By using the Leggett-Williams fixed theorem, we establish the existence of multiple positive solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with linear functional boundary conditions. One explicit example with singularity is presented to demonstrate the application of our main results.

With the aid of fixed-point theorem (an equivalent version for the linear case) and biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a linear Fredholm-Volterra integro-differential equation is turned into a numerical algorithm, so that it can be solved numerically.

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.

The hybrid algorithms for constructing fixed points of nonlinear mappings have been studied extensively in recent years. The advantage of this methods is that one can prove strong convergence theorems while the traditional iteration methods just have weak convergence. In this paper, we propose two types of hybrid algorithm to find a common fixed point of a finite family of asymptotically nonexpansive mappings in Hilbert spaces. One is cyclic Mann's iteration scheme, and the other is cyclic Halpern's iteration scheme. We prove the strong convergence theorems for both iteration schemes.

Under some weaker conditions, we prove the strong convergence of the sequence generated by a modified regularization method of finding a zero for a maximal monotone operator in a Hilbert space. In addition, an example is also given in order to illustrate the effectiveness of our generalizations. The results presented in this paper can be viewed as the improvement, supplement, and extension of the corresponding results.

We investigate the global existence of the delayed nonlinear evolutionary equation ${\partial }_{t}u+Au=f(u(t),u(t-\tau ))$. Our work space is the fractional powers space $X^{\alpha }$. Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.

This paper deals with bimatrix games in uncertainty environment based on several types of ordering, which Maeda proposed. But Maeda’s models was just made based on symmetrical triangle fuzzyvariable. In this paper, we generalized Maeda’s model to the non-symmetrical environment. In other words, we investigated the fuzzy bimatrix games based on nonsymmetrical $L\text{-}R$ fuzzy variables. Then the pseudoinverse of a nonconstant monotone function was given and the concept of crisp parametric bimatrix games was introduced. At last, the existence condition of Nash equilibrium strategies of the fuzzy bimatrix games is proposed and (weak) Pareto equilibrium of the fuzzy bimatrix games was obtained through the Nash equilibrium of the crisp parametric bimatrix.

We introduce and consider a new class of equilibrium problems and variational inequalities involving bifunction, which is called the nonconvex bifunction equilibrium variational inequality. We suggest and analyze some iterative methods for solving the nonconvex bifunction equilibrium variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only monotonicity. Some special cases are also considered. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

We introduce an iterative method for finding a common element of set of fixed points of nonexpansive mappings, the set of solutions of a finite family of variational inclusion with set-valued maximal monotone mappings and inverse strongly monotone mappings, and the set of solutions of a mixed equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of Plubtemg and Sripard and many others.

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mapping $T$ of a closed convex subset $C$ of a CAT(0) space $X$. Suppose that the set Fix$\left(T\right)$ of fixed points of $T$ is nonempty. For a contraction $f$ on $C$ and $t\in \left(\mathrm{0,1}\right)$, let $x_t\in C$ be the unique fixed point of the contraction $x\mapsto tf\left(x\right)\oplus (1-t)Tx$. We will show that if $X$ is a CAT(0) space satisfying some property, then $\left\{x_t\right\}$ converge strongly to a fixed point of $T$ which solves some variational inequality. Consider also the iteration process $\left\{x_n\right\}$, where $x_0\in C$ is arbitrary and $x_{n+1}={\alpha }_{n}f\left(x_n\right)\oplus (1-{\alpha }_n)T{x}_n$ for $n\ge 1$, where $\left\{{\alpha }_n\right\}\subset \left(\mathrm{0,1}\right)$. It is shown that under certain appropriate conditions on ${\alpha }_n,\left\{x_n\right\}$ converge strongly to a fixed point of $T$ which solves some variational inequality.

This paper deals with the existence results of integral solutions for nondensely defined fractional evolution differential inclusions. Our approach is based on integrated semigroup theory and a fixed point theorem for condensing map due to Martelli. An example is also given to illustrate our results.

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

We proved that the modified implicit Mann iteration process can be applied to approximate the fixed point of strictly hemicontractive mappings in smooth Banach spaces.

Four fixed point theorems for nonlinear set-valued contractive mappings in complete metric spaces are proved. The results presented in this paper are extensions of a few well-known fixed point theorems. Two examples are also provided to illustrate our results.

We study a new system of nonlinear set-valued variational inclusions involving a finite familyof $H(·,·)$-accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite familyof $H(·,·)$-accretive operators, we prove the existence of the solution for the system of nonlinear set-valued variationalinclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions for this system.

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

An explicit iterative method for solving the variational inequalities on Hadamard manifold is suggested and analyzed using the auxiliary principle technique. The convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. Results can be viewed as refinement and improvement of previously known results.

Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.

The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalized f-projection operator for finding a common solution of fixed-point problem of a totally quasi-$\varphi$-asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth andstrictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improvethe result of Chang et al., (2012), and some others.

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.

We prove an equilibrium existence result for vector functions defined on noncompact domain and we give some applications in optimization and Nash equilibrium in noncooperative game.

We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.

This paper deals with the split feasibility problem that requires to find a point closest to a closed convex set in one space such that its image under a linear transformation will be closest to another closed convex set in the image space. By combining perturbed strategy with inertial technique, we construct an inertial perturbed projection algorithm for solving the split feasibility problem. Under some suitable conditions, we show the asymptotic convergence. The results improve and extend the algorithms presented in Byrne (2002) and in Zhao and Yang (2005) and the related convergence theorem.

We introduce a new iterative method for finding a common element of the set of fixed points of a strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of a variational inequality problem for an inverse-strongly-monotone mapping in Hilbert spaces and then show that the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets under suitable control conditions. The results in this paper substantially improve, develop, and complement the previous well-known results in this area.

We study the existence of positive solutions for second-order nonlinear differential equations with nonseparated boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved.

We use the auxiliary principle technique to suggest and analyze an implicit method for solving the equilibrium problems on Hadamard manifolds. The convergence of this new implicit method requires only the speudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered.

We define and study some generalized classes of Bazilevic functions associated with convex domains. These convex domains are formed by conic regions which are included in the right half plane. Such results as inclusion relationships and integral-preserving properties are proved. Some interesting special cases of the main results are also pointed out.

We consider and study a new class of variational inequality, which is called the general mixed quasivariational inequality. We use the auxiliary principle technique to study the existence of a solution of the general mixed quasivariational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

We establish a tripled fixed point result for a mixed monotone mapping satisfying nonlinear contractions in ordered generalized metric spaces. Also, some examples are given to support our result.

Let $n$ and $r$ be two integers such that ; we denote by $\gamma (n,r)$ $\left[\eta \right(n,r\left)\right]$ the minimum [maximum] number of the nonnegative partial sums of a sum ${\sum }_{1=1}^n{a}_i\ge 0$, where $a_1,\dots ,a_n$ are $n$ real numbers arbitrarily chosen in such a way that $r$ of them are nonnegative and the remaining $n-r$ are negative. We study the following two problems: $\left(P1\right)$ which are the values of $\gamma (n,r)$ and $\eta (n,r)$ for each $n$ and $r$, ? $\left(P2\right)$ if $q$ is an integer such that $\gamma (n,r)\le q\le \eta (n,r)$ , can we find $n$ real numbers $a_1,\dots ,a_n$, such that $r$ of them are nonnegative and the remaining $n-r$ are negative with ${\sum }_{1=1}^n{a}_i\ge 0$, such that the number of the nonnegative sums formed from these numbers is exactly $q$ ?

We introduce and study a class of a system of random set-valued variational inclusionproblems. Some conditions for the existence of solutions of such problems are provided, when the operatorsare contained in the classes of generalized monotone operators, so-called ($A,m,\eta$)-monotone operator. Further,the stability of the iterative algorithm for finding a solution of the considered problem is also discussed.

Three hundred patients were randomly selected as the research object, of which 150 were on Chinese medical treatment; rehabilitation of 150 people was recorded for each patient before and after treatment by the three outcome measures (Fugle-Meyer baseline, NHISS baseline, and baseline BI), coupled with the treatment process spent in a variety of costs (mainly medicine costs, medicine,laboratory test, treatment, bed, care, diagnosis and examination fees, inspection fees). By combining the meaning of data with its practicality we get the definition of the efficacy. Via using the softwares of Excel, Matlab, and Eviews for data processing and fitting, it can be found that there exsists the fractal nature between efficacy and cost of treatment during the cerebral diseases. Then combined with the fractal theory, the application of chaotic time series, and two Fractal Indexes, the largest Lyapunov exponent and correlation dimension were extracted under two conditions of Chinese medical treatment and rehabilitation, and in the comparison of significance the brain vascular disease in traditional Chinese medicine treatment and rehabilitation was found. There were significant difference in fractal indicators of the time series of effective unit cost. At the same time, there were similar significant differences in the three outcome measures. This paper studied the fractal nature of cerebrovascular disease between the efficacy and cost and draw some fractal relationships and conclusions, so as to find better medical treatment to provide a theoretical basis for the hope of the treatment of cerebral vascular disease to provide some valuable reference.

We prove the existence of common fixed points for three relatively asymptotically regular mappings defined on an orbitally complete ordered metric space using orbital continuity of one of the involved maps. We furnish a suitable example to demonstrate the validity of the hypotheses of our results.

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.

We consider the analytic modelling of wireless systems with multiple access technologies in the perspective of teletraffic engineering and provide a framework for the performance analysis and evaluation of a wireless HetNet (heterogeneous network) system with both cellular and WLAN access technologies. In particular, an approach with Erlang's fixed-point approximation to calculate the new call blocking and handover call dropping probabilities in such systems is introduced. The model is versatile enough to cover not only cellular/WLAN HetNet systems but other wireless HetNets with difference access technologies in general.

We prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation $f\left(x+2y\right)+f\left(x-2y\right)=4f\left(x+y\right)+4f\left(x-y\right)-6f\left(x\right)+f\left(2y\right)+f\left(-2y\right)-4f\left(y\right)-4f\left(-y\right)$ in various complete random normed spaces.

$p\left(\ge 2\right)$-cyclic and contractive self-mappings on a set of subsets of a metric space which are simultaneously accretive on the whole metric space are investigated. The joint fulfilment of the $p$-cyclic contractiveness and accretive properties is formulated as well as potential relationships with cyclic self-mappings in order to be Kannan self-mappings. The existence and uniqueness of best proximity points and fixed points is also investigated as well as some related properties of composed self-mappings from the union of any two adjacent subsets, belonging to the initial set of subsets, to themselves.

We introduce and study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the SGNMVLIP. First, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Second, an iterative algorithm for solving the SGNMVLIP is constructed by using this existence result. Finally, not only the existence of solutions of the SGNMVLIP is shown but also the convergence of iterative sequences generated by the algorithm is also proven. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in the literature.

Let $C$ be a nonempty closed convex subset of a real uniformly smooth Banach space $X$, $\{{T}_{k}{\}}_{k=1}^{\infty }:C\to C$ an infinite family of nonexpansive mappings with the nonempty set of common fixed points ${\bigcap }_{k=1}^{\infty }\mathrm{Fix}({T}_{k})$, and $f:C\to C$ a contraction. We introduce an explicit iterative algorithm ${x}_{n+1}={\alpha }_{n}f({x}_{n})+(1-{\alpha }_{n}){L}_{n}{x}_{n}$, where ${L}_{n}={\sum }_{k=1}^{n}\left({\omega }_{k}/{\text{s}}_{n}\right){T}_{k},{S}_{n}={\sum }_{k=1}^{n}{\omega }_{k},\mathrm{ }$ and ${w}_{k}>0$ with ${\sum }_{k=1}^{\infty }{\omega }_{k}=1$. Under certain appropriate conditions on $\{{\alpha }_{n}\}$, we prove that $\{{x}_{n}\}$ converges strongly to a common fixed point ${x}^{*}$ of $\{{T}_{k}{\}}_{k=1}^{\infty }$, which solves the following variational inequality: $〈{x}^{*}-f({x}^{*}),J({x}^{*}-p)〉\le 0,\mathrm{ }\mathrm{ }p\in {\bigcap }_{k=1}^{\infty }\text{Fix}({T}_{k})$, where $J$ is the (normalized) duality mapping of $X$. This algorithm is brief and needs less computational work, since it does not involve $W$-mapping.

Let $G=\left(V,E\right)$ be an undirected graph with a weight function and a costfunction on edges. The constrained minimum spanning tree problem is to find aminimum cost spanning tree T in G such that the total weight in T is at most a given bound B. In this paper, we present two polynomial time approximation schemes (PTASs) for the constrained minimum spanning tree problem.

Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixed-point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

We introduce several types of the Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. Criteria and characterizations of these types of the Levitin-Polyak well-posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper unify, generalize, and extend some known results in the literature.

Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others.

The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others.

We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem.

We prove the existence of fixed point and uniqueness of quasi-contractive mappings in modular metric spaces which was introduced by Ćirić

We study strong convergence of the sequence generated by implicit andexplicit general iterative methods for a one-parameter nonexpansive semigroupin a reflexive Banach space which admits the duality mapping $J_{\phi }$, where $\phi$ is a gauge function on $\left[0,\infty \right)$. Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

The projected-gradient method is a powerful tool for solving constrained convex optimization problems and has extensively been studied. In the present paper, a projected-gradient method is presented for solving the minimization problem, and the strong convergence analysis of the suggested gradient projection method is given.

The main purpose of this paper is to investigate the existence of solutions of BVPs for a very general case in which both the system of ordinary differential equations and the boundary conditions are nonlinear. By employing the implicit function theorem, sufficient conditions for the existence of three-point boundary value problems are established.

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan (2010) and many others.

We introduce an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.

Kirk and Xu studied the existence of fixed points of asymptotic pointwise nonexpansive mappings in the Banach space. In this paper, we investigate these kinds of mappings in modular spaces. Moreover, we prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular spaces. The results improve and extend the corresponding results of Kirk and Xu (2008) to modular spaces.

We introduce the new generalized mixed equilibrium problem with respect to relaxed semimonotone mappings. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in Banach spaces. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the fixed point set of an asymptotically nonexpansive mapping. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.

The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets ${(A+M_2)}^{-1}$(0) and ${(B+M_1)}^{-1}$(0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.

We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.

We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in the literature.

Let T be a time scale. We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with $p$-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with $p$-Laplacian operator is obtained.

Recently, Yao et al. (2011) introduced two algorithms for solving a system of nonlinear variational inequalities. In this paper, we consider two general algorithms and obtain the extension results for computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points solve the same system of nonlinear variational inequalities.

This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result on ${\sigma }_k$ Yamabe problem of negative admissible metrics by virtue of the degree theory in nonlinear functional analysis and the maximum principle and then establish an existence and uniqueness theorem for the solutions to the problem.

Of concern is a class of nonlinear neutral fractional integrodifferential inclusions with infinite delay in Banach spaces. A theorem about the existence of mild solutions to the fractional integrodifferential inclusions is obtained based on Martelli’s fixed point theorem. An example is given to illustrate the existence result.

We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and some well-known results in the literature.

We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.

This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.

We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically I-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

In this paper we prove the equivalence and the strong convergence of an explicit Mann iterative process and a modified implicit iterative process for asymptotically $\varphi$-strongly pseudocontractive mappings in a uniformly smooth Banach space.

We introduce the generalized split common fixed point problem (GSCFPP) and show that the GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover, we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some strong convergence theorems under suitable assumptions.

Based on the recent work by Censor and Segal (2009 J. Convex Anal.16), and inspired by Moudafi (2010 Inverse Problems 26), we modify the algorithm of demicontractive operators proposed by Moudafi and study the modified algorithm for the class of firmly pseudodemicontractive operators to solve the split common fixed-point problem in a Hilbert space. We also give the strong convergence theorem under some appropriate conditions. Our work improves and/or develops the work of Moudafi, Censor and Segal, and other results.

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

We introduce and consider a proximal point algorithm for solving minimization problems using the technique of Güler. This proximal point algorithm is obtained by substituting the usual quadratic proximal term by a class of convex nonquadratic distance-like functions. It can be seen as an extragradient iterative scheme. We prove the convergence rate of this new proximal point method under mild assumptions. Furthermore, it is shown that this estimate rate is better than the available ones.

The multiple-set split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. It generalizes the convex feasibility problem as well as the two-set split feasibility problem. In this paper, we will review and report some recent results on iterative approaches to the multiple-set split feasibility problem.

An equilibrium problem is investigated based on a hybrid projection iterative algorithm. Strong convergence theorems for solutions of the equilibrium problem are established in a strictly convex and uniformly smooth Banach space which also enjoys the Kadec-Klee property.

We introduce and consider a new class of complementarity problems, which is called the absolute value complementarity problem. We establish the equivalence between the absolute complementarity problems and the fixed point problem using the projection operator. This alternative equivalent formulation is used to discuss the existence of a solution of the absolute value complementarity problem. A generalized AOR method is suggested and analyzed for solving the absolute the complementarity problems. We discuss the convergence of generalized AOR method for the L-matrix. Several examples are given to illustrate the implementation and efficiency of the method. Results are very encouraging and may stimulate further research in this direction.

Some fixed point theorems for $(\phi ,\psi ,p)$-contractive maps and $(\phi ,k,p)$-contractive maps on a complete metric space are proved. Presented fixed point theorems generalize many results existing in the literature.

We generalize the Hausdorff fuzzy metric in the sense of Rodríguez-López and Romaguera, and we introduce a new $M_{\infty }$-fuzzy metric, where $M_{\infty }$-fuzzymetric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, under $\varphi$-contraction condition, in the fuzzy metric space, we give some common fixed point theorems for fuzzy mappings.

A general hierarchical problem has been considered, and an explicit algorithm has been presented for solving this hierarchical problem. Also, it is shown that the suggested algorithm converges strongly to a solution of the hierarchical problem.

A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

Under the assumption of two coupled parallel subsuper solutions, the existence of at least six solutions for a kind of second-order m-point differential equations system is obtained using the fixed point index theory. As an application, an example to demonstrate our result is given.

The purpose of this paper is to introduce a new iterative scheme for finding a commonelement of the set of solutions of generalized mixed equilibrium problems, the set of solutions ofvariational inequality problems, the zero point of maximal monotone operators, and the set of twocountable families of quasi-$\varphi$-nonexpansive mappings in Banach spaces. Moreover, the strong convergencetheorems of this method are established under the suitable conditions of the parameter imposedon the algorithm. Finally, we apply our results to finding a zero point of inverse-strongly monotoneoperators and complementarity problems. Our results presented in this paper improve and extendthe recently results by many others.

We introduce a Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.

We suggest and analyze some implicit iterative methods for solving the extended general nonconvex variational inequalities using the projection technique. We show that the convergence of these iterative methods requires only the gh-pseudomonotonicity, which is a weaker condition than gh-monotonicity. We also discuss several special cases. Our method of proof is very simple as compared with other techniques.

The purpose of this paper is to propose an algorithm for solving the split common fixed point problems for total asymptotically strictly pseudocontractive mappings in infinite-dimensional Hilbert spaces. The results presented in the paper improve and extend some recent results of Moudafi (2011 and 2010), Xu (2010 and 2006), Censor and Segal (2009), Censor et al. (2005), Masad and Reich (2007), Censor et al. (2007), Yang (2004), and others.

The purpose of this paper is first to introduce the concept of total quasi-$\varphi$-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme forfinding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi-$\varphi$-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

A modified spectral PRP conjugate gradient method is presented for solving unconstrained optimization problems. The constructed search direction is proved to be a sufficiently descent direction of the objective function. With an Armijo-type line search to determinate the step length, a new spectral PRP conjugate algorithm is developed. Under some mild conditions, the theory of global convergence is established. Numerical results demonstrate that this algorithm is promising, particularly, compared with the existing similar ones.

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality.

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

We introduce an iterative for finding the zeros point of the sum of two monotone operators. We prove that the suggested method converges strongly to the zeros point of the sum of two monotone operators.

The purpose of this paper is to investigate the problems of the well-posednessfor a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept of $\alpha$-well-posedness to the system of mixed quasivariational-like inequalities,which includes symmetric quasi-equilibrium problems as a special case. Second, we establishsome metric characterizations of $\alpha$-well-posedness for the system of mixed quasivariational-likeinequalities. Under some suitable conditions, we prove that the $\alpha$-well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-likeinequalities. The corresponding concept of $\alpha$-well-posedness in the generalized sense is alsoconsidered for the system of mixed quasivariational-like inequalities having more than onesolution. The results presented in this paper generalize and improve some known results inthe literature.

We prove the equivalence and the strong convergence of iterative processes involving generalized strongly asymptotically $\varphi$-pseudocontractive mappings in uniformly smooth Banach spaces.

It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered under some suitable conditions. Results proved in this paper may stimulate further research in this direction.

We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed.

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let $f:K\to K$ a contractive mapping and $T:K\to K$ be a uniformly continuous pseudocontractive mapping with $F(T)\ne \varnothing$. Let $\left\{{\lambda }_n\right\}\subset (0,1/2)$ be a sequence satisfying the following conditions: (i) ${\lim }_{n\to \infty }{\lambda }_n=0$; (ii) ${\displaystyle {\sum }_{n=0}^{\infty }{\lambda }_n=\infty }$. Define the sequence $\left\{x_n\right\}$ in K by $x_0\in K$, $x_{n+1}={\lambda }_{n}f(x_n)+(1-2{\lambda }_n)x_n+{\lambda }_{n}T{x}_n$, for all $n\ge 0$. Under some appropriate assumptions, we prove that the sequence $\left\{x_n\right\}$ converges strongly to a fixed point $p\in F(T)$ which is the unique solution of the following variational inequality: $\langle f(p)-p,j(z-p)\rangle \le 0$, for all $z\in F(T)$.

Let $C$ be a nonempty bounded closed convex subset of a complete CAT(0) space $X$. We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on $C$ is nonempty closed and convex. We also show that, under some suitable conditions, the sequence $\{x_k{\}}_{k=1}^{\infty }$ defined by $x_{k+1}=(1-t_{mk})x_k\oplus t_{mk}{T}_m^{n_k}{y}_{(m-1)k},\hspace{0.17em}{y}_{(m-1)k}=(1-t_{(m-1)k})x_k\oplus t_{(m-1)k}{T}_{m-1}^{n_k}{y}_{(m-2)k},y_{(m-2)k}=(1-t_{(m-2)k})x_k\oplus t_{(m-2)k}{T}_{m-2}^{n_k}{y}_{(m-3)k},\dots ,y_{2k}=(1-t_{2k})x_k\oplus t_{2k}{T}_2^{n_k}{y}_{1k},y_{1k}=(1-t_{1k})x_k\oplus t_{1k}{T}_1^{n_k}{y}_{0k},y_{0k}=x_k,k\in \mathbb{N}$, converges to a common fixed point of $T_1,T_2,\dots ,T_m$ where they are asymptotic pointwise nonexpansive mappings on $C$, $\{t_{ik}{\}}_{k=1}^{\infty }$ are sequences in $\left[\mathrm{0,1}\right]$ for all $i=\mathrm{1,2},\dots ,\mathrm{m,}$ and $\left\{n_k\right\}$ is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

We investigate the following regularized gradient projectionalgorithm $x_{n+\text{1}}=P_c\left(I-{\gamma }_n\left(\nabla f+{\alpha }_{n}I\right)\right)x_n$, $n\ge \text{0}$. Under some different control conditions, we prove that this gradient projection algorithmstrongly converges to the minimum norm solution of the minimization problem ${\text{min}}_{x\in C}f\left(x\right)$.

Let $H$ be a real Hilbert space. Consider on $H$ a nonexpansive semigroup with a common fixed point, a contraction $f$ with the coefficient , and a strongly positive linear bounded self-adjoint operator $A$ with the coefficient $\overline{\gamma }$> 0. Let /$\alpha$. It is proved that the sequence $\left\{x_n\right\}$ generated by the iterative method $x_0\in H,\hspace{0.17em}{x}_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\right(1-{\beta }_n)I-{\alpha }_{n}A)\left(1/s_n\right){\int }_0^{s_n}T\left(s\right)x_{n}ds,\hspace{0.17em}n\ge 0$ converges strongly to a common fixed point $x^{*}\in F\left(S\right)$, where $F\left(S\right)$ denotes the common fixed point of the nonexpansive semigroup. The point $x^{*}$ solves the variational inequality $\langle (\gamma f-A)x^{*},x-x^{*}\rangle \le 0$ for all $x\in F\left(S\right)$.

The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result of $\text{Đ}$orić and Lazović (2011) for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić (1974). Existence of a common solution for a class of functional equations arising in dynamic programming is also discussed.

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for a $\beta$-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

Let $X$ be a uniformly convex Banach space and be a nonexpansive semigroup such that $F\left(\mathcal{S}\right)={\bigcap }_{s>0}F\left(T\right(s\left)\right)\ne \varnothing$. Consider the iterative method that generates the sequence $\left\{x_n\right\}$ by the algorithm $x_{n+1}={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+(1-{\alpha }_n-{\beta }_n)(1/s_n){\int }_0^{s_n}T\left(s\right)x_{n}ds,n\ge 0$, where $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$, and $\left\{s_n\right\}$ are three sequences satisfying certain conditions, $f:C\to C$ is a contraction mapping. Strong convergence of the algorithm $\left\{x_n\right\}$ is proved assuming $X$ either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.

A class of three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration, which is important in engineering problems. One method of the class is established analytically. To test the derived methods from the class, we apply them to a lot of nonlinear scalar equations. Numerical examples suggest that the novel class of derivative-free methods is better than the existing methods of the same type in the literature.

We use auxiliary principle technique coupled with iterative regularization method to suggest and analyze some new iterative methods for solving mixed variational-like inequalities. The convergence analysis of these new iterative schemes is considered under some suitable conditions. Some special cases are also discussed. Our method of proofs is very simple as compared with other methods. Our results represent a significant refinement of the previously known results.