P-Primitives and Explicit Solutions of Polynomial Differential Equations in LΦ(T)

Let T = [ − π , π ] , Φ be an arbitrary Young function and P ( x ) be a polynomial. In this paper we introduce a notion called P -primitive of a function in S ′ ( ℝ ) and apply it to examine the existence and uniqueness of solutions in Orlicz spaces L Φ ( T ) of the equation P ( D ) f = ψ ∈ L Φ ( T...

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Bibliographic Details
Published in:Vietnam journal of mathematics 2023, Vol.51 (2), p.245-261
Main Authors: Bang, Ha Huy, Huy, Vu Nhat
Format: Article
Language:English
Subjects:
Differential equations
Fourier transforms
Functions (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Operators (mathematics)
Original Article
Polynomials
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Summary:Let T = [ − π , π ] , Φ be an arbitrary Young function and P ( x ) be a polynomial. In this paper we introduce a notion called P -primitive of a function in S ′ ( ℝ ) and apply it to examine the existence and uniqueness of solutions in Orlicz spaces L Φ ( T ) of the equation P ( D ) f = ψ ∈ L Φ ( T ) . The explicit solutions of the equation are given. In particular, we show that the condition P ( x )≠ 0 for all x ∈ supp ψ ̂ is a criterion for existence of a P -primitive in L Φ ( T ) of f . We also describe behavior of the sequences of norm of P -primitives of functions in L Φ ( T ) based on its spectrum (the support of its Fourier transform). Moreover, the behavior of the sequences of norm of functions generated by polynomial integral operators in L Φ ( T ) is also given.
ISSN:2305-221X
2305-2228
DOI:10.1007/s10013-021-00500-z