Fractional calculus and fractional differential equations in nonreflexive Banach spaces

doi:10.1016/j.cnsns.2013.10.010

Communications in Nonlinear Science and Numerical Simulation

Volume 20, Issue 1, January 2015, Pages 59-73

https://doi.org/10.1016/j.cnsns.2013.10.010 Get rights and content

Highlights

•
We introduce the notion of pseudo-fractional derivatives.
•
We introduce the notion of fractional Pettis integrals.
•
We discuss functions in non-reflexive Banach spaces equipped with the weak topology.
•
We discuss fractional differential equations in an abstract setting.

Abstract

In this paper we establish an existence result for the fractional differential equation $\{\begin{matrix} D_{p}^{α} y (t) = f (t, y (t)), \\ y (0) = y_{0}, \end{matrix}$ where $D_{p}^{α} y (\cdot)$ is a fractional pseudo-derivative of a weakly absolutely continuous and pseudo-differentiable function y(·) : T → E, the function f (t, ·) : T × E → E is weakly–weakly sequentially continuous for every t ∈ T and f (·, y(·)) is Pettis integrable for every weakly absolutely continuous function y(·) : T → E, T is a bounded interval of real numbers and E is a nonreflexive Banach space.

Introduction

The study of first order ordinary differential equations in Banach spaces (reflexive or not) equipped with the weak topology was initiated in the 1950s. Let E be a Banach space and let f (·,·) : [a, b] × E → E be continuous. It is well known that if E is finite dimensional, then for each (t₀, y₀) ∈ [a, b) × E, there exists a continuous differentiable function y(·) which is a solution of the Cauchy problem $y^{'} (t) = f (t, y (t)), y (t_{0}) = y_{0},$ on some open interval which contains t₀. In 1950, Dieudonné [15] showed that when E = c₀ the Cauchy problem (1) has no solutions for every continuous function f (·, ·). The notion of the measure of noncompactness was introduced by Kuratowski [29] in 1930. Ambrosetti [1] used the Kuratowski noncompactness measure and Darbo’s fixed point theorem to prove an existence result for (1) in infinite dimensional Banach spaces. Szep [49] first established the existence of weak solutions for (1) if f (·, ·) : [a, b] × E → E is weakly continuous and E is a reflexive Banach space. Chow and Schuur [10] treated the case where E is separable and reflexive and f (·, ·) is a weakly continuous function with bounded range. Kato in [26] showed that if f (·, ·) : [a, b] × B_E[y₀, r] → E is weakly continuous, then all that is needed to assure the existence of solutions to (1) is the relatively weak compactness of f ([a, b] × B_E[y₀, r]). Pianigiani [39] showed that in every nonreflexive retractive Banach space there exists a weakly continuous function f (·, ·) such that (1) does not have a weak solution, and Perri [36] showed that this property is true in every nonreflexive Banach space. The measure of weak noncompactness was introduced by De Blasi [11], and it was used by Cramer et al. [9] to obtain an existence result for weak solutions of (1) in nonreflexive Banach spaces. Using the measure of weak noncompactness, Cichoń [6], Cichoń and Kubiaczyk [8], Dutkiewicz and Szufla [17], Gomaa [23], O’Regan [33], [34], improved and generalized previous results in the literature. For a review of this topic we refer the reader to Cichoń [7], Deimling [12], Hashem [24] and Teixeira [51]. In recent years fractional differential equations in Banach spaces were studied. The general literature on fractional differential equations in finite or infinite dimensional Banach spaces is extensive and different topics on the existence and qualitative properties of solutions are considered. Concerning the literature on fractional differential equations we cite the books [30], [43] and the references therein. Only a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology. Salim and El-Sayed [41] were the first authors to discuss the existence of weak solutions for fractional differential equation. Recent progress in this direction can be found in [4], [5], [40] and for a recent review of this topic we refer the reader to [24].

In this paper, we develop fractional calculus for functions with values in a nonreflexive Banach space equipped with the weak topology. Using the Pettis integral, we introduce the notions of fractional Pettis integrals and pseudo-fractional derivatives. Then we present a very general theory for fractional calculus and fractional differential equations in a nonreflexive Banach spaces equipped with the weak topology.

Section snippets

Preliminaries

Let E be a Banach space with the norm ∥ · ∥ and let E^∗ be the topological dual of E. If x^∗ ∈ E^∗, then its value on an element x ∈ E will be denoted by 〈x^∗, x〉. The space E endowed with the weak topology σ(E, E^∗) will be denoted by E_w. Consider an interval T = [0, b] of $R$ endowed with the Lebesgue σ-algebra $L$ (T) and the Lebesgue measure λ. We will denote by L¹(T) the space of all measurable and Lebesgue integrable real functions defined on T, and by L^∞(T) the space of all measurable and essential bounded

Fractional Pettis integral and Abel integral equation

Let us denote by P^∞(T, E) the space of all weakly measurable and Pettis integrable functions x(·) : T → E with the property that 〈x^∗, x(·)〉 ∈ L^∞(T) for every x^∗ ∈ E^∗. Since for each t ∈ T the real valued function s ↦ (t − s)^α−1 is Lebesgue integrable on [0, t] for every α > 0 then, by Proposition 2.4, the fractional Pettis integral $I^{α} x (t) ≔ \int_{0}^{t} \frac{{(t - s)}^{α - 1}}{Γ (α)} x (s) ds, t \in T$ exists for every function x(·) ∈ P^∞(T, E) as a function from T into E. Moreover, we have that $〈 x^{*}, I^{α} x (t) 〉 = \int_{0}^{t} \frac{{(t - s)}^{α - 1}}{Γ (α)} 〈 x^{*}, x (s) 〉 ds, t \in T,$ for every x^∗ ∈ E^∗, and

Fractional pseudo-derivatives

In the following, consider a ∈ (0, 1). If y(·) : T → E is a pseudo-differentiable function with a pseudo-derivative x(·) ∈ P^∞(T, E) on T, then the following fractional Pettis integral $I^{1 - α} x (t) = \int_{0}^{t} \frac{{(t - s)}^{- α}}{Γ (1 - α)} x (s) ds$ exists on T. The fractional Pettis integral I^1−α x(·) is called a fractional pseudo-derivative of y(·) on T and it will be denoted by $D_{p}^{α} y (\cdot)$ ; that is, $D_{p}^{α} y (t) = I^{1 - α} x (t), t \in T .$ If y(·) : T → E is an a.e. weakly differentiable function with the weak derivative $y_{w}^{'} (\cdot) \in P^{\infty} (T, E)$ on T, then $D_{w}^{α} y (t) ≔ I^{1 - α} y_{w}^{'} (t), t$

Fractional differential equations

In this section we establish an existence result for the following fractional differential equation $\{\begin{matrix} D_{p}^{α} y (t) = f (t, y (t)), \\ y (0) = y_{0}, \end{matrix}$ where $D_{p}^{α} y (\cdot)$ is a fractional pseudo-derivative of the function y(·) : T → E and f (·, ·) : T × E → E is a given function. Along with the Cauchy problem (18) consider the following integral equation $y (t) = y_{0} + \int_{0}^{t} \frac{{(t - s)}^{α - 1}}{Γ (α)} f (s, y (s)) ds, t \in T,$ where the integral is in the sense of Pettis.

A wAC function (or an AC function, if E is a weakly sequentially complete space) y(·) : T → E is said to be

Conclusions

In this paper, we developed fractional calculus for functions with values in a nonreflexive Banach space equipped with the weak topology. Using the Pettis integral, we introduced and studied the notions of fsractional Pettis integrals and pseudo-fractional derivatives. Then we obtained an existence result for fractional differential equations in a nonreflexive Banach space equipped with the weak topology.

Acknowledgments

The authors wish to thank the referees for their valuable comments and suggestions.

References (51)

M. Cichoń
On solutions of differential equations in Banach spaces
Nonlinear Anal
(2005)
A.M. Gomaa
Weak and strong solutions for differential equations in Banach spaces
Chaos Solutions Fract
(2003)
D. O’Regan
Fixed point theory for weakly sequentially continuous mapping
Math Comput Model.
(1998)
D. O’Regan
Weak solutions of ordinary differential equations in Banach spaces
Appl Math Lett
(1999)
H.A.H. Salem
On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional orders
Comput Math Appl
(2010)
E.V. Teixeira
Strong solutions for differential equations in abstract spaces
J Differ Equ
(2005)
A. Ambrosetti
Un teorema di esistenza per le equazioni differenziali negli spazi di Banach
Rend Sem Mat Univ Padova
(1967)
W. Arendt et al.
Vector-valued laplace transforms and cauchy problems
O. Arino et al.
A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations
Funkcialaj Ekvacioj
(1984)
M. Benchohra et al.
Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces
Opuscula Math
(2012)

M. Benchohra et al.

Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces

Electron J Qual Theory Differ Equ

(2010)

M. Cichoń

Weak solutions of ordinary differential equations in Banach spaces

Discuss Math Differ Inc Control Optimal

(1995)

M. Cichoń et al.

Kneser’s theorem for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

Ann Polon Math

(1995)

E. Cramer et al.

On the existence of weak solutions of differential equations in nonreflexive Banach spaces

Nonlinear Anal

(1978)

S. Chow et al.

An existence theorem for ordinary differential equations in Banach spaces

Bull Am Math Soc

(1971)

F. De Blasi

On a property of the unit sphere in a Banach space

Bull Math Soc Sci Math RS Roumanie

(1977)

K. Deimling

Ordinary differential equations in Banach spaces

J. Diestel et al.

Vector measures

(1977)

S.J. Dilworth et al.

Nowhere weak differentiability of the Pettis integral

Quaestiones Math

(1995)

J. Dieudonné

Deux exemples singuliers d’équations différentielles

Acta Sci Math

(1950)

N. Dinuleanu

Vector measures

(1967)

A. Dutkiewicz et al.

Kneser’s theorem for weak solution of and integral equation with weakly singular kernal

Publ Inst Math

(2005)

R.E. Edwards

Functional analysis

(1965)

Gautier S, Penot JP. Existence des courbes integrales pour les champs de vecteurs faiblement continus, Publications...

R.F. Geitz

Geometry and the Pettis integral

Trans Am Math Soc

(1982)

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Fractional calculus and fractional differential equations in nonreflexive Banach spaces

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Fractional Pettis integral and Abel integral equation

Fractional pseudo-derivatives

Fractional differential equations

Conclusions

Acknowledgments

Nonlinear Anal

Chaos Solutions Fract

Math Comput Model.

Appl Math Lett

Comput Math Appl

J Differ Equ

Un teorema di esistenza per le equazioni differenziali negli spazi di Banach

Rend Sem Mat Univ Padova

Vector-valued laplace transforms and cauchy problems

A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations

Funkcialaj Ekvacioj

Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces

Opuscula Math

Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces

Electron J Qual Theory Differ Equ

Weak solutions of ordinary differential equations in Banach spaces

Discuss Math Differ Inc Control Optimal

Kneser’s theorem for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

Ann Polon Math

On the existence of weak solutions of differential equations in nonreflexive Banach spaces

Nonlinear Anal

An existence theorem for ordinary differential equations in Banach spaces

Bull Am Math Soc

On a property of the unit sphere in a Banach space

Bull Math Soc Sci Math RS Roumanie

Ordinary differential equations in Banach spaces

Vector measures

Nowhere weak differentiability of the Pettis integral

Quaestiones Math

Deux exemples singuliers d’équations différentielles

Acta Sci Math

Vector measures

Kneser’s theorem for weak solution of and integral equation with weakly singular kernal

Publ Inst Math

Functional analysis

Geometry and the Pettis integral

Trans Am Math Soc