Abstract
We present the existence and uniqueness of global and local \({{\rm \Phi}}\)-bounded variation (\({{\rm \Phi}BV}\)) solutions as well as continuous \({{\rm \Phi}BV}\)-solutions of nonlinear Hammerstein and Volterra–Hammerstein integral equations formulated in terms of the Lebesgue integral.
Similar content being viewed by others
References
Alexiewicz A.: Functional Analysis. PWN, Warsaw (1969)
Appell J., Chen C.-J.: How to solve Hammerstein equations. J. Integral Equations Appl. 18, 287–296 (2006)
J. Appell, J. Banaś and N. Merentes, Bounded Variation and Around. De Gruyter Ser. Nonlinear Anal. Appl. 17, De Gruyter, Berlin, 2014.
Banaś J.: Integrable solutions of Hammerstein and Urysohn integral equations. J. Aust. Math. Soc. Ser. A 46, 61–68 (1989)
Banaś J., Knap Z.: Integrable solutions of a functional-integral equation. Rev. Mat. Univ. Complut. Madrid 2, 31–38 (1989)
Bugajewski D.: On BV-solutions of some nonlinear integral equations. Integral Equations Operator Theory 46, 387–398 (2003)
Bugajewska D., Bugajewski D., Hudzik H.: \({BV_{\Phi}}\)-solutions of some nonlinear integral equations. J. Math. Anal. Appl. 287, 265–278 (2003)
Bugajewska D., O’Regan D.: On nonlinear integral equations and \({{\rm \Lambda}}\)-bounded variation. Acta Math. Hungar. 107, 295–306 (2005)
Emmanuele G.: About the existence of integrable solutions of a functionalintegral equation. Rev. Mat. Univ. Complut. Madrid 4, 65–69 (1991)
Emmanuele G.: Integrable solutions of a functional-integral equation. J. Integral Equations Appl. 4, 89–94 (1992)
Krasnosel’skiĭ M. A., Rutitskiĭ Y. B.: Convex Functions and Orlicz Spaces. Hindustan Publishing Corp. (India), Delhi (1962)
Pachpatte B. G.: Applications of the Leray-Schauder alternative to some Volterra integral and integrodiferential equations. Indian J. Pure Appl. Math. 26, 1161–1168 (1995)
B. G. Pachpatte, Multidimensional Integral Equations and Inequalities. Atlantis Stud. Math. Eng. Sci. 9, Atlantis Press, Paris, 2011.
O’Regan D.: Fixed point theorems for nonlinear operators. Electron. J. Math. Anal. Appl. 202, 413–432 (1996)
O’Regan D.: Existence theory for nonlinear Volterra integrodifferential and integral equations. Nonlinear Anal. 31, 317–341 (1998)
O’Regan D., Precup R.: Theorems of Leray-Schauder Type and Applications. Gordon and Breach Science Publishers, Amsterdam (2001)
Schramm M.: Functions of \({{\rm \Phi}}\)-bounded variation and Riemann-Stieltjes integration. Trans. Amer. Math. Soc. 287, 49–63 (1985)
Š. Schwabik, M. Tvrdý and O. Vejvoda, Differential and Integral Equations: Boundary Value Problems and Adjoints. D. Reidel Publishing Co., Dordrecht, Boston, Mass., London, 1979.
Waterman D.: On \({\Lambda}\)-bounded variation. Studia Math. 57, 33–45 (1976)
Young L. C.: 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, 470–472 (1937)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aziz, W., Guerrero, J.A. & Merentes, N. On nonlinear integral equations in the space \({{\rm \Phi}BV (I)}\) . J. Fixed Point Theory Appl. 18, 351–366 (2016). https://doi.org/10.1007/s11784-015-0280-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-015-0280-x