# zbMATH — the first resource for mathematics

Perturbations of existence families for abstract Cauchy problems. (English) Zbl 1032.47025
The authors consider the multiplicative and additive Desh-Schappacher type perturbations for the C-existence family of operators for an arbitrary order abstract Cauchy problem $u^{(n)}(t) = A u(t) (t\geq 0); u^{(j)}(0) = x_j (0\leq j\leq n-1).\tag{1}$ Under a C-existence family for (1) is understood the strongly continuous family of operators $$\{S(t)\}_{t\geq 0} \subset {\mathbf L}(X)$$ such that for all $$x$$ from a Banach space $$X$$ and $$t\geq 0$$ $S(\cdot)x \in C^{n-1}({\mathbb R}_{+}, X),$ $A \int_{0}^{t} S(s)x ds \in C({\mathbb R}_{+}, X),$ $S(t)x = \frac{t^{n-1}}{(n-1)!} Cx + A \int_{0}^{t} \frac{(t-s)^{n-1}}{(n-1)!} S(s)x ds,$ where $$C\in {\mathbf L}(X)$$ is an injective operator. The uniqueness of the solution to the correspondingly perturbed problem (1) is obtained. Illustrative examples of interest for applications are presented.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces
##### Keywords:
abstract Cauchy problem; C-existence family; perturbation
Full Text:
##### References:
 [1] Proceedings of the Second World Congress of Nonlinear Analysts. Part 6, Elsevier Ltd, Oxford, 1997. Held in Athens, July 10 – 17, 1996; Nonlinear Anal. 30 (1997), no. 6. [2] G. Da Prato, Semigruppi di crescenza \?, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 753 – 782 (Italian). · Zbl 0198.16801 [3] G. Da Prato, Semigruppi regolarizzabili, Ricerche Mat. 15 (1966), 223 – 248 (Italian). · Zbl 0195.41001 [4] E. B. Davies and M. M. H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc. (3) 55 (1987), no. 1, 181 – 208. · Zbl 0651.47026 · doi:10.1112/plms/s3-55.1.181 · doi.org [5] Ralph deLaubenfels, Existence and uniqueness families for the abstract Cauchy problem, J. London Math. Soc. (2) 44 (1991), no. 2, 310 – 338. · Zbl 0766.47011 · doi:10.1112/jlms/s2-44.2.310 · doi.org [6] Ralph deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics, vol. 1570, Springer-Verlag, Berlin, 1994. · Zbl 0811.47034 [7] Ralph deLaubenfels and Fuyuan Yao, Regularized semigroups of bounded semivariation, Semigroup Forum 53 (1996), no. 3, 369 – 383. · Zbl 0859.47028 · doi:10.1007/BF02574151 · doi.org [8] Wolfgang Desch and Wilhelm Schappacher, Some generation results for perturbed semigroups, Semigroup theory and applications (Trieste, 1987) Lecture Notes in Pure and Appl. Math., vol. 116, Dekker, New York, 1989, pp. 125 – 152. · Zbl 0701.47021 [9] Odo Diekmann, Mats Gyllenberg, and Horst R. Thieme, Perturbing semigroups by solving Stieltjes renewal equations, Differential Integral Equations 6 (1993), no. 1, 155 – 181. · Zbl 0770.47012 [10] Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. · Zbl 0952.47036 [11] Jerome A. Goldstein, Ralph deLaubenfels, and James T. Sandefur Jr., Regularized semigroups, iterated Cauchy problems and equipartition of energy, Monatsh. Math. 115 (1993), no. 1-2, 47 – 66. · Zbl 0784.34046 · doi:10.1007/BF01311210 · doi.org [12] M. Hieber, A. Holderrieth, and F. Neubrander, Regularized semigroups and systems of linear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 3, 363 – 379. · Zbl 0789.35075 [13] M. Jung, Multiplicative perturbations in semigroup theory with the (\?)-condition, Semigroup Forum 52 (1996), no. 2, 197 – 211. · Zbl 0841.47023 · doi:10.1007/BF02574096 · doi.org [14] Hermann Kellerman and Matthias Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), no. 1, 160 – 180. · Zbl 0689.47014 · doi:10.1016/0022-1236(89)90116-X · doi.org [15] J.-L. Lions, Les semi groupes distributions, Portugal. Math. 19 (1960), 141 – 164 (French). · Zbl 0103.09001 [16] Chung-Cheng Kuo and Sen-Yen Shaw, \?-cosine functions and the abstract Cauchy problem. I, II, J. Math. Anal. Appl. 210 (1997), no. 2, 632 – 646, 647 – 666. · Zbl 0881.34071 · doi:10.1006/jmaa.1997.5420 · doi.org [17] Isao Miyadera, \?-semigroups and semigroups of linear operators, Differential equations (Plovdiv, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 133 – 143. · Zbl 0817.47052 [18] Hirokazu Oka, Linear Volterra equations and integrated solution families, Semigroup Forum 53 (1996), no. 3, 278 – 297. · Zbl 0862.45017 · doi:10.1007/BF02574144 · doi.org [19] S. Piskarëv and S.-Y. Shaw, Multiplicative perturbations of \?$$_{0}$$-semigroups and some applications to step responses and cumulative outputs, J. Funct. Anal. 128 (1995), no. 2, 315 – 340. · Zbl 0823.47036 · doi:10.1006/jfan.1995.1034 · doi.org [20] S. Piskarëv and S.-Y. Shaw, Perturbation and comparison of cosine operator functions, Semigroup Forum 51 (1995), no. 2, 225 – 246. · Zbl 0828.47037 · doi:10.1007/BF02573631 · doi.org [21] N. Tanaka, $$C$$-semigroups of linear operators in Banach spaces - a generalization of the Hille-Yosida theorem, thesis, Waseda University, 1992. [22] Ti-Jun Xiao and Jin Liang, The Cauchy problem for higher-order abstract differential equations, Lecture Notes in Mathematics, vol. 1701, Springer-Verlag, Berlin, 1998. · Zbl 0915.34002 [23] T. J. Xiao and J. Liang, Higher order abstract Cauchy problems and their existence, uniqueness families, J. London Math. Soc., to appear. · Zbl 1073.34072
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.