Nonlinear fractional differential equations in nonreflexive Banach spaces and fractional calculus

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.

We denote by L p (T) the space of all real measurable functions f : T → R, whose absolute value raised to the pth power has finite integral, or equivalently, that where  ≤ p < ∞. Moreover, by L ∞ (T) we denote the space of all measurable and essential bounded real functions defined on T. Let C(T, E) denote the space of all strong continuous functions y(·) : T → E, endowed with the supremum norm y(·) c = sup t∈T y(t) . Also, we consider the space C(T, E) with its weak topology σ (C(T, E), C(T, E) * ). It is well known that (see [ for all m(·) ∈ M(T, E * ). In [], Lemma , it is shown that a sequence {y n (·)} n≥ converges weakly to y(·) in C(T  , E) if and only if y n (t) tends weakly to y(t) for each t ∈ 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 ∈ L(T) is called a null set if λ(N) = .
A function x(·) : T → E is said to be pseudo-differentiable on T to a function y(·) : T → E if, for every x * ∈ E * , there exists a null set N(x * ) ∈ L(T) such that the real function t → x * , x(t) is differentiable on T N(x * ) and The function y(·) is called a pseudo-derivative of x(·) and it will be denoted by x p (·) or by d p dt x(·). A pseudo-derivative x p (·) of a pseudo-differentiable function x(·) : T → E is weakly measurable on T (see []). We recall that a function x(·) : T → E is said to be weakly differentiable on T if there exists a function x w (·) : T → E such that for every x * ∈ E * . If it exists, x w (·) is uniquely determined and it is called the weak derivative of x(·) on T. Obviously, if x(·) : T → E is a weakly differentiable function on T, then the real function t → x * , x(t) is differentiable on T. Moreover, in this case we have for every x * ∈ E * . It is easy to see that, if x(·) : T → E is a function a.e. weakly differentiable on T, then x(·) is pseudo-differentiable on T and x p (·) = x w (·) a.e. on T.
The concept of a Bochner integral and a Pettis integral are well known [-]. We recall that a weakly measurable function for every x * ∈ E * . The element x A ∈ E is called the Pettis integral on A and it will be denoted by A x(s) ds. It is easy to show that a Bochner integrable function x(·) : T → E is Pettis integrable and both integrals of x(·) are equal on each Lebesgue measurable subset A of T ([], Proposition ..). The best result for a descriptive definition of the Pettis integral is that given by Pettis in [].

Proposition  Let x(·) : T → E be a weakly measurable function. (a) If x(·) is Pettis integrable on T, then the indefinite Pettis integral
is AC on T and x(·) is a pseudo-derivative of y(·). (b) If y(·) : T → E is an AC function on T and it has a pseudo-derivative x(·) on T, then x(·) is Pettis integrable on T and It is well known that the Pettis integrals of two strongly measurable functions x(·) : T → E and y(·) : T → E coincide over every Lebesgue measurable set in T if and only if x(·) = y(·) a.e. on T ([], Theorem .). Since a pseudo-derivative of a pseudo-differentiable function x(·) : T → E is not unique (see []) and two pseudo-derivatives of x(·) need not be a.e. equal, the concept of weakly equivalence plays an important role in the following.
Two weak measurable functions x(·) : T → E and y(·) : T → E are said to be weakly equivalent on T if, for every x * ∈ E * , we have x * , x(t) = x * , y(t) for a.e. t ∈ T. In the following, if two weak measurable functions x(·) : T → E and y(·) : T → E are weakly equivalent on T, then we will write x(·) y(·) or x(t) y(t), t ∈ T.
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 → (ts) α- is Lebesgue integrable on [, t], the fractional Pettis integral exists, for every function x(·) ∈ P ∞ (T, E), as a function from T into E (see []). Moreover, we have for every x * ∈ E * , and the real function t → x * , I α x(t) is continuous (in fact, bounded and uniformly continuous on In the following, consider α ∈ (, ) and for a given function x(·) ∈ P ∞ (T, E) we also denote by x -α (t) the fractional Pettis integral

Lemma  ([, ]) The fractional Pettis integral is a linear operator from P
If y(·) : T → E is a pseudo-differentiable function on T with a pseudo-derivative x(·) ∈ P ∞ (T, E), then the fractional Pettis integral I -α x(t) exists on T. The fractional Pettis integral I -α x(·) is called a fractional pseudo-derivative of y(·) on T and it will be denoted by D α p y(·); that is, on T, and so D α p y(·) does not depend on the choice of a pseudo-derivatives of the function y(·). Therefore, we can write () as where y p (·) is a given pseudo-derivatives of y(·).
We recall that a function x(·) : T → E is said to be weakly absolutely continuous (wAC, for short) on T if, for every x * ∈ E * , the real valued function t → x * , x(t) is absolutely continuous on T.

Lemma  ([]) If y(·) ∈ P ∞ (T, E) is a pseudo-differentiable function on T with a pseudoderivative x(·) ∈ P ∞ (T, E), then the function
is wAC and it has a pseudo-derivative Remark  Relation () can be written as Note () suggests us that we can extend the definition of the fractional pseudo-derivative Therefore, the Caputo fractional pseudo-derivative D α p y(·) exists together with the Riemann-Liouville fractional pseudo-derivative D α p y(·) and they satisfy (). It is easy to see that if y() = , then Remark  Let y(·) : T → E be a pseudo-differentiable function with a pseudo-derivative y p (·) ∈ P ∞ (T, E). Then from Lemma  we find that the function is wAC and has a pseudo-derivative is continuous on T, it follows that x * , I α y() = , for every x * ∈ E * , and thus I α y() = . Then by Remark  we have D β p I α y(t) = D β p I α y(t), and so by Lemma  and Proposition  we have (b) By Lemma  and Proposition  we have Lemma  Let y(·) : T → E be a pseudo-differentiable function on T with y p (·) ∈ P ∞ (T, E) and  < α ≤ β < . Then we have (a) and Proof If y(·) : T → E is a pseudo-differentiable function on T, then by Lemma  we have If y() = , then by Remark  and () we have Also, since y() = , then by Lemma  and Proposition  we have I β D α p y(t) = I β I -α y p (t) = I β-α+ y p (t) = I β-α I  y p (t) = I β-α y(t), t ∈ T.

Differential equations with fractional pseudo-derivatives
The existence of weak solutions or pseudo-solutions for ordinary differential equations in Banach spaces were investigated in many papers (see [-]). In reflexive Banach spaces, the existence of weak solutions or pseudo-solutions for fractional differential equations were studied in [-]. In this section we establish an existence result for the following fractional differential equation: where D α p y(·) 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 () consider the following integral equation: where the integral is in the sense of Pettis.
A continuous function y(·) : T → E is said to be a solution of () if y(·) has a pseudoderivative belonging to P ∞ (T, E), D α p y(t) f (t, y(t)) for t ∈ T and y() = y  . To prove a result on the existence of solutions for () we need some preliminary results.

Lemma  Let f (·, ·) : T × E → E be a function such that f (·, y(·)) ∈ P ∞ (T, E), for every continuous function y(·) : T → E. Then a continuous function y(·) : T → E is a solution of () if and only if it satisfies the integral equation ().
Proof Indeed, if a continuous function y(·) : T → E is a solution of (), then y(·) has a pseudo-derivative belonging to P ∞ (T, E), D α p y(t) f (t, y(t)) for t ∈ T and y() = y  . Then we have I α D α p y(t) = I α f (t, y(t)) on T, and thus from Lemma (b) it follows that y(t)y() = I α f (t, y(t)) on T; that is, y(·) satisfies the integral equation (). Conversely, suppose that a continuous function y(·) : T → E satisfies the integral equation (). Then the function z(·) := f (·, y(·)) ∈ P ∞ (T, E) satisfies the Abel equation where v(t) := y(t)y  , t ∈ T. Then from [], Theorem ., and Lemma  it follows that v -α (·) has a pseudo-derivative on T and Then by Remark  we have z(t) D α p y(t) for t ∈ T; that is, D α p y(t) f (t, y(t)) on T.
In this section we shall discuss the existence of solutions of fractional differential equations in nonreflexive Banach spaces. We recall that a function f (·) : E → 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≥ ⊂ E, the sequence {f (x n )} n≥ is weakly convergent in E.
By a Gripenberg function we mean a function g : R + → R + such that g(·) is continuous, nonincreasing with g() = , and u ≡  is the only continuous solution of The problem of uniqueness of the null solution of () was studied by Gripenberg in []. Let us denote by P wk (E) the set of all weakly compact subset of E. The weak measure of noncompactness [] is the set function β : where B  is the closed unit ball in E. The properties of the weak noncompactness measure are analogous to the properties of the measure of noncompactness, namely (see []):

Lemma  ([]) Let E be a metrizable locally convex topological vector space and let K be a closed convex subset of E, and let Q be a weakly sequentially continuous map of K into itself. If for some y ∈ K the implication
⇒ V is relatively weakly compact holds, for every subset V of K , then Q has a fixed point.

where g(·) is a Gripenberg function. Then () admits a solution y(·) on an interval T
Proof In our proof we shall use some ideas from [] and []. We define the nonlinear operator Q(·) : If y(·) ∈ C(T  , E), then by (H) we have f (·, y(·)) ∈ P ∞ (T, E) and so the operator Q makes sense. To show that Q is well defined, let t  , t  ∈ T  with t  > t  . Without loss of generality, assume that (Qy)(t  ) -(Qy)(t  ) = . Then by the Hahn-Banach theorem, there exists a y * ∈ E * with y * =  and (Qy)(t  ) -(Qy)(t  ) = | y * , (Qy)(t  ) -(Qy)(t  ) |. Then so Q maps C(T  , E) into itself. Let K be the convex, closed, and equicontinuous set defined by We will show that Q maps K into itself and Q restricted to the set K is weakly-weakly sequentially continuous. To show that Q : K → K , let y(·) ∈ K and t ∈ T  . Again, without loss of generality, assume that (Qy)(t) = . By the Hahn-Banach theorem, there exists a y * ∈ E * with y * =  and (Qy)(t) = | y * , (Qy)(t) |. Then by (H), we have and using () it follows that Q maps K into K . Next, we show that Q is weakly-weakly sequentially continuous. First, we recall that the weak convergence in K ⊂ C(T  , E) is exactly the weak pointwise convergence. Let {y n (·)} n≥ be a sequence in K such that y n (·) converges weakly to y(·) in K . Then y n (t) converges weakly to y(t) in E for each t ∈ T  .
Since K is a closed convex set, by Mazur's lemma we have y(·) ∈ K . Further, by (H) it follows that f (t, y n (t)) converges weakly to f (t, y(t)) for each t ∈ T  . Then the Lebesgue dominated convergence theorem for the Pettis integral (see []) yields I α y n (t) converging weakly to I α y(t) in E for each t ∈ T  . Since K is an equicontinuous subset of C(T  , E) it follows that Q(·) is weakly-weakly sequentially continuous. Suppose that V ⊂ K is such that V = co(Q(V ) ∪ {y(·)}) for some y(·) ∈ K . We will show that V is relatively weakly compact in (α) f (s, y(s)) ds =  then, by the Hahn-Banach theorem, there exists a y * ∈ E * with y * =  and and thus using property () of the noncompactness measure we infer Hence, there exists δ >  such that for all τ , s, ξ ∈ [, tξ ] with |τ -s| < δ and |τξ | < δ. Consider a partition of the interval [, tη] into n parts  = t  < t  < · · · < t n = tη such that t it i- < δ, i = , , . . . , n.
and so Using () we have This implies that Thus we obtain g v(s) ds + ε t αη α +  .
As the last inequality is true, for every ε > , we infer Since g(·) is a Gripenberg function, it follows that v(t) =  for t ∈ T  . Since V as a subset of K is equicontinuous, by Lemma  we infer Thus, by Arzelá-Ascoli's theorem we find that V is weakly relatively compact in C(T  , E). Using Lemma  there exists a fixed point of the operator Q which is a solution of ().
If E is reflexive and f (·, ·) : T × E → E is bounded, then (H) 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 α =  we put D  p y(·) = y p (·), then from Theorem  we obtain the following result (see [, ]).
Corollary  If f (·, ·) : T × E → E is a function that satisfies the conditions (H)-(H) in Theorem , then the differential equation y p (t) = f (t, y(t)), y() = y  , has a solution on [, a] with a = min{b, /M}.

Multi-term fractional differential equation
The case of multi-term fractional differential equations in reflexive Banach spaces was recently considered in [-]. Consider the following multi-term fractional differential