Displaying similar documents to “Multiplicity solutions of a class fractional Schrödinger equations”

Multiplicity results for a class of fractional boundary value problems

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: ⎧ - d / d t ( 1 / 2 0 D t - σ ( u ' ( t ) ) + 1 / 2 t D T - σ ( u ' ( t ) ) ) - λ β ( t ) f ( u ( t ) ) - μ γ ( t ) g ( u ( t ) ) = 0 , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where 0 D t - σ and t D T - σ 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].

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

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 Y κ ( t ) , t 0 is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of Y κ ( t ) via the measure of its codependence r(θ₁,θ₂,t). We prove that Y κ ( t ) 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...

Functions of finite fractional variation and their applications to fractional impulsive equations

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 σ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ -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.

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

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, T Ω , α A 1 , A 2 , ... , A k , 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, T Ω , α A 1 , A 2 , ... , A k , from [...] Mp,φ1wptoMp,φ2wq M p , ϕ 1 w p to M p , ϕ 2 w q for 1 < p < q < ∞. In all cases the conditions...

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

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 u ( x , t ) = t α f ( | x | t β ) with suitable and β . 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...

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

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We prove that a function belonging to a fractional Sobolev space L α , p ( ) may be approximated in capacity and norm by smooth functions belonging to C m , λ ( ) , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Set-valued fractional order differential equations in the space of summable functions

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 ( D α - i = 1 n - 1 a i D α i ) x ( t ) F ( t , x ( φ ( t ) ) ) , a.e. on (0,1), I 1 - α x ( 0 ) = c , 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.

Trace inequalities for fractional integrals in grand Lebesgue spaces

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 L p ) , θ ( X , μ ) to L q ) , q θ / p ( X , ν ) (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...

Fractional domination in prisms

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 γ f ( G × K ) > γ f ( G ) .

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u t = · ( u ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x n , t } with nonnegative initial data u ( x , 0 ) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and C α regularity of such weak solutions. Finally, we extend...

Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

Luchko, Yury, Trujillo, Juan (2007)

Fractional Calculus and Applied Analysis

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2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05 The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the Grünwald- Letnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville...

Commutators with fractional integral operators

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 μ , λ A p , q and α/n + 1/q = 1/p, the norm | | [ b , I α ] : L p ( μ p ) L q ( λ q ) | | is equivalent to the norm of b in the weighted BMO space BMO(ν), where ν = μ λ - 1 . 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,...

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

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 L p ( ) -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable...

Left general fractional monotone approximation theory

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...

A new characteristic property of Mittag-Leffler functions and fractional cosine functions

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 E α ( a t α ) 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.

Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics

Saxena, R., Saxena, Ravi, Kalla, S. (2010)

Fractional Calculus and Applied Analysis

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Dedicated to Professor A.M. Mathai on the occasion of his 75-th birthday. Mathematics Subject Classi¯cation 2010: 26A33, 44A10, 33C60, 35J10. The object of this article is to present the computational solution of one-dimensional space-time fractional Schrödinger equation occurring in quantum mechanics. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the H-function....

Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces

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 P C -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 P C -mild solution of a two-dimensional impulsive...

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

Mahsa Fatehi, Bahram Khani Robati (2012)

Czechoslovak Mathematical Journal

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In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator C ϕ , when ϕ is a linear-fractional self-map of 𝔻 . In this paper first, we investigate the essential normality problem for the operator T w C ϕ on the Hardy space H 2 , where w is a bounded measurable function on 𝔻 which is continuous at each point of F ( ϕ ) , ϕ 𝒮 ( 2 ) , and T w is the Toeplitz operator with symbol w . Then we use these results and characterize the essentially...

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

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Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts a / n n = 1 is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length c N - 0 . 475 contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

Mainardi, Francesco, Gorenflo, Rudolf (2007)

Fractional Calculus and Applied Analysis

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2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05, The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classica theory of linear viscoelasticity, we contrast these two types of fractiona derivatives in their...