Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals

• Autorzy: A. Sikorska-Nowak
• Języki publikacji: EN

Abstrakty

EN

We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness.

Słowa kluczowe

EN

pseudo-solution; Pettis integral; Henstock-Kurzweil integral; Henstock-Kurzweil-Pettis integral; Cauchy problem

Kategorie tematyczne

• 28B05: Vector-valued set functions, measures and integrals
• 34G20: Nonlinear equations

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2007
• Tom: 27
• Numer: 2
• Strony: 315-327

Daty

wydano

2007

otrzymano

2006-06-22

Twórcy

autor

A. Sikorska-Nowak

• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

Bibliografia

• [1] Z. Artstein, Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations 23 (1977), 224-243.
• [2] J.M. Ball, Weak continuity properties of mappings and semi-groups, Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280.
• [3] J. Banaś, Demicontinuity and weak sequential continuity of operators in the Lebesgue space, Proceedings of the 1th Polish Symposium on Nonlinear Analysis, Łódź (1997), 124-129.
• [4] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Dekker, New York and Basel, 1980.
• [5] F.S. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259-262.
• [6] S.S. Cao, The Henstock integral for Banach valued functions, SEA Bull. Math. 16 (1992), 36-40.
• [7] T.S. Chew, On Kurzweil generalized ordinary differential equations, J. Differential Equations 76 (1988), 286-293.
• [8] T.S Chew and F. Flordeliza, On x' = f(t,x) and Henstock-Kurzweil integrals, Differential and Integral Equations 4 (1991), 861-868.
• [9] T.S. Chew, W. van Brunt and G.C. Wake, On retarded functional differential equations and Henstock-Kurzweil integrals, Differential and Integral Equations 9 (1996), 569-580.
• [10] T.S. Chew and T.L. Toh, On functional differential equation with unbounded delay and Henstock-Kurzweil integrals, New Zeland Journal of Mathematics 28 (1999), 111-123.
• [11] M. Cichoń, Convergence theorems for the Henstock-Kurzweil-Pettis integral, Acta Math. Hungarica 92 (2001), 75-82.
• [12] M. Cichoń, Weak solutions of differential equations in Banach spaces, Disc. Math. Differ. Incl. 15 (1995), 5-14.
• [13] M. Cichoń, I. Kubiaczyk and A. Sikorska, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, Czech. Math. J. 54 (129) (2004), 279-289.
• [14] M.C. Deflour and S.K. Mitter, Hereditary differential systems with constant delays, I General case, J. Differential Equations 9 (1972), 213-235.
• [15] R.F.Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1991), 81-86.
• [16] R.A. Gordon, Riemann integration in Banach spaces, Rocky Mountain J. Math. 21 (1991), 923-949.
• [17] R.A. Gordon, The Denjoy extension of the Bochner, Pettis and Dunford integrals, Studia Math. 92 (1989), 73-91.
• [18] R.A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Amer. Math. Soc., Providence, R.I. 1994.
• [19] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. Math. 34 (1990), 557-567.
• [20] J. Hale, Functional Differential Equations, Springer-Verlag, 1971.
• [21] R. Henstock, The General Theory of Integration, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1991.
• [22] W.J. Knight, Solutions of differential equations in Banach spaces, Duke Math. J. 41 (1974), 437-442
• [23] I. Kubiaczyk, On a fixed point theorem for weakly sequentially continuous mappings, Disc. Math. Differ. Incl. 15 (1995), 15-20.
• [24] I. Kubiaczyk, A. Sikorska, Differential equations in Banach spaces and Henstock-Kurzweil integrals, Disc. Math. Differ. Incl. 19 (1999), 35-43.
• [25] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (1957), 642-659.
• [26] A.R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, Nonlinear Equations in Abstract Spaces, (V. Lakshmikantham, ed.), 1978, 378-404.
• [27] P.Y. Lee, Lanzhou Lectures on Henstock Integration, Ser. Real Anal. 2, World Sci., Singapore, 1989.
• [28] B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
• [29] A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals, Demonstratio Math. 35 (2002), 49-60.

Identyfikatory

DOI

10.7151/dmdico.1087