Fractional-order Problems Research Articles

Existence of solutions for impulsive integral boundary value problems of fractional order

In this paper, we prove some existence results for a boundary value problem of nonlinear impulsive differential equations of fractional-order q ∈ ( 1 , 2 ] with integral boundary conditions by applying the contraction mapping principle and Krasnoselskii’s fixed point theorem.

On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order

In the following pages, based on the linear functional over a Banach space E and on the definition of fractional integrals of real-valued functions, we define the fractional Pettis-integrals of E -valued functions and the corresponding fractional derivatives. Also, we show that the well-known properties of fractional calculus over the domains of the Lebesgue integrable also hold in the Pettis space. To encompass the full scope of the paper, we apply this abstract result to investigate the existence of Pseudo-solutions to the following fractional-order boundary value problem { D α x ( t ) + λ a ( t ) f ( t , x ( t ) ) = 0 , t ∈ [ 0 , 1 ] , α ∈ ( n − 1 , n ] , n ≥ 2 , x ( 1 ) + ∫ 0 1 u ( τ ) x ( τ ) d τ = l , x ( k ) ( 0 ) = 0 , k = 0 , 1 , … , n − 2 , in the Banach space C [ I , E ] under Pettis integrability assumptions imposed on f . Our results extend all previous results of the same type in the Bochner integrability setting and in the Pettis integrability one. Here, λ ∈ R , u ∈ L p , a ∈ L q and l ∈ E .

Solutions to a class of nonlinear differential equations of fractional order

In this paper we investigate the formulation of a class of boundary value problems of fractional order with the Riemann-Liouville fractional derivative and integral-type boundary conditions. The existence of solutions is established by applying a fixed point theorem of Krasnosel’skiuo and Zabreiko for asymptotically linear mappings.

On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order

In this paper, we study the existence of solutions of the operator equations p + λ G f x = x in the Banach space C [ I , E ] . It is assumed the vector-valued function f is nonlinear Pettis-integrable. Some additional assumptions imposed on f are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type − d α d t α x ( t ) = λ f ( t , x ( t ) ) , a.e. on [ 0 , 1 ] , x ( 0 ) = x ( 1 ) = 0 , α ∈ ( 1 , 2 ] , under the Pettis integrability assumption imposed on f .

EXISTENCE OF SOLUTION FOR A BOUNDARY VALUE PROBLEM OF FRACTIONAL ORDER

This article considers the existence of solution for a boundary value problem of fractional order, involving Caputo's derivative { C 0 D t δ u ( t ) = g ( t , u ( t ) ) , 0 < t < 1 , 1 < δ < 2 , u ( 0 ) = α ≠ 0 , u ( 1 ) = β ≠ 0

Cauchy problems of fractional order and stable processes

We consider the Cauchy problem (∂/∂t)α u(t,x)=(∂/∂x)β u(t,x) for 1≤α, β≤2. The aim of this paper is threefold. First, we show the existence and the uniqueness of the solution, and determine its structure. Second, we give a representation of the solution by the distributions of several stable processes. Third, we show the positivity of the fundamental solution for β=2. These results offer an interpretation to phenomena between the heat equation (α=1, β=2) and the wave equation (α=β=2).