Nonlinear Hammerstein equations and functions of bounded Riesz-Medvedev variation

Jürgen Appell, Tomas Dominguez Benavides

DOI: http://dx.doi.org/10.12775/TMNA.2016.008

Abstract


In this paper we study the solvability of a nonlinear Hammerstein type integral equation in the space of functions of bounded Riesz-Medvedev variation. To this end, we derive a compactness criterion and apply Schauder's fixed point theorem to a suitable operator whose fixed points coincide with the solutions of the integral equation.

Keywords


Bounded variation; nonlinear integral equation; fixed point theorem; Orlicz space

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R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators (Russian), Nauka, Novosibirsk 1986; Engl. transl.: Birkhauser, Basel 1992.

J. Appell, Measures of noncompactness, condensing operators and fixed points: An application-oriented survey, Fixed Point Theory (Cluj) 6 (2005), 157-229.

J. Appell, J. Banas and N. Merentes, Bounded Variation and Around, De Gruyter Ser. Nonlinear Anal. Appl. 17, De Gruyter, Berlin 2014.

J.M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser, Basel 1997.

J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math. 60, M. Dekker, New York 1980.

D. Bugajewska, On the superposition operator in the space of functions of bounded variation, revisited, Math. Comp. Modelling 52 (2010), 791-796.

D. Bugajewska and D. Bugajewski, On nonlinear integral equations and nonabsolutely convergent integrals, J. Dynam. Systems Appl. 14 (2005), 135-148.

D. Bugajewska, D. Bugajewski and H. Hudzik, BV_phi-solutions of nonlinear integral equations, J. Math. Anal. Appl. 287 (2003), 265-278.

D. Bugajewska, D. Bugajewski and G. Lewicki, On nonlinear integral equations in spaces of functions of bounded generalized '-variation, J. Integral Equtions Appl. 21 (1) (2009), 1-20.

D. Bugajewska and D. O'Regan, On nonlinear integral equations and Lambda-bounded variation, Acta Math. Hung. 107 (4) (2005), 295-306.

D. Bugajewski, On BV-solutions of some nonlinear integral equations, Integral Equations Operator Theory 46 (2003), 387-398.

D. Bugajewski and D. O'Regan, Existence results for BV-solutions of nonlinear integral equations, J. Integral Equations Appl. 15 (4) (2003), 343-357.

G. Darbo, Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92.

Yu.I. Gribanov and P.K. Belobrov, On a class of Banach spaces of functions, Izv. Vysh. Uchebn. Zaved. Mat. 35 (4) (1963), 44-55 (Russian).

C. Jordan, Sur la serie de Fourier, C.R. Acad. Sci. Paris 2 (1881), 228-230.

M.A. Krasnosel'skii and Ya.B. Rutitskii, Convex Functions and Orlicz Spaces (Russian), Fizmatgiz., Moscow 1958; Engl. transl.: Noordhoff, Groningen 1961.

K. Kuratowski, Sur les espaces completes, Fund. Math. 15 (1934), 301-335.

J.J. Levin, On a nonlinear Volterra equations, J. Math. Anal. Appl. 39 (1972), 458-476.

Yu. T. Mevdvedev, A generalization of a certain theorem of Riesz (Russian), Uspekhi Mat. Nauk. 6 (1953), 115-118.

N. Merentes, On a characterization of Lipschitzian operators of substitution in the space of bounded Riesz varphi-variation, Ann. Univ. Sci. Budapest 34 (1991), 139-144.

N. Merentes, On the composition operator in RV'[a; b], Collect. Math. 46 (3) (1995), 231-238.

N. Merentes and S. Rivas, On characterization of the Lipschitzian composition operator between spaces of functions of bounded p-variation, Czechosl. Math. J. 45, (4) (1995), 627-637.

M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, M. Dekker, New York 1991.

F. Riesz, Untersuchungen uber Systeme integrierbarer Funktionen, Math. Annalen 69 (1910), 449-497.

F. Riesz, Sur certain systemes singulieres d'equations integrales, Ann. Sci. Ecole Norm. Sup. Paris 28 (1911), 33-68.

F. Riesz, Sur les ensembles compacts de fonctions sommables, Acta Sci. Math. (Szeged) 6 (2-3) (1932-34), 136-142.

D. Waterman, On the summability of Fourier series of functions of Lambda-bounded variation, Studia Math. 55 (1976), 87-95.

D. Waterman, On Lambda-bounded variation, Studia Math. 57 (1976), 33-45.

D. Waterman, Fourier series of functions of Lambda-bounded variation, Proc. Amer. Math. Soc. 74 (1) (1979), 119-123.

D. Waterman, Estimating functions by partial sums of their Fourier series, J. Math. Anal. Appl. 87 (1982), 51-57.

N. Wiener, The quadratic variation of a function and its Fourier coeffcients, J. Math. Phys. MIT 3 (1924), 73-94.


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