We show that Halász's theorem holds for Beurling numbers under the following two mild hypotheses on the generalized number system: existence of a positive density for the generalized integers and a Chebyshev upper bound for the generalized primes.∞ 1 − x −s dG(x) and we use the notation s = σ + it for complex variables. Zhang's version of Halász's theorem reads as follows. His result generalizes [5, Theorem 3.1], where the set of hypotheses (1.1), (1.2), and (1.3) were actually introduced.Theorem 1.1 (Zhang [17]). Suppose that the generalized number system satisfies a Chebyshev upper estimate
We study the uniqueness of some inverse source problems arising in thermoelastic models of type-III. We suppose that the source terms can be decomposed as a product of a time dependent and a space dependent function, i.e. g ( t ) 𝐟 ( 𝐱 ) {g(t)\mathbf{f}(\mathbf{x})} for the load source and g ( t ) f ( 𝐱 ) {g(t)f(\mathbf{x})} for the heat source. In the first inverse source problem, the source 𝐟 ( 𝐱 ) {\mathbf{f}(\mathbf{x})} has to be determined from the final in time measurement of the displacement 𝐮 ( 𝐱 , T ) {\mathbf{u}(\mathbf{x},T)} , or from the time-average measurement ∫ 0 T 𝐮 ( 𝐱 , t ) d t {\int_{0}^{T}\mathbf{u}(\mathbf{x},t)\,\mathrm{d}t} . In the second inverse source problem, the source f ( 𝐱 ) {f(\mathbf{x})} has to be determined from the time-average measurement of the temperature ∫ 0 T θ ( 𝐱 , t ) d t {\int_{0}^{T}\theta(\mathbf{x},t)\,\mathrm{d}t} . We show the uniqueness of a solution to these problems under suitable assumptions on the function g ( t ) {g(t)} . Moreover, we provide some examples showing the necessity of these assumptions. Finally, we conclude the article by studying two combined problems of determining both sources.
The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the form h(t)f(x) with a known function h(t). The unknown space dependent source f(x) is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions on h(t) (monotonically increasing character of h(t) was removed) in a case of dominant parabolic behavior. The proof technique was based on spectral analysis. Section 4 shows that an analogy of Theorem 2 for dominant hyperbolic behavior (fractional Cattaneo–Vernotte equation) is not possible.
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