Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A method of lines for a nonlinear abstract functional evolution equation


Authors: A. G. Kartsatos and M. E. Parrott
Journal: Trans. Amer. Math. Soc. 286 (1984), 73-89
MSC: Primary 34K30
DOI: https://doi.org/10.1090/S0002-9947-1984-0756032-2
MathSciNet review: 756032
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a real Banach space with $ {X^\ast}$ uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E)

$\displaystyle u\prime(t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T].$

The operators $ A(t):D \subset X \to X$ are $ m$-accretive and $ G(t,\phi )$ is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators $ \hat A(t)$ and the corresponding use of nonlinear evolution operator theory.


References [Enhancements On Off] (What's this?)

  • [1] H. Banks and J. Burns, Hereditary control problems: numerical methods based on averaging approximations, SIAM J. Control Optim. 16 (1978), 169-208. MR 0483428 (58:3430)
  • [2] H. Banks and F. Kappel, Spline approximations for functional differential equations, J. Differential Equations 34 (1974), 496-522. MR 555324 (81c:65031)
  • [3] M. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57-94. MR 0300166 (45:9214)
  • [4] C. Cryer and L. Tavernini, The numerical solution of Volterra differential equations by Euler's method, SIAM J. Numer. Anal. 9 (1972), 105-129. MR 0312756 (47:1311)
  • [5] J. Dyson and R. Villella Bressan, Functional differential equations and nonlinear evolution operators, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975/76), 223-234. MR 0442402 (56:784)
  • [6] -, Semigroups of translations associated with functional and functional differential equations, Proc. Roy. Soc. Edinburgh Sect. A 82 (1979), 171-188. MR 532900 (80i:34113)
  • [7] L. Evans, Nonlinear evolution equations in an arbitrary Banach space, Israel J. Math. 26 (1977), 1-42. MR 0440431 (55:13306)
  • [8] W. Fitzgibbon, Approximations of nonlinear evolution equations, J. Math. Soc. Japan 25 (1973), 211-221. MR 0326515 (48:4859)
  • [9] F. Kappel and W. Schappacher, Non-linear functional differential equations and abstract integral equations, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 71-91. MR 549872 (80m:34067)
  • [10] A. G. Kartsatos and M. E. Parrott, Convergence of the Kato approximants for evolution equations involving functional perturbations, J. Differential Equations 47 (1983), 358-377. MR 692836 (84f:34088)
  • [11] A. G. Kartsatos and W. Zigler, Rothe's method and weak solutions of perturbed evolution equations in reflexive Banach spaces, Math. Ann. 219 (1976), 159-166. MR 0390856 (52:11679)
  • [12] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520. MR 0226230 (37:1820)
  • [13] J. Mermin, Accretive operators and nonlinear semigroups, Ph.D. Thesis, Univ. of California, Berkeley, 1968.
  • [14] J. Nečas, Application of Rothe's method to abstract parabolic equations, Czechoslovak Math. J. 24 (1974), 496-500. MR 0348571 (50:1069)
  • [15] M. E. Parrott, Representation and approximation of generalized solutions of a nonlinear functional differential equation, Nonlinear Anal. 6 (1982), 307-318. MR 654808 (83d:34127)
  • [16] E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann. 102 (1930), 650-670. MR 1512599
  • [17] R. Thompson, On some functional differential equations: Existence of solutions and difference approximations, SIAM J. Numer. Anal. 5 (1968), 475-487. MR 0239783 (39:1140)
  • [18] G. Webb, Asymptotic stability for abstract nonlinear functional differential equations, Proc. Amer. Math. Soc. 54 (1976), 225-230. MR 0402237 (53:6058)
  • [19] -, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl. 46 (1974), 1-12. MR 0348224 (50:722)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34K30

Retrieve articles in all journals with MSC: 34K30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0756032-2
Keywords: Functional evolution equation, uniformly convex dual, method of lines, $ m$-accretive operator
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society