Single projection method for pseudo-monotone variational inequality in Hilbert spaces

ABSTRACT

In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given.     

A hybrid steepest-descent method for variational inequalities in Hilbert spaces

A Mann-type hybrid steepest-descent method for solving the variational inequality ?F(u*), v ? u*? ≥ 0, v ∈ C is proposed, where F is a Lipschitzian and strong monotone operator in a real Hilbert space H and C is the intersection of the fixed point sets of finitely many non-expansive mappings in H. This method combines the well-known Mann's fixed point method with the hybrid steepest-descent method. Strong convergence theorems for this method are established, which extend and improve certain corresponding results in recent literature, for instance, Yamada (The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Amsterdam, Holland, 2001, pp. 473–504), Xu and Kim (Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theor. Appl. 119 (2003), pp. 185–201), and Zeng, Wong and Yao (Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theor. Appl. 132 (2007), pp. 51–69).     

Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type

The Hilbert matrix induces a bounded operator on most Hardy and Bergman spaces, as was shown by Diamantopoulos and Siskakis. We generalize this for any Hankel operator on Hardy spaces by using a result of Hollenbeck and Verbitsky on the Riesz projection and also compute the exact value of the norm of the Hilbert matrix. Using a new technique, we determine the norm of the Hilbert matrix on a wide range of Bergman spaces.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

Adjoints of composition operators on Hilbert spaces of analytic functions

We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.     

A characterization of an order ideal in Riesz spaces

In this paper we give a characterization of order ideals in Riesz spaces.

     

Existence of solutions for fractional differential equation with p-Laplacian through variational method

In this paper, a class of fractional differential equation with p-Laplacian operator is studied based on the variational approach. Combining the mountain pass theorem with iterative technique, the existence of at least one nontrivial solution for our equation is obtained. Additionally, we demonstrate the application of our main result through an example.     

Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Haar wavelet operational matrix has been widely applied in system analysis, system identification, optimal control and numerical solution of integral and differential equations. In the present paper we derive the Haar wavelet operational matrix of the fractional order integration, and use it to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations. The results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply.     

An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

In this paper, a new monotonicity, M$M$-monotonicity, is introduced, and the resolvent operator of an M$M$-monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI(C,F+G)$(C,F+G)$ and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F$F$ in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping _∏C(⋅)${\prod }_C(\cdot )$ is semismooth, is given for calculating ε$\epsilon$-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.     

Convergence theorems for fixed point problems and variational inequality problems in Hilbert spaces

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an α ‐inverse strongly monotone mapping in a Hilbert space. We show that the sequence converges strongly to a common element of two sets under some mild conditions on parameters (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and estimate of solutions of some nonlinear Volterra difference equations in Hilbert spaces

In this paper we consider the Volterra difference equation

     

Estimates for norms of resolvents of~operators on tensor products of Hilbert spaces

A class of linear operators on tensor products of Hilbert spaces is considered. That class contains integro-differential operators arising in various applications. Estimates for the norm of the resolvent of considered operators are derived. By virtue of the obtained estimates, the spectrum of perturbed operators is investigated. These results are new even in the finite-dimensional case. Applications to integro-differential operators are also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43-52], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506-515], and many others.     

Norm estimates for functions of two operators on tensor products of Hilbert spaces

Analytic operator valued functions of two operators on tensor products of Hilbert spaces are considered. A precise norm estimate is established. Applications to operator differential equations are also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reverses of the continuous triangle inequality for Bochner integral in complex Hilbert spaces

Some reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in complex Hilbert spaces are given. Applications for complex-valued functions are provided as well.     

Inequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces

In this paper we have studied the separation for the Laplace-Beltrami differential operator of the form

     

Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

Necessary and sufficient conditions are given for the fractional integral operator to be bounded from weighted strong and weak spaces within the range into suitable weighted and Lipschitz spaces. We also characterize the weights for which can be extended to a bounded operator from weighted into a weighted Lipschitz space of order . Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of between weighted Lipschitz spaces.

     

A general multiplicity theorem for certain nonlinear equations in Hilbert spaces

In this paper, we prove the following general result. Let be a real Hilbert space and a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that

Then, for each for which the set is not convex and for each convex set dense in , there exist and 0$"> such that the equation

has at least three solutions.

     

Solution of nonlinear fractional differential equations using homotopy analysis method

In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.