Nonlinear fractional differential equations in nonreflexive Banach spaces and fractional calculus
- Ravi P Agarwal^{1, 2}Email author,
- Vasile Lupulescu^{3},
- Donal O’Regan^{2, 4} and
- Ghaus ur Rahman^{5}
https://doi.org/10.1186/s13662-015-0451-5
© Agarwal et al.; licensee Springer. 2015
Received: 12 January 2015
Accepted: 25 March 2015
Published: 4 April 2015
Abstract
The aim of this paper is to correct some ambiguities and inaccuracies in Agarwal et al. (Commun. Nonlinear Sci. Numer. Simul. 20(1):59-73, 2015; Adv. Differ. Equ. 2013:302, 2013, doi:10.1186/1687-1847-2013-302) and to present new ideas and approaches for fractional calculus and fractional differential equations in nonreflexive Banach spaces.
1 Introduction
One of the sections, Section 5, of our paper [1] contains a number of ambiguities (and inaccuracies) which we correct here. The notion of pseudo-solution in [1] is not adequately defined and assumption (h2) in Theorem 5.1 is strong. Also, there is some ambiguity regarding the use of the space \(C(T,E)\) of all continuous functions from T into E with its weak topology \(\sigma(C(T,E),C(T,E)^{\ast})\) and the space \(C_{w}(T,E)\) of all weakly continuous functions from T into \(E_{w}\) endowed with the topology of weak uniform convergence. Parts of Corollaries 5.1-5.6 are no longer valid in their current form. Similar comments also apply to [2]. In [3] the authors developed fractional calculus for vector-valued functions using the weak Riemann integral and they established the existence of weak solutions for a class of fractional differential equations with fractional weak derivatives. In this paper we present new ideas in fractional calculus and we present a new approach to establishing existence to some fractional differential equations in nonreflexive Banach spaces. References [4–6], and [7] were helpful in presenting these new ideas.
2 Preliminaries
In the following we outline some aspects of fractional calculus in a nonreflexive Banach space. This subject has been treated extensively in [1, 3]. Let E be a Banach space with norm \(\Vert \cdot \Vert \) and let \(E^{\ast}\) be the topological dual of E. If \(x^{\ast}\in E^{\ast}\), then its value on an element \(x\in E\) will be denoted by \(\langle x^{\ast},x \rangle\). The space E endowed with the weak topology \(\sigma ( E,E^{\ast} ) \) will be denoted by \(E_{w}\). Consider an interval \(T= [ 0,b ] \) of ℝ, the set of real numbers, endowed with the Lebesgue σ-algebra \(\mathcal{L} ( T ) \) and the Lebesgue measure λ. A function \(x(\cdot):T\rightarrow E\) is said to be strongly measurable on T if there exists a sequence of simple functions \(x_{n}(\cdot):T\rightarrow E\) such that \(\lim_{n\rightarrow\infty}x_{n}(t)=x(t)\) for a.e. \(t\in T\). Also, a function \(x(\cdot):T\rightarrow E\) is said to be weakly measurable (or scalarly measurable) on T if, for every \(x^{\ast}\in E^{\ast}\), the real valued function \(t\mapsto \langle x^{\ast },x(t) \rangle\) is Lebesgue measurable on T.
Let \(C_{w}(T,E)\) denote the space of all weakly continuous functions from T into \(E_{w}\) endowed with the topology of weak uniform convergence. A set \(N\in\mathcal{L} ( T ) \) is called a null set if \(\lambda ( N ) =0\).
The concept of a Bochner integral and a Pettis integral are well known [12–14].
- (a)
\(x(\cdot)\) is scalarly integrable; that is, for every \(x^{\ast}\in E^{\ast}\), the real function \(t\mapsto \langle x^{\ast },x(t) \rangle\) is Lebesgue integrable on T;
- (b)for every set \(A\in\mathcal{L} ( T ) \), there exists an element \(x_{A}\in E\) such thatfor every \(x^{\ast}\in E^{\ast}\). The element \(x_{A}\in E\) is called the Pettis integral on A and it will be denoted by \(\int_{A}x(s)\,ds\).$$ \bigl\langle x^{\ast},x_{A} \bigr\rangle =\int _{A} \bigl\langle x^{\ast },x(s) \bigr\rangle \,ds, $$(3)
It is easy to show that a Bochner integrable function \(x(\cdot ):T\rightarrow E\) is Pettis integrable and both integrals of \(x(\cdot )\) are equal on each Lebesgue measurable subset A of T ([14], Proposition 2.3.1). The best result for a descriptive definition of the Pettis integral is that given by Pettis in [15].
Proposition 1
- (a)If \(x(\cdot)\) is Pettis integrable on T, then the indefinite Pettis integralis AC on T and \(x(\cdot)\) is a pseudo-derivative of \(y(\cdot)\).$$ y(t):=\int_{0}^{t}x(s)\,ds,\quad t\in T $$
- (b)If \(y(\cdot):T\rightarrow E\) is an AC function on T and it has a pseudo-derivative \(x(\cdot )\) on T, then \(x(\cdot)\) is Pettis integrable on T and$$ y(t)=y(0)+\int_{0}^{t}x(s)\,ds,\quad t\in T. $$
It is well known that the Pettis integrals of two strongly measurable functions \(x(\cdot):T\rightarrow E\) and \(y(\cdot):T\rightarrow E\) coincide over every Lebesgue measurable set in T if and only if \(x(\cdot )=y(\cdot)\) a.e. on T ([15], Theorem 5.2). Since a pseudo-derivative of a pseudo-differentiable function \(x(\cdot ):T\rightarrow E\) is not unique (see [11]) and two pseudo-derivatives of \(x(\cdot) \) need not be a.e. equal, the concept of weakly equivalence plays an important role in the following.
Two weak measurable functions \(x(\cdot):T\rightarrow E\) and \(y(\cdot):T\rightarrow E\) are said to be weakly equivalent on T if, for every \(x^{\ast}\in E^{\ast}\), we have \(\langle x^{\ast },x(t) \rangle= \langle x^{\ast},y(t) \rangle\) for a.e. \(t\in T\). In the following, if two weak measurable functions \(x(\cdot ):T\rightarrow E\) and \(y(\cdot):T\rightarrow E\) are weakly equivalent on T, then we will write \(x(\cdot)\eqsim y(\cdot)\) or \(x(t)\eqsim y(t)\), \(t\in T\).
Proposition 2
([15])
A weakly measurable function \(x(\cdot):T\rightarrow E\) is Pettis integrable on T and \(\langle x^{\ast},x(\cdot) \rangle\in L^{\infty }(T)\), for every \(x^{\ast}\in E^{\ast}\), if and only if the function \(t\mapsto\varphi(t)x(t)\) is Pettis integrable on T, for every \(\varphi(\cdot)\in L^{1}(T)\).
Lemma 1
([1], Lemma 3.1)
If \(x(\cdot ),y(\cdot)\in P^{\infty}(T,E)\) are weakly equivalent on T, then \(I^{\alpha}x(t)=I^{\alpha}y(t)\) on T.
Lemma 2
- (a)
\(I^{\alpha}I^{\beta}x(t)=I^{\alpha+\beta}x(t)\), \(t\in T\);
- (b)
\(\lim_{\alpha\rightarrow1}I^{\alpha }x(t)=I^{1}x(t)=x(t)-x(0)\) weakly uniformly on T;
- (c)
\(\lim_{\alpha\rightarrow0}I^{\alpha}x(t)=x(t)\) weakly on T.
Remark 1
We recall that a function \(x(\cdot):T\rightarrow E\) is said to be weakly absolutely continuous (\(wAC\), for short) on T if, for every \(x^{\ast}\in E^{\ast}\), the real valued function \(t\mapsto \langle x^{\ast},x(t) \rangle\) is absolutely continuous on T.
Lemma 3
([1])
Remark 2
Remark 3
Lemma 4
- (a)If \(y(\cdot)\in P^{\infty}(T,E)\), then$$ D_{p}^{\alpha}I^{\alpha}y(t)=y(t),\quad t\in T. $$
- (b)If \(y(\cdot)\in P^{\infty}(T,E)\) and \(y_{1-\alpha }(\cdot)\) is pseudo-differentiable with a pseudo-derivative \(\frac{d_{p}}{dt}y_{1-\alpha}(\cdot)\in P^{\infty}(T,E)\), then$$ I^{\alpha}D_{p}^{\alpha}y(t)=y(t)-y(0),\quad t\in T. $$
Proof
Lemma 5
- (a)$$ I^{\alpha}D_{p}^{\beta}y(t)=D_{p}^{\beta-\alpha}y(t) \quad\textit{on } T. $$(10)
- (b)If \(y(0)=0\), thenand$$ D_{p}^{\beta}I^{\alpha}y(t)=D_{p}^{\beta-\alpha}y(t) \quad\textit{on }T $$(11)$$ I^{\beta}D_{p}^{\alpha}y(t)=I^{\beta-\alpha}y(t)\quad \textit{on }T. $$(12)
Proof
3 Differential equations with fractional pseudo-derivatives
A continuous function \(y(\cdot):T\rightarrow E\) is said to be a solution of (13) if \(y(\cdot)\) has a pseudo-derivative belonging to \(P^{\infty}(T,E)\), \(D_{p}^{\alpha}y(t)\eqsim f(t, y(t))\) for \(t\in T\) and \(y(0)=y_{0}\).
To prove a result on the existence of solutions for (13) we need some preliminary results.
Lemma 6
Let \(f(\cdot,\cdot):T\times E\rightarrow E\) be a function such that \(f(\cdot,y(\cdot))\in P^{\infty}(T,E)\), for every continuous function \(y(\cdot ):T\rightarrow E\). Then a continuous function \(y(\cdot ):T\rightarrow E\) is a solution of (13) if and only if it satisfies the integral equation (14).
Proof
In this section we shall discuss the existence of solutions of fractional differential equations in nonreflexive Banach spaces. We recall that a function \(f(\cdot):E\rightarrow E\) is said to be sequentially continuous from \(E_{w}\) into \(E_{w}\) (or weakly-weakly sequentially continuous) if, for every weakly convergent sequence \(\{x_{n}\}_{n\geq1}\subset E\), the sequence \(\{f(x_{n})\}_{n\geq1}\) is weakly convergent in E.
- (1)
\(A\subseteq B\) implies that \(\beta(A)\leq\beta(B)\);
- (2)
\(\beta(A)=\beta(cl_{w}A)\), where \(cl_{w}A\) denotes the weak closure of A;
- (3)
\(\beta(A)=0\) if and only if \(cl_{w}A\) is weakly compact;
- (4)
\(\beta(A\cup B)=\max\{\beta(A),\beta(B)\}\);
- (5)
\(\beta(A)=\beta(\operatorname{conv}(A))\);
- (6)
\(\beta(A+B)\leq\beta(A)+\beta(B)\);
- (7)
\(\beta(x+A)=\beta(A)\), for all \(x\in E\);
- (8)
\(\beta(\lambda A)=|\lambda|\beta(A)\), for all \(\lambda\in \mathbb{R}\);
- (9)
\(\beta(\bigcup_{0\leq r\leq r_{0}}rA)=r_{0}\beta(A)\);
- (10)
\(\beta(A)\leq2\operatorname{diam}(A)\).
Lemma 7
([39])
- (i)
the function \(t\rightarrow\beta(H(t)) \) is continuous on T,
- (ii)
\(\beta_{c}(H)=\sup_{t\in T}\beta(H(t))\),
Lemma 8
([21])
Theorem 1
- (H1)
\(f(t,\cdot)\) is weakly-weakly sequentially continuous, for every \(t\in T\);
- (H2)
\(f(\cdot,y(\cdot))\in P^{\infty}(T,E)\), for every continuous function \(y(\cdot):T\rightarrow E\);
- (H3)
\(\|f(t,y)\|\leq M\), for all \((t,y)\in T\times E\);
- (H4)for every bounded set \(A\subseteq E\) we have$$ \beta\bigl(f(T\times A)\bigr)\leq g\bigl(\beta(A)\bigr), $$
Proof
If E is reflexive and \(f(\cdot,\cdot):T\times E\rightarrow E\) is bounded, then (H4) is automatically satisfied since a subset of a reflexive Banach space is weakly compact iff it is closed in the weak topology and bounded in the norm topology.
If for \(\alpha=1\) we put \(D_{p}^{1}y(\cdot)=y_{p}^{\prime}(\cdot)\), then from Theorem 1 we obtain the following result (see [18, 23]).
Corollary 1
4 Multi-term fractional differential equation
- (i)
\(y(\cdot)\) has Caputo fractional pseudo-derivatives of orders \(\alpha _{i}\in(0,1)\), \(i=1,2,\ldots,m\),
- (ii)
\(( D^{\alpha_{m}}-\sum_{i=1}^{m-1}a_{i}D^{\alpha _{i}} ) y(t)\eqsim f(t, y(t))\), for all \(t\in T\),
- (iii)
\(y(0)=0\).
Lemma 9
Assume that \(f(\cdot,\cdot ):T\times E\rightarrow E\) satisfy the assumptions (H2) and (H3) in Theorem 1. Then every continuous function \(y(\cdot):T\rightarrow E\) which satisfies the integral equation (22) is a solution of (21).
Proof
Lemma 10
([24], Theorem 2.2)
Let K be a nonempty, bounded, convex, closed set in a Banach space E. Assume \(Q:K\rightarrow K\) is weakly sequentially continuous and β-contractive (that is, there exists \(0\leq k_{0}<1\) such that \(\beta(Q(A))\leq k_{0}\beta(A)\), for all bounded sets \(A\subset E\)). Then Q has a fixed point.
Remark 4
Since the function \(\sigma\mapsto\Gamma (\sigma)\) is convex and \(\Gamma(\sigma)\geq\Gamma(3/2)\approx0.8856032\) for \(\sigma\in(1,2)\), for every \(r\in(0,\Gamma(3/2))\) we have \(\Gamma(\alpha_{m}+1)>r\).
Next we establish an existence result for the multi-term fractional integral equation (22) in nonreflexive Banach spaces.
Theorem 2
Proof
Using Lemma 9 we obtain the following result.
Declarations
Acknowledgements
The authors wish to thank Professor Mieczysław Cichoń for his helpful comments.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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