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In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces. We propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike, respectively. We also propose the idea of (α,F)\left(\alpha ,F)-Geraghty-type generalized FF-contraction mappings utilizing fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study, we consider an application to dynamic market equilibrium.

This article presents the notions of extended b-gauge space (U,Qφ;Ω)\left(U,{Q}_{\varphi ;\Omega }) and extended Jφ;Ω{{\mathcal{J}}}_{\varphi ;\Omega }-families of generalized extended pseudo-b-distances on UU. Furthermore, we look at these extended Jφ;Ω{{\mathcal{J}}}_{\varphi ;\Omega }-families on UU and define the extended Jφ;Ω{{\mathcal{J}}}_{\varphi ;\Omega }-sequential completeness. We also look into some fixed and periodic point theorems for set-valued mappings in the new space with a graph that does not meet the completeness condition of the space. Additionally, this article includes some examples to explain the corresponding results and highlights some important consequences of our obtained results.

This manuscript is concerned with obtaining results for fixed points that arise from new contractive mappings on controlled metric spaces. These mappings are a mixture of Wardowski's contractions with both multivalued, nonlinear mappings and auxiliary functions. It is also proved that the obtained fixed-point outcomes are well-posed. Additionally, a data-dependence result for fixed points is given. To aid with understanding, several illustrative examples are also provided. Numerous findings that are currently in the literature are specific instances of the findings that were made. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski's and Banach's fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example. [ABSTRACT FROM AUTHOR]

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations. [ABSTRACT FROM AUTHOR]

The paper deals with a source identification problem of the anomalous diffusion equations from nonlocal final data observations where the nonlinearity probably takes values in Hilbert scales. The existence and uniqueness results are proved by establishing some estimates for resolvent operators and using the embedding theorems. We also study regularity results for this equation in terms of the Hölder continuity of mild solutions. Finally, the multi-term fractional diffusion equations with polynomial nonlinearities and the ultra-slow diffusions are considered as illustrative applications. [ABSTRACT FROM AUTHOR]

The present paper is concerned with a fractional parabolic reaction-diffusion system posed in a regular bounded open subset of R N , where the gradients of the unknowns act as source terms (see (S) below). First, we establish some nonexistence and blow-up in finite time results. Second, we prove some new weighted regularity results. Such results are interesting in themselves and play a crucial role to study local existence of nonnegative solutions to our system under suitable assumptions on the data. This work also highlights a substantial difference between the nonlocal case and the local case already studied by the fourth author and his coworkers. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we introduced a new type of cyclic (noncyclic) mappings, called Jaggi relatively nonexpansive mappings and use to investigate the existence of best proximity points (pairs) in the framework of reflexive (strictly convex) Banach spaces by using a geometric concept of proximal normal structure. We also present a best proximity version of Schauder’s fixed point problem for non-self mappings. Finally, we prove the existence of a common best proximity point for an arbitrary family of cyclic relatively u-continuous mappings which are compact and then we obtain a generalization of Markov-Kakutani’s theorem. [ABSTRACT FROM AUTHOR]

In the context of hyperbolic spaces, our study presents a novel iterative approach for approximating common fixed points satisfying general contractive condition involving a pair of mappings with weak compatibility. Also, we notice that our iterative procedure approximates to a point of coincidence if the weak compatibility condition is violated. We provide theorems to demonstrate the Δ - convergence, stability, and efficiency of this iteration process. Additionally, we provided some immediate corollaries that involve mappings with contractive condition, instead of general contractive condition. Furthermore, we demonstrate with examples and graphs that our iteration process is faster than all previous procedures, including those of Jungck-SP, Jungck-CR, and Jungck-DK, utilizing MATLAB software. Also, we compare the impact of the initial values and the parameters on the convergence behavior of the proposed iterative process with existing iterative schemes using an example. Finally, we focus on using our iterative technique to approximate the solution of a non-linear integral equation with two delays. [ABSTRACT FROM AUTHOR]

This paper presents novel fixed-point results for two distinct families of fuzzy-dominated operators satisfying a generalized nonlinear contraction condition on a closed ball in a complete strong b-metric-like space. Our research introduces innovative fixed-point theorems for separate families of ordered fuzzy-dominated mappings in ordered complete strong b-metric-like spaces. Two different kinds of mappings are used in our methodology: a class of fuzzy-dominated mappings and a class of strictly non-decreasing mappings. Furthermore, we establish new fixed-point results for fuzzy-graph-dominated contractions. To substantiate our findings, we provide both rigorous and illustrative examples. We demonstrate the uniqueness of our results by applying them to obtain common solutions for fractional differential equations and fuzzy Volterra integral equations. [ABSTRACT FROM AUTHOR]

This work emphasizes the computational and analytical analysis of integral-differential equations, with a particular application in modeling avoidance learning processes. Firstly, we suggest an approach to determine a unique solution to the given model by employing methods from functional analysis and fixed-point theory. We obtain numerical solutions using the approach of Picard iteration and evaluate their stability in the context of minor perturbations. In addition, we explore the practical application of these techniques by providing two examples that highlight the thorough analysis of behavioral responses using numerical approximations. In the end, we examine the efficacy of our suggested ordinary differential equations (ODEs) for studying the avoidance learning behavior of animals. Furthermore, we investigate the convergence and error analysis of the proposed ODEs using multiple numerical techniques. This integration of theoretical and practical analysis enhances the domain of applied mathematics by providing important insights for behavioral science research. [ABSTRACT FROM AUTHOR]

The paper focuses on establishing sufficient conditions for the existence of the solutions in some functional q-integral equations, particularly in Banach spaces. In this method, the technique of measures of noncompactness and Petryshyn's fixed point theorem in Banach space is used. We provide some examples of equations, which confirm that our result is applicable to a wide class of integral equations. [ABSTRACT FROM AUTHOR]

Let Ω be a nonempty closed convex subset of a real Hilbert space H . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences { ψ n } n = 1 ∞ and { ϕ n } n = 1 ∞ as follows: { ψ n + 1 = π n ψ n + (1 − π n) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , for n ∈ N , where 0 ≤ π n ≤ 1 , and π n → 0 . In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of (η , β) -enriched strictly pseudononspreading ((η , β) -ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature. [ABSTRACT FROM AUTHOR]

The concept of weak orthogonality of order p ( 1 ≤ p ≤ ∞ ) in Banach lattices is introduced in order to obtain spaces with the weak fixed point property of order p. Moreover, various connections between a number of Banach space properties to imply the weak fixed point property, such as Opial condition, weak normal structure and property (M) are investigated. In particular, it is established that for each Banach space X and a suitable Banach lattice F, a Banach lattice M ⊂ K (X , F) has the weak fixed point property of order p, if each evaluation operator ψ y ∗ on M is a p-convergent operator for y ∗ ∈ F ∗ . [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence theorems for elliptic boundary value problems within the p(x)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer’s and Schauder’s theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a Hadamard-type fractional differential equation on the interval (1 , ∞) equipped with a new class of logarithmic type integro-initial conditions. We apply the Leggett–Williams fixed point theorem and the concept of iterative positive solutions to establish the existence of solutions for the problem at hand. Our results are new and enrich the literature on Hadamard-type fractional differential equations on the infinite domain. Examples illustrating the main results are presented. [ABSTRACT FROM AUTHOR]

This article explores the existence and asymptotic stability in the p-th moment of mild solutions to a class of Hilfer fractional neutral stochastic differential equations with infinite delay in Hilbert spaces. To prove our main results, we use fractional calculus, stochastic analysis, semigroup theory, and the Krasnoselskii-Schaefer type fixed point theorem. Moreover, a set of novel sufficient conditions is derived for achieving the required result. Following that, we extend the given system to the Sobolev type and provided the existence results of the considered system. After that, we provided an example to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

The main goal of this study is to use the ψ -Caputo fractional derivative of order ϑ ∈ (0 , 1) to construct the criteria for controllability in non-instantaneous impulsive dynamical systems. To obtain the necessary and sufficient conditions for the controllability of linear fractional systems by incorporating the positiveness of the Grammian matrices. To obtain sufficient conditions for controllability criteria for nonlinear systems, we have used Schaefer’s fixed point theorem. To enhance comprehension of the theoretical findings, several numerical examples have been provided. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability for Hilfer fractional stochastic hemivariational inequalities of order 1 < μ < 2 in Hilbert spaces. Initially, we define the concept of a mild solution for our problem in terms of fractional calculus, cosine families, stochastic analysis, and generalized Clarke subdifferential. Then, the existence and approximate controllability for Hilfer fractional stochastic evolution hemivariational inequalities are formulated and proven under appropriate conditions using fixed point theorems for multivalued maps. Finally, an example is presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

Recently, wavelet analysis over the p-adic fields are widely used in physics, biology and geophysics. In this paper, p-adic wavelets are used to study various p-adic pseudo-differential equations. p-Adic analogue of wave equation and general linear second order pseudo-differential equation are solved using both Haar and non-Haar wavelets. Finally, the existence of solutions of nonlinear p-adic pseudo differential equation and p-adic analogue of Navier Stokes equation are proved using the Schauder fixed point theorem together with wavelet functions. [ABSTRACT FROM AUTHOR]

It is known that the global convergence of the method of successive approximations is obtained by means of the Banach contraction principle. In this paper, we study the global convergence of the method by means of a technique that uses auxiliary points and, as a consequence of this study, we obtain fixed-point type results on closed balls. We apply the study to nonlinear Fredholm integral equations of the second kind. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new two-step inertial method for approximating a solution to a generalized split common fixed point problem, which is a unique solution to a variational inequality problem. We establish a strong convergence theorem for the sequence generated by the algorithm. We explore various special cases related to fundamental problems, including the split feasibility problem, the split common null point problem, and the constrained convex minimization problem. To demonstrate the efficacy and performance of our proposed algorithm, we apply it to a practical scenario involving support vector machines for binary classification. The algorithm is employed on diverse datasets sourced from the UC Irvine Machine Learning Repository, serving as the training set. [ABSTRACT FROM AUTHOR]

The purpose of our paper is to present a few convergence results for the Krasnoselskii–Mann iteration and also the modified Krasnoselskii–Mann iteration used for approximating the fixed point of k-strictly pseudo contractive map in the Banach space, which can be derived from the corresponding convergence theorems in the class of non-expansive maps. Also, using the enrichment techniques for contractive type map T, i.e., the averaged operator T λ (u) = (1 - λ) u + λ T u , where λ ∈ (0 , 1 ] , we identified that pseudo-contractive and demi-contractive maps are an unsaturated class of mappings in the Banach space. Illustrative examples and numerical calculations are given to support the obtained results, which are in fact the generalizations of approximations results in the Banach space from the Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the strong b-suprametric spaces in which we prove the fixed point principles of Banach and Edelstein. Moreover, we prove a variational principle of Ekeland and deduce a Caristi fixed point theorem. Furthermore, we introduce the strong b-supranormed linear spaces in which we establish the fixed point principles of Brouwer and Schauder. As applications, we study the existence of solutions to an integral equation and to a third-order boundary value problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the sufficient conditions for the existence of fixed points of Perov type T-contractive mappings in the setup of complete cone b-metric space associated with generalized c-distance. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we modify the Thakur three-step iterative algorithm and provide sufficient conditions and a useful lemma to ensure the convergence of a best proximity pair for a noncyclic relatively Suzuki’s nonexpansive mapping in uniformly convex Banach spaces. Additionally, we present an example to illustrate the results, accompanied by a numerical simulation for the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a (p , q) -fractional nonlinear difference equation of different orders is considered and discussed. With the help of (p , q) -calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241:46–55, 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences (x n) generated by the discrete viscosity approximation (DDS) [ABSTRACT FROM AUTHOR]

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker L p -Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities. [ABSTRACT FROM AUTHOR]

The note is concerned with the functional equation λ 1 H 1 (f (x)) + λ 2 H 2 (f 2 (x)) + ⋯ + λ n H n (f n (x)) = F (x) , which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of variational inequality problems the constraint set of which is the set of common solutions of a finite family of operator equations, involving hemi-continuous accretive operators on a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. We present a sequential regularization method of Lavrentiev type and an iterative regularization one in combination with an inertial term to speed up convergence. The strong convergence of the methods is proved without the co-coercivity imposed on any operator in the family. An application of our results to solving the split common fixed point problem with pseudocontractive and nonexpansive operators is given with computational experiments for illustration. [ABSTRACT FROM AUTHOR]

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution. [ABSTRACT FROM AUTHOR]

In this article, we study the controllability of dynamical systems with (k , ψ) -Hilfer fractional derivative. The Gramian matrix is used to get a necessary and sufficient controllability requirement for linear systems, which are characterized by the Mittag–Leffler (M–L) functions, while the fixed point approach is used to arrive at adequate controllability criteria for nonlinear systems. The novel feature of this study is to inquire into the controllability notion by using (k , ψ) -Hilfer fractional derivative, the most generalized variant of the Hilfer derivative. The advantage of this type of fractional derivative is that it recovers the majority of earlier studies on fractional differential equations (FDEs). Finally, we provide numerical examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In the literature, several methods have been proposed for solving the split variational inequality problems and most of these methods require that the underlying operators are co-coercive while some of them require that the problem is transformed into a product space. These restrictive conditions affect the feasibility of these existing methods. In order to overcome these setbacks, we propose two new inertial projection and contraction methods for solving the split variational inequality problem in real Hilbert spaces without the co-coercive condition and without the product space formulation, which does not fully exploit the attractive splitting structure of the split variational inequality problem. The sequences generated by these methods converge strongly to the solution of the split varitional inequality problems in real Hilbert spaces under the assumptions that the operators are pseudomonotone, Lipschitz continuous and without the sequentially weakly continuity condition. Furthermore, we present several numerical experiments for the proposed methods and compare their performance with other related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point results for a pair of relatively nonexpansive mappings using Thakur iteration in uniformly convex W-hyperbolic spaces. Our results extend some of the existing results in the literature. We illustrate the results with an example. Also, we compare the convergence of two different types of Thakur iterations for a pair of relatively nonexpansive mappings and establish that one converges faster than the other. We also establish the existence of a best proximity pair using the main theorem. [ABSTRACT FROM AUTHOR]

This article is concerned with an appropriate definition of the fractional derivative and integral based on the concept of new generalized Caputo-type fractional derivative introduced recently by Odibat and Baleanu (Appl Numer Math 156:94-105, 2020), hence the abbreviated name OBC, in fuzzy environment. Using this concept, we provide some results on the existence and uniqueness of solutions for a class of fractional fuzzy initial value problems of order m - 1 < κ < m under the newly introduced OBC derivative. Further, we present some applications of the established derivative in the real world with the help of numerical examples based on Euler's method of integration. Along with a theoretical example, we study an example depicting Allee's effect and another model of the growth of technological innovations. [ABSTRACT FROM AUTHOR]

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra [Formula: see text]. Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.