Abstract
In this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders \(0<\vartheta \leq 1\) and \(1<\vartheta \leq 2\). We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.
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1 Introduction
Fractional calculus [1–3] has continued to attract the attention of many authors in the past three decades. Recently, new fractional derivatives (FDs) which interpolate the Riemann–Liouville, Caputo, Hilfer, Hadamard, and generalized FDs have appeared, see [4–9]. Some investigators have recognized that innovation for novel FDs with various nonsingular or singular kernels is necessary to address the need to model more realistic problems in various areas of engineering and science. Caputo and Fabrizio [10] introduced a new kind of FDs where the kernel is based on the exponential function. Losada and Nieto [11] studied some properties of this new operator. In [12, 13], the authors presented new interesting FDs where the kernel relies on Mittag-Leffler function, the so-called Atangana–Baleanu–Caputo (AB–Caputo) which is basically a generalization of the Caputo FD. Then in [14, 15], the authors deliberated the discrete versions of those new operators. For modeling in the framework of nonsingular kernels and fractal-fractional derivatives, we refer to [16–18]. There are many works pertinent to ABC problem in medical science and engineering. Hence we highlight medical, as well as engineering, applications by referring to [19–21].
On the other hand, the fixed point theory is a collection of results saying that a mapping T will have at least one fixed point (i.e., \(T(x) = x\)), under some conditions on T. Results of this kind are of paramount importance in many areas of mathematics, other sciences, and engineering. So, some recent articles which are pertinent to the fixed point theory can found in [22–29]. The existence and uniqueness of solutions for different classes of fractional differential equations (FDEs) with initial or boundary conditions have been studied by several researchers; see [30–38] and the references therein. Some recent contributions on FDEs involving ABC-FDs can be found in the following articles series: [39–49]. For instance, AB–Caputo fractional IVP is one of the studied problems by Jarad et al. [39], and has the form
The BVP of AB-Caputo FD, presented by Abdeljawad in [40], is also one of the recent problems through which the higher fractional orders are addressed:
Motivated by the above arguments, the intent of this work is to investigate two AB–Caputo-type implicit FDEs with nonlinear integral conditions described by
and
where \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\) is the AB–Caputo FD of order ϑ, while \(f:[a,T]\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) and \(g:[a,T]\times \mathbb{R} \rightarrow \mathbb{R} \) are continuous functions.
Some fixed point theorems (FPTs) are applied to establish the existence and uniqueness theorems for the problems (1.1) and (1.2). The proposed problems are more general, and the results generalize those obtained in recent studies; we also provide an extension of the development of FDEs involving this new operator. Moreover, the analysis of the results was limited to the minimum assumptions.
Many other recent works have investigated similar topics using the same concepts; one can see [50–55].
The rest of the paper is structured as follows. In Sect. 2, we give some useful preliminaries related to main consequences. Section 3 is devoted to obtaining formulas of solution to the proposed problems. Moreover, the existence and uniqueness theorems for the problems at hand are proved by means of various techniques for FPTs. Ultimately, illustrative examples are offered in Sect. 4.
2 Background materials and preliminaries
Here we recollect some requisite definitions and preliminary concepts related to our work.
Let \(\mathfrak{Z}=[a,T]\subset \mathbb{R} \), \(C(\mathfrak{Z},\mathbb{R} )\) be the space of continuous functions \(\varsigma :\mathfrak{Z}\rightarrow \mathbb{R} \) with the norm
Clearly, \(C(\mathfrak{Z},\mathbb{R} )\) is a Banach space with the norm \(\Vert \varsigma \Vert \).
Definition 2.1
Let \(\vartheta \in (0,1]\) and \(\mathfrak{p}\in H^{1}(\mathfrak{Z})\). Then the AB–Caputo and AB–Riemann–Liouville FDs of order ϑ for a function \(\mathfrak{p}\) are described by
and
respectively, where \(\mathbb{E}_{\vartheta }\) is called the Mittag-Leffler function and described by
The associated AB fractional integral is specified by
where \(\mathfrak{N}(\vartheta )>0\) is a normalization function satisfying \(\mathfrak{N}(0)=\mathfrak{N}(1)=1\).
Definition 2.2
([13])
In particular, if \(a=0\), the Laplace transform of AB–Caputo FD of \(\mathfrak{p}(\mathfrak{r})\) is specified by
Lemma 2.1
([14])
Let \(\vartheta \in (0,1]\) and \(\mathfrak{p}\in H^{1}(\mathfrak{Z})\), if AB–Caputo FD exists, then we have
and
Definition 2.3
([40])
Let \(\vartheta \in (n,n+1]\) and \(\mathfrak{p}\) be such that \(\mathfrak{p}^{n}\in H^{1}(\mathfrak{Z})\). Set \(\mathfrak{v}= \vartheta -n\) where \(\mathfrak{v}\in (0,1]\). Then the AB–Caputo and AB–Riemann–Liouville FDs of order ϑ for a function \(\mathfrak{p}\) are described by
and
respectively. The associated AB fractional integral is specified by
Remark 2.1
If \(\vartheta \in (0,1]\), we have \(\vartheta =\mathfrak{v}\). Hence
Definition 2.4
([40])
The relation between the AB–Riemann–Liouville and AB–Caputo FDs is
Lemma 2.2
([40])
For \(n-1<\vartheta \leq n\), \(n\in \mathbb{N} _{0}\), and \(\mathfrak{p}(\mathfrak{r})\) defined on \(\mathfrak{Z}\), we have:
-
(i)
\({}^{ABR}\mathbb{D}_{a^{+}}^{\vartheta }{}^{AB}\mathbb{I}_{a^{+}}^{\vartheta }\mathfrak{p}(\mathfrak{r})=\mathfrak{p}( \mathfrak{r})\);
-
(ii)
\({}^{AB}\mathbb{I}_{a^{+}}^{\vartheta }{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\mathfrak{p}(\mathfrak{r})=\mathfrak{p}( \mathfrak{r})-\sum_{k=0}^{n}\frac{\mathfrak{p}^{(k)}(a)}{k!}(\mathfrak{r}-a)^{k}\);
-
(iii)
\({}^{AB}\mathbb{I}_{a^{+}}^{\vartheta }{}^{ABR}\mathbb{D}_{a^{+}}^{\vartheta }\mathfrak{p}(\mathfrak{r})=\mathfrak{p}( \mathfrak{r})-\sum_{k=0}^{n-1}\frac{\mathfrak{p}^{(k)}(a)}{k!}(\mathfrak{r}-a)^{k}\).
Remark 2.2
With the help of (2.1), for any ϑ, it can be shown that
Hence, under the condition that \(\mathfrak{p}(a)=0\), we get the identity
Lemma 2.3
([40])
Let \(n<\vartheta \leq n+1\). Then \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\mathfrak{p}(\mathfrak{r})=0\), if \(\mathfrak{p}(\mathfrak{r})\) is constant function.
Lemma 2.4
([13])
Let \(\vartheta >0\). Then \({}^{AB}\mathbb{I}_{a^{+}}^{\vartheta }\) is bounded from \(C(\mathfrak{Z}, \mathbb{R})\) into \(C(\mathfrak{Z},\mathbb{R})\).
Lemma 2.5
Let \(n<\vartheta \leq n+1\). Then \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }(\mathfrak{r}-a)^{k}=0\), for \(k=0,1,\ldots,n\).
Proof
Let \(\mathfrak{p}(\mathfrak{r})=(\mathfrak{r}-a)^{k}\). By Definition 2.3, we have
Since \(k< n\in \mathbb{N} \), we have \(( \frac{d}{d\mathfrak{r}} ) ^{n}(\mathfrak{r}-a)^{k}=0 \). It follows from Lemma 2.3 that
□
Lemma 2.6
([39])
Let \(\vartheta \in (0,1]\) and \(\varpi \in C(\mathfrak{Z},\mathbb{R} )\) with \(\varpi (a)=0\). Then the solution of the following problem
is given by
Lemma 2.7
([40])
Let \(\vartheta \in (1,2]\) and \(\varpi \in C(\mathfrak{Z},\mathbb{R} )\) with \(\varpi (a)=0\). Then the solution of the following problem
is given by
Definition 2.5
([56])
Let \(\mathcal{J}\) be a Banach space. The operator \(\mathfrak{B}:\mathcal{J}\rightarrow \mathcal{J}\) is a contraction if
Theorem 2.1
(Banach FPT, [56])
Let \(\mathcal{J}\) be a Banach space, and \(\mathfrak{K}\) be a nonempty closed subset of \(\mathcal{J}\). If \(\mathfrak{B}:\mathfrak{K}\longrightarrow \mathfrak{K}\) is a contraction, then there exists a unique fixed point of \(\mathfrak{B}\).
Theorem 2.2
(Krasnoselskii FPT, [56])
Let \(\mathfrak{K}\) be a nonempty, closed, convex subset of a Banach space \(\mathcal{J}\). Let \(\mathfrak{B}_{1}\), \(\mathfrak{B}_{2}\) be two operators such that (i) \(\mathfrak{B}_{1}u+\mathfrak{B}_{2}v\in \mathfrak{K}\), \(\forall u,v\in \mathfrak{K}\); (ii) \(\mathfrak{B}_{1}\) is compact and continuous; (iii) \(\mathfrak{B}_{2}\) is a contraction mapping. Then, there exists \(w\in \mathfrak{K}\) such that \(\mathfrak{B}_{1}w+\mathfrak{B}_{2}w=w\).
3 Main results
This section is devoted to obtaining formulas of solutions to linear problems corresponding to (1.1) and (1.2). Moreover, we prove the existence and uniqueness theorems to suggested problems by applying Theorems 2.1 and 2.2.
3.1 Solution formulas
Theorem 3.1
Let \(0<\vartheta \leq 1\), and let \(\varpi ,g\in C(\mathfrak{Z},\mathbb{R} )\) with \(\varpi (a)=\varpi ^{\prime }(\mathfrak{a})=0\). A function \(\varsigma \in C(\mathfrak{Z},\mathbb{R} )\) is a solution of the fractional integral equation (FIE)
if and only if ς is a solution of the ABC-problem
Proof
Assume ς satisfies the first equation of (3.2). From Lemma 2.6, we have
Also,
Taking \(\mathfrak{r}\rightarrow a\) on both sides of (3.4), we have
Using the integral condition, we obtain
From (3.3) and (3.5), and from fact that \(\varpi ^{\prime }(\mathfrak{a})=0\), we get
Thus (3.1) is satisfied.
Conversely, suppose that ς satisfies equation (3.1). Applying \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\) on both sides of (3.1), then using Remark 2.2 and Lemma 2.3, we find that
Thus, we can simply infer that
□
Theorem 3.2
Let \(1<\vartheta \leq 2\), and let \(\varpi ,g\in C(\mathfrak{Z},\mathbb{R} )\) with \(\varpi (a)=0\). A function \(\varsigma \in C(\mathfrak{Z},\mathbb{R} )\) is a solution of the FIE
if and only if ς is a solution of the ABC-problem
Proof
Assume ς satisfies the first equation of (3.7). From Lemma 2.7, we have
for some \(c_{1},c_{2}\in \mathbb{R} \). Since \(\varsigma (a)=0\), we get \(c_{1}=0\). Hence
Using the integral condition \(\varsigma (T)=\int _{a}^{T}g(\mathfrak{s})\,d\mathfrak{s}\), we get
Substituting the values of \(c_{1}\) and \(c_{2}\) into (3.8), we obtain
Thus (3.6) is satisfied.
Conversely, assume that ς satisfies (3.6). Applying \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\) on both sides of (3.6), then using Lemmas 2.2, 2.3, and 2.5, we find that
Clearly, \(\varsigma (a)=0\). Thus, we can simply infer that
□
Before proceeding with the main findings, we are obligated to provide the following assumptions:
- \((A_{1})\):
-
\(f:\mathfrak{Z}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) is continuous and there exist \(L_{f}>0\) and \(0< K_{f}<1\) such that
$$ \bigl\vert f(\mathfrak{r},u,\overline{u})-f(\mathfrak{r},v, \overline{v}) \bigr\vert \leq L_{f} \vert u-v \vert +K_{f} \vert \overline{u}-\overline{v} \vert , \quad \mathfrak{r}\in \mathfrak{Z} \text{ and } u,v,\overline{u},\overline{v}\in \mathbb{R} ; $$ - \((A_{2})\):
-
\(g:\mathfrak{Z}\times \mathbb{R} \rightarrow \mathbb{R} \) is continuous and there exist constant \(L_{f}>0\) such that
$$ \bigl\vert g(\mathfrak{r},u)-g(\mathfrak{r},v) \bigr\vert \leq L_{g} \vert u-v \vert , \quad \mathfrak{r}\in \mathfrak{Z} \text{ and }u,v\in \mathbb{R} ; $$ - \((A_{3})\):
-
$$ \biggl[ L_{g}(T-a)+ \biggl( \frac{1-\vartheta }{\mathfrak{N}(\vartheta )}+ \frac{(T-a)^{\vartheta }}{\mathfrak{N}(\vartheta )\Gamma (\vartheta )} \biggr) \frac{L_{f}}{1-K_{f}} \biggr] < 1. $$
3.2 Existence and uniqueness theorems for (1.1)
As a result of Theorem 3.1, we have the following theorem:
Theorem 3.3
Let \(0<\vartheta \leq 1\), and let \(f:\mathfrak{Z}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) and \(g:\mathfrak{Z}\times \mathbb{R} \rightarrow \mathbb{R} \) be continuous with \(f(a,\varsigma (a),{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (a))=f^{\prime }(a,\varsigma (a),{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (a))=0\). If \(\varsigma \in C(\mathfrak{Z},\mathbb{R} )\) then ς satisfies (1.1) if and only if ς fulfills
Theorem 3.4
Let \((A_{1})\) and \((A_{3})\) be fulfilled. Then the ABC-problem (1.1) has a unique solution.
Proof
Set
By Theorem 3.3, we define the operator \(\mathbb{T}:\mathfrak{D}\rightarrow \mathfrak{D}\) by
This \(\mathbb{T}\) is well defined, that is, \(\mathbb{T}(\mathfrak{D})\subseteq \mathfrak{D}\). Indeed, for any \(\varsigma \in C(\mathfrak{Z},\mathbb{R} )\), \(f(\cdot ,\varsigma (\cdot ),{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (\cdot ))\) is continuous. Besides, by Lemma 2.4, \(\mathbb{T}\varsigma \in C(\mathfrak{Z},\mathbb{R} )\). Also, by Lemma 2.1 and Remark 2.1, we end up with
Since \(f(\mathfrak{r},\cdot ,\cdot )\) is continuous on \([a,T]\), one has \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta } ( \mathbb{T}\varsigma ) ( \mathfrak{r})\in C(\mathfrak{Z},\mathbb{R} )\).
Now, we need to prove that \(\mathbb{T}\) is a contraction. Let \(\varsigma ,\overline{\varsigma }\in \mathfrak{D}\) and \(\mathfrak{r}\in \mathfrak{Z}\). Then
Using \((A_{1})\) and the fact that \({}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (\mathfrak{r})=f ( \mathfrak{r},\varsigma (\mathfrak{r}),{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (\mathfrak{r}) ) \), we obtain
which implies
By \((A_{2})\) and (3.11), for \(\mathfrak{r}\in \mathfrak{Z}\),
Condition \((A_{3})\) shows that \(\mathbb{T}\) is a contraction. Hence, by Theorem 2.1, \(\mathbb{T}\) has a unique fixed point. □
Theorem 3.5
Suppose \((A_{1})\) and \((A_{3})\) are fulfilled. Then there exists at least one solution of the problem (1.1).
Proof
Choose \((\mathbb{T}\varsigma )(\mathfrak{r})=(\mathbb{T}_{1}\varsigma )(\mathfrak{r})+(\mathbb{T}_{2}\varsigma )(\mathfrak{r})\), where
and
Set \(\mu _{f}:=\max \{ \vert f(\mathfrak{r},0,0) \vert ;\mathfrak{r}\in \mathfrak{Z} \} <\infty \) and \(\mu _{g}:=\max \{ \vert g(\mathfrak{r},0) \vert ; \mathfrak{r}\in \mathfrak{Z} \} <\infty \). Let
with the radius
We will complete the proof in several steps.
Step 1. We show that \(\mathbb{T}_{1}\varsigma +\mathbb{T}_{2}\upsilon \in B_{\xi }\), for all \(\varsigma ,\upsilon \in B_{\xi }\).
By (3.12),
From \((A_{1})\) and \((A_{2})\), we have
which gives
and
By substituting (3.17) and (3.18) into (3.16), we have for \(\varsigma \in B_{\xi }\),
Also, by (3.13),
From (3.17), then for \(\upsilon \in B_{\xi }\),
Inequalities (3.19) and (3.20) give
Using \((A_{3})\) and (3.15), for \(\mathfrak{r}\in \mathfrak{Z}\) and \(\varsigma ,\upsilon \in B_{\xi }\),
Thus, \(\mathbb{T}_{1}\varsigma +\mathbb{T}_{2}\upsilon \in B_{\xi }\), for all \(\varsigma ,\upsilon \in B_{\xi }\).
Step 2. We prove that \(\mathbb{T}_{1}\) is a contraction.
From \((A_{1})\), we have
From \((A_{2})\) and (3.21), for \(\varsigma ,\varsigma ^{\ast }\in B_{\xi }\),
Since \((A_{3})\) holds, \(( L_{g}(T-a)+\frac{1-\vartheta }{\mathfrak{N}(\vartheta )}\frac{L_{f}}{1-K_{f}} ) <1\). Hence, \(\mathbb{T}_{1}\) is a contraction.
Step 3. \(\mathbb{T}_{2}\) is compact and continuous.
The map \(\mathbb{T}_{2}:B_{\xi }\rightarrow B_{\xi }\) is continuous due to the continuity of f. Next, \(\mathbb{T}_{2}\) is uniformly bounded on \(B_{\xi }\) by (3.20), because for any \(\varsigma \in B_{\xi }\) and \(\mathfrak{r}\in \mathfrak{Z}\), we have
This leads to a conclusion that \(\mathbb{T}_{2}\) is uniformly bounded on \(B_{\xi }\).
Now, we show that \(\mathbb{T}_{2}(B_{\xi })\) is equicontinuous. In order to establish that, let \(\varsigma \in B_{\xi }\) and \(a\leq \mathfrak{r}_{1}<\mathfrak{r}_{2}\leq T\). Then
Using (3.17), for \(\varsigma \in B_{\xi }\),
Observe that \(\vert (\mathbb{T}_{2}\varsigma )(\mathfrak{r}_{2})-(\mathbb{T}_{2}\varsigma )(\mathfrak{r}_{1}) \vert \rightarrow 0\) as \(t_{2}\rightarrow t_{1}\). In light of the former steps, together with Arzela–Ascoli theorem, we derive that \((\mathbb{T}_{2}B_{\xi })\) is relatively compact, and hence \(\mathbb{T}_{2}\) is completely continuous. So, Theorem 2.2 shows that (1.1) has at least one solution. □
3.3 Existence and uniqueness theorems for (1.2)
As a result of Theorem 3.2, we have the following theorem:
Theorem 3.6
Let \(1<\vartheta \leq 2\), and let \(f:\mathfrak{Z}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) and \(g:\mathfrak{Z}\times \mathbb{R} \rightarrow \mathbb{R} \) be continuous with \(f(a,\varsigma (a),{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (a))=0\). If \(\varsigma \in C(\mathfrak{Z},\mathbb{R} )\), then ς satisfies (1.2) if and only if ς fulfills
Theorem 3.7
Suppose \((A_{1})\) and \((A_{2})\) are satisfied. If
then the problem (1.2) has a unique solution.
Proof
In view of Theorem 3.6, we consider \(\mathbb{T}^{\ast }:\mathfrak{D}\rightarrow \mathfrak{D}\) defined by
From the continuity of g and f, \(\mathbb{T}^{\ast }\) is well defined, that is, \(\mathbb{T}^{\ast }(\mathfrak{D})\subseteq \mathfrak{D}\).
Now, let \(\varsigma ,\overline{\varsigma }\in \mathfrak{D}\) and \(\mathfrak{r}\in \mathfrak{Z}\). Then
Using \((A_{1}) \) and same arguments used to get (3.11), we obtain
By \((A_{2})\) and (3.23), for \(\mathfrak{r}\in \mathfrak{Z}\),
Condition (3.22) shows that \(\mathbb{T}^{\ast }\) is a contraction. Hence, by Theorem 2.1, \(\mathbb{T}^{\ast }\) has a unique fixed point. □
Theorem 3.8
Suppose that \((A_{1})\) and \((A_{2})\) are satisfied. If
then there exists at least one solution of the problem (1.2).
Proof
Choose \((\mathbb{T}^{\ast }\varsigma )(\mathfrak{r})=(\mathbb{T}_{1}^{\ast } \varsigma )(\mathfrak{r})+(\mathbb{T}_{2}^{\ast }\varsigma )( \mathfrak{r})\) where
and
Let \(B_{\xi }\) be defined by (3.14) with the radius
where \(\mu _{g}\) is as in Theorem 3.5. The proof will be complete in several steps:
Claim 1. \(\mathbb{T}_{1}^{\ast }\varsigma +\mathbb{T}_{2}^{\ast }\upsilon \in B_{ \xi }\), for all \(\varsigma ,\upsilon \in B_{\xi }\).
By (3.25),
From \((A_{1})\), \((A_{2})\), and for \(\varsigma \in B_{\xi }\), we get \(\vert f ( \mathfrak{r},\varsigma (\mathfrak{r}),{}^{ABC} \mathbb{D}_{a^{+}}^{\vartheta }\varsigma (\mathfrak{r}) ) \vert \leq \frac{L_{f}\xi +\mu _{f}}{1-K_{f}}\) (where \(\mu _{f}\) is as in Theorem 3.5) and \(\vert g(\mathfrak{s},\varsigma (\mathfrak{s})) \vert \leq ( L_{g}\xi +\mu _{g} ) \). Hence,
Also, by (3.26),
For \(\upsilon \in B_{\xi }\),
From (3.28), (3.29), and for \(\mathfrak{r}\in \mathfrak{Z}\), we get
Here we used fact that \((T-a)<(T-a)^{\vartheta }\) for all \(1<\vartheta \leq 2\). By (3.24), we conclude that \(L_{g}(T-a)<1\), it follows from (3.27) that
Thus, \(\mathbb{T}_{1}^{\ast }\varsigma +\mathbb{T}_{2}^{\ast }\upsilon \in B_{ \xi }\), for all \(\varsigma ,\upsilon \in B_{\xi }\).
Claim 2. \(\mathbb{T}_{1}^{\ast }\) is a contraction.
From \((A_{1})\) and \((A_{2})\). Then for each \(\varsigma ,\varsigma ^{\ast }\in B_{\xi }\) and \(\mathfrak{r}\in \mathfrak{Z}\),
Condition (3.24) shows that \(\mathbb{T}_{1}^{\ast }\) is a contraction.
Claim 3. \(\mathbb{T}_{2}^{\ast }\) is compact and continuous.
The map \(\mathbb{T}_{2}^{\ast }:B_{\xi }\rightarrow B_{\xi }\) is continuous due to the continuity of f. Next, let \(\varsigma \in B_{\xi }\) and \(\mathfrak{r}\in \mathfrak{Z}\). Then by using (3.29), we have
This leads to the conclusion that \(\mathbb{T}_{2}^{\ast }\) is uniformly bounded on \(B_{\xi }\).
Now, we show that \(\mathbb{T}_{2}^{\ast }(B_{\xi })\) is equicontinuous. Let \(\varsigma \in B_{\xi }\) and \(a\leq \mathfrak{r}_{1}<\mathfrak{r}_{2}\leq T\). Then
Since \(\vert f ( \mathfrak{s},\varsigma (\mathfrak{s}),{}^{ABC}\mathbb{D}_{a^{+}}^{\vartheta }\varsigma (\mathfrak{s}) ) \vert \leq \frac{L_{f}\xi +\mu _{f}}{1-K_{f}}\), for \(\varsigma \in B_{\xi }\), we have
Observe that \(\vert (\mathbb{T}_{2}^{\ast }\varsigma )(\mathfrak{r}_{2})-(\mathbb{T}_{2}^{\ast }\varsigma )(\mathfrak{r}_{1}) \vert \rightarrow 0\) as \(t_{2}\rightarrow t_{1}\). In view of the preceding claims, together with Arzela–Ascoli theorem, we infer that \((\mathbb{T}_{2}^{\ast }B_{\xi })\) is relatively compact. Hence, Claim 3 holds. So, Theorem 2.2 shows that (1.2) has at least one solution. □
4 Examples
Example 4.1
In this example, we justify the validity of Theorem 3.4. For \(\vartheta \in (0,1]\), we consider the following ABC fractional problem:
Set \(f(\mathfrak{r},\varsigma (\mathfrak{r}),\overline{\varsigma }( \mathfrak{r}))=\frac{\mathfrak{r}^{2}}{9e^{\mathfrak{r}-1}} ( \frac{\varsigma (\mathfrak{r})}{1+\varsigma (\mathfrak{r})}+ \frac{\overline{\varsigma }(\mathfrak{r})}{1+\overline{\varsigma }(\mathfrak{r})} ) \) and \(g(\mathfrak{r},\varsigma (\mathfrak{r}))= \frac{\varsigma (\mathfrak{r})}{10+\varsigma (\mathfrak{r})}\), for \(\mathfrak{r}\in {}[ 0,1]\), \(\varsigma ,\overline{\varsigma }\in \mathbb{R} \).
Clearly, \(f(0,\varsigma (0),\overline{\varsigma }(0))=f^{\prime }(0,\varsigma (0), \overline{\varsigma }(0))=0\). Let \(\mathfrak{r}\in {}[ 0,1]\) and \(\varsigma ,\overline{\varsigma },\upsilon ,\overline{\upsilon }\in \mathbb{R} \). Then
and
Therefore, the hypotheses \((A_{1})\) and \((A_{2})\) hold with \(L_{f}=K_{f}=\frac{1}{9}\) and \(L_{g}=\frac{1}{10}\). We shall examine that the condition \((A_{3})\) holds too, with \(\vartheta =\frac{1}{2}\) and \(\mathfrak{N}(\vartheta )=1\). Indeed,
Thus by Theorem 3.4, the problem (4.1) has a unique solution on \([0,1]\).
Example 4.2
The following example validates Theorem 3.7. For \(\vartheta \in (1,2]\), we consider the following ABC fractional problem:
Set \(f(\mathfrak{r},\varsigma (\mathfrak{r}),\overline{\varsigma }( \mathfrak{r}))=\frac{\mathfrak{r}-1}{9+e^{\mathfrak{r}-1}} ( \frac{\varsigma (\mathfrak{r})}{1+\varsigma (\mathfrak{r})}+ \frac{\overline{\varsigma }(\mathfrak{r})}{1+\overline{\varsigma }(\mathfrak{r})} ) \) and \(g(\mathfrak{r},\varsigma (\mathfrak{r}))= \frac{\varsigma (\mathfrak{r})}{100+\varsigma (\mathfrak{r})}\), for \(\mathfrak{r}\in {}[ 1,2]\), \(\varsigma ,\overline{\varsigma }\in \mathbb{R} \).
Clearly, \(f(1,\varsigma (1),\overline{\varsigma }(1))=0\). Let \(\mathfrak{r}\in {}[ 1,2]\) and \(\varsigma ,\overline{\varsigma },\upsilon ,\overline{\upsilon }\in \mathbb{R} \). Then
and
Therefore, the hypotheses \((A_{1})\) and \((A_{2})\) hold with \(L_{f}=K_{f}=\frac{1}{9}\) and \(L_{g}=\frac{1}{100}\). Also, for \(1<\vartheta \leq 2\), \(a=1\), \(T=2\), and \(\mathfrak{N}(\vartheta )=1\), the condition \((A_{3})\) holds, that is, \(L_{g}(T-a)=\frac{1}{100}<1\). Thus by Theorem 3.7, the problem (4.2) has a unique solution on \([1,2]\).
Remark 4.1
In Theorems 3.3, 3.4, and 3.5, if \(f^{\prime } ( a,\varsigma (a),{}^{ABC}\mathbb{D}_{a^{+}}^{ \vartheta }\varsigma (a) ) \neq 0\), then the formula of solution of the problem (1.1) becomes
for \(\mathfrak{r}\in \mathfrak{Z}\). Accordingly, the analysis of the results remains valid with the addition of the Lipschitz-type condition on \(f^{\prime }\), that is,
- \((A_{4})\):
-
\(f^{\prime }:\mathfrak{Z}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) is continuous and there exist \(L^{\ast }>0\) and \(0< K^{\ast }<1\) such that
$$ \bigl\vert f^{\prime }(\mathfrak{r},u,\overline{u})-f^{\prime }( \mathfrak{r},v,\overline{v}) \bigr\vert \leq L^{\ast } \vert u-v \vert +K^{\ast } \vert \overline{u}-\overline{v} \vert , \quad \mathfrak{r}\in \mathfrak{Z}\text{ and }u,v,\overline{u}, \overline{v}\in \mathbb{R} . $$
5 Conclusions
The theory of fractional operators containing nonsingular kernels is novel and of considerable recent interest, thus there is a need to study the qualitative properties of differential equations involving such operators. In this work, we have considered two classes of boundary value problems for fractional implicit differential equations with nonlinear integral conditions in the framework of Atangana–Baleanu–Caputo derivatives. Krasnoselskii and Banach fixed point theorems were applied to establish the existence and uniqueness theorems for the problems (1.1) and (1.2). For problem (1.1), we realized that one must always have the necessary conditions \(f(a,\varsigma (a),{}^{ABC}\mathbb{D} _{a^{+}}^{\vartheta }\varsigma (a))=0\) and \(f^{\prime }(a,\varsigma (a),{}^{ABC}\mathbb{D} _{a^{+}}^{\vartheta } \varsigma (a))=0\) to guarantee a unique solution, whereas for problem (1.2) we needed \(f(a,\varsigma (a),{}^{ABC}\mathbb{D} _{a^{+}}^{\vartheta }\varsigma (a))=0\) to confirm the initial data for the solution. To avoid the condition \(f^{\prime }(a,\varsigma (a),{}^{ABC}\mathbb{D} _{a^{+}}^{\vartheta } \varsigma (a))=0\) in Theorem 3.3, one can use condition \((A_{4})\) mentioned in Remark 4.1 to obtain the same results. The proposed problems are more general, also the results obtained generalize the recent studies and offer an extension of the development of FDEs that involve this new operator. Moreover, the analysis of the results was limited to the minimum assumptions. The problems scrutinized include some special cases, in other words, they could be reduced to the corresponding problems that contain Caputo–Fabrizio operator. We are certain that the communicated results here are rather interesting, and will have a positive effect on the development of more applications in engineering and sciences.
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Acknowledgements
The authors are very thankful to the reviewers for their useful suggestions. Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.
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Abdo, M.S., Abdeljawad, T., Ali, S.M. et al. On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions. Adv Differ Equ 2021, 37 (2021). https://doi.org/10.1186/s13662-020-03196-6
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DOI: https://doi.org/10.1186/s13662-020-03196-6