On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations

Abstract

A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alternative and the Banach contraction principle, the main results are established. In the case of a local Lipschitz right hand side part function, the existence of a bounded mild solution is proved. Some examples illustrating the main results are provided.

1. Introduction

Differential equations of fractional order have recently obtained good reputation and copious contributions from those interested due to their wide applications in numerous fields of science and engineering [1,2,3,4,5,6,7]. Moreover, the fractional calculus approach has been widely used in the modeling of some real-world phenomena, such as economic processes with dynamic memory [8,9] and financial processes [10].

Multi-term time-fractional differential equations have been used for describing important physical phenomena, such as the multi-term time-fractional diffusion-wave/diffusion equations; see [11].

In addition to the most popular fractional derivatives, the Liouville–Caputo, Riemann–Liouville, and Hadamard fractional operators, the emergence of new ones, such as the Hilfer, Katugampola, Caputo-Fabrizio, and generalized proportional fractional operators, has enriched the research in the topic; see [12,13,14,15,16,17] and the references therein.

In the literature, several interesting articles have studied initial and boundary value problems for fractional differential equations in the frame of the traditional fractional derivatives, such as the Liouville–Caputo, Riemann–Liouville, and Hadamard fractional derivatives. We claim that our approach in the current paper is totally different from the previous contributions in the sense of the generalized proportional vector-order fractional derivatives.

The essential stimulus of the current paper is to investigate the existence and uniqueness of solutions of initial value problems for Caputo-type generalized proportional vector-order fractional differential equations, namely

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C{\mathcal{D}}_a^{\overline{\alpha },\rho }z\left(t\right)=K\left(t,z\left(t\right)\right),t\in \mathbf{J}:=[a,b],\hfill \\ z\left(a\right)=z_a,\hfill \end{array}\right.\end{array}$$

(1)

where $z_a\in {\mathbb{R}}^r,z_a={(z_{a,1},z_{a,2},\dots ,z_{a,r})}^T,z_{a,j}\in \mathbb{R},j=1,2,\dots ,r$, the proportionality parameter $\rho \in (0,1]$, ${}^C{\mathcal{D}}_a^{\overline{\alpha },\rho }$ denotes the Caputo-type generalized proportional fractional derivatives of the vector-order $\overline{\alpha }={({\alpha }_1,\dots ,{\alpha }_r)}^T$, $0<{\alpha }_j<1$, $K:\mathbf{J}\times {\mathbb{R}}^r\to {\mathbb{R}}^r,K(t,z)=(K_1(t,z),K_2(t,z),\dots ,K_r(t,z))$ for $t\in \mathbf{J},z\in {\mathbb{R}}^r$, is a given vector-valued function, and the unknown vector-valued function $z:\mathbf{J}\to {\mathbb{R}}^r$ is continuous on $\mathbf{J}$, where $z\left(t\right)={(z_1\left(t\right),\dots ,z_r\left(t\right))}^T$.

The structure of the current paper is as follows. In Section 2, basic definitions, properties and lemmas are given. The fundamental results of the present work are established in Section 3. Finally, some illustrative examples are proposed in Section 4.

2. Preliminaries

Let ${\mathbb{R}}^r$ be the standard r-dimensional Euclidean space equipped with the norm

$${\parallel z\parallel }_{\infty }=\underset{j=1,2,\dots ,r}{max}\left|z_j\right|,$$

for every $z\in {\mathbb{R}}^r,z={(z_1,z_2,\dots ,z_r)}^T,z_j\in \mathbb{R},j=1,2,\dots ,r$ and $r\in \mathbb{N}$.

Denote by ${\mathcal{C}}_r=C(\mathbf{J},{\mathbb{R}}^r)$ the space of continuous functions $z:\mathbf{J}\to {\mathbb{R}}^r$ with the norm

$${\parallel z\parallel }_{{\mathcal{C}}_r}=\underset{t\in \mathbf{J}}{sup}{\parallel z\left(t\right)\parallel }_{\infty }.$$

It is well known that $({\mathcal{C}}_r,\parallel ·{\parallel }_{{\mathcal{C}}_r})$ is a Banach space.

Let $\ell >0$ be a real number. Define the set

$${\mathbf{S}}_{\ell }:=\{z\in {\mathcal{C}}_r{:\parallel z\parallel }_{{\mathcal{C}}_r}\le \ell \}\subset {\mathcal{C}}_r.$$

By $L^1(\mathbf{J},{\mathbb{R}}^r)$, we indicate the space of Lebesgue integrable functions $z:\mathbf{J}\to {\mathbb{R}}^r$ with the norm ${\parallel z\parallel }_{L^1}={\int }_a^b\parallel z\left(t\right)\parallel dt$.

Now, we recall the basic definitions of the generalized proportional fractional and vector-order fractional operators.

Definition 1 

([17,18]). Let $\rho \in (0,1]$ and $x\in L^1(\mathbf{J},\mathbb{R})$. The generalized proportional fractional integral of order $\alpha >0$ is defined by

$$\begin{array}{cc}\hfill {\mathcal{I}}_a^{\alpha ,\rho }x\left(t\right)& =\frac{1}{{\rho }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{\alpha -1}x\left(\tau \right)d\tau ,t\in (a,b].\hfill \end{array}$$

Definition 2 

([17,18]). Let $\rho \in (0,1]$ and $x\in C^1(\mathbf{J},\mathbb{R})$. The Caputo-type generalized proportional fractional derivative of order $\alpha \in (0,1)$ is defined by

$$\begin{array}{cc}\hfill {}^C{\mathcal{D}}^{\alpha ,\rho }x\left(t\right)& ={\mathcal{I}}_a^{1-\alpha ,\rho }\left({\mathcal{D}}^{1,\rho }x\right)\left(t\right)\hfill \\ \hfill & =\frac{1}{{\rho }^{\alpha }\mathrm{\Gamma }(1-\alpha )}{\int }_a^t{e}^{\frac{\rho -1}{\rho }\left(t-\tau \right)}{\left(t-\tau \right)}^{-\alpha }\left({\mathcal{D}}^{1,\rho }x\right)\left(\tau \right)d\tau ,t\in (a,b],\hfill \end{array}$$

where $\left({\mathcal{D}}^{1,\rho }x\right)\left(t\right)=\left({\mathcal{D}}^{\rho }x\right)\left(t\right)\equiv (1-\rho )x\left(t\right)+\rho x^{\prime }\left(t\right)$.

Lemma 1 

([17,18]). Let $\rho \in (0,1)$, $0<\alpha <1$ and $\gamma >0$. Then we have the following properties:

$$\begin{array}{cc}\hfill & \left({\mathcal{I}}_a^{\alpha ,\rho }{\mathcal{I}}_a^{\gamma ,\rho }x\right)\left(t\right)=\left({\mathcal{I}}_a^{\gamma ,\rho }{\mathcal{I}}_a^{\alpha ,\rho }x\right)\left(t\right)=\left({\mathcal{I}}_a^{\alpha +\gamma ,\rho }x\right)\left(t\right),withx\in L^1(\mathbf{J},\mathbb{R});\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \left({}^C{\mathcal{D}}_a^{\alpha ,\rho }{\mathcal{I}}_a^{\alpha ,\rho }x\right)\left(t\right)=x\left(t\right),withx\in L^1(\mathbf{J},\mathbb{R});\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \left({\mathcal{I}}_a^{\alpha ,\rho }{}^C{\mathcal{D}}_a^{\alpha ,\rho }x\right)\left(t\right)=x\left(t\right)-e^{\frac{\rho -1}{\rho }(t-a)}x\left(a\right),withx\in C^1(\mathbf{J},\mathbb{R}).\hfill \end{array}$$

(4)

Following Definitions 1 and 2, we will define generalized proportional vector-order fractional integrals and Caputo derivatives.

Definition 3. 

Let $z={(z_1,\dots ,z_r)}^T,z\in L^1(\mathbf{J},{\mathbb{R}}^r)$ be a vector-valued function and $\overline{\alpha }={({\alpha }_1,\dots ,{\alpha }_r)}^T$, with $r\in \mathbb{N},{\alpha }_j>0$ for $j=1,\dots ,r$. The generalized proportional vector-order fractional integral of vector order $\overline{\alpha }$ is defined by

$${\mathcal{I}}_a^{\overline{\alpha },\rho }z\left(t\right)={\left({\mathcal{I}}_a^{{\alpha }_1,\rho }{z}_1\left(t\right),\dots ,{\mathcal{I}}_a^{{\alpha }_r,\rho }{z}_r\left(t\right)\right)}^T.$$

Definition 4. 

Let $z={(z_1,\dots ,z_r)}^T,z\in C^1(\mathbf{J},{\mathbb{R}}^r)$ be a vector-valued function and $\overline{\alpha }={({\alpha }_1,\dots ,{\alpha }_r)}^T$, with $r\in \mathbb{N},{\alpha }_j\in (0,1)$ for $j=1,\dots ,r$. The generalized proportional vector-order fractional derivative of vector order $\overline{\alpha }$ is defined by

$${}^C{\mathcal{D}}_a^{\overline{\alpha },\rho }z\left(t\right)={\left({}^C{\mathcal{D}}_a^{{\alpha }_1,\rho }{z}_1\left(t\right),\dots ,{}^C{\mathcal{D}}_a^{{\alpha }_r,\rho }{z}_r\left(t\right)\right)}^T.$$

Remark 1. 

Throughout this paper, all operators related to $\overline{\alpha }$ and $z\in {\mathbb{R}}^r$ are element-wise.

Straightforwardly, the aforesaid properties (2), (3) and (4) are verified in the vector-order case. That is, one can obtain the following statements:

$$\begin{array}{cc}\hfill & \left({\mathcal{I}}_a^{\overline{\alpha },\rho }{\mathcal{I}}_a^{\overline{\gamma },\rho }z\right)\left(t\right)=\left({\mathcal{I}}_a^{\overline{\gamma },\rho }{\mathcal{I}}_a^{\overline{\alpha },\rho }z\right)\left(t\right)=\left({\mathcal{I}}_a^{\overline{\alpha }+\overline{\gamma },\rho }z\right)\left(t\right),\mathrm{with}z\in L^1(\mathbf{J},{\mathbb{R}}^r);\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \left({}^C{\mathcal{D}}_a^{\overline{\alpha },\rho }{\mathcal{I}}_a^{\overline{\alpha },\rho }z\right)\left(t\right)=z\left(t\right),\mathrm{with}z\in L^1(\mathbf{J},{\mathbb{R}}^r);\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \left({\mathcal{I}}_a^{\overline{\alpha },\rho }{}^C{\mathcal{D}}_a^{\overline{\alpha },\rho }z\right)\left(t\right)=z\left(t\right)-e^{\frac{\rho -1}{\rho }(t-a)}z\left(a\right),\mathrm{with}z\in C^1(\mathbf{J},{\mathbb{R}}^r).\hfill \end{array}$$

(7)

To end this section, we recall the Leray–Schauder nonlinear alternative.

Theorem 1 

([19]). Let E be a Banach space and $\mathbf{Z}$ be a bounded open subset of E. If $\mathcal{A}:\overline{\mathbf{Z}}\to E$ is a completely continuous operator. Then, either there exist $z\in \partial \mathbf{Z}$ and $\lambda >1$ such that $\mathcal{A}z=\lambda z$, or there exists a fixed point $z^{*}\in \overline{\mathbf{Z}}$.

3. Main Results

Definition 5. 

A function $z\in C(\mathbf{J},{\mathbf{R}}^r)$ is said to be a solution of the Cauchy problem (1) if z satisfies ${}^C{\mathcal{D}}_a^{\overline{\alpha },\rho }z\left(t\right)=K\left(t,z\left(t\right)\right)$ for a.e. $t\in \mathbf{J}$ and the initial condition $z\left(a\right)=z_a$.

The following lemma plays an essential role in the forthcoming discussions.

Lemma 2 

([17], Lemma 4.1). For any function $\mathrm{\Psi }\in C(\mathbf{J},\mathbb{R})$ the solution of the linear scalar generalized proportional fractional differential equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_a^C{\mathcal{D}}^{\alpha ,\rho }x\left(t\right)=\mathrm{\Psi }\left(t\right),t\in \mathbf{J},0<\alpha <1\hfill \\ x\left(a\right)=x_a\in \mathbb{R}\hfill \end{array}\right.\end{array}$$

(8)

is given by the following integral equation

$$x\left(t\right)=x_a{e}^{\frac{\rho -1}{\rho }(t-a)}+\frac{1}{{\rho }^{\alpha }\mathrm{\Gamma }\left(\alpha \right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{\alpha -1}\mathrm{\Psi }\left(\tau \right)d\tau ,t\in \mathbf{J}.$$

(9)

From Definition 4 for the generalized proportional vector-order fractional derivative of vector order $\overline{\alpha }$, we could easily generalize Lemma 2 for the vector case:

Lemma 3. 

For any function $\varphi \in C(\mathbf{J},{\mathbb{R}}^r)$, $\varphi ={({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_r)}^T$, the solution $z:\mathbf{J}\to {\mathbb{R}}^r,z={(z_1,z_2,\dots ,z_r)}^T,$ of the linear generalized proportional vector-order fractional differential equation

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_a^C{\mathcal{D}}^{\overline{\alpha },\rho }z\left(t\right)=\varphi \left(t\right),t\in \mathbf{J},\hfill \\ z\left(a\right)=\overline{b},\hfill \end{array}\right.\end{array}$$

(10)

with $\overline{b}={(b_1,b_2,\dots ,b_r)}^T\in {\mathbb{R}}^r$, is given by the following integral equations

$$\begin{array}{cc}\hfill z_i\left(t\right)& =b_i{e}^{\frac{\rho -1}{\rho }(t-a)}+\frac{1}{{\rho }^{{\alpha }_i}\mathrm{\Gamma }\left({\alpha }_i\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_i-1}{\varphi }_i\left(\tau \right)d\tau ,\hfill \\ \hfill & fort\in \mathbf{J},i=1,2,\dots ,r.\hfill \end{array}$$

(11)

Remark 2. 

The integral equations (11) could be symbolically written in a vector form,

$$z\left(t\right)=\overline{b}{e}^{\frac{\rho -1}{\rho }(t-a)}+\frac{1}{{\rho }^{\overline{\alpha }}\mathrm{\Gamma }\left(\overline{\alpha }\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{\overline{\alpha }-1}\varphi \left(\tau \right)d\tau ,t\in \mathbf{J},$$

(12)

where

$$\begin{array}{cc}& \frac{1}{{\rho }^{\overline{\alpha }}\mathrm{\Gamma }\left(\overline{\alpha }\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{\overline{\alpha }-1}\varphi \left(\tau \right)d\tau \hfill \\ \hfill & =(\frac{1}{{\rho }^{{\alpha }_1}\mathrm{\Gamma }\left({\alpha }_1\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_1-1}{\varphi }_1\left(\tau \right)d\tau ,\frac{1}{{\rho }^{{\alpha }_2}\mathrm{\Gamma }\left({\alpha }_2\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_2-1}{\varphi }_2\left(\tau \right)d\tau ,\hfill \\ \hfill & \dots ,\frac{1}{{\rho }^{{\alpha }_r}\mathrm{\Gamma }\left({\alpha }_r\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_r-1}{\varphi }_r\left(\tau \right)d\tau {)}^T.\hfill \end{array}$$

In accordance with Lemma 3 we shall introduce the concept of mild solutions to the Cauchy problem (1):

Definition 6. 

A function $z\in C(\mathbf{J},{\mathbb{R}}^r)$ is called a mild solution of the Cauchy problem (1) if it fulfills the integral equation

$$z\left(t\right)=z_a{e}^{\frac{\rho -1}{\rho }(t-a)}+\frac{1}{{\rho }^{\overline{\alpha }}\mathrm{\Gamma }\left(\overline{\alpha }\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{\overline{\alpha }-1}K(\tau ,z\left(\tau \right))d\tau ,t\in \mathbf{J},$$

(13)

where

$$\begin{array}{cc}\hfill & \frac{1}{{\rho }^{\overline{\alpha }}\mathrm{\Gamma }\left(\overline{\alpha }\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{\overline{\alpha }-1}K(\tau ,z\left(\tau \right))d\tau \hfill \\ \hfill & =(\frac{1}{{\rho }^{{\alpha }_1}\mathrm{\Gamma }\left({\alpha }_1\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_1-1}{K}_1(\tau ,z\left(\tau \right))d\tau ,\hfill \\ \hfill & \frac{1}{{\rho }^{{\alpha }_2}\mathrm{\Gamma }\left({\alpha }_2\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_2-1}{K}_2(\tau ,z\left(\tau \right))d\tau ,\hfill \\ \hfill & \dots ,\frac{1}{{\rho }^{{\alpha }_r}\mathrm{\Gamma }\left({\alpha }_r\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_r-1}{K}_r(\tau ,z\left(\tau \right))d\tau {)}^T.\hfill \end{array}$$

Now, we are ready to present the first main result of the current paper via the Banach contraction principle. We will introduce the following assumption:

(A1) 

The function $K\in C(\mathbf{J}\times {\mathbb{R}}^r,{\mathbb{R}}^r),K={(K_1,K_2,\dots ,K_r)}^T$ is globally Lipschitz in its second variable, i.e., there exist bounded continuous functions ${\mathrm{\Phi }}_j:\mathbf{J}\to [0,\infty ),j=1,2,\dots ,r:$

$${\mathrm{\Phi }}_j^{*}<\frac{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}{{(b-a)}^{{\alpha }_j}}\mathrm{for\; all}j=1,2,\dots ,r,$$

(14)

where ${\mathrm{\Phi }}_j^{*}={sup}_{t\in \mathbf{J}}{\mathrm{\Phi }}_j\left(t\right)$, such that

$$|K_j(t,z)-K_j(t,w)|\le {\mathrm{\Phi }}_j\left(t\right)\parallel z-w\parallel ,\mathrm{for\; all}t\in \mathbf{J},z,w\in {\mathbb{R}}^r,j=1,2,\dots ,r.$$

Theorem 2. 

Let the assumption (A1) be satisfied. Then, for any initial value $z_a\in {\mathbb{R}}^r$, the Cauchy problem (1) possesses a unique mild solution on $\mathbf{J}$.

Proof. 

Define the operator $\mathcal{A}:{\mathcal{C}}_r\to {\mathcal{C}}_r$ by the equalities

$$\left(\mathcal{A}z\right)\left(t\right)={\left({\left(\mathcal{A}z\right)}_1\left(t\right),{\left(\mathcal{A}z\right)}_2\left(t\right),\dots ,{\left(\mathcal{A}z\right)}_r\left(t\right)\right)}^T,z\in {\mathcal{C}}_r,$$

where ${\mathcal{A}}_j:{\mathcal{C}}_r\to {\mathcal{C}}_r$ is defined by

$${\left(\mathcal{A}z\right)}_j\left(t\right):=z_{a,j}{e}^{\frac{\rho -1}{\rho }(t-a)}+\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_j-1}{K}_j(\tau ,z\left(\tau \right))d\tau ,j=1,2,\dots ,r.$$

(15)

By virtue of (A1), the operator $\mathcal{A}z$ is well-defined for any $z\in {\mathcal{C}}_r$. Thus, the existence of a mild solution of the Cauchy problem (1) is equivalent to the fixed point problem $z=\mathcal{A}z$.

We show that $\mathcal{A}$ is a contraction. Let $z,w\in {\mathcal{C}}_r,z=(z_1,z_2,\dots ,z_r),w=(w_1,w_2,\dots ,w_r)$. For any $t\in \mathbf{J}$ and $j=1,2,\dots ,r$ by (A1), Equation (15) and $\rho \in (0,1]$, one obtains

$$\begin{array}{cc}\hfill {|\left(\mathcal{A}z\right)}_j\left(t\right)-{\left(\mathcal{A}w\right)}_j\left(t\right)|& \le \frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_j-1}|K_j(\tau ,z\left(\tau \right))-K_j(\tau ,w\left(\tau \right))|d\tau \hfill \\ \hfill & \le \frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{(t-\tau )}^{{\alpha }_j-1}{\mathrm{\Phi }}_j\left(\tau \right)\parallel z\left(\tau \right)-w\left(\tau \right)\parallel d\tau \hfill \\ \hfill & \le \underset{j=1,2,\dots ,r}{max}\left\{\frac{{\mathrm{\Phi }}_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\right\}{\parallel z-w\parallel }_{{\mathcal{C}}_r}.\hfill \end{array}$$

Hence,

$${\parallel \mathcal{A}z-\mathcal{A}w\parallel }_{{\mathcal{C}}_r}\le \underset{j=1,2,\dots ,r}{max}\left\{\frac{{\mathrm{\Phi }}_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\right\}{\parallel z-w\parallel }_{{\mathcal{C}}_r}.$$

By the condition (14), we infer that $\mathcal{A}$ is a contraction. Thus, the Banach contraction principle guarantees that the operator $\mathcal{A}$ possesses a unique fixed point $z\in {\mathcal{C}}_r$, which is the unique mild solution for the Cauchy problem (1) on $\mathbf{J}$. The proof is finished. □

In the case the function K is only locally Lipschitz, the existence result of Theorem 2 could not be applied. We will prove the next existence result, based on the Leray–Schauder nonlinear alternative.

Introduce the following assumptions:

(A1*) 

The function $K\in C(\mathbf{J}\times {\mathbb{R}}^r,{\mathbb{R}}^r),K={(K_1,K_2,\dots ,K_r)}^T,$ is globally Lipschitz in its second variable, i.e., there exist a constant $L>0$ and continuous functions ${\mathrm{\Phi }}_j:\mathbf{J}\to [0,\infty ),j=1,2,\dots ,r,$ such that

$$|K_j(t,z)-K_j(t,w)|\le {\mathrm{\Phi }}_j\left(t\right)\parallel z-w\parallel ,\mathrm{for\; all}t\in \mathbf{J},z,w\in {\mathbf{S}}_L,j=1,2,\dots ,r.$$

(A2) 

There exist continuous functions ${\mu }_j,{\nu }_j:\mathbf{J}\to [0,\infty )$ such that

$$|K_j(t,z)|\le {\mu }_j\left(t\right)+{\nu }_j\left(t\right)\parallel z\parallel ,t\in \mathbf{J},z\in {\mathbf{S}}_{L}j=1,2,\dots ,r,$$

where the functions ${\nu }_j$ are bounded:

$$\underset{t\in \mathbf{J}}{sup}{\nu }_j\left(t\right)\le \frac{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}{{(b-a)}^{{\alpha }_j}},\mathrm{for\; all}j=1,2,\dots ,r.$$

(16)

Remark 3. 

Note if $K(t,0)\equiv 0$ for all $t\in \mathbf{J}$, then from assumption(A1*)it follows the validity of(A2)with $\mu \left(t\right)\equiv 0,\nu \left(t\right)\equiv \mathrm{\Phi }\left(t\right)$ and inequality (14) coincides (16).

Theorem 3. 

Suppose that(A1*)and(A2)are fulfilled. Then for any initial value $z_a\in {\mathbb{R}}^r$, the Cauchy problem (1) possesses at least one mild bounded solution $z\in {\mathbf{S}}_{\ell }$, where

$$\ell =max\left\{L,\frac{\parallel z_a\parallel +{max}_{j=1,2,\dots ,r}\left\{\frac{{\mu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\right\}}{1-{max}_{j=1,2,\dots ,r}\left\{\frac{{\nu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\right\}}\right\}$$

with ${\mu }_j^{*}:={sup}_{t\in \mathbf{J}}{\mu }_j\left(t\right)$, ${\nu }_j^{*}:={sup}_{t\in \mathbf{J}}{\nu }_j\left(t\right)$.

Proof. 

Consider the bounded set ${\mathbf{S}}_{\ell }\subset {\mathcal{C}}_r$.

Consider the operator ${\mathcal{A}}_j:{\mathcal{C}}_r\to {\mathcal{C}}_r$, defined by (15). We will show that the operator $\mathcal{A}:{\mathbf{S}}_{\ell }\to {\mathcal{C}}_r$ is completely continuous in ${\mathbf{S}}_{\ell }$.

With the aim of showing that the conditions of Theorem 1 are verified, the proof will be split into three claims.

Claim 1.$\mathcal{A}$ is continuous in ${\mathbf{S}}_{\ell }$.

Let ${\left\{z_n\right\}}_{n=0}^{\infty },z_n\in {\mathbf{S}}_{\ell }$ be a sequence convergent to z in ${\mathbf{S}}_{\ell }$, i.e., $\left|\right|z_n-z{\left|\right|}_{{\mathcal{C}}_r}\to 0$. Let $t\in \mathbf{J}$ be fixed. For any $j=1,2,\dots ,r$ from assumption (A1*) we have

$$\begin{array}{cc}\hfill |{\left(\mathcal{A}{z}_n\right)}_j\left(t\right)-{\left(\mathcal{A}z\right)}_j\left(t\right)|& \le \frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_j-1}|K_j(\tau ,z_n\left(\tau \right))-K_j(\tau ,z\left(\tau \right))|d\tau \hfill \\ \hfill & \le \frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{(t-\tau )}^{{\alpha }_j-1}{\mathrm{\Phi }}_j\left(\tau \right)\parallel z_n\left(\tau \right)-z\left(\tau \right)\parallel d\tau \hfill \\ \hfill & \le \underset{j=1,2,\dots ,r}{max}\left\{\frac{{\mathrm{\Phi }}_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\right\}{\parallel z_n-z\parallel }_{{\mathcal{C}}_r},\hfill \end{array}$$

(17)

where ${\mathrm{\Phi }}_j^{*}:={sup}_{t\in \mathbf{J}}{\mathrm{\Phi }}_j\left(t\right)$.

Therefore, from inequality (17) it follows $\parallel \mathcal{A}{z}_n{-\mathcal{A}z\parallel }_{{\mathcal{C}}_r}\to 0asn\to \infty$. Thus, the operator $\mathcal{A}$ is continuous.

Claim 2.$\mathcal{A}\left({\mathbf{S}}_{\ell }\right)$ is relatively compact.

We show firstly that $\mathcal{A}\left({\mathbf{S}}_{\ell }\right)$ is uniformly bounded.

Let $z\in {\mathbf{S}}_{\ell },z=(z_1,z_2,\dots ,z_r)$. For any $t\in \mathbf{J}$ and $j=1,2,\dots ,r$ by (A2), Equation (15) and $\rho \in (0,1]$, one has

$$\begin{array}{cc}\hfill {|\left(\mathcal{A}z\right)}_j\left(t\right)|& \le |z_{a,j}|e^{\frac{\rho -1}{\rho }(t-a)}+\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{e}^{\frac{\rho -1}{\rho }(t-\tau )}{(t-\tau )}^{{\alpha }_j-1}\left|K_j(\tau ,z\left(\tau \right))\right|d\tau \hfill \\ \hfill & \le |z_a|+\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^t{(t-\tau )}^{{\alpha }_j-1}\left({\mu }_j\left(\tau \right)+{\nu }_j\left(\tau \right)\left|z\left(\tau \right)\right|\right)d\tau \hfill \\ \hfill & \le \parallel z_a\parallel +\left({\mu }_j^{*}+{\nu }_j^{*}\ell \right)\frac{{(t-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}\hfill \\ \hfill & \le \parallel z_a\parallel +\frac{{\mu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}+\ell \frac{{\nu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}\hfill \\ \hfill & \le \parallel z_a\parallel +\underset{j=1,2,\dots ,r}{max}\frac{{\mu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}+\ell \underset{j=1,2,\dots ,r}{max}\left(\frac{{\nu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}\right).\hfill \end{array}$$

(18)

From inequalities (16) it follows

$$\parallel z_a\parallel +\underset{j=1,2,\dots ,r}{max}\frac{{\mu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}\le \ell \left[1-\underset{j=1,2,\dots ,r}{max}\left(\frac{{\nu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}\right)\right].$$

Thus, from inequality (18) we get ${\parallel \mathcal{A}z\parallel }_{{\mathcal{C}}_r}\le \ell ,$ which implies that $\mathcal{A}\left({\mathbf{S}}_{\ell }\right)$ is uniformly bounded.

Next, we shall show that $\mathcal{A}\left({\mathbf{S}}_{\ell }\right)$ is equicontinuous.

For $t_1<t_2,t_1,t_2\in \mathbf{J}$ and $z\in {\mathbf{S}}_{\ell }$, for any $j=1,2,\dots ,r$, we have

$$\begin{array}{cc}\hfill & {|\left(\mathcal{A}z\right)}_j\left(t_2\right)-{\left(\mathcal{A}z\right)}_j\left(t_1\right)|\hfill \\ \hfill & \le |z_{a,j}\left|\right|e^{\frac{\rho -1}{\rho }(t_2-a)}-e^{\frac{\rho -1}{\rho }(t_1-a)}|\hfill \\ \hfill & +|\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_2}{e}^{\frac{\rho -1}{\rho }(t_2-\tau )}{(t_2-\tau )}^{{\alpha }_j-1}{K}_j(\tau ,z\left(\tau \right))d\tau \hfill \\ \hfill & -\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}{e}^{\frac{\rho -1}{\rho }(t_1-\tau )}{(t_1-\tau )}^{{\alpha }_j-1}{K}_j(\tau ,z\left(\tau \right))d\tau |\hfill \\ \hfill & \le \left|\right|z_a\left|\right||e^{\frac{\rho -1}{\rho }(t_2-a)}-e^{\frac{\rho -1}{\rho }(t_1-a)}|+\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_{t_1}^{t_2}{e}^{\frac{\rho -1}{\rho }(t_2-\tau )}{(t_2-\tau )}^{{\alpha }_j-1}\left|K_j(\tau ,z\left(\tau \right))\right|d\tau \hfill \\ \hfill & +\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}{e}^{\frac{\rho -1}{\rho }(t_2-\tau )}\left({(t_2-\tau )}^{{\alpha }_j-1}-{(t_1-\tau )}^{{\alpha }_j-1}\right)\left|K_j(\tau ,z\left(\tau \right))\right|d\tau \hfill \\ \hfill & +\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}|e^{\frac{\rho -1}{\rho }(t_2-\tau )}-e^{\frac{\rho -1}{\rho }(t_1-\tau )}|{(t_1-\tau )}^{{\alpha }_j-1}\left|K_j(\tau ,z\left(\tau \right))\right|d\tau \hfill \\ \hfill & \le \left|\right|z_a\left|\right||e^{\frac{\rho -1}{\rho }(t_2-a)}-e^{\frac{\rho -1}{\rho }(t_1-a)}|+\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_{t_1}^{t_2}{(t_2-\tau )}^{{\alpha }_j-1}({\mu }_j\left(\tau \right)+{\nu }_j\left(\tau \right)\parallel z\left(\tau \right))\parallel )d\tau \hfill \\ \hfill & +\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}\left({(t_2-\tau )}^{{\alpha }_j-1}-{(t_1-\tau )}^{{\alpha }_j-1}\right)({\mu }_j\left(\tau \right)+{\nu }_j\left(\tau \right)\parallel z\left(\tau \right))\parallel )d\tau \hfill \\ \hfill & +\frac{1}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}|e^{\frac{\rho -1}{\rho }(t_2-\tau )}-e^{\frac{\rho -1}{\rho }(t_1-\tau )}|{(t_1-\tau )}^{{\alpha }_j-1}({\mu }_j\left(\tau \right)+{\nu }_j\left(\tau \right)\parallel z\left(\tau \right))\parallel )d\tau \hfill \\ \hfill & \le \left|\right|z_a\left|\right||e^{\frac{\rho -1}{\rho }(t_2-a)}-e^{\frac{\rho -1}{\rho }(t_1-a)}|+\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}\{{\int }_{t_1}^{t_2}{(t_2-\tau )}^{{\alpha }_j-1}d\tau \hfill \\ \hfill & +{\int }_a^{t_1}\left({(t_2-\tau )}^{{\alpha }_j-1}-{(t_1-\tau )}^{{\alpha }_j-1}\right)d\tau +{\int }_a^{t_1}|e^{\frac{\rho -1}{\rho }(t_2-\tau )}-e^{\frac{\rho -1}{\rho }(t_1-\tau )}|{(t_1-\tau )}^{{\alpha }_j-1}d\tau \}.\hfill \end{array}$$

By applying the mean value theorem to the function $e^{\frac{\rho -1}{\rho }(t-\tau )}$ on $(t_1,t_2)$ for any $\tau \in (a,t_1)$ we get

$$|e^{\frac{\rho -1}{\rho }(t_2-\tau )}-e^{\frac{\rho -1}{\rho }(t_1-\tau )}|=\left|\frac{\rho -1}{\rho }{e}^{\frac{\rho -1}{\rho }(\xi -\tau )}(t_2-t_1)\right|\le \frac{1-\rho }{\rho }(t_2-t_1),\forall \xi \in (t_1,t_2),$$

since $0<\xi -\tau ,\rho \le 1$.

Thus, from the definition of the constant ℓ ad the above inequality we get

$$\begin{array}{cc}& {|\left(\mathcal{A}z\right)}_j\left(t_2\right)-{\left(\mathcal{A}z\right)}_j\left(t_1\right)|\hfill \\ \hfill & \le \left|\right|z_a\left|\right||e^{\frac{\rho -1}{\rho }(t_2-a)}-e^{\frac{\rho -1}{\rho }(t_1-a)}|+\underset{j=1,2,\dots ,r}{max}\{\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}{(t_2-t_1)}^{{\alpha }_j}\hfill \\ \hfill & +\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}\left({(t_2-\tau )}^{{\alpha }_j-1}-{(t_1-\tau )}^{{\alpha }_j-1}\right)d\tau +(t_2-t_1)\frac{1-\rho }{\rho }{(t_1-a)}^{{\alpha }_j}\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\}.\hfill \end{array}$$

(19)

By taking the maximum over j on (19), we obtain that

$$\begin{array}{cc}& \parallel \left(\mathcal{A}z\right)\left(t_2\right)-\left(\mathcal{A}z\right)\left(t_1\right)\parallel \hfill \\ \hfill & \le \left|\right|z_a\left|\right||e^{\frac{\rho -1}{\rho }(t_2-a)}-e^{\frac{\rho -1}{\rho }(t_1-a)}|+\underset{j=1,2,\dots ,r}{max}\left\{\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}{(t_2-t_1)}^{{\alpha }_j}\right\}\hfill \\ \hfill & +\underset{j=1,2,\dots ,r}{max}\left\{\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}{\int }_a^{t_1}\left({(t_2-\tau )}^{{\alpha }_j-1}-{(t_1-\tau )}^{{\alpha }_j-1}\right)d\tau \right\}\hfill \\ \hfill & +\underset{j=1,2,\dots ,r}{max}\left\{(t_2-t_1)\frac{1-\rho }{\rho }{(t_1-a)}^{{\alpha }_j}\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }({\alpha }_j+1)}\right\}\hfill \end{array}$$

(20)

As $|t_2-t_1|\to 0$, $\parallel \left(\mathcal{A}z\right)\left(t_2\right)-\left(\mathcal{A}z\right)\left(t_1\right){\parallel }_{{\mathcal{C}}_r}\to 0$. Consequently $\mathcal{A}\left({\mathbf{S}}_{\ell }\right)$ is uniformly bounded and equicontinuous. From the Arzelá–Ascoli theorem, we infer that $\mathcal{A}$ is completely continuous in ${\mathcal{C}}_r$.

Claim 3.The a priori bounds

Let $\mathbf{Z}=\{z\in {\mathcal{C}}_r:\parallel z{\parallel }_{{\mathcal{C}}_r}<\ell \}$ and $\partial \mathbf{Z}=\{z\in {\mathcal{C}}_r:\parallel z{\parallel }_{{\mathcal{C}}_r}=\ell \}$. For any $z\in \partial \mathbf{Z}$, such that $z=\lambda \mathcal{A}z,0<\lambda <1$, by (18) we get

$$\begin{array}{cc}\hfill |z_j{\left(t\right)|=|\lambda \left(\mathcal{A}z\right)}_j\left(t\right)|& {<|\left(\mathcal{A}z\right)}_j\left(t\right)|\le \parallel z_a\parallel +\frac{{\mu }_j^{*}+{\nu }_j^{*}\ell }{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right)}\frac{{(t-a)}^{{\alpha }_j}}{{\alpha }_j}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill {\parallel z\parallel }_{{\mathcal{C}}_r}& <\parallel z_a\parallel +\underset{j=1,2,\dots ,r}{max}\frac{{\mu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}+\ell \underset{j=1,2,\dots ,r}{max}\left(\frac{{\nu }_j^{*}{(b-a)}^{{\alpha }_j}}{{\rho }^{{\alpha }_j}\mathrm{\Gamma }\left({\alpha }_j\right){\alpha }_j}\right)\le \ell .\hfill \end{array}$$

Therefore, ${\parallel z\parallel }_{{\mathcal{C}}_r}<\ell$ which contradicts the fact that $z\in \partial \mathbf{Z}$. Hence, by virtue of the Leray–Schauder nonlinear alternative (Theorem 1), we infer that operator $\mathcal{A}$ possesses at least one fixed point $z^{*}\in {\mathbf{S}}_{\ell }$, which is a mild solution $z^{*}\in {\mathcal{C}}_r$ for the Cauchy problem (1). The proof is finished. □

4. Examples

Example 1. 

Consider the system

$$\begin{array}{cc}\hfill & {}^C{\mathcal{D}}_0^{{\alpha }_1,\rho }{z}_1\left(t\right)=0.8sin\left(z_2\left(t\right)\right),t\in [0,1],\hfill \\ \hfill & {}^C{\mathcal{D}}_0^{{\alpha }_2,\rho }{z}_2\left(t\right)=0.7cos\left(z_1\left(t\right)\right),t\in [0,1],\hfill \\ \hfill & z_1\left(0\right)=a,z_2\left(0\right)=b;\hfill \end{array}$$

(21)

then $K_1(t,z)=0.8sin\left(z_2\right),K_2(t,z)=0.7cos\left(z_1\right)$ and ${\mathrm{\Phi }}_1\left(t\right)=0.8,{\mathrm{\Phi }}_2\left(t\right)=0.7$.

Then the assumption (A1) could be satisfied for different values of ${\alpha }_1,{\alpha }_2,\rho$. The graphs of the functions in the inequality (14) are graphed on Figure 1 and Figure 2. For example, for $\rho =0.2$ and ${\alpha }_i\in (0,0.106)$, or for $\rho =0.3$ and ${\alpha }_i\in (0,0.133)$, or for $\rho =0.6$ and ${\alpha }_i\in (0,0.2462)$, or for $\rho =0.8$ and ${\alpha }_i\in (0,0.456),i=1,2$ the inequalities (14) hold, i.e., condition (A1) is satisfied. Thus, for the corresponding values of ρ and ${\alpha }_i,i=1,2,$ according to Theorem 2 the initial value problem (21) has a unique solution for any initial value ${(a,b)}^T$.

Example 2. 

Consider the system

$$\begin{array}{cc}\hfill & {}^C{\mathcal{D}}_0^{0.3,0.8}{z}_1\left(t\right)=2+0.8z_2^2\left(t\right),t\in [0,1],\hfill \\ \hfill & {}^C{\mathcal{D}}_0^{0.7,0.8}{z}_2\left(t\right)=2+0.75z_1^2\left(t\right),t\in [0,1],\hfill \\ \hfill & z_1\left(0\right)=1,z_2\left(0\right)=1;\hfill \end{array}$$

(22)

then ${\alpha }_1=0.3,{\alpha }_2=0.7,\rho =0.8,K_1(t,z)=2+0.8z_2^2,K_2(t,z)=2+0.75z_1^2$.

The quadratic function is not globally Lipschitz, therefore, Theorem 2 could not be applied.

But the assumption (A1*) is satisfied because $0.8z_2^2-0.8u_2^2\left(t\right)\le 1.6|z_2-u_2|$ and $0.75z_1^2-0.75u_1^2\le 1.5|z_1-u_1|$ for $u_1,u_2,z_1,z_2\in \{y\in \mathbb{R}:|y|\le 1\}$, i.e., ${\mathrm{\Phi }}_1\left(t\right)=1.6,{\mathrm{\Phi }}_2\left(t\right)=1.5$.

The assumption (A2) is satisfied with ${\mu }_1\left(t\right)={\mu }_2\left(t\right)=2$ and ${\nu }_1=0.8,{\nu }_2\left(t\right)=0.75$. It is clear that $0.8<\frac{0.8^{0.3}\mathrm{\Gamma }\left(1.3\right)}{{(1-0)}^{0.3}}=0.839358$ and $0.75<\frac{0.8^{0.7}\mathrm{\Gamma }\left(1.7\right)}{{(1-0)}^{0.3}}=0.77738$.

According to Theorem 3 the initial value problem (22) has at least one solution such that $\parallel z\parallel \le \ell$ where $\ell =min\{1,\frac{1+2.57321}{1-0.964955}\}=min\{1,101.96\}=1$.

5. Conclusions

Vector-order fractional initial value problems for the generalized proportional fractional differential equations in the Caputo sense are investigated. By means of the Leray-Schauder nonlinear alternative and the contraction principle, two types of existence results for the mild solutions are proved. These results refer to systems with different Lipschitz functions—global and local ones. Two illustrative examples are given in order to enlighten the theoretical results.