Literature cited
G. A. Baker, V. A. Dougalis, and O. Karakashian, “On multistep-Galerkin discretizations of semilinear hyperbolic and parabolic equations,” Nonlinear Anal.,4, No. 3, 579–597 (1980).
W. Dickmeis, “On Lax-type theorems with orders for semilinear initial value problems,” Abh. Math. Sem. Univ. Hamburg,51, 13–22 (1981).
V. Thomée and L. Wahlbin, “On Galerkin methods in semilinear parabolic problems,” SIAM J. Numer. Anal.,12, No. 3, 378–389 (1975).
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math, No. 1054, Springer, Berlin (1984).
N. M. Spirina, “On the convergence of the Tonelli method in a Banach space,” Manuscript deposited at VINITI, Voronezh, No. 2017-B86 (1986).
N. Pavel, “Invariant sets for a class of semilinear equations of evolution,” Nonlinear Anal.,1, No. 2, 187–196 (1977).
J. H. Lightbourne, III and R. H. Martin, Jr., “Relatively continuous nonlinear perturbations of analytic semigroups,” Nonlinear Anal.,1, No. 3, 277–292 (1977).
H. C. Sager, “An evolution equation perturbed by an unbounded semilinear operator and a boundary condition,” Nonlinear Anal.,8, No. 10, 1151–1166 (1984).
A. Schiaffino, “Compactness methods for quasilinear evolution-equations,” Rend. Sem. Mat. Univ. Padova,55, 151–166 (1976).
R. Redlinger, “Compactness results for time-dependent parabolic systems,” J. Differential Equations,64, No. 2, 133–153 (1986).
J.-L. Lions, Quelques methodes de Resolution des Problemes aux Limites Nonlineaires, Dunod, Gauthier-Villars, Paris (1969).
D. Henry, Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Math., No. 840), Springer, Berlin (1981).
M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden (1976).
H. O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, Mass. (1983).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).
J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York (1985).
J. Pruss, “Periodic solutions of semilinear evolution equations,” Nonlinear Anal.,3, No. 5, 601–612 (1979).
M. G. Crandall, “A remark on semilinear perturbations of abstract parabolic equations,” Nonlinear Anal.,9, No. 12, 1331–1335 (1985).
M. Miklavčic, “Stability for semilinear parabolic equations with noninvertible linear operator,” Pac. J. Math.,118, No. 1, 199–214 (1985).
J. T. Sandefur, Jr., “Existence and uniqueness of solutions of second order nonlinear differential equations,” SIAM J. Math. Anal.,14, No. 3, 477–487 (1983).
G. Da Prato and P. Grisvard, “Maximal regularity for evolution equations by interpolation and extrapolation,” J. Funct. Anal.,58, No. 2, 107–124 (1984).
G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner, Leipzig (1976).
S. Piskarev, “Approximation of holomorphic semigroups,” Uch. Zap. Tartusk. Univ., No. 492, 3–14 (1979).
Yu. A. Smirnitskii and P. E. Sobolevskii, A pointwise estimate of the Green function of the resolvent of a difference elliptic operator with variable coefficients in Rn. Manuscript deposited at VINITI, Voronezh, No. 1519-82 (1982).
Kh. A. Alibekov and P. E. Sobolevskii, “On the stability and the convergence of difference schemes of higher order of approximation for parabolic equations. II,” Manuscript deposited at VINITI, Voronezh, No. 3645-76 (1976).
A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).
M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin (1984).
S. Piskarev, “Solution of a nonhomogeneous abstract linear hyperbolic eqations,” Uch. Zap Tartusk. Univ., No. 500, 24–32 (1979).
S. I. Piskarev, “Almost-periodic solutions of second-order differential equations,” Sib. Mat. Zh.,25, No. 3, 137–147 (1984).
S. I. Piskarev, “On the compact convergence of the resolvent and of semigroups,” in: Materials of the Conference Methods of Algebra and Functional Analysis in the Investigation of Families of Operators [in Russian], Tartu (1978), pp. 71–74.
S. I. Piskarev, “The solution of second-order evolution equations under the Krein-Fattorini conditions,” Differents. Uravn.,21, No. 9, 1604–1612 (1985).
S. Piskarev, “Error estimates in the approximation of semigroups of operators by Pade fractions,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 33–38 (1979).
N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Interscience, New York (1963).
J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).
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Translated from Matematicheskie Zametki, Vol. 44, No. 1, pp. 112–123, July, 1988.
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Piskarev, S.I. Convergence of difference schemes for solving nonlinear parabolic equations. Mathematical Notes of the Academy of Sciences of the USSR 44, 549–556 (1988). https://doi.org/10.1007/BF01158123
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DOI: https://doi.org/10.1007/BF01158123