Wyniki wyszukiwania - BazTech

1

Generalized split null point of sum of monotone operators in Hilbert spaces

Mebawondu Akindele A., Abass Hammed A., Oyewole Olawale K., Aremu Kazeem O., Narain Ojen K.

Demonstratio Mathematica

|

2021

|

Vol. 54, nr 1

359--376

EN

In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.

2

A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces

Jolaoso Lateef Olakunle

Demonstratio Mathematica

|

2021

|

Vol. 54, nr 1

527--547

EN

In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.

3

On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

Ogwo Grace N., Izuchukwu Chinedu, Aremu Kazeem O., Mewomo Oluwatosin T.

Demonstratio Mathematica

|

2020

|

Vol. 53, nr 1

95--111

EN

Our main interest in this article is to introduce and study the class of θ-generalized demimetric mappings in Hadamard spaces. Also, a Halpern-type proximal point algorithm comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. Furthermore, we apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems in Hadamard spaces.

4

Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

Rouhani Behzad Djafari, Piranfar Mohsen Rahimi

Demonstratio Mathematica

|

2019

|

Vol. 52, nr 1

274--282

EN

We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü(t) +𝛾 ů(t) + Au(t) = f(t), where A is a maximal monotone andα-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on 𝛾 and f(t) we show that A-1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A-1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A-1(0). As a discrete version of (SEE), we consider the following second order difference equation un+1-un-αn(un-un-1)+λnAun+1 ∋ f(t), where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417], we prove ergodic, weak and strong convergence theorems for the sequence un, and show that the limit is the asymptotic center of un and belongs to A−1(0). This again shows that A−1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3-11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465-476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289-1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648-654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369-4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836-857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3-11].

5

Koiter shell governed by strongly monotone constitutive equations

Kalita P.

International Journal of Applied Mathematics and Computer Science

|

2004

|

Vol. 14, no 2

127-137

EN

In this paper we use the theory of monotone operators to generalize the linear shell model presented in (Blouza and Le Dret, 1999) to a class of physically nonlinear models. We present a family of nonlinear constitutive equations, for which we prove the existence and uniqueness of the solution of the presented nonlinear model, as well as the convergence of the Galerkin method. We also present the physical discussion of the model.

6

Dynamic Contact Problems With Velocity Conditions

Chau O., Motreanu V. V.

International Journal of Applied Mathematics and Computer Science

|

2002

|

Vol. 12, no 1

17-26

EN

We consider dynamic problems which describe frictional contact between a body and a foundation. The constitutive law is viscoelastic or elastic and the frictional contact is modelled by a general subdifferential condition on the velocity, including the normal damped responses. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of second-order evolution variational inequalities. We show that the solutions of the viscoelastic problems converge to the solution of the corresponding elastic problem as the viscosity tensor tends to zero and when the frictional potential function converges to the corresponding function in the elastic problem

7

Proximal-based regularization methods and successive approximation of variational inequalities in Hilbert spaces

Kaplan A., Tichatschke R.

Control and Cybernetics

|

2002

|

Vol. 31, no 3

521-544

EN

For variational inequalities with multi-valued, maximal monotone operators in Hilbert spaces we study proximal-based methods with an improvement of the data approximation after each (approximately performed) proximal iteration. The standard conditions on a distance functional of Bregman's type are weakened, depending on a "reserve of monotonicity" of the operator in the variational inequality, and the enlargement concept is used for approximating the operator. Weak convergence of the proxinnal iterates to a solution of tire original problem is proved. The construction of the [epsilon]-enlargement of monotone operators is analyzed for some particular cases.