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This paper presents and analyzes a new viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive mappings and then its strong convergence theorem is established under some suitable conditions. As a consequence, we employ our proposed algorithm for solving some convex bilevel optimization problems and then apply it for solving regression of a graph of cosine function and classification of some noncommunicable diseases by using the extreme learning machine model. We perform a comparative analysis with other algorithms to demonstrate the performance of our approach. Our numerical experiments confirm that our proposed algorithm outperforms other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified general inertial Mann algorithm and prove that it generates a sequence which converges weakly to a fixed point of a nonexpansive mapping in Hilbert spaces. Moreover, by using the viscosity method, we introduce a general inertial viscosity algorithm and prove that it generates a sequence which converges strongly to a common fixed point of a countable family of nonexpansive operators. We also derive schemes for solving constrained convex optimization, monotone inclusion, and nonsmooth convex optimization problems. Finally, we apply one of our proposed algorithms to solve image restoration problem. [ABSTRACT FROM AUTHOR]

In this paper, we modified proximal point algorithm with some convex combination technique to approximate a minimizer, equilibrium point and a common fixed point of a family of nonexpansive mappings in Hilbert spaces. We establish a weak convergence theorem under some mild conditions. Moreover, we also provide a numerical example to illustrate the convergence behavior of the proposed iterative method. [ABSTRACT FROM AUTHOR]

In the present paper we study the relationships between different concepts of convex structures in metric spaces that that are related to the works of K. Menger [Menger, K. Untersuchungen über allgemeine Metrik. Math. Ann. 100 (1928), 75-163], H. Busemann [Busemann, H. The geometry of geodesics, Academic Press, 1955], I. N. Herstein; J. Milnor [Herstein, I. N.; Milnor, J. An axiomatic approach to measurable utility. Econometrica 21 (1953), 291-297], E. Michael [Michael, E. Convex structures and continuous selections. Canad. J. Math. 11 (1959), 556-575], A. Nijenhuis [Nijenhuis, A. A note on hyperconvexity in Riemannian manifolds. Canad. J. Math. 11 (1959), 576-582.], W. Takahashi; T. Shimizu [Shimizu, T.; Takahashi, W. Fixed points of multivalued mappings in certain metric spaces. Topol. Methods Nonlinear Anal. 8 (1996), no. 1, 197-203 and Takahashi, W. A convexity in metric space and nonexpansive mappings I. Kodai Math. Sem. Rep. 22 (1970), 142-149], M. Taskovi'c [Taskovi'c, M. General convex topological spaces and fixed points. Math. Moravica 1 (1997), 127-134], Yu. A. Aminov [Aminov, Yu. A. Two-Dimensional Surfaces in 3-Dimensional and 4-Dimensional Euclidean Spaces. Results and Unsolved Problems. Ukr. Math. J. 71 (2019), no. 1, 1-38.], H. V. Machado [Machado, H. V. A characterization of convex subsets of normed spaces. Kodai Math. Sem. Rep. 25 (1973), 307-320], and many other papers. Some well known examples of concrete convex structures are reexamined and the possibilities of different embeddings of metric spaces with convex structures are also studied. Corollary 6.1 states that the Bolyai-Lobachevskii plane and the Bolyai-Lobachevskii half-plane are not isometrically embedding in some strictly convex normed space. A characteristic of the invariant metric generated by a norm is presented (Proposition 4.6). [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the intermixed algorithm in p-uniformly convex metric spaces, and then we prove Δ-convergence of the proposed iterative method for finding a common element of the sets of fixed points of finite families of nonexpansive mappings in the framework of complete p-uniformly convex metric spaces. Furthermore, we apply our main theorem to prove Δ-convergence to solve the minimization problems in the framework of complete p-uniformly convex metric spaces. Finally, we give two examples in Lp spaces and numerical examples to support our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative process for solving simultaneously one-parameter nonexpansive semigroup, convex minimization and fixed point problems involving set-valued operators in real Hilbert spaces and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption. Our results improve important recent results. [ABSTRACT FROM AUTHOR]

In this paper, new geometric inequalities are established in real Banach spaces. As an application, a new iterative algorithm is proposed for approximating a solution of a split equality fixed point problem (SEFPP) for a quasi-ϕ-nonexpansive semigroup. It is proved that the sequence generated by the algorithm converges strongly to a solution of the SEFPP in p-uniformly convex and uniformly smooth real Banach spaces, p > 1. Furthermore, the theorem proved is applied to approximate a solution of a variational inequality problem. All the theorems proved are applicable, in particular, in Lp, lp and the Sobolev spaces, Wmp (Ω), for p such that 2 < p < ∞. [ABSTRACT FROM AUTHOR]

We prove that the convergence theorems for Mann iteration used for approximation of the fixed points of demicontractive mappings in Hilbert spaces can be derived from the corresponding convergence theorems in the class of quasi-nonexpansive mappings. Our derivation is based on an important auxiliary lemma (Lemma 3.2), which shows that if T is k-demicontractive, then for any λ ∈ (0, 1 − k), Tλ is quasi-nonexpansive. In this way we obtain a unifying technique of proof for various well known results in the fixed point theory of demicontractive mappings. We illustrate this reduction technique for the case of two classical convergence results in the class of demicontractive mappings: [Măruşter, Şt. The solution by iteration of nonlinear equations in Hilbert spaces. Proc. [ABSTRACT FROM AUTHOR]

The purpose of this study is to introduce an inertial algorithm for approximating a solution of the split equality Hammerstein type equation problem in general reflexive real Banach spaces. Strong convergence results are established under the assumption that the associated mappings are monotone and uniformly continuous. The results in this paper generalize and improve many of the existing results in the literature in the sense that the underlying mappings are relaxed from Lipschitz continuous to uniformly continuous and the spaces under consideration are extended from Hilbert spaces to reflexive real Banach spaces with a more general problem which includes the Hammerstein type equation problems. [ABSTRACT FROM AUTHOR]

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij- Mann type associated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generalizations of the corresponding ones in [Berinde, V., Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293-304.], from the setting of Hilbert spaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), No. 2, 375-380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.], by considering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related results in literature can be obtained as particular instances of our results. [ABSTRACT FROM AUTHOR]

In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel'skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of -strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized nonexpansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new accelerated iterative method for finding a common fixed point of a countable family of nonexpansive mappings in the Hilbert spaces framework. Using our main result, we obtain a new accelerated image restoration iterative method for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply our algorithm to solving image restoration problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to invent an accelerated algorithm for the convex minimization problem which can be applied to the image restoration problem. Theoretically, we first introduce an algorithm based on viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive operators. Under some suitable assumptions, a strong convergence theorem of the proposed algorithm is established. Subsequently, we utilize our proposed algorithm to solving a convex minimization problem of the sum of two convex functions. As an application, we apply and analyze our algorithm to image restoration problems. Moreover, we compare convergence behavior and efficiency of our algorithm with other well-known methods such as the forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm. By using image quality metrics, numerical experiments show that our algorithm has a higher efficiency than the mentioned algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative method for nonexpansive mappings in CAT(κ) spaces. First, the rate of convergence of proposed method and comparison with recently existing method is proved. Second, strong and Δ-convergence theorems of the proposed method in such spaces under some mild conditions are also proved. Finally, we provide some non-trivial examples to show efficiency and comparison with many previously existing methods. [ABSTRACT FROM AUTHOR]

Recently Berinde and Păcurar [Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22 (2020), no. 2., 1--10], first introduced the idea of enriched contraction mappings and proved the existence of a fixed point of an enriched contraction mapping using the well-known fact that any fixed point of the averagedmapping Tλ, where λ ∈ (0, 1], is also a fixed point of the initialmapping T. In this work, we introduce the idea of weak enriched contraction mappings, and a new generalization of an averaged mapping called double averaged mapping. The first attempt is to prove the existence and uniqueness of the fixed point of a double averaged mapping associated with a weak enriched contraction mapping. Based on this result on Banach spaces, we give some sufficient conditions for the equality of all fixed points of a double averaged mapping and the set of all fixed points of a weak enriched contraction mapping. Moreover, our results show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a weak enriched contraction mapping. An illustrative example for showing the efficiency of our results is given. [ABSTRACT FROM AUTHOR]

In the setting of Hilbert spaces, we show that a hybrid steepest-descent algorithm converges strongly to a solution of a convex minimization problem over the fixed point set of a finite family of multivalued demicontractive mappings. We also provide numerical results concerning the viability of the proposed algorithm with possible applications. [ABSTRACT FROM AUTHOR]

A class of demicontractive mappings was first introduced in [Hicks, T. L. and Kubicek, J. D., On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl., 59 (1977) 498-504 and Maruster, S., The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc., 63 (1977), 69-73] and was first mentioned in the case of multi-valued mappings in [Chidume, C. E., Bello, A. U. and Ndambomve, P., Strong and δ-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal., 2014 (2014), https://doi.org/10.1155/2014/805168 and Isiogugu, F. O. and Osilike, M. O., Convergence theorems for new classes of multivalued hemicontractive-type mappings, Fixed Point Theory Appl., 2014 (2014), https://doi.org/10.1186/1687-1812-2014-93]. The demicontractivity with some weak smoothness conditions ensures only weak convergence of Mann iteration. In 2015, Marus,ter and Rus [Kannan contractions and strongly demicontractive mappings, Creat. Math. Inform., 24 (2015), No. 2, 173-182], introduced a class of strongly demicontractive mappings, and also discussed some relationships between strongly demicontractive mappings and Kannan contractions. In this paper, we introduce a new class of strongly demicontractive multivalued mappings in Hilbert spaces. Strong convergence theorems of Picard and Mann iterative methods for strongly demicontractive multi-valued mappings are established under some suitable coefficients and control sequences. [ABSTRACT FROM AUTHOR]

In this paper, a novel algorithm, called parallel inertial S-iteration forward-backward algorithm (PISFBA) is proposed for finding a common fixed point of a countable family of nonexpansive mappings and convergence behavior of PISFBA is analyzed and discussed. As applications, we apply PISFBA to estimate the weight connecting the hidden layer and output layer in a regularized extreme learning machine. Finally, the proposed learning algorithm is applied to solve regression and data classification problems. [ABSTRACT FROM AUTHOR]

Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, presented here for the first time, we introduce and study the class of enriched nonexpansive mappings in Hilbert spaces. In order to approximate the fixed points of enriched nonexpansive mappings we use the Krasnoselskij iteration for which we prove strong and weak convergence theorems. Examples to illustrate the richness of the new class of contractive mappings are also given. Our results in this paper extend some classical convergence theorems established by Browder and Petryshyn in [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228] from the case of nonexpansive mappings to that of enriched nonexpansive mappings, thus including many other important related results from literature as particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified inertial subgradient extragradient algorithm in a 2-uniformly convex and uniformly smooth real Banach space and prove a strong convergence theorem for approximating a common solution of fixed point equation with a demigeneralized mapping and a variational inequality problem of a monotone and Lipschitz mapping. We present an example to validate our new findings. This work substantially improves and generalizes some well-known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new modified proximal point algorithm involving fixed point iteration for nonexpansive mappings in CAT(1) spaces. Under some mild conditions, we prove that the sequence generated by our iterative algorithm Δ-converges to a common solution between certain convex optimization and fixed point problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to combine and extend some recent fixed point results of Suzuki, T., [A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313-5317] and Secelean, N. A. & Wardowski, D., [ψF-contractions: not necessarily nonexpansive Picard operators, Results Math., 70 (2016), 415-431]. The continuity and the completeness conditions are replaced by orbitally continuity and orbitally completeness respectively. It is given an illustrative example of a Picard operator on a non complete metric space which is neither nonexpansive nor expansive and has a unique continuity point. [ABSTRACT FROM AUTHOR]

In this paper, an iterative process for a hybrid pair of a finite family of generalized I-asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multi-valued mappings is established. Moreover, the weak convergence theorems and strong convergence theorems of the proposed iterative process in Banach spaces are proven. The examples are established for supporting our main results. The obtained results can be viewed as an improvement and extension of the several results in the literature. [ABSTRACT FROM AUTHOR]

In this work, we analyse the class of strictly pseudocontractive mappings in general metric spaces by providing a comprehensive and appropriate definition of a strictly pseudocontractive mapping, which serves as a natural extension of the existing notion. Moreover, we establish its various characterizations and explore several significant properties of these mappings in relation to fixed point theory in CAT(0) spaces. Specifically, we establish that these mappings are Lipschitz continuous, satisfying the demiclosedness-type property, and possessing a closed convex fixed point set. Furthermore, we show that the fixed points of the mappings can be effectively approximated using an iterative scheme for fixed points of nonexpansive mappings. The results in this work contribute to a deeper understanding of strictly pseudocontractive mappings and their applicability in the context of fixed point theory in metric spaces. [ABSTRACT FROM AUTHOR]

In this work, we propose and study an inertial Krasnosel'skii-Mann iterative algorithm with step-size parameters involving nonexpansive mapping to find a solution of a fixed point problem of a nonexpansive mapping in the frame work of Hilbert spaces. Strong convergence of the new proposed algorithm is proved under some useful properties of a nonexpansive mapping and inequalities on real Hilbert spaces together with the appropriate conditions of scalar controls without relying on the concept of viscosity methods. For the applications, we employ the obtained results to find a zero point of some monotone inclusion problems and then apply to solve image restoration problems. For representing the advantage of the new algorithm, the signal to noise ratio (SNR) with various types of blurring operators and some numerical experiments are presented to compare and illustrate the behavior of the new algorithm with numerical results of some previous existing algorithms. [ABSTRACT FROM AUTHOR]

In this work, two inertial algorithms for approximating common elements of the sets of solutions of three important problems are constructed. The first problem is a generalized mixed equilibrium one involving relaxed monotone mapping, the second is a zero problem of inverse strongly monotone mappings, while the third one is a fixed point problem of a family of relatively nonexpansive mappings. The first algorithm is a shrinking projection type for a common solution of all the three problems. The second is a generalized Alber projection free method for the second and the third problems. Each of the devised algorithms uses the conjugate gradient-like direction, which allows it to accelerate its iterates toward a solution of the problems. The strong convergence theorem for each of the algorithms is formulated and proved in a real 2 - uniformly convex and uniformly smooth Banach space. Additionally, the applications of our algorithms to convex optimization problems and image recovery problems are studied. The advantages and computational efficiency of our methods are analyzed based on their numerical performance in comparison to some of the existing and recently proposed methods using numerical example. [ABSTRACT FROM AUTHOR]

In this article, we introduce a Halpern iterative method together with an inertial projection and contraction method for finding an approximate solution of variational inequality problem involving pseudomonotone mapping which also solves split common fixed point problem of Bregman demigeneralized mapping and Bregman strongly nonexpansive mapping in the framework of p-uniformly convex and uniformly smooth real Banach spaces. Using our iterative method, we establish a strong convergence result for approximating the solution of the aforementioned problems and state some consequences of our main result. The result discussed in this article extends and complements many related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative process to approximate fixed points of nonexpansive maps in real Banach spaces having weakly continuous duality map and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption on K or on T. Our results improve important recent results. [ABSTRACT FROM AUTHOR]

In this paper, we study the M-iteration process in hyperbolic spaces and prove some strong and Δ-convergence theorems of this iteration process for generalized nonexpansive mappings. Moreover, we establish the weak w²-stability and data dependence theorems for a class of contractive-type mappings by using M-iteration process. The results presented here extend and improve some recent results announced in the current literature. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space E with dual space E∗. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a common solution of a family of variational inequality problems and a common element of fixed points of a family of a general class of nonlinear nonexpansive maps. The sequence of this new method is proved to converge strongly to a common element of the families. Our theorem and its applications complement, generalize, and extend various results in literature. [ABSTRACT FROM AUTHOR]

We apply the concept of quasilinearization to introduce some enriched classes of Banach contraction mappings and analyse the fixed points of such mappings in the setting of Hadamard spaces. We establish existence and uniqueness of the fixed point of such mappings. To approximate the fixed points, we use an appropriate Krasnoselskij-type scheme for which we establish ∆ and strong convergence theorems. Furthermore, we discuss the fixed points of local enriched contractions and Maia-type enriched contractions in Hadamard spaces setting. In addition, we establish demiclosedness-type property of enriched nonexpansive mappings. Finally, we present some special cases and corresponding fixed point theorems. [ABSTRACT FROM AUTHOR]

In this work, we introduce and study the modified inertial Mann’s algorithm and inertial hybrid algorithm for approximating some fixed points of a k-strict pseudo-contractive mapping in Hilbert spaces. Weak convergence to a solution of fixed-point problems for a k-strict pseudo-contractive mapping is obtained by using the modified inertial Mann’s algorithm. In order to obtain strong convergence, we introduce an inertial hybrid algorithm by using the inertial extrapolation method mixed with the convex combination of three iterated vectors and forcing for strong convergence by the hybrid projection method for a k-strict pseudo-contractive mapping in Hilbert spaces. The strong convergence theorem of the proposed method is proved under mild assumptions on the scalars. For illustrating the performance of the proposed algorithms, we provide some new nonlinear k-strict pseudo-contractive mappings which are not nonexpansive to create some numerical experiments to show the advantage of the two new inertial algorithms for a k-strict pseudo-contractive mapping. [ABSTRACT FROM AUTHOR]

In this paper, we study a zero point problem of the sum of two accretive operators based on a viscosity forward-backward iterative algorithm with computational errors. Strong convergence results are established in the framework of q-uniformly smooth Banach spaces. We also apply the strong convergence results to solve variational inequality problems, convex minimization problems and fixed point problems. [ABSTRACT FROM AUTHOR]

In this paper we study the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings in Banach spaces endowed with a graph. In particular, we prove that under certain conditions, this set is a monotone nonexpansive retract. [ABSTRACT FROM AUTHOR]

This article considers a split generalized equilibrium problem and fixed point problem for infinite family of nonexpansive mapping in Hilbert space. We propose an algorithm for finding a common solution of these problems. Under mild assumptions, we establish a strong convergence theorem for the sequence generated by the proposed algorithm using viscosity technique. We present the implementation of the proposed algorithm by considering some numerical illustrations and comparison of the proposed algorithm with an existing algorithm in the literature. [ABSTRACT FROM AUTHOR]

Using the concept of the CATp(0) spaces proposed by Khamsi et al. [Khamsi, M. A. and Shukri, S. A., Generalized CAT(0) spaces. Bull. Belg. Math. Soc. Simon Stevin, 24 (2017), No. 3, 417-426], we establish Δ-convergence and strong convergence of a perturbed variant of Agarwal et al. S-iteration process for nonexpansive mapping. Finally, from them we deduce results valid for α-nonexpansive mappings. [ABSTRACT FROM AUTHOR]

A probabilistic version of the Meir-Keeler type fixed point theorem, which characterizes completeness of the metric space is established. In addition to it, a fixed point theorem for non-expansive mappings satisfying (β - δ) type condition in Menger probabilistic metric space (Menger PM-space) is proved. As a byproduct we find an affirmative answer to the open question on the existence of contractive mappings which admit discontinuity at the fixed point (see Rhoades, B. E., Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245, p. 242) in the setting of Menger PM-space. [ABSTRACT FROM AUTHOR]

We prove that a cyclic coordinate fixed point algorithm for nonexpansive mappings when the underlying Hilbert space is decomposed into a Cartesian product of finitely many block spaces is weakly convergent to a fixed point of the mapping under investigation. Our result relaxes a condition imposed on the stepsizes of Theorem 3.4 of Chow, et al [Chow, Y. T., Wu, T. and Yin, W., Cyclic coordinate-update algorithms for fixed-point problems: analysis and applcations, SIAM J. Sci. Comput., 39 (2017), No. 4, A1280-A1300]. [ABSTRACT FROM AUTHOR]