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In this paper, we are concerned with the study of the existence and uniqueness of fixed points for the class of functions $ f: C\to C $ satisfying the inequality$ \ell\left(\alpha f(t)+(1-\alpha)f(s)\right)\leq \sigma \ell(\alpha t+(1-\alpha)s) $for every $ t, s\in C $ with $ f(t)\neq f(s) $, where $ C $ is a closed subset of $ [0, \infty) $, $ \alpha, \sigma\in (0, 1) $ are constants, and $ \ell: [0, \infty)\to [0, \infty) $ is a function satisfying the condition $ \inf_{t > 0} \frac{\ell(t)}{t^\rho} > 0 $ for some constant $ \rho > 0 $. Namely, under a weak continuity condition imposed on $ f $, we show that $ f $ possesses a unique fixed point, and for every $ t_0\in C $, the Picard sequence defined by $ t_{n+1} = f(t_n) $, $ n\geq 0 $, converges to this fixed point. Next, we study the special cases when $ C $ is a closed interval and $ \ell $ is a convex or concave function. Namely, making use of the Hermite-Hadamard inequalities, we obtain several new fixed point theorems. To the best of our knowledge, the considered class of functions was never previously investigated in the literature.

Abstract In this article, we introduce the structure of a new extended cone b-metric-like space over a Banach algebra. In this generalized space, we define the notion of generalized Reich-type mappings and prove a fixed point result. We also prove a fixed point result using α ∗ − ψ $\alpha^{\ast}-\psi$ multivalued contraction and provide some of its consequences. Finally, we furnish with applications to establish the validity of our results.

Abstract The survey of the available literature shows that a lot of important invariant point problems of Banach and Heilpern types have been examined in both metric and quasimetric spaces. However, a handful of the existing results employed the recent approaches of interpolative contractions. Therefore, based on the new idea of interpolation techniques in fixed point theory, this article studies new notions of L-fuzzy contractions and investigates conditions for the existence of L-fuzzy fixed points for such mappings. On the fact that fixed points of point-to-point mappings satisfying interpolative-type contraction are not always unique, whence making the concepts more fitted for invariant point results of crisp set-valued maps, new multi-valued analogues of the key findings put forward in this work are derived. Comparative illustrations, which indicate the preeminence of the results presented herein, are constructed. From application viewpoint, one of the theorems so obtained is employed to introduce new solvability conditions of Fredholm-type integral inclusions.

We consider the class of functions $ u\in C^2((0, \infty)) $ satisfying second-order differential inequalities in the form $ u''(x)+\frac{k}{x^2}u(x)\geq 0 $ for all $ x > 0 $. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ($ k=\frac{1}{4} $ and $ 0 < k < \frac{1}{4} $). We next extend our obtained results to the two-dimensional case. In the limit case $ k\rightarrow 0^+ $ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.

This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixed-point method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes.

This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.

We unified several kinds of convexity by introducing the class $ \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ of $ (\zeta, w) $-admissible functions $ F: [0, 1]\times I\times I\to \mathbb{R} $. Namely, we proved that most types of convexity from the literature generate functions $ F\in \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ for some $ \zeta\in C([0, 1]) $ and $ w\in C^1(I) $ with $ w(I)\subset I $ and $ w' > 0 $. We also studied some properties of $ (\zeta, w) $-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.

We introduce the class of $ \psi $-convex functions $ f:[0, \infty)\to \mathbb{R} $, where $ \psi\in C([0, 1]) $ satisfies $ \psi\geq 0 $ and $ \psi(0)\neq \psi(1) $. This class includes several types of convex functions introduced in previous works. We first study some properties of such functions. Next, we establish a double Hermite-Hadamard-type inequality involving $ \psi $-convex functions and a Simpson-type inequality for functions $ f\in C^1([0, \infty)) $ such that $ |f'| $ is $ \psi $-convex. Our obtained results are new and recover several existing results from the literature.

This article delves deeply into some mathematical basic theorems and their diverse applications in a variety of domains. The major issue of interest is the Banach Fixed Point Theorem (BFPT), which states the existence of a unique fixed point in fractional metric spaces. The significance of this theorem stems from its utility in a variety of mathematical situations for approximating solutions and resolving iterative problems. On this foundational basis, the study expands by introducing the concept of fractional geometric contraction mappings, which provide a new perspective on how convergence develops in fractional metric spaces.

The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended b-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended b-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature.

Abstract The purpose of this study is to establish fixed-point results for new interpolative contraction mappings in the setting of Busemann space involving a convex hull. To illustrate our findings, we also offer helpful and compelling examples. Finally, the theoretical results are applied to study the existence of solutions to fractional Navier–Stokes and fractional-functional differential equations as applications.

In this work, our aim is to ensure the existence of mild solutions and study the controllability of fractional evolution inclusions of Clarke's subdifferential type of order q∈(1,2) with nonlocal conditions in the setting of Hilbert spaces. Using fixed point techniques, fractional calculus, multivalued maps, cosine and sine function operators, we discuss the existence of mild solutions for the considered system. Moreover, under suitable conditions, we investigate the controllability of the proposed system. Finally, we present an illustrated example.

Abstract In this article, we obtain some fixed-point results involving α-admissibility for multivalued F-contractions in the framework of partial b $\mathfrak{b}$ -metric spaces. Appropriate illustrations are provided to support the main results. Finally, an application is developed by demonstrating the existence of a solution to an integral equation.

Abstract The logical progression from the constant order differential equations is the field of variable-order differential equations. Such equations can frequently give a more succinct description of problems in the real world. In light of this, we therefore take into account a class of coupled boundary value problems under variable-order differentiation. By utilizing the fixed-point techniques of Banach and Schauder, we investigate the existence and uniqueness of solutions to the proposed problem. Also, sufficient results are documented for the necessary needs. Furthermore, some stability results based on the ideas of Ulam, Hyers, and Rassias are elaborated upon. Ultimately, appropriate examples and in-depth analysis are presented to support our results.

In this manuscript, using the idea of $ b $-simulation functions, certain common fixed point results via generalized $ \Im_b $-contractions are investigated in the context of b-metric spaces. These findings generalize and supplement various numbers of established results from the existing literature. Examples and applications are also provided for the authenticity of the presented work. Besides, we ensure the existence of a unique common solution for systems of Volterra-Hammerstein integral and Urysohn integral equations, respectively, by applying the established results.

In this manuscript, we give strong convergence results for a fixed point of a subfamily of an evolution family on a convex and closed subset $ \mathcal{D} $ of a Banach space $ \mathsf{B} $. An example is also provided which shows the applications of evolution families and our main results. At the end, an open problem is given.

In this article, we introduce the concept of generalized rational type $ F $ -contractions on relation theoretic m-metric spaces (denoted as $ F_{R}^{m} $-contractions, where $ R $ is a binary relation) and some related fixed point theorems are provided. Then, we achieve some fixed point results for cyclic rational type $ F_{R}^{m} $- generalized contraction mappings. Moreover, we state some illustrative numerically examples to show our results are true and meaningful. As an application, we discuss a positive definite solution of a nonlinear matrix equation of the form $ \Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i} $.

In this article, the notion of Hausdorff fuzzy $ b $-metric space is studied. Some fixed point results for multivalued mappings using Geraghty type contractions in $ G $-complete fuzzy $ b $-metric spaces are established. To strengthen the results, an illustrative example is furnished. A fuzzy integral inclusion is constructed as an application of fixed point result which shows the validity of the proved results. The presented outcomes are the generalization of the existing results in literature.

In this study, we utilize the direct method (Hyers approach) to examine the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $ \Delta _{2} $-condition. We also use the direct approach to discuss the Ulam stability in $ 2 $-Banach spaces. Ultimately, we ensure that stability of above equations does not hold in a particular scenario by utilizing appropriate counter-examples.

In this manuscript, fixed point results without continuity via triangular notion on fuzzy bipolar metric spaces are established. The paper includes tangible examples which display the motivation for such investigations as those presented here. We solve an integral equation in this setting. The present work is a generalization of some published works.

In this paper, we present some generalized multi-valued contraction results on cone metric spaces. We use some maximum and sum types of contractions for a pair of multi-valued mappings to prove some common fixed point theorems on cone metric spaces without the condition of normality. We present an illustrative example for multi-valued contraction mappings to support our work. Moreover, we present a supportive application of nonlinear integral equations to validate our work. This new theory, can be modified in different directions for multi-valued mappings to prove fixed point, common fixed point and coincidence point results in the context of different types of metric spaces with the application of different types of integral equations.

Abstract In this manuscript, a novel general class of contractions, called Jaggi–Suzuki-type hybrid (G-α-ϕ)-contraction, is introduced and some fixed point theorems that cannot be deduced from their akin in metric spaces are proved. The dominance of this family of contractions is that its contractive inequality can be specialized in various manners, depending on multiple parameters. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. Consequently, a number of corollaries that reduce our result to some prominent results in the literature are highlighted and analyzed. Additionally, we examine Ulam-type stability and well-posedness for the new contraction proposed herein. Finally, one of our obtained corollaries is applied to set up unprecedented existence conditions for solution to a class of integral equations. For future aspects of our results, an open problem is noted concerning the discretized population balance model, whose solution may be analyzed using the techniques established herein.

In this paper, a general framework for the fractional boundary value problems is presented. The problem is created by Riemann-Liouville type two-term fractional differential equations with a fractional bi-order setup. Moreover, the boundary conditions of the suggested system are considered as mixed Riemann-Liouville integro-derivative conditions with four different orders, which it cover a variety of specific instances previously researched. Further, the provided problem's Hyers-Ulam stability and the possibility of a fixed-point approach solution are both investigated. Finally, to support our theoretical findings, an example is developed.

Using Atangana-Baleanu (AB) fractional integral operators, we first establish some fractional Hermite-Hadamard-Mercer inclusions for interval-valued functions in this study. In this, some fresh developments of the Hermite-Hadamard inequality for fractional integral operators are presented. A few instances are also given to support our conclusions. The resulting results could provide new insight into a variety of integral inequalities for fuzzy interval-valued functions, fractional interval-valued functions, and the optimization issues they raise. Finally, matrices-related applications are also shown.

The goal of this paper is to obtain some tripled coincidence point results for generalized contraction mappings in the setting of JS-metric spaces endowed with a partial order. Furthermore, illustrative examples to support the theoretical results and the application are obtained. Finally, some theoretical results are applied to discuss the existence of a solution for a system of non-homogeneous and homogeneous integral equations as applications.

The goal of this paper is to present a new class of contraction mappings, so-called $ \eta _{\theta }^{\ell } $-contractions. Also, in the context of partially ordered metric spaces, some coupled fixed-point results for $ \eta _{\theta }^{\ell } $-contraction mappings are introduced. Furthermore, to support our results, two examples are provided. Finally, the theoretical results are applied to obtain the existence of solutions to coupled fractional differential equations with a Mittag-Leffler kernel.

In this work, we initiate the notion of a fuzzy cyclic (α,β)-admissibility to establish some fixed point results for contraction mappings involving a generalized simulation function in the class of fuzzy b-metric spaces. We give some illustrative examples to validate the new concepts and obtained results. At the end, we present an application on a Fredholm integral equation.

The purpose of this study is to give some findings on the existence, uniqueness, and Hyers-Ulam stability of the solution of an implicit coupled system of impulsive fractional differential equations possessing a fractional derivative of the Hadamard type. The existence and uniqueness findings are obtained using a fixed point theorem of the type of Kransnoselskii. In keeping with this, many forms of Hyers-Ulam stability are examined. Ultimately, to support main results, an example is provided.

The aim of this article is to obtain common fixed point results on complex-valued extended b-metric spaces for rational contractions involving control functions of two variables. Our theorems generalize some famous results in the literature. We supply an example to show the originality of our main result. As an application, we develop common fixed point results for rational contractions involving control functions of one variable in the class of complex-valued extended b-metric spaces.

This article investigates the existence and uniqueness (EU) of positive solutions to the tripled system of multi-point boundary value problems (M-PBVPs) for fractional order differential equations (FODEs). The topological degree theory technique is employed to derive sufficient requirements for the (EU) of positive solutions to the proposed system. To justify the efficiency and validity of our study, an illustrative example is considered.

Abstract The objective of this article is to incorporate the concept of the Ostrowski inequality with the Atangana–Baleanu fractional integral operator. A novel integral identity for twice-differentiable functions is established after a rigorous investigation of several basic definitions and existing ideas related to inequalities and fractional calculus. Following that, numerous Ostrowski-type inequalities are provided based on this identity, which uses Mittag–Leffler as its kernel structure. Some specific applications, such as q-digamma functions and modified Bessel functions, are also investigated. Choosing s = 1 $s=1$ , we also analyze new results for convex functions as special cases. Our findings corroborate some well-documented inequalities.

In this paper, a novel and more general type of sequence of non-linear multivalued mappings as well as the corresponding contractions on a metric space equipped with a graph is initiated. Fixed point results of a single-valued mapping and the new sequence of multivalued mappings are examined under suitable conditions. A non-trivial comparative illustration is provided to support the assumptions of our main theorem. A few important results in ϵ-chainable metric space and cyclic contractions are deduced as some consequences of the concepts obtained herein. As a result of our findings, new criteria for solving a broader form of Fredholm integral equation are established. An open problem concerning discretized population balance model whose solution may be investigated using any of the ideas proposed in this note is highlighted as a future assignment.

Abstract The objective of the present research is to establish and prove some new common fuzzy fixed-point theorems for fuzzy set-valued mappings involving Θ-contractions in a complete metric space. For this purpose, a novel integral-type contraction condition is applied to obtain these results. In this way, several useful and classical results have been generalized. Moreover, a concrete example is created to furnish our results. An application to stochastic Volterra integral equations has been given to enhance the validity of our results.

In this paper, we introduce $ \mathcal{J}_{s; \Omega} $-families of generalized pseudo-$ b $-distances in $ b $-gauge spaces $ (U, {Q}_{s; \Omega}) $. Moreover, by using these $ \mathcal{J}_{s; \Omega} $-families on $ U $, we define the $ \mathcal{J}_{s; \Omega} $-sequential completeness and construct an $ F $-type contraction $ T:U\rightarrow U $. Furthermore, we develop novel periodic and fixed point results for these mappings in the setting of $ b $-gauge spaces using $ \mathcal{J}_{s; \Omega} $-families on $ U $, which generalize and improve some of the results in the corresponding literature. The validity and importance of our theorems are shown through an application via an existence solution of an integral equation.

By combining the concept of orthogonality and the Geraghty type contraction, we give some fixed point results in the class of O-metric spaces. Our obtained results extend the existing results in the literature. We also resolve an ordinary type differential equation.

The purpose of this manuscript is to present some fixed point results for a Λ-Ćirić mapping in the setting of non-triangular metric spaces. Also, two numerical examples are given to support the theoretical study. Finally, under suitable conditions, the existence and uniqueness of a solution to a general Fredholm integral equation, a Riemann-Liouville fractional differential equation and a Caputo non-linear fractional differential equation are discussed as applications.

This manuscript was built to generalize Ekeland variational principle for mixed monotone functions in the setting of partially ordered complete metric spaces. The results obtained are applied to give different proofs for tripled fixed points of mixed monotone mappings in the mentioned space by using a variational technique. The results presented in our manuscripts generalize and expand many of the findings presented in the earlier period. For the sobriety and enhancement of our paper, two examples are given and the existence and uniqueness of the solution to a periodic boundary value problem are studied as applications.

The article presents a systematic investigation of an extension of the developments concerning $ F $-contraction mappings which were proposed in 2012 by Wardowski. We develop the notion of $ F $-contractions to the case of non-linear ($ F $, $ F_{H} $)-dynamic-iterative scheme for Branciari Ćirić type-contractions and prove multi-valued fixed point results in controlled-metric spaces. An approximation of the dynamic-iterative scheme instead of the conventional Picard sequence is determined. The paper also includes a tangible example and a graphical interpretation that displays the motivation for such investigations. The work is illustrated by providing an application of the proposed non-linear ($ F $, $ F_{H} $)-dynamic-iterative scheme to the Liouville-Caputo fractional derivatives and fractional differential equations.

In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). Besides, the proposed GFPDEs represent a great generalization of a significant type of FPDEs and their applications, which contain many previous reports as a special case. Moreover, the proposed scheme uses shifted Chebyshev sixth-kind (SCSK) polynomials with spectral collocation approach. The fractional differential derivatives are expressed in terms of the Caputo's definition. Furthermore, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. In addition, the classical fourth-order Runge-Kotta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple and effective numerical method.

The goal of this manuscript is to obtain some tripled fixed point results under a new contractive condition and triangular property in the context of fuzzy cone metric spaces (F$ _{\text{CM}} $-spaces). Moreover, two examples and corollaries are given to validate our work. Ultimately, as applications, the notion of Lebesgue integral is represented by the fuzzy method to discuss the existence of fixed points. Also, the existence and uniqueness solution for a system of Volterra integral equations are studied by the theoretical results.

The purpose of this paper is to present a new concept of a Banach algebra in a fuzzy metric space (FM-space). We define an open ball, an open set and prove that every open ball in an FM-space over a Banach algebra A is an open set. We present some more topological properties and a Hausdorff metric on FM-spaces over A. Moreover, we state and prove a fuzzy Banach contraction theorem on FM-spaces over a Banach algebra A. Furthermore, we present an application of an integral equation and will prove a result dealing with the integral operators in FM-spaces over a Banach algebra.

The present work is about the existence of best proximity points for Prešić type nonself operators in b-metric spaces. In order to elaborate the results an example is presented. Moreover, some interesting formulations of Prešić fixed point results are also established. In addition a result in double controlled metric type space is also formulated.

Our aim is to prove some new fixed point theorems for a hybrid pair of multivalued α∗-dominated mappings involving a generalized Q-contraction in a complete modular-like metric space. Further results involving graphic contractions for a pair of multi-graph dominated mappings have been considered. Applying our obtained results, we resolve a system of nonlinear integral equations.

In this paper we study the positivity analysis problems for discrete fractional operators with exponential kernel, namely the discrete Caputo-Fabrizio operators. The results are applied to a discrete Caputo-Fabrizio-Caputo fractional operator of order ω of another discrete Caputo-Fabrizio-Riemann fractional operator of order β. Furthermore, the results are obtained for these operators with having the same orders. The conditions for the discrete fractional operators with respect to negative lower bound conditions are expressed in terms of a positive epsilon.

The aim of this manuscript is to present some new fixed point results in complete partially order metric spaces and to derive some extended forms of Suzuki and Banach fixed point theorems via a τ-distance by applying some new control functions. Our results are extensions of several existing fixed point theorems in the literature. To show the dominance of the established results, some examples and an application are studied.

Since the supposed Hermite-Hadamard inequality for a convex function was discussed, its expansions, refinements, and variations, which are called Hermite-Hadamard type inequalities, have been widely explored. The main objective of this article is to acquire new Hermite-Hadamard type inequalities employing the Riemann-Liouville fractional operator for functions whose third derivatives of absolute values are convex and quasi-convex in nature. Some special cases of the newly presented results are discussed as well. As applications, several estimates concerning Bessel functions and special means of real numbers are illustrated.

In this article, the concept of a Hausdorff fuzzy b-metric space is introduced. The new notion is used to establish some fixed point results for multivalued mappings in G-complete fuzzy b-metric spaces satisfying a suitable requirement of contractiveness. An illustrative example is formulated to support the results. Eventually, an application for the existence of a solution for an integral inclusion is established which involves showing the materiality of the obtained results. These results are more general and some theorems proved by of Shehzad et al. are their special cases.

Over time, hybrid fixed point results have been examined merely in the framework of classical mathematics. This one way research has clearly dropped-off a great amount of important results, considering the fact that a fuzzy set is a natural enhancement of a crisp set. In order to entrench hybrid fixed notions in fuzzy mathematics, this paper focuses on introducing a new idea under the name intuitionistic fuzzy p-hybrid contractions in the realm of -metric spaces. Sufficient conditions for the existence of common intuitionistic fuzzy fixed points for such maps are established. In the instance where our presented results are slimmed down to their equivalent nonfuzzy counterparts, the concept investigated herein unifies and generalizes a significant number of well-known fixed point theorems in the setting of both single-valued and multivalued mappings in the corresponding literature. A handful of these special cases are highlighted and analysed as corollaries. A nontrivial example is put together to indicate that the hypotheses of our results are valid.

Abstract The purpose of this paper is to consider some F-contraction mappings in a dualistic partial metric space and to provide sufficient related conditions for the existence of a fixed point. The obtained results are extensions of several ones existing in the literature. Moreover, we present examples and an application to support our results.

Abstract The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.