On the Hammerstein equation in the space of functions of bounded ϕ -variation in the plane

Luis Azócar; Hugo Leiva; Jesús Matute; Nelson Merentes

Archivum Mathematicum (2013)

  • Volume: 049, Issue: 1, page 51-64
  • ISSN: 0044-8753

Abstract

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In this paper we study existence and uniqueness of solutions for the Hammerstein equation u ( x ) = v ( x ) + λ I a b K ( x , y ) f ( y , u ( y ) ) d y , x I a b : = [ a 1 , b 1 ] × [ a 2 , b 2 ] , in the space B V ϕ ( I a b ) of function of bounded total ϕ - variation in the sense of Riesz, where λ , K : I a b × I a b and f : I a b × are suitable functions.

How to cite

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Azócar, Luis, et al. "On the Hammerstein equation in the space of functions of bounded $\varphi $-variation in the plane." Archivum Mathematicum 049.1 (2013): 51-64. <http://eudml.org/doc/252518>.

@article{Azócar2013,
abstract = {In this paper we study existence and uniqueness of solutions for the Hammerstein equation \[ u(x) = v(x) + \lambda \int \_\{I\_\{a\}^\{b\}\} K(x,y) f\big (y,u(y)\big )\, dy\,, \quad x \in I\_\{a\}^\{b\} := [a\_\{1\},b\_\{1\}] \times [a\_\{2\},b\_\{2\}]\,, \] in the space $BV_\{\varphi \}^\{\mathbb \{R\}\}(I_\{a\}^\{b\})$ of function of bounded total $\varphi -$variation in the sense of Riesz, where $ \lambda \in \mathbb \{R\} $, $ K \colon I_\{a\}^\{b\} \times I_\{a\}^\{b\} \rightarrow \mathbb \{R\} $ and $ f\colon I_\{a\}^\{b\} \times \mathbb \{R\} \rightarrow \mathbb \{R\}$ are suitable functions.},
author = {Azócar, Luis, Leiva, Hugo, Matute, Jesús, Merentes, Nelson},
journal = {Archivum Mathematicum},
keywords = {existence and uniqueness of solutions of the Hammerstein integral equation in the plane; $\varphi $-bounded total variation norm on a rectangle; existence; uniqueness; Hammerstein integral equation in the plane; -bounded total variation norm on a rectangle; nonlinear Leray-Schauder type alternative},
language = {eng},
number = {1},
pages = {51-64},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the Hammerstein equation in the space of functions of bounded $\varphi $-variation in the plane},
url = {http://eudml.org/doc/252518},
volume = {049},
year = {2013},
}

TY - JOUR
AU - Azócar, Luis
AU - Leiva, Hugo
AU - Matute, Jesús
AU - Merentes, Nelson
TI - On the Hammerstein equation in the space of functions of bounded $\varphi $-variation in the plane
JO - Archivum Mathematicum
PY - 2013
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 049
IS - 1
SP - 51
EP - 64
AB - In this paper we study existence and uniqueness of solutions for the Hammerstein equation \[ u(x) = v(x) + \lambda \int _{I_{a}^{b}} K(x,y) f\big (y,u(y)\big )\, dy\,, \quad x \in I_{a}^{b} := [a_{1},b_{1}] \times [a_{2},b_{2}]\,, \] in the space $BV_{\varphi }^{\mathbb {R}}(I_{a}^{b})$ of function of bounded total $\varphi -$variation in the sense of Riesz, where $ \lambda \in \mathbb {R} $, $ K \colon I_{a}^{b} \times I_{a}^{b} \rightarrow \mathbb {R} $ and $ f\colon I_{a}^{b} \times \mathbb {R} \rightarrow \mathbb {R}$ are suitable functions.
LA - eng
KW - existence and uniqueness of solutions of the Hammerstein integral equation in the plane; $\varphi $-bounded total variation norm on a rectangle; existence; uniqueness; Hammerstein integral equation in the plane; -bounded total variation norm on a rectangle; nonlinear Leray-Schauder type alternative
UR - http://eudml.org/doc/252518
ER -

References

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  2. Aziz, W. A., Algunas Extensiones a 2 de la Noción de Funciones con ϕ –Variación Acotada en el Sentido de Riesz y Controlabilidad de las RNC, Ph.D. thesis, Universidad Central de Venezuela, Facultad de Ciencias, Postgrado de Matemática, Caracas, 2009, in Spanish. (2009) 
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  6. Bugajewski, D., 10.1007/s00020-001-1146-8, Integral Equations Operator Theory 46 (2003), 387–398. (2003) Zbl1033.45002MR1997978DOI10.1007/s00020-001-1146-8
  7. Pachpatte, B. G., Multidimensional Integral Equations and Inequalities, Atlantis Studies in Mathematics for Engineering and Science, Atlantis Press, 2011. (2011) Zbl1232.45001MR2882942
  8. Schwabik, Š., Tvrdý, M., Vejvoda, O., Differential and integral equations. Boundary value problems and adjoints, D. Reidel Publishing Co., Dordrecht–Boston, Mass.–London, 1979. (1979) Zbl0417.45001MR0542283
  9. Vaz, P. T., Deo, S. G., 10.1155/S104895339000017X, J. Appl. Math. Stochastic Anal. 3 (1990) 3 (3) (1990), 177–191. (1990) MR1070899DOI10.1155/S104895339000017X

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