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Existence and Ulam stability for implicit fractional q-difference equations

Abstract

This paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

Introduction

Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics, and other applied sciences [33]. For some fundamental results in the theory of fractional calculus and fractional differential equations we refer the reader to the monographs [4,5,6, 23, 32, 39], the papers [24, 34, 36,37,38, 40] and the references therein. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations and inclusions with Caputo fractional derivative; [5, 22]. Implicit fractional differential equations were analyzed by many authors; see for instance [4, 5, 11,12,13] and the references therein.

Considerable attention has been given to the study of the Ulam stability of functional differential and integral equations; see the monographs [6, 19], the papers [1,2,3, 20, 28, 30, 31] and the references therein.

Fractional q-difference equations were initiated in the beginning of the 19th century [7, 15], and received significant attention in recent years. Some interesting results about initial and boundary value problems of q-difference and fractional q-difference equations can be found in [9, 10, 16, 17] and the references therein. An implicit fractional q-integral equation is considered in [18].

In this paper we discuss the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions for the following implicit fractional q-difference equation:

$$ \bigl({}^{c}D_{q}^{\alpha}u\bigr) (t)=f \bigl(t,u(t),\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t) \bigr),\quad t\in I:=[0,T], $$
(1)

with the initial condition

$$ u(0)=u_{0}, $$
(2)

where \(q\in(0,1)\), \(\alpha\in(0,1]\), \(T>0\), \(f:I\times{\Bbb {R}}\times {\Bbb {R}}\to{\Bbb {R}}\) is a given continuous function, and \({}^{c}D_{q}^{\alpha}\) is the Caputo fractional q-difference derivative of order α.

This paper initiates the study of implicit Caputo fractional q-difference equations.

Preliminaries

Consider the complete metric space \(C(I):=C(I,{\Bbb {R}})\) of continuous functions from I into \({\Bbb {R}}\) equipped with the usual metric

$$d(u,v):=\max _{t\in I} \bigl\vert u(t)-v(t) \bigr\vert . $$

Notice that \(C(I)\) is a Banach space with the supremum (uniform) norm

$$\Vert u \Vert _{\infty}:=\sup _{t\in I} \bigl\vert u(t) \bigr\vert . $$

As usual, \(L^{1}(I)\) denotes the space of measurable functions \(v:I\rightarrow{\Bbb {R}}\) which are Lebesgue integrable with the norm

$$\Vert v \Vert _{1}= \int_{I} \bigl\vert v(t) \bigr\vert \, dt. $$

Let us recall some definitions and properties of fractional q-calculus. For \(a\in\Bbb {R}\), we set

$$[a]_{q}=\frac{1-q^{a}}{1-q}. $$

The q analogue of the power \((a-b)^{n}\) is

$$(a-b)^{(0)}=1,\quad\quad (a-b)^{(n)}=\prod_{k=0}^{n-1} \bigl(a-bq^{k}\bigr); \quad a,b\in{\Bbb {R}}, n\in{\Bbb {N}}. $$

In general,

$$(a-b)^{(\alpha)}=a^{\alpha}\prod_{k=0}^{\infty} \biggl( \frac {a-bq^{k}}{a-bq^{k+\alpha}} \biggr);\quad a,b,\alpha\in{\Bbb {R}}. $$

Definition 2.1

([21])

The q-gamma function is defined by

$$\varGamma_{q}(\xi)=\frac{(1-q)^{(\xi-1)}}{(1-q)^{\xi-1}};\quad \xi\in{\Bbb {R}}-\{0,-1,-2,\ldots\}. $$

Notice that the q-gamma function satisfies \(\varGamma_{q}(1+\xi)=[\xi ]_{q}\varGamma_{q}(\xi)\).

Definition 2.2

([21])

The q-derivative of order \(n\in{\Bbb {N}}\) of a function \(u:I\to{\Bbb {R}}\) is defined by \((D_{q}^{0}u)(t)=u(t)\),

$$(D_{q}u) (t):=\bigl(D_{q}^{1}u\bigr) (t)= \frac{u(t)-u(qt) }{(1-q)t} ;\quad t\neq0,\qquad (D_{q}u) (0)=\lim_{t\to0}(D_{q}u) (t), $$

and

$$\bigl(D_{q}^{n}u\bigr) (t)=\bigl(D_{q}D_{q}^{n-1}u \bigr) (t);\quad t\in I, n\in\{1,2,\ldots\}. $$

Set \(I_{t}:=\{tq^{n}:n\in{\Bbb {N}}\}\cup\{0\}\).

Definition 2.3

([21])

The q-integral of a function \(u:I_{t}\to{\Bbb {R}}\) is defined by

$$(I_{q}u) (t)= \int_{0}^{t}u(s)\,d_{q}s= \sum _{n=0}^{\infty}t(1-q)q^{n}f\bigl(tq^{n} \bigr), $$

provided that the series converges.

We note that \((D_{q}I_{q}u)(t)=u(t)\), while if u is continuous at 0, then

$$(I_{q}D_{q}u) (t)=u(t)-u(0). $$

Definition 2.4

([8])

The Riemann–Liouville fractional q-integral of order \(\alpha\in{\Bbb {R}}_{+}:=[0,\infty)\) of a function \(u:I\to{\Bbb {R}}\) is defined by \((I_{q}^{0}u)(t)=u(t)\), and

$$\bigl(I_{q}^{\alpha}u\bigr) (t)= \int_{0}^{t}\frac{(t-qs)^{(\alpha-1)}}{\varGamma _{q}(\alpha)}u(s) \,d_{q}s;\quad t\in I. $$

Lemma 2.5

([26])

For \(\alpha\in{\Bbb {R}}_{+}:=[0,\infty)\)and \(\lambda\in(-1,\infty)\)we have

$$\bigl(I_{q}^{\alpha}(t-a)^{(\lambda)}\bigr) (t)= \frac{\varGamma_{q}(1+\lambda )}{\varGamma(1+\lambda+\alpha)}(t-a)^{(\lambda+\alpha)};\quad 0< a< t< T. $$

In particular,

$$\bigl(I_{q}^{\alpha}1\bigr) (t)=\frac{1}{\varGamma_{q}(1+\alpha)}t^{(\alpha)}. $$

Definition 2.6

([27])

The Riemann–Liouville fractional q-derivative of order \(\alpha\in {\Bbb {R}}_{+}\) of a function \(u:I\to{\Bbb {R}}\) is defined by \((D_{q}^{0}u)(t)=u(t)\), and

$$\bigl(D_{q}^{\alpha}u\bigr) (t)=\bigl(D_{q}^{[\alpha]}I_{q}^{[\alpha]-\alpha}u \bigr) (t);\quad t\in I, $$

where \([\alpha]\) is the integer part of α.

Definition 2.7

([27])

The Caputo fractional q-derivative of order \(\alpha\in{\Bbb {R}}_{+}\) of a function \(u:I\to{\Bbb {R}}\) is defined by \(({}^{C}D_{q}^{0}u)(t)=u(t)\), and

$$\bigl({}^{C}D_{q}^{\alpha}u\bigr) (t)= \bigl(I_{q}^{[\alpha]-\alpha}D_{q}^{[\alpha]}u\bigr) (t);\quad t \in I. $$

Lemma 2.8

([27])

Let \(\alpha\in{\Bbb {R}}_{+}\). Then the following equality holds:

$$\bigl(I_{q}^{\alpha} {}^{C}D_{q}^{\alpha}u \bigr) (t)=u(t)-\sum _{k=0}^{[\alpha ]-1}\frac{t^{k}}{\varGamma_{q}(1+k)} \bigl(D_{q}^{k}u\bigr) (0). $$

In particular, if \(\alpha\in(0,1)\), then

$$\bigl(I_{q}^{\alpha} {}^{C}D_{q}^{\alpha}u \bigr) (t)=u(t)-u(0). $$

From the above lemma, and in order to define the solution for the problem (1)–(2), we need the following lemma.

Lemma 2.9

Let \(f:I\times{\Bbb {R}}\times{\Bbb {R}}\rightarrow{\Bbb {R}}\)such that \(f(\cdot,u,v)\in C(I)\), for each \(u,v\in{\Bbb {R}}\). Then the problem (1)(2) is equivalent to the problem of obtaining the solutions of the integral equation

$$ g(t)=f\bigl(t,u_{0}+\bigl(I_{q}^{\alpha}g \bigr) (t),g(t)\bigr), $$
(3)

and if \(g(\cdot)\in C(I)\), is the solution of this equation, then

$$u(t)=u_{0}+\bigl(I_{q}^{\alpha}g\bigr) (t). $$

Proof

Let u be a solution of problem (1)–(2), and let \(g(t)=({}^{C}D_{q}^{\alpha}u)(t)\); for \(t\in I\). We will prove that \(u(t)=u_{0}+(I_{q}^{\alpha}g)(t)\), and g satisfies Eq. (3). From Lemma 2.8, we have \(u(t)=u_{0}+(I_{q}^{\alpha}g)(t)\), and it is easy to see that Eq. (1) implies (3). Reciprocally, if u satisfies the integral equation \(u(t)=u_{0}+(I_{q}^{\alpha}g)(t)\), and if g satisfies Eq. (3), then u is a solution of the problem (1)–(2). □

Now, we consider the Ulam stability for the problem (1)–(2). Let \(\epsilon>0\) and \(\varPhi:I\to{\Bbb {R}}_{+}\) be a continuous function. We consider the following inequalities:

$$\begin{aligned}& \bigl|\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t)-f \bigl(t,u(t),\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t) \bigr)\bigr|\leq \epsilon;\quad t\in I, \end{aligned}$$
(4)
$$\begin{aligned}& \bigl|\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t)-f \bigl(t,u(t),\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t) \bigr)\bigr|\leq \varPhi(t);\quad t\in I, \end{aligned}$$
(5)
$$\begin{aligned}& \bigl|\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t)-f \bigl(t,u(t),\bigl({}^{c}D_{q}^{\alpha}u\bigr) (t) \bigr)\bigr|\leq \epsilon\varPhi(t);\quad t\in I. \end{aligned}$$
(6)

Definition 2.10

([5, 30])

The problem (1)–(2) is Ulam–Hyers stable if there exists a real number \(c_{f}>0\) such that for each \(\epsilon>0\) and for each solution \(u\in C(I)\) of the inequality (4) there exists a solution \(v\in C(I)\) of (1)–(2) with

$$\bigl\vert u(t)-v(t) \bigr\vert \leq\epsilon c_{f};\quad t\in I. $$

Definition 2.11

([5, 30])

The problem (1)–(2) is generalized Ulam–Hyers stable if there exists \(c_{f}:C({\Bbb {R}}_{+},{\Bbb {R}}_{+})\) with \(c_{f}(0)=0\) such that for each \(\epsilon>0\) and for each solution \(u\in C(I)\) of the inequality (4) there exists a solution \(v\in C(I)\) of (1)–(2) with

$$\bigl\vert u(t)-v(t) \bigr\vert \leq c_{f}(\epsilon);\quad t\in I. $$

Definition 2.12

([5, 30])

The problem (1)–(2) is Ulam–Hyers–Rassias stable with respect to Φ if there exists a real number \(c_{f,\varPhi}>0\) such that for each \(\epsilon>0\) and for each solution \(u\in C(I)\) of the inequality (6) there exists a solution \(v\in C(I)\) of (1)–(2) with

$$\bigl\vert u(t)-v(t) \bigr\vert \leq\epsilon c_{f,\varPhi}\varPhi(t);\quad t\in I. $$

Definition 2.13

([5, 30])

The problem (1) is generalized Ulam–Hyers–Rassias stable with respect to Φ if there exists a real number \(c_{f,\varPhi}>0\) such that for each solution \(u\in C_{\gamma,\ln}\) of the inequality (5) there exists a solution \(v\in C_{\gamma,\ln}\) of (1)–(2) with

$$\bigl\vert u(t)-v(t) \bigr\vert \leq c_{f,\varPhi}\varPhi(t);\quad t\in I. $$

Remark 2.14

It is clear that

  1. (i)

    Definition 2.10 Definition 2.11,

  2. (ii)

    Definition 2.12 Definition 2.13,

  3. (iii)

    Definition 2.12 for \(\varPhi(\cdot)=1\) Definition 2.10.

One can have similar remarks for the inequalities (4) and (6).

Definition 2.15

([29])

A nondecreasing function \(\phi:{\Bbb {R}}_{+}\to{\Bbb {R}}_{+}\) is called a comparison function if it satisfies one of the following conditions:

  1. (1)

    For any \(t>0\) we have

    $$\lim _{n\to\infty} \phi^{(n)}(t)=0, $$

    where \(\phi^{(n)}\) denotes the nth iteration of ϕ.

  2. (2)

    The function ϕ is right-continuous and satisfies

    $$\phi(t)< t\quad \forall t>0. $$

Remark 2.16

The choice \(\phi(t)=kt\) with \(0< k<1\) gives the classical Banach contraction mapping principle.

For our purpose we will need the following fixed point theorems.

Theorem 2.17

([14, 25])

Let \((X,d)\)be a complete metric space and \(T:X\to X\)be a mapping such that

$$d\bigl(T(x),T(y)\bigr)\leq\phi\bigl(d(x,y)\bigr), $$

whereϕis a comparison function. ThenThas a unique fixed point inX.

Theorem 2.18

(Schauder fixed point theorem [35])

LetXbe a Banach space, Dbe a bounded closed convex subset ofXand \(T:D\to D\)be a compact and continuous map. ThenThas at least one fixed point inD.

Existence results

In this section, we are concerned with the existence and uniqueness of solutions of the problem (1)–(2).

Definition 3.1

By a solution of the problem (1)–(2) we mean a continuous function \(u\in C(I)\) that satisfies Eq. (1) on I and the initial condition (2).

The following hypotheses will be used in the sequel.

\((H_{1})\):

The function f satisfies the generalized Lipschitz condition:

$$\bigl\vert f(t,u_{1},v_{1})-f(t,u_{2},v_{2}) \bigr\vert \leq\phi_{1}\bigl( \vert u_{1}-u_{2} \vert \bigr)+\phi_{2}\bigl( \vert v_{1}-v_{2} \vert \bigr), $$

for \(t\in I\) and \(u_{1},u_{2},v_{1},v_{2}\in{\Bbb {R}}\), where \(\phi_{1}\) and \(\phi_{2}\) are comparison functions.

\((H_{2})\):

There exist functions \(p,d,r\in C(I,[0,\infty))\) with \(r(t)<1\) such that

$$\bigl\vert f(t,u,v) \bigr\vert \leq p(t)+d(t) \vert u \vert +r(t) \vert v \vert ,\quad \text{for each } t\in I \text{ and } u,v\in{\Bbb {R}}. $$
Set

$$p^{\ast}=\sup _{t\in I}p(t),\qquad d^{\ast}=\sup _{t\in I}d(t),\qquad r^{\ast }=\sup _{t\in I}r(t). $$

First, we prove an existence and uniqueness result for the problem (1)–(2).

Theorem 3.2

Assume that the hypothesis \((H_{1})\)holds. Then there exists a unique solution of problem (1)(2) onI.

Proof

By using Lemma 2.9, we transform the problem (1)–(2) into a fixed point problem. Consider the operator \(N:C(I)\rightarrow C(I)\) defined by

$$ (Nu) (t)=u_{0}+\bigl(I_{q}^{\alpha}g \bigr) (t);\quad t\in I, $$
(7)

where \(g\in C(I)\) such that

$$g(t)=f\bigl(t,u(t),g(t)\bigr), \quad\text{or}\quad g(t)=f\bigl(t,u_{0}+ \bigl(I_{q}^{\alpha}g\bigr) (t),g(t)\bigr). $$

Let \(u,v\in C(I)\). Then, for \(t\in I\), we have

$$ \bigl\vert (Nu) (t)-(Nv) (t) \bigr\vert \leq \int_{0}^{t}\frac{(t-qs)^{(\alpha-1)}}{\varGamma _{q}(\alpha)} \bigl\vert g(s)-h(s) \bigr\vert \,d_{q}s, $$
(8)

where \(g,h\in C(I)\) such that

$$g(t)=f\bigl(t,u(t),g(t)\bigr) $$

and

$$h(t)=f\bigl(t,v(t),h(t)\bigr). $$

From \((H_{1})\), we obtain

$$\bigl\vert g(t)-h(t) \bigr\vert \leq\phi_{1}\bigl(\bigl|u(t)-v(t)\bigr|\bigr)+ \phi_{2}\bigl(\bigl|g(t)-h(t)\bigr|\bigr). $$

Thus

$$\bigl\vert g(t)-h(t) \bigr\vert \leq(\mathit{Id}-\phi_{2})^{-1} \phi_{1}\bigl(\bigl|u(t)-v(t)\bigr|\bigr), $$

where Id is the identity function.

Set

$$L:=\sup _{t\in I} \int_{0}^{T}\frac{(t-qs)^{(\alpha-1)}}{\varGamma _{q}(\alpha)}\,d_{q}s, $$

and \(\phi:=L(\mathit{Id}-\phi_{2})^{-1}\phi_{1}\). From (8), we get

$$\begin{aligned} \bigl\vert (Nu) (t)-(Nv) (t) \bigr\vert &\leq\phi \bigl( \bigl\vert u(t)-v(t) \bigr\vert \bigr) \\ &\leq\phi\bigl(d(u,v)\bigr). \end{aligned} $$

Hence, we get

$$d\bigl(N(u),N(v)\bigr)\leq\phi\bigl(d(u,v)\bigr). $$

Consequently, from Theorem 2.17, the operator N has a unique fixed point, which is the unique solution of the problem (1)–(2). □

Theorem 3.3

Assume that the hypothesis \((H_{2})\)holds. If

$$r^{\ast}+Ld^{\ast}< 1, $$

then the problem (1)(2) has at least one solution defined onI.

Proof

Let N be the operator defined in (7). Set

$$R\geq\frac{Lp^{\ast}}{1-r^{\ast}-Ld^{\ast}}, $$

and consider the closed and convex ball \(B_{R}=\{u\in C(I):\|u\| _{\infty}\leq R\}\).

Let \(u\in B_{R}\). Then, for each \(t\in I\), we have

$$\bigl\vert (Nu) (t) \bigr\vert \leq \int_{0}^{t}\frac{(t-qs)^{(\alpha-1)}}{\varGamma _{q}(\alpha)} \bigl\vert g(s) \bigr\vert \,d_{q}s, $$

where \(g\in C(I)\) such that

$$g(t)=f\bigl(t,u(t),g(t)\bigr). $$

By using \((H_{2})\), for each \(t\in I\) we have

$$\begin{aligned} \bigl\vert g(t) \bigr\vert &\leq p(t)+d(t) \bigl\vert u(t) \bigr\vert +r(t) \bigl\vert g(t) \bigr\vert \\ &\leq p^{\ast}+d^{\ast} \Vert u \Vert _{\infty}+r^{\ast} \bigl\vert g(t) \bigr\vert \\ &\leq p^{\ast}+d^{\ast}R+r^{\ast} \bigl\vert g(t) \bigr\vert . \end{aligned} $$

Thus

$$\bigl\vert g(t) \bigr\vert \leq\frac{p^{\ast}+d^{\ast}R}{1-r^{\ast}}. $$

Hence

$$\bigl\Vert N(u) \bigr\Vert _{\infty}\leq\frac{L(p^{\ast}+d^{\ast}R)}{1-r^{\ast}}, $$

which implies that

$$\bigl\Vert N(u) \bigr\Vert _{\infty}\leq R. $$

This proves that N maps the ball \(B_{R}\) into \(B_{R}\). We shall show that the operator \(N:B_{R}\to B_{R}\) is continuous and compact. The proof will be given in three steps.

Step 1: N is continuous.

Let \(\{u_{n}\}_{n\in\mathbb{N}}\) be a sequence such that \(u_{n}\rightarrow u\) in \(B_{R}\). Then, for each \(t\in I\), we have

$$\begin{aligned} \bigl\vert (Nu_{n}) (t)-(Nu) (t) \bigr\vert &\leq \int_{0}^{t}\frac{(t-qs)^{(\alpha -1)}}{\varGamma_{q}(\alpha)} \bigl\vert \bigl(g_{n}(s)-g(s)\bigr) \bigr\vert \,d_{q}s, \end{aligned} $$

where \(g_{n},g\in C(I)\) such that

$$g_{n}(t)=f\bigl(t,u_{n}(t),g_{n}(t)\bigr) $$

and

$$g(t)=f\bigl(t,u(t),g(t)\bigr). $$

Since \(u_{n}\rightarrow u\) as \(n\rightarrow\infty\) and f is continuous function, we get

$$g_{n}(t)\rightarrow g(t)\quad \text{as } n\rightarrow\infty, \text{ for each } t\in I. $$

Hence

$$\bigl\Vert N(u_{n})-N(u) \bigr\Vert _{\infty}\leq \frac{p^{\ast}+d^{\ast}R}{1-r^{\ast}} \Vert g_{n}-g \Vert _{\infty}\rightarrow0\quad \text{as }n\rightarrow\infty. $$

Step 2: \(N(B_{R})\) is bounded. This is clear since \(N(B_{R})\subset B_{R}\) and \(B_{R}\) is bounded.

Step 3: N maps bounded sets into equicontinuous sets in \(B_{R}\).

Let \(t_{1},t_{2}\in I\), such that \(t_{1}< t_{2}\) and let \(u\in B_{R}\). Then, we have

$$\begin{aligned} \bigl\vert (Nu) (t_{1})-(Nu) (t_{2}) \bigr\vert &\leq \int_{0}^{t_{1}}\frac{ \vert (t_{2}-qs)^{(\alpha -1)}-(t_{1}-qs)^{(\alpha-1)} \vert }{\varGamma_{q}(\alpha)} \bigl\vert g(s) \bigr\vert \,d_{q}s \\ &\quad{}+ \int_{t_{1}}^{t_{2}}\frac{ \vert (t_{2}-qs)^{(\alpha-1)} \vert }{\varGamma _{q}(\alpha)} \bigl\vert g(s) \bigr\vert \,d_{q}s, \end{aligned} $$

where \(g\in C(I)\) such that \(g(t)=f(t,u(t),g(t))\). Hence

$$\begin{aligned} \bigl\vert (Nu) (t_{1})-(Nu) (t_{2}) \bigr\vert &\leq \frac{ p^{\ast}+d^{\ast}R}{1-r^{\ast}} \int_{0}^{t_{1}}\frac { \vert (t_{2}-qs)^{(\alpha-1)}-(t_{1}-qs)^{(\alpha-1)} \vert }{\varGamma_{q}(\alpha)}\, d_{q}s \\ &\quad{}+\frac{p^{\ast}+d^{\ast}R}{1-r^{\ast}} \int _{t_{1}}^{t_{2}}\frac{ \vert (t_{2}-qs)^{(\alpha-1)} \vert }{\varGamma_{q}(\alpha)}\,d_{q}s. \end{aligned} $$

As \(t_{1}\rightarrow t_{2}\) and since G is continuous, the right-hand side of the above inequality tends to zero.

As a consequence of the above three steps with the Arzelá–Ascoli theorem, we can conclude that \(N:B_{R}\to B_{R}\) is continuous and compact.

From an application of Theorem 2.18, we deduce that N has at least a fixed point which is a solution of problem (1)–(2). □

Ulam stability results

In this section, we are concerned with the generalized Ulam–Hyers–Rassias stability results of the problem (1)–(2).

The following hypotheses will be used in the sequel.

\((H_{3})\):

There exist functions \(p_{1},p_{2},p_{3}\in C(I,[0,\infty))\) with \(p_{3}(t)<1\) such that

$$\begin{gathered} \bigl(1+ \vert u \vert + \vert v \vert \bigr) \bigl\vert f(t,u,v) \bigr\vert \\\quad \leq p_{1}(t)\varPhi(t)+p_{2}(t)\varPhi(t) \vert u \vert +p_{3}(t) \vert v \vert , \quad\text{for each } t\in I \text{ and } u,v\in{\Bbb {R}}.\end{gathered} $$
\((H_{4})\):

There exists \(\lambda_{\varPhi}>0\) such that for each \(t\in I\), we have

$$\bigl(I_{q}^{\alpha}\varPhi\bigr) (t)\leq\lambda_{\varPhi}\varPhi(t). $$
Set \(\varPhi^{\ast}=\sup _{t\in I}\varPhi(t)\) and

$$p_{i}^{\ast}=\sup _{t\in I}p_{i}(t),\quad i\in \{1,2,3\}. $$

Theorem 4.1

Assume that the hypotheses \((H_{3})\)and \((H_{4})\)hold. If

$$p_{3}^{\ast}+Lp_{2}^{\ast} \varPhi^{\ast}< 1, $$

then the problem (1)(2) has at least one solution and it is generalized Ulam–Hyers–Rassias stable.

Proof

Consider the operator N defined in (7). We can see that hypothesis \((H_{3})\) implies \((H_{2})\) with \(p\equiv p_{1}\varPhi\), \(d\equiv p_{2}\varPhi\) and \(r\equiv p_{3}\).

Let u be a solution of the inequality (5), and let us assume that v is a solution of the problem (1)–(2). Thus, we have

$$v(t)=u_{0}+\bigl(I_{q}^{\alpha}h\bigr) (t), $$

where \(h\in C(I)\) such that \(h(t)=f(t,v(t),h(t))\).

From the inequality (5) for each \(t\in I\), we have

$$\bigl\vert u(t)-u_{0}-\bigl(I_{q}^{\alpha}g\bigr) (t) \bigr\vert \leq\bigl(I_{q}^{\alpha}\varPhi\bigr) (t), $$

where \(g\in C(I)\) such that \(g(t)=f(t,u(t),g(t))\).

From the hypotheses \((H_{3})\) and \((H_{4})\), for each \(t\in I\), we get

$$\begin{aligned} \bigl\vert u(t)-v(t) \bigr\vert \leq& \bigl\vert u(t)-u_{0}- \bigl(I_{q}^{\alpha}g\bigr) (t) +\bigl(I_{q}^{\alpha}(g-h) \bigr) (t) \bigr\vert \\ \leq&\bigl(I_{q}^{\alpha}\varPhi\bigr) (t) + \int_{0}^{t}\frac{(t-qs)^{(\alpha-1)}}{\varGamma_{q}(\alpha )}\bigl( \bigl\vert g(s) \bigr\vert + \bigl\vert h(s) \bigr\vert \bigr) \,d_{q}s \\ \leq&\bigl(I_{q}^{\alpha}\varPhi\bigr) (t) +\frac{p_{1}^{\ast}+p_{2}^{\ast}}{1-p_{3}^{\ast}} \bigl(I_{q}^{\alpha}\varPhi \bigr) (t) \\ \leq&\lambda_{\phi}\varPhi(t)+2\lambda_{\phi}\frac{p_{1}^{\ast }+p_{2}^{\ast}}{1-p_{3}^{\ast}} \varPhi(t) \\ \leq& \biggl[1+2\frac{p_{1}^{\ast}+p_{2}^{\ast}}{1-p_{3}^{\ast}} \biggr]\lambda_{\phi}\varPhi(t) \\ :=&c_{f,\varPhi}\varPhi(t). \end{aligned}$$

Hence, the problem (1)–(2) is generalized Ulam–Hyers–Rassias stable. □

Examples

Example 1

Consider the following problem of implicit fractional \(\frac{1}{4} \)-difference equations:

$$ \textstyle\begin{cases} ({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}u)(t)=f(t,u(t),({}^{c}D_{\frac {1}{4}}^{\frac{1}{2}}u)(t));\quad t\in[0,1],\\ u(0)=1, \end{cases} $$
(9)

where

$$f\bigl(t,u(t),\bigl({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}u\bigr) (t) \bigr)=\frac{t^{2}}{1+ \vert u(t) \vert + \vert {}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}u(t) \vert } \biggl(e^{-7}+\frac {1}{e^{t+5}} \biggr)u(t);\quad t\in[0,1]. $$

The hypothesis \((H_{1})\) is satisfied with

$$\phi_{1}(t)=\phi_{2}(t)=t^{2} \biggl(e^{-7}+\frac{1}{e^{t+5}} \biggr)t. $$

Hence, Theorem 3.2 implies that our problem (9) has a unique solution defined on \([0,1]\).

Example 2

Consider now the following problem of implicit fractional \(\frac {1}{4} \)-difference equations:

$$ \textstyle\begin{cases} ({}^{c}D_{\frac{1}{4}}^{\frac{1}{2}}u)(t)=f(t,u(t),({}^{c}D_{\frac {1}{4}}^{\frac{1}{2}}u)(t));\quad t\in[0,1],\\ u(0)=2, \end{cases} $$
(10)

where

$$\textstyle\begin{cases} f(t,x,y)=\frac{t^{2}}{1+ \vert x \vert + \vert y \vert } (e^{-7}+\frac{1}{e^{t+5}} )(t^{2}+xt^{2}+y);\quad t\in(0,1], \\ f(0,x,y)=0. \end{cases} $$

The hypothesis \((H_{3})\) is satisfied with \(\varPhi(t)=t^{2}\) and \(p_{i}(t)= (e^{-7}+\frac{1}{e^{t+5}} )t\); \(i\in\{1,2,3\}\). Hence, Theorem 3.3 implies that our problem (10) has at least a solution defined on \([0,1]\).

Also, the hypothesis \((H_{4})\) is satisfied. Indeed, for each \(t\in(0,1]\), there exists a real number \(0<\epsilon <1\) such that \(\epsilon< t\leq1\), and

$$\begin{aligned} \bigl(I_{q}^{\alpha}\varPhi\bigr) (t) \leq&\frac{t^{2}}{\epsilon^{2}(1+q+q^{2})} \\ \leq&\frac{1}{\epsilon^{2}}\varPhi(t) \\ =&\lambda_{\varPhi}\varPhi(t). \end{aligned}$$

Consequently, Theorem 4.1 implies that the problem (10) is generalized Ulam–Hyers–Rassias stable.

References

  1. 1.

    Abbas, S., Albarakati, W., Benchohra, M., N’Guérékata, G.M.: Existence and Ulam stabilities for Hadamard fractional integral equations in Fréchet spaces. J. Fract. Calc. Appl. 7(2), 1–12 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Abbas, S., Albarakati, W.A., Benchohra, M., Henderson, J.: Existence and Ulam stabilities for Hadamard fractional integral equations with random effects. Electron. J. Differ. Equ. 2016, Article ID 25 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Abbas, S., Albarakati, W.A., Benchohra, M., Sivasundaram, S.: Dynamics and stability of Fredholm type fractional order Hadamard integral equations. Nonlinear Stud. 22(4), 673–686 (2015)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Abbas, S., Benchohra, M., Graef, J.R., Henderson, J.: Implicit Fractional Differential and Integral Equations: Existence and Stability. de Gruyter, Berlin (2018)

    Google Scholar 

  5. 5.

    Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)

    Google Scholar 

  6. 6.

    Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Publ. (Nova Science Publishers), New York (2015)

    Google Scholar 

  7. 7.

    Adams, C.R.: On the linear ordinary q-difference equation. Ann. Math. 30, 195–205 (1928)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Agarwal, R.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ahmad, B.: Boundary value problem for nonlinear third order q-difference equations. Electron. J. Differ. Equ. 2011, Article ID 94 (2011)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Ahmad, B., Ntouyas, S.K., Purnaras, L.K.: Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations. Adv. Differ. Equ. 2012, Article ID 140 (2012)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Benchohra, M., Berhoun, F., N’Guérékata, G.M.: Bounded solutions for fractional order differential equations on the half-line. Bull. Math. Anal. Appl. 146(4), 62–71 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Benchohra, M., Bouriah, S., Darwish, M.: Nonlinear boundary value problem for implicit differential equations of fractional order in Banach spaces. Fixed Point Theory 18(2), 457–470 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Benchohra, M., Bouriah, S., Henderson, J.: Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses. Commun. Appl. Nonlinear Anal. 22(1), 46–67 (2015)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Browder, F.: On the convergence of successive approximations for nonlinear functional equations. Indag. Math. 30, 27–35 (1968)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Carmichael, R.D.: The general theory of linear q-difference equations. Am. J. Math. 34, 147–168 (1912)

    MathSciNet  Article  Google Scholar 

  16. 16.

    El-Shahed, M., Hassan, H.A.: Positive solutions of q-difference equation. Proc. Am. Math. Soc. 138, 1733–1738 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Etemad, S., Ntouyas, S.K., Ahmad, B.: Existence theory for a fractional q-integro-difference equation with q-integral boundary conditions of different orders. Mathematics 7, Article ID 659 (2019)

    Article  Google Scholar 

  18. 18.

    Jleli, M., Mursaleen, M., Samet, B.: Q-integral equations of fractional orders. Electron. J. Differ. Equ. 2016, Article ID 17 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    Google Scholar 

  20. 20.

    Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)

    Google Scholar 

  21. 21.

    Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    Google Scholar 

  22. 22.

    Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  24. 24.

    Li, M., Wang, J.: Finite time stability of fractional delay differential equations. Appl. Math. Lett. 64, 170–176 (2017)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Matkowski, J.: Integrable solutions of functional equations. Diss. Math. 127, 1–68 (1975)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1, 311–323 (2007)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S.: On q-analogues of Caputo derivative and Mittag-Leffler function. Fract. Calc. Appl. Anal. 10, 359–373 (2007)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Rassias, Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Rus, I., Petrusel, A., Petrusel, G.: Fixed Point Theory. Cluj University Press, Cluj (2008)

    Google Scholar 

  30. 30.

    Rus, I.A.: Ulam stability of ordinary differential equations. Stud. Univ. Babeş–Bolyai, Math. LIV(4), 125–133 (2009)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Rus, I.A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10, 305–320 (2009)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Amsterdam (1987) (English translation from the Russian)

    Google Scholar 

  33. 33.

    Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg; Higher Education Press, Beijing (2010)

    Google Scholar 

  34. 34.

    Tenreiro Machado, J.A., Kiryakova, V.: The chronicles of fractional calculus. Fract. Calc. Appl. Anal. 20, 307–336 (2017)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Toledano, J.M.A., Benavides, T.D., Acedo, G.L.: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel (1997)

    Google Scholar 

  36. 36.

    Wang, J., Ibrahim, A.G., Feckan, M.: Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Appl. Math. Comput. 257, 103–118 (2015)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Wang, J., Li, X.: Ulam–Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258, 72–83 (2015)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Wang, J., Li, X.: A uniformed method to Ulam–Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Google Scholar 

  40. 40.

    Zhou, Y., He, J.W., Ahmad, B., Tuan, N.H.: Existence and regularity results of a backward problem for fractional diffusion equations. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.5781

    Article  Google Scholar 

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The work was supported by the Fundo para o Desenvolvimento das Ciências e da Tecnologia (FDCT) of Macau under Grant 0074/2019/A2 and the National Natural Science Foundation of China (No. 11671339).

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Abbas, S., Benchohra, M., Laledj, N. et al. Existence and Ulam stability for implicit fractional q-difference equations. Adv Differ Equ 2019, 480 (2019). https://doi.org/10.1186/s13662-019-2411-y

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MSC

  • 26A33

Keywords

  • Fractional q-difference equation
  • Implicit
  • Ulam–Hyers–Rassias stability
  • Fixed point