Analysis of nonlinear implicit fractional differential equations with the Atangana-Baleanu derivative via measure of non-compactness

1.   Introduction

Riemann-Liouville and Caputo fractional derivatives are most commonly used in the analysis of differential equations of non-integer order. The research monographs ^[1,2,3,4,5] present the fundamental calculus of Riemann-Liouville and Caputo fractional derivatives, along with key studies on the existence, uniqueness, and various qualitative properties of solutions to differential equations involving these operators.

According to researchers ^[6,7,8,9], fractional derivatives with singular kernels may not adequately describe certain natural phenomena with nonlocal characteristics. As a result, efforts are being made by mathematicians to define new, non local fractional derivatives involving a non-singular kernel. In this sense, Caputo-Fabrizio ^[10] and Atangana-Baleanu ^[11] have defined new fractional derivative operators with a non-singular kernel in the form of exponential and Mittag-Leffler functions, respectively. Similar to other fractional derivatives, the Caputo version of the Atangana-Baleanu (AB) derivative ^[11] also has some drawbacks, though it overcomes some of the limitations due to conventional fractional derivatives.

The AB-derivative has garnered much attention because it effectively models certain phenomena that cannot be modeled as FDEs involving other conventional fractional derivatives. Atangana and Araz used the AB derivative ^[12] to analyse nonlinear differential equations in order to investigate the existence and uniqueness of solutions. The approximate controllability of fractional neutral stochastic systems with indefinite delay was investigated by Dineshkumar et al. ^[13] using the AB derivative. Initial value problems for several classes of AB FDEs have been studied by Kucche and Sutar ^[14,15,16], who have demonstrated comparative results as well as findings on the existence and uniqueness of solutions, extremal solutions, and data dependency of solutions.

To know more about mathematical modeling utilizing the AB derivative for various outbreaks such as dengue fever, a tumor-immune surveillance mechanism, optimal control of diabetes and tuberculosis co-existence etc., one can refer to ^[6,7,8,9] and ^[17].

Byszewski ^[18] pioneered the investigation of differential equations with nonlocal conditions. Nonlocal conditions play a crucial role in modeling many phenomena where the state at a point depends on the state at distant points. Nonlocal conditions generalize classical initial conditions to include additional information and provide more accurate solutions. For more details on nonlocal conditions and analysis of associated differential equations, we recommend the foundational works of Byszewski ^[19,20,21] and Balachandran ^[22,23].

The measures of non-compactness (MNC) provides another approach to manage differential equations by leveraging fixed point theorems that are proved in the form of MNC. For a comprehensive understanding and applications of MNC in the analysis of differential equations, we refer to ^{[24,25,26,27]} and the works cited therein. Researchers working in this field are actively exploring the application of MNC to investigate a wide range of differential equations subject to local and nonlocal constraints. Here, we include a few relevant research works. Sarwar et al.^[28] analyzed controllability results for semi-linear non-instantaneous impulsive neutral stochastic differential equations with the AB derivative. In another study, Thilakraj et al. ^[29] analyzed the sobolev-type Volterra-Fredholm functional integro-differential equation with non-local conditions.

On the other hand, implicit FDEs are essential for modeling various physical phenomena, in situations when it is not feasible to express fractional derivatives of a dependent variable explicitly. Since it might not always be possible to obtain analytical solutions for these equations, researchers are still working to develop the theory, methods, and techniques to analyze implicit FDEs. Kucche et al. ^[30,31,32] have examined different classes of implicit FDEs that involve the Caputo fractional derivative, and proved results about the existence and uniqueness of solutions as well as the interval of solution existence. The authors also investigated global existence results and Ulam-type stabilities via successive approximations for implicit AB FDEs. Sutar et al. ^[33] have studied implicit FDEs involving Caputo fractional derivatives through Picard and weakly Picard operator theory, along with the Pompeiu-Hausdorff functional to examine the existence, uniqueness, and dependence of the solution on the initial condition and the nonlinear-functions involved in the FDEs.

Inspired by the significance of implicit differential equations and the non-local, non-singular nature of the AB fractional derivative, we examine the boundary value problem for nonlinear implicit AB-FDEs of the following form:

and implicit AB-FDEs with non local condition of the form:

in a Banach space $\left(E, \|\cdot\|\right)$ over a field $\mathbb{R}$. Here, $^\mathfrak{AB}{_0\mathfrak{D}}_{\eta}^\mu$ denotes the AB fractional derivative of order $\mu \; (0 < \mu < 1)$, $\mathfrak{F}\in C^1(J\times E \times E, E)$ is any nonlinear function, $\mathfrak{g}:\mathcal{C}\to E$ is a continuous function, $\alpha, \beta \in \mathbb{R}$ such that $\alpha+\beta\ne 0$ and $\gamma, \, \zeta_0 \in E$. We assume that $\mathcal{C} = C(J, E)$ is the Banach space equipped with the supremum norm $\|\zeta\|_{\mathcal{C}} = \sup\left\lbrace \|\zeta(\eta)\| : \eta \in J\right\rbrace$.

Proving the existence results to the boundary value problem for nonlinear implicit FDEs (1.3) subject to boundary condition (1.4), and nonlinear implicit FDEs (1.1) subject to nonlocal condition (1.2) using the AB derivative, is our primary objective. Our primary findings are derived from the fixed point theorems of Darbo, Mönch, and the theory of the Kuratowski's measure of non-compactness.

We note that, the results obtained for nonlinear implicit FDEs (1.1)-(1.2) and (1.3)-(1.4) covers the study of the following implicit FDEs subject to following different types of initial and boundary conditions:

● The boundary value problem (1.1)-(1.2) reduces to an initial value problem for implicit FDEs if $\alpha = 1$ and $\beta = 0$.

● The non-local implicit FDEs (1.3)-(1.4) reduce to an initial value problem for implicit FDEs if $\mathfrak{g}(\zeta) = 0$.

● If $\mathfrak{g}(\zeta) = \sum_{k = 1}^{n}c_k \, \zeta(\eta_k)$, where $\eta_k\in J$ with $0\leq \eta_k<\eta_{k+1}\leq T$ for $k = 0, 1, \ldots n-1$, and $c_k\in E$ for $k = 1, 2, \ldots n$, then implicit FDEs (1.3)-(1.4) reduces to a non-local problem where we are excepting a solution to the problem (1.3)-(1.4) passing through $n$ number of points.

The key highlights can be summarized as follows:

● Established several significant existence results for nonlinear implicit FDEs that incorporate AB derivative.

● The investigation of nonlinear implicit FDEs was subject to both boundary and nonlocal conditions, shedding light on the complexities of such equations.

● The foundational tools employed are Kuratowski's measure of non-compactness and associated fixed point theorems, namely Darbo's fixed point theorem and Mönch's fixed point theorem.

● The results were illustrated by an example where we also obtained the solution to the problem for boundary conditions and non local conditions.

The remaining structure of this article is as follows: In Section 2, we provide the definitions and results regarding the AB fractional derivative, Kuratowski's measure of non-compactness, and fixed point theorems. In Section 3, we prove different existence results for the boundary value problem of the implicit FDEs (1.1)-(1.2). In Section 4, we establish existence results for implicit FDE (1.3) subject to the nonlocal condition (1.4). Finally, in Section 5, we provide examples of nonlinear implicit FDEs subject to boundary and initial conditions that illustrate the existence results we obtained.

2.   Preliminaries

In this section, we introduce preliminary information that will be used throughout the paper.

2.1.   AB fractional derivative and integral

In this section, we provide the definitions and few properties of the AB fractional derivative and integral.

Definition 2.1. ^[11] Let $\zeta\in H^1(0, 1)$ and $0 < \mu < 1$, the left AB fractional derivative of $\zeta$ of order $\mu$ is defined by

where $\mathfrak{A}(\mu) > 0$ is a normalization function satisfying $\mathfrak{A}(0) = \mathfrak{A}(1) = 1$ and $\mathbb{E}_\mu$ is one parameter Mittag-Leffler function.

Definition 2.2. ^[11] For any $\zeta\in H^1(0, 1)$ and $0 < \mu < 1$, the fractional integral in sense of Atangana-Baleanu fractional derivative is given by

Lemma 2.1. ^[34] For $0 < \mu < 1$,

(i) $^{\mathfrak{AB}}{_0}\mathfrak{I}_\eta^\mu\; \left[ ^\mathfrak{AB}{_0}\mathfrak{D}_{\eta}^\mu \zeta(\eta)\right] = \zeta(\eta)-\zeta(0)$,

(ii) $^\mathfrak{AB}{_0}\mathfrak{D}_{\eta}^\mu \; \left[^{\mathfrak{AB}}{_0}\mathfrak{I}_\eta^\mu \zeta(\eta)\right] = \zeta(\eta).$

Lemma 2.2. ^[35] Suppose that $\mu> 0, c(\eta)(1 -\frac{1-\mu}{\mathfrak{A} (\mu)} d(\eta))^ {-1}$ is a non-negative, non decreasing and locally integrable function on $[ a, b)$, $\frac{ \mu d(\eta)} {\mathfrak{A} (\mu) }\left(1 -\frac{1 -\mu} {\mathfrak{A} (\mu)} d(\eta)\right) ^ {-1}$ is non negative and bounded on $[ a, b)$ and $\zeta (\eta)$ is non negative and locally integrable on $[ a, b)$ with $\zeta (\eta) \leq c(\eta) + d(\eta)(^{\mathfrak{AB}}_ a \mathfrak{I}^\mu \upsilon)(\eta).$ Then,

2.2.   Kuratowski measure of non compactness

In this section, we provide the definition and properties of the Kuratowski measure of non compactness and the associated fixed point theorems.

Definition 2.3. ^[24] (The Kuratowski measure of non compactness) Let $\mathfrak{E}$ be a Banach space and $\Omega_\mathfrak{E}$ the set of bounded subsets of $\mathfrak{E}$. The Kuratowski measure of non compactness is the map $\nu : \Omega_\mathfrak{E}\to [0, \infty]$ defined by

where ${ \mathfrak{E}_i\in \Omega_\mathfrak{E}}$ and $diam(\mathfrak{E}_i) = \sup\left\lbrace \|a - b\| : a, b \in \mathfrak{E}_i\right\rbrace$.

Lemma 2.3. ^[24] Let $U$ and $V$ bounded sets.

$(a)$ $\nu (U) = 0\iff \bar{U}$ is compact, i. e., $U$ is relatively compact, where $\bar{U}$ denotes closure of $U$.

$(b)$ Non singularity: $\nu$ is equal to zero on every one element set.

$(c)$ $\nu (U) = \nu(\bar{U}) = \nu({ conv\; } U)$, where conv $U$ is the convex hull of $U$.

$(d)$ Monotonicity: $U\subset V\implies \nu(U)\leq \nu(V)$.

$(e)$ Algebraic semi-additivity: $\nu(U+V)\leq \nu(U)+\nu(V)$, where

$(f)$ Semi-homogencity: $\nu(\lambda V) = |\lambda|\nu(V); \lambda\in \mathbb{R}$ where $\lambda V = \left\lbrace \lambda v : v\in V\right\rbrace.$

$(g)$ Semi-additivity: $\nu(U \cup V) = \max\left\lbrace \nu(U), \nu(V)\right\rbrace$.

$(h)$ Invariance under translations: $\nu(U + \zeta_0) = \nu(U)$ for any $\zeta_0 \in E.$

The following fixed-point theorems and subsequent lemmas that are connected to Kuratowski's measure of non-compactness serve as the foundation for our findings.

Lemma 2.4. ^[25] (Darbo's fixed point theorem) Let $\mathfrak{X}$ be a Banach space and $\mathfrak{C}$ be a bounded, closed, convex and nonempty subset of $\mathfrak{X}$. Suppose a continuous mapping $\mathfrak{N} : \mathfrak{C} \to \mathfrak{C}$ is such that for all closed subsets $\mathfrak{D}$ of $\mathfrak{C}$,

where $0<\mathfrak{K}<1$. Then, $\mathfrak{N}$ has a fixed point in $\mathfrak{C}$.

Lemma 2.5. ^[26] (Mönch's fixed point theorem) Let $\mathfrak{D}$ be a bounded, closed and convex subset of a Banach space such that $0 \in \mathfrak{D},$ and let $\mathfrak{N}$ be a continuous mapping of $\mathfrak{D}$ into itself. If the implication

holds for every subset $\mathfrak{V}$ of $\mathfrak{D}$, then, $\mathfrak{N}$ has a fixed point.

Lemma 2.6. If $\mathfrak{V}\subset C(J, \mathbb{E})$ is a bounded and equicontinious set, then:

(i) The function $\eta \to \nu(\mathfrak{V}(\eta))$ is continuous on $J$, and

(ii)

where $\mathfrak{D}(\eta) = \left\lbrace \zeta(\eta):\zeta\in \mathfrak{D} \right\rbrace, \; \eta \in J.$

3.   Boundary value problem for implicit FDEs

Lemma 3.1. Let $\zeta, \; ^\mathfrak{AB}_0\mathfrak{D}_{\eta}^\mu \zeta\in C(J, E)$ and $\mathfrak{F}(0, \zeta(0), 0) = 0$. Then, $\zeta\in C(J, E)$ is a solution of implicit AB-FDEs (1.1)-(1.2) if and only if

where $\upsilon(\eta)$ is a solution of the functional equation

and

Proof. Define

Operating the AB-fractional integral operator $^{\mathfrak{AB}}{_0}\mathfrak{I}_\eta^\mu$ on both sides of (3.4) and utilizing Lemma 2.1, we obtain

Using the boundary condition given in Eq (1.2), from Eq (3.5), we get

This gives

From Eqs (3.5) and (3.6), we obtain Eq (3.1). Further, using Eqs (3.1) and (3.4) in the Eq (1.1), we obtain Eq (3.2).

On the other hand, assume that $\zeta$ satisfy functional Eq (3.1). Operating the AB-fractional derivative operator $^\mathfrak{AB}_0\mathfrak{D}_{\eta}^\mu$ on both sides of Eq (3.1) and using the Lemma 2.1, provides the following equation:

Equations (3.1) and (3.7) are used in Eq (3.2), to get

which is Eq (1.1).

Next, we show that $\zeta$ defined in Eq (3.1) also fulfills the boundary condition (1.2). In fact, based on Eqs (3.1) and (3.5), we have

Using the value of $\mathfrak{A}_\upsilon$ defined in (3.6), from above equation, we have

Remark 3.1. In the view of Lemma 3.1, to demonstrate the existence of the solution for the nonlinear implicit AB-FDEs (1.1)-(1.2), we need to demonstrate the existence of the solution $\zeta(\eta)$ for the functional fractional integral equation Eq (3.2). The function $\zeta(\eta)$ when substituted into Eq (3.1) provides the solution to the nonlinear implicit AB-FDEs (1.1)-(1.2).

We require the subsequent assumption to demonstrate the existence results for functional fractional integral equation (3.2).

(H1) $\|\mathfrak{F}(\eta _1, \zeta_1, \delta_1)-\mathfrak{F}(\eta_2, \zeta_2, \delta_2) \| \leq \mathfrak{L} |\eta_1-\eta_2|+\mathfrak{M} \|\zeta_1-\zeta_2 \| +\mathfrak{N} \|\delta_1 -\delta_2\|;$ $\mathfrak{L}, \mathfrak{M} \in\mathbb{R}^+, 0<\mathfrak{N} <1.$

Remark 3.2. ^[27] The condition (H1) is equivalent to the following inequality:

for any bounded sets $\mathfrak{A}_1, \mathfrak{A}_2 \subseteq E$ and for each $\eta \in J$.

The proof of the first existence result is provided through the fixed point theorem of Darbo.

Theorem 3.1. Let the function $\mathfrak{F}\in C(J\times E\times E, E)$ satisfies Lipschitz type condition (H1). Then, the nonlinear implicit AB-FDEs (1.1)-(1.2) has at least one solution, provided

Proof. We define the mapping $\mathfrak{T}:\mathcal{C} \to\mathcal{C}$, $\mathcal{C} = C(J, E)$ by

in the context of comments in Remark 3.1, where $\mathfrak{A}_\upsilon$ refers to the constant specified in Eq (3.3). We demonstrate that the mapping $\mathfrak{T}$ fulfills every condition of Lemma 2.4.

First, we establish that the mapping $\mathfrak{T}$ is continuous. Let the sequence $\left\lbrace \upsilon_m\right\rbrace$ of points in $E$ convergent to $\upsilon\in E$. Then for any $\eta\in J$, from Eq (3.9), we obtain

This gives

Since

we have

We have established the mapping $\mathfrak{T}$ is continuous.

Next, consider the number $R$ defined by

where $\mathfrak{M} _\mathfrak{F} = \sup_{ \eta \in J}\|\mathfrak{F}(\eta, 0, 0)\|$. Condition (3.8) implies $R>0$. Define

Note that $B_R$ is non-empty, convex, closed and bounded subset of $E$. Our aim is to prove that $\mathfrak{T}(B_R)\subseteq B_R$. To prove this, let's consider any $\eta\in J$ and $\upsilon\in B_r$. Then, by employing hypothesis (H1), we obtain

Using Eq (3.10), we have

Applying it in the aforementioned inequality leads us to

Consequently, $\|\mathfrak{T}(\upsilon)\|_\mathcal{C}\leq R$, implying $\mathfrak{T}\upsilon\in B_R$, and the proof of $\mathfrak{T}(B_R)\subseteq B_R$ is completed. Moreover, for each $\upsilon\in B_R$, $\|\mathfrak{T}\upsilon\|_\mathcal{C}\leq R$ indicates that $\mathfrak{T}(B_R)$ is bounded.

To demonstrate $\mathfrak{T}(B_R)$ is equicontinuous, consider any $\eta_1, \eta_2\in J$ with $\eta_1\leq \eta_2$. Then,

From the above inequality, it can be concluded that

Therefore, the demonstration of the equicontinuity of $\mathfrak{T}(B_R)$ is complete. To finalize the proof, it is necessary to confirm that the mapping $\mathfrak{T}:B_R\to B_R$ satisfies the following condition:

for any closed subset $\mathfrak{V}$ of $B_R$ and $0<k<1$. Employing the properties $\nu$ as described in Lemma 2.3, we have

This gives

Since

we get

The condition (3.8) allows us write

Inequality (3.11) leads us to conclude that $\nu$ is a set contraction. In light of Lemma 2.4, the mapping $T$ possesses a fixed point, which is the solution to the functional fractional integral equation (3.2). After adding this to Eq (3.1) the solution of problem (1.1)-(1.2) is provided.

In the subsequent theorem, we demonstrated that, with the same assumptions as given in Theorem 3.1, the existence result for nonlinear implicit FDEs (1.1)-(1.2) can be proved using the fixed point theorem of Mönch.

Theorem 3.2. Assume that the hypothesis (H1) hold. Then, the nonlinear implicit AB-FDEs (1.1)-(1.2) has a solution.

Proof. Consider the set $B_R$ we used in the Theorem 3.1. Clearly $B_R$ is a convex, closed and bounded subset of a Banach space $E$ containing $0$. Consider the operator $\mathfrak{T}:B_R\to B_R$ defined as in Theorem 3.1. We show that $\mathfrak{T}$ satisfies the requirement of M$\ddot{o}$nach's fixed point theorem. As established in the proof of Theorem 3.1, $\mathfrak{T}$ is bounded and continuous. Hence, it suffices to prove that the implication

holds for every subset $\mathfrak{V}$ of $B_R$.

Let $\mathfrak{V}$ be an equicontinuous subset of $B_R$ such that $\mathfrak{V} \subset conv(\mathfrak{T}(\mathfrak{V}) \cup \left\lbrace 0\right\rbrace).$ Define

Using Lemma 2.6, it follows that $x$ is continuous. Further, utilizing Remark 3.2, Lemma 2.5 and the properties of the measure $\nu$, for each $\eta \in J$, we obtain

This gives

By an application of Lemma 2.2 with $c(\eta) = 0$ and $d(\eta) = \dfrac{\mathfrak{M} }{1-\mathfrak{N} }$, we get $x(\eta) = 0$ for all $\eta \in J$. This gives $\mathfrak{V} (\eta) = 0$ for all $\eta\in J.$ Consequently, this gives $\nu_c(\mathfrak{V}) = 0$. Applying Theorem 2.5, we conclude that $\mathfrak{T}$ has a fixed point $\upsilon\in B_R$, which act as a solution of functional equation (3.1). Utilizing this $\upsilon$, we are able to obtain the required solution of nonlinear implicit AB-FDEs (1.1)-(1.2).

4.   Implicit FDEs with non-local condition

Essential conditions are established in this section for the solution to exist about the following implicit FDE:

subject to nonlocal initial condition

Lemma 4.1. Let $\zeta, \; ^\mathfrak{AB}_0\mathfrak{D}_{\eta}^\mu \zeta\in E$ and $\mathfrak{F}(0, \zeta(0), 0) = 0$. Then, $\zeta\in E$ is a solution of nonlinear implicit AB-FDEs (4.1)-(4.2) if and only if

where $\upsilon_\zeta$ is a solution of functional equation

Proof. Corresponding to $\zeta\in E$, we put

Operating $^\mathfrak{AB} _0\mathfrak{I}_\eta^\mu$ on both sides of Eq (4.5), and using Lemma 2.1 and the nonlocal initial condition (1.3), we obtain (4.3).

Utilizing Eqs (4.3) and (4.5) in Eq (4.1), one can see that $\upsilon_\zeta(\eta)$ satisfies the functional equation (4.4).

On the other hand, suppose that $\zeta\in E$ is a solution of functional equation (4.3). Operating $^\mathfrak{AB}_0\mathfrak{D}_{\eta}^\mu$ on both sides of Eq (4.3) and using Lemma 2.1, we get

By utilizing Eqs (4.3) and (4.6), the functional represented by Eq (4.10) reduces to (4.1). Next, we verify that non-local initial condition stated in Eq (4.2). Since $\mathfrak{F}(0, \zeta(0), 0) = 0,$ from Eq (4.3), we have

which is the nonlocal condition (4.2).

The subsequent theorem establishes both the existence and uniqueness results for non-local initial value problem (4.1)-(4.2).

Theorem 4.1. Let the function $\mathfrak{F}\in C^1(J\times E\times E, E)$ satisfies hypothesis (H1) and the function $\mathfrak{g}$ satisfies the condition

(H2) $\|\mathfrak{g}(\zeta)-\mathfrak{g}(\zeta_1) \|\le\mathfrak{K} \|\zeta -\zeta_1 \|_\mathcal{C}, \; \; \mathfrak{K}\in R.$

Then, the non-local initial value problem (4.1)-(4.2) has a solution provided

where

Proof. Considering Lemma 4.1, we define the mapping $\tilde{\mathfrak{T}}:\mathcal{C}\to \mathcal{C}$, $\mathcal{C} = C(J, E)$ as follows:

where $\upsilon_\zeta$ is a solution of functional equation

We demonstrate that the mapping $\tilde{\mathfrak{T}}$ fulfills all the conditions outlined in Lemma 2.4. For the continuity of $\tilde{\mathfrak{T}}$, consider any sequence $\left\lbrace \zeta_m\right\rbrace$ in $\mathcal{C}$ converging to $\zeta\in \mathcal{C}$. Then, for any $\eta\in J$, using hypothesis (H2), we obtain

where $\upsilon_{\zeta_m}$ and $\upsilon_\zeta$ are the solutions of functional equations

and

respectively. Using the hypothesis (H1), for any $\eta\in J$, we obtain

This gives

Application of Lemma 2.2 with $c(\eta) = \dfrac{\mathfrak{M} \mathfrak{K}\|\zeta_m -\zeta \|_\mathcal{C}}{1-\mathfrak{N} }$ and $d(\eta) = \dfrac{\mathfrak{M} }{1-\mathfrak{N} }$ gives

This gives

Since $\|\zeta_m- \zeta\|_\mathcal{C} \to 0$, from above inequality we get $\|\upsilon_{\zeta_m} - \upsilon_\zeta \|_\mathcal{C}\to 0$. Taking into account the previous estimates, using the inequality (4.11), we obtain $\|\tilde{\mathfrak{T}}\zeta_m- \tilde{\mathfrak{T}}\zeta\|_\mathcal{C} \to 0$. Therefore, $\tilde{\mathfrak{T}}$ is continuous.

Define

Condition (3.8) leads us to conclude $\tilde{R}>0$. With this particular choice of $\tilde{R}>0$, we define the set $B_{\tilde{R}} = \left\lbrace \zeta\in \mathcal{C}:\|\zeta\|_\mathcal{C}\leq \tilde{R}\right\rbrace.$

Next, we demonstrate that $\tilde{\mathfrak{T}}(B_{\tilde{R}})\subseteq B_{\tilde{R}}$. Let any $\eta\in J$ and $\zeta\in B_r$. Then applying hypothesis (H2), we obtain

For any $\eta\in J$, we discover that

Therefore,

As an application of Lemma 2.2 with $c(\eta) = \frac{\mathfrak{M} _\mathfrak{F}+\mathfrak{M} \left(\|\zeta_0\|+\mathfrak{K}R+\|\mathfrak{g}(0)\| \right)}{1-\mathfrak{N} }$ and $d(\eta) = \frac{\mathfrak{M} }{1-\mathfrak{N} }$, we get

This gives

Utilizing the value of $\mathfrak{J}$ defined in (4.8), from above inequality, we obtain

Thus from inequality (4.12), we have

Thus, $\|\tilde{\mathfrak{T}}(\zeta)\|_\mathcal{C}\leq \tilde{R}$ for any $\zeta\in B_{\tilde{R}}$ and the proof of $\tilde{\mathfrak{T}}(B_{\tilde{R}})\subseteq B_{\tilde{R}}$ is completed. For any $\zeta\in B_{\tilde{R}}$, we have $\|\tilde{\mathfrak{T}}\zeta\|_\mathcal{C}\leq \tilde{R}$ and hence $\tilde{\mathfrak{T}}(B_{\tilde{R}})$ is bounded.

To prove the equicontinuity of $\tilde{\mathfrak{T}}(B_{\tilde{R}})$, take any $\eta_1, \eta_2\in J$ with $\eta_1< \eta_2$. Then, we have

Since $\zeta \in B_R$, from above inequality it follows that $\|(\tilde{\mathfrak{T}}\zeta)\eta_2- (\tilde{\mathfrak{T}}\zeta)\eta_1\|\to 0$ whenever $\|\eta_2- \eta_1\|\to 0$. This proves that $\tilde{\mathfrak{T}}(B_R)$ is equicontinuous.

Finally, we show that function $\tilde{\mathfrak{T}}:B_R\to B_R$ satisfies the following requirement:

for any closed subset $\mathfrak{V}$ of $B_R$ and $0<k<1$. Utilizing the properties of $\mu$ as outlined in Lemma 2.3, we deduce

Thus,

Using functional relation (4.10), we find

This gives

An application of Lemma 2.2 with $c(\eta) = 0$ and $d(\eta) = \dfrac{\mathfrak{M} }{1-\mathfrak{N} }$ gives

Using the Eq (4.14), the inequality (4.13) becomes

and hence we obtain

For any $\zeta\in \mathfrak{V}$, we have $\nu(\zeta(\eta))\geq 0, \; \eta \in J$, hence for any $k\in (0, 1)$, we get $k\nu(\zeta(\eta))\geq 0, \; \eta \in J,$ which gives

Combining (4.15) and (4.16), we obtain

This proves that $\nu$ is a set contraction. Accordingly, Lemma 2.4 states that $\tilde{\mathfrak{T}}$ has a fixed point, which is the desired solution of the problem (1.1) subject to non-local initial condition (1.3).

Theorem 4.2. Assume that the hypotheses (H1) and (H2) hold. Then, the nonlinear implicit AB-FDEs (4.1)-(4.2) has a solution if $\mathfrak{K} < 1$.

Proof. Consider the set $B_{\tilde{R}}$ defined in Theorem 4.1. Note that $B_{\tilde{R}}$ is bounded, closed and convex subset of a Banach space $\mathcal{C}$ such that $0 \in B_{\tilde{R}}$. Further, we consider the same operator $\tilde{\mathfrak{T}} :B_{\tilde{R}}\to B_{\tilde{R}}$ defined in the proof of Theorem 4.1. We prove that $\tilde{\mathfrak{T}}$ satisfies the requirement of Monach fixed point theorem.

The boundedness and continuity of $\tilde{\mathfrak{T}}$ have previously been demonstrated in the proof of Theorem 4.1. Thus, all we have to do is demonstrate that the implication

holds for every subset $\mathfrak{V}$ of $B_{\tilde{R}}$. Let $\mathfrak{V}$ equicontiniuos subset of $B_{\tilde{R}}$ such that $\mathfrak{V} \subset conv(\tilde{\mathfrak{T}}(\mathfrak{V}) \cup \left\lbrace 0\right\rbrace).$ Define

It is obvious from Lemma 2.6 that $x$ is continuous. Lemma 2.5, Remark 3.2, and the properties of the measure $\mu$ allow us to get, for any $\eta \in J$,

As mentioned in Remark 3.2, based on hypothesis (H2), we can write

Therefore,

This gives

Applying Lemma 2.2 with $c(\eta) = 0$ and $d(\eta) = \dfrac{1}{1-\mathfrak{K}}$, we obtain $x(\eta) = 0$ for all $\eta \in J$. Theorem 2.5 is then used to determine that $\tilde{\mathfrak{T}}$ has a fixed point $x\in B_{\tilde{R}}$, which forms a solution of (1.1) subject to nonlocal initial condition (1.3).

5.   Examples

Example 5.1 Consider the following non-linear implicit AB-FDE:

with boundary condition

Choose the normalizing function $\mathfrak{A}(\mu) = \mu-\mu^2+1, \; \mu\in[0, 1]$. Then $B(0) = B(1) = 1$. Define, $\mathfrak{F}:J\times \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ such that

Let any $(\eta_i, \zeta_i, y_i)\in J\times \mathbb{R} \times \mathbb{R}, (i = 1, 2)$. Then,

Note that

which exists for all $\eta \in [0, 1]$. Thus, $45\eta^2\mathbb{E}_{\frac{1}{2}, 2}(-\sqrt{\eta})$ is continuous and differentiable on $[0, 1]$, by mean value theorem there exists $\xi$ lying between $\eta_1, \eta_2$ such that

Let $\mathfrak{L} = 1+\dfrac{p}{10}, \text{where}\; p = \max_{ \eta \in J} p(\eta).$ Then,

This proves that the function $\mathfrak{F}$ satisfies hypothesis (H1) with Lipschitz constants

Then,

Since the function $\mathfrak{F}$ meets every requirement of Theorem 3.1, the non-linear implicit AB-FDE (5.1) subject to boundary condition (5.2) has a solution according to Theorem 3.1. One can verify that

is the solution of the problem (5.1)-(5.2). Note that with this solution we find $\zeta(0) = 1,$ which yields

Example 5.2 Consider the non-linear implicit AB-FDE (5.1) subject to nonlocal initial condition

where $g: C(J, \mathbb{R})\to \mathbb{R}$ is the function defined by

For any $\zeta, \zeta_1 \in C(J, \mathbb{R})$, we have

This proves $\mathfrak{g}$ satisfies the hypothesis with $\mathfrak{K} = 0.5$. Next, consider

Using expansion of one parameter Mittag-Leffler function, we have

Utilizing estimation (5.5) in Eq (5.4), we obtain

Therefore,

Meeting all the conditions outlined in Theorem 4.1, it follows that the non-linear implicit AB-FDE given by Eq (5.1), accompanied by the non-local initial condition (5.3), have a solution. Further, one can verify that

is the solution of non-linear implicit AB-FDEs (5.1) with non-local initial condition (5.3).

6.   Conclusions

We successfully proved significant existence results for nonlinear implicit fractional differential equations involving the nonsingular version of the Caputo fractional derivative defined by Atangana and Baleanu. The investigation of implicit fractional differential equations encompassed both boundary and nonlocal conditions, shedding light on the complexities of such equations. The foundational tools employed in this analysis were Kuratowski's measure of non-compactness and associated fixed point theorems, namely Darbo's fixed point theorem and Mönch's fixed point theorem. To illustrate our findings, we provided concrete examples and solutions to the considered problems with both boundary and non-local initial conditions.

Authors contributions

K. D. Kucche, S. T. Sutar and K. S. Nisar: Conceptualization, Formal analysis, Investigations, Methodology, Validation, Visualization, Writing–original drat, Writing–review & editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2024/01/99519).

Conflict of interest

The authors declare no conflicts of interest.