Abstract
This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the extended fractional snap problem by means of the \(\mathbb{G}\)-operators. After finding its equivalent solution as a form of the integral equation, we establish the existence criterion of this reformulated model with respect to some known fixed point techniques. Then we analyze its stability and further investigate the inclusion version of the problem with the help of some special contractions. We present numerical simulations for solutions of several examples regarding the fractional \(\mathbb{G}\)-snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders.
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1 Introduction
Fractional calculus is one of the most important branches of applied mathematics. The main importance of this field can be observed in many published papers regarding different fractional differential equations and inclusions in recent years. In this direction, different generalizations of derivatives have been introduced by some researchers. For example, recently, Lazreg et al. [1] investigated the Cauchy problem of Caputo–Fabrizio impulsive fractional differential equations
where \(\mathbb{I}_{0}= [0, a_{1}]\), \(\mathbb{I}_{k}= (a_{k}, a_{k+1}]\), \(k=1,2, \dots, \mathrm{m}\), \(0 = a_{0}< a_{1}< a_{2}< \cdots < a_{\mathrm{m}}< a_{\mathrm{m}+1}= \tau \), \(\mathrm{v}_{0} \in \mathbb{R}\), \(f: \mathbb{I}_{k}\times \mathbb{R}\to \mathbb{R} \ (k=0, 1, \dots, \mathrm{m})\) and \(\varrho _{k}: \mathbb{R} \to \mathbb{R} \ (k=1, \dots, \mathrm{m})\) are given continuous functions, and \({}^{\mathrm{CF}}\mathcal{D}_{a_{k}}^{r}\) is the Caputo–Fabrizio derivative of order \(r \in (0,1)\). Also, Krim et al. [2] considered the class of terminal value problems of Katugampola implicit differential equations of noninteger orders
where the function \(f: \mathbb{I}\times \mathbb{R}^{2}\to \mathbb{R}\) is continuous, and \({}^{K}\mathcal{D}_{0^{+}}^{r}\) is the Katugampola fractional derivative of order \(r\in (0,1]\). In 2020, Baitiche et al. [3] generalized the fractional settings and studied the existence of solutions of the following ψ-Caputo fractional differential equation:
where \({}^{C}\mathcal{D}_{a^{+}}^{q,\psi }\) is the ψ-Caputo fractional derivative of order \(q\in (2, 3]\), \(\mathrm{w}: \mathbb{J}\times \mathbb{R} \to \mathbb{R}\) is a given continuous function, and \(\lambda _{i}\) are real constants satisfying \(\Delta = \sum_{i=1}^{m} \lambda _{i} (\psi (\eta _{i}) - \psi (a) )^{2} - (\psi (b) - \psi (a))^{2} \neq 0\). Also, Wahash et al. [4] investigated the existence and interval of existence, uniqueness, estimates of solutions, and different types of Ulam stability results of solutions on a subinterval of \([0, b]\) for the nonlinear fractional differential equation involving generalized Caputo fractional derivatives with respect to the function ψ given by \({}^{C}\mathcal{D}_{a^{+}}^{q,\psi } \mathrm{v}(\mathfrak{t})= f( \mathfrak{t}, \mathrm{v}(\mathfrak{t})), \mathfrak{t} \in [0, b]\), with nonlocal condition \(\mathrm{v}(0) = \hslash (\mathrm{v}) = \mathrm{v}_{0}\), where \(q \in (0,1)\), \(\mathrm{v}_{0} \in \mathbb{R}\), \({}^{C}\mathcal{D}_{a^{+}}^{q,\psi }\) denotes the ψ-Caputo fractional derivative of order q, \(f: [0, b]\times \mathbb{R} \to \mathbb{R}\) and \(\hslash: C([0, b], \mathbb{R}) \to \mathbb{R}\) are nonlinear continuous functions, and \(\mathrm{v}\in C([0,b], \mathbb{R} )\) is such that the operator \({}^{C}\mathcal{D}_{a^{+}}^{q,\psi }\) exists and \({}^{C}\mathcal{D}_{a^{+}}^{q,\psi }\in C([0,b], \mathbb{R} )\).
In 2019, Pham et al. [5] introduced a chaotic integer-order system, called a snap system, which involves only one quadratic nonlinear term and takes the following mathematical form:
where \(\mathcal{T}(\mathrm{v}_{1}, \mathrm{v}_{2},\mathrm{v}_{3},\mathrm{v}_{4})=-a \mathrm{v}_{1}-\mathrm{v}_{2}-\mathrm{v}_{4}+b\mathrm{v}_{1} \mathrm{v}_{3}\). Equation (1) can be transformed into a fourth-order differential equation
The new equation (2) contains a fourth-order derivative of the variable \(\mathrm{v}_{1}\), which in physics stands for a second derivative of acceleration in a mechanical system. Equation (2) is called a snap or jounce equation and describes a fourth-order dynamical model.
Many researchers have investigated sufficient conditions for the uniqueness, existence, stability, and attractivity of solutions for a wide domain of fractional nonlinear ordinary differential equations (ODEs) or mathematical models containing different fractional derivatives by using numerous types of methods including standard fixed point theory, T-degree theory, variational methods, monotone iterative approaches, MNC technique, and so on. For more detail, see [6–23]. However, to the best of our knowledge, limited results can be found on the existence and stability of solutions of fractional snap systems via the generalized \(\mathbb{G}\)-Caputo derivative.
The authors in [24] studied the fractional snap model
where \(a=2\), \(b=1\), and the Caputo fractional order \(q=0.95\).
In view of the above facts, in this paper, we focus our attention on the problem of the existence and uniqueness along with the Hyers–Ulam stability of solutions for different forms of fractional nonlinear snap systems in the \(\mathbb{G}\)-Caputo sense with initial conditions. Namely, we study the following problem:
where \({}^{c}\mathcal{D}_{a^{+}}^{ \eta;\mathbb{G}}\) are the \(\mathbb{G}\)-Caputo derivatives, η belong to \(\{q, p, r, k \}\) such that \(0< q, p, r, k \leq 1\), the increasing function \(\mathbb{G}\in C^{1}([a,b])\) is such that \(\mathbb{G}^{\prime }(\mathfrak{t})\neq 0\), \(\mathfrak{t}\in {}[ a,b]\), \(h\in C([a,b] \times \mathbb{R}^{4},\mathbb{R})\), and \(\mathrm{v}_{0}, \mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}\in \mathbb{R}\). It is obvious that this system can be rewritten as
It is natural that if we set \(\mathbb{G}(\mathfrak{t})=\mathfrak{t}\), \(a=0\), and \(q =p =r=k =1\), then we obtain the standard 4th-order ODE (2) with initial conditions. Our method in this paper is based on fixed point approaches. Also, we can find more ideas on fractional calculus and its applications in [3, 25–41].
The summary of our work in this research is as follows. In Sect. 2, we recall several assembled concepts of fractional calculus, useful lemmas, and some theorems about the fixed points. In Sect. 3, we give the proof of the fundamental theorems of this paper by utilizing fixed point approaches such as Banach’s principle and Schauder’s theorem. In Sect. 4, we discuss the stability in the context of the Ulam–Hyers stability, its generalized version along with Ulam–Hyers–Rassias stability, and its generalized version for solutions of the fractional \(\mathbb{G}\)-snap system (4). In Sect. 5, we utilize a special form of contractions to prove the existence results for an inclusion version of (4). Appropriate applications with numerical simulation are provided in Sect. 6 to illustrate and analyze the obtained results. Finally, in Sect. 7, we give the conclusion of our article.
2 Preliminaries
Here we recall some initial notions, definitions and notations.
Let \(\mathbb{G}:[a, b] \to \mathbb{R}\) be increasing via \(\mathbb{G}' (\mathfrak{t}) \neq 0\) for all \(\mathfrak{t}\). We start this part by defining the \(\mathbb{G}\)-Riemann–Liouville fractional (\(\mathbb{G}\)-FRL) integrals and derivatives. In this section, we set
Definition 2.1
For \(\eta > 0\), the ηth \(\mathbb{G}\)-FRL integral of an integrable function \(\mathrm{v}: [a, b] \to \mathbb{R}\) with respect to \(\mathbb{G}\) is given as follows:
where \(\Gamma (\eta ) = \int _{0}^{+\infty } e^{-\mathfrak{t}} \mathfrak{t}^{ \eta -1} \,\mathrm{d}\mathfrak{t}, \eta >0\).
Let \(n\in \mathbb{N,}\) and let \(\mathbb{G}, \mathrm{v} \in C^{n}([a,b], \mathbb{R})\) be such that \(\mathbb{G}\) has the same properties mentioned above. The ηth \(\mathbb{G}\)-FRL derivative of v is defined by
where \(n=[\eta ]+1\) [42, 43]. The ηth \(\mathbb{G}\)-fractional Caputo derivative of v is defined by \({}^{c}\mathcal{D}_{a^{+}}^{\eta;\mathbb{G}} \mathrm{v}( \mathfrak{t}) = \mathcal{I}_{a^{+}}^{ n - \eta; \mathbb{G}} A^{(n)} \mathrm{v}( \mathfrak{t})\), where \(n=[\eta ]+1\) for \(\eta \notin \mathbb{N}\) and \(n=\eta \) for \(\eta \in \mathbb{N}\) [44]. In other words,
Extension (6) gives the Caputo derivative when \(\mathbb{G} (\mathfrak{t}) = \mathfrak{t}\). Also, in the case \(\mathbb{G} (\mathfrak{t})= \ln \mathfrak{t}\), it yields the Caputo–Hadamard derivative. If \(\mathrm{v} \in C^{n}([a,b], \mathbb{R})\), then the ηth \(\mathbb{G}\)-fractional Caputo derivative of v is specified as [44, Theorem 3]
The composition rules for the above \(\mathbb{G}\)-operators are recalled in the following lemma.
Lemma 2.2
([45])
Let \(n-1<\eta <n\) and \(\mathrm{v}\in C^{n} ([a,b],\mathbb{R})\). Then
for all \(\mathfrak{t}\in [a,b]\). Moreover, if \(m\in \mathbb{N}\) and \(\mathrm{v}\in C^{n+m}([a,b],\mathbb{R})\), then
From equation (7) observe that if \(A^{(j)}\mathrm{v}(a) = 0\) for \(j=n,n+1,\dots,n+m-1\), then \(A^{(m)} ({}^{c}\mathcal{D}_{a^{+}}^{\eta;\mathbb{G} } \mathrm{v} )(\mathfrak{t}) = ^{c}\mathcal{D}_{a^{+}}^{\eta +m; \mathbb{G}} \mathrm{v}(\mathfrak{t})\), \(\mathfrak{t}\in [a,b]\).
Lemma 2.3
([45])
Let \(\eta, \nu >0\) and \(\mathrm{v}\in C( [a, b], \mathbb{R})\). Then for all \(\mathfrak{t} \in [a,b]\), denoting \(F_{a}(\mathfrak{t}) = \mathbb{G}( \mathfrak{t}) - \mathbb{G}(a)\), we have
-
1.
\(\mathcal{I}_{a^{+}}^{\eta;\mathbb{G} } ( \mathcal{I}_{a^{+}}^{\nu; \mathbb{G} } \mathrm{v} )(\mathfrak{t}) = \mathcal{I}_{a^{+}}^{\eta + \nu;\mathbb{G} } \mathrm{v}(\mathfrak{t})\),
-
2.
\({}^{c}\mathcal{D}_{a^{+}}^{\eta;\mathbb{G} } ( \mathcal{I}_{a^{+}}^{\eta; \mathbb{G} }\mathrm{v} ) (\mathfrak{t}) = \mathrm{v}(\mathfrak{t})\),
-
3.
\(\mathcal{I}_{a^{+}}^{\eta; \mathbb{G} } ( F_{a}( \mathfrak{t}))^{\nu -1}= \frac{\Gamma (\nu )}{ \Gamma (\nu +\eta )}( F_{a} (\mathfrak{t}))^{\nu + \eta -1}\),
-
4.
\({}^{c}\mathcal{D}_{a^{+}}^{\eta;\mathbb{G} }( F_{a}( \mathfrak{t}))^{\nu -1}=\frac{\Gamma (\nu )}{\Gamma (\nu -\eta )}( F_{a}( \mathfrak{t}))^{\nu -\eta -1}\),
-
5.
\({}^{c}\mathcal{D}_{a^{+}}^{\eta;\mathbb{G} }( F_{a}( \mathfrak{t}))^{j} = 0, (j=0,1,\dots,n-1), n\in \mathbb{N},n-1\leq \eta \leq n\).
To end this part of the paper, we state the following fixed point theorems.
Theorem 2.4
(Banach contraction principle [46])
Let \((\mathbb{V}, \rho )\) be a nonempty complete metric space, and let \(\Psi: \mathbb{V}\to \mathbb{V}\) be a contraction, that is,
and for some \(\mu \in (0, 1)\). Then Ψ admits a unique fixed point.
Theorem 2.5
(Leray–Schauder [46])
Let \(\mathbb{V}\) be a Banach space, let Σ be a bounded convex closed subset of \(\mathbb{V}\), and let \(\mathbb{U}\) be an open set contained in Σ with \(0\in \mathbb{U}\). Let \(\Psi: \overline{\mathbb{U}} \to \Sigma \) be a continuous and compact mapping. Then either (i) Ψ admits a fixed point belonging to \(\bar{\mathbb{U}}\), or (ii) there exist \(\mathrm{v}\in \partial \mathbb{U}\) and \(\mu \in (0,1)\) such that \(\mathrm{v} = \mu \Psi (\mathrm{v})\).
Consider normed space \((\mathcal{C}, \|\cdot \|)\). The collection of all closed, bounded, compact and convex subsets of \(\mathcal{C}\) are denoted by \({\mathcal{P}}_{\mathrm{CL}}(\mathcal{C})\), \({\mathcal{P}}_{\mathrm{BN}}( \mathcal{C})\), \({\mathcal{P}}_{\mathrm{CP}}( \mathcal{C})\), and \({\mathcal{P}}_{\mathrm{CV}}( \mathcal{C})\), respectively.
Definition 2.6
([47])
Consider \(\mathrm{v}: \mathbb{R} \to \mathbb{R}\) as a real-valued function and \(\mathfrak{H}\) as a multifunction. (i) \(\mathfrak{H}\) is u.s.c on \(\mathcal{C} \) if \(\mathfrak{H}(\mathrm{v}^{*})\in \mathcal{P}_{\mathrm{CL}}(\mathcal{C})\) for any \(\mathrm{v}^{*} \in \mathcal{C} \), and also there exists a neighborhood \(\mathfrak{N}_{0}^{*}\) of \(\mathrm{v}^{*}\) subject to \(\mathfrak{H}(\mathfrak{N}_{0}^{*}) \subseteq \mathbb{O}\) for \(\mathbb{O} \subseteq \mathcal{C}\), where \(\mathbb{O}\) is an arbitrary open set. (ii) A real-valued map \(\mathrm{v}: \mathbb{R} \to \mathbb{R}\) is upper semicontinuous such that \(\lim \sup_{n\to \infty } \mathrm{v}( r_{n})\leq \mathrm{v}(r)\) for each \(\{ r_{n} \}_{n\geq 1}\) with \(r_{n} \to r\).
A Pompeiu–Hausdorff metric \(\mathcal{H}_{\rho }: (\mathcal{P}(\mathcal{C}) )^{2} \to \mathbb{R} \cup \{\infty \}\) is defined as
where ρ is the metric of \(\mathcal{M,}\) and [47] \(\rho (\mathcal{A}_{1}^{*},a_{2}^{*}) = \inf_{a_{1}^{*}\in \mathcal{A}_{1}^{*}} \rho ( a_{1}^{*}, a_{2}^{*})\) and \(\rho ( a_{1}^{*},\mathcal{A}_{2}^{*}) = \inf_{A_{2}^{*}\in \mathcal{A}_{2}^{*}} \rho (a_{1}^{*},a_{2}^{*})\). Suppose for \(\mathfrak{H}: \mathcal{C} \to {\mathcal{P}}_{\mathrm{CL}} (\mathcal{C})\) and \(\mathrm{v}_{1}, \mathrm{v}_{2} \in \mathcal{M}\), we have the inequality
Then \(\mathfrak{H}\) is said to be (H1) a Lipschitz map if \(L > 0\) and (H2) a contraction if \(0 < L < 1\) [47].
Definition 2.7
([47])
(i) \(\mathfrak{H}: [a, b]\times \mathbb{R} \to {\mathcal{P}}( \mathbb{R})\) is Carathéodory if \(\mathfrak{t} \mapsto \mathfrak{H}(\mathfrak{t},\mathrm{v})\) is measurable for any \(\mathrm{v} \in \mathbb{R}\) and \(\mathrm{v} \mapsto \mathfrak{H}(\mathfrak{t},\mathrm{v})\) is u.s.c for a.e. \(\mathfrak{t} \in [a, b]\). (ii) A Carathéodory multifunction \(\mathfrak{H}: [a, b]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})\) is \(L^{1}\)-Carathéodory if for any \(\epsilon >0\), there exists \(\kappa _{\epsilon }\in L^{1} ([a, b], \mathbb{R}_{+})\) such that
for all \(| \upsilon |\leq \epsilon \) and almost all \(\mathfrak{t} \in [a, b]\).
Definition 2.8
([48])
Let \(\uppsi: \mathbb{R}_{\geq 0}\to \mathbb{R}_{\geq 0}\) be a nondecreasing map belonging to class Π such that for all \(\mathfrak{t}>0\), \(\sum_{j=1}^{\infty }\uppsi ^{j} (\mathfrak{t}) < \infty \) and \(\uppsi ( \mathfrak{t}) < \mathfrak{t}\). Let \(\Phi ^{*}: \mathcal{C}\to \mathcal{C}\) and \(\alpha: \mathcal{C}^{2} \to \mathbb{R}_{\geq 0}\). Then
-
(i)
\(\Phi ^{*}\) is α-ψ-contraction if for \(\mathrm{v}_{1}, \mathrm{v}_{2}\in \mathcal{C}\),
$$\begin{aligned} \alpha (\mathrm{v}_{1}, \mathrm{v}_{2}) \rho \bigl( \Phi ^{*} \mathrm{v}_{1}, \Phi ^{*} \mathrm{v}_{2} \bigr) \leq \uppsi \bigl(\rho ( \mathrm{v}_{1}, \mathrm{v}_{2})\bigr). \end{aligned}$$ -
(ii)
\(\Phi ^{*}\) is α-admissible if \(\alpha (\mathrm{v}_{1}, \mathrm{v}_{2}) \geq 1\) gives \(\alpha (\Phi ^{*}\mathrm{v}_{1}, \Phi ^{*}\mathrm{v}_{2}) \geq 1\).
-
(iii)
\(\mathcal{C}\) has property (B) if for every sequence \(\{\mathrm{v}_{n}\}_{n\geq 1}\) of \(\mathcal{C}\) with \(\alpha ( \mathrm{v}_{n}, \mathrm{v}_{n+1})\geq 1\) and \(\mathrm{v}_{n}\to \mathrm{v}\), we have \(\alpha ( \mathrm{v}_{n}, \mathrm{v})\geq 1\) for all \(n \geq 1\).
Definition 2.9
([49])
Let \(\uppsi: \mathbb{R}_{\geq 0}\to \mathbb{R}_{\geq 0}\) be a nondecreasing map belonging to class Π such that for all \(\mathfrak{t}>0\), \(\sum_{j=1}^{\infty }\uppsi ^{j}(\mathfrak{t}) < \infty \) and \(\uppsi (\mathfrak{t}) < \mathfrak{t}\). Let \(\mathfrak{H}: \mathcal{C}\to \mathcal{P}( \mathcal{C})\) and \(\alpha: \mathcal{C}^{2} \to \mathbb{R}_{\geq 0}\). Then
-
(i)
\(\mathfrak{H}: \mathcal{C} \to \mathcal{P}_{\mathrm{CL},\mathrm{BN}} (\mathcal{C})\) is α-ψ-contraction if for all \(\mathrm{v}_{1}, \mathrm{v}_{2}\in \mathcal{C}\),
$$\begin{aligned} \alpha (\mathrm{v}_{1}, \mathrm{v}_{2}) \mathcal{H}_{\rho } ( \mathfrak{H} \mathrm{v}_{1}, \mathfrak{H} \mathrm{v}_{2} ) \leq \uppsi \bigl(\rho ( \mathrm{v}_{1}, \mathrm{v}_{2})\bigr). \end{aligned}$$ -
(ii)
\(\mathfrak{H}\) is α-admissible if for all \(\mathrm{v}_{1}\in \mathcal{C}\) and \(\mathrm{v}_{2}\in \mathfrak{H}\mathrm{v}_{1}\), the inequality \(\alpha (\mathrm{v}_{1}, \mathrm{v}_{2}) \geq 1\) gives \(\alpha ( \mathrm{v}_{2}, \mathrm{v}_{3}) \geq 1\) for each \(\mathrm{v}_{3} \in \mathfrak{H} \mathrm{v}_{2}\).
-
(iii)
\(\mathcal{C}\) has property \((C_{\alpha })\) if for every sequence \(\{ \mathrm{v}_{n} \}_{n \geq 1} \) of \(\mathcal{C}\) with \(\mathrm{v}_{n} \to \mathrm{v}\) and \(\alpha (\mathrm{v}_{n}, \mathrm{v}_{n+1}) \geq 1\), there exists a subsequence \(\{ \mathrm{v}_{n_{k}} \}\) of \(\{ \mathrm{v}_{n} \}\) such that \(\alpha (\mathrm{v}_{n_{k}}, \mathrm{v})\geq 1\) for all \(k\in \mathbb{N}\).
Theorem 2.10
([48])
Let \((\mathcal{C},\rho )\) a complete metric space, and let \(\uppsi \in \Pi \), \(\alpha: \mathcal{C}^{2}\to \mathbb{R,}\) and \(\Phi ^{*}: \mathcal{C} \to \mathcal{C}\). Assume that: (i) \(\Phi ^{*}\) is α-admissible and α-ψ-contraction, (ii) \(\alpha (\mathrm{v}_{0}, \Phi ^{*}\mathrm{v}_{0} )\geq 1\) for some \(\mathrm{v}_{0} \in \mathcal{C}\), and (iii) \(\mathcal{C}\) has property (B). Then \(\Phi ^{*}\) has a fixed point.
Theorem 2.11
([50])
Let \(\mathcal{C}\) be a Banach space, and let \(\mathbb{A} \neq \emptyset \) belong to \(\mathcal{P}_{\mathrm{CL},\mathrm{BN},\mathrm{CV}}(\mathcal{C})\). Suppose that for \(\mathfrak{T}_{1}\) and \(\mathfrak{T}_{1}\) defined on \(\mathbb{A}\), (i) \(\mathfrak{T}_{1} \mathrm{v} + \mathfrak{T}_{2} \mathrm{v}' \in \mathbb{A}\) for \(\mathrm{v}, \mathrm{v}'\in \mathbb{A}\), (ii) \(\mathfrak{T}_{1}\) is compact-continuous, and (iii) \(\mathfrak{T}_{2}\) is a contraction. Then there exists \(\mathrm{v}_{*}\in \mathbb{A}\) such that \(\mathrm{v}_{*} = \mathfrak{T}_{1} \mathrm{v}_{*} + \mathfrak{T}_{2} \mathrm{v}_{*}\).
Theorem 2.12
([49])
Let \((\mathcal{C},\rho )\) be a complete metric space, and let \(\uppsi \in \Pi \), \(\alpha: \mathcal{C}^{2}\to \mathbb{R}_{\geq 0}\), and \(\mathfrak{H}: \mathcal{C} \to \mathcal{P}_{\mathrm{CL},\mathrm{BN}} (\mathcal{C})\). Assume that (i) \(\mathfrak{H}\) is an α-admissible α-ψ-contraction, (ii) \(\alpha (\mathrm{v}_{0}, \mathrm{v}_{1} )\geq 1\) for some \(\mathrm{v}_{0} \in \mathcal{C}\) and \(\mathrm{v}_{1} \in \mathfrak{H} \mathrm{v}_{0}\), and (iii) \(\mathcal{C}\) has property \((C_{\alpha })\). Then \(\mathfrak{H}\) has a fixed point.
Theorem 2.13
([47])
Let \((\mathcal{C}, \rho )\) be a complete metric space. Assume that (i) \(\uppsi \in \Pi \) is u.s.c such that \(\liminf_{ \mathfrak{t} \to \infty } ( \mathfrak{t} - \uppsi ( \mathfrak{t})) > 0\) for \(\mathfrak{t}>0\) and (ii) \(\mathfrak{H}: \mathcal{C} \to \mathcal{P}_{\mathrm{CL},\mathrm{BN}}(\mathcal{C})\) satisfies the property
Then \(\mathfrak{H}\) has a unique end-point iff \(\mathfrak{H}\) has the (AEP)-property.
3 Existence and uniqueness results
Here we analyze the existence properties of solutions and their uniqueness for the proposed fractional \(\mathbb{G}\)-snap problem (4). We need the following lemma, which specifies the corresponding integral equation.
Lemma 3.1
Let \(q,p,r,k \in (0,1]\) and \(\mathrm{v}_{0}, \mathrm{v}_{1},\mathrm{v}_{2},\mathrm{v}_{3}\in \mathbb{R}\). If \(g\in C([a,b], \mathbb{R})\), then the linear \(\mathbb{G}\)-snap FBVP
has the solution
Proof
Consider \(\mathrm{v}(\mathfrak{t})\) satisfying the linear fractional \(\mathbb{G}\)-snap problem (3.1). Applying the kth \(\mathbb{G}\)-integral operator \(\mathcal{I}_{a^{+}}^{k;\mathbb{G}}\) to both sides of equation (8), by the 4th boundary condition we obtain
Similarly, by the 3rd boundary condition, applying the r-th \(\mathbb{G}\)-integral operator \(\mathcal{I}_{a^{+}}^{r;\mathbb{G}}\), we get
By the 2nd boundary condition, applying the pth \(\mathbb{G}\)-integral operator \(\mathcal{I}_{a^{+}}^{p;\mathbb{G}}\), we get
and finally, applying the qth \(\mathbb{G}\)-integral operator \(\mathcal{I}_{a^{+}}^{q;\mathbb{G}}\) to both sides of (10), by the 1st boundary condition, we get
We see that \(\mathrm{v}(\mathfrak{t})\) fulfills (9), and the proof is complete. □
At present, we aim to verify the existence of a unique solution of the fractional \(\mathbb{G}\)-snap system (4) by relying on Theorem 2.4. Note that \(C([a,b],\mathbb{R})\) is a Banach space with norm
Theorem 3.2
Let \(h\in C([a,b]\times \mathbb{R}^{4},\mathbb{R})\), and let
-
(C1)
\(\exists L >0\) such that \(\forall \mathfrak{t}\in {}[ a,b]\) and \(\mathrm{v}_{j},\mathrm{v}^{*}_{j}\in C([a,b],\mathbb{R})\), \(j=1,2,3\), 4,
$$\begin{aligned} & \bigl\vert h\bigl( \mathfrak{t}, \mathrm{v}_{1}( \mathfrak{t}), \mathrm{v}_{2}( \mathfrak{t}), \mathrm{v}_{3}( \mathfrak{t}),\mathrm{v}_{4}( \mathfrak{t})\bigr) - h\bigl( \mathfrak{t},\mathrm{v}^{*}_{1}( \mathfrak{t}), \mathrm{v}^{*}_{2}(\mathfrak{t}), \mathrm{v}^{*}_{3}( \mathfrak{t}), \mathrm{v}^{*}_{4}(\mathfrak{t})\bigr) \bigr\vert \\ & \quad\leq L \sum_{j=1}^{4} \bigl\vert \mathrm{v}_{j}( \mathfrak{t})-\mathrm{v}^{*}_{j}( \mathfrak{t}) \bigr\vert . \end{aligned}$$(11)
Then the fractional \(\mathbb{G}\)-snap system (4) admits a unique solution on \([a,b]\) if \(L \mathcal{O} <1\), where
Proof
To prove the desired result, we first let
for some constant \(\ell >0\) satisfying
where \(h_{0}^{*} = \sup_{\mathfrak{t}\in [a,b]} \vert h(t, 0,0,0,0)\vert \), and
To apply the Banach principle, we verify that \(\Psi:C([a,b],\mathbb{R})\rightarrow C([a,b],\mathbb{R})\) given as
where
admits a unique fixed point, which is the same solution of the fractional \(\mathbb{G}\)-snap BVP (4). First, we show \(\Psi \Omega _{\ell }\subset \Omega _{\ell }\), that is, Ψ maps \(\Omega _{\ell }\) into itself. For each \(\mathrm{v} \in \Omega _{r}\), we have
Also,
and
From (16), (17), (18), (19), and (13) we get
which implies that \(\Vert \Psi \mathrm{v}\Vert \leq \ell \) for \(\mathrm{v}\in \Omega _{\ell }\), and so \(\Psi \Omega _{\ell }\subset \Omega _{\ell }\). Next, we investigate the contractivity property of the operator Ψ. For \(\mathrm{v},\mathrm{w} \in C([a,b], \mathbb{R})\), we estimate
and
From (20), (21), (22), and (23) we obtain
Thus \(\Vert \Psi \mathrm{v} - \Psi \mathrm{w} \Vert \leq L \mathcal{O} \Vert \mathrm{v} - \mathrm{w} \Vert \). Since \(L \mathcal{O} < 1\), Ψ is a contraction on \(C([a,b], \mathbb{R})\). This, together with Theorem 2.4, guarantees the existence of a unique fixed point for Ψ and accordingly the existence of a unique solution for the fractional \(\mathbb{G}\)-snap BVP (4). The proof is complete. □
The next existence property for possible solutions of the fractional \(\mathbb{G}\)-snap BVP (4) is checked based on the hypotheses of Theorem 2.5.
Theorem 3.3
Let \(h\in C([a,b]\times \mathbb{R}^{4}, \mathbb{R})\) and assume that:
-
(C2)
there exist \(\varrho \in L^{1}([a,b], \mathbb{R}^{+})\) and an increasing function \(f \in C([0,\infty ),(0,\infty ))\) such that for all \(\mathfrak{t}\in {}[ a,b]\) and \(\mathrm{v}_{j}\in C([a,b],\mathbb{R})\), \(j=1,2,3,4\),
$$\begin{aligned} \bigl\vert h\bigl(\mathfrak{t}, \mathrm{v}_{1}( \mathfrak{t}), \mathrm{v}_{2} ( \mathfrak{t}), \mathrm{v}_{3} ( \mathfrak{t}), \mathrm{v}_{4} ( \mathfrak{t}) \bigr) \bigr\vert \leq \varrho (\mathfrak{t}) f \Biggl(\sum_{j=1}^{4} \bigl\vert \mathrm{v}_{j}(\mathfrak{t}) \bigr\vert \Biggr); \end{aligned}$$ -
(C3)
there exists \(B>0\) such that
$$\begin{aligned} \frac{B}{\Lambda + \mathcal{O} \varrho _{0}^{\ast } f (B)} > 1, \end{aligned}$$(24)where \(\varrho _{0}^{\ast }= \sup_{ \mathfrak{t} \in {}[ a,b]}| \varrho (\mathfrak{t})|\), and \(\mathcal{O}\) and Λ are represented in (12) and (14).
Then the fractional \(\mathbb{G}\)-snap system (4) has at least one solution on \([a,b]\).
Proof
consider \(\Psi: C([a,b], \mathbb{R}) \rightarrow C([a,b],\mathbb{R})\) defined by (15) and the ball \(N_{\epsilon }= \{ \mathrm{v} \in C([a,b], \mathbb{R}): \Vert \mathrm{v}\Vert \leq \epsilon \}\) for some \(\epsilon >0\). The continuity of h yields that of the operator Ψ. Now by (C2) we have
for \(\mathrm{v}\in N_{\epsilon }\). In a similar way, we get that
and
As a consequence, by (25), (26), (27), and (28) we obtain
where \(\mathcal{O}\) and Λ are represented by (12) and (14). Hence Ψ is uniformly bounded on \(C([a,b], \mathbb{R})\). Now let us check the equicontinuity of Ψ. Choose arbitrary \(\mathfrak{t},\mathfrak{t}^{*} \in [a,b]\) with \(t< t^{*}\) and \(\mathrm{v}\in N_{\epsilon }\). We have
By letting
this becomes
Obviously, the right-hand side of (30) does not depend on v and approaches 0 as \(\mathfrak{t}^{*}\) tends to \(\mathfrak{t}\). In the same way,
Again, the right-hand side of (31) goes to zero as \(\mathfrak{t}^{*} \to \mathfrak{t}\) independently of v. Finally,
and
which independent of v. The right-hand sides of (34) and (33) approach 0 as \(\mathfrak{t}^{*} \to \mathfrak{t}\). Therefore relations (30), (31), (32), and (34) imply that
as \(\mathfrak{t}^{*} \to \mathfrak{t}\). Thus the equicontinuity of Ψ is confirmed. Hence Ψ is compact on \(N_{\epsilon }\) by the Arzelá–Ascoli theorem. Until now, we saw that the hypotheses of Theorem 2.5 are fulfilled for the operator Ψ. Thus one of two cases (i) or (ii) is valid. By (C3) we build
for \(B>0\) via \(\Lambda + \mathcal{O}\varrho _{0}^{*} f(B) < B\). With the help of (C2), by (29) we write
Now we assume the existence of \(\mathrm{v}\in \partial \mathbb{U}\) and \(\mu \in (0,1)\) subject to \(\mathrm{v} = \mu \Psi \mathrm{v}\). For such a selection of v and μ, we may write by (34) that
a contradiction. Therefore case (ii) does not hold, and by Theorem 2.5 Ψ admits a fixed point in \(\overline{\mathbb{U}}\), which is regarded as a solution of the fractional \(\mathbb{G}\)-snap system (4), and this concludes the proof. □
4 Stability criterion
In this part, we review the stability criterion in the context of the Ulam–Hyers stability, its generalized version along with Ulam–Hyers–Rassias stability, and its generalized version for solutions of the fractional \(\mathbb{G}\)-snap system (4).
Definition 4.1
The fractional \(\mathbb{G}\)-snap BVP (4) is Ulam–Hyers stable if there exists \(0< c_{h}^{*} \in \mathbb{R} \) such that for all \(\epsilon > 0\) and \(\mathrm{v}^{*}\in C([a,b],\mathbb{R}) \) satisfying
there exists \(\mathrm{v}\in C([a,b], \mathbb{R}) \) satisfying the fractional \(\mathbb{G}\)-snap BVP (4) with
Definition 4.2
The fractional \(\mathbb{G}\)-snap BVP (4) is generalized Ulam–Hyers stable if there exists \(c_{h}^{*} \in C(\mathbb{R}^{+}, \mathbb{R}^{+})\) with \(c_{h}^{*}(0)=0\) such that for all \(\epsilon >0\) and \(\mathrm{v}^{*}\in C([a,b], \mathbb{R}) \) satisfying the inequality
there exists a solution \(\mathrm{v}\in C([a,b], \mathbb{R}) \) of the fractional \(\mathbb{G}\)-snap BVP (4) such that
Definition 4.3
The fractional \(\mathbb{G}\)-snap BVP (4) is Ulam–Hyers–Rassias stable with respect to Φ if there exists \(0 < c_{h,\Phi }^{*} \in \mathbb{R} \) such that for all \(\epsilon >0\) and \(\mathrm{v}^{*}\in C([a,b], \mathbb{R})\) satisfying
there exists a solution \(\mathrm{v} \in C([a,b],\mathbb{R})\) of the fractional \(\mathbb{G}\)-snap BVP (4) such that
Definition 4.4
The fractional \(\mathbb{G}\)-snap BVP (4) is generalized Ulam–Hyers–Rassias stable with respect to Φ if there exists \(0< c_{h,\Phi }^{*} \in \mathbb{R}\) such that for all \(\mathrm{v}^{*}\in C([a,b], \mathbb{R})\) satisfying
there exists a solution \(\mathrm{v}\in C([a,b], \mathbb{R})\) of the fractional \(\mathbb{G}\)-snap BVP (4) such that
Remark 4.1
\((a_{1})\) Def. 4.1 ⇒ Def. 4.2; \((a_{2})\) Def. 4.3 ⇒ Def. 4.4; and \((a_{3})\) for \(\Phi (\mathfrak{t}) = 1\), Def. 4.3 ⇒ Def. 4.1.
Remark 4.2
Note that \(\mathrm{v}^{*}\in C([a,b], \mathbb{R}) \) is called a solution ofinequality (35) iff there exists \(g\in C([a,b],\mathbb{R})\) depending on \(\mathrm{v}^{*}\) such that for all \(\mathfrak{t} \in [a,b]\), (i) \(\vert g(\mathfrak{t}) \vert < \epsilon \); and (ii)
Remark 4.3
Note that \(\mathrm{v}^{*}\in C([a,b], \mathbb{R}) \) is called a solution off inequality (36) iff there exists \(g\in C([a,b],\mathbb{R})\) depending on \(\mathrm{v}^{*}\) such that for all \(\mathfrak{t}\in [a,b]\), (i) \(\vert g(\mathfrak{t})\vert < \epsilon \Phi (\mathfrak{t})\); and (ii)
Here we discuss the Ulam–Hyers stability of the fractional \(\mathbb{G}\)-snap BVP (4).
Theorem 4.5
If all assumptions (C1) are fulfilled, then the fractional \(\mathbb{G}\)-snap BVP (4) is Ulam–Hyers stable on \([a,b]\) and is generalized Ulam–Hyers stable if \(L\mathcal{O} <1\).
Proof
For every \(\epsilon >0\) and all \(\mathrm{v}^{*}\in C([a,b], \mathbb{R})\) satisfying
we can find a function \(g(\mathfrak{t})\) satisfying
with \(\vert g(\mathfrak{t}) \vert \leq \epsilon \). It follows that
Let \(\mathrm{v}\in C([a,b], \mathbb{R}) \) be the unique solution of the fractional \(\mathbb{G}\)-snap BVP (4). Then it is given by
and
Also,
and
From (37), (38), (39), and (40) we get
where \(\mathcal{O}\) is defined in (12). As a consequence, it follows that
If we let \(c_{h}^{*}=\frac{\mathcal{O}}{1-L\mathcal{O}} \), then the Ulam–Hyers stability is fulfilled. Next, for
with \(c_{h}^{*}(0)=0 \), the generalized Ulam–Hyers stability is fulfilled. □
The Ulam–Hyers–Rassias stability for the fractional \(\mathbb{G}\)-snap BVP (4) is checked in the following:
Theorem 4.6
Let conditions (C1) be satisfied, and assume that
-
(C4)
there exist an increasing map \(\Phi \in C([a,b],\mathbb{R}^{+}) \) and \(\lambda _{\Phi }>0\) such that for all \(\mathfrak{t}\in [a,b]\),
$$\begin{aligned} \mathcal{I}_{a^{+}}^{q +p+r+k;\mathbb{G}} \Phi ( \mathfrak{t}) + \mathcal{I}_{a^{+}}^{p +r+k;\mathbb{G}} \Phi ( \mathfrak{t})+ \mathcal{I}_{a^{+}}^{r+k +;\mathbb{G}} \Phi (\mathfrak{t}) + \mathcal{I}_{a^{+}}^{ k;\mathbb{G}} \Phi (\mathfrak{t})< \lambda _{\Phi }\Phi (\mathfrak{t}). \end{aligned}$$(41)
Then the fractional \(\mathbb{G}\)-snap BVP (4) is Ulam–Hyers–Rassias stable and is generalized Ulam–Hyers–Rassias stable.
Proof
For every \(\epsilon >0\) and all \(\mathrm{v}^{*}\in C([a,b],\mathbb{R}) \) satisfying
we can find a function \(g(\mathfrak{t})\) satisfying
with \(\vert g(\mathfrak{t}) \vert \leq \epsilon \Phi (\mathfrak{t})\). It follows that
If \(\mathrm{v}\in C([a,b],\mathbb{R}) \) is a unique solution of (4), then we have
Then
Also,
and
From (42), (43), (44), and (45) we get
where \(\mathcal{O}\) is defined in (12). Accordingly, it gives
If we let \(c_{h,\Phi }^{*} = \frac{\lambda _{\Phi }}{1-L\mathcal{O}}\), then the fractional \(\mathbb{G}\)-snap BVP (4) is stable in the Ulam–Hyers–Rassias sense. Along with this, setting \(\epsilon =1\), the fractional \(\mathbb{G}\)-snap BVP (4) is generalized Ulam–Hyers–Rassias stable. □
5 Inclusion version of (4)
Here we will derive the existence of solutions to the inclusion version of fractional nonlinear snap system of the \(\mathbb{G}\)-Caputo sense with initial conditions (4), which takes the form
where \(\mathfrak{H}\) ia a multifunction on the product space \([a, b]\times \mathbb{R}^{4}\). The function \(\mathrm{v} \in \mathcal{C}:= C([a, b], \mathbb{R})\) is called a solution of system (46) if it satisfies the boundary conditions and there is \(\wp \in L^{1}([a,b])\) such that \(\wp (\mathfrak{t}) \in \widehat{\mathfrak{H}}_{\mathrm{v}}( \mathfrak{t})\) for almost all \(\mathfrak{t}\in [a,b]\), where
and
for all \(\mathfrak{t}\in [a,b]\). For each \(\mathrm{v} \in \mathcal{C}\), we define the set of selections of the operator \(\mathfrak{H}\) as
and define the operator \(\mathfrak{U}: \mathcal{C} \to \mathcal{P}(\mathcal{C})\) by
where
Theorem 5.1
Let \(\mathfrak{H}: [a,b] \times \mathcal{C}^{4} \to \mathcal{P}_{\mathrm{CP}}( \mathcal{C})\) be a multifunction. Suppose that the following conditions are satisfied:
-
(C5)
The multifunction \(\mathfrak{H}\) is integrable and bounded, and
$$\begin{aligned} \mathfrak{H}( \cdot, \mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}, \mathrm{v}_{4}): [a,b] \to \mathcal{P}_{\mathrm{CP}}(\mathcal{C}) \end{aligned}$$is measurable for \(\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}, \mathrm{v}_{4} \in \mathcal{C}\);
-
(C6)
There exist \(\upphi \in C([a, b], [0, \infty ))\) and a nondecreasing function \(\uppsi \in \Pi \) such that
$$\begin{aligned} \mathcal{H}_{d} \bigl( \mathfrak{H}( \mathfrak{t}, \mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}, \mathrm{v}_{4}), \mathfrak{H}( \mathfrak{t}, \acute{ \mathrm{v}_{1}},\acute{\mathrm{v}_{2}}, \acute{ \mathrm{v}_{3}}, \acute{\mathrm{v}_{4}} ) \bigr) \leq \frac{\upphi (\mathfrak{t}) \lambda ^{*}}{ \Vert \upphi \Vert } \uppsi \Biggl( \sum_{k=1}^{4} \vert \mathrm{v}_{k} - \acute{\mathrm{v}_{k}} \vert \Biggr) \end{aligned}$$for all \(\mathfrak{t} \in [a,b]\) and \(\mathrm{v}_{1}\), \(\mathrm{v}_{2}\), \(\mathrm{v}_{3}\), \(\mathrm{v}_{4}\), \(\acute{\mathrm{v}_{1}}\), \(\acute{\mathrm{v}_{2}}\), \(\acute{\mathrm{v}_{3}}\), \(\acute{\mathrm{v}_{4}} \in \mathcal{C}\), where \(\mathcal{O}^{*} = \mathcal{O}^{-1}\);
-
(C7)
There is \(\chi ^{*}:\mathbb{R}^{4} \times \mathbb{R}^{4}\to \mathbb{R}\) such that
$$\begin{aligned} \chi ^{*} \bigl( (\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}, \mathrm{v}_{4}), (\acute{ \mathrm{v}_{1}}, \acute{\mathrm{v}_{2}}, \acute{ \mathrm{v}_{3}}, \acute{\mathrm{v}_{4}}) \bigr) \geq 0 \end{aligned}$$for all \(\mathrm{v}_{k}, \acute{\mathrm{v}_{k}} \in \mathcal{C}\ (k=1,2,3,4)\);
-
(C8)
If \(\{\mathrm{v}_{n}\}\) is a sequence in \(\mathcal{C}\) with \(\mathrm{v}_{n} \to \mathrm{v}\) and
$$\begin{aligned} &\chi ^{*} \bigl( \bigl(\mathrm{v}_{n}(\mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{q;\mathbb{G}} \mathrm{v}_{n} ( \mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{p; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathrm{v}_{n}(\mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}_{n}( \mathfrak{t}) \bigr) \bigr) \bigr), \\ &\quad \bigl( \mathrm{v}_{n+1}(\mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}_{n+1}( \mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{p ; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathrm{v}_{n+1}( \mathfrak{t}) \bigr), \\ &\quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{p;\mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}_{n+1}( \mathfrak{t}) \bigr) \bigr) \bigr) \bigr) \geq 0 \end{aligned}$$for all \(\mathfrak{t}\in [a,b]\) and natural numbers n, then there exists a subsequence \(\{ \mathrm{v}_{n_{j}} \}\) of \(\{\mathrm{v}_{n}\}\) such that
$$\begin{aligned} &\chi ^{*} \bigl( \bigl(\mathrm{v}_{n_{j}}(\mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{q;\mathbb{G}} \mathrm{v}_{n_{j}} ( \mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{p; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathrm{v}_{n_{j}} (\mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}_{n_{j}} (\mathfrak{t}) \bigr) \bigr) \bigr), \\ &\quad \bigl( \mathrm{v}(\mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}( \mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{p; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathrm{v}( \mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl({^{c} \mathcal{D}}_{a^{+}}^{p;\mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}( \mathfrak{t}) \bigr) \bigr) \bigr) \bigr) \geq 0 \end{aligned}$$for all \(\mathfrak{t}\in [a,b]\) and \(j\geq 1\);
-
(C9)
There exist \(\mathrm{v}_{0} \in \mathcal{C}\) and \(\mathfrak{p}\in \mathfrak{U}(\mathrm{v}_{0})\) such that
$$\begin{aligned} &\chi ^{*} \bigl( \bigl(\mathrm{v}_{0}(\mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{q;\mathbb{G}} \mathrm{v}_{0} ( \mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{p; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathrm{v}_{0} (\mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v}_{0} (\mathfrak{t}) \bigr) \bigr) \bigr), \\ & \quad\bigl( \mathfrak{p}(\mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathfrak{p}( \mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{p ; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathfrak{p}( \mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl({^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{ q;\mathbb{G}} \mathfrak{p}( \mathfrak{t}) \bigr) \bigr) \bigr) \bigr) \geq 0 \end{aligned}$$for \(\mathfrak{t}\in [a,b]\), where \(\mathfrak{U}: \mathcal{C} \rightarrow P(\mathcal{C})\) is defined by (48);
-
(C10)
For any \(\mathrm{v}\in \mathcal{C}\) and \(\mathfrak{p}\in \mathfrak{U}(\mathrm{v})\) with
$$\begin{aligned} &\chi ^{*} \bigl( \bigl(\mathrm{v}(\mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v} ( \mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{p; \mathbb{G}} \bigl({^{c} \mathcal{D}}_{a^{+}}^{ q; \mathbb{G}} \mathrm{v} (\mathfrak{t}) \bigr), \\ &\quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathrm{v} ( \mathfrak{t}) \bigr) \bigr) \bigr), \\ & \quad\bigl( \mathfrak{p}(\mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathfrak{p}( \mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{p ; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathfrak{p}( \mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl({^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{ q;\mathbb{G}} \mathfrak{p}( \mathfrak{t}) \bigr) \bigr) \bigr) \bigr) \geq 0, \end{aligned}$$there exists \(\mathfrak{p}^{*}\in \mathfrak{U}(\mathrm{v})\) such that
$$\begin{aligned} &\chi ^{*} \bigl( \bigl( \mathfrak{p}(\mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{q;\mathbb{G}} \mathfrak{p}( \mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{p; \mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathfrak{p}( \mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl({^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c}\mathcal{D}}_{a^{+}}^{ q;\mathbb{G}} \mathfrak{p}( \mathfrak{t}) \bigr) \bigr) \bigr), \\ & \quad\bigl(\mathrm{v}(\mathfrak{t}), {^{c}\mathcal{D}}_{a^{+}}^{q; \mathbb{G}} \mathfrak{p}^{*} ( \mathfrak{t}), {^{c} \mathcal{D}}_{a^{+}}^{p ; \mathbb{G}} \bigl({^{c} \mathcal{D}}_{a^{+}}^{ q; \mathbb{G}} \mathfrak{p}^{*} ( \mathfrak{t}) \bigr), \\ & \quad {^{c}\mathcal{D}}_{a^{+}}^{ r; \mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{ p;\mathbb{G}} \bigl( {^{c} \mathcal{D}}_{a^{+}}^{q ;\mathbb{G}} \mathfrak{p}^{*} (\mathfrak{t}) \bigr) \bigr) \bigr) \bigr) \geq 0 \end{aligned}$$for all \(\mathfrak{t}\in [ a, b]\).
Then the inclusion problem (46) has at least one solution.
Proof
Obviously, the fixed point of \(\mathfrak{U}: \mathcal{C} \rightarrow \mathcal{P}(\mathcal{C})\) is a solution of BVP (46). Since the multivalued map \(\mathfrak{t} \rightarrow \widehat{\mathfrak{H}}_{ \mathrm{v}}( \mathfrak{t})\) is closed-valued and measurable for all \(\mathrm{v} \in \mathcal{C}\), \(\mathfrak{H}\) has measurable selection, and \(\mathfrak{S}_{ \mathfrak{H},\mathrm{v}}\) is nonempty. We have to prove that \(\mathfrak{U}(\mathrm{v})\) is closed in \(\mathcal{C}\) for \(\mathrm{v} \in \mathcal{C}\). Take \(\{\mathrm{v}_{n}\}\) in \(\mathfrak{U}(\mathrm{v})\) such that \(\mathrm{v}_{n} \to \mathrm{v}\). For each n, \(\wp _{n} \in \mathfrak{S}_{\mathfrak{H}, \mathrm{v}}\) is chosen such that
for all \(\mathfrak{t} \in [a, b ]\). Since \(\mathfrak{H}\) has compact values, we define a subsequence of \(\{\wp _{n}\}\) (again by the same notation) that converges to \(\wp \in L^{1}([0,1])\). Hence \(\wp \in \mathfrak{S}_{\mathfrak{H},\mathrm{v}}\) and
for all \(\mathfrak{t} \in [a, b ]\), which gives that \(\mathrm{v} \in \mathfrak{U}(\mathrm{v})\) and \(\mathfrak{U}\) is closed valued. As \(\mathfrak{H}\) is compact-valued, it is a simple task to affirm the boundedness of \(\mathfrak{U}(\mathrm{v})\) for arbitrary \(\mathrm{v}\in \mathcal{C}\). We have to prove that \(\mathfrak{U}\) is an α-ψ-contraction. For such a goal, we define \(\alpha ( \mathrm{v}, \acute{\mathrm{v}} )=1\) whenever
otherwise \(\alpha ( \mathrm{v}, \acute{\mathrm{v}} )=0\) for all \(\mathrm{v}, \acute{\mathrm{v}} \in \mathcal{C}\). Let \(\mathrm{v}, \acute{\mathrm{v}} \in \mathcal{C}\) and \(\hslash _{1}^{*} \in \mathfrak{U}(\acute{\mathrm{v}})\) and choose \(\wp _{1}\in \mathfrak{S}_{\mathfrak{H}, \acute{\mathrm{v}}}\) such that
for all \(\mathfrak{t}\in [a, b]\). We estimate
for all \(\mathrm{v},\acute{\mathrm{v}} \in \mathcal{C}\) with
for almost all \(\mathfrak{t}\in [a, b]\). Thus there exists \(\Upsilon \in \widehat{ \mathfrak{H}}_{\mathrm{v}}\) such that
Now let \(\mathfrak{N}^{*}:[0,1] \to \mathcal{P}(\mathcal{C})\) be a multivalued map defined as
for all \(\mathfrak{t}\in [a, b]\). As \(\wp _{1}\) and
are measurable, so is the multivalued function \(\mathfrak{N}^{*}(\cdot ) \cap \widehat{\mathfrak{H}}_{\mathrm{v}}( \mathfrak{\cdot })\). Now let \(\wp _{2} \in \widehat{\mathfrak{H}}_{\mathrm{v}}( \mathfrak{t})\) be such that
for all \(\mathfrak{t}\in [a, b]\). Let us define \(\hslash _{2}^{*}\in \mathfrak{U}(\mathfrak{t})\) by
for all \(\mathfrak{t}\in [ a, b]\). Let \(\sup_{\mathfrak{t} \in [a, b]} |\upphi (\mathfrak{t})| = \|\upphi \|\). Then
Also,
and
for all \(\mathfrak{t} \in [a, b]\). Hence
and thus
for all \(\mathrm{v}, \acute{\mathrm{v}} \in \mathcal{C,}\) which implies that \(\mathfrak{U}\) is an α-ψ-contraction. Now, let \(\mathrm{v}\in \mathcal{C}\) and \(\acute{\mathrm{v}} \in \mathfrak{U} ( \mathrm{v})\) be two functions such that \(\alpha (\mathrm{v}, \acute{\mathrm{v}}) \geq 1\). In this case,
so there exists \(\Upsilon \in \mathfrak{U}(\acute{\mathrm{v}})\) such that
From this it follows that \(\alpha (\acute{\mathrm{v}}, \Upsilon ) \geq 1\), which means that the operator \(\mathfrak{U}\) is an α-admissible. Now suppose that \(\mathrm{v}_{0}\in \mathcal{C}\) and \(\acute{\mathrm{v}} \in \mathfrak{U}(\mathrm{v}_{0})\) are such that
for all \(\mathfrak{t} \in [a, b]\). Subsequently, we have \(\alpha ( \mathrm{v}_{0}, \acute{\mathrm{v}}) \geq 1\). Consider \(\{\mathrm{v}_{n}\} \subseteq \mathcal{C}\) such that \(\mathrm{v}_{n} \to \mathrm{v}\) and \(\alpha (\mathrm{v}_{n}, \mathrm{v}_{n+1}) \geq 1\). Then we get
By hypothesis (C8) there is a subsequence \(\{\mathrm{v}_{n_{j}}\}\) of \(\{\mathrm{v}_{n}\}\) such that
Thus \(\alpha (\mathrm{v}_{n_{j}}, \mathrm{v}) \geq 1 (\forall j)\), that is, \(\mathcal{C}\) has the property \(C_{\alpha }\). Theorem 2.12 guarantees that \(\mathfrak{N}\) has a fixed point, which is the solution of the inclusion BVP (46). □
Theorem 5.2
Consider a multifunction \(\mathfrak{H}: [a, b] \times \mathcal{C}\times \mathcal{C} \to \mathcal{P}(\mathcal{C})\). Assume that:
-
(C11)
\(\uppsi: \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}\) is u.s.c nondecreasing map with \(\liminf_{\mathrm{v} \to \infty }(\mathrm{v} - \uppsi (\mathrm{v})) > 0\) and \(\uppsi (\mathrm{v}) < \mathrm{v}\) for all \(\mathrm{v}>0\);
-
(C12)
The operator \(\mathfrak{H}: [a, b] \times \mathcal{C} \times \mathcal{C} \to \mathcal{P}_{\mathrm{CP}}(\mathcal{C})\) is integrable and bounded, and \(\mathfrak{H}( \cdot,\mathrm{v}_{1}',\mathrm{v}_{2}', \mathrm{v}_{3}', \mathrm{v}_{4}'): [a, b] \to \mathcal{P}_{\mathrm{CP}}(\mathcal{C})\) is measurable for all \(\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}, \mathrm{v}_{4}\in \mathcal{C}\);
-
(C13)
There is \(\upphi \in C( [a, b], [0,\infty ))\) such that
$$\begin{aligned} \mathcal{H}_{d} \bigl( \mathfrak{H}( \mathfrak{t}, \mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}, \mathrm{v}_{4}), \mathfrak{H}( \mathfrak{t}, \acute{ \mathrm{v}_{1}},\acute{\mathrm{v}_{2}}, \acute{ \mathrm{v}_{3}}, \acute{\mathrm{v}_{4}} ) \bigr) \leq \upphi (\mathfrak{t}) \mathcal{O}^{*} \uppsi \Biggl( \sum _{k=1}^{4} \vert \mathrm{v}_{k} - \acute{\mathrm{v}_{k}} \vert \Biggr) \end{aligned}$$for all \(\mathrm{v}_{k}, \acute{\mathrm{v}}_{k} \in \mathcal{C}\ (k=1,2,3,4)\), where \(\mathcal{O}^{*} =\mathcal{O}^{-1}\);
-
(xv)
\(\mathfrak{U}\) has the (AEP)-property.
Then the inclusion BVP (46) has a solution.
Proof
We have to prove that \(\mathfrak{U}: \mathcal{C} \to \mathcal{P}(\mathcal{C})\) includes end points. Firstly, we must prove that \(\mathfrak{U}(\mathrm{v})\) is closed for every \(\mathrm{v} \in \mathcal{C}\). Since the mapping
is closed-valued and measurable for \(\mathrm{v} \in \mathcal{C}\), it has a measurable selection, and \(\mathfrak{S}^{*}_{\mathfrak{H},\mathrm{v}} \neq \emptyset \). By applying the same deduction as in the proof of Theorem 5.1, we may simply verify that \(\mathfrak{U}(\mathrm{v})\) is closed. Also, \(\mathfrak{U}(\mathrm{v})\) is bounded because of the compactness of \(\mathfrak{H}\). Finally, it is simple to prove that
Suppose that \(\mathrm{v}, \Upsilon \in \mathcal{C}\) and \(\hslash _{1}^{*} \in \mathfrak{U}(\Upsilon )\). Choose \(\wp _{1} \in \mathfrak{S}_{\mathfrak{H},\Upsilon }\) such that
for all \(\mathfrak{t} \in [a, b]\). As
for all \(\mathfrak{t}\in {}[ a, b]\), there exists \(\upphi ^{*} \in \widehat{\mathfrak{H}}_{\mathrm{v}}( \mathfrak{t})\) such that
for all \(\mathfrak{t}\in [a, b]\). Consider the multivalued map \(\mathfrak{O}^{*}: [a, b] \to \mathcal{P}(\mathcal{C})\) defined by
By the measurability of \(\wp _{1}\) and
it is obvious that that multifunction \(\mathfrak{O}^{*}(\cdot ) \cap \widehat{\mathfrak{H}}_{\mathrm{v}}( \mathfrak{\cdot })\) is also measurable. Now we take \(\wp _{2} \in \widehat{\mathfrak{H}}_{\mathrm{v}}( \mathfrak{t})\) such that
for all \(\mathfrak{t}\in [a, b]\). Choose \(\hslash _{2}^{*} \in \mathfrak{U}(\mathrm{v})\) such that
for all \(\mathfrak{t}\in [a, b]\). By the same argument as in Theorem 5.1 we get
Hence
for all \(\mathrm{v}, \Upsilon \in \mathcal{C}\). By using hypothesis (xv) we can easily find that \(\mathfrak{U}\) has the (AEP)-property. By Theorem 2.13 there exists \(\mathrm{v}^{*} \in \mathcal{C}\) such that \(\mathfrak{U}(\mathrm{v}^{*}) = \{\mathrm{v}^{*}\}\). This implies that \(\mathrm{v}^{*}\) satisfies the given problem (46), and the proof is completed. □
6 Numerical applications
Here we give some examples of fractional \(\mathbb{G}\)-snap systems based on numerical simulations to analyze their solutions. In these examples, we consider different cases of the function \(\mathbb{G}\) to cover the Caputo, Caputo–Hadamard, and Katugampola versions. For numerical computations, one can use Algorithms 1, 2 and 3.
Example 6.1
Based on system (4), we consider the nonlinear fractional ψ-snap BVP
where
for \(\mathfrak{t} \in [1.1, 2.6]\). It is clear that \(a=1.1\), \(b=2.6\), \(q=0.34\in (0,1]\), \(\mathrm{v}(0)= \mathrm{v}_{0} =2.25\), \(p=0.86\in (0,1]\), \(\mathrm{u}(0) = \mathrm{v}_{1}=-1.69\), \(r=0.54 \in (0,1]\), \(\mathrm{w}(0) = \mathrm{v}_{2} = 3.12\), \(k=0.25 \in (0,1]\), \(\mathrm{x}(0) = \mathrm{v}_{3}=-4.71\), and
Thus we can rewrite the above system as
Now we have
So we can choose \(L=\frac{1}{30}\). Additionally,
Now we consider four cases for \(\mathbb{G}\):
Note that \(\mathbb{G}_{2}\), \(\mathbb{G}_{3}\), and \(\mathbb{G}_{4}\) give the Caputo, Caputo–Hadamard, and Katugampola (for \(\rho = 0.5\)) derivatives. By using equation (12) in the first case \(\mathbb{G}_{1}(\mathfrak{t})= 2^{\mathfrak{t}}\), we have
Thus \(L \mathcal{O}_{1} = 0.791561<1\), and (C1) holds. Also, using equation (14), we obtain
Hence
Table 1 shows the numerical results of \(\mathcal{O}_{1}\), \(\Lambda _{1}\), and \(\ell _{1}\) for \(\mathfrak{t}\in [1.1,2.6]\). These values are also shown in Fig. 1.
Graphical representation of \(L \mathcal{O}_{1}\) and \(\ell _{1} \) for \(\mathfrak{t} \in [0,2]\) in Example 6.1
In the second case \(\mathbb{G}_{2}(\mathfrak{t})=\mathfrak{t}\) (Caputo type), we have
Thus \(L \mathcal{O}_{2} = 0.176894<1\), and (C1) holds. Also, using equation (14), we obtain
Hence
In the third case \(\mathbb{G}_{3}(\mathfrak{t})= \ln \mathfrak{t}\) (Caputo–Hadamard type), we have
Thus \(L \mathcal{O}_{3} = 0.082363<1\), and (C1) holds. Also, using equation (14), we obtain
Hence
In the fourth case \(\mathbb{G}_{4}(\mathfrak{t})= \sqrt{\mathfrak{t}}\) (Katugampola type for \(\rho =0.5\)), we have
Thus \(L \mathcal{O}_{4} = 0.047713<1\), and (C1) holds. Also, using equation (14), we obtain
Hence
Table 2 shows the numerical values of \(\mathcal{O}_{j}\), \(\Lambda _{j}\), and \(\ell _{j}\), \(j=2,3,4\), for \(\mathfrak{t}\in [1.1,2.6]\). These values are also shown in Fig. 2. Figure 3 shows a 3D-graph of the numerical values of \(\ell _{j}\) based on \(\mathcal{O}_{j}\) and \(\Lambda _{j}\), \(j=2,3,4\), for \(\mathfrak{t} \in [1.1, 2.6]\).
Graphical representation of \(\mathcal{O}_{j}\), \(\Lambda _{j}\), and \(\ell _{j}\) for \(\mathfrak{t} \in [1.1, 2.6]\) and \(j=2,3,4\) in Example 6.1 where \(\mathbb{G}_{2}(t)=\mathfrak{t}\), \(\mathbb{G}_{3}(\mathfrak{t})=\ln \mathfrak{t,}\) and \(\mathbb{G}_{4}(\mathfrak{t}) = \sqrt{\mathfrak{t}}\)
3D-graph of \(\ell \geq \frac{\Lambda + h_{0}^{*} \mathcal{O}}{1 - L \mathcal{O}}\) for \(\mathfrak{t} \in [1.1, 2.6]\) in Example 6.1
In all four cases for the function \(\mathbb{G}\), we saw that all requirements of Theorem 3.2 are fulfilled. Therefore this guarantees that for all four different cases in terms of the function \(\mathbb{G}\), the fractional \(\mathbb{G}\)-snap system (56) admits a unique solution on the interval \([1.1, 2.6]\).
In the next example, we examine the correctness of the results caused by Theorem 3.3. In that example, we consider the case \(\mathbb{G}(\mathfrak{t}) = \mathfrak{t}\) (Caputo type) for three different orders \(q_{1}\), \(q_{2}\), and \(q_{3}\) and show the obtained results computationally and graphically.
Example 6.2
Based on the given system (4) for \(\mathbb{G}(\mathfrak{t}) = \mathfrak{t}\) (Caputo type), we consider the nonlinear fractional \(\mathbb{G}\)-snap BVP
where
for \(\mathfrak{t} \in [0.02, 0.99]\). Clearly, \(a=0.02\), \(b=0.99\), \(\mathrm{v}(0)= \mathrm{v}_{0} =-1.07\), \(p=0.37\in (0,1]\), \(\mathrm{u}(0) = \mathrm{v}_{1}=4.46\), \(r=0.27 \in (0,1]\), \(\mathrm{w}(0) = \mathrm{v}_{2} = -3.8\), \(k=0.8 \in (0,1]\), \(\mathrm{x}(0) = \mathrm{v}_{3}=-2.15\), and
for \(\mathfrak{t} \in [0.02,0.99]\). Thus we can rewrite the above system as
Now we have
So we can choose \(\varrho (\mathfrak{t}) = \frac{1}{10} \mathfrak{t}\) and \(f(\mathrm{v}) = \frac{1}{15} \mathrm{v}\). Thus for \(j=1,2,3,4\),
and (C2) holds. In addition,
Now we consider three cases for \(q\in \{ q_{1} = 0.28, q_{2} = 0.53, q_{3} = 0.89\}\). By equation (12), in the first case \(q = q_{1} = 0.28\), we have
Also, by equation (14) we obtain
We consider \(B=100\). Then, substituting (69), (70), and (71) into inequality (24), we obtain
Hence (C3) holds for \(q = q_{1}=0.28\).
In the second case for \(q = q_{2} = 0.53\), we get
Also, by equation (14) we obtain
We consider \(K=100\). Then, substituting (69), (72), and (73) into inequality (24), we obtain
Hence (C3) holds for \(q = q_{2}=0.53\).
In the third case for \(q =q_{3} = 0.89\), we get
Also, using equation (14), we obtain
We consider \(B=100\). Then, substituting (69), (74), and (75) into inequality (24), we obtain
Hence (C3) holds for \(q=q_{3}=0.89\). Tables 3, 4, and 5 show the numerical values of \(\mathcal{O}_{j}\), \(\Lambda _{j}\), and \(\frac{B}{\Lambda _{j} + \mathcal{O}_{j} \varrho _{0}^{*} f (B)}\) for \(\mathfrak{t}\in [0.02,0.99]\) and \(q_{j} \in \{0.28, 0.53, 0.89 \}\), \(j= 1,2,3\).
These results are also plotted in Fig. 4. In all three cases for the order \(q_{i}\), we see that all requirements of Theorem 3.3 are fulfilled. Therefore this guarantees that for all three different cases by terms of the order q, the fractional \(\mathbb{G}\)-snap system (67) admits at least one solution on the interval \([0.02,0.99]\).
Graphical representation of \(\Delta _{j}\), \(\Lambda _{j}\), and \(\frac{K}{\Lambda _{j} + \Delta _{j} \phi _{0}^{*} \varphi (K)}\) for \(\mathfrak{t} \in [0.05,0.95]\), \(j=1,2,3\), in Example 6.2 where \(q_{1}=0.28\), \(q_{2}=0.53\), and \(q_{3}=0.89\)
Example 6.3
Based on system (46), we consider the nonlinear fractional inclusion system
for \(\mathfrak{t} \in [0.2, 0.85]\). It is clear that \(a=0.2\), \(b=0.85\), \(q=0.61\in (0,1]\), \(\mathrm{v}(0.2)= \mathrm{v}_{0} =3.92\), \(p=0.49\in (0,1]\), \(\mathrm{u}(0.2) = \mathrm{v}_{1}=-5.23\), \(r=0.35 \in (0,1]\), \(\mathrm{w}(0.2) = \mathrm{v}_{2} = 4.08\), \(k=0.73 \in (0,1]\), \(\mathrm{x}(0) = \mathrm{v}_{3} = -1.15\), and
For, \(\mathrm{v}_{j}, \acute{\mathrm{v}}_{j} \in \mathcal{C}\ (j=1,2,3,4)\), we have
Now we consider four cases for \(\mathbb{G}\):
Note that \(\mathbb{G}_{2}\), \(\mathbb{G}_{3}\), and \(\mathbb{G}_{4}\) give the Caputo, Caputo–Hadamard, and Katugampola (for \(\rho = 0.5\)) derivatives in this example. By equation (12) we have
Therefore
for \(\mathbb{G}_{j}(\mathfrak{t})\) \((j=1,2,3,4)\), respectively. Choose the nonnegative function \(\upphi \in C([a, b], [0,\infty ))\) defined by \(\upphi (\mathfrak{t}) =\frac{\mathfrak{t}}{4}\) for \(\mathfrak{t}\in [a, b]\). Then \(\Vert \upphi \Vert = 0.2125\). Also, we consider the nonnegative nondecreasing u.s.c map \(\uppsi: \mathbb{R}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}\) defined by \(\uppsi (\mathfrak{t}) = \frac{\mathfrak{t}}{2}\) for almost all \(\mathfrak{t} > 0\). Note that \(\lim_{\mathfrak{t} \to \infty } \inf (\mathfrak{t} -\uppsi ( \mathfrak{t})) > 0\) with \(\uppsi (\mathfrak{t}) <\mathfrak{t} (\forall \mathfrak{t} > 0)\). Finally, consider \(\mathfrak{U}: \mathcal{C}\to \mathcal{P}( \mathcal{C})\) by
where we have
Considering \(\wp =\frac{\mathfrak{t}}{10}\), we can see the results of \(\Upsilon (\mathfrak{t})\) in Table 6. These results are plotted in Fig. 5. Since the operator \(\mathfrak{U}\) has the (AEP)-property, by Theorem 5.2 system (76) has at least one solution.
Graphical representation of \(\mathcal{O}_{j}\) and \(\Upsilon _{j}\) for \(\mathfrak{t} \in [0.2,0.85]\), \(j=1,2,3,4\), in Example 6.3 where \(\mathbb{G}_{1}(\mathfrak{t}) = 2^{\mathfrak{t}}\), \(\mathbb{G}_{2}(\mathfrak{t})= \mathfrak{t}\), \(\mathbb{G}_{3}(\mathfrak{t}) = \ln \mathfrak{t}\), \(\mathbb{G}_{4}( \mathfrak{t} )= \sqrt{\mathfrak{t}}\)
MATLAB function for calculating the fractional integral \(\int _{a}^{\mathfrak{t}}\mathbb{G}'(\xi ) \frac{(\mathbb{G}(\mathfrak{t}) - \mathbb{G}(\xi ))^{q + p + r + k -1}}{\Gamma (q + p + r + k)} \wp (\xi ) \,\mathrm{d}\xi \) in Example 6.3 for \(\mathfrak{t} \in [a,b]\)
MATLAB lines for calculating values of \(\mathcal{O}\), \(L\mathcal{O}\), Λ, and ℓ in Example 6.1 for \(\mathfrak{t} \in [1.1, 2.6]\) and \(\mathbb{G}(\mathfrak{t}):=\{ 2^{\mathfrak{t}}, \mathfrak{t}, \ln \mathfrak{t}, \sqrt{\mathfrak{t}} \}\)
7 Conclusion
In this paper, we defined a new fractional mathematical model of a BVP consisting of the snap equation in the framework of the generalized sequential \(\mathbb{G}\)-operators and turned to the investigation of the qualitative behaviors of its solutions including the existence, uniqueness, stability, and inclusion version. To obtain an existence criterion, we used the Leray–Schauder theorem, and to obtain a uniqueness criterion, we utilized the Banach theorem. We studied different kinds of stability criteria based on the standard definitions of these notions. With the help of some special contractions, we established some theorems regarding the inclusion structure of the \(\mathbb{G}\)-snap problem. In the final step, we designed three examples, and considering different cases of the function \(\mathbb{G}\) and order q, we obtained numerical results of these two suggested fractional \(\mathbb{G}\)-snap systems in Caputo, Caputo–Hadamard, and Katugampola versions. Note that in this paper, by assuming \(\mathbb{G}(\mathfrak{t}) =\mathfrak{t}\) and \(q=p=r=k = 1\) we derived the standard 4th-order ODE of snap equation. Therefore we will be able to review other properties of this extended fractional \(\mathbb{G}\)-snap BVP by designing new generalized models based on nonsingular operators in the future works.
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Acknowledgements
The second and fifth authors were supported by Azarbaijan Shahid Madani University. The third author was supported by Bu-Ali Sina University. The authors express their gratitude to unknown referees for their helpful suggestions, which improved the final version of this paper.
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MATLAB lines for calculating values of \(\mathcal{O}\), Λ, and \(\frac{B}{\Lambda + \mathcal{O}\varrho _{0}^{*} f (B)}\) in Example 6.2 for \(\mathfrak{t} \in [0.02,0.99]\) and \(q \in \{0.28, 0.53, 89 \}\)
MATLAB lines for calculating values of \(\mathcal{O}^{*}\) and ϒ in Example 6.3 for \(\mathfrak{t} \in [0.2,0.85]\) and \(\mathbb{G}(\mathfrak{t}):=\{ 2^{\mathfrak{t}}, \mathfrak{t}, \ln \mathfrak{t}, \sqrt{\mathfrak{t}} \}\)
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Samei, M.E., Matar, M.M., Etemad, S. et al. On the generalized fractional snap boundary problems via G-Caputo operators: existence and stability analysis. Adv Differ Equ 2021, 498 (2021). https://doi.org/10.1186/s13662-021-03654-9
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DOI: https://doi.org/10.1186/s13662-021-03654-9