On mappings of higher order and their applications to nonlinear equations

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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

Abstract
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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 and 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.
[2]	J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators," Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, 1990.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	G. M. Fichtenholz, "Differential and Integral Calculus," vol. 2, Fizmatgiz, Moscow, 1959, (in Russian).
[10]	S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 8 (1970), 457-474.
[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.
[12]	J.-P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math., 56 (1989), 409-424. doi: 10.1007/BF01396646.
[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.
[14]	L. Maligranda and W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math., 104 (1987), 53-65. doi: 10.1007/BF01540525.
[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.
[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.
[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.
[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.
[19]	J. Musielak, "Orlicz Spaces and Modular Spaces," Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
[20]	J. Musielak and W. Orlicz, On generalized variations (I), Studia Math., 18 (1959), 11-41.
[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.
[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.
[23]	W. H. Schikhof, "Ultrametric Calculus. An introduction to $p$-adic Analysis," Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, 2006.

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.
[2]	J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators," Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, 1990.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	G. M. Fichtenholz, "Differential and Integral Calculus," vol. 2, Fizmatgiz, Moscow, 1959, (in Russian).
[10]	S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 8 (1970), 457-474.
[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.
[12]	J.-P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math., 56 (1989), 409-424. doi: 10.1007/BF01396646.
[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.
[14]	L. Maligranda and W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math., 104 (1987), 53-65. doi: 10.1007/BF01540525.
[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.
[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.
[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.
[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.
[19]	J. Musielak, "Orlicz Spaces and Modular Spaces," Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
[20]	J. Musielak and W. Orlicz, On generalized variations (I), Studia Math., 18 (1959), 11-41.
[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.
[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.
[23]	W. H. Schikhof, "Ultrametric Calculus. An introduction to $p$-adic Analysis," Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, 2006.

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