Global solutions for Volterra ordinary and retarded integral equations

• Autorzy: Satco B.
• Języki publikacji: EN

Abstrakty

EN

Using a generalization of Darbo's fixed point theorem, we obtain the existence of global solutions for nonlinear Volterra-type integral equations in Banach spaces. The involved functions are supposed to be continuous only with respect to some variables, integrability or essential boundedness conditions being also imposed. Our result improves the similar result given in [10] (where uniform continuity was required), as well as those referred by the authors of the cited paper. Finally, following the same ideas, the existence of continuous solutions is proved for a Volterra-type retarded integral equation, under less restrictive assumptions than in the others related results in literature.

Słowa kluczowe

EN

nonlinear Volterra equation; retarded equation; fixed point theorem

PL

równanie Volterra; twierdzenie o punkcie stałym

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2008
• Tom: Vol. 41, nr 3
• Strony: 627--637
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Satco B.

• Faculty of Electrical Engineering and Computer Science, "Stefan Cel Mare" University of Suceava, Universitatii 13 - Suceava, Romania,

Bibliografia

• [1] A. Ambrosetti, Un teorema di existenza per le equazioni differenziali neglispazi di Banach, Rend. Sem. Univ. Padova 39 (1967), 349-360.
• [2] J. Banás, K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
• [3] M. C. Delfour, S. K. Mitter, Hereditary differential systems with constant delays, I General case, J. Differential Equations 9 (1972), 213-235.
• [4] C. Gori, V. Obukhovskii, M. Ragni, P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal. 51 (2002), 765-782.
• [5] D. J. Guo, Solutions of nonlinear integro-differential equations of mixed type in Banach spaces, J. Appl. Math. Simulation 2 (1989), 1-11.
• [6] D. J. Guo, V. Lakshmikantham, X. Z. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht, 1996.
• [7] J. Hale, Theory of Functional Differential Equations, Springer, Berlin, 1977.
• [8] H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371.
• [9] L. S. Liu, Existence of global solutions of initial value problem for nonlinear integro-differential equations of mixed type in Banach spaces, J. Systems Sci. Math. Sci. 20 (2000), 112-116 (in Chinese).
• [10] L. Liu, F. Guo, C. Wu, Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), 638-649.
• [11] L. S. Liu, C. X. Wu, F. Guo, Existence theorems of global solutions of initial value problem for nonlinear integro-differential equations of mixed type in Banach spaces and applications, Comput. Math. Appl. 47 (2004), 13-22.
• [12] D. O'Regan and R. Precup, Fixed Point Theorems for Set-Valued Maps and Existence Principles for Integral Inclusions, J. Math. Anal. Appl. 245 (2000), 594-612.
• [13] D. O'Regan and R. Precup, Existence Criteria for Integral Equations in Banach Spaces, J. Inequal. Appl. 6 (2001), 77-97.