Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
existence and uniqueness of solutions of the Hammerstein integral equation in the plane; $\varphi $-bounded total variation norm on a rectangle
existence and uniqueness of solutions of the Hammerstein integral equation in the plane; $\varphi $-bounded total variation norm on a rectangle
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.
References:
[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
[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.
[3] Bugajeswska, D., Bugajewski, D., Hudzik, H.: $BV_{ \varphi } $–solutions of nonlinear integral equations. J. Math. Anal. Appl. 287 (2003), 265–278. DOI 10.1016/S0022-247X(03)00550-X | MR 2010270
[4] Bugajewska, D.: On the superposition operator in the space of functions of bounded variation, revisited. Math. Comput. Modelling 52 (2010), 791–796. DOI 10.1016/j.mcm.2010.05.008 | MR 2661764 | Zbl 1202.45005
[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. DOI 10.1007/s10474-005-0197-8 | MR 2150792 | Zbl 1085.45005
[6] Bugajewski, D.: On BV–solutions of some nonlinear integral equations. Integral Equations Operator Theory 46 (2003), 387–398. DOI 10.1007/s00020-001-1146-8 | MR 1997978 | Zbl 1033.45002
[7] Pachpatte, B. G.: Multidimensional Integral Equations and Inequalities. Atlantis Studies in Mathematics for Engineering and Science, Atlantis Press, 2011. MR 2882942 | Zbl 1232.45001
[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. MR 0542283 | Zbl 0417.45001
[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. DOI 10.1155/S104895339000017X | MR 1070899
Search
Partner of
