| Title:
|
On the Hammerstein equation in the space of functions of bounded $\varphi $-variation in the plane (English) |
| Author:
|
Azócar, Luis |
| Author:
|
Leiva, Hugo |
| Author:
|
Matute, Jesús |
| Author:
|
Merentes, Nelson |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
51-64 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
existence and uniqueness of solutions of the Hammerstein integral equation in the plane |
| Keyword:
|
$\varphi $-bounded total variation norm on a rectangle |
| MSC:
|
45G10 |
| idZBL:
|
Zbl 06321148 |
| idMR:
|
MR3073016 |
| DOI:
|
10.5817/AM2013-1-51 |
| . |
| Date available:
|
2013-05-28T13:31:29Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143300 |
| . |
| Reference:
|
[1] Azis, W., Leiva, H., Merentes, N., Sánchez, J. L.: Functions of two variables with bounded $\varphi $–variation in the sense of Riesz.J. Math. Appl. 32 (2010), 5–23. MR 2664252 |
| Reference:
|
[2] Aziz, W. A.: Algunas Extensiones a $\mathbb{R}^{2}$ de la Noción de Funciones con $\varphi $–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. |
| Reference:
|
[3] Bugajeswska, D., Bugajewski, D., Hudzik, H.: $BV_{ \varphi } $–solutions of nonlinear integral equations.J. Math. Anal. Appl. 287 (2003), 265–278. MR 2010270, 10.1016/S0022-247X(03)00550-X |
| Reference:
|
[4] Bugajewska, D.: On the superposition operator in the space of functions of bounded variation, revisited.Math. Comput. Modelling 52 (2010), 791–796. Zbl 1202.45005, MR 2661764, 10.1016/j.mcm.2010.05.008 |
| Reference:
|
[5] Bugajewska, D., O ' Regan, D.: On nonlinear integral equations and $\Lambda $–bounded variation.gan, D., On nonlinear integral equations and $\Lambda $–bounded variation, Acta Math. Hungar. 107 (4) (2005), 295–306. Zbl 1085.45005, MR 2150792, 10.1007/s10474-005-0197-8 |
| Reference:
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[6] Bugajewski, D.: On BV–solutions of some nonlinear integral equations.Integral Equations Operator Theory 46 (2003), 387–398. Zbl 1033.45002, MR 1997978, 10.1007/s00020-001-1146-8 |
| Reference:
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[7] Pachpatte, B. G.: Multidimensional Integral Equations and Inequalities.Atlantis Studies in Mathematics for Engineering and Science, Atlantis Press, 2011. Zbl 1232.45001, MR 2882942 |
| Reference:
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[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. Zbl 0417.45001, MR 0542283 |
| Reference:
|
[9] Vaz, P. T., Deo, S. G.: On a Volterra–Stieltjets Integral Equation.J. Appl. Math. Stochastic Anal. 3 (1990) 3 (3) (1990), 177–191. MR 1070899, 10.1155/S104895339000017X |
| . |
