Solutions of the Hammerstein equations in $BV_\varphi(I_{a}^{b},\, \mathbb{R})

Aziz, Wadie ; Guerrero, José ; Azócar, L. ; Merentes, Nelson

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 36 (2016) no. 2, pp. 207-229.

Résumé

In this paper we study existence and uniqueness of solutions for the Hammerstein equation u(x)= v(x) + λ∫_I_a^bK(x,y)f(y,u(y))dy in the space of function of bounded total φ-variation in the sense of Hardy-Vitali-Tonelli, where λ∈ℝ, K:I_a^b× I_a^b ⟶ℝ and f:I_a^b×ℝ⟶ℝ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

Détail

Keywords: Hammerstein integral equation, Banach spaces, bounded $\varphi$-variation in the sense of Hardy-Vitali-Tonelli, Banach's contraction principle, Leray-Schauder nonlinear alternative principle

Bibliographie

[1] A. Alexiewicz, Functional Analysis (PWN 49, Warsaw, 1969).

[2] J. Appell and C.-J. Chen, How to solve Hammerstein equations, J. Int. Equ. Appl 18 (2006), 287-296. doi: 10.1216/jiea/1181075392

[3] R.P. Agarwal, D. O'Regan and P.J. Wong, Positive solutions of differential, difference and integral equations (Kluwer Academic Publishers , Dordrecht, 1999). doi: 10.1007/978-94-015-9171-3

[4] W. Aziz, H. Leiva and N. Merentes, Solutions of Hammerstein equations in the space $BV(I_a^b)$, Quaestiones Mathematicae 37 (2014), 1-12. doi: 10.2989/16073606.2014.894675

[5] L.A. Azocar, H. Leiva, J. Matute and N. Merentes, On the Hammerstein equation in the space of functions of bounded $\varphi$-variation in the plane, Archivum Mathematicum (Brno) 49 (2013), 51-64. doi: 10.5817/AM2013-1-51

[6] J. Banaś, Integrable solutions of Hammerstein and Urysohn integral equations, J. Austral. Math. Soc. 46 (1989), 61-68. doi: 10.1017/S1446788700030378

[7] J. Banaś and Z. Knap, Integrable solutions of a functional-integral equation, Revista Matem\'atica de la Universidad Complutense de Madrid} 2 (1989), 31-38.

[8] D. Bugajewska, D. Bugajewski and H. Hudzik, On $BV_\phi$-solutions of some nonlinear integral equations, J. Math. Anal. Appl.} 287 (2003), 265-278. doi: 10.1016/S0022-247X(03)00550-X

[9] D. Bugajewska, D. Bugajewski and G. Lewicki, On nonlinear integral equations in the space of functions of bounded generalized $\phi$-variation, J. Int. Equ. Appl.} 21 (2009), 1-20. doi: 10.1216/JIE-2009-21-1-1

[10] D. Bugajewska and D. O'Regan, On nonlinear integral equations and $\Gamma$-bounded variation, Acta Math. Hungar.} 107 (2005), 295-306. doi: 10.1007/s10474-005-0197-8

[11] T.A. Burton, Volterra Integral and Differential Equations (Academic Press, New York, 1983).

[12] V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (2004), 1-82. doi: 10.1515/JAA.2004.1

[13] C. Corduneanu, Integral Equations and Applications (Cambridge University Press, New York, 1973).

[14] J. Diestel, Sequences and Series in Banach Spaces ({Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1984). doi: 10.1007/978-1-4612-5200-9

[15] G. Emmanuele, About the existence of integrable solutions of a functional-integral equation, Revista de Matemática de la Universidad Complutense de Madrid 4 (1991), 65-69. doi: 10.5209/rev_rema.1991.v4.n1.18000

[16] G. Emmanuele, Integrable solutions of a functional-integral equation, J. Int. Equ. Appl. 4 (1992), 89-94. doi: 10.1216/jiea/1181075668

[17] J.A. Guerrero, Extensi\'{o}n a $\mathbb{R}^2$ de la Noci\'{o}n de Funci\'{o}n de Variaci\'{o}n Acotada en el Sentido Hardy-Vitali-Wiener, Ph.D. Thesis,Universidad Central de Venezuela, Facultad de Ciencias, Postgrado de Matematica, Caracas - Venezuela, 2010, in Spanish.

[18] G.H. Hardy, On double {Fourier} series, and especially those which represent the double zeta-function with real and incommesurable parameters, Q.J. Math. Oxford 37 (1905), 53-79.

[19] C. Jordan, Sur la série de Fourier, C.R. Acad. Sci. 92 (1881), 228-230.

[20] H. Leiva, J. Matute and N. Merentes, On the Hammerstein-Volterra equation in the space of the absolutely continuous functions, I.J. Math. Anal. 6 (2012), 2977-2999.

[21] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. (1034) (Springer-Verlag, 1983).

[22] B.G. Pachpatte, Applications of the Leray-Schauder alternative to some Volterra integral and integro-diferential equations, Indian J. Pure Appl. Math. 26 (1995), 1161-1168.

[23] B.G. Pachpatte, Multidimensional Integral Equations and Inequalities, Atlantis Studies in Mathematics for Engineering and Science (Atlantis Press, 2011).

[24] D. O'Regan, Fixed point theorems for nonlinear operators, JMAA 212 (1996), 413-432. doi: 10.1006/jmaa.1996.0324

[25] D. O'Regan, Existence theory for nonlinear Volterra integro-differential and integral equations, Nonlinear Anal. 31 (1998), 317-341. doi: 10.1016/S0362-546X(96)00313-6

[26] R. Precup, Theorems of Leray-Schauder Type and Application (Gordon and Breach Science Publishers, 2001).

[27] Š. Schwabik, M. Tvrdý and O. Vejvoda, Differential and integral equations, Boundary value problems and adjoints, Academia Praha and Reidel D. (1979), 239-246.

[28] L. Tonelli, Sulle funzioni di due variabili generalmente a variazione limitata, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1936), 315-320.

[29] G. Vitali, Sulle funzione integrali, Atti Accad. Sci. Torino CI Sci Fis Mat Natur 40 (1904/1905) 1021-1034 and (1984) Opere sull'analisi teale Cremonese 205-220.

[30] L.C. Young, Sur une généralisation de la notion de variation de puissance pieme borneé au sens de N. Wiener et sur la convergence des séries de Fourier, C.R. Acad. Sci. Paris Sér. A 204 (1937), 470-472.

[31] P.P. Zabrejko, A.I. Koshelev, M.A. Krasnosel'skii, S.G. Mikhlin, L.S. Rakovschik and V.J. Stetsenko, Integral Equations (Noordhoff, Leyden, 1975).