Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a family of bounded operators to rough approximate data that do not necessarily lie within the domain of unbounded operator. In this paper we shall be concerned with the stable method of computing values of unbounded operators having perturbations and the stability is established for this method.
References
[1]
Tikhonov, A.N. and Arsenin, V.Y. (1978) Solution of Ill-Posed Problems. John Wiley & Sons, Hoboken.
Groetsch, C.W. and Scherzer, O. (1993) The Optimal Order of Convergence for Stable Evaluation of Differential Operators. Electronic Journal of Differential Equations, 3, 1-12.
Van Kinh, N. (2001) On the Stable Method of Computing Values of Unbounded Operators. Proceedings of Science of Quinhon University of Educations, 14, 27-38. http://www.ictp.trieste.it/pub_off
[6]
Van Kinh, N. (2014) On the Stable Method of Computing Values of Unbounded Operators. Journal Science of Ho Chi Minh City University of Food Industry, 2, 21-30.
[7]
Van Kinh, N., Chuong, N.M. and Gorenflo, R. (1996) Regularization Method for Nonlinear Variational Inequalities. Proceedings of the First National Workshop “Optimization and Control”, Freie Universitat, Berlin.
[8]
Vaiberg, I.M. (1972) The Variational Method and the Monotonic-Operator Method. Nauka, Moscow.
[9]
Zeidler, E. (1989) Nonlinear Functional Analysis and Its Applications II/B (Nonlinear Monotone Operators). Springer-Verlag, Berlin.
[10]
Al’ber, Y.I. and Ryazantseva, P. (1979) Solution of Nonlinear Problems Involving Monotonic Discontinuous Mapping. Differentsial’nyje Uravnenija, 15, 31-342.
[11]
Riesz, S. and Sz-Nagy, B. (1955) Functional Analysis. Ungar, New York.
[12]
Browder, F.E. (1966) On the Unification of the Calculus of Variations and the Theory Monotone Nonlinear in Banach Spaces. Proceedings of the National Academy of Sciences of the United States of America, 56, 419-425. https://doi.org/10.1073/pnas.56.2.419
[13]
Browder, F.E. (1966) Existence and Approximation of Solution of Nonlinear Inequalities. Proceedings of the National Academy of Sciences of the United States of America, 56, 1080-1086. https://doi.org/10.1073/pnas.56.4.1080
Liskovets, O.A. (1983) Regularizaion Problem with Monotone Discontinuous Perturbations of Operators. Proceedings of the USSR Academy of Sciences, 272, 30-34.
[16]
Liskovets, O.A. (1983) Solution of the First Kind Operator Equations with Non- Monotone Perturbations. Proceedings of the USSR Academy of Sciences, 272, 101-104.
[17]
Abramov, A. and Gaipova, A.N. (1972) On the Solvability of Certain Equations Containing Monotonic Discontinuous Transformations. USSR Computational Mathematics and Mathematical Physics, 12, 320-324. https://doi.org/10.1016/0041-5553(72)90191-7
[18]
Lions, J.L. (1972) Methods of Solution of Nonlinear Boundary-Values Problems. Mir, Moscow.
[19]
Rockafellar, R.T. (1970) On the Maximality of Sum of Nonlinear Monotone Operators. American Mathematical Society, 149, 75-88. https://doi.org/10.1090/S0002-9947-1970-0282272-5