Search

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces. [ABSTRACT FROM AUTHOR]

We introduce and study the concept of φ-metric modular space and, then define φ-α-Meir-Keeler contraction on it and explore its fixed point. Further, we define the Hausdorff distance between two non-empty compact subsets of the considered space. Some topological properties of φ-metric modular space are also explored. Additionally, we prove the existence of the attractor (fractal) of the IFS consisting of φ-α-Meir-Keeler contractions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two types of weak enriched contractions, namely weak enriched F -contraction, weak enriched F ′ -contraction, and k-fold averaged mapping based on Kirk's iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated. [ABSTRACT FROM AUTHOR]

We introduce a large class of contractive mappings, called Suzuki–Berinde type contraction. We show that any Suzuki–Berinde type contraction has a fixed point and characterizes the completeness of the underlying normed space. A fixed point theorem for multivalued mappings is also obtained. These results unify, generalize and complement various known comparable results in the literature. [ABSTRACT FROM AUTHOR]

Let A and B be two nonempty subsets of a normed space X, and let T : A ∪ B → A ∪ B be a cyclic (resp., noncyclic) mapping. The objective of this paper is to establish weak conditions on T that ensure its relative nonexpansiveness. The idea is to recover the results mentioned in two papers by Matkowski (Banach J. Math. Anal. 2:237–244, 2007; J. Fixed Point Theory Appl. 24:70, 2022), by replacing the nonexpansive mapping f : C → C with a cyclic (resp., noncyclic) relatively nonexpansive mapping to obtain the best proximity pair. Additionally, we provide an application to a functional equation. [ABSTRACT FROM AUTHOR]

In this paper, using a new shrinking projection method, we study the strong convergence theorem for finding a common point of the set of common fixed points of a finite family of Bregman demigeneralized mappings, common zero points of a finite family of Bregman inverse strongly monotone mappings and zero points of maximal monotone mapping in reflexive Banach space. Using this result, we get a new strong convergence theorem in Banach space. Our result extends and improves important recent results presented by many authors. [ABSTRACT FROM AUTHOR]

We present enriched Kannan and enriched Bianchini mappings in the framework of unique geodesic spaces. For such mappings, we establish the existence and uniqueness of a fixed point in the setting of CAT(0) spaces and show that an appropriate Krasnoselskij scheme converges with certain rate to the fixed point. We proved some inclusion relations between enriched Kannan mapping and some applicable mappings such as strongly demicontractive mapping. Finally, we give an example in a nonlinear CAT(0) space and perform numerical experiments to support the theoretical results. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

In this article, we propose an iterative method, called the GA iterative method, to approximate the fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. Further, we obtain some convergence results of the new iterative method. Also, we provide a nontrivial example of a generalized α-nonexpansive mapping and with the example, we carry out a numeral experiment to show that our new iterative algorithm is more efficient than some existing iterative methods. Again, we present an interesting strategy based on the GA iterative method to solve nonlinear third-order boundary value problems (BVPs). For this, we derive a sequence named the GA–Green iterative method and show that the sequence converges strongly to the fixed point of an integral operator. Finally, the approximation of the solution for a nonlinear integrodifferential equation via our new iterative method is considered. We present some illustrative examples to validate our main results in the application sections of this article. Our results are a generalization and an extension of several prominent results of many well-known authors in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the problem of finding fixed points of nonexpansive mappings on Hadamard manifolds. To solving this kind of problem, a modified Riemannian Halpern algorithm, which is a natural generalization of a modified Halpern algorithm in Euclidean space is proposed. By giving some mild assumptions and necessary lemmas, the global convergence of the proposed algorithm is established. Finally, when the problem is solved in the framework of Hadamard manifolds, the numerical experiments show the effectiveness of the proposed algorithm, especially in computational time and number of iterations. [ABSTRACT FROM AUTHOR]

In this paper, an inertial S-iteration iterative process for approximating a common fixed point of a finite family of quasi-Bregman nonexpansive mappings is introduced and studied in a reflexive Banach space. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, extend, and generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]

Let (X, D) be a complete metric space and H, F be single and multivalued mappings on X respectively. In this paper, we investigate two new types of fixed point theorems from a set of single and multivalued mappings in a complete metric spaces endowed with a graph G such that the set of vertices of G, V(G) = X and the set of edges of G, E(G) ⊆ X × X. [ABSTRACT FROM AUTHOR]

The purpose of this study is to demonstrate results on fixed point theory in quasi-partial b-metric space recognizing a new type of mapping, which is a blend of F-contraction and Kannan contraction, and to establish the fixed point results in F-expanding type mappings. Additionally, the obtained results are the application of the contractive mappings to functional equations. Furthermore, Mathematica software is used to demonstrate the 3D shapes of the examples discussed here. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of generalized scale of Banach spaces. The spaces of a generalized scale of Banach spaces define a projective limit space denoted by X Δ . In that space we will define a class of operators based on the degree of nondensifiability. Such operators will be called DND-condensing operators. It is shown that these operators have a fixed point in X Δ . In this manner we obtain another method for the theory of fixed point that joins the already existing one of measures of noncompactness. Some applications of the DND-condensing operator method will be illustrated. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding the resolvent of the sum of two maximal monotone operators. Such a problem arises frequently in practice, but it seems that computation of the solution of the problem is not necessarily easy. It is assumed that both the resolvents of two maximal monotone operators can be easily computed. This enables us to consider the case in which a solution to the problem cannot be computed easily. This paper introduces a new mapping, which satisfies the nonexpansivity property, from the individual resolvents of two maximal monotone operators and investigates some of its properties. In particular, we show that the mapping has a fixed point if and only if the problem has a solution. Then, using this mapping, we propose a splitting method for solving the problem in a real Hilbert space. In particular, we show that the sequences generated by the method converge strongly to the solution to the problem under certain assumptions. Convergence rate analysis of the methods is also provided to illustrate the method's efficiency. Finally, we apply the results to a class of optimization problems. [ABSTRACT FROM AUTHOR]

In this article, we present an inertial subgradient extragradient-type method that uses a non-monotone step size rule to find a numerical solution to equilibrium problems in real Hilbert spaces. The presented iterative scheme is based on an extragradient subgradient method and an inertial-type scheme. In fact, the proposed iterative scheme is effective in terms of performance, and the key advantage derives directly from the use of the variable step size rule, which is revised by each iteration on the basis of the Lipschitz-type constants as well as certain prior iterations. We obtain a weak convergence theorem for a new method by using mild conditions on a bifunction. Applications of the main results are given to solve various nonlinear problems. Several numerical findings are given in order to illustrate the numerical behaviour of the proposed method and to compare it to others. [ABSTRACT FROM AUTHOR]

In this chapter we present a brief survey of theory of measures of noncompactness and discuss some fixed point theorems of Darbo's type. We describe some applications in the solvability of infinite systems of differential equations in classical sequence spaces. [ABSTRACT FROM AUTHOR]

We introduce a new subgradient extragradient algorithm utilizing the concept of the set of solutions of the split modified system of variational inequality problems (SMSVIP). Our main theorem is weak convergence theorem for such an algorithm for approximating the fixed point problem in a real Hilbert space. We also apply these results to approximate the split minimization problem. In the last section, we provide an example to illustrate the potential of our main theorem. [ABSTRACT FROM AUTHOR]

In this paper, we define a quadrilateral inequality, Browder type and implicit iterative method in a CAT(1) space. Furthermore, we prove a convergence theorem of the implicit iterative method in the same space used the quadrilateral inequality. [ABSTRACT FROM AUTHOR]

In this paper, we establish an approximate fixed point theorem for Γ-τ-sequentially continuous countably Φ τ -condensing mappings. As application, we obtain an approximate fixed point result for demicontinuous mappings and use this to prove new results in asymptotic fixed point theory. Moreover, we obtain new fixed point results for some countably Φ τ -condensing mappings and multivalued mappings and fixed point results of Krasnoselskii-Daher type for the sum of two τ-sequentially continuous mappings in non-separable Hausdorff topological vector spaces. We also present a multivalued version of an approximation result of Ky Fan for τ-Γ-s.l.sc multivalued mappings. Apart from that, we show the applicability of our results to the theory of Volterra integral equations in Banach spaces and we prove the existence of limiting-weak solutions for differential equations in Banach spaces not necessarily reflexive. Our results extend the results of Banaś and Ben Amar, Barroso and Seda. [ABSTRACT FROM AUTHOR]

We study the split common fixed point problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose a new algorithm and establish a strong convergence theorem for it. Moreover, using our method, we also remove the assumptions imposed on the norms of the transfer operators. [ABSTRACT FROM AUTHOR]

The objective of this article is to study a three-step iteration process in the framework of Banach spaces and to obtain convergence results for Suzuki generalized nonexpansive mappings. We also provide numerical examples that support our main results and illustrate the convergence behavior of the proposed process. Further, we present a data-dependence result that is also supported by a nontrivial numerical example. Finally, we discuss the solution of a nonlinear fractional differential equation by utilizing our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a distributed optimization problem of minimizing sum of convex functions over the intersection of fixed-point constraints. We propose a distributed method for solving the problem. We prove the convergence of the generated sequence to the solution of the problem under certain assumption. We further discuss the convergence rate with an appropriate positive stepsize. A numerical experiment is given to show the effectiveness of the obtained theoretical result. [ABSTRACT FROM AUTHOR]

We present several fixed point and convergence results for nonexpansive set-valued mappings which map a closed subset of a complete metric space into the space. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

In this paper, △ -convergence and strong convergence theorems for asymptotically quasi-nonexpansive sequences in Hadamard spaces are derived. The results extend and improve recent results in the literature and some of them seem new even in Hilbert spaces. Applications to convex minimization and common fixed point problems are discussed. [ABSTRACT FROM AUTHOR]

We discuss fixed points of Chatterjea mappings and Kannan mappings on Vector-b-metric spaces endowed with a graph. This work should be seen as a generalization of Kannan mappings and Chatterjea mappings on metric spaces with graph. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki's condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated. [ABSTRACT FROM AUTHOR]

Every non-convex pair (C , D) may not have proximal normal structure even in a Hilbert space. In this article, we use cyclic relatively nonexpansive maps with respect to orbits to show the presence of best proximity points in C ∪ D , where C ∪ D is a cyclic T-regular set and (C , D) is a non-empty, non-convex proximal pair in a real Hilbert space. Moreover, we show the presence of best proximity points and fixed points for non-cyclic relatively nonexpansive maps with respect to orbits defined on C ∪ D , where C and D are T-regular sets in a uniformly convex Banach space satisfying T (C) ⊆ C , T (D) ⊆ D wherein the convergence of Kranoselskii's iteration process is also discussed. [ABSTRACT FROM AUTHOR]

In this manuscript we introduce the notion of (α , β , ψ , ϕ) -interpolative contraction that unifies and generalizes significant concepts: Proinov type contractions, interpolative contractions, and ample spectrum contraction. We investigate the necessary and sufficient conditions to guarantee existence and uniqueness of the fixed point of such mappings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a new viscosity iterative algorithm for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping and the set of solutions of a new variational inequality problem involving inverse-strongly monotone operators in Hilbert spaces. We prove some strong convergence theorems under some suitable assumptions imposed on the parameters by using a modified extragradient method. The results obtained in this paper may be an improvement of many recent ones in this fields. [ABSTRACT FROM AUTHOR]

A convergence rate of Mann iteration has been studied by many mathematicians. Recently, its rate was improved in a Hilbert space. In this paper, we consider a convergence rate in a geodesic space and generalize the known result. [ABSTRACT FROM AUTHOR]

The variable Lebesgue spaces are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p (⋅). Beside their intrinsic interest, they are also very important for their applications to partial differential equations and variational integrals with non-standard growth conditions. In this paper, we study some properties related to the existence of a fixed point for several classes of mappings defined on these spaces, considering both the modular and its induced norm. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate a descent-type iterative algorithm for finding a common element of fixed-point sets of a finite family of nonexpansive mappings and zero-point sets of pseudomonotone mappings in Hilbert spaces. With suitable assumptions, we derive necessary and sufficient conditions for the strong convergence of the algorithm. Numerical examples and applications to signal processing problems are provided to support the main results. [ABSTRACT FROM AUTHOR]

In this article, we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form X s + A ∗ X - t A + B ∗ X - p B = Q , where s , t , p ≥ 1 , A, B are nonsingular matrices and Q is a Hermitian positive definite matrix. We derive some iterations to compute the solutions followed by some examples. In this context, we also discuss about the maximal and the minimal Hermitian positive definite solution of this particular nonlinear matrix equation. [ABSTRACT FROM AUTHOR]

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is E = l p , 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0 < κ ≤ 1 2 , then E is necessarily l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study three fixed point problems. The first one is a fixed point problem with a set-valued mapping, while the second and third problems are convex feasibility problems with infinitely many constraints solved by the subgradient projection algorithm. We show that an approximate solution is reached after a finite number of iterations under the presence of computational errors. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in 𝑝-uniformly convex metric spaces, and prove both Δ-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete 𝑝-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the class of monotone ρ-nonexpansive semigroups and give existence and convergence results for common fixed points. First, we prove that the set of common fixed points is nonempty in uniformly convex modular spaces and modular spaces. Then we introduce an iteration algorithm to approximate a common fixed point for the same class of semigroups. [ABSTRACT FROM AUTHOR]

For a certain class of mean nonexpansive self-mappings of complete metric spaces, we show that all the iterates of each mapping in this class converge, uniformly on bounded subsets, to its unique fixed point. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mapping given by a finite family of multivalued monotone mappings in a CAT(0) space. We further propose a modified Halpern-type algorithm for the mapping and prove a strong convergence theorem for approximating a common solution of finite family of monotone inclusion problems in a complete CAT(0) space. We also applied our results to solve a finite family of minimization problems in a complete CAT(0) space. Some non-trivial examples of monotone mappings in complete CAT(0) space setting were also studied and a numerical example of our algorithm to further show its applicability is also presented. Our numerical experiment shows that our algorithm converges faster than that proposed in [42] by Takahashi and Shimoji, Mathematical and Computer Modelling32 (2000), 1463–1471. [ABSTRACT FROM AUTHOR]

In this paper, using the hybrid method defined by Nakajo and Takahashi, we first obtain a strong convergence theorem for two noncommutative generic skew generalized nonspreading mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove another strong convergence theorem for the mappings in a Banach space. Using these results, we get new strong convergence theorems by the hybrid method and the shrinking projection method in a Hilbert space and a Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of asymptotically demicontractive multivalued mappings and establish a strong convergence theorem of the modified Mann iteration to a common fixed point of a finite family of asymptotically demicontractive multivalued mappings in a complete CAT(0) space. We also give a numerical example of our iterative method to show its applicability. [ABSTRACT FROM AUTHOR]

Recently, Chang et al. (2015) constructed an algorithm that converges weakly to the solution of the split equality fixed point problem for quasi-pseudo-contractive mappings under some suitable conditions. They also showed that strong convergence is obtained in the case when the quasi-pseudo-contractive mappings are semi-compact. In this article, we construct an algorithm for quasi-pseudo-contractive mappings that always converge strongly to some solution of the split equality fixed point problem under mild conditions. We mention that we do not require the quasi-pseudo-contractive mappings to be semi-compact to obtain strong convergence. The algorithm does not require any prior knowledge of operator norms. The result of this article provides a unified framework for this type of problems. [ABSTRACT FROM AUTHOR]

Let B be a uniformly convex and uniformly smooth real Banach space with dual space B ∗ . Let F : B → B ∗ , K : B ∗ → B be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u + K F u = 0 . This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce iterative regularization methods for solving the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a bounded linear mapping will be closest to another family of closed convex subsets in the image space. We consider the cases, when the families are either finite or infinite. We also give two numerical examples for illustrating our main method. [ABSTRACT FROM AUTHOR]

In this paper, we establish strong and Δ-convergence theorems of the modified hybrid-CR three steps iteration with perturbations for total asymptotically non-expansive mapping in CAT (0) spaces. Our results improve and extend the corresponding results from the current literature. We also provide three examples to illustrate the convergence behaviour of the proposed algorithm and numerically compare the convergence of the proposed iteration scheme with the existing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study some fixed point results for ϕ -Kannan contractions in metric spaces with w-distance. Among other things, we have recovered Suzuki's (Yokohama Math 44:61–72, 1997) results for p-Kannan contractions. Some examples and consequences are also presented to illustrate our results. [ABSTRACT FROM AUTHOR]

Let C be a closed affine subset of a real Hilbert space H and T : C → C be a nonexpansive mapping. In this paper, for any fixed u ∈ C, a general Halpern iteration process: x 0 ∈ C , x n + 1 = t n u + (1 − t n) T x n , n ≥ 0 , is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, ℝ . The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities. [ABSTRACT FROM AUTHOR]