. The present work examines the solvability of a tripled system of fractional Langevin differential equations with cyclic antiperiodic boundary conditions. The Krasnoselskii fixed point theorem, the Banach contraction mapping theorem, and specific properties of the Mittag-Leffler functions are employed to establish sufficient conditions for the existence and uniqueness of solutions. The feasibility of the primary findings is illustrated through the discussion of several numerical examples.

In this study, we investigate the solvability of a tripled system of fractional Langevin differential equations subject to cyclic anti-periodic boundary conditions. We do this through the systematic derivation of explicit solution formulas involving Mittag-Leffler functions. Using fixed-point methods alongside the key properties of Mittag-Leffler functions, we establish existence and uniqueness results under weaker sufficient conditions than those used in previous studies, thereby extending the current theoretical framework. Additionally, we provide numerical examples that demonstrate the effectiveness and practical advantages of the proposed solutions.

The paper examines the solvability and stability of a particular set of fractional Langevin equations under anti-periodic boundary conditions. Utilizing the Krasnoselskii fixed point theorem, the Banach contraction mapping theorem, and properties of the Mittag-Leffler function, we establish less stringent criteria for the existence and uniqueness of solutions compared to previous findings in the literature. Furthermore, we present illustrative examples with specific parameters that highlight the reduced conditions necessary for ensuring the existence of a unique solution.

In the present paper, the solvability and stability of a class of fractional Langevin equations with integral and anti-periodic boundary conditions are considered. By applying the Krasnoselskii fixed point theorem, the Banach contraction mapping theorem, and specific properties of the Mittag–Leffler function, we derive weaker sufficient conditions for the existence, uniqueness, and stability of results, building upon prior work in this literature. Some illustrative numerical examples are also discussed to demonstrate the superiority of our results.

Inspired by the paper of Fazli and Nieto in [Open Math. 17 (2019) 499–512], we establish new existence and uniqueness result for a type of fractional Bagley–Torvik differential equation. Reported result not only generalizes previous results but also adopts different technique. We finish this study by concluding remarks which discuss the preference of our theorem compared to previous results. An example is constructed with specific parameters that requires weaker conditions for the existence of a unique solution. Meanwhile, we construct an iterative sequence that converges to the unique solution and can not be commented via the results of Fazli and Nieto.

This article investigates a nonlinear fractional Caputo-Langevin equation D-beta(D-alpha + lambda) x(t) = f(t, x(t)), 0 < t < 1, 0 < alpha <= 1, 1 < ss <= 2, subject to the multi-point boundary conditions x(0) = 0, D(2 alpha)x(1) + lambda D(a)x(1) = 0, x(1) = integral(eta)(0) x(tau) d tau for some 0 < eta < 1, where D-alpha is the Caputo fractional derivative of order alpha, f : [0, 1] x R -> R is a given continuous function, and lambda is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

In this paper, we introduce an Caputo fractional high-order problem with a new boundary condition including two orders gamma is an element of (n(1) - 1, n(1)] and eta is an element of (n(2) - 1, n(2)] for any n(1), n(2) is an element of N. We deals with existence and uniqueness of solutions for the problem. The approach is based on the Krasnoselskii's fixed point theorem and contraction mapping principle. Moreover, we present several examples to show the clarification and effectiveness.

We present a novel generalization of the Hyers–Ulam–Rassias stability definition to study a generalized cubic set-valued mapping in normed spaces. In order to achieve our goals, we have applied a brand new fixed point alternative. Meanwhile, we have obtained a practicable example demonstrating the stability of a cubic mapping that is not defined as stable according to the previously applied methods and procedures.

Abstract In this paper, we obtain some fixed point theorems for multivalued mappings in incomplete metric spaces. Moreover, as motivated by the recent work of Olgun, Minak and Altun [M. Olgun, G. Minak and I. Altun, A new approach to Mizoguchi–Takahashi type fixed point theorems, J. Nonlinear Convex Anal. 17 2016, 3, 579–587], we improve these theorems with a new generalization contraction condition for multivalued mappings in incomplete metric spaces. This result is a significant generalization of some well-known results in the literature. Also, we provide some examples to show that our main theorems are a generalization of previous results. Finally, we give an application to a boundary value differential equation.

In this paper, firstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some fixed point theorems for single-valued mappings in incomplete metric spaces. Secondly, as motivated by the recent work of A.H. Ansari et al. [J. Fixed Point Theory Appl. (2017), 1145-1163], we obtain that an existence and uniqueness result for the following problem: finding x is an element of X such that x = Tx, Ax R-1 Bx and Cx R-2 Dx, where (X, d) is an incomplete metric space equipped with the two binary relations R-1 and R-2,( )A, B, C, D : X -> X are discontinuous mappings and T : X -> X satisfies in a new contractive condition. This result is a real generalization of main theorem of A.H. Ansari's. Finally, we provide some examples for our results and as an application, we find that the solutions of a differential equation.

In this paper, by using a new fixed point alternative, we establish a new generalization of the Hyers-Ulam-Rassias stability concerning a Cauchy-Jensen functional equation in normed spaces which are not necessarily Banach spaces. Moreover, our paper consists of several non-trivial examples which signify the motivation of such investigations.

In this paper, we introduce the notion of orthogonal relation on a soft set (F, A) and some related concepts. This notion allows us to consider fixed point theorem in SO-complete instead of complete soft metric spaces introduced by Yazar et.al. (Filomat 30:2 (2016), 269-279). Then, the existence and uniqueness of soft fixed points for a generalized soft contractive mapping are proved. Also, some examples are given to support that our main theorem is a real extension of Yazar et.al.

In this paper, we provide an extension result for the existence and uniqueness of solutions for a coupled system of fractional Langevin differential equations with antiperiodic boundary conditions. The system involves two different Caputo fractional derivatives defined on different intervals and associated with boundary conditions described by sequential fractional derivatives. As a conclusion for our main result, we deduce the results of H. Fazli, J.J. Nieto, Fractional Langevin equation with anti‐periodic boundary conditions, Chaos Soliton Fract , 114 (2018),332–337 under less restrictive conditions. The consistency of the main results is demonstrated by two numerical examples. For the sake of completeness, we end the paper by a concluding discussion.

We present the notion of orthogonal F -metric spaces and prove some fixed and periodic point theorems for orthogonal ⊥ Ω -contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness of the solution of a nonlinear fractional differential equation.

In their remarkable paper, Fazli et al. (Int J Comput Math, 2019. https://doi.org/10.1080/00207160.2019.1658870 ) have recently established existence results for fractional Basset–Boussinesq–Oseen equation in an appropriate partially ordered Banach space. In this paper, we improve this study by proposing different approach that provides existence criterion for the addressed equation. The main assumptions are less restrictive and easily verifiable. Examples with graphical representations are constructed to demonstrate consistency to theoretical findings. Besides, we construct an iterative sequence that converges to the unique solution. We end the paper by a conclusion that demonstrates the advantage of our theorem compared to the previous results.

An open problem proposed by Rhoades is the following. Is there a contractive condition which guarantees the existence of a fixed point, but does not require the mapping to be continuous at the point? In this paper, we generalize a celebrated result of Eshaghi et al., [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578], which allows us to find a new solution to this open problem. Furthermore we show that a claim of the aforementioned paper, that Banach’s fixed point theorem cannot be applied in their application, is incorrect. Finally, as an application, we prove that a multivalued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation.

In this paper, as motivated by a work of Daffer et al. [6], we state and prove some theorems for set valued mappings and by them we conclude the existence of coincidence points and fixed points of a general class of set valued mappings satisfying a new generalized contractive condition which extends some well-known results in the literature. For this reason, firstly, by using a recent work of Eshaghi et al [11], we define the notion of orthogonal sets and by the notion, we consider our results in strongly orthogonal complete metric spaces (not necessarily complete metric spaces). In addition, this article has a new and different view on the subject and consists of several non-trivial examples which signify the motivation of such investigations. Also, in the end of this paper, by using our examples, we show that our results are real generalization of the previous results in the literature.

In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical improvement for Fazli–Nieto’s results with some conditions removed based on a new technique is the main objective of this paper. In addition, we introduce an infinite system of nonlinear sequential fractional differential equations and discuss the existence of a solution for them in the classical Banach sequence spaces $c_{0}$ and $\ell_{p}$ by applying the Darbo fixed point theorem. Moreover, the proposed method is applied to several examples to show the clarity and effectiveness.

In this paper, we consider a coupled system of sequential fractional differential equations associated with initial conditions. The main theorems provide new existence and uniqueness conditions for solutions of the proposed coupled system. We conclude an immediate consequence that establishes weaker conditions to ensure the existence and uniqueness of solutions for the corresponding sequential fractional differential equation. Meanwhile, an iterative sequence is constructed in terms of solution operator that converges to the unique fixed point which corresponds to the unique solution. The consistency of the main results is verified by presenting two numerical examples. For the sake of completeness, we end the paper with a concluding remark.

In this paper, using the conditions of Taleb-Hanebaly’s theorem in a modular space where the modular is s-convex and symmetric with respect to the ordinate axis, we prove a new generalized modular version of the Schauder and Petryshyn fixed point theorems for nonexpansive mappings in s-convex sets. Our results can be applied to a nonlinear integral equation in Musielak-Orlicz space L p where 0 < p ≤ 1 and 0 < s ≤ p .