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We attempt to find fixed points of a general quadratic polynomial in the algebra of split quaternion. In some cases, we characterize fixed points in terms of the coefficients of these polynomials and also give the cardinality of these points. As a consequence, we give some simple examples to strengthen the infinitude of these points in these cases. We also find the roots of quadratic polynomials as simple consequences. [ABSTRACT FROM AUTHOR]

This paper deals with the second-order nonlinear neutral delay difference equation Δ(anh(xn-τ1n,xn-τ2n,…,xn-τmn)Δ(xn-qnxn-τ0))+f(n,xn-σ1n,xn-σ2n,…,xn-σkn)=bn, n≥n0. Using the Banach fixed point theorem, Mann iterative method with errors, and some new techniques, we prove the existence of uncountably many positive solutions and the convergence of the sequences generated by the Mann iterative method with errors relative to these solutions for the above equation. Six examples are included. Our results extend and improve essentially the known results in this field. [ABSTRACT FROM AUTHOR]

We establish new fixed point results in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) by using Jungck Noor iteration with s-convexity. [ABSTRACT FROM AUTHOR]

We use a modified S-iterative process to prove some strong and Δ-convergence results for asymptotically nonexpansive type mappings in uniformly convex hyperbolic spaces, which includes Banach spaces and CAT(0) spaces. Thus, our results can be viewed as extension and generalization of several known results in Banach spaces and CAT(0) spaces (see, e.g., Abbas et al. (2012), Abbas et al. (2013), Bruck et al. (1993), and Xin and Cui (2011)) and improve many results in the literature. [ABSTRACT FROM AUTHOR]

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this paper, two classes of first-order neutral functional differential equations with periodic delays are considered. Some results on the existence of positive periodic solutions for the equations are obtained by using the Krasnoselskii fixed point theorem. Four examples are included to illustrate our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation ?(a(n, ya1n, . . ., yarn)?(yn + bnyn-t )) + f(n, yf1n, . . ., yfkn) = cn, ∀n = n0. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this work is to study the nonlinear functional-integral equation Using the technique of the measure of noncompactness and the Darbo fixed-point theorem, we prove the existence and asymptotic stability of solutions for the equation. A nontrivial example is given to support our result. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following minimization problem where is a bounded linear operator, is some constant, is a potential function for , is the set of fixed points of nonexpansive mapping and is the solution set of an equilibrium problem. This paper introduces two new algorithms (one implicit and one explicit) that can be used to find the solution of the above minimization problem. [Copyright &y& Elsevier]

Abstract: In this paper, Moudafi’s viscosity approximations with continuous strong pseudocontractions for a pseudocontraction semigroup are considered. A strong convergence theorem of fixed points is established in the framework of Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study the following first-order nonlinear neutral delay differential equation where , for , and . By using the Schauder and Krasnoselskii fixed point theorems, we establish the existence of uncountably many bounded nonoscillatory solutions for the above equation. To dwell upon the importance and advantages of our results, seven nontrivial examples are included. [Copyright &y& Elsevier]

Abstract: In this paper, the Banach fixed-point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference equationwhere , , with for , and withA few Mann type iterative approximation schemes with errors are suggested, and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed. Seven nontrivial examples are given to illustrate the advantages of our results. [Copyright &y& Elsevier]

Abstract: In this paper we consider the second order nonlinear neutral delay differential equation where with for , , , and with Under suitable conditions, by making use of the Banach fixed point theorem, we show the existence of infinitely many nonoscillatory solutions, which are uncountable, for the above equation, suggest several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. Five nontrivial examples are given to illustrate the advantages of our results. [Copyright &y& Elsevier]

In this paper, we introduce a new class of completely generalized nonlinear quasivariational inequalities and obtain its equivalence with a class of fixed point problems by using the resolvent operator technique. Using this equivalence, we develop perturbed three-step iterative algorithm for this class of completely generalized quasivariational inequalities. We establish a few existence theorems of unique solution for the class of completely generalized quasivariational inequalities involving relaxed monotone, generalized pseudocontractive and strongly monotone mappings and prove some convergence and stability results of iterative sequence generated by perturbed three-step iterative algorithm. [Copyright &y& Elsevier]

The present paper deals with two types of topologies on the set of integers, Z : a quasi-discrete topology and a topology satisfying the T 1 2 -separation axiom. Furthermore, for each n ∈ N , we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on Z n , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on. [ABSTRACT FROM AUTHOR]

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y ∖ { p } . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of an α -admissible nonself-mapping, an α -F-contractive nonself-mapping, a weak α -F-contractive nonself-mapping, and a generalized α -F-contractive nonself-mapping and prove some P P F (past-present-future)-dependent fixed point theorems for the proposed contractive nonself-mappings in certain Razumikhin classes. By using our results, we derive some P P F -dependent fixed point theorems for an α -F-contractive nonself-mapping endowed with a graph or a partial order. Finally, we give some applications to illustrate the main results. [ABSTRACT FROM AUTHOR]

Abstract: Some fixed point theorems for a few nonlinear multi-valued contraction mappings in complete metric spaces with w-distance are proved. The results presented in this paper generalize the recent results due to Ćirić (2009) [3], Feng and Liu (2006) [4], Klim and Wardowski (2007) [8], Latif and Abdou (2011) [12] and Liu et al. (2010) [14]. Seven examples are given to explain our results. [Copyright &y& Elsevier]

Abstract: It is shown that in a Banach space with weak uniform normal structure, every demicontinuous asymptotically regular nearly Lipschitzian self-mapping with defined on a weakly compact convex subset of satisfies the -fixed point property. We show that if has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically pseudocontractive nearly nonexpansive mapping is nonempty and a sunny nonexpansive retract of . Further, we study the approximation of fixed points of by Halpern type iteration process: , where and is a sequence in (0,1) satisfying appropriate conditions. Our results improve several known existence and convergence fixed point theorems in general Banach spaces for a wider class of nonlinear mappings which are not necessarily Lipschitzian. [Copyright &y& Elsevier]

Abstract: In this work, a projection algorithm is considered for treating strongly continuous semigroups of demicontinuous pseudocontractions. Theorems of strong convergence of fixed points are established in the framework of real Hilbert spaces. [Copyright &y& Elsevier]

Abstract: In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality. [Copyright &y& Elsevier]

Abstract: This paper deals with the second order nonlinear neutral delay system of difference equationsBy virtue of the Banach fixed point theorem and some natural modifications of the techniques in the literature, we prove the existence results of uncountably many bounded positive solutions for the system of difference equations. Seven examples are constructed to explain the main results presented in this paper. [Copyright &y& Elsevier]

Abstract: In this work, strongly continuous semigroups of pseudocontractions are studied based on an implicit iterative algorithm. Strong convergence theorems of fixed points are obtained in an arbitrary Banach space. [Copyright &y& Elsevier]

Abstract: This paper deals with the first order nonlinear neutral delay differential equationwhere and with lim t→+∞ σ l (t)=+∞ for l ∈{1,2,…, n}. By using the Banach fixed point theorem, we prove the global existence of uncountably many bounded positive solutions for the above equation relative to all ranges of the function p, construct some Mann type iterative algorithms with errors to approximate these positive solutions and discuss several error estimates between the sequences generated by the iterative algorithms and these positive solutions. Seven examples are presented to illuminate the results obtained in this paper. [Copyright &y& Elsevier]

Abstract: This paper deals with the second order nonlinear neutral delay difference equation By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain the existence results of uncountably many bounded positive solutions for the above equation. Examples that cannot be solved by known results are given to illustrate the results presented in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an iterative method for finding a common element in the solution set of generalized equilibrium problems, in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings. Strong convergence theorems are established in the framework of Hilbert spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to consider the problem of approximating a common fixed point of two asymptotically quasi--nonexpansive mappings based on hybrid projection methods. Strong convergence theorems are established in a real Banach space. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems, the set of common fixed points of a family of infinite non-expansive mappings. Strong convergence theorems are established in a real Hilbert space under suitable conditions. As some applications, we consider convex feasibility problems and equilibrium problems. The results presented improve and extend the corresponding results of many others. [Copyright &y& Elsevier]

Abstract: In an infinite-dimensional Hilbert space, the normal Mann’s iteration has only weak convergence, in general, even for nonexpansive mappings. The purpose of this paper is to modify the normal Mann’s iteration to have strong convergence for a family of Lipschitz quasi-pseudocontractions in the framework of Hilbert spaces. Our results improve and extend the corresponding ones announced by many others. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi--nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others. [Copyright &y& Elsevier]

Abstract: In this paper, we modify the normal Mann’s iterative process to have strong convergence for a -strictly pseudo-contractive non-self mapping in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others. [Copyright &y& Elsevier]

Abstract: Let C be a nonempty closed convex subset of a real strictly convex and reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Let be a given contractive mapping and be an infinite family of nonexpansive mappings such that the common fixed point sets . Let and be two real sequences in [0,1]. For given arbitrarily, let the sequence be generated iteratively bywhere is the W-mapping generated by the mappings and . Suppose the iterative parameters and satisfy the following control conditions: [(C1)] ; [(C2)] ; [(B5)] . Then the sequence converges strongly to where p is the unique solution in F to the following variational inequality: [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the third order two-point generalized right focal boundary value problem A few new results are given for the existence of at least one, two, three and infinitely many monotone positive solutions of the above boundary value problem by using the Krasnosel’skii fixed-point theorem in cones, and the Leggett–Williams fixed-point theorem. [Copyright &y& Elsevier]

Abstract: In this work, an implicit iterative process is considered for asymptotically pseudocontractive nonself-mappings. Weak and strong convergence theorems for common fixed points of a family of asymptotically pseudocontractive nonself-mappings are established in the framework of Hilbert spaces. [Copyright &y& Elsevier]

Abstract: This paper deals with the solvability of the third order nonlinear neutral delay difference equation relative to the sequence . Using the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem, a few sufficient conditions of the existence of uncountably many bounded positive solutions for the equation are presented. Seven examples are included to demonstrate the advantages and effectiveness of the results presented in this paper. [Copyright &y& Elsevier]

Abstract: In an infinite-dimensional Hilbert space, the normal Mann’s iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. In order to get a strong convergence result, we modify the normal Mann’s iterative process for an infinite family of nonexpansive mappings in the framework of Banach spaces. Our results improve and extend the recent results announced by many others. [Copyright &y& Elsevier]