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Product integrals of continuous resolvents: existence and nonexistence. (English) Zbl 0537.34060
Of concern is the initial value problem (*) \(dy(t)/dt=A(t)y(t)\), 0\(\leq t\leq T\), \(y(0)=y_ 0\). Assume, for simplicity, that each A(t) is densely defined and m-dissipative on a Banach space X. If A is independent of t, then existence and uniqueness holds for (*), whether A is linear or not. When A depends on t nicely, then a solution can be constructed by a product integral formula, namely \[ y(t)=\lim_{n\to \infty}J_{t/n}(((n-1)/n)t)...J_{t/n}((2/n)t)J_{t/n}((1/n)t)y_ 0 \] where \(J_{\lambda}(s)=(I-\lambda A(s))^{-1}\). The usual smoothness condition in t is that for \(y_ 0\in X,\) \[ (+)\quad \| J_{\lambda}(t)y_ 0-J_{\lambda}(s)y_ 0\| \leq \lambda \| f(t)-f(s)\| L(\| y_ 0\|) \] where L is increasing and \(f:[0,T]\to X\) is continuous. Condition \((+)\) can be weakened slightly, but the author shows that replacing \((+)\) by the condition that \((\lambda,t,y_ 0)\to J_{\lambda}(t)y_ 0\) is continuous on \((0,\infty)\times [0,T]\times X\) is not adequate to make the product integral converge or (*) have a solution, even when each A(t) is a bounded linear operator. The construction of the counterexample is quite clever.
Reviewer: J.A.Goldstein

MSC:
34G10 Linear differential equations in abstract spaces
47D03 Groups and semigroups of linear operators
47B44 Linear accretive operators, dissipative operators, etc.
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[1] M. G. Crandall and A. Pazy,Nonlinear evolution equations in Banach spaces, Isr. J. Math.11 (1972), 57–94. · Zbl 0249.34049 · doi:10.1007/BF02761448
[2] J. Dieudonné,Deux exemples singuliers d’équations différentielles, Acta Sci. Math. (Szeged)12 (1950), B38–40. · Zbl 0037.06002
[3] J. D. Dollard and C. N. Friedman,On strong product integration, J. Funct. Anal.28 (1978), 309–354. · Zbl 0404.34047 · doi:10.1016/0022-1236(78)90091-5
[4] J. D. Dollard and C. N. Friedman,Product Integration, Addison-Wesley, Massachusetts, 1979. · Zbl 0454.28002
[5] J. Dyson and R. Villella Bressan,Functional differential equations and nonlinear evolution operators, Edinburgh J. Math.75A, 20 (1975/76), 223–234. · Zbl 0361.34055
[6] L. C. Evans,Nonlinear evolution equations in an arbitrary Banach space, Technical Report No. 1568, Mathematics Research Center. · Zbl 0349.34043
[7] L. C. Evans,Nonlinear evolution equations in an arbitrary Banach space, Isr. J. Math.26 (1977), 1–42. · Zbl 0349.34043 · doi:10.1007/BF03007654
[8] L. A. Lusternik and V. J. Sobolev,Elements of Functional Analysis, Hindustan, Delhi, 1961. · Zbl 0293.46001
[9] R. H. Martin,Differential equations on closed subsets of a Banach space, Trans. Am. Math. Soc.179 (1973), 399–414. · Zbl 0293.34092 · doi:10.1090/S0002-9947-1973-0318991-4
[10] R. H. Martin,Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976. · Zbl 0333.47023
[11] M. Pierre,Enveloppe d’une famille de semi-groupes dans un espace de Banach, C.R. Acad. Sci. Paris284 (1977), 401–404. · Zbl 0355.47040
[12] E. Schechter,Existence and limits of Carathéodory-Martin evolutions, J. Nonlin. Anal. Theory Methods Appl.5 (1981), 897–930. · Zbl 0465.34036 · doi:10.1016/0362-546X(81)90093-6
[13] E. Schechter,Interpolation of nonlinear partial differential operators and generation of differentiable evolutions, J. Differ. Equ.46 (1982), 78–102. · Zbl 0489.34069 · doi:10.1016/0022-0396(82)90111-5
[14] G. F. Webb and M. Badii,Nonlinear nonautonomous functional differential equations in L p spaces, J. Nonlin. Anal. Theory Methods Appl.5 (1981), 203–223. · Zbl 0451.34062 · doi:10.1016/0362-546X(81)90045-6
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