×

zbMATH — the first resource for mathematics

On a functional equation associated with \((a, k)\)-regularized resolvent families. (English) Zbl 1250.39012
Summary: Let \(a \in L^1_{\text{loc}}(\mathbb R_+)\) and \(k \in C(\mathbb R_+)\) be given. We study the functional equation \(R(s)(a \ast R)(t) - (a \ast R)(s)R(t) = k(s)(a \ast R)(t) - k(t)(a \ast R)(s)\), for bounded operator valued functions \(R(t)\) defined on the positive real line \(\mathbb R_+\). We show that, under some natural assumptions on \(a(\cdot)\) and \(k(\cdot)\), every solution of the above mentioned functional equation gives rise to a commutative \((a, k)\)-resolvent family \(R(t)\) generated by \(Ax = \lim_{t \rightarrow 0^+}(R(t)x - k(t)x/(a \ast k)(t))\) defined on the domain \(D(A) : = \{x \in X \lim_{t \rightarrow 0^+}(R(t)x - k(t)x/(a \ast k)(t))\) exists in \(X\}\) and, conversely, that each \((a, k)\)-resolvent family \(R(t)\) satisfies the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems.

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Aczél, “On history, applications and theory of functional equations,” in Functional Equations: History, Applications and Theory, vol. 312 of Mathematics and its Applications, pp. 3-12, Reidel, Dordrecht, The Netherlands, 1984. · Zbl 0583.39004 · doi:10.1007/978-94-009-6320-7_1
[2] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, vol. 31 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1957. · Zbl 0078.10004 · www.ams.org
[3] M. Sova, “Cosine operator functions,” Rozprawy Mathematyczne, vol. 49, pp. 1-14, 1966. · Zbl 0156.15404
[4] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1993. · Zbl 0793.45014 · doi:10.1007/978-3-0348-8570-6
[5] J. Peng and K. Li, “A novel characteristic of solution operator for the fractional abstract Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 786-796, 2012. · Zbl 1231.47039 · doi:10.1016/j.jmaa.2011.07.009
[6] J. Peng and K. Li, “A note on property of the Mittag-Leffler function,” Journal of Mathematical Analysis and Applications, vol. 370, no. 2, pp. 635-638, 2010. · Zbl 1194.30002 · doi:10.1016/j.jmaa.2010.04.031
[7] C. Chen and M. Li, “On fractional resolvent operator functions,” Semigroup Forum, vol. 80, no. 1, pp. 121-142, 2010. · Zbl 1185.47040 · doi:10.1007/s00233-009-9184-7
[8] C. Lizama, “Regularized solutions for abstract Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 243, no. 2, pp. 278-292, 2000. · Zbl 0952.45005 · doi:10.1006/jmaa.1999.6668
[9] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, vol. 96 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 2001. · Zbl 0978.34001
[10] S.-Y. Shaw and J.-C. Chen, “Asymptotic behavior of (a,k)-regularized resolvent families at zero,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 531-542, 2006. · Zbl 1106.45004
[11] S.-Y. Shaw and H. Liu, “Continuity of restrictions of (a,k)-regularized resolvent families to invariant subspaces,” Taiwanese Journal of Mathematics, vol. 13, no. 2, pp. 535-544, 2009. · Zbl 1191.47057
[12] M. Kostić, “(a,k)-regularized C-resolvent families: regularity and local properties,” Abstract and Applied Analysis, vol. 2009, Article ID 858242, 27 pages, 2009. · Zbl 1200.47059 · doi:10.1155/2009/858242
[13] C. Lizama and P. J. Miana, “A Landau-Kolmogorov inequality for generators of families of bounded operators,” Journal of Mathematical Analysis and Applications, vol. 371, no. 2, pp. 614-623, 2010. · Zbl 1205.47018 · doi:10.1016/j.jmaa.2010.05.065
[14] C. Lizama and H. Prado, “Rates of approximation and ergodic limits of regularized operator families,” Journal of Approximation Theory, vol. 122, no. 1, pp. 42-61, 2003. · Zbl 1032.47024 · doi:10.1016/S0021-9045(03)00040-6
[15] C. Lizama and H. Prado, “On duality and spectral properties of (a,k)-regularized resolvents,” Proceedings of the Royal Society of Edinburgh, Section A, vol. 139, no. 3, pp. 505-517, 2009. · Zbl 1206.47037 · doi:10.1017/S0308210507000364
[16] C. Lizama and J. Sánchez, “On perturbation of K-regularized resolvent families,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 217-227, 2003. · Zbl 1051.45009
[17] C. Lizama and G. M. N’Guérékata, “Mild solutions for abstract fractional differential equations,” Applicable Analysis. In press. · Zbl 1276.34004 · doi:10.1080/00036811.2012.698271
[18] H. Kellerman and M. Hieber, “Integrated semigroups,” Journal of Functional Analysis, vol. 84, no. 1, pp. 160-180, 1989. · Zbl 0689.47014 · doi:10.1016/0022-1236(89)90116-X
[19] W. Arendt and H. Kellermann, “Integrated solutions of Volterra integrodifferential equations and applications,” in Volterra Integrodifferential Equations in Banach Spaces and Applications (Trento, 1987), vol. 190 of Pitman Research Notes in Mathematics Series, pp. 21-51, Longman, Harlow, UK, 1989. · Zbl 0675.45017
[20] C. Lizama, “On approximation and representation of K-regularized resolvent families,” Integral Equations and Operator Theory, vol. 41, no. 2, pp. 223-229, 2001. · Zbl 1011.45006 · doi:10.1007/BF01295306
[21] Y.-C. Li and S.-Y. Shaw, “Mean ergodicity and mean stability of regularized solution families,” Mediterranean Journal of Mathematics, vol. 1, no. 2, pp. 175-193, 2004. · Zbl 1164.47306 · doi:10.1007/s00009-004-0010-x
[22] B. Nagy, “On cosine operator functions in Banach spaces,” Acta Universitatis Szegediensis, vol. 36, pp. 281-289, 1974. · Zbl 0273.47008
[23] M. Li and Q. Zheng, “On spectral inclusions and approximations of \alpha -times resolvent families,” Semigroup Forum, vol. 69, no. 3, pp. 356-368, 2004. · Zbl 1096.47516 · doi:10.1007/s00233-004-0128-y
[24] C. Lizama and J. De Souza, “The complex inversion formula in UMD spaces for families of bounded operators,” Applicable Analysis, vol. 91, no. 15, pp. 937-946, 2012. · Zbl 1252.34065 · doi:10.1080/00036811.2011.556625
[25] V. Keyantuo, C. Lizama, and P. J. Miana, “Algebra homomorphisms defined via convoluted semigroups and cosine functions,” Journal of Functional Analysis, vol. 257, no. 11, pp. 3454-3487, 2009. · Zbl 1190.47045 · doi:10.1016/j.jfa.2009.07.017
[26] D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3692-3705, 2008. · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004
[27] E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, 2001. · Zbl 0989.34002
[28] M. Kostić and S. Pilipović, “Global convoluted semigroups,” Mathematische Nachrichten, vol. 280, no. 15, pp. 1727-1743, 2007. · Zbl 1147.47028 · doi:10.1002/mana.200510574
[29] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra Integral and Functional Equations, vol. 34 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York, NY, USA, 1990. · Zbl 0695.45002 · doi:10.1017/CBO9780511662805
[30] M. Kostic, “(a, k)-regularized (C1;C2)-existence and uniqueness families,” preprint. · Zbl 1340.47093
[31] B. Kaltenbacher, I. Lasiecka, and R. Marchand, “Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equations arising in high intensity ultrasound,” Control and Cybernetics. In press. · Zbl 1318.35080
[32] G. C. Gorain, “Stabilization for the vibrations modeled by the ’standard linear model’ of viscoelasticity,” Proceedings of the Indian Academy of Sciences, vol. 120, no. 4, pp. 495-506, 2010. · Zbl 1202.35026 · doi:10.1007/s12044-010-0038-8
[33] T. Diagana, “Existence of almost periodic solutions to some third-order nonautonomous differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 65, pp. 1-15, 2011. · Zbl 1205.35015
[34] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of Mathematical Studies, North-Holland, Amsterdam, The Netherlands, 1985. · Zbl 0564.34063
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.