Existence, Uniqueness and the Multi-Stability Results for a $\mathcal{W}$-Hilfer Fractional Differential Equation

Equations (19) and (20) have multi-stability with respect to if there exists $\underset{i=1,\cdots ,15}{\underbrace{c_i}}>0$, such that for every $\underset{i=1,\cdots ,15}{\underbrace{{\theta }_i}}>0$ and every solution $\mathcal{L}\in C_{1-𝒲;\mathcal{W}}(\mathsf{{\rm Y}},\mathbb{R})$ to (45) and $I_{{\mathsf{\Theta }}^{+}}^{1-𝒲;\mathcal{W}}\mathcal{L}\left({\mathsf{\Theta }}^{+}\right)=J_{\mathsf{\Theta }}$, there exists a solution $\mathcal{J}\in C_{1-𝒲;\mathcal{W}}(\mathsf{{\rm Y}},\mathbb{R})$ to (19) and (20) with for every $\mathcal{S}\in \delta .$

([21]). Let a continuous function $\rho (\mathcal{S},\kappa ,\upsilon ):\mathsf{{\rm Y}}\times {\mathbb{R}}^2\to \mathbb{R}$. Then, Equation (1) is equivalent to where $⋌\in C(\mathsf{{\rm Y}},\mathbb{R})$ satisfies the equality below:

Let $\mathcal{L}\in C_{1-𝒲;\mathcal{W}}(\mathsf{{\rm Y}},\mathbb{R})$ be a solution of (45), and $I_{{\mathsf{\Theta }}^{+}}^{1-𝒲;\mathcal{W}}\mathcal{L}\left({\mathsf{\Theta }}^{+}\right)=J_{\mathsf{\Theta }}$. Then, $\mathcal{L}$ is a solution of the inequality below: where ${⋌}_{\mathcal{L}}\in C(\mathsf{{\rm Y}},\mathbb{R})$ satisfies the equality below:

Let $\left({\mathcal{F}}_1\right)$ and $\left({\mathcal{F}}_2\right)$ be satisfied. If $\mathcal{L}\in C_{1-𝒲;\mathcal{W}}(\mathsf{{\rm Y}},\mathbb{R})$ satisfies (45) and $I_{{\mathsf{\Theta }}^{+}}^{1-𝒲;\mathcal{W}}$$\mathcal{L}\left({\mathsf{\Theta }}^{+}\right)=J_{\mathsf{\Theta }}$, then there is a single function $\mathcal{J}$ satisfying (19) and (20), s.t., for every $\mathcal{S}\in \mathsf{{\rm Y}}$.

Set $\varrho =C_{1-𝒲;\mathcal{W}}(\mathsf{{\rm Y}},\mathbb{R})$, and define a mapping, $\chi :\varrho \times \varrho \to {[0,\infty ]}^n$, by We show $(\varrho ,\chi )$ is a complete metric space. Let $\chi ({\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2)>\chi ({\mathsf{\Phi }}_1,\nu )+\chi (\nu ,{\mathsf{\Phi }}_2)$, for some ${\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2$ and $\nu \in \varrho$. Thus, there exists ${\mathcal{S}}_{\circ }\in \delta$ with Thus, from the definition of $\chi$, we obtain which is contradictory. We now show $(\varrho ,\chi )$ is complete. Presume ${\omega }_k$ is a Cauchy sequence in $(\varrho ,\chi )$. Thus, for every $\underset{i=1,\cdots ,15}{\underbrace{{ϵ}_i}}>0,$ there exists a $\underset{i=1,\cdots ,15}{\underbrace{{\aleph }_{{ϵ}_i}}}\in \mathbb{N}$, s.t. $\chi ({\omega }_m,{\omega }_k)\le ({ϵ}_1,\cdots ,{ϵ}_{15}),$ for every $m,k\ge \underset{i=1,\cdots ,15}{\underbrace{{\aleph }_{{ϵ}_i}}}$. According to (25), we obtain for each $\mathcal{S}\in \mathsf{{\rm Y}}$. If $\mathcal{S}$ is fixed, (26) concludes that $\left\{{\omega }_k\left(\mathcal{S}\right)\right\}$ is a Cauchy sequence in $\mathbb{R}$. As the set of real numbers $\mathbb{R}$ is complete, $\left\{{\omega }_k\left(\mathcal{S}\right)\right\}$ converges for any $\mathcal{S}\in \mathsf{{\rm Y}}$. Then, we obtain a function $\omega$, defined by which gives us $\omega \in \varrho$, (because $\left\{{\omega }_k\left(\mathcal{S}\right)\right\}$ is Cauchy in complete space $\mathbb{R}$, so they are uniformly convergent on the mapping $\omega$ defined in (27). The uniform convergence leads us to the fact that $\omega$ is continuous, and is an element of $\varrho$). If we set $m\to \infty$, it follows from (26) that Considering (25), we obtain This confirms that the Cauchy sequence $\left\{{\omega }_k\right\}$ converges to $\omega$ in $(\varrho ,\chi )$. Thus, $(\varrho ,\chi )$ is complete. In view of Lemma 2, Equations (19) and (20) are equivalent to the system below: in which $⋌\in C(\mathsf{{\rm Y}},\mathbb{R})$ satisfies the equality, for all $\mathcal{S}\in \mathsf{{\rm Y}}$. To prove this note, by using $I_{{\mathsf{\Theta }}^{+}}^{\mathcal{X};\mathcal{W}}(.)$ on both sides of (1), and utilizing Theorem 2, we obtain Thus, In addition, if $\mathcal{J}$ satisfies (30), then it satisfies (1). To see this, apply ${}^{\mathcal{H}}{D}_{{\mathsf{\Theta }}^{+}}^{\mathcal{X},\mathcal{Z};\mathcal{W}}(.)$ on both sides of Equation (30). Then, according to Theorem 1, we obtain

${}^{\mathcal{H}}{D}_{{\mathsf{\Theta }}^{+}}^{\mathcal{X},\mathcal{Z};\mathcal{W}}\mathcal{J}\left(\mathcal{S}\right)={}^{\mathcal{H}}{D}_{{\mathsf{\Theta }}^{+}}^{\mathcal{X},\mathcal{Z};\mathcal{W}}{\displaystyle \frac{{(\mathcal{W}\left(\mathcal{S}\right)-\mathcal{W}\left(\mathsf{\Theta }\right))}^{𝒲-1}}{\Gamma \left(𝒲\right)}}{J}_{\mathsf{\Theta }}+{}^{\mathcal{H}}{D}_{{\mathsf{\Theta }}^{+}}^{\mathcal{X},\mathcal{Z};\mathcal{W}}{I}_{{\mathsf{\Theta }}^{+}}^{\mathcal{X};\mathcal{W}}⋌\left(\mathcal{S}\right)=⋌\left(\mathcal{S}\right),$ where, for $𝒲\in (0,1),$ we apply ${}^{\mathcal{H}}{D}_{{\mathsf{\Theta }}^{+}}^{\mathcal{X},\mathcal{Z};\mathcal{W}}{(\mathcal{W}\left(\mathcal{S}\right)-\mathcal{W}\left(\mathsf{\Theta }\right))}^{𝒲-1}=0.$ We deduce that $\mathcal{J}\left(\mathcal{S}\right)$ satisfies (1) if $\mathcal{J}\left(\mathcal{S}\right)$ satisfies (29).

Let $\propto :\varrho ⟶\varrho$, such that ${\mathsf{\Phi }}_1\in \varrho$ where ${⋌}_{{\mathsf{\Phi }}_1}\in C(\mathsf{{\rm Y}},\mathbb{R})$ satisfies the following equality: