Generalized function fractional integration operators in some classes of analytic functions
V. Kiryakova (1988)
Matematički Vesnik
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V. Kiryakova (1988)
Matematički Vesnik
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Celso Martínez, Antonia Redondo, Miguel Sanz (2011)
Studia Mathematica
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We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable...
Nassim Guerraiche, Samira Hamani, Johnny Henderson (2016)
Archivum Mathematicum
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We establish sufficient conditions for the existence of solutions of a class of fractional functional differential inclusions involving the Hadamard fractional derivative with order . Both cases of convex and nonconvex valued right hand side are considered.
Dariusz Idczak (2017)
Czechoslovak Mathematical Journal
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We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.
M. Magdziarz, A. Weron (2007)
Studia Mathematica
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We introduce a fractional Langevin equation with α-stable noise and show that its solution is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of via the measure of its codependence r(θ₁,θ₂,t). We prove that is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of...
Nemat Nyamoradi (2013)
Annales Polonici Mathematici
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We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where and are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].
Ali Akbulut, Amil Hasanov (2016)
Open Mathematics
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In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, from [...] Mp,φ1wptoMp,φ2wq for 1 < p < q < ∞. In all cases the conditions...
Vakhtang Kokilashvili, Alexander Meskhi (2012)
Studia Mathematica
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rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from to (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace...
Juan Luis Vázquez (2014)
Journal of the European Mathematical Society
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We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form with suitable and . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov...
Hussein A.H. Salem (2008)
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type , a.e. on (0,1), , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.
Zhinan Xia (2017)
Czechoslovak Mathematical Journal
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In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic -mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic -mild solution of a two-dimensional impulsive...
A. Gatto, Stephen Vági (1999)
Studia Mathematica
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We introduce Sobolev spaces for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in with fractional derivative of order α, , as introduced in [2], in . We show that for small α, coincides with the continuous version of the Triebel-Lizorkin space as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity...
George A. Anastassiou (2016)
Applicationes Mathematicae
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We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative...
Bartłomiej Dyda, Rupert L. Frank (2012)
Studia Mathematica
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We prove a fractional version of the Hardy-Sobolev-Maz’ya inequality for arbitrary domains and norms with p ≥ 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.
Irina Holmes, Robert Rahm, Scott Spencer (2016)
Studia Mathematica
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We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for and α/n + 1/q = 1/p, the norm is equivalent to the norm of b in the weighted BMO space BMO(ν), where . This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...
Matthew Walsh (2007)
Discussiones Mathematicae Graph Theory
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Mynhardt has conjectured that if G is a graph such that γ(G) = γ(πG) for all generalized prisms πG then G is edgeless. The fractional analogue of this conjecture is established and proved by showing that, if G is a graph with edges, then .
Zhan-Dong Mei, Ji-Gen Peng, Jun-Xiong Jia (2014)
Studia Mathematica
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A new characteristic property of the Mittag-Leffler function with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.
Li-Jiang Jia, Bin Ge, Ying-Xin Cui, Liang-Liang Sun (2017)
Open Mathematics
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In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, where [...] (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.
David Swanson (2002)
Studia Mathematica
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We prove that a function belonging to a fractional Sobolev space may be approximated in capacity and norm by smooth functions belonging to , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].
Chacón, G.A., Chacón, G.R. (2005)
Acta Mathematica Universitatis Comenianae. New Series
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Alejandra Perini (2011)
Commentationes Mathematicae Universitatis Carolinae
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In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces for , and . Specifically, we show that, for suitable values of and , if (Sawyer’s classes of weights) then the one-sided fractional integral can be extended to a bounded operator from to . The result is a consequence of the pointwise inequality where denotes the Calderón maximal function.