This paper deals with new methods for approximating a solution to the fixed point problem; find $\tilde{x}\in F(T)$, where $H$ is a Hilbert space, $C$ is a closed convex subset of $H$, $f$ is a $\rho$-contraction from $C$ into $H$, , $A$ is a strongly positive linear-bounded operator with coefficient $\bar{\gamma }>0$, , $T$ is a nonexpansive mapping on $\mathrm{C,}$ and $P_{F(T)}$ denotes the metric projection on the set of fixed point of $T$. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality $\langle (A-\gamma f)\tilde{x}+\tau (I-S)\tilde{x},x-\tilde{x}\rangle \ge 0$ for $x\in F(T)$, where $\tau \in [0,\infty )$. Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.

The relationships between nonexpansive, weakly nonexpansive, $*$-nonexpansive, proximally nonexpansive, proximally continuous, almost lower semicontinuous, and $\varepsilon$-semicontinuous mappings in $ℝ$-trees are studied. Convergence theorems for the Ishikawa iteration processes are also discussed.

Inspired by the very recent results of Wang and Xu (2010), we study properties of the approximating curve with 1-norm regularization method for the split feasibility problem (SFP). The concept of the minimum-norm solution set of SFP in the sense of 1-norm is proposed, and the relationship between the approximating curve and the minimum-norm solution set is obtained.

Let $\{T_i{\}}_{i=1}^N$ be $N$ strictly pseudononspreading mappings defined on closed convex subset $C$ of a real Hilbert space $H$. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality $\langle (\gamma f-\mu B)x^{*},v-x^{*}\rangle \le 0,\forall v\in {\bigcap }_{i=1}^N{F}_{ix}(T_i)$.

We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

It is well known that the gradient-projection algorithm (GPA) is very useful in solving constrained convex minimization problems. In this paper, we combine a general iterative method with the gradient-projection algorithm to propose a hybrid gradient-projection algorithm and prove that the sequence generated by the hybrid gradient-projection algorithm converges in norm to a minimizer of constrained convex minimization problems which solves a variational inequality.

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

We propose long-time convergent numerical integration processes for delay differential equations. We first construct an integration process based on modified Laguerre functions. Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. We also developed a technique for refinement of modified Laguerre-Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

We discuss the existence of solutions, under the Pettis integrability assumption, for a class of boundary value problems for fractional differential inclusions involving nonlinear nonseparated boundary conditions. Our analysis relies on the Mönch fixed point theorem combined with the technique of measures of weak noncompactness.

We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping in $q$-uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others.

We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.

In all polynomial zerofinding algorithms, a good convergence requires a very good initial approximation of the exact roots. The objective of the work is to study the conditions for determining the initial approximations for an iterative matrix zerofinding method. The investigation is based on the Newbery's matrix construction which is similar to Fiedler's construction associated with a characteristic polynomial. To ensure that convergence to both the real and complex roots of polynomials can be attained, three methods are employed. It is found that the initial values for the Fiedler's companion matrix which is supplied by the Schmeisser's method give a better approximation to the solution in comparison to when working on these values using the Schmeisser's construction towards finding the solutions. In addition, empirical results suggest that a good convergence can still be attained when an initial approximation for the polynomial root is selected away from its real value while other approximations should be sufficiently close to their real values. Tables and figures on the errors that resulted from the implementation of the method are also given.

This paper is concerned with the numerical approximations of the Cahn-Hilliard-type equation with concentration-dependent mobility. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method for the space and the implicit Euler method for the time. Numerical experiments are carried out to illustrate the theoretical analysis.

Stochastic systems with Markovian switching have been used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In this paper, we study the Euler-Maruyama (EM) method for neutral stochastic functional differential equations with Markovian switching. The main aim is to show that the numerical solutions will converge to the true solutions. Moreover, we obtain the convergence order of the approximate solutions.

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

A new class of generalized dynamical systems involving generalized f-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

The purpose of this paper is to study the existence and convergence analysis of the solutions of the system of mixed variational inequalities in Banach spaces by using the generalized f projection operator. The results presented in this paper improve and extend important recent results of Zhang et al. (2011) and Wu and Huang (2007) and some recent results.

Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubic B-spline. Usual finite difference scheme is used for time and space integrations. Cubic B-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

Many applications of various self-adjoint differential equations, whose solutions are complex, are produced (Arfken, 1985; Gandarias, 2011; and Delkhosh, 2011). In this work we propose a method for the solving some self-adjoint equations with variable change in problem, and then we obtain a analytical solutions. Because this solution, an exact analytical solution can be provided to us, we benefited from the solution of numerical Self-adjoint equations (Mohynl-Din, 2009; Allame and Azal, 2011; Borhanifar et al. 2011; Sweilam and Nagy, 2011; Gülsu et al. 2011; Mohyud-Din et al. 2010; and Li et al. 1996).

We develop a new-two-stage finite difference method for computing approximate solutions of a system of third-order boundary value problems associated with odd-order obstacle problems. Such problems arise in physical oceanography (Dunbar (1993) and Noor (1994), draining and coating flow problems (E. O. Tuck (1990) and L. W. Schwartz (1990)), and can be studied in the framework of variational inequalities. We show that the present method is of order three and give numerical results that are better than the other available results. Numerical example is presented to illustrate the applicability and efficiency of the new method.