# Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

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## Abstract

**:**

## 1. Introduction and Some Preliminaries

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

## 3. Main Results

- (I)
- $T\in L(\mathcal{E},\mathcal{E})$.
- (II)
- $h\in C(J\times \mathcal{E},\mathcal{E})$ and there exist a ${p}_{h}\in {L}^{1}([0,1],{R}^{+})$ such that $\parallel h(\mu ,v)\parallel \le {p}_{h}(\mu )\parallel v\parallel $ for $\mu \in J$ and each $v\in \mathcal{E}$.
- (III)
- $H:J\to \mathcal{E}$, $H(.)=h(.,\omega (.))$ is a differentiable function, for any $\omega \in {C}^{1}(J,\mathcal{E})$.
- (IV)
- There exists a constant L such that $\parallel T\parallel +L<1-\beta $ and:$$\parallel h(\xi ,v)-h(\xi ,\overline{v})\parallel \le L\parallel v-\overline{v}\parallel \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{every}v,\overline{v}\in \mathcal{E}.$$

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Keten, A.; Yavuz, M.; Baleanu, D.
Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces. *Fractal Fract.* **2019**, *3*, 27.
https://doi.org/10.3390/fractalfract3020027

**AMA Style**

Keten A, Yavuz M, Baleanu D.
Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces. *Fractal and Fractional*. 2019; 3(2):27.
https://doi.org/10.3390/fractalfract3020027

**Chicago/Turabian Style**

Keten, Ayşegül, Mehmet Yavuz, and Dumitru Baleanu.
2019. "Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces" *Fractal and Fractional* 3, no. 2: 27.
https://doi.org/10.3390/fractalfract3020027