Existence of Mild Solutions for Impulsive Fractional Functional Differential Equations of Order α ∈ (1, 2)

Abstract

This paper investigates the existence result for fractional order functional differential equations subject to non-instantaneous impulsive condition by applying the classical fixed point technique. At last, an example involving partial derivatives is presented to verify the uniqueness result.

1 Introduction

In this paper, we investigate the existence and uniqueness result of mild solutions for the following non-instantaneous impulsive fractional functional differential equation of the form

$$\displaystyle\begin{array}{rcl} ^{C}D_{ t}^{\alpha }y(t)& =& Ay(t) + f(t,y_{\rho (t,y_{t})}),\;t \in (s_{i},t_{i+1}] \subset J,\;i = 0,1,\ldots,N,\qquad {}\end{array}$$

(1)

$$\displaystyle\begin{array}{rcl} y(t)& =& g_{i}(t,y(t)),\;y'(t) = q_{i}(t,y(t)),\;t \in (t_{i},s_{i}],\;i = 1,2,\ldots,N,{}\end{array}$$

(2)

$$\displaystyle\begin{array}{rcl} y(t)& =& \phi (t),\;y'(t) =\varphi (t),\;t \in [-d,0],{}\end{array}$$

(3)

where^C D _t ^α denotes the Caputo’s fractional derivative of order α ∈ (1, 2) and A: D(A) ⊂ X → X is the sectorial operator defined on a complex Banach space X. Functions f: J × PC ₀ → X;  ρ: J × PC ₀ → [−d, T] are continuous and satisfy some assumptions, where PC ₀ is an abstract space defined in the next section. The map y _t is the element of PC ₀ and defined as y _t (θ) = y(t +θ),  θ ∈ [−d, 0]. J = [0, T] is operational interval such that 0 = t ₀ = s ₀ < t ₁ ≤ s ₁ ≤ t ₂ < ⋯ < t _N  ≤ s _N  ≤ t _N+1 = T are prefixed numbers. Here y ^′ denotes the derivative of y with respect to t and g _i , q _i  ∈ C((t _i , s _i ] × X; X) for all i = 1, 2, …, N. The functions ϕ, φ belong to PC ₀ respectively.

The impulsive differential equations have been appeared as in natural description evolution processes. The impulsive effects may be instantaneous or non-instantaneous which is shown in many disciplines. Instantaneous impulse is characterized by abrupt changes of the state at certain moments, but in case of non-instantaneous impulse, it starts abruptly at the fixed moments as the points t _i , and their action continues on the finite interval [t _i , s _i ]. For the future development and recent update of theory for fractional functional differential equations, we refer the papers [1, 2, 4–6, 9, 10] for state-dependent delay, and for non-instantaneous impulse, one can see the papers [7, 8, 11, 12] and the references therein.

On the available of literature, we found that Hernandez et al. [11] used the first time non-instantaneous impulsive condition for abstract differential equations for order one and established the existence results. Kumar et al. [12] have studied the fractional order problem with non-instantaneous impulse, and by using the Banach fixed point theorem with condensing map, they established the existence and uniqueness results. Motivated by the work [11, 12], we have studied the problem considered in [8] for the order α ∈ (0, 1) and established the existence results of mild solution of problem. Shu et al. [14] gave the definition of mild solution for fractional differential equations of order α ∈ (1, 2) and then established the existence results of mild solutions using the Krasnoselskii’s fixed point theorem and analytic operator theory.

Inspired by the work [11, 12, 14] and by the survey, we found that there is no literature on fractional functional differential equation with state-dependent delay subject to non-instantaneous impulsive condition of order (1, 2). This is the reason to investigate the problems (1)–(3) and establish the existence of uniqueness result. For further information, we have divided our work in four sections.

2 Preliminary

In this section, we have introduced some notations, basic definitions, and preliminary result, which were required to establish our main results. Let (X, ∥ ⋅ ∥  _X ) be a complex Banach space of functions with the sup-norm ∥ u ∥  _X  = sup _t ∈ J { | u(t) | : u ∈ X}, and let L(X) denote the space of bounded linear operators from X into X endowed with the natural norm of operators denoted by ∥ ⋅ ∥  _L(X).

As usual, PC ₀ = C([−d, 0], X) (with [−d, 0] ⊂ R) is the space formed by all the continuous functions defined from [−d, 0] into X, endowed with the norm

$$\displaystyle{\|u(t)\|_{PC_{0}} =\sup _{t\in [-d,0]}\{\vert u(t)\vert _{X}\}.}$$

In the case of impulsive conditions, we consider

$$\displaystyle{PC_{T} = PC([-d,T];X),\;0 <T <\infty,}$$

which is a Banach space of all such functions u: [−d, T] → X, which are absolutely continuous everywhere except for a finite number of points t _i  ∈ (0, T),  i  =  1, 2, …, N, at which u(t _i ⁺) and u(t _i ⁻) = u(t _i ) exists and endowed with the norm

$$\displaystyle{\|u\|_{PC_{T}} =\sup _{t\in [-d,T]}\{\|u(t)\|_{X},u \in PC_{T}\}.}$$

For a function u ∈ PC _T and i ∈ { 0, 1, …, N}, we introduce the function $\bar{u}_{i} \in A([t_{i},t_{i+1}];X)$ given by

$$\displaystyle\begin{array}{rcl} \bar{u}_{i}(t) = \left \{\begin{array}{lllllll} u(t),\;\mbox{ for}\;\;t \in (t_{i},t_{i+1}], \\ u(t_{i}^{+}),\;\mbox{ for}\;\;t = t_{i}. \end{array} \right.& & {}\\ \end{array}$$

For further analysis, again consider

$$\displaystyle{PC_{T}^{1} = PC([-d,T];X),\;0 <T <\infty,}$$

which is a Banach space of all such functions u: [−d, T] → X, which are absolutely continuously differentiable everywhere except for a finite number of points t _i  ∈ (0, T),  i = 1, 2, …, N, at which u′(t _i ⁺) and u′(t _i ⁻) = u′(t _i ) exists and endowed with the norm

$$\displaystyle{\|u\|_{PC_{T}^{1}} =\sup _{t\in [-d,T]}\left \{\sum _{j=0}^{1}\|u^{j}(t)\|_{ X},u \in PC_{T}^{1}\right \}.}$$

For a function u ∈ PC _T ¹ and i ∈ { 0, 1, …, N}, we introduce the function $\bar{u}_{i} \in C^{1}([t_{i},t_{i+1}];X)$ given by

$$\displaystyle\begin{array}{rcl} \bar{u}_{i}(t) = \left \{\begin{array}{lllllll} u'(t),\;\mbox{ for}\;\;t \in (t_{i},t_{i+1}], \\ u'(t_{i}^{+}),\;\mbox{ for}\;\;t = t_{i}. \end{array} \right.& & {}\\ \end{array}$$

Definition 1.

[13] Caputo’s derivative of order α > 0 with lower limit a, for a function g: [a, ∞) → R such that g ∈ C ⁿ([a, ∞), X) is defined as

$$\displaystyle\begin{array}{rcl} _{a}^{C}D_{ t}^{\alpha }g(t) = \frac{1} {\varGamma (n-\alpha )}\int _{a}^{t}(t - s)^{n-\alpha -1}g^{(n)}(s)ds = _{ a}J_{t}^{n-\alpha }g^{(n)}(t),& & {}\\ \end{array}$$

where n − 1 < α < n,  a ≥ 0,   n ∈ N.

Definition 2.

The Riemann–Liouville fractional integral operator of order α > 0 with lower limit a, for a function g ∈ L _loc ¹([a, ∞), X) is defined by

$$\displaystyle\begin{array}{rcl} _{a}J_{t}^{0}g(t) = g(t),\;_{ a}J_{t}^{\alpha }g(t) = \frac{1} {\varGamma (\alpha )}\int _{a}^{t}(t - s)^{\alpha -1}g(s)ds,\quad \alpha> 0,\;t> 0,& & {}\\ \end{array}$$

where a ≥ 0,   n ∈ N and Γ(⋅ ) denotes the Gamma function.

Definition 3.

Let A: D(A) ⊂ X → X be a closed and linear operator and α, β > 0. We can say that A is the generator of (α, β) operator function if there exists ω ≥ 0 and a strongly continuous function W _α, β : R⁺ → L(X) such that {λ ^α: Reλ > ω} ⊂ ρ(A) and

$$\displaystyle{\lambda ^{\alpha -\beta }(\lambda ^{\alpha }I - A)^{-1}u =\int _{ 0}^{\infty }e^{-\lambda t}W_{\alpha,\beta }(t)udt,\;\mathrm{Re}\lambda>\omega,\;u \in X.}$$

Here W _α, β (t) is called the operator function generated by A.

Remark 1.

The operator function W _α, β (t) is a general case of α-resolvent family and solution operator. In the case β = 1, operator function corresponds to solution operator S _α (t) by Definition 2.1 in [2], whereas in the case β = α, operator function corresponds to α-resolvent family defined in [3] in Definition (2.3), and operator function corresponds to K _α (t) in [14] in the case β = 2.

The following result is based on Definition 2.1 in [11].

Definition 4.

A function y: [−d, T] → X s.t. y ∈ PC _T ¹ is called a mild solution of the problems (1)–(3) if y(0) = ϕ(0), y′(0) = φ(0),  y(t) = g _j (t, y(t)), y′(t) = q _j (t, y(t)) for t ∈ (t _j , s _j ] for each j = 1, 2, …, N, and satisfying the following integral equation

$$\displaystyle\begin{array}{rcl} y(t) = \left \{\begin{array}{llllll} \phi (0)S_{\alpha }(t) +\varphi (0)K_{\alpha }(t) \\ +\int _{ 0}^{t}T_{\alpha }(t)f(s,y_{\rho (s,y_{s})})ds, &t \in [0,t_{1}], \\ g_{i}(s_{i},y(s_{i}))S_{\alpha }(t - s_{i}) + q_{i}(s_{i},y(s_{i}))K_{\alpha }(t - s_{i}) \\ +\int _{ s_{i}}^{t}T_{\alpha }(t - s)f(s,y_{\rho (s,y_{s})})ds, &t \in [s_{i},t_{i+1}], \end{array} \right.& & {}\\ \end{array}$$

for i = 1, 2, …, N.

3 Main Results

In this section, we have established the existence result of solution for the problems (1)–(3). Let A be a sectorial operator and then strongly continuous functions ∥ S _α (t) ∥ ≤ M; ∥ K _α (t) ∥ ≤ M; ∥ T _α (t) ∥ ≤ M. Let us assume the function ρ: [0, T] × PC ₀ → [−d, T] is continuous. Now, we introduce the following assumption:

(H ₁):

The function f is continuous and $\exists$ positive constants L _f1 such that

$$\displaystyle\begin{array}{rcl} & & \|f(t,\psi ) - f(t,\xi )\|_{X} \leq L_{f1}\|\psi -\xi \|_{PC_{0}},\;\forall \;\psi,\xi \in PC_{0}. {}\\ \end{array}$$

(H ₂):

The functions g _i , q _i are continuous and $\exists$ positive constants $L_{g_{i}},L_{q_{i}}$ such that

$$\displaystyle{\|g_{i}(t,x) - g_{i}(t,y)\|_{X} \leq L_{g_{i}}\|x - y\|_{X};\;\|q_{i}(t,x) - q_{i}(t,y)\|_{X} \leq L_{q_{i}}\|x - y\|_{X}}$$

for all x, y ∈ X, t ∈ (t _i , s _i ] and each i = 1, 2, …, N.

Theorem 1.

Let the assumptions (H ₁ ) and (H ₂ ) hold and are constant:

$$\displaystyle{\varDelta =\max \{ MTL_{f_{1}},L_{g_{i}}M + L_{q_{i}}M + MTL_{f1}\} <1,}$$

for i = 1,…,N. Then there exists a unique mild solution y(t) of problems ( 1 )–( 3 ) on J.

Proof.

We convert problems (1)–(3) in to the fixed point problem. Consider $\mathcal{B} =\{ y: y \in PC_{T}^{1},y(0) =\phi (0),y'(0) =\varphi (0)\}$. Define an operator $\mathcal{P}: \mathcal{B}\rightarrow \mathcal{B}$ as

$$\displaystyle\begin{array}{rcl} \mathcal{P}y(t) = \left \{\begin{array}{llllll} \phi (0)S_{\alpha }(t) +\varphi (0)K_{\alpha }(t) \\ +\int _{ 0}^{t}T_{\alpha }(t - s)f(s,y_{\rho (s,y_{s})})ds, &t \in [0,t_{1}], \\ g_{i}(s_{i},y(s_{i}))S_{\alpha }(t - s_{i}) + q_{i}(s_{i},y(s_{i}))K_{\alpha }(t - s_{i}) \\ +\int _{ s_{i}}^{t}T_{\alpha }(t - s)f(s,y_{\rho (s,y_{s})})ds, &t \in [s_{i},t_{i+1}]. \end{array} \right.& &{}\end{array}$$

(4)

It is obvious that $\mathcal{P}$ is well defined. Now, we will express that the operator $\mathcal{P}$ has a unique fixed point. So let $y(t),y^{{\ast}}(t) \in \mathcal{B}$ and t ∈ [0, t ₁]; we get

$$\displaystyle\begin{array}{rcl} \|\mathcal{P}y -\mathcal{P}y^{{\ast}}\|_{ X}& \leq & \int _{0}^{t}\|T_{\alpha }(t - s)\|_{ L(X}\|f(s,y_{\rho (s,y_{s})}) - f(s,y_{\rho (s,y_{s}^{{\ast}})}^{{\ast}})\|_{ X}ds {}\\ & \leq & TML_{f1}\|y - y^{{\ast}}\|_{ X}. {}\\ \end{array}$$

For t ∈ [s _i , t _i+1], we have

$$\displaystyle\begin{array}{rcl} \|\mathcal{P}y -\mathcal{P}y^{{\ast}}\|_{ X}& \leq & \|g_{i}(s_{i},y(s_{i})) - g_{i}(s_{i},y^{{\ast}}(s_{ i}))\|_{X}\|S_{\alpha }(t - s_{i})\|_{L(X} {}\\ & & +\|q_{i}(s_{i},y(s_{i})) - q_{i}(s_{i},y^{{\ast}}(s_{ i}))\|_{X}\|K_{\alpha }(t - s_{i})\|_{L(X} {}\\ & & +\int _{s_{i}}^{t}\|T_{\alpha }(t - s)\|_{ L(X}\|f(s,y_{\rho (s,y_{s})}) - f(s,y_{\rho (s,y_{s}^{{\ast}})}^{{\ast}})\|_{ X}ds {}\\ & \leq & (L_{g_{i}}M + L_{q_{i}}M + TML_{f1})\|y - y^{{\ast}}\|_{ X}. {}\\ \end{array}$$

For t ∈ (t _j , s _j ], we get

$$\displaystyle\begin{array}{rcl} \|\mathcal{P}y -\mathcal{P}y^{{\ast}}\|_{ X}& \leq & L_{g_{j}}\|y - y^{{\ast}}\|_{ X},\;j = 1,2,\ldots,N. {}\\ \end{array}$$

Gathering above results, we obtain

$$\displaystyle\begin{array}{rcl} \|\mathcal{P}y -\mathcal{P}y^{{\ast}}\|_{ X}& \leq & \max \{MTL_{f_{1}},L_{g_{i}}M + L_{q_{i}}M + MTL_{f1}\}\|y - y^{{\ast}}\|_{ X} {}\\ & \leq & \varDelta \|y - y^{{\ast}}\|_{ X}. {}\\ \end{array}$$

Since Δ < 1, which implies that $\mathcal{P}$ is a contraction map, there exists a unique fixed point which is the mild solution of problems (1)–(3) on J.

4 Example

In this section, we gave an example to illustrate our main result. Consider the following fractional order functional differential equation:

$$\displaystyle\begin{array}{rcl} \frac{\partial ^{\alpha }u(t,x)} {\partial t^{\alpha }} & =& \frac{\partial ^{2}u(t,x)} {\partial y^{2}} + \frac{u(t -\sigma (\|u\|),x)} {49},(t,x) \in \cup _{i=1}^{N}[s_{ i},t_{i+1}] \times [0,\pi ],{}\end{array}$$

(5)

$$\displaystyle\begin{array}{rcl} u(t,0)& =& u(t,\pi ) = 0,\;\;t \geq 0,{}\end{array}$$

(6)

$$\displaystyle\begin{array}{rcl} u(t,x)& =& \phi (t,x),u'(t,x) =\varphi (t,x),t \in [-d,0],x \in [0,\pi ],{}\end{array}$$

(7)

$$\displaystyle\begin{array}{rcl} u(t,x)& =& G_{i}(t,y);\;u'(t,x) = H_{i}(t,y),\;t \in (t_{i},s_{i}].{}\end{array}$$

(8)

where $\frac{\partial ^{\alpha }} {\partial t^{\alpha }}$ denotes the partial Caputo’s fractional derivative of order α ∈ (1, 2), 0 = t ₀ = s ₀ < t ₁ ≤ s ₁ < ⋯ < t _N  ≤ sN < t _N+1 = 1 are prefixed numbers, and ϕ, φ ∈ PC ₀. Let X = L ²[0, π] be a Banach space and define the operator A: D(A) ⊂ X → X by Ay = y″ with the domain D(A): = { y ∈ X:   y, y′ to be absolutely continuous, y″ ∈ X, y(0) = 0 = y(π)}. Then

$$\displaystyle{Ay =\sum _{ n=1}^{\infty }n^{2}(y,y_{ n})y_{n},\;y \in D(A),}$$

where set $y_{n}(x) = \sqrt{\frac{2} {\pi }} \sin (nx),\;n \in N$ is the space of eigenvectors of A in which element is orthogonal. It is clear that that the operator A stays the infinitesimal generator of an analytic semigroup operator (T(t)) _t ≥ 0 in Banach space X and is defined as

$$\displaystyle{T(t)\omega =\sum _{ n=1}^{\infty }e^{-n^{2}t }(\omega,\omega _{n})\omega _{n},\;\mbox{ for all}\;\omega \in X,\;\mbox{ and every}\;t> 0.}$$

The subordination opinion of solution operator implies that A stays the infinitesimal generator of K(t), S(t). Since K(t), S(t) are strongly continuous operators on interval [0, ∞) by the theorem of uniformly boundedness, there exists a constant M > 0 such that ∥ S(t) ∥ ≤ M, ∥ K(t) ∥ ≤ M for t ∈ [0, 1]. We have for (t, ϕ) ∈ [0, 1] × PC ₀.

Setting u(t)(x) = u(t, x), and

$$\displaystyle{\rho (t,\phi ) = t -\sigma (\|\phi (0)\|),\;(t,\phi ) \in J \times PC_{0},}$$

we have

$$\displaystyle{f(t,\phi ) = \frac{\phi } {49};\;g_{i}(t,y) = G_{i}(t,y);\;q_{i}(t,y) = H_{i}(t,y),}$$

then by the above Eqs. (5)–(8) can be composed in the given abstract form as (1)–(3). Furthermore, we can see that for (t, ϕ), (t, ψ) ∈ J × PC ₀, we may verify that

$$\displaystyle\begin{array}{rcl} \|f(t,\phi ) - f(t,\psi )\|_{L^{2}}& \leq & \left [\int _{0}^{\pi }\left \{\| \frac{\phi } {49} - \frac{\psi } {49}\|\right \}^{2}dy\right ]^{1/2} \leq \frac{\sqrt{\pi }} {49}\|\phi -\psi \|. {}\\ \end{array}$$

Hence function f satisfies (H ₁). Similarly we can show that the functions g _i ,  q _i satisfy (H ₂). All the conditions of Theorem 1 have been satisfied, so we can drive that the system (5)–(8) has a unique mild solution on [0, 1].