• Title/Summary/Keyword: strongly quasi-convex

Search Result 7, Processing Time 0.025 seconds

QUASI STRONGLY E-CONVEX FUNCTIONS WITH APPLICATIONS

  • Hussain, Askar;Iqbal, Akhlad
    • Nonlinear Functional Analysis and Applications
    • /
    • v.26 no.5
    • /
    • pp.1077-1089
    • /
    • 2021
  • In this article, we introduce the quasi strongly E-convex function and pseudo strongly E-convex function on strongly E-convex set which generalizes strongly E-convex function defined by Youness [10]. Some non trivial examples have been constructed that show the existence of these functions. Several interesting properties of these functions have been discussed. An important characterization and relationship of these functions have been established. Furthermore, a nonlinear programming problem for quasi strongly E-convex function has been discussed.

GENERALIZED BI-QUASI-VARIATIONAL-LIKE INEQUALITIES ON NON-COMPACT SETS

  • Cho, Yeol Je;Chowdhury, Mohammad S.R.;Ha, Je Ai
    • Communications of the Korean Mathematical Society
    • /
    • v.32 no.4
    • /
    • pp.933-957
    • /
    • 2017
  • In this paper, we prove some existence results of solutions for a new class of generalized bi-quasi-variational-like inequalities (GBQVLI) for (${\eta}-h$)-quasi-pseudo-monotone type I and strongly (${\eta}-h$)-quasi-pseudo-monotone type I operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. To obtain our results on GBQVLI for (${\eta}-h$)-quasi-pseudo-monotone type I and strongly (${\eta}-h$)-quasi-pseudo-monotone type I operators, we use Chowdhury and Tan's generalized version of Ky Fan's minimax inequality as the main tool.

WEAK AND STRONG CONVERGENCE FOR QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Kim, Gang-Eun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.49 no.4
    • /
    • pp.799-813
    • /
    • 2012
  • In this paper, we first show that the iteration {$x_n$} defined by $x_{n+1}=P((1-{\alpha}_n)x_n +{\alpha}_nTP[{\beta}_nTx_n+(1-{\beta}_n)x_n])$ converges strongly to some fixed point of T when E is a real uniformly convex Banach space and T is a quasi-nonexpansive non-self mapping satisfying Condition A, which generalizes the result due to Shahzad [11]. Next, we show the strong convergence of the Mann iteration process with errors when E is a real uniformly convex Banach space and T is a quasi-nonexpansive self-mapping satisfying Condition A, which generalizes the result due to Senter-Dotson [10]. Finally, we show that the iteration {$x_n$} defined by $x_{n+1}={\alpha}_nSx_n+{\beta}_nT[{\alpha}^{\prime}_nSx_n+{\beta}^{\prime}_nTx_n+{\gamma}^{\prime}_n{\upsilon}_n]+{\gamma}_nu_n$ converges strongly to a common fixed point of T and S when E is a real uniformly convex Banach space and T, S are two quasi-nonexpansive self-mappings satisfying Condition D, which generalizes the result due to Ghosh-Debnath [3].

GLOBAL CONVERGENCE PROPERTIES OF TWO MODIFIED BFGS-TYPE METHODS

  • Guo, Qiang;Liu, Jian-Guo
    • Journal of applied mathematics & informatics
    • /
    • v.23 no.1_2
    • /
    • pp.311-319
    • /
    • 2007
  • This article studies a modified BFGS algorithm for solving smooth unconstrained strongly convex minimization problem. The modified BFGS method is based on the new quasi-Newton equation $B_k+1{^s}_k=yk\;where\;y_k^*=yk+A_ks_k\;and\;A_k$ is a matrix. Wei, Li and Qi [WLQ] have proven that the average performance of two of those algorithms is better than that of the classical one. In this paper, we prove the global convergence of these algorithms associated to a general line search rule.

REGULARIZED MIXED QUASI EQUILIBRIUM PROBLEMS

  • Noor Muhammad Aslam
    • Journal of applied mathematics & informatics
    • /
    • v.23 no.1_2
    • /
    • pp.183-191
    • /
    • 2007
  • In this paper, we introduce and study a new class of equilibrium problems, known as regularized mixed quasi equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving equilibrium problems and variational inequalities involving the convex sets.

ITERATIVE PROCESS FOR FINDING FIXED POINTS OF QUASI-NONEXPANSIVE MULTIMAPS IN CAT(0) SPACES

  • Pitchaya Kingkam;Jamnian Nantadilok
    • Korean Journal of Mathematics
    • /
    • v.31 no.1
    • /
    • pp.35-48
    • /
    • 2023
  • Let 𝔼 be a CAT(0) space and K be a nonempty closed convex subset of 𝔼. Let T : K → 𝓟(K) be a multimap such that F(T) ≠ ∅ and ℙT(x) = {y ∈ Tx : d(x, y) = d(x, Tx)}. Define sequence {xn} by xn+1 = (1 - α)𝜈n⊕αwn, yn = (1 - β)un⊕βwn, zn = (1-γ)xn⊕γun where α, β, γ ∈ [0; 1]; un ∈ ℙT (xn); 𝜈n ∈ ℙT (yn) and wn ∈ ℙT (zn). (1) If ℙT is quasi-nonexpansive, then it is proved that {xn} converges strongly to a fixed point of T. (2) If a multimap T satisfies Condition(I) and ℙT is quasi-nonexpansive, then {xn} converges strongly to a fixed point of T. (3) Finally, we establish a weak convergence result. Our results extend and unify some of the related results in the literature.