# Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

**Definition**

**1**

**.**The Riemann-Liouville fractional integral of order $\alpha >0$ for a measurable function $f:(0,+\infty )\to \mathbb{R}$ is defined as

**Definition**

**2**

**.**The Riemann-Liouville fractional derivative of order $\alpha >0$ for a measurable function $f:(0,+\infty )\to \mathbb{R}$ is defined as

## 3. Linear Problem

**Definition**

**3**

**.**A two-parameter function of the Mittag−Leffler ${E}_{\alpha ,\beta}\left(x\right)$ is defined by the series expansion

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

- 1.
- G is a continuous function on$(0,1]\times [0,1)$.
- 2.
- If $\lambda >{\lambda}_{1}^{*}$ then $G(t,s)>0$ for all $t,s\in (0,1)$
- 3.
- Consider the function$m\left(t\right)$and the positive content M, introduced in Lemma 2. Then the following inequality holds:$$m\left(t\right)\le \frac{{t}^{2-\alpha}G(t,s)}{s{(1-s)}^{\alpha -2}}\le {M}^{\prime},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathit{for}\mathit{all}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t,s\in (0,1),$$$${M}^{\prime}=M\left(1+\frac{L}{(1-\mu \theta -\eta \sigma )(\alpha -1)}\right)$$$$L=\mu \phantom{\rule{0.166667em}{0ex}}\parallel {v}_{1}{\parallel}_{2-\alpha}+\eta \phantom{\rule{0.166667em}{0ex}}{\parallel {v}_{2}\parallel}_{2-\alpha}.$$

**Proof.**

- It is obvious from the continuity of ${G}_{1}$, ${v}_{1}$ and ${v}_{2}$.
- From Lemma 2 and for $t\in (0,1]$ and $s\in (0,1)$, since $(1-\mu \theta -\eta \sigma )>0$, we have$$\begin{array}{ccc}\hfill {t}^{2-\alpha}G(t,s)& =& {t}^{2-\alpha}{G}_{1}(t,s)+{t}^{2-\alpha}\frac{(\mu {v}_{1}\left(t\right)+\eta {v}_{2}\left(t\right))}{(1-\mu \theta -\eta \sigma )}\left({\int}_{0}^{1}{G}_{1}(t,s)dt\right)\hfill \\ & \ge & {t}^{2-\alpha}{G}_{1}(t,s)\hfill \\ & \ge & s{(1-s)}^{\alpha -2}m\left(t\right).\hfill \end{array}$$Now, using again Lemma 2, from equation (8) we obtain$$\begin{array}{ccc}\hfill {t}^{2-\alpha}G(t,s)& \le & s{(1-s)}^{\alpha -2}M+\frac{L}{(1-\mu \theta -\eta \sigma )}s{(1-s)}^{\alpha -2}M{\int}_{0}^{1}{r}^{\alpha -2}dr\hfill \\ & =& s{(1-s)}^{\alpha -2}M\left(1+\frac{L}{(1-\mu \theta -\eta \sigma )(\alpha -1)}\right).\hfill \end{array}$$Which completes the proof. ☐

## 4. Nonlinear Problem

#### 4.1. Existence of Solutions

**Lemma**

**4.**

- (1)
- If there exists $e\in K\backslash \left\{0\right\}$ such that $x\ne Tx+\mu e$ for all $x\in \partial {D}_{K}$ and all $\mu >0$, then ${i}_{K}(T,{D}_{K})=0$.
- (2)
- If $\gamma x\ne Tx$ for all $x\in \partial {D}_{K}$ and all $\gamma \ge 1$, then ${i}_{K}(T,{D}_{K})=1$.
- (3)
- Let ${D}^{1}$ be open in X such that $\overline{{D}^{1}}\subset {D}_{K}$. Then$${i}_{K}(T,{D}_{K})={i}_{K}(T,{D}^{1})+{i}_{K}(T,{D}_{K}\backslash \overline{{D}^{1}}).$$
- (4)
- If ${i}_{K}(T,{D}_{K})\ne 0$ then there exists $u\in {D}_{K}$ such that $u=Tu$.

- (H
_{1}) - $f:[0,1]\times [0,\infty )\u27f6[0,\infty )$ is a continuous function.

**Definition**

**4.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 4.2. Non-Existence Results

**Theorem**

**4.**

- (i)
- $f(t,u)\le \tilde{m}u$ for $u\ge 0$ and $t\in I$, where $0<\tilde{m}<\frac{\alpha (\alpha -1)}{{M}^{\prime}}$.
- (ii)
- $f(t,u)\ge \tilde{M}u$ for $u\ge 0$ and $t\in [{c}_{1},1]$, with $\tilde{M}>{m}_{1}$ (${m}_{1}$ given in Lemma 6).

**Proof.**

- (i)
- Suppose, on the contrary, that there exists $u\in {C}_{2-\alpha}\left(I\right)$, $u\ge 0$ on I, u not identically zero on I, that solves (1). As we have seen, this property is equivalent to the fact that $u=Tu$. As a consequence, since ${\parallel u\parallel}_{2-\alpha}>0$, for $t\in I$, we have$$\begin{array}{ccc}\hfill 0\le {t}^{2-\alpha}u\left(t\right)& =& {t}^{2-\alpha}{\int}_{0}^{1}G(t,s)f(s,{s}^{2-\alpha}u\left(s\right))ds\hfill \\ & \le & {M}^{\prime}{\int}_{0}^{1}s{(1-s)}^{\alpha -2}f(s,{s}^{2-\alpha}u\left(s\right))ds\hfill \\ & \le & {M}^{\prime}\tilde{m}{\int}_{0}^{1}s{(1-s)}^{\alpha -2}{s}^{2-\alpha}u\left(s\right)ds\hfill \\ & \le & \frac{{M}^{\prime}\tilde{m}}{\alpha (\alpha -1)}{\parallel u\parallel}_{2-\alpha}\hfill \\ & <& {\parallel u\parallel}_{2-\alpha}.\hfill \end{array}$$Therefore, we get ${\parallel u\parallel}_{2-\alpha}<{\parallel u\parallel}_{2-\alpha}$, which is a contradiction.
- (ii)
- In this case, it the result is false, we have that there exists $u\in {C}_{2-\alpha}\left(I\right)$, $u\ge 0$ on I, with ${\parallel u\parallel}_{2-\alpha}>0$, such that $u=Tu$.Then, for $t\in [{c}_{1},1]$, we have$$\begin{array}{ccc}\hfill {t}^{2-\alpha}u\left(t\right)& \ge & {t}^{2-\alpha}{\int}_{{c}_{1}}^{1}G(t,s)f(s,{s}^{2-\alpha}u\left(s\right))ds\hfill \\ & \ge & \tilde{M}{t}^{2-\alpha}{\int}_{{c}_{1}}^{1}G(t,s){s}^{2-\alpha}u\left(s\right)ds.\hfill \end{array}$$Using that ${t}^{2-\alpha}\phantom{\rule{0.166667em}{0ex}}G(t,s)>0$ for all $t,s\in [{c}_{1},1]$ and, since ${s}^{2-\alpha}u\left(s\right)$ is a continuous, non-negative and non-trivial function on $[{c}_{1},1]$, we have that$$\underset{t\in [{c}_{1},1]}{min}\left\{{t}^{2-\alpha}{\int}_{{c}_{1}}^{1}G(t,s){s}^{2-\alpha}u\left(s\right)ds\right\}>0.$$In particular, previous inequalities show us that$$\overline{u}=\underset{t\in [{c}_{1},1]}{min}\left\{{t}^{2-\alpha}u\left(t\right)\right\}>0.$$Moreover$$\overline{u}\ge \tilde{M}\underset{t\in [{c}_{1},1]}{min}\left\{{t}^{2-\alpha}{\int}_{{c}_{1}}^{1}G(t,s){s}^{2-\alpha}u\left(s\right)ds\right\}\ge \tilde{M}\overline{u}\underset{t\in [{c}_{1},1]}{min}\left\{{\int}_{{c}_{1}}^{1}{t}^{2-\alpha}G(t,s)ds\right\}>\overline{u},$$

## 5. A Particular Example

**Example**

**1.**

## Author Contributions

## Funding

## Conflicts of Interest

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$\mathit{\alpha}$ | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 | 1.8 | 1.9 | 2 |
---|---|---|---|---|---|---|---|---|---|---|

$\lambda $ | −5 | −4.5 | −4 | −3.5 | −3 | −2.5 | −2 | −1.5 | −1 | −0.5 |

$\theta (\alpha ,\lambda )$ | 12.647 | 5.62054 | 2.83501 | 1.47577 | 0.657725 | −0.0336574 | −1.21724 | 54.6033 | 2.13745 | 1.20846 |

$\sigma (\alpha ,\lambda )$ | −4.52049 | −1.94567 | −0.99204 | −0.688973 | −0.638201 | −0.777128 | −1.40827 | 38.7227 | 1.22996 | 0.630732 |

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Cabada, A.; Wanassi, O.K. Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions. *Mathematics* **2020**, *8*, 255.
https://doi.org/10.3390/math8020255

**AMA Style**

Cabada A, Wanassi OK. Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions. *Mathematics*. 2020; 8(2):255.
https://doi.org/10.3390/math8020255

**Chicago/Turabian Style**

Cabada, Alberto, and Om Kalthoum Wanassi. 2020. "Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions" *Mathematics* 8, no. 2: 255.
https://doi.org/10.3390/math8020255