Search

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ > 0 by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper solves fractional differential equations using the Shehu transform in combination with the q-homotopy analysis transform method (q-HATM). As the Shehu transform is only applicable to linear equations, q-HATM is an efficient technique for approximating solutions to nonlinear differential equations. In nonlinear systems that explain the emergence of stripes in 2D systems, the Newell–Whitehead–Segel equation plays a significant role. The findings indicate that the outcomes derived from the tables yield superior results compared to the existing LTDM in the literature. Maple is utilized to depict three-dimensional surfaces and find numerical values that are displayed in a table. [ABSTRACT FROM AUTHOR]

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions { (C D 0 + α u) (t) + ∑ i = 1 m λ i (t) (C D 0 + α i u) (t) + ∑ j = 1 n μ j (t) (C D 0 + β j u) (t) + ∑ k = 1 p ξ k (t) (C D 0 + γ k u) (t) + ∑ l = 1 q ω l (t) (C D 0 + δ l u) (t) + σ (t) u (t) + f (t , u (t)) = 0 , t ∈ [ 0 , 1 ] , u ″ (0) = u ‴ (0) = 0 , u ′ (0) = η 1 ∫ 0 1 u (s) d s , u (1) = η 2 ∫ 0 1 u (s) d s , where 0 < δ 1 < δ 2 < ⋯ < δ q < 1 < γ 1 < γ 2 < ⋯ < γ p < 2 < β 1 < β 2 < ⋯ < β n < 3 < α 1 < α 2 < ⋯ < α m < α < 4 and η 1 + 2 (1 − η 2) ≠ 0 . Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel'skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively. [ABSTRACT FROM AUTHOR]

This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result. [ABSTRACT FROM AUTHOR]

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder's and Banach's fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model of a coronavirus disease involving the Caputo–Fabrizio fractional derivative by dividing the total population into the susceptible population S (t) , the vaccinated population V (t) , the infected population I (t) , the recovered population R (t) , and the death class D (t) . A core goal of this study is the analysis of the solution of a proposed mathematical model involving nonlinear systems of Caputo–Fabrizio fractional differential equations. With the help of Lipschitz hypotheses, we have built sufficient conditions and inequalities to analyze the solutions to the model. Eventually, we analyze the solution for the formed mathematical model by employing Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, the Banach contraction principle, and Ulam–Hyers stability theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the results. [ABSTRACT FROM AUTHOR]

We prove the existence of the solutions for the new mixed differential equations, which is characteristic of the right-sided Caputo and the left-sided Riemann–Liouville fractional derivatives. There are four major ingredients. The first is composed of some basic definitions and lemmas. The second is the Green's function of the new mixed fractional differential equations. We calculate the corresponding Green's functions as well as their properties. The third, which is the main new ingredient of this paper, is demonstration of the existence of the solutions for fractional equations by the fixed-point theorem in cone expansion and compression of norm type. The fourth, as applications, is the example provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of fractional differential equations (FDEs) by using the fixed-point theory (FPT). We discuss also the Ulam–Hyers–Rassias (UHR) stability of some generalized FDEs according to some classical mathematical techniques and the FPT. Finally, two illustrative examples are presented to show the validity of our results. [ABSTRACT FROM AUTHOR]

In this article, we present an iterative transformation method for solving fractional partial differential equations that combines the Elzaki transform and iterative methods. By this iterative transformation method, numerical solutions in the form of series are obtained. When we apply this method to the fractional linear Klein–Gordon equation, we find that it yields the same results, just like the Homotopy perturbation method. The procedures and results of this method for solving the new generalized fractional Hirota–Satsuma coupled KdV equation are given in the paper. [ABSTRACT FROM AUTHOR]

In this present manuscript, by applying fractional quantum calculus, we study a nonlinear fractional pantograph q-difference equation with nonlocal boundary conditions. We prove the existence and uniqueness results by using the well-known fixed-point theorems of Schaefer and Banach. We also discuss the Ulam–Hyers stability of the mentioned pantograph q-difference problem. Lastly, the paper includes pertinent examples to support our theoretical analysis and justify the validity of the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of ρ-concave positive solutions for a p-Laplacian boundary value problem of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. Using Guo–Krasnoselskii fixed point theorem and under some additional assumptions, we prove some important results and obtain the existence of at least three solutions. To establish the results, Green functions are used to transform the considered two-sided generalized Katugampola and Caputo fractional derivatives. Finally, applications with illustrative examples are presented to show the validity and correctness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper aims to consider the multiplicity of solutions for a kind of boundary value problem to a fractional quasilinear differential model with impulsive effects. By establishing a new variational structure and overcoming the difficulties brought by the influence of impulsive effects, some new results are acquired via the symmetry mountain-pass theorem, which extend and enrich some previous results. [ABSTRACT FROM AUTHOR]

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green's function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones. [ABSTRACT FROM AUTHOR]

This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a p-Laplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed. [ABSTRACT FROM AUTHOR]

In this paper, a kind of αth $(3<\alpha\leq4)$ order differential equation with two-point boundary conditions is considered. The existence result of a sign-changing solution is given by the topological degree theory and the fixed point index theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of multiple solutions for Kirchhoff-type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions as follows: where Ω is a smooth bounded set in $\mathbb{R}^{n}$ , $n>ps$ with $s\in(0,1)$ fixed, ${\lambda,\mu}>0$ are two parameters, $1< q< p< p(\tau+1)<\alpha+\beta0$ , $l,\tau\geq0$ , and $(-\triangle)_{p}^{s}$ is the fractional p-Laplacian operator. We will prove that the problem has at least two solutions by using the Nehari manifold method and fibering maps. [ABSTRACT FROM AUTHOR]

In this paper, we use the Banach contraction mapping principle and Leray-Schauder degree theory to obtain some results of the existence and uniqueness of solution for a class of fractional boundary value problem with integral and anti-periodic boundary conditions. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence of mild solutions for fractional differential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using fractional calculus, Hausdorff measure of noncompactness, and the multivalued fixed point theorem. The results obtained in the present paper extend some related results on this topic. [ABSTRACT FROM AUTHOR]

In this paper, without requiring the complete continuity of integral operators and the existence of upper–lower solutions, by means of the sum-type mixed monotone operator fixed point theorem based on the cone P h , we investigate a kind of p-Laplacian differential equation Riemann–Stieltjes integral boundary value problem involving a tempered fractional derivative. Not only the existence and uniqueness of positive solutions are obtained, but also we can construct successively sequences for approximating the unique positive solution. As an application of our fundamental aims, we offer a realistic example to illustrate the effectiveness and practicability of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem. Furthermore, we discuss the Ulam–Hyers stability of the above fractional differential equation. Three examples illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiplicity results of some positive solutions for a system of Hadamard fractional differential equations with parameters and p-Laplacian operator subject to three-point boundary conditions which contains fractional derivatives. The proofs of our main result, multiplicity of positive solutions, are derived in terms of different values of parameters by using Guo–Krasnosel'skii's fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions for boundary value problems of high-order Riemann–Liouville fractional differential equations involving the p-Laplacian operator. Not only new existence conclusions of two positive solutions are obtained by employing functional-type cone expansion-compression fixed point theorem, but also some sufficient conditions for existence of at least three positive solutions are established by applying the Leggett–Williams fixed point theorem. In addition, we demonstrate the effectiveness of the main result by using an example. [ABSTRACT FROM AUTHOR]

In this paper, we concerned the existence of solutions of the following nonlinear mixed fractional differential equation with the integral boundary value problem: { D 1 − α C D 0 + β u (t) = f (t , u (t) , D 0 + β + 1 u (t) , D 0 + β u (t)) , 0 < t < 1 , u (0) = u ′ (0) = 0 , u (1) = ∫ 0 1 u (t) d A (t) , where D 1 − α C is the left Caputo fractional derivative of order α ∈ (1 , 2 ] , and D 0 + β is the right Riemann–Liouville fractional derivative of order β ∈ (0 , 1 ] . The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems. [ABSTRACT FROM AUTHOR]

In the present paper, the existence of nontrivial solutions of impulsive fractional differential equations with Dirichlet boundary conditions is studied. We apply Morse theory coupled with local linking arguments to solve the topic, and we prove the existence of at least one nontrivial solution for the impulsive fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we approximate the solution of fractional Bagley–Torvik equation by using the exponential spline function and the shifted Grünwald difference operator. The proposed methods reduce to the system of algebraic equations. The convergence analysis of the methods has been discussed. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. [ABSTRACT FROM AUTHOR]

The uniqueness of positive solution for a class of singular fractional differential system with integral boundary conditions is considered in this paper and many types of equation system are contained in this equation system because there are many parameters which can be changeable in this equation system. The fractional orders are involved in the nonlinearity of the boundary value problem and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable. The existence of uniqueness of positive solution is mainly obtained by fixed point theorem of mixed monotone operator and the positive solution of equation system is dependent on λ. An iterative sequence and convergence rate are given which are important for practical application and an example is given to demonstrate the validity of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of singular fractional differential equations with infinite-point boundary conditions. The fractional orders are involved in the nonlinearity of the boundary value problem, and the nonlinearity is allowed to be singular with respect to not only the time variable but also to the space variable. Firstly, we give Green's function and establish its properties. Then, we utilize the sequential technique and regularization to investigate the existence of positive solutions. Finally, we give an example of application of our result. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained. Our results are based on some fixed point theorems. Some examples are also presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of symmetric solutions for fractional differential equations with multi-order fractional integral boundary conditions, by means of standard fixed point theorems. Examples which support our theoretical results are also presented. [ABSTRACT FROM AUTHOR]

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results. [ABSTRACT FROM AUTHOR]

In this work, we introduce a novel idea of generalized (k , ψ) -Hilfer proportional fractional operators. The proposed operator combines the (k , ψ) -Riemann–Liouville and (k , ψ) -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under (k , ψ) -Hilfer proportional fractional operators is proved by using Picard's iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems. [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]

This research article introduces a novel approach based on the fuzzy Adomian decomposition method (FADM) to solve specific time fuzzy fractional partial differential equations with initial and boundary conditions (IBCs). The proposed approach addresses the challenge of incorporating both initial and boundary conditions into the FADM framework by employing a modified approach. This approach iteratively generates a new initial solution using the decomposition method. The method presented here offers a significant contribution to solving fuzzy fractional partial differential equations (FFPDEs) with fuzzy IBCs, a topic that has received limited attention in the literature. Furthermore, it satisfies a high convergence rate with minimal computational complexity, establishing a novel aspect of this research. By providing a series solution with a small number of recursive formulas, this method enhances accuracy and emerges as a preferred choice for tackling FFPDEs with mixed initial and boundary conditions. The effectiveness of the proposed technique is further supported by the inclusion of several illustrative examples. [ABSTRACT FROM AUTHOR]

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results. [ABSTRACT FROM AUTHOR]

We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 $3

This paper concerns a new kind of fractional differential equation of arbitrary order by combining a multi-point boundary condition with an integral boundary condition. By solving the equation which is equivalent to the problem we are going to investigate, the Green's functions are obtained. By defining a continuous operator on a Banach space and taking advantage of the cone theory and some fixed point theorems, the existence of multiple positive solutions for the BVPs is proved based on some properties of Green's functions and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]