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New attitude on sequential ΨCaputo differential equations via concept of measures of noncompactness
Journal of Inequalities and Applications volume 2024, Article number: 116 (2024)
Abstract
In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integrodifferential equations comprising of the ΨCaputo fractional derivative and the ΨRiemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko’s attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples.
1 Introduction
There exists a profound and extensive historical background pertaining to the subject matter of fractional calculus, which traces its origins back to the emergence of classical calculus. In previous times, certain scholars dedicated their efforts to the exploration of this particular field; nevertheless, contemporary researchers have displayed a heightened level of enthusiasm towards the study of the novel dynamic equations. Caputo fractional derivatives stand as prominent concepts commonly employed within various classes of fractional derivatives. The Riemann–Liouville derivative is accompanied by a certain degree of mathematical abstraction, whereas the Caputo fractional derivatives are predominantly favored by engineers [4, 15, 20, 25, 26, 36].
The domain of fractional differential equations has seemingly experienced significant growth, thereby serving as a testament to the prominent position and status that fractional calculus has attained within the realms of science and engineering. It is noteworthy to mention that fractional calculus finds wideranging applications in naturally occurring fields such as porous media, chemical physics, viscoelasticity, electrical networks and fluid dynamics. Consequently, scientists underscore the significance and relevance of this particular field [15–17].
The boundary value problems pertaining to fractional differential equations have attracted significant interest from numerous scholars as of late, emerging as an important field of research due to their wide range of uses in the fields of science. These applications encompass control theory, mechanics, biology and wave propagation, among others [1, 3–5, 8, 11–13, 22–24, 27–35].
Many effective theoretical studies have been published by several researchers focusing on the result of existence, uniqueness and the stability for differential equations involving a fractional derivative with various conditions, see [9, 10].
In [30], Tariboon and colleagues explored the existence and uniqueness of solutions for the subsequent FDE:
subject to
as nonlocal fractional integral boundary conditions, where \(S_{1},S_{2}\in \mathbb{R}\), \(\digamma : [ A,\mathfrak{b}]\times \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function, for \(i = 1, 2, . . . ,\mathfrak{m}\), \(j =1,2, . . . , \mathfrak{n}\) considering \(\eta _{i},\vartheta _{j}\in \mathbb{R}\), and using Banach’s contractive principle, Krasnoselskii’s fixedpoint theorem and Leray–Schauder’s nonlinear alternative [19].
In 2013, Yan et al. [32] conducted a study on the existence and uniqueness of solutions for the ensuing boundary value problems of fractional differential equations using multiple customary fixed point theorems:
with the nonlocal boundary condition:
where \(\digamma : C^{2}[0,\mathfrak{b}]\rightarrow \mathbb{R}\) is a \(C^{2}\) continuous functional.
Just recently, the existence and uniqueness of a nonlinear integral differential equation via a boundary condition was examined by Li et al. [21]. This study employed several fixed point theorems:
in which \(0 \leq \mathfrak{p} < P < +\infty \) and μ is a constant.
Let \(f : C[ A,\mathfrak{b}]\rightarrow \mathbb{R}\), \(\digamma : [ A,\mathfrak{b}]\times \mathbb{R}\rightarrow \mathbb{R}\) and \(A(\mathcal{Q}) \in C[ A,\mathfrak{b}]\).
We will examine the existence and uniqueness of solutions for the subsequent nonlinear Ψ integral differential equation with nonlocal boundary condition and varying coefficients when \(l<\varphi \leq l+1\) and \(\beta \geq 0\)
where \(A(\mathcal{Q})\) is a variable coefficient, η is constant, \({}^{c}\mathfrak{D}^{\varphi ;\Psi}_{ A^{+}}\) and \(\mathfrak{I}^{\kappa ;\Psi}_{ A^{+}}\) are considered the ΨRiemann–Liouville fractional integral operators and ΨCaputo fractional, in the state order. Babenko’s attitude [11] and topological degree theory for condensing map are powerful tools for solving differential and integrodifferential equations with initial conditions by treating bounded integral operators as normal variables. In particular, for \(\Psi (\mathcal{Q})=\mathcal{Q}\), problem (1.1) arises; as a result,
and for \(\Psi (\mathcal{Q})=\mathcal{Q}\), \(l=1\), problem (1.1) turns out to be
Very limited information is available in contemporary literature regarding the boundary value problem of Ψfractional integrodifferential equations with integral boundary conditions and coefficients that vary. The paper is structured as follows. Section 2 encompasses several fundamental definitions, topological degree theory, an introductory overview of fractional calculus, and a collection of lemmas that are further elaborated upon in this article. In Sect. 3, the outcomes associated with the existence and uniqueness of solutions for ΨCaputo (1.1) are presented, employing the theory of topological degree coincidences for the contraction principle and the curtailing maps. Two specific examples of the study results are provided in Sect. 4 to illustrate its functionality and demonstrate its efficience.
2 Preliminaries
In this particular section, we shall revisit a few of the fundamental outcomes and concepts that will find applications within the context of this manuscript.
Definition 2.1
The Mittag–Leffler function with two parameters is stated as [26]
Babenko’s methodology [11] is an efficient instrument for resolving differential and integrodifferential equations featuring initial conditions through the treatment of bounded integral operators as ordinary variables.
We hereby present the results that are furnished in the subsequent from [2, 14].
Definition 2.2
Authorizing θ represent the collection of bounded subsets of X which X denote a Banach space. The measure of noncompactness known as the Kuratowski measure, is a mapping \(\ell : \theta \rightarrow [0,\infty )\), which is defined as follows:
Definition 2.3
Supposing \(G: Z \rightarrow X\) and \(Z\subset X\) is a map that are both continuous and bounded, it is possible to state that G is ℓLipschitz if \(\exists P\geq 0\) such that
In the event that \(P< 1\), we classify G as a strict ℓcontraction. It is possible to state that G is ℓcondensing if
for every bounded and nonprecompact subset \(\theta \subset Z\).
Definition 2.4
Assuming \(Z\subset X\) and letting \(G: Z \rightarrow X\), it is worth noting that G is said to be Lipschitz if \(\exists P \geq 0\) such that
and G is a strict contraction as \(P>1\).
We direct the interested reader for the subsequent results to reference [18].
Proposition 2.5
If \(F, G: A\rightarrow X\) represent mappings that are \(\ell Lipschitz \) and possess constants \(P_{1}\) and \(P_{2}\), for specified, it follows that the mapping \(F + G: A\rightarrow X\) is also \(\ell Lipschitz\) containing \(P_{1}+ P_{2}\).
Proposition 2.6
In case \(G : A\rightarrow X\) represent, mappings that are compact, it follows that the mapping G is \(\ell Lipschitz\) featuring \(P=0\).
Proposition 2.7
In case \(G: A\rightarrow X\) represent mappings that are \(Lipschitz \) and possess constant P, it follows that the mapping \(G: A\rightarrow X\) is also \(\ell Lipschitz\) with a constant of P.
Similarly, Isaia [18] derived the following findings using topological degree theory.
Theorem 2.8
Assuming the mapping \(\mathcal{K}: X\rightarrow X\) to be ℓcondensing and considering the set
If ζ is a bounded set in X, whenever there exists a positive constant r such that ζ is contained in the ball \(B_{r} (0)\) centered at the origin. In this case, for all \(k \in [0,1]\), there exist
Afterwards, we can suggest that the mapping \(\mathcal{K}\) has at least one fixed point and the set of fixed points of \(\mathcal{K}\) is contained within the ball \(B_{r} (0)\).
Later, we will provide a detailed explanation of the properties and conclusions related to the field of fractional calculus. This explanation will begin by introducing a definition of ΨRiemann–Liouville fractional integrals and derivatives. Furthermore, we will delve deeper into the subject matter.
Definition 2.9
[6] Regarding \(\varphi > 0\), the leftsided ΨRiemann–Liouville fractional integral of variable order \(l(\mathcal{Q})\) for a function \(Z\in L( H,\mathbb{R})\) due to a different function \(\Psi : H \rightarrow \mathbb{R}\), which is an increasing differentiable function in such a way that \(\Psi '(\mathcal{Q})\neq 0\), may be elucidated as follows:
for all \(\mathcal{Q}\in H\) in such a way that \(l:[ A,\mathfrak{b}]\rightarrow (0,1]\) is a continuous function.
It is imperative to acknowledge that the decline of (2.1) can be observed in relation to the Riemann–Liouville and Hadamard fractional integrals, provided that \(\Psi (\mathcal{Q})=\mathcal{Q}\) and \(\Psi (\mathcal{Q})=\ln \mathcal{Q}\), in the sequence offered.
Definition 2.10
[6] Considering n as a natural number and Z and Ψ as two functions belonging to \(C^{n}(H;\mathbb{R})\), where Ψ is increasing and \(\Psi '(\mathcal{Q})\) is not equal to zero for all \(\mathcal{Q}\) in H, we can elaborate the leftsided ΨCaputo of Z of order φ
From the equation (2.2), it is reduced to the CFD operator as long as \(\Psi (\mathcal{Q})=\mathcal{Q}\). Moreover, if \(\Psi (\mathcal{Q}) = ln \mathcal{Q}\), therefore it gives rise to the CaputoHadamard fractional derivative.
Lemma 2.11
[7] In the event that both φ and ϕ are greater than zero, and Z belongs to the space of integrable functions \(L^{1}( H,\mathbb{R})\), with \(\mathcal{Q}\) being an element of H, it follows that
Especially, if Z belongs to \(C( H,\mathbb{R})\), then \(\mathfrak{I}^{\varphi ;\Psi}_{ A^{+}}\mathfrak{I}^{\phi ;\Psi}_{ A^{+}} Z(\mathcal{Q})= \mathfrak{I}^{\varphi +\phi ;\Psi}_{ A^{+}} Z( \mathcal{Q})\), \(\mathcal{Q}\in H\).
Lemma 2.12
[7] Assuming that φ is greater than zero, if Z belongs to \(C( H,\mathbb{R})\), then for \(\mathcal{Q}\) belongs to H
For \(n1<\varphi <n\), if \(Z\in C^{n}\left ( H,\mathbb{R}\right ) \), then
Notation 1
For the remaining stages, in order to streamline and enhance the ease of computation, it is necessary that the subsequent notations be posited
3 Main results
Here, we obtain our primary findings concerning the existence and uniqueness for the given problem (1.1).
Lemma 3.1
Given a postulated function \(\mathbf{ H}\in C( H,\mathbb{R})\), the solution to the fractional BVP
is determined by
Proof
Utilizing Lemma 2.12, the comprehensive solution of the Ψfractional differential equation
may be expressed as follows
or
where \(\mathfrak{c}_{1}\in \mathbb{R}\). Obviously,
Similarly,
Thus,
Now, from conditions in (3.1), we get
Thus,
by noting that \(f(\mathcal{Q})\in \mathbb{R}\). So, from the relation (3.3),
Considering the factor \(\left (1+\mathfrak{I}^{\varphi ;\Psi}_{ A^{+}} A(\mathcal{Q}) \mathfrak{I}^{\kappa ;\Psi}_{ A^{+}}\right )\) as a variable, we can infer that through Babenko’s attitude (3.2), the inverse form of the lemma can be derived through a straightforward computation. Consequently, it is possible to consider the proof as completed. □
In the subsequent discussion, we shall expound upon the primary findings pertaining to the presence of resolutions for the aforementioned issue (1.1). In this section, it is appropriate to posit the following hypotheses: (H1) a constant \(t > 0\) is posited such that for each \(\mathcal{Q}\in H \) and for each \(Z, Z^{*}\in \mathbb{R} \):
(H2) There exists \(t_{1}\) such that for each \(\mathcal{Q}\in H \) and for each \(Z, Z^{*}\in \mathbb{R} \):
(H3) The functions Ϝ fulfills the next rising concessions for \(\mu ,\nu >0\):
In view of Lemma 3.1, we assume two operators \(G_{1}; G_{2}: C( H,\mathbb{R})\rightarrow C( H,\mathbb{R}) \) as follows:
Then, it is possible to rephrase the integral equation mentioned in reference (3.2) as stated in Lemma 3.1, \(\mathcal{K} Z(\mathcal{Q})= G_{1} Z(\mathcal{Q})+ G_{2} Z( \mathcal{Q})\) for \(\mathcal{Q}\in H\).
The continuous Ϝ implies that the operator \(\mathcal{K}\) is wellfounded and fixed points of the operator equation can be considered as solutions of the integral equations (3.2) in Lemma 3.1.
Lemma 3.2
\(G_{1}\) is a continuous function that satisfies the growth condition mentioned below:
Proof
To establish the continuity of \(G_{1}\), let us consider the scenario where \(Z_{n}, Z\in C( H,\mathbb{R})\) and the \(\lim _{n\rightarrow +\infty}\ Z_{n} Z\\rightarrow 0\) holds. It is evident that the collection \(\{ Z_{n}\}\) can be categorized as a bounded subset of \(C( H,\mathbb{R})\). Consequently, there exists a constant \(r > 0\) such that the norm of \(Z_{n}\) is bounded by r for all \(n\geq 1\). Upon evaluating the limit, it becomes evident that \(\ Z\\leq r\).
Subsequently, it is not difficult to observe that as n approaches infinity, the function \(\digamma \left (F, Z_{n}(F)\right )\) converges to \(\digamma \left (F, Z(F)\right )\), given the continuity of the function Ϝ.
Taking into consideration another viewpoint, when we consider (H3), we will encounter the subsequent inequality:
It is observable that the function below
is integrated over \([\mathcal{Q}_{i1},\mathcal{Q}]\) using Lebesgue integration. As \(n\rightarrow +\infty \), it can be inferred from the present argument along with the Lebesgue dominated convergence theorem.
As n approaches infinity, the expression \(\ G_{1} Z_{n} G_{1} Z\\rightarrow 0\).
Which also shows that the operator \(G_{1}\) possesses a continuous attribute. For the growth condition, utilizing the assumption (H2), the result will be
Therefore,
This serves as proof of the fulfillment of the lemma (3.3). □
Lemma 3.3
\(G_{2}\) is Lipschitz via constant \({k}_{2}= T E_{\varphi +\kappa ,1}\left (M\left (\Psi (\mathfrak{b}) \Psi ( A)\right )^{\varphi +\kappa}\right )\). Plus, \(G_{2}\) fulfills the growth condition stated below
Proof
In order to demonstrate that the operator \(G_{2}\) is Lipschitz via constant \(l_{\digamma}= t\eta _{2}\), and authorizing \(Z, Z^{*}\in C( H,\mathbb{R})\), and \(\mathfrak{M}= \sup _{ A\leq \mathcal{Q}\leq \mathfrak{b}} \frac{\mathfrak{k}S_{2}}{S_{1}}e^{\mathfrak{k}\Psi (\mathcal{Q})}\), we will encounter for all \(\mathcal{Q}\in H\):
In relation to the supremum of \(\mathcal{Q}\), the next inequality will be accomplished.
Hence, the operator \(G_{2}\), which maps from \(C( H,\mathbb{R})\) to \(C( H,\mathbb{R})\), is a Lipschitzian operator on \(C( H,\mathbb{R})\) with a Lpischitz constant \({k}_{2} = T E_{\varphi +\kappa ,1}\left (M\left (\Psi (\mathfrak{b}) \Psi ( A)\right )^{\varphi +\kappa}\right )\).
According to Proposition 2.7, G is ℓLipschitz with constant \({k}_{2}\). Furthermore, considering the growth condition, we obtain
Hence it follows that
This serves as proof of the fulfillment of the lemma 3.3. □
Lemma 3.4
The operator \(G_{1}\) is compact, regarding \(G_{1}:C( H,\mathbb{R})\rightarrow C( H,\mathbb{R})\). As a consequence, \(G_{1}\) is ℓLipschitz through zero consistent.
Proof
Consider a bounded category \(\Omega \subset B_{r} = \{ Z \in C( H,\mathbb{R}) :  Z \leq r \}\). We must show that \(G_{1}( Z)\) is relatively compact in \(C( H,\mathbb{R})\). For any \(Z \in \Omega \subset B_{r}\), we will achieve this by using the estimations referred to in (3.4)
and \(G_{1}( Z)\) remains uniformly bounded. Moreover, for any \(Z\in C( H,\mathbb{R})\) and \(\mathcal{Q}\in H\), to demonstrate the equicontinuity of \(G_{1}\), consider \(\mathcal{Q}_{1}, \mathcal{Q}_{2}\in H\) with \(\mathcal{Q}_{1}<\mathcal{Q}_{2}\), and let \(Z\in \Omega \). Subsequently, we will proceed
Based on our latest estimate, we can infer that as \(\mathcal{Q}_{2} \rightarrow \mathcal{Q}_{1}\) the expression \( G_{1} Z(\mathcal{Q}_{2}) G_{1} Z(\mathcal{Q}_{1})\) goes to 0, which means that \(G_{1}\) is equicontinuous. Thus, using the AscoliArzela theorem, we conclude that the operator \(G_{1}\) is compact. Also, from Proposition 2.6, it follows that \(G_{1}\) is ℓLipschitz via zero constant. □
Theorem 3.5
Presuming that conditions (H1)–(H3) are satisfied, it follows that the BVP (1.1) possesses at least one solution denoted by Z and belonging to the set of continuous functions from H to \(\mathbb{R}\), provided that the constant \({k}_{2}<1\). Moreover, the set of solutions is encompassed within the space \(C\left ( H,\mathbb{R}\right )\).
Proof
Assume that the operators \(G_{1}\), \(G_{2}\), \(\mathcal{K}\), and \(\mathcal{K}\) have been introduced as described in the preceding section. These operators possess a continuous nature and are encompassed within their respective spaces. Plus

operator \(G_{2}\) is ℓLipschitz via constant \({k}_{2}\), through Lemma 3.3,

operator \(G_{1}\) is ℓLipschitz with constant 0 through Lemma 3.4. Thus, \(\mathcal{K}\) is ℓLipschitz with constant \({k}_{2}\), through Lemma 2.5.
Furthermore, the operator \(\mathcal{K}\) can be characterized as a strict contraction with respect to the constant \({k}_{2}\). Given that \({k}_{2}\) is less than 1, it can be deduced that \(\mathcal{K}\) is ℓcondensing. The subsequent category is then considered
The boundary of category ζ can be demonstrated. For \(Z\in \zeta \), we have
which implies that
In this manner, the set ζ is bounded, and the operator \(\mathcal{K}\) possesses at least one fixed point that corresponds to the solution of the BVP (1.1). □
Theorem 3.6
In the context of the assumption labeled as (H1)–(H2), the BVP cited as (1.1) has a unique solution given that the prescribed condition is satisfied:
Proof
Assuming that both Z and \(Z^{*}\) belong to \(C( H,\mathbb{R})\), and that \(\mathcal{Q}\) is an element of H, we will encounter
In light of the aforementioned concession \(P<1\), it can be comprehended that the mapping denoted by \(\mathcal{K}\) exhibits the property of contraction. Consequently, by virtue of the Banach fixedpoint theorem, \(\mathcal{K}\) possesses a distinct fixed point, which can be regarded as the unique solution to problem (1.1). □
4 Illustrative examples
In this section, certain issues are approached as a means of illustrating our findings:
Test example 4.1
Examine the problem below:
It is important to acknowledge that the issue at hand represents a particular instance of problem (1.1), encompassing the subsequent information
the continuous function \(\digamma :\mathbf{ J}\times \mathbb{R}\rightarrow \mathbb{R}\) is shown below
Point 1: Hypothesis (H1) is confirmed:
For every \(\mathcal{Q}\in \mathbf{\mathfrak{I}}\) and \(Z, W\in \mathbb{R}\), we obtain the following
Point 2: Hypothesis (H2) holds:
For each \(Z, W\in \mathbb{R}\), we gain
Point 3: Hypothesis (H3) is confirmed:
For each \(\mathcal{Q}\in \mathbf{\mathfrak{I}}\) and \(Z\in \mathbb{R}\), we obtain the following equation
that shows here \(\mu =\nu =0.01\), \(\mathfrak{p}=1\). Due to the theorem 3.5
is the solution set; then, we gain
where \(\mu ^{*}=0.050525\), \(\nu ^{*}=0.007217\), \({k}_{1}=0.001129\), \(\ E_{4.5,1}(0.04)=1.00076\). And so, theorem 3.5 confirms that the system is resolved. Furthermore
Theorem 3.6 proves that system (4.1) possesses a unique solution.
Test example 4.2
Allow us to examine the next problem:
It is important to acknowledge that the issue at hand represents a particular instance of problem (1.1), encompassing the subsequent information
the operators denoted by \({}^{cH}\mathfrak{D}{{1^{+}}}^{4.7}\) and \(\ ^{H}\mathfrak{I}{{1^{+}}}^{1.2}\) represent the Caputo–Hadamard derivative and Hadamard integral, respectively, and continuous function \(\digamma :\mathbf{ J}\times \mathbb{R}\rightarrow \mathbb{R}\) is stated as
Point 1: Hypothesis (H1) is confirmed: \(\forall \mathcal{Q}\in J\) and \(Z, W\in \mathbb{R}\), we gain
Point 2: Hypothesis (H2) hold: For each \(Z, Z^{*}\in \mathbb{R}\), we gain
Point 3: Condition \(P<1\) is confirmed:
Theorem 3.6 guarantees that system (4.2) possesses a unique solution.
5 Conclusions
We have investigated the existence and uniqueness of solutions to the nonlinear Ψ fractional integral differential equation (1.1), where nonlocal boundary conditions and variable coefficients are present. To achieve this, we used the Mittag–Leffler function, Babenko’s approach, and Banach’s contraction principle and topological degree theory. The proof of existence results is based on the fixed point theorem due to Isaia [18], who pretty recently obtained such a fixedpoint theorem via coincidence degree theory for condensing maps and that of uniqueness of the solution is proven by applying the Banach contraction principle. Of course, two representative examples have been given to illustrate the efficiency and performance of the results of the present study.
The above technique, obviously, opens up the doors for further study involving various other types of boundary conditions or with different fractional derivatives. The scope of this study encompasses the investigation of the BVP related to the nonlinear fractional partial IntegralDifferential Equations with varying coefficients, along with the scrutiny of nonlinear integrodifferential equations that incorporate the Hilfer fractional derivatives.
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References
Agarwal, R.P., Ahmad, B., Nieto, J.J.: Fractional differential equations with nonlocal (parametric type) antiperiodic boundary conditions. Filomat 31(5), 1207–1214 (2017)
Agarwal, R.P., O’Regan, D., Cho, Y.J., Chen, Y.Q.: Toplogical Degree Theory and Its Applications. Tylor & Francis, London (2006)
Agarwal, R.P., O’Regan, D., Stanek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371(1), 57–68 (2010)
Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: HadamardType Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham (2017)
Ahmad, B., Sivasundaram, S.: On fourpoint nonlocal boundary value problems of nonlinear integrodifferential equations of fractional order. Appl. Math. Comput. 217(2), 480–487 (2010)
Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)
Almeida, R., Malinowska, A.B., Monteiro, M.T.T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods Appl. Sci. 41(1), 336–352 (2018)
Bai, Z., Lu, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311(2), 495–505 (2005)
Beddani, M., Beddani, H.: Compactness of boundary value problems for impulsive integrodifferential equation. Filomat 37(20), 6855–6866 (2023)
Beddani, M., Beddani, H., Feckan, M.: Qualitative study for impulsive pantograph fractional integrodifferential equation via ΨHilfer derivative. Miskolc Math. Notes 24(2), 635–651 (2023)
Bergman, T.L.: Fundamentals of Heat and Mass Transfer. Wiley, New York (2011)
Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)
Chen, P., Gao, Y.: Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions. Positivity 22, 761–772 (2018)
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (2013)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Furati, K.M.: A Cauchytype problem with a sequential fractional derivative in the space of continuous functions. Bound. Value Probl. 2012, 58 (2012)
Hilfer, R. (ed.): Applications of Fractional Calculus in Physics World Scientific, Singapore (2000)
Isaia, F.: On a nonlinear integral equation without compactness. Acta Math. Univ. Comen. 75(2), 233–240 (2006)
Istratescu, V.I.: Fixed Point Theory: An Introduction, vol. 7. Springer, Berlin (2001)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)
Li, C., Saadati, R., Srivastava, R., Beaudin, J.: On the boundary value problem of nonlinear fractional integrodifferential equations. Mathematics 10(12), 1971 (2022)
Ntouyas, S.K., AlSulami, H.H.: A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions. Adv. Differ. Equ. 2020, 73 (2020)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, New York (1998)
Qi, T., Liu, Y., Zou, Y.: Existence result for a class of coupled fractional differential systems with integral boundary value conditions. J. Nonlinear Sci. Appl. 10(7), 4034–4045 (2017)
Ross, B. (ed.): The Fractional Calculus and Its Application. Lecture Notes in Mathematics. Springer, Berlin (1975)
Smoko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Application. Gordon & Breach, New York (1993)
Sudsutad, W., Tariboon, J.: Existence results of fractional integrodifferential equations with mpoint multiterm fractional order integral boundary conditions. Bound. Value Probl. 2012, 94 (2012)
Sun, Y., Zeng, Z., Song, J.: Existence and uniqueness for the boundary value problems of nonlinear fractional differential equation. Appl. Math. 8(3), 312–323 (2017)
Sun, Y., Zhao, M.: Positive solutions for a class of fractional differential equations with integral boundary conditions. Appl. Math. Lett. 34, 17–21 (2014)
Tariboon, J., Ntouyas, S.K., Singubol, A.: Boundary value problems for fractional differential equations with fractional multiterm integral conditions. J. Appl. Math. 2014(1), 806156 (2014)
Wang, X., Wang, L., Zeng, Q.: Fractional differential equations with integral boundary conditions. J. Nonlinear Sci. Appl. 8(4), 309–314 (2015)
Yan, R., Sun, S., Sun, Y., Han, Z.: Boundary value problems for fractional differential equations with nonlocal boundary conditions. Adv. Differ. Equ. 2013, 176 (2013)
Yang, C., Guo, Y., Zhai, C.: An integral boundary value problem of fractional differential equations with a signchanged parameter in Banach spaces. Complexity 2021(1), 9567931 (2021)
Zhang, H., Li, Y., Lu, W.: Existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with fractional integral boundary conditions. J. Nonlinear Sci. Appl. 9(05), 2434–2447 (2016)
Zhao, K.: Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions. Bound. Value Probl. 2015, 181 (2015)
Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2023)
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Agheli, B., Darzi, R. New attitude on sequential ΨCaputo differential equations via concept of measures of noncompactness. J Inequal Appl 2024, 116 (2024). https://doi.org/10.1186/s13660024031880
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DOI: https://doi.org/10.1186/s13660024031880