Abstract
We consider the elliptic operator Lu(x):= xu″(x)+β(x)u′(x) + γ (x)u(x) with Wentzell-type boundary condition, in spaces of continuous function on [0,+∞[. We prove that such operators generate positive C 0-semigroup which can be approximated by means of iterated of modified Szász-Mirakjan operators here introduced.
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Altomare, F., Amiar, R.: Approximation by positive operators of the C 0-semigroups associated with one-dimensional diffusion equations-Part I, Numer. Funct. Anal. Optim., 26 (2005), 1–15
Altomare, F., Attalienti, A.: Degenerate evolution equations in weighted continuous function spaces, Markov processes and the Black-Scholes equation-Part I, Result. Math., 42 (2002), 193–211
Altomare, F., Attalienti, A.: Degenerate evolution equations in weighted continuous function spaces, Markov processes and the Black-Scholes equation-Part II, Result. Math., 42 (2002), 212–228
Altomare, F., Carbone, I.: On some degenerate operators on weighted function spaces, J. Math. Anal. Appl., 213 (1997) 308–333
Altomare, F., Leonessa, V., Milella, S.: Cores for second-order differential operators on real intervals, Commun. Appl. Anal., 13 (2009), no.4, 477–496
Becker, M.: Global approximation theorem for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27 (1978), no. 1, 127–142
Bony, J.M., Courrège, P., Priouret, P.: Semi-groupes de Feller sur une variété a borde compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier, Grenoble, 18 (1968), 369–521
Brezis, H., Rosenkrantz, W., Singer, B.: On a degenerate elliptic-parabolic equation occurring in theory of probability, Comm. Pure Appl. Math., 24 (1971), 395–416
Carbone, I.: Shape preserving properties of some positive linear operators on unbounded intervals, J. Approx. Theory, 93 (1998), 140–156
Carbone, I.: Trotter’s type theorems for generalized Szász-Mirakjan operators in polynomial weighted functions spaces, Indian J. Math., 44 (2002), 1–20
Clément, Ph., Timmermans, C.A.: On C 0-semigroups generated by differential operators satisfying Ventcel’s boundary conditions, Indag. Math., 89 (1986), 379–387
Timmermans, C.A.: On C 0-semigroups in a space of bounded continuous functions in the case of entrance or naturally boundary points. (Approximation and Optimization, J.A. Gómez Fernández et al. (eds.), Lecture Notes in Math. 1354) Berlin: Springer (1988), 209–216
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. (Graduate Texts in Mathematics 194 Berlin: Springer-Verlag (2000)
Feller, W.: The parabolic differential equations and the associated semi-groups of transformations, Ann. Math., 55 (1952), 468–519
Sato, R., Ueno, T.: Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto University, 4 (1965), 529–605
Taira, K.: Diffusion Processes and Partial Differential Equations. San Diego, CA: Academic Press (1988)
Trotter, H.: Approximation of semigroups of operators, Pacific J.Math., 8 (1958), 887–919
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Altomare, F., Milella, S. Degenerate differential equations and modified Szász-Mirakjan operators. Rend. Circ. Mat. Palermo 59, 227–250 (2010). https://doi.org/10.1007/s12215-010-0017-z
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DOI: https://doi.org/10.1007/s12215-010-0017-z
Keywords
- Positive semigroup
- Parabolic equation
- Weighted continuous function space
- Approximation by positive operators