Evolution system approximations of solutions to closed linear operator equations

Author:
Seaton D. Purdom

Journal:
Trans. Amer. Math. Soc. **215** (1976), 63-79

MSC:
Primary 47B44; Secondary 47A50

MathSciNet review:
0420331

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Abstract | References | Similar Articles | Additional Information

Abstract: With *S* a linearly ordered set with the least upper bound property, with *g* a nonincreasing real-valued function on *S*, and with *A* a densely defined dissipative linear operator, an evolution system *M* is developed to solve the modified Stieljes integral equation . An affine version of this equation is also considered. Under the hypothesis that the evolution system associated with the linear equation is strongly (resp. weakly) asymptotically convergent, an evolution system is used to approximate strongly (resp. weakly) solutions to the closed operator equation .

**[1]**F. E. Browder and W. V. Petryshyn,*The solution by iteration of nonlinear functional equations in Banach spaces*, Bull. Amer. Math. Soc.**72**(1966), 571–575. MR**0190745**, 10.1090/S0002-9904-1966-11544-6**[2]**Frank Gilfeather,*Asymptotic convergence of operators in Hilbert space*, Proc. Amer. Math. Soc.**22**(1969), 69–76. MR**0247508**, 10.1090/S0002-9939-1969-0247508-2**[3]**J. V. Herod,*A pairing of a class of evolution systems with a class of generators.*, Trans. Amer. Math. Soc.**157**(1971), 247–260. MR**0281059**, 10.1090/S0002-9947-1971-0281059-8**[4]**-,*Generators for evolution systems with quasicontinuous trajectories*, Pacific J. Math. (to appear).**[5]**J. S. MacNerney,*Integral equations and semigroups*, Illinois J. Math.**7**(1963), 148–173. MR**0144179****[6]**J. S. MacNerney,*A linear initial-value problem*, Bull. Amer. Math. Soc.**69**(1963), 314–329. MR**0146613**, 10.1090/S0002-9904-1963-10905-2**[7]**J. S. MacNerney,*A nonlinear integral operation*, Illinois J. Math.**8**(1964), 621–638. MR**0167815****[8]**Robert H. Martin Jr.,*Product integral approximations of solutions to linear operator equations*, Proc. Amer. Math. Soc.**41**(1973), 506–512. MR**0380463**, 10.1090/S0002-9939-1973-0380463-4**[9]**Kôsaku Yosida,*Functional analysis*, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York Inc., New York, 1968. MR**0239384**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1976-0420331-X

Keywords:
Dissipative,
evolution system,
product integral,
asymptotically convergent

Article copyright:
© Copyright 1976
American Mathematical Society