Existence results for impulsive fractional differential equations with $p$-Laplacian via variational methods

John R. Graef; Shapour Heidarkhani; Lingju Kong; Shahin Moradi

Displaying similar documents to “Existence results for impulsive fractional differential equations with $p$ -Laplacian via variational methods”

Duality for a fractional variational formulation using $η$ -approximated method

Sony Khatri, Ashish Kumar Prasad (2023)

Kybernetika

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The present article explores the way $η$ -approximated method is applied to substantiate duality results for the fractional variational problems under invexity. $η$ -approximated dual pair is engineered and a careful study of the original dual pair has been done to establish the duality results for original problems. Moreover, an appropriate example is constructed based on which we can validate the established dual statements. The paper includes several recent results as special cases. ...

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].

A spatially sixth-order hybrid $L 1$ -CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang, Jun-Wei Jin, Hai-Wei Sun, Qin Sheng (2021)

Applications of Mathematics

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We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L 1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L 1$ -CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2 - γ$ in time, where $0 < γ < 1$ is the order of the Caputo fractional derivative...

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 $(D^{α ₙ} - \sum_{i = 1}^{n - 1} a_{i} D^{α_{i}}) x (t) \in F (t, x (φ (t)))$ , a.e. on (0,1), $I^{1 - α ₙ} x (0) = c$ , 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.

Fractional $q$ -difference equations on the half line

Saïd Abbas, Mouffak Benchohra, Nadjet Laledj, Yong Zhou (2020)

Archivum Mathematicum

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This article deals with some results about the existence of solutions and bounded solutions and the attractivity for a class of fractional $q$ -difference equations. Some applications are made of Schauder fixed point theorem in Banach spaces and Darbo fixed point theorem in Fréchet spaces. We use some technics associated with the concept of measure of noncompactness and the diagonalization process. Some illustrative examples are given in the last section.

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

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For α ∈ (1,2) we consider the equation $\partial_{t} u = Δ^{α / 2} u + b \cdot \nabla u$ , where b is a time-independent, divergence-free singular vector field of the Morrey class $M ₁^{1 - α}$ . We show that if the Morrey norm ${| | b | |}_{M ₁^{1 - α}}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Fractional integral operators on $B^{p, λ}$ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

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We introduce function spaces $B^{p, λ}$ with Morrey-Campanato norms, which unify $B^{p, λ}$ , $C M O^{p, λ}$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{α}$ on these spaces.

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 $u_{t} = \nabla \cdot (u \nabla {(- Δ)}^{- s} u), 0 < s < 1$ . The problem is posed in ${x \in ℝ^{n}, t \in ℝ}$ 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...

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 ${a ⁿ / n}_{n = 1}^{\infty}$ 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.

Existence Results for a Fractional Boundary Value Problem via Critical Point Theory

A. Boucenna, Toufik Moussaoui (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper, we consider the following boundary value problem $\{\begin{matrix} D_{T^{-}}^{α} (D_{0^{+}}^{α} (D_{T^{-}}^{α} (D_{0^{+}}^{α} u (t)))) = f (t, u (t)), t \in [0, T], \\ u (0) = u (T) = 0 \\ D_{T^{-}}^{α} (D_{0^{+}}^{α} u (0)) = D_{T^{-}}^{α} (D_{0^{+}}^{α} u (T)) = 0, \end{matrix}$ where $0 < α \leq 1$ and $f : [0, T] \times ℝ \to ℝ$ is a continuous function, $D_{0^{+}}^{α}$ , $D_{T^{-}}^{α}$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball

Vladimir Varlamov (2000)

Studia Mathematica

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The nonlinear heat equation with a fractional Laplacian $[u_{t} + {(- Δ)}^{α / 2} u = u^{2}, 0 < α \leq 2]$ , is considered in a unit ball $B$ . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space $C ⁰ ([0, \infty), H ₀^{κ} (B))$ with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.

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 $μ, λ \in A_{p, q}$ and α/n + 1/q = 1/p, the norm $| | [b, I_{α}] : L^{p} (μ^{p}) \to L^{q} (λ^{q}) | |$ 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,...

Generalized fractional integrals on central Morrey spaces and generalized λ-CMO spaces

Katsuo Matsuoka (2014)

Banach Center Publications

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We introduce the generalized fractional integrals $I ̃_{α, d}$ and prove the strong and weak boundedness of $I ̃_{α, d}$ on the central Morrey spaces $B^{p, λ} (ℝ ⁿ)$ . In order to show the boundedness, the generalized λ-central mean oscillation spaces $Λ_{p, λ}^{(d)} (ℝ ⁿ)$ and the generalized weak λ-central mean oscillation spaces $W Λ_{p, λ}^{(d)} (ℝ ⁿ)$ play an important role.

Stability of nonlinear $h$ -difference systems with $n$ fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

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In the paper we study the subject of stability of systems with $h$ -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$ -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$ -orders.

Fractional global domination in graphs

Subramanian Arumugam, Kalimuthu Karuppasamy, Ismail Sahul Hamid (2010)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g (N [v]) = \sum_{u \in N [v]} g (u) \geq 1$ and $g (\bar{N (v)}) = \sum_{u \notin N (v)} g (u) \geq 1$ . A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{f g} (G)$ is defined as follows: $γ_{f g} (G)$ = min|g|:g is an MGDF of G where $| g | = \sum_{v \in V} g (v)$ . In this paper we initiate a study of this parameter.

A uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Given a two-dimensional fractional multiplicative process ${(F_{t})}_{t \in [0, 1]}$ determined by two Hurst exponents $H_{1}$ and $H_{2}$ , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0, 1]$ by $F$ if and only if $H_{1} = H_{2}$ .

Some applications of subordination theorems associated with fractional $q$ -calculus operator

Wafaa Y. Kota, Rabha Mohamed El-Ashwah (2023)

Mathematica Bohemica

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Using the operator $𝔇_{q, ϱ}^{m} (λ, l)$ , we introduce the subclasses $𝔜_{q, ϱ}^{* m} (l, λ, γ)$ and $𝔎_{q, ϱ}^{* m} (l, λ, γ)$ of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.

On the variation of certain fractional part sequences

Michel Balazard, Leila Benferhat, Mihoub Bouderbala (2021)

Communications in Mathematics

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Let $b > a > 0$ . We prove the following asymptotic formula $\sum_{n ⩾ 0} | {x / (n + a)} - {x / (n + b)} | = \frac{2}{π} ζ (3 / 2) \sqrt{c x} + O (c^{2 / 9} x^{4 / 9}),$ with $c = b - a$ , uniformly for $x ⩾ 40 c^{- 5} {(1 + b)}^{27 / 2}$ .

Displaying similar documents to “Existence results for impulsive fractional differential equations with p -Laplacian via variational methods”

Displaying similar documents to “Existence results for impulsive fractional differential equations with $p$ -Laplacian via variational methods”