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Existence and continuation of solutions of Hilfer fractional differential equations

In this paper we study the existence and continuation of solution to general fractional differential equation with Hilfer fractional derivative. First we establish new local existence theorems. Then we derive the continuation theorems. With the help of continuation theorems derived in this paper, several global existence results are constructed.


Introduction
Theory of fractional order systems has gained remarkable significance during last few decades due to its real world applications in ostensibly diverse and wide spread fields of applied mathematics, physics and engineering. The monographs [11,13,15,16,19] are devoted to such practical problems in control theory, modeling, relaxations and serve as a foundation of fractional order theory in physics and applied sciences. Recently, Hilfer [11,12], Mainardi [16] discussed various applications of fractional differential equations in their works. Many complex phenomena in nature can be described more accurately using various fractional operators and are characterized by rapid change in their state.
Nowadays, numerous fractional differential operators are present in literature but Riemann-Liouville (R-L) [13] and Caputo [2,16] are universally accepted approaches. R-L operator places less constraints on concerned function but fails to physically sound with practically applicable initial conditions. To avoid such a difficulty, scientists accepted Caputo approach which admits lot of properties from classical calculus. But in [7], it has shown that, Caputo derivative also has some defects in applications. Concretely, as in [7], one has Observe that, if x (n) (a) = 0 near an integer n, a very small error of measurement on fractional order may result in totally unlike results, which implies a common case of fractional dynamic system when system starts from non-constant state.
Such a problem does not arises in R-L sense. Additionally, when the Caputo fractional derivative is applied to describe the Nutting's law [8,16,18], says a constant strain ǫ implies stress is independent of time t, i.e. σ ≡ 0. This violates the physical properties of real viscoelastic materials. While in the R-L theory, the constant strain ǫ does not lead to constant stress [21]. Analogously, we prefer the Hilfer (generalized Riemann-Liouville) derivative operator which interpolates the both R-L as well as Caputo sense.
In the recent investigations, many researchers studied the existence and uniqueness of solution of nonlinear fractional differential equations, see [1,4,6,8,9,14,17,19,20,22] and references therein. Infact, the global existence of solution of fractional differential equation is one of the elementary property, see [13,19]. In this paper we mainly focus on developing the theory of existence and uniqueness. First we obtain the local existences followed by continuation theorems to extend the existence of solutions globally. The results obtained in this paper generalizes the existing results [3,5] in the literature. Equivalently, the works by C. Kou, H. Zhou, C. P. Li [3] and C. Li, S. Sarwar [5] and references therein follow as particular cases of our main results.
The rest of the article is organized as follows: in section 2, we collect all the useful definitions and previously known lemmas which are used in construction of our main results. Section 3 devoted to local existence of solutions followed by continuation results with global existence theorems in section 4. Concluding remarks are given in the last section.

Prerequisites
This section is devoted to basic definitions and lemmas from [13] the theory of fractional calculus which are used in subsequent sections. Let C 1−γ [0, T ] is a complete metric space of all continuous functions mapping [0, T ] into R with the metric d defined by [9] d(x 1 , Definition 2.1 [4] Let Ω = (0, T ] and f : (0, ∞) → R is a real valued continuous function. The Riemann-Liouville fractional integral of a function f of order α ∈ R + is denoted as I α 0 + f and defined by where Γ(α) is the Euler's Gamma function.

Remark 2.1 The Hilfer fractional derivative interpolates between the R-L and Caputo fractional derivative since
For the analysis we consider the initial value problem x), t ∈ (0, +∞), 4) and the initial value problem (IVP) for the system of differential equations ., x n ), have weak singularities with respect to t and satisfies the Lipschitz conditions respectively to ensure the existence of unique solutions. Furthermore, the equivalence of Hilfer fractional IVP and it's equivalent integral equation is established in [9] the following lemma.
, then x satisfies IVP (2.4) if and only if x satisfies the Volterra fractional integral equation of the second kind (2.6) Then M is precompact if and only if the following conditions hold: Then Lemma 2.4 Schauder Fixed Point Theorem: [10] Let U be a closed bounded convex subset of a Banach space X and T : U → U is completely continuous. Then T has a fixed point in U.

Local existence
In this section, we obtain the local existence of solutions of IVPs (2.4) and (2.5). For this, let us make the following two hypothesis.

) be a continuous functions and there exists constants 0
closed bounded and convex subset. Note that h ≤ T, define the operator B as follows: (3.1) It follows from (H 1 ) and Lemma 2.5 that we have On the other hand by relation (3.1), for any .
Next we show that operators B i are continuous. Let x m , x i ∈ D ih , m > n, i = 1, 2, .., n such that x m − x i → 0 as m → +∞. In view of continuity of Furthermore, we prove that operators B i D ih are continuous. Let x i ∈ D ih and 0 ≤ t 1 < t 2 ≤ h. For any ǫ > 0, note that holds. In this case, for t 1 , t 2 ∈ [0,δ i ], we have In the case for t 1 , t 2 ∈ [δ i 2 , h], we get We see from the fact that if 0 ≤ µ 1 < µ 2 ≤ h, then for 0 ≤ s < µ 1 , we have (µ 2 − s) α−1 and we obtain from the first term on right hand side of inequality (3.6) that On the other hand from second term on right hand side of equation (3.6) From above discussion, there exist a λ, (δ i 2 >)λ > 0 such that for t 1 , t 2 ∈ [δ i 2 , h] and |t 1 − t 2 | < λ implies It follows from equations (3.5) and (3.7) that

Continuation and global existence
In this section, we concerned with continuation of solution of IVP (2.4) and then we obtain the global existence. Initially, we need the following definition. Then Theorem 4.1 (Continuation Theorem I) Assume that (H 1 ) hold. Then x = x(t), t ∈ (0, ν) is noncontinuable if and only if for some τ ∈ (0, ν 2 ) and any bounded closed subset D ⊂ [τ, +∞) × R, there exists a t * ∈ [τ, ν) such that (t * , x(t * )) / ∈ D.
Step:2. Now we show that x(t) is continuable. Since Ω is closed subset, we have (ν, where y ∈ [ν, ν + 1] and In view of Lemma 2.5 and Lemma 4. In view of continuity of f on E b , we denote M = max (t,y)∈E b |f (t, y)|. Let We claim that N is completely continuous on E h . First we show the operator N is continuous. In fact, let {y n } ⊆ C[ν, ν + h], y n − y [ν,ν+h] → 0 as n → +∞. Then we have By virtue of continuity of f on E b , we have N y n − N y [ν,ν+h] → 0 as n → +∞, which implies that N is continuous. Secondly, we prove that N E h is euqicontinuous. For any y ∈ E h , for which (N y)(ν) = x 1 (ν) and In view of uniform continuity of I(t) on [ν, ν + h] and relation (4.3), we obtain {(N y)(t) : y ∈ E h } is equicontinuous. Therefore N is completely continuous. By Schauder's fixed point theorem, N has a fixed pointx(t) ∈ E h . i.e.
Since {x(t n )} is bounded convergent subsequence, without loss of generality, set lim n→+∞ x(t n ) = x * . Now we show that, for any given ǫ > 0, there exists T ∈ (0, ν), such that |x(t) − x * | < ǫ, t ∈ (T, ν), we have lim t→ν − x(t n ) = x * .  The existence of global solution can be obtained by an arguments similar as above. From the Lipschitz-type condition and Lemma 4.2, we can conclude the uniqueness of global solution. We omit the proof here.

Concluding remarks
In this paper, the global existence of a unique solution of nonlinear IVP with Hilfer fractional derivative is proved with the help of fixed point technique and continuation theorems. Continuation theorem 2 is conveniently more applicable in practical problems. Our results in this paper generalizes the existing results in the literature.