. doi: 10.3934/cpaa.2013.12.939 [12] Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929 [13] Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038 [14] Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51 [15] Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119 [16] S. Mohamad, K. Gopalsamy. Neuronal dynamics in time varying enviroments: Continuous and discrete time models. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 841-860. doi: 10.3934/dcds.2000.6.841 [17] Yoshikazu Giga, Robert V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 509-535. doi: 10.3934/dcds.2011.30.509 [18] Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28 [19] Mahmoud M. El-Borai. On some fractional differential equations in the Hilbert space. Conference Publications, 2005, 2005 (Special) : 233-240. doi: 10.3934/proc.2005.2005.233 [20] Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273

2017 Impact Factor: 1.179