Delayed Langevin type equations with two fractional derivatives

N. I. Mahmudov

arXiv:1907.01746·math.DS·July 4, 2019

Delayed Langevin type equations with two fractional derivatives

N. I. Mahmudov

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TL;DR

This paper introduces a new class of delayed Mittag-Leffler functions to explicitly solve linear fractional time-delay Langevin equations with two Riemann-Liouville derivatives, establishing existence, uniqueness, and stability of solutions.

Contribution

It develops explicit solutions for fractional Langevin equations with delays using novel delayed Mittag-Leffler functions, extending the analytical tools for such equations.

Findings

01

Explicit solutions derived using delayed Mittag-Leffler functions

02

Proved existence and uniqueness of solutions

03

Established Ulam-Hyers stability results

Abstract

In this paper, we introduce a delayed Mittag-Leffler type function. With the help of the delayed Mittag-Leffler type functions, we give an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville fractional derivatives. The existence and uniqueness of solutions are obtained by using an estimation of delayed Mittag-Leffler type functions in terms of exponential functions and a weighted norm via fixed point theorems. Further, we present Ulam--Hyers stability results.

Equations100

\left\{\begin{tabular}[c]{ll}${}^{RL}D_{-h^{+}}^{\alpha}y\left(t\right)-\lambda\ ^{RL}D_{-h^{+}}^{\beta}y\left(t\right)=\mu y\left(t-h\right)+f\left(t,y\left(t\right)\right),\ \ t\in\left(0,T\right],\ \ h>0,$&\\ $y\left(t\right)=\varphi\left(t\right),\ \ \left(I_{-h^{+}}^{2-\alpha}y\right)\left(-h^{+}\right)=\varphi\left(-h\right),\ \ -h\leq t\leq 0,\ 0<\beta<1,\ 1<\alpha\leq 2,\ \alpha-\beta>1,$&\\ ${}^{RL}D_{-h^{+}}^{\alpha-1}y\left(t\right)=\ ^{RL}D_{-h^{+}}^{\alpha-1}\varphi\left(t\right),$ $\ \ \ \ \left({}^{RL}D_{-h+}^{\alpha-1}y\right)\left(-h^{+}\right)=\ \ ^{RL}D_{-h+}^{\alpha-1}\varphi\left(-h\right),$&\end{tabular}\ \right.

\left\{\begin{tabular}[c]{ll}${}^{RL}D_{-h^{+}}^{\alpha}y\left(t\right)-\lambda\ ^{RL}D_{-h^{+}}^{\beta}y\left(t\right)=\mu y\left(t-h\right)+f\left(t,y\left(t\right)\right),\ \ t\in\left(0,T\right],\ \ h>0,$&\\ $y\left(t\right)=\varphi\left(t\right),\ \ \left(I_{-h^{+}}^{2-\alpha}y\right)\left(-h^{+}\right)=\varphi\left(-h\right),\ \ -h\leq t\leq 0,\ 0<\beta<1,\ 1<\alpha\leq 2,\ \alpha-\beta>1,$&\\ ${}^{RL}D_{-h^{+}}^{\alpha-1}y\left(t\right)=\ ^{RL}D_{-h^{+}}^{\alpha-1}\varphi\left(t\right),$ $\ \ \ \ \left({}^{RL}D_{-h+}^{\alpha-1}y\right)\left(-h^{+}\right)=\ \ ^{RL}D_{-h+}^{\alpha-1}\varphi\left(-h\right),$&\end{tabular}\ \right.

e_{α, β} (λ; t) := t^{β - 1} E_{α, β} (λ t^{α}) := t^{β - 1} k = 0 \sum \infty \frac{λ ^{k} t ^{α k}}{Γ ( k α + β )}, α, β > 0, t \in R .

e_{α, β} (λ; t) := t^{β - 1} E_{α, β} (λ t^{α}) := t^{β - 1} k = 0 \sum \infty \frac{λ ^{k} t ^{α k}}{Γ ( k α + β )}, α, β > 0, t \in R .

{}_{p}\Psi_{q}\left(z\right)=\ _{p}\Psi_{q}\left[\left.\begin{array}[c]{c}\left(\lambda_{l},\alpha_{l}\right)_{1,p}\\ \left(b_{l},\beta_{l}\right)_{1,q}\end{array}\right|z\right]={\displaystyle\sum_{k=0}^{\infty}}\frac{{\displaystyle\prod_{l=1}^{p}}\Gamma\left(\lambda_{l}+\alpha_{l}k\right)}{{\displaystyle\prod_{j=1}^{q}}\Gamma\left(b_{j}+\beta_{j}k\right)}\frac{z^{k}}{k!}

{}_{p}\Psi_{q}\left(z\right)=\ _{p}\Psi_{q}\left[\left.\begin{array}[c]{c}\left(\lambda_{l},\alpha_{l}\right)_{1,p}\\ \left(b_{l},\beta_{l}\right)_{1,q}\end{array}\right|z\right]={\displaystyle\sum_{k=0}^{\infty}}\frac{{\displaystyle\prod_{l=1}^{p}}\Gamma\left(\lambda_{l}+\alpha_{l}k\right)}{{\displaystyle\prod_{j=1}^{q}}\Gamma\left(b_{j}+\beta_{j}k\right)}\frac{z^{k}}{k!}

G_{\alpha,\beta}\left(\lambda,\mu;t\right)={\displaystyle\sum_{n=0}^{\infty}}\frac{\lambda^{n}}{n!}t^{\alpha n}\ _{1}\Psi_{1}\left[\left.\begin{array}[c]{c}\left(n+1,1\right)\\ \left(\alpha n+\alpha,\alpha-\beta\right)\end{array}\right|\mu t^{\alpha-\beta}\right]={\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=0}^{\infty}}\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)\lambda^{n}\mu^{k}\frac{t^{\alpha n+k\left(\alpha-\beta\right)}}{\Gamma\left(\alpha n+\left(\alpha-\beta\right)k+\alpha\right)}.

G_{\alpha,\beta}\left(\lambda,\mu;t\right)={\displaystyle\sum_{n=0}^{\infty}}\frac{\lambda^{n}}{n!}t^{\alpha n}\ _{1}\Psi_{1}\left[\left.\begin{array}[c]{c}\left(n+1,1\right)\\ \left(\alpha n+\alpha,\alpha-\beta\right)\end{array}\right|\mu t^{\alpha-\beta}\right]={\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=0}^{\infty}}\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)\lambda^{n}\mu^{k}\frac{t^{\alpha n+k\left(\alpha-\beta\right)}}{\Gamma\left(\alpha n+\left(\alpha-\beta\right)k+\alpha\right)}.

\mathfrak{E}_{\alpha,\beta}^{h,\gamma}\left(\lambda,\mu;t\right)={\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=0}^{\infty}}\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)\lambda^{n}\mu^{k}\dfrac{\left(t-kh\right)^{k\gamma+n\alpha+\beta-1}}{\Gamma\left(k\gamma+n\alpha+\beta\right)}H\left(t-kh\right),

\mathfrak{E}_{\alpha,\beta}^{h,\gamma}\left(\lambda,\mu;t\right)={\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=0}^{\infty}}\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)\lambda^{n}\mu^{k}\dfrac{\left(t-kh\right)^{k\gamma+n\alpha+\beta-1}}{\Gamma\left(k\gamma+n\alpha+\beta\right)}H\left(t-kh\right),

H\left(t\right)=\left\{\begin{array}[c]{c}1,\ \ t\geq 0,\\ 0,\ \ \ t<0.\end{array}\right.

H\left(t\right)=\left\{\begin{array}[c]{c}1,\ \ t\geq 0,\\ 0,\ \ \ t<0.\end{array}\right.

E^{h,\alpha,\beta}\left(\mu;t\right):=\left\{\begin{tabular}[c]{ll}$\Theta,$&$-\infty<t\leq-h,$\\ $I\frac{\left(h+t\right)^{\beta-1}}{\Gamma\left(\beta\right)},$&$-h<t\leq 0,$\\ $I\frac{\left(h+t\right)^{\beta-1}}{\Gamma\left(\beta\right)}+\mu\frac{t^{\alpha+\beta-1}}{\Gamma\left(\alpha+\beta\right)}+\mu^{2}\frac{\left(t-h\right)^{2\alpha+\beta-1}}{\Gamma\left(2\alpha+\beta\right)}+...+\mu^{k}\frac{\left(t-\left(k-1\right)h\right)^{k\alpha+\beta-1}}{\Gamma\left(k\alpha+\beta\right)},$&$\left(k-1\right)h<t\leq kh.$\end{tabular}\ \ \ \ \ \ \ \ \ \ \ \ \ \right.

E^{h,\alpha,\beta}\left(\mu;t\right):=\left\{\begin{tabular}[c]{ll}$\Theta,$&$-\infty<t\leq-h,$\\ $I\frac{\left(h+t\right)^{\beta-1}}{\Gamma\left(\beta\right)},$&$-h<t\leq 0,$\\ $I\frac{\left(h+t\right)^{\beta-1}}{\Gamma\left(\beta\right)}+\mu\frac{t^{\alpha+\beta-1}}{\Gamma\left(\alpha+\beta\right)}+\mu^{2}\frac{\left(t-h\right)^{2\alpha+\beta-1}}{\Gamma\left(2\alpha+\beta\right)}+...+\mu^{k}\frac{\left(t-\left(k-1\right)h\right)^{k\alpha+\beta-1}}{\Gamma\left(k\alpha+\beta\right)},$&$\left(k-1\right)h<t\leq kh.$\end{tabular}\ \ \ \ \ \ \ \ \ \ \ \ \ \right.

^{R L} D_{- h^{+}}^{α} E_{α - β, α}^{h, α - j} (λ, μ; t) - λ^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α - j} (λ, μ; t) = μ E_{α - β, α}^{h, α - j} (λ, μ; t - h) .

^{R L} D_{- h^{+}}^{α} E_{α - β, α}^{h, α - j} (λ, μ; t) - λ^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α - j} (λ, μ; t) = μ E_{α - β, α}^{h, α - j} (λ, μ; t - h) .

E_{α - β, α}^{h, α} (λ, μ; t)

E_{α - β, α}^{h, α} (λ, μ; t)

^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α} (λ, μ; t)

^{R L} D_{- h^{+}}^{α} E_{α - β, α}^{h, α} (λ, μ; t)

^{R L} D_{- h^{+}}^{α} E_{α - β, α}^{h, α} (λ, μ; t)

^{R L} D_{- h^{+}}^{α} E_{α - β, α}^{h, α} (λ, μ; t)

\displaystyle={\displaystyle\sum_{n=1}^{\infty}}\lambda^{n}\dfrac{t^{n\left(\alpha-\beta\right)-1}}{\Gamma\left(n\left(\alpha-\beta\right)\right)}+{\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=1}^{\infty}}\left(\begin{array}[c]{c}n+k-1\\ k\end{array}\right)\lambda^{n}\mu^{k}\dfrac{\left(t-kh\right)^{k\alpha+n\left(\alpha-\beta\right)-1}}{\Gamma\left(k\alpha+n\left(\alpha-\beta\right)\right)}H\left(t-kh\right)

\displaystyle+{\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=1}^{\infty}}\left(\begin{array}[c]{c}n+k-1\\ k-1\end{array}\right)\lambda^{n}\mu^{k}\dfrac{\left(t-kh\right)^{k\alpha+n\left(\alpha-\beta\right)-1}}{\Gamma\left(k\alpha+n\left(\alpha-\beta\right)\right)}H\left(t-kh\right)

\displaystyle=\lambda{\displaystyle\sum_{n=0}^{\infty}}\lambda^{n}\dfrac{t^{n\left(\alpha-\beta\right)+\alpha-\beta-1}}{\Gamma\left(n\left(\alpha-\beta\right)+\alpha-\beta\right)}+{\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=1}^{\infty}}\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)\lambda^{n+1}\mu^{k}\dfrac{\left(t-kh\right)^{k\alpha+n\left(\alpha-\beta\right)+\alpha-\beta-1}}{\Gamma\left(k\alpha+n\left(\alpha-\beta\right)+\alpha-\beta\right)}H\left(t-kh\right)+

\displaystyle+{\displaystyle\sum_{n=0}^{\infty}}{\displaystyle\sum_{k=0}^{\infty}}\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)\lambda^{n}\mu^{k+1}\dfrac{\left(t-h-kh\right)^{k\alpha+n\left(\alpha-\beta\right)+\alpha-1}}{\Gamma\left(k\alpha+n\left(\alpha-\beta\right)+\alpha\right)}H\left(t-h-kh\right)

= λ^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α} (λ, μ; t) + μ E_{α - β, α}^{h, α} (λ, μ; t - h) .

y (t)

y (t)

+ \int_{- h}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) (^{R L} D_{- h^{+}}^{α} φ (s) - λ^{R L} D_{- h^{+}}^{β} φ (s)) d s .

y (t)

y (t)

y (t)

^{R L} D_{- h^{+}}^{α - 1} y (t)

^{R L} D_{- h^{+}}^{α - j} (\int_{- h}^{\cdot} E_{α - β, α}^{h, α} (λ, μ; \cdot - s) g (s) d s) (- h^{+}) = 0, j = 0, 1,

^{R L} D_{- h^{+}}^{α - j} (\int_{- h}^{\cdot} E_{α - β, α}^{h, α} (λ, μ; \cdot - s) g (s) d s) (- h^{+}) = 0, j = 0, 1,

^{R L} D_{- h^{+}}^{α} (\int_{- h}^{\cdot} E_{α - β, α}^{h, α} (λ, μ; \cdot - s) g (s) d s) (t) = λ \int_{- h}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) g (s) d s + g (t),

^{R L} D_{- h^{+}}^{α} E_{α - β, α}^{h, α - j} (λ, μ; t + h) = λ^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α - j} (λ, μ; t + h), j = 0, 1.

^{R L} D_{- h^{+}}^{α - 2} y (- h^{+})

^{R L} D_{- h^{+}}^{α - 2} y (- h^{+})

^{R L} D_{- h^{+}}^{α - 1} y (- h^{+})

θ^{R L} D_{- h^{+}}^{α} φ (t)

θ^{R L} D_{- h^{+}}^{α} φ (t)

+ λ^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α - 1} (λ, μ; t + h) c_{2}

+ λ \int_{- h}^{t}^{R L} D_{- h^{+}}^{β} E_{α - β, α}^{h, α} (λ, μ; t - s) g (s) d s + g (t), - h \leq t \leq 0.

y (t) = \int_{0}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) f (s) d s, t \geq 0.

y (t) = \int_{0}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) f (s) d s, t \geq 0.

y (t)

y (t)

+ \int_{- h}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) (^{R L} D_{- h^{+}}^{α} φ (s) - λ^{R L} D_{- h^{+}}^{β} φ (s)) d s + \int_{0}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) f (s) d s .

(F y) (t)

(F y) (t)

+ \int_{- h}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) (^{R L} D_{- h^{+}}^{α} φ (s) - A φ (s)) d s + \int_{0}^{t} E_{α - β, α}^{h, α} (λ, μ; t - s) f (s, y (s)) d s

E_{α - β, α}^{h, α} (λ, μ; t) \leq t^{α - 1} exp (∣ λ ∣ t^{α - β} + ∣ μ ∣ t^{α}) .

E_{α - β, α}^{h, α} (λ, μ; t) \leq t^{α - 1} exp (∣ λ ∣ t^{α - β} + ∣ μ ∣ t^{α}) .

E_{α - β, α}^{h, α} (λ, μ; t)

E_{α - β, α}^{h, α} (λ, μ; t)

\leq n = 0 \sum \infty k = 0 \sum \infty \frac{( n + k ) !}{n ! k !} ∣ λ ∣^{n} ∣ μ ∣^{k} \frac{t ^{k α + n (α - β) + α - 1}}{Γ ( k α + n ( α - β ) + α )} .

Γ (k α + n (α - β) + α) > Γ (k + n + 1) .

Γ (k α + n (α - β) + α) > Γ (k + n + 1) .

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TopicsFractional Differential Equations Solutions · Nonlinear Differential Equations Analysis · Stability and Controllability of Differential Equations

Full text

Delayed Langevin type equations with two fractional derivatives

N. I . Mahmudov

Department of Mathematics, Eastern Mediterranean University

Famagusta, T.R. North Cyprus, via Mersin 10, Turkey

Abstract

In this paper, we introduce a delayed Mittag-Leffler type function. With the help of the delayed Mittag-Leffler type functions, we give an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville fractional derivatives. The existence and uniqueness of solutions are obtained by using an estimation of delayed Mittag-Leffler type functions in terms of exponential functions and a weighted norm via fixed point theorems. Further, we present Ulam–Hyers stability results.

Keywords: fractional Langevin type equation, Wright type function, Mittag-Leffler type matrix function.

1 Introduction

One century ago, Langevin offered a detailed description of Brownian motion due to collisions with much smaller fluid molecules. Many of the stochastic problems in fluctuating media can be described by the Langevin equation [3], [4]. However, for some complex systems, the classical Langevin equation cannot offer an adequate description of the problem. As a result, various generalizations have been proposed that constitute the inadequacy of the classical case and describe more physical phenomena in disordered regions [5]. One of them is so called fractional Langevin type equation, which is obtained from the classical Langevin equation by replacing classical derivative by fractional derivative. The nonlinear Langevin type equations involving two fractional orders was introduced and investigated in [10]-[15].

Our goal is to study the following time-delay Langevin type equations with two different fractional derivatives

[TABLE]

where ${}^{RL}D_{-h^{+}}^{\alpha}$ and ${}^{RL}D_{-h^{+}}^{\beta}$ are Riemann-Liouville fractional derivatives of order $1<\alpha\leq 2$ and $0<\beta<1,$ $T=lh,$ $l\in\mathbb{N},$ $\varphi:\left[-h,T\right]\rightarrow\mathbb{R}$ is a two times continuously differentiable function.

The main contributions are as follows: (i) we introduce a delayed Mittag-Leffler type function; (ii) we derive the representation of solutions of nonhomogeneous equation (1); (iii) we give fundamental estimation for delayed Mittag-Leffler type function in terms of exponential function; (iv) we introduce a weighted norm (10) in $C(\left[0,T\right],\mathbb{R}$ ) (the Banach space of all continuous functions from $\left[0,T\right]$ into $\mathbb{R}$ with the norm $\left\|x\right\|_{\infty}=\sup\left\{\left|x\left(t\right)\right|:t\in\left[0,T\right]\right\}$ ) and establish sufficient conditions to guarantee the existence and uniqueness of global solution on $\left[0,T\right]$ for fractional time-delay Langevin type equation (1) and (v) we study the Ulam-Hyers stability of (1) in a weighted space.

2 Delayed Mittag-Leffler type function

Definition 1

[1]** Mittag-Leffler type function of two parameters $e_{\alpha,\beta}\left(\lambda,\cdot\right):\mathbb{R}\rightarrow\mathbb{R}$ is defined by

[TABLE]

Definition 2

[1]** Let $\lambda_{l},b_{j}\in C,$ $\alpha_{l},\beta_{j}\in\mathbb{R},\ l=1,...,p;$ $j=1,...,q$ . Generalized Wright function ${}_{p}\Psi_{q}\left(\cdot\right):C\rightarrow C$ is defined by

[TABLE]

It is known that, see [1] Theorem 1.5, if ${\displaystyle\sum_{j=1}^{q}}\beta_{j}-{\displaystyle\sum_{l=1}^{p}}\alpha_{l}>-1$ , then the series ${}_{p}\Psi_{q}\left(z\right)$ is absolutely convergent for all $z\in C$ .

The following function is defined in [1]

[TABLE]

Motivated by this we introduce so called delayed Mittag-Leffler type function generated by $\lambda,\mu$ as follows:

Definition 3

Let $\alpha,\beta,\gamma>0.$ Delayed Mittag-Leffler type function generated by $\lambda,\mu$ of two parameters $\mathfrak{E}_{\alpha,\beta}^{h,\gamma}\left(\lambda,\mu;\cdot\right):\mathbb{R}\rightarrow\mathbb{R}$ is defined by

[TABLE]

where $H$ is the Heaviside function

[TABLE]

Definition 4

[2]** Delayed Mittag-Leffler type function of two parameters $E^{h,\alpha,\beta}\left(\mu;\cdot\right):\mathbb{R}\rightarrow\mathbb{R}$ is defined by

[TABLE]

Lemma 5

We have

(i)

if $\mu=0$ , then $\mathfrak{E}_{\alpha-\beta,\beta}^{h,\gamma}\left(\lambda,\mu;t\right)={\displaystyle\sum_{n=0}^{\infty}}\lambda^{n}\dfrac{t^{n\left(\alpha-\beta\right)+\beta-1}}{\Gamma\left(n\left(\alpha-\beta\right)+\beta\right)}H\left(t\right)=t^{\beta-1}E_{\alpha-\beta,\beta}\left(\lambda;t\right)=e_{\alpha-\beta,\beta}\left(\lambda;t\right).$

(ii)

if $\lambda=0$ , then $\mathfrak{E}_{\alpha,\beta}^{h,\gamma}\left(\lambda,\mu;t\right)={\displaystyle\sum_{k=0}^{\infty}}\mu^{k}\dfrac{\left(t-kh\right)^{k\gamma+\beta-1}}{\Gamma\left(k\gamma+\beta\right)}H\left(t-kh\right)=E^{h,\gamma,\beta}\left(\mu;t-h\right).$

Lemma 6

$\mathfrak{E}_{\alpha-\beta,\alpha}^{h,\alpha-j}\left(\lambda,\mu;t\right)$ , $j=0,1,$ satisfies the following equation

[TABLE]

Proof. We prove lemma just for $j=0$ . Direct calculations show that

[TABLE]

Using the binomial identity $\left(\begin{array}[c]{c}n+k\\ k\end{array}\right)=\left(\begin{array}[c]{c}n+k-1\\ k\end{array}\right)+\left(\begin{array}[c]{c}n+k-1\\ k-1\end{array}\right)$ we get

[TABLE]

Lemma is proved.

Theorem 7

A solution $y\in C\left(\left(\left(p-1\right)h,ph\right],\mathbb{R}\right),$ $0\leq p\leq l,$ of (1) with $f=0$ has a form

[TABLE]

Proof. We are looking for a solution of the form

[TABLE]

where $c_{1}$ and $c_{2}$ are unknown constants, $g(t)$ is an unknown continuously differentiable function. Direct calculations and Lemma 6 show that

[TABLE]

Taking Riemann-Liouville derivative ${}^{RL}D_{-h^{+}}^{\alpha-2}y\left(-h^{+}\right)$ and ${}^{RL}D_{-h^{+}}^{\alpha-1}y\left(-h^{+}\right)$ of $y\left(t\right)$ at $-h$ we get

[TABLE]

On the other hand differentiating (3), we obtain

[TABLE]

Therefore, $g\left(t\right)=\ ^{RL}D_{-h^{+}}^{\alpha}\varphi\left(t\right)-\lambda\ ^{RL}D_{-h^{+}}^{\beta}\varphi\left(t\right)$ and the desired result holds.

Theorem 8

The solution $y(t)$ of (1) satisfying zero initial condition $y\left(t\right)=0,\ -h\leq t\leq 0,$ has a form

[TABLE]

Proof. Taking the fractional derivative $\ {}^{RL}D_{-h^{+}}^{\alpha}$ of $\int_{0}^{t}\mathfrak{E}_{\alpha-\beta,\alpha}^{h,\alpha}\left(\lambda,\mu;t-s\right)f\left(s\right)ds$ we can easily get the result.

Combining Theorems 7 and 8, we have the following result.

Corollary 9

A solution $y\in C\left(\left[-h,T\right]\cap\left(\left(p-1\right)h,ph\right],\mathbb{R}\right),$ $p\in\mathbb{N},$ of (1) has a form

[TABLE]

3 Existence, uniqueness and Ulam-Hyers stability

To introduce a fixed point problem associated with (1) we define an integral operator $\mathfrak{F:}C\left(\left[0,T\right],\mathbb{R}\right)\rightarrow C\left(\left[0,T\right],\mathbb{R}\right)$ by

[TABLE]

Lemma 10

We have

[TABLE]

Proof. Firstly, we estimate $\mathfrak{E}_{\alpha-\beta,\alpha}^{h,\alpha}\left(\lambda,\mu;t\right)$ as follows.

[TABLE]

Since $1<\alpha<2,$ $\alpha-\beta>1$ we have

[TABLE]

From estimations (8) and (9), it follows that

[TABLE]

Theorem 11

Let $f:\left[0,T\right]\times\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that the following conditions hold:

(A1)

there exists $L_{f}>0$ such that

[TABLE]

Then the problem (1) has a unique solution in $C\left(\left[0,T\right],\mathbb{R}\right).$

Proof. We will apply Contraction mapping principle to show that $\mathfrak{F}$ has a unique fixed point. At first glance it seems natural to use the maximum norm on $C\left(\left[0,T\right],\mathbb{R}\right)$ , but this choice would lead us only to a local solution defined on a subinterval of $\left[0,T\right]$ . The trick is to use the weighted maximum norm

[TABLE]

on $C\left(\left[0,T\right],\mathbb{R}\right)$ . Observe that $C\left(\left[0,T\right],\mathbb{R}\right)$ is a Banach space with this norm since it is equivalent to the maximum norm.

We now show that $\mathfrak{F}$ is a contraction on $\left(C\left(\left[0,T\right],\mathbb{R}\right),\left\|\cdot\right\|_{\omega}\right)$ . To see this let $y,z\in C\left(\left[0,T\right],\mathbb{R}\right)$ and notice that

[TABLE]

Thus for $t\in\left[0,T\right],$ from Lemma 10 and Lipschitz condition (A1) it follows that

[TABLE]

On the other hand, it is known that

[TABLE]

Combining the last two inequalities we get

[TABLE]

Taking maximum over $\left[0,T\right]$ we get

[TABLE]

Choose $\omega>0$ so that $\omega>L_{f}\Gamma\left(\alpha\right)\exp\left(\left|\lambda\right|T^{\alpha-\beta}+\left|\mu\right|T^{\alpha}\right).\$ So $\mathfrak{F}$ is a contraction. Thus by the Banach fixed point theorem has $\mathfrak{F}$ a unique fixed point in $C\left(\left[0,T\right],\mathbb{R}\right)$ .

Next, we are going to discuss Ulam–Hyers stability of the equation (1) on the time interval $\left[0,T\right]$ .

Let $\varepsilon>0$ .. Consider the equation (1) and the inequality

[TABLE]

Definition 12

We say that the equation (1) is Ulam–Hyers stable if there exists $c>0$ such that for each $\varepsilon>0$ and for each solution $x\in C\left(\left[0,T\right],\mathbb{R}\right)$ of the inequality (12) there exists a solution $y\in C\left(\left[0,T\right],\mathbb{R}\right)$ of the equation (1) with

[TABLE]

Let

[TABLE]

The solution of

[TABLE]

can be represented by

[TABLE]

Then we have the following estimation.

[TABLE]

Now we are ready to state our Ulam–Hyers stability result.

Theorem 13

Assume that (A1) is satisfied. Then the equation (1) is Ulam–Hyers stable on $\left[0,T\right]$ .

Proof. Let $x\in C\left(\left[0,T\right],\mathbb{R}\right)$ be a solution of the inequality (12) and let $y$ be a unique solution of the Cauchy problem (1), that is,

[TABLE]

By using estimation (11) and inequality (13), we have

[TABLE]

By choosing $\omega>L_{f}\Gamma\left(\alpha\right)\exp\left(\left|\lambda\right|T^{\alpha-\beta}+\left|\mu\right|T^{\alpha}\right)$ which yields that

[TABLE]

The proof is completed.

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