Identifications for General Degenerate Problems of Hyperbolic Type in Hilbert Spaces

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Abstract


In a Hilbert space X, we consider the abstract problem M∗ddt(My(t))=Ly(t)+f(t)z,0≤t≤τ,My(0)=My0, where L is a closed linear operator in X and M∈L(X) is not necessarily invertible, z∈X. Given the additional information Φ[My(t)]=g(t) wuth Φ∈X∗, g∈C1([0,τ];C). We are concerned with the determination of the conditions under which we can identify f∈C([0,τ];C) such that y be a strict solution to the abstract problem, i.e., My∈C1([0,τ];X), Ly∈C([0,τ];X). A similar problem is considered for general second order equations in time. Various examples of these general problems are given.

About the authors

A Favini

Universita` di Bologna

Email: angelo.favini@unibo.it

G Marinoschi

Institute of Statistical Mathematics and Applied Mathematics

Email: gabimarinoschi@yahoo.com

H Tanabe

Hirai Sanso

Email: h7tanabe@jttk.zaq.ne.jp

Ya Yakubov

Tel-Aviv University

Email: yakubov@post.tau.ac.il

References

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  2. Favini A., Marinoschi G. Identification for degenerate problems of hyperbolic type// Appl. Anal. - 2012. - 91, № 8. - С. 1511-1527.
  3. Favini A., Marinoschi G. Identification for general degenerate problems of hyperbolic type// Bruno Pini Math. Anal. Semin. Univ. Bologna - 2016. - 7. - С. 175-188.
  4. Favini A., Yagi A. Degenerate differential equations in Banach spaces. - New York: Marcel Dekker, 1999.
  5. Lorenzi A. An introduction to identification problems via functional analysis. - Berlin: De Gruyter, 2001.
  6. Pazy A. Semigroup of linear operators and applications to partial differential equations. - New York: Springer-Verlag, 1983.

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