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In the present work, we have introduced a new type of (ξ , ω) -weakly cyclic generalized contraction in the setting of metric spaces and established some fixed-point results. Fixed-point results are useful in establishing the existence of unique solution to differential equations. We have supplemented the derived results with suitable non-trivial examples with an application to the Boundary Value Problem, generalizing some known results. The analytical result has been verified with numerical simulation. [ABSTRACT FROM AUTHOR]

In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in L p (·) (Ω) obtained under the assumptions p + < ∞ and the property (R) satisfied by ρ will force p − > 1 . Therefore, the conclusion is well known. In this note, we establish said conclusion without the assumption p + < ∞ . [ABSTRACT FROM AUTHOR]

The goal of this paper is to consider a differential equation system written as an interesting equivalent form that has not been used before. Using Perov's fixed point theorem in generalized metric spaces, the existence and uniqueness of the solution are obtained for the proposed system. The approximation of the solution is given, and as a novelty, the approximation of its derivative is also obtained using the same iteration steps. [ABSTRACT FROM AUTHOR]

In this paper, we present an extraordinary method to calculate the Gao-type constant of absolute normalized norms on R 2 . By using our method, we not only provide the values of the Gao-type constant in new specific spaces, but also improve well-known results from the existing literature. [ABSTRACT FROM AUTHOR]

We discuss the solvability of a (p , q) -difference equation of fractional order α ∈ (1 , 2 ] , equipped with anti-periodic boundary conditions involving the first-order (p , q) -difference operator. The desired results are accomplished with the aid of standard fixed point theorems. Examples are presented for illustrating the obtained results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to approximate the fixed points of generalized α -nonexpansive mappings using A A -iterative algorithm. We establish some weak and strong convergence results for generalized α -nonexpansive mappings in uniformly convex Banach spaces. A numerical example is also given to show that the A A -iterative algorithm converges faster than some others algorithms for generalized α -nonexpansive mappings. Lastly, using the A A -iterative algorithm, we approximate the weak solution of delay composite functional differential equation of the Volterra–Stieltjes type. [ABSTRACT FROM AUTHOR]

In this paper, we present extensions of some classic results regarding the existence and uniqueness of fixed points of operators fulfilling generalized contractive conditions defined by sums involving some functions with suitable properties in the setting of Jleli–Samet generalized metric spaces. As a consequence, some known results in the literature are obtained. Examples are provided to prove the usability of our developments. [ABSTRACT FROM AUTHOR]

We introduce the class of enriched φ -contractions in Banach spaces as a natural generalization of φ -contractions and study the existence and approximation of the fixed points of mappings in this new class, which is shown to be an unsaturated class of mappings in the setting of a Banach space. We illustrated the usefulness of our fixed point results by studying the existence and uniqueness of the solutions of some second order (p , q) -difference equations with integral boundary value conditions. [ABSTRACT FROM AUTHOR]

The point of this research is to present a new iterative procedure for approximating common fixed points of nonexpansive mappings in Hadamard manifolds. The convergence theorem of the proposed method is discussed under certain conditions. For the sake of clarity, we provide some numerical examples to support our results. Furthermore, we apply the suggested approach to solve inclusion problems and convex feasibility problems. [ABSTRACT FROM AUTHOR]

This paper deals with the question of achieving a suitable extension of the notion of Suzuki-type contraction to the framework of quasi-metric spaces, which allows us to obtain reasonable fixed point theorems in the quasi-metric context. This question has no an easy answer; in fact, we here present an example of a self map of Smyth complete quasi-metric space (a very strong kind of quasi-metric completeness) that fulfills a simple and natural contraction of Suzuki-type but does not have fixed points. Despite it, we implement an approach to obtain two fixed point results, whose validity is supported with several examples. Finally, we present a general method to construct non- T 1 quasi-metric spaces in such a way that it is possible to systematically generate non-Banach contractions which are of Suzuki-type. Thus, we can apply our results to deduce the existence and uniqueness of solution for some kinds of functional equations which is exemplified with a distinguished case. [ABSTRACT FROM AUTHOR]

This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure. [ABSTRACT FROM AUTHOR]

The subject of this paper is the existence, uniqueness and stability of solutions for a new sequential Van der Pol–Duffing (VdPD) jerk fractional differential oscillator with Caputo–Hadamard derivatives. The arguments are based upon the Banach contraction principle, Krasnoselskii fixed-point theorem and Ulam–Hyers stabilities. As applications, one illustrative example is included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In the present work, we establish fixed-point results for a pair of mappings satisfying some contractive conditions on rational expressions with coefficients as point-dependent control functions in the setting of tricomplex-valued metric spaces. The proven results are extension and generalisation of some of the literature's well-known results. We also explore some of the applications to our key results. [ABSTRACT FROM AUTHOR]

Partial metrics constitute a generalization of classical metrics for which self-distance may not be zero. They were introduced by S.G. Matthews in 1994 in order to provide an adequate mathematical framework for the denotational semantics of programming languages. Since then, different works were devoted to obtaining counterparts of metric fixed-point results in the more general context of partial metrics. Nevertheless, in the literature was shown that many of these generalizations are actually obtained as a corollary of their aforementioned classical counterparts. Recently, two fuzzy versions of partial metrics have been introduced in the literature. Such notions may constitute a future framework to extend already established fuzzy metric fixed point results to the partial metric context. The goal of this paper is to retrieve the conclusion drawn in the aforementioned paper by Haghia et al. to the fuzzy partial metric context. To achieve this goal, we construct a fuzzy metric from a fuzzy partial metric. The topology, Cauchy sequences, and completeness associated with this fuzzy metric are studied, and their relationships with the same notions associated to the fuzzy partial metric are provided. Moreover, this fuzzy metric helps us to show that many fixed point results stated in fuzzy metric spaces can be extended directly to the fuzzy partial metric framework. An outstanding difference between our approach and the classical technique introduced by Haghia et al. is shown. [ABSTRACT FROM AUTHOR]

In this paper, we investigate contractions in a rational form in the context of the supermetric space, which is a very interesting generalization of the metric space. We consider an illustrative example to support this new result on supermetric space. [ABSTRACT FROM AUTHOR]

Modular metric space is one of the most interesting spaces in the framework of the metric fixed point theory. The main goal of the paper is to provide some certain fixed point results in the context of modular metric spaces and non-Archimedean modular metric spaces. In particular, we examine the existence of interpolative Meir–Keeler contraction types via admissible mappings for fixed point theory. Our results bring together several results available in the current corresponding literature. [ABSTRACT FROM AUTHOR]

This paper introduces a complete G n -Menger space and defines the Hausdorff–Pompeiu distance in the space. Furthermore, we show a novel fixed-point theorem for G n -Menger- θ -contractions in fractal spaces. [ABSTRACT FROM AUTHOR]

Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the properties of the operators satisfying this type of condition; in particular, we study the case where the space is normed, and the maps are linear. Then we generalize some theorems proposed by this author on the existence of a fixed point of one operator or a common fixed point for two operators. Our results first prove the existence of a common fixed point of a set of self-maps of any cardinal number (countable or uncountable) satisfying the conditions of Kannan type in metric spaces. The same is proved for a set of maps satisfying the mutual relations of classical contractivity. We prove in both cases the convergence of iterative schemes involving operators with mutual relations of contractivity, proposing sufficient conditions for the iteration of the operators on any element of the space to converge to the common fixed point when a different operator is taken in each step. The results obtained are applied to operators acting on real functions, coming from the fractal convolution with the null function. [ABSTRACT FROM AUTHOR]

Much has been written on variable exponent spaces in recent years. Most of the literature deals with the normed space structure of such spaces. However, because of the variability of the exponent, the underlying modular structure of these spaces is radically different from that induced by the norm. In this article, we focus our attention on the progress made toward the study of the modular structure of the sequence Lebesgue spaces of variable exponents. In particular, we present a survey of the state of the art regarding modular geometric properties in variable exponent spaces. [ABSTRACT FROM AUTHOR]

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving (k , ψ) -Hilfer fractional derivative operator of order in (1 , 2 ] , equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel'skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler's fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a new concept of Z-Z-B spaces, which is a generalization of Z-C-X spaces. Meanwhile, the new concept of the superior cone is introduced. Secondly, we study some new problems for semi-closed 1-set-contractive operators in the Z-Z-B space and obtain some new results. These new theorems are proven by combining partial order theory with fixed point index theory. Regarding these theorems, in the latter part of the paper, the proofs are omitted since the methods of proving these theorems are similar. Moreover, two important inequality lemmas are proven. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notion of Perov–Ćirić–Prešić-type Θ -contractions and to obtain some generalized fixed point theorems in the setting of vector-valued metric spaces. We derive some fixed point results as consequences of our main results. A nontrivial example is also provided to support the validity of our established results. As an application, we investigate the solution of a semilinear operator system in Banach space. [ABSTRACT FROM AUTHOR]

In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we establish existence, uniqueness and approximation results for the common fixed point. In the same framework, the second problem is to study various stability properties. More precisely, we will obtain sufficient conditions assuring that the common fixed point problem is well-posed and has the Ulam–Hyers stability, as well as the Ostrowski property for the considered problem. Some examples and applications are finally given in order to illustrate the abstract theorems proposed in the first part of the paper. Our results extend and complement some theorems in the recent literature. [ABSTRACT FROM AUTHOR]

Considering the ω -distance function defined by Kostić in proximity space, we prove the Matkowski and Boyd–Wong fixed-point theorems in proximity space using ω -distance, and provide some examples to explain the novelty of our work. Moreover, we characterize Edelstein-type fixed-point theorem in compact proximity space. Finally, we investigate an existence and uniqueness result for solution of a kind of second-order boundary value problem via obtained Matkowski-type fixed-point results under some suitable conditions. [ABSTRACT FROM AUTHOR]

The split feasibility problem (SFP) has many practical applications, which has attracted the attention of many authors. In this paper, we propose a different method to solve the SFP and the fixed-point problem involving quasi-nonexpansive mappings. We relax the conditions of the operator as well as consider the inertial iteration and the adaptive step size. For example, the convergence generated by our new method is better than that of other algorithms, and the convergence rate of our algorithm greatly improves that of previous algorithms. [ABSTRACT FROM AUTHOR]

The exploration of contractive inequalities which do not imply the continuity of the mapping at fixed points was an interesting open problem for quite some time. A significant amount of progress was made in the last two decades towards the solution of this problem. In the current paper, we attempt to address the question of discontinuity at fixed point with the help of F-contractions in a metric space. We establish a common fixed-point (CFP) result for such contractive mappings and investigate its discontinuity at the CFP. A fixed-circle result is also obtained consequently. [ABSTRACT FROM AUTHOR]

In this paper, we present a new type of rational contraction in double controlled metric-like spaces and improve recent results of such spaces. Moreover, there is an example to verify the correctness of our results. Finally, we also obtain some new fixed point results, which can be derive directly from our main theorem. [ABSTRACT FROM AUTHOR]

In this work, we prove a fixed-point theorem in the variable exponent spaces ℓ p (.) , when p − = 1 without further conditions. This result is new and adds more information regarding the modular structure of these spaces. To be more precise, our result concerns ρ -nonexpansive mappings defined on convex subsets of ℓ p (.) that satisfy a specific condition which we call "condition of uniform decrease". [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of a fixed point of almost contractions via simulation functions in metric spaces. Moreover, some examples and an application to integral equations are given to support availability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized nonlinear stability of the first and second of the following ρρ -functional equations, G(ÖaÖΔ*AÖbÖ)Δ*BG(ÖaÖΔ**AÖbÖ)-G(ÖaÖ)Δ**BG(ÖbÖ)=ρ(2[G,(ÖaÖΔ*AÖbÖ2),Δ*B,G,(ÖaÖΔ**AÖbÖ2)]-G(ÖaÖ)Δ**BG(ÖbÖ))G(ÖaÖΔA*ÖbÖ)ΔB*G(ÖaÖΔA**ÖbÖ)-G(ÖaÖ)ΔB**G(ÖbÖ)=ρ(2GÖaÖΔA*ÖbÖ2ΔB*GÖaÖΔA**ÖbÖ2-G(ÖaÖ)ΔB**G(ÖbÖ)) , and 2[G,(ÖaÖΔ*AÖbÖ2),Δ*B,G,(ÖaÖΔ**AÖbÖ2)]-G(ÖaÖ)Δ**BG(ÖbÖ)=ρ(G(ÖaÖ,Δ*A,ÖbÖ),Δ*B,G(ÖaÖ,Δ**A,ÖbÖ),-,G,(ÖaÖ),Δ**B,G,(ÖbÖ))2GÖaÖΔA*ÖbÖ2ΔB*GÖaÖΔA**ÖbÖ2-G(ÖaÖ)ΔB**G(ÖbÖ)=ρG(ÖaÖΔA*ÖbÖ)ΔB*G(ÖaÖΔA**ÖbÖ)-G(ÖaÖ)ΔB**G(ÖbÖ) in latticetic random Banach lattice spaces, where ρρ is a fixed real or complex number with ρ≠1ρ≠1 . View Full-Text [ABSTRACT FROM AUTHOR]

In this paper, the existence of fixed point for Pata type Zamfirescu mapping in a complete metric space is proved. Our result give existence of fixed point for a wider class of functions and also prove the existence of best proximity point to the result on "A fixed point theorem in metric spaces", given by vittorino Pata. [ABSTRACT FROM AUTHOR]

We prove the generalized Hyers–Ulam stability of a mean value type functional equation f (x) - g(y) = (x - y)h(x + y) by applying a method originated from fixed point theory. [ABSTRACT FROM AUTHOR]

In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster-Tarski theorem has also been established using our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new notion of Suzuki-type (a, b, q, g)-contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak Pl-property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we give some remarks on the recent papers on generalized F-contractions. Our results unify and generalize the previous results in the existing literature. Moreover, we give an example to support our results. As an application, we give the existence and uniqueness of a solution to a class of differential equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some common fixed point theorems for generalized contractions involving certain control functions in controlled metric space and derive some generalized fixed point results as a consequence of our main results. We also prove some common fixed point theorems in controlled metric spaces endowed with a graph. Our results will generalize and amend many famous results from the literature. We also provide an example to show the authenticity of the established results. As an application of our main result, we investigate the solution of integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove a fixed point theorem of Sehgal type (see Sehgal, V.M., Proc Amer Math Soc 23: 631–634, 1969) in a more general setting of ρ -space (see Secelean, N.A. and Wardowski, D., Results Math, 72: 919–935, 2017). In this way, we can find, as particular cases, some results of Sehgal type in metric, b-metric and rectangular b-metric spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a new faster iterative scheme (called A A -iteration) to approximate the fixed point of (b , η) -enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using A A -iterative scheme. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation. [ABSTRACT FROM AUTHOR]

A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the ψ -Caputo fractional derivative in Banach spaces. Besides that, an example is included to verify our main results. Moreover, the outcomes obtained in this research paper ameliorate and expand some previous findings in this area. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of extended cone b-metric-like spaces over Banach algebra and investigated some fixed point results. We also provide examples to illustrate the results presented herein. Finally, as an application of our main result, we examine the existence and uniqueness of solution for a Fredholm integral equation. [ABSTRACT FROM AUTHOR]

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper "A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces". We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article. [ABSTRACT FROM AUTHOR]

We develop a class of Steffensen-like schemes for approximating solution of Banach space valued equations. The sequences generated by these schemes are, converging to the solution under certain hypotheses that are weaker than in earlier studies. Hence, extending the region of applicability of these schemes without additional hypotheses. Benefits include: more choices for initial points; the computation of fewer iterates to reach a certain accuracy in the error distances, and a more precise knowledge of the solution. Technique is applicable on other schemes our due to its generality. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to improve, generalize, unify, extend and enrich the recent results established by Dung and Hang (2015), Piri and Kumam (2014, 2016), and Singh et al. (2018). In our proofs, we only use the property (F1) of Wardowski's F-contraction, while the many authors in their papers still use all tree properties of F-contraction as well as two new properties introduced by Piri and Kumam. Our approach in this paper indicates that for most results with F-contraction, property (F1) is sufficient. It is interesting to investigate whether (F1) is sufficient in the case of multi-valued mappings. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Sehgal–Guseman-type fixed point theorem in b-rectangular metric spaces which provides a complete solution to an open problem raised by Zoran D. Mitrović (A note on a Banach's fixed point theorem in b-rectangular metric space and b-metric space). The result presented in the paper generalizes and unifies some results in fixed point theory. [ABSTRACT FROM AUTHOR]

Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form ψ (d (T x , T y)) ≤ φ (d (x , y)) , Proinov proved some fixed-point theorems, which extended and unified many existing results in the literature. Accordingly, inspired by Proinov-type contraction conditions, Roldán López de Hierro et al. introduced a novel family of contractions in fuzzy metric spaces (in the sense of George and Veeramani), whose main advantage is the very weak constraints imposed on the auxiliary functions that appear in the contractivity condition. They also proved the existence and uniqueness of fixed points for the discussed family of fuzzy contractions in the setting of non-Archimedean fuzzy metric spaces. In this paper, we introduce a new family of fuzzy contractions based on Proinov-type contractions for which the involved auxiliary functions are not supposed to satisfy any monotonicity assumptions; further, we establish some new results about the existence and uniqueness of fixed points. Furthermore, we show how the main results in the above-mentioned paper can be deduced from our main statements. In this way, our conclusions provide a positive partial solution to one of the open problems posed by such authors for deleting or weakening the hypothesis of the nondecreasingness character of the auxiliary functions. [ABSTRACT FROM AUTHOR]

We establish fixed point, stability and genericity theorems for strict contractions on complete metric spaces with graphs. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the Hyers–Ulam stability results of the quadratic functional equation in Banach spaces and non-Archimedean Banach spaces by utilizing two different techniques in terms of direct and fixed point techniques. [ABSTRACT FROM AUTHOR]

In this paper, we define ordered vectorial quasi contractions. We show that ordered quasi contractions are ordered vectorial quasi contractions, but the reverse is not true. We also define ordered vectorial almost contractions and present fixed point theorems for this type of contractions. Hence, we disclose many results in the literature. With the help of examples, we illustrate the relationship between these two types of contractions and some others in the literature. [ABSTRACT FROM AUTHOR]