# American Institute of Mathematical Sciences

• Previous Article
On the critical exponents for porous medium equation with a localized reaction in high dimensions
• CPAA Home
• This Issue
• Next Article
The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy
March  2012, 11(2): 627-647. doi: 10.3934/cpaa.2012.11.627

## On mappings of higher order and their applications to nonlinear equations

 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.
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.   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.  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.  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.  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.  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.  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.  doi: 10.1016/0022-0396(79)90080-9.  Google Scholar [9] G. M. Fichtenholz, "Differential and Integral Calculus," vol. 2,, Fizmatgiz, (1959).   Google Scholar [10] S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems,, J. Differential Equations, 8 (1970), 457.   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.  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.  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.  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.  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.  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.  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.  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.  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, (1983).   Google Scholar [20] J. Musielak and W. Orlicz, On generalized variations (I),, Studia Math., 18 (1959), 11.   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.   Google Scholar [22] C. Petalas and T. Vidalis, A fixed point theorem in non-Archimedean vector spaces,, Proc. Amer. Math. Soc., 118 (1993), 819.  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.   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.  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.  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.  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.  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.  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.  doi: 10.1016/0022-0396(79)90080-9.  Google Scholar [9] G. M. Fichtenholz, "Differential and Integral Calculus," vol. 2,, Fizmatgiz, (1959).   Google Scholar [10] S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems,, J. Differential Equations, 8 (1970), 457.   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.  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.  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.  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.  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.  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.  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.  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.  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, (1983).   Google Scholar [20] J. Musielak and W. Orlicz, On generalized variations (I),, Studia Math., 18 (1959), 11.   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.   Google Scholar [22] C. Petalas and T. Vidalis, A fixed point theorem in non-Archimedean vector spaces,, Proc. Amer. Math. Soc., 118 (1993), 819.  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
 [1] 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 [2] Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527 [3] Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205 [4] Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004 [5] Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051 [6] 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 [7] T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure & Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217 [8] Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241 [9] Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229 [10] Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826 [11] T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277 [12] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [13] Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043 [14] Franco Obersnel, Pierpaolo Omari. Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 305-320. doi: 10.3934/dcds.2013.33.305 [15] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [16] Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347 [17] Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555 [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] Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511 [20] Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129

2018 Impact Factor: 0.925