# zbMATH — the first resource for mathematics

Perturbations of an abstract Euler-Poisson-Darboux equation. (English. Russian original) Zbl 0898.34057
Math. Notes 60, No. 3, 269-273 (1996); translation from Mat. Zametki 60, No. 3, 363-369 (1996).
Summary: The stability of the uniform correctness of the Cauchy problem $u''(t)+ {k\over t} u'(t)= \mathbb{A} u(t), \quad t>0,\;u(0) =u_0,\;u'(0)=0,$ for $$k>0$$ is studied with respect to perturbations of the operator $$\mathbb{A}$$.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 35Q05 Euler-Poisson-Darboux equations
##### Keywords:
stability; uniform correctness; Cauchy problem; perturbations
Full Text:
##### References:
 [1] V. V. Vasil’ev, S. G. Krein, and S. I. Piskarev, ”Operator semigroups, cosine operator functions, and linear differential equations,” in:Itogi Nauki i Tekhniki. Matem. Analiz [in Russian], VINITI, Moscow (1990), pp. 87–102. [2] A. V. Glushak, V. I. Kononenko, and S. D. Shmulevich, ”A singular abstract Cauchy problem,”Izv. Vyssh. Uchebn. Zaved. Mat., [Soviet Math. J. (Iz. VUZ)], No. 6, 55–56 (1986). · Zbl 0613.34047 [3] V. A. Kostin, ”Analytic semigroups and cosine functions,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],307, No. 4, 796–799 (1989). [4] T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, New York-Berlin (1966). · Zbl 0148.12601 [5] A. V. Glushak and S. D. Shmulevich, ”Integral representations of solutions of a singular equation containing a sum of commuting operators,”Differentsial’nye Uravneniya [Differential Equations],28, No. 5, 831–838 (1992). · Zbl 0829.34046 [6] B. M. Levitan, ”Fourier series and integral expansions in Bessel functions,”Uspekhi Mat. Nauk [Russian Math. Surveys],1, No. 2(42), 102–143 (1951). · Zbl 0043.07002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.