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## Studia Mathematica

2000 | 140 | 2 | 117-139
Tytuł artykułu

### Degenerate evolution problems and Beta-type operators

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
The present paper is concerned with the study of the differential operator Au(x):=α(x)u''(x)+β(x)u'(x) in the space C([0,1)] and of its adjoint Bv(x):=((αv)'(x)-β(x)v(x))' in the space $L^1(0,1)$, where α(x):=x(1-x)/2 (0≤x≤1). A careful analysis of their main properties is carried out in view of some generation results available in [6, 12, 20] and [25]. In addition, we introduce and study two different kinds of Beta-type operators as a generalization of similar operators defined in [18]. Among the corresponding approximation results, we show how they can be used in order to represent explicitly the solutions of the Cauchy problems associated with the operators A and Ã, where Ã is equal to B up to a suitable bounded additive perturbation.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
117-139
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-01-19
poprawiono
2000-02-04
Twórcy
autor
autor
• Department of Mathematics, Polytechnic of Bari, Via E. Orabona, 4 70125 Bari, Italy, campiti@dm.uniba.it
Bibliografia
• [1] F. Altomare, Limit semigroups of Bernstein-Schnabl operators associated with positive projections, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 259-279.
• [2] F. Altomare and A. Attalienti, Forward diffusion equations and positive operators, Math. Z. 225 (1997), 211-229.
• [3] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter Stud. in Math. 17, de Gruyter, Berlin, 1994.
• [4] F. Altomare and I. Carbone, On some degenerate differential operators on weighted function spaces, J. Math. Anal. Appl. 213 (1997), 308-333.
• [5] A. Attalienti, Generalized Bernstein-Durrmeyer operators and the associated limit semigroup, J. Approx. Theory 99 (1999), 289-309.
• [6] A. Attalienti and M. Campiti, On the generation of $C_0$-semigroups in $L^1(I)$, preprint, Bari University, 1998.
• [7] M. Campiti and G. Metafune, Approximation properties of recursively defined Bernstein-type operators, J. Approx. Theory 87 (1996), 243-269.
• [8] M. Campiti and G. Metafune, Evolution equations associated with recursively defined Bernstein-type operators, ibid., 270-290.
• [9] M. Campiti and G. Metafune, Approximation of solutions of some degenerate parabolic problems, Numer. Funct. Anal. Optim. 17 (1996), 23-35.
• [10] M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math. (Basel) 70 (1998), 377-390.
• [11] M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum 57 (1995), 1-36.
• [12] P. Clément and C. A. Timmermans, On $C_0$-semigroups generated by differential operators satisfying Ventcel's boundary conditions, Indag. Math. 89 (1986), 379-387.
• [13] N. S. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, 1986.
• [14] W. Feller, Diffusion processes in genetics, in: Proc. 2 nd Berkeley Sympos. Math. Statist. and Probab., Univ. of California Press, 1951, 227-246.
• [15] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. 55 (1952), 468-519.
• [16] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1-31.
• [17] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966.
• [18] A. Lupaş, Die Folge der Beta Operatoren, Dissertation, Universität Stuttgart, 1972.
• [19] R. G. Mamedov, The asymptotic value of the approximation of differentiable functions by linear positive operators, Dokl. Akad. Nauk SSSR 128 (1959), 471-474 (in Russian).
• [20] G. Metafune, Analyticity for some degenerate one-dimensional evolution equations, Studia Math. 127 (1998), 251-276.
• [21] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986.
• [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.
• [23] B. Sendov and V. Popov, The Averaged Moduli of Smoothness, Pure Appl. Math., Wiley, 1988.
• [24] N. Shimakura, Existence and uniqueness of solutions for a diffusion model of intergroup selection, J. Math. Kyoto Univ. 25 (1985), 775-788.
• [25] C. A. Timmermans, On $C_0$-semigroups in a space of bounded continuous functions in the case of entrance or natural boundary points, in: Approximation and Optimization, J. A. Gómez Fernández et al. (eds.), Lecture Notes in Math. 1354, Springer, Berlin, 1988, 209-216.
• [26] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887-919.
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Bibliografia
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