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Results reported in this paper establish the existence of solutions for a class of generalized fractional inclusions based on the Caputo–Hadamard jerk system. Under some inequalities between multi-functions and with the help of special contractions and admissible maps, we investigate the existence criteria. Fixed points and end points are key roles in this manuscript, and the approximate property for end points helps us to derive the desired result for existence theory. An example is prepared to demonstrate the consistency and correctness of analytical findings. [ABSTRACT FROM AUTHOR]

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems. [ABSTRACT FROM AUTHOR]

We study two hybrid and non-hybrid fractional boundary value problems via the Caputo–Hadamard type derivatives. We seek the existence criteria for these two problems separately. By utilizing the generalized Dhage's theorem, we derive desired results for an integral structure of solutions for the hybrid problems. Also by considering the special case as a non-hybrid boundary value problem (BVP), we establish other results based on the existing tools in the topological degree theory. In the end of the article, we examine our theoretical results by presenting some numerical examples to show the applicability of the analytical findings. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for a fractional hybrid multi-term integro-differential inclusion with four-point sum and integral boundary value conditions. By using Dhage's fixed point results, we prove our main existence result. Finally, we give an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for two nonlinear fractional multi-term integro-differential inclusions in two hybrid and non-hybrid versions. The boundary value conditions are in the form of three-point integral hybrid conditions. In this way, we define a new operator based on the integral solution of the given boundary value inclusion problem and then we use assumptions of a Dhage's fixed point result for this fractional operator in the hybrid case. Also, the approximate endpoint property is applied for the corresponding set-valued maps in the non-hybrid case. Finally, we provide two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings. [ABSTRACT FROM AUTHOR]

Our main purpose in this work is to derive an existence criterion for a Caputo conformable hybrid multi-term integro-differential equation equipped with initial conditions. In this way, we consider a partially ordered Banach space, and, by applying the lower solution property, the existence and successive approximations of solutions for the mentioned hybrid initial problem are investigated. Eventually, we formulate an illustrative example for this hybrid IVP to support our findings from a numerical point of view. Moreover, we plot the sequence of the obtained approximate solutions for different values of noninteger orders. [ABSTRACT FROM AUTHOR]

We study a fractional-order model for the anthrax disease between animals based on the Caputo–Fabrizio derivative. First, we derive an existence criterion of solutions for the proposed fractional CF -system of the anthrax disease model by utilizing the Picard–Lindelof technique. By obtaining the basic reproduction number R 0 of the fractional CF -system we compute two disease-free and endemic equilibrium points and check the asymptotic stability property. Moreover, by applying an iterative approach based on the Sumudu transform we investigate the stability of the fractional CF -system. We obtain approximate series solutions of this system by means of the homotopy analysis transform method, in which we invoke the linear Laplace transform. Finally, after the convergence analysis of the numerical method HATM, we present a numerical simulation of the CF -fractional anthrax disease model and review the dynamical behavior of the solutions of this CF -system during a time interval. [ABSTRACT FROM AUTHOR]

We discuss some existence criteria for a new category of the Caputo conformable differential inclusion furnished with four-point mixed Riemann–Liouville conformable integro-derivative boundary conditions. In this way, we employ some analytical techniques on α-ψ-contractive mappings and operators having the approximate endpoint property to reach desired theoretical results. Finally, we provide an example to illustrate our last main result. [ABSTRACT FROM AUTHOR]

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 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 G -snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders. [ABSTRACT FROM AUTHOR]

Our basic purpose is to derive several existence aspects of solutions for a novel class of the fractional inclusion problem in terms of the well-defined generalized φ-Caputo and φ-Riemann–Liouville operators. The existing boundary conditions in such an inclusion problem are endowed with mixed generalized φ-Riemann–Liouville conditions. To reach this goal, we utilize the analytical methods on α-ψ-contractive maps and multifunctions involving approximate endpoint specification to derive the required results. In the final part, we formulate an illustrative simulation example to examine obtained theoretical outcomes by computationally and numerically. [ABSTRACT FROM AUTHOR]

In this research, we first investigate the existence of solutions for a new fractional boundary value problem in the Liouville–Caputo setting with mixed integro-derivative boundary conditions. To do this, Kuratowski's measure of noncompactness and Sadovskii's fixed point theorem are our tools to reach this aim. In the sequel, we discuss the continuous dependence of solutions on parameters by means of the generalized Gronwall inequality. Moreover, we consider an inclusion version of the given boundary problem in which we study its existence results by means of the endpoint theory. Finally, we prepare two simulative numerical examples to confirm the validity of the analytical findings. [ABSTRACT FROM AUTHOR]

In this research paper, we introduce a general structure of a fractional boundary value problem in which a 2-term fractional differential equation has a fractional bi-order setting of Riemann–Liouville type. Moreover, we consider the boundary conditions of the proposed problem as mixed Riemann–Liouville integro-derivative conditions with four different orders which cover many special cases studied before. In the first step, we investigate the existence and uniqueness of solutions for the given multi-order boundary value problem, and then the Hyers–Ulam stability is another notion in this regard which we study. Finally, we provide two illustrative examples to support our theoretical findings. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to concern with a new category of the sequential hybrid inclusion boundary value problem with three-point integro-derivative boundary conditions. In this direction, we employ various novel analytical techniques based on α-ψ-contractive mappings, endpoints, and the fixed points of the product operators to obtain the main results. Finally, we provide two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of non-hybrid single-valued fractional boundary value problems equipped with integro-non-hybrid-multiterm-multipoint-multistrip conditions and a fully hybrid integro-multi-valued fractional boundary value problem by some new methods including the Kuratowski measures based on Sadovskii's theorem, Krasnoselskii–Zabreiko criterion, and Dhage's technique. We generalize the Gronwall inequality in relation to a non-hybrid single-valued fractional boundary value problem, and then we investigate the stability notions in two versions. To examine the correctness of the results, we provide some examples. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is obtained by using the Banach principle. For obtaining two positive solutions, we apply another fixed point criterion due to Avery–Anderson–Henderson on cones by establishing some inequalities. An illustrative example is presented to indicate the validity of the obtained results. The results are new and provide a generalization to some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for a generalized sequential Caputo-type fractional neutral differential inclusion with generalized integral conditions. The used fractional operator has the generalized kernel in the format of (ϑ (t) − ϑ (s)) along with differential operator 1 ϑ ′ (t) d d t . We obtain existence results for two cases of convex-valued and nonconvex-valued multifunctions in two separated sections. We derive our findings by means of the fixed point principles in the context of the set-valued analysis. We give two suitable examples to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage's technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results. [ABSTRACT FROM AUTHOR]

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result. [ABSTRACT FROM AUTHOR]

In this research study, we are concerned with the existence and stability of solutions of a boundary value problem (BVP) of the fractional thermostat control model with ψ-Hilfer fractional operator. We verify the uniqueness criterion via the Banach fixed-point principle and establish the existence by using the Schaefer and Krasnoselskii fixed-point results. Moreover, we apply the arguments related to the nonlinear functional analysis to discuss various types of stability in the format of Ulam. Finally, by several examples we demonstrate applications of the main findings. [ABSTRACT FROM AUTHOR]

The main intention of this article is that new techniques of existence theory are used to derive some required criteria pertinent to a given fractional multi-term problem and its inclusion version. In such an approach, we do our research on a fractional integral equation corresponding to the mentioned BVPs. In more precise words, by virtue of this integral equation, we construct new operators which belong to a special category of functions named α-admissible and α-ψ-contraction maps coupled with operators having (AEP)-property. Next, by considering some new properties on the existing Banach space having properties (B) and (C α) , our argument for ensuring the existence of solutions is completed. In addition, we also add two simulative examples to review our findings by a numerical view. [ABSTRACT FROM AUTHOR]

In this research article, we turn to studying the existence and different types of stability such as generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability of solutions for a new modeling of a boundary value problem equipped with the fractional differential equation which contains the multi-order generalized Caputo type derivatives furnished with four-point mixed generalized Riemann–Liouville type integro-derivative conditions. At the end of the current paper, we formulate two illustrative examples to confirm the correctness of theoretical findings from computational aspects. [ABSTRACT FROM AUTHOR]

In the present research manuscript, we formulate a new generalized structure of the nonlinear Caputo fractional quantum multi-integro-differential equation in which such a multi-order structure of quantum integrals is considered for the first time. In fact, in the light of this type of boundary value problem equipped with the multi-integro-differential setting, one can simply study different cases of the existing usual integro-differential problems in the literature. In this direction, we utilize well-known analytical techniques to derive desired criteria which guarantee the existence of solutions for the proposed multi-order quantum multi-integro-differential problem. Further, some numerical examples are considered to examine our theoretical and analytical findings using the proposed methods. [ABSTRACT FROM AUTHOR]

In the present research article, we find some important criteria on the existence of solutions for a class of the hybrid fractional Caputo–Hadamard differential equations and its corresponding inclusion problem supplemented with hybrid Hadamard integral boundary conditions. In this direction, we utilize some theorems due to Dhage's fixed point results in our proofs. Finally, we demonstrate two numerical examples to confirm the validity of the main obtained results. [ABSTRACT FROM AUTHOR]