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March  2012, 11(2): 627-647. doi: 10.3934/cpaa.2012.11.627

On mappings of higher order and their applications to nonlinear equations

Dariusz Bugajewski 1, and  Piotr Kasprzak 2,

1. 

Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Matejki 48/49, 60-769 Poznań, Poland
2. 

Optimization and Control Theory Department, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland

Received  September 2010 Revised  February 2011 Published  October 2011

One of the main goals of this paper is to investigate mappings of higher order which possess ``good'' properties, in particular, when we treat them as perturbations of nonlinear differential as well as integral equations. We draw a particular attention to nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan or in the sense of Young. We provide sufficient conditions which guarantee that nonlinear Hammerstein operators are of higher order in such spaces. We also prove a few extensions of Lovelady's fixed point theorem in Archimedean as well as non-Archimedean setting. Finally, we apply our results to prove the existence and uniqueness results to some commonly known nonlinear equations with perturbations.
Keywords: non-Archimedean normed space, Lipschitz condition, superposition operator., Banach contraction principle, mapping of higher order, function of bounded variation in the sense of Jordan, Hammerstein integral operator, functional-integral equation, function of bounded $\phi$-variation, fixed point theorem, mixed Volterra-Fredholm integral equation.
Mathematics Subject Classification: 47H09, 47H10, 26A4.
Citation: Dariusz Bugajewski, Piotr Kasprzak. On mappings of higher order and their applications to nonlinear equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 627-647. doi: 10.3934/cpaa.2012.11.627
References:
[1]

K. Asano and W. Tutschke, An extended Cauchy-Kovalevskaya problem and its solution in associated spaces, Z. Anal. Anwend., 21 (2002), 1055-1060.  Google Scholar

[2]

J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators," Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, 1990.  Google Scholar

[3]

M. Borkowski, D. Bugajewski and M. Zima, On some fixed point theorems for generalized contractions and their perturbations, J. Math. Anal. Appl., 367 (2010), 464-475. doi: 10.1016/j.jmaa.2010.01.014.  Google Scholar

[4]

H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal., 27 (1990), 987-1000. doi: 10.1137/0727057.  Google Scholar

[5]

D. Bugajewska, D. Bugajewski and H. Hudzik, $BV_{\phi}$ -solutions of nonlinear integral equations, J. Math. Anal. Appl., 287 (2003), 265-278. doi: 10.1016/S0022-247X(03)00550-X.  Google Scholar

[6]

D. Bugajewska, D. Bugajewski and G. Lewicki, On nonlinear integral equations in the space of functions of bounded generalized $\phi$-variation, J. Integral Equations Appl., 21 (2009), 1-20. doi: 10.1216/JIE-2009-21-1-1.  Google Scholar

[7]

D. Bugajewski, On $BV$-solutions of some nonlinear integral equations, Integral Equations Operator Theory, 46 (2003), 387-398. doi: 10.1007/s00020-001-1146-8.  Google Scholar

[8]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9.  Google Scholar

[9]

G. M. Fichtenholz, "Differential and Integral Calculus," vol. 2, Fizmatgiz, Moscow, 1959, (in Russian). Google Scholar

[10]

S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 8 (1970), 457-474.  Google Scholar

[11]

S. Heikkilä, M. Kumpulainen and S. Seikkala, On functional improper Volterra integral equations and impulsive differential equations in ordered Banach spaces, J. Math. Anal. Appl., 341 (2008), 433-444. doi: 10.1016/j.jmaa.2007.10.015.  Google Scholar

[12]

J.-P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math., 56 (1989), 409-424. doi: 10.1007/BF01396646.  Google Scholar

[13]

D. L. Lovelady, A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation, Trans. Amer. Math. Soc., 182 (1973), 71-83. doi: 10.1090/S0002-9947-1973-0336263-9.  Google Scholar

[14]

L. Maligranda and W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math., 104 (1987), 53-65. doi: 10.1007/BF01540525.  Google Scholar

[15]

J. Matkowski and J. Miś, On a characterization of Lipschitzian operators of substitution in the space $BV$ , Math. Nachr., 117 (1984), 155-159. doi: 10.1002/mana.3211170111.  Google Scholar

[16]

R. K. Miller, On the linearization of Volterra integral equations, J. Math. Anal. Appl., 23 (1968), 198-208. doi: 10.1016/0022-247X(68)90127-3.  Google Scholar

[17]

R. K. Miller, Admissibility and nonlinear Volterra integral equations, Proc. Amer. Math. Soc., 25 (1970), 65-71. doi: 10.1090/S0002-9939-1970-0257674-9.  Google Scholar

[18]

R. K. Miller, J. A. Nohel and J. S. W. Wong, A stability theorem for nonlinear mixed integral equations, J. Math. Anal. Appl., 25 (1969), 446-449. doi: 10.1016/0022-247X(69)90247-9.  Google Scholar

[19]

J. Musielak, "Orlicz Spaces and Modular Spaces," Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.  Google Scholar

[20]

J. Musielak and W. Orlicz, On generalized variations (I), Studia Math., 18 (1959), 11-41.  Google Scholar

[21]

M. A. Nashed and J. S. W. Wong, Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech., 18 (1969), 767-777.  Google Scholar

[22]

C. Petalas and T. Vidalis, A fixed point theorem in non-Archimedean vector spaces, Proc. Amer. Math. Soc., 118 (1993), 819-821. doi: 10.1090/S0002-9939-1993-1132421-2.  Google Scholar

[23]

W. H. Schikhof, "Ultrametric Calculus. An introduction to $p$-adic Analysis," Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, 2006.  Google Scholar

show all references

References:
[1]

K. Asano and W. Tutschke, An extended Cauchy-Kovalevskaya problem and its solution in associated spaces, Z. Anal. Anwend., 21 (2002), 1055-1060.  Google Scholar

[2]

J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators," Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, 1990.  Google Scholar

[3]

M. Borkowski, D. Bugajewski and M. Zima, On some fixed point theorems for generalized contractions and their perturbations, J. Math. Anal. Appl., 367 (2010), 464-475. doi: 10.1016/j.jmaa.2010.01.014.  Google Scholar

[4]

H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal., 27 (1990), 987-1000. doi: 10.1137/0727057.  Google Scholar

[5]

D. Bugajewska, D. Bugajewski and H. Hudzik, $BV_{\phi}$ -solutions of nonlinear integral equations, J. Math. Anal. Appl., 287 (2003), 265-278. doi: 10.1016/S0022-247X(03)00550-X.  Google Scholar

[6]

D. Bugajewska, D. Bugajewski and G. Lewicki, On nonlinear integral equations in the space of functions of bounded generalized $\phi$-variation, J. Integral Equations Appl., 21 (2009), 1-20. doi: 10.1216/JIE-2009-21-1-1.  Google Scholar

[7]

D. Bugajewski, On $BV$-solutions of some nonlinear integral equations, Integral Equations Operator Theory, 46 (2003), 387-398. doi: 10.1007/s00020-001-1146-8.  Google Scholar

[8]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9.  Google Scholar

[9]

G. M. Fichtenholz, "Differential and Integral Calculus," vol. 2, Fizmatgiz, Moscow, 1959, (in Russian). Google Scholar

[10]

S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 8 (1970), 457-474.  Google Scholar

[11]

S. Heikkilä, M. Kumpulainen and S. Seikkala, On functional improper Volterra integral equations and impulsive differential equations in ordered Banach spaces, J. Math. Anal. Appl., 341 (2008), 433-444. doi: 10.1016/j.jmaa.2007.10.015.  Google Scholar

[12]

J.-P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math., 56 (1989), 409-424. doi: 10.1007/BF01396646.  Google Scholar

[13]

D. L. Lovelady, A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation, Trans. Amer. Math. Soc., 182 (1973), 71-83. doi: 10.1090/S0002-9947-1973-0336263-9.  Google Scholar

[14]

L. Maligranda and W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math., 104 (1987), 53-65. doi: 10.1007/BF01540525.  Google Scholar

[15]

J. Matkowski and J. Miś, On a characterization of Lipschitzian operators of substitution in the space $BV$ , Math. Nachr., 117 (1984), 155-159. doi: 10.1002/mana.3211170111.  Google Scholar

[16]

R. K. Miller, On the linearization of Volterra integral equations, J. Math. Anal. Appl., 23 (1968), 198-208. doi: 10.1016/0022-247X(68)90127-3.  Google Scholar

[17]

R. K. Miller, Admissibility and nonlinear Volterra integral equations, Proc. Amer. Math. Soc., 25 (1970), 65-71. doi: 10.1090/S0002-9939-1970-0257674-9.  Google Scholar

[18]

R. K. Miller, J. A. Nohel and J. S. W. Wong, A stability theorem for nonlinear mixed integral equations, J. Math. Anal. Appl., 25 (1969), 446-449. doi: 10.1016/0022-247X(69)90247-9.  Google Scholar

[19]

J. Musielak, "Orlicz Spaces and Modular Spaces," Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.  Google Scholar

[20]

J. Musielak and W. Orlicz, On generalized variations (I), Studia Math., 18 (1959), 11-41.  Google Scholar

[21]

M. A. Nashed and J. S. W. Wong, Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech., 18 (1969), 767-777.  Google Scholar

[22]

C. Petalas and T. Vidalis, A fixed point theorem in non-Archimedean vector spaces, Proc. Amer. Math. Soc., 118 (1993), 819-821. doi: 10.1090/S0002-9939-1993-1132421-2.  Google Scholar

[23]

W. H. Schikhof, "Ultrametric Calculus. An introduction to $p$-adic Analysis," Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, 2006.  Google Scholar

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