• Kyungpook Mathematical Journal
• /
• v.59 no.3
• /
• pp.445-464
• /
• 2019

In this paper, we propose a new iterative algorithm generated by a finite family of accretive operators in a q-uniformly smooth Banach space. We prove the strong convergence of the proposed iterative algorithm. The results presented in this paper are interesting extensions and improvements of known results of Qin et al. [Fixed Point Theory Appl. 2014(2014): 166], Kim and Xu [Nonlinear Anal. 61(2005), 51-60] and Benavides et al. [Math. Nachr. 248(2003), 62-71].

• East Asian mathematical journal
• /
• v.36 no.5
• /
• pp.571-581
• /
• 2020

In this paper, an equilibrium problems involving a finite family of maximal monotone operators and inverse-strongly monotone operators are introduced and investigated. A strong convergence theorem of common solutions is obtained in Hilbert spaces.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.4
• /
• pp.717-728
• /
• 2008

Let H be a real Hilbert space and S = {T(s) : $0\;{\leq}\;s\;<\;{\infty}$} be a nonexpansive semigroup on H such that $F(S)\;{\neq}\;{\emptyset}$ For a contraction f with coefficient 0 < $\alpha$ < 1, a strongly positive bounded linear operator A with coefficient $\bar{\gamma}$ > 0. Let 0 < $\gamma$ < $\frac{\bar{\gamma}}{\alpha}$. It is proved that the sequences {$x_t$} and {$x_n$} generated by the iterative method $$x_t\;=\;t{\gamma}f(x_t)\;+\;(I\;-\;tA){\frac{1}{{\lambda}_t}}\;{\int_0}^{{\lambda}_t}\;T(s){x_t}ds,$$ and $$x_{n+1}\;=\;{\alpha}_n{\gamma}f(x_n)\;+\;(I\;-\;{\alpha}_nA)\frac{1}{t_n}\;{\int_0}^{t_n}\;T(s){x_n}ds,$$ where {t}, {${\alpha}_n$} $\subset$ (0, 1) and {${\lambda}_t$}, {$t_n$} are positive real divergent sequences, converges strongly to a common fixed point $\tilde{x}\;{\in}\;F(S)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)\tilde{x},\;x\;-\;\tilde{x}{\rangle}\;{\leq}\;0$ for $x\;{\in}\;F(S)$.

• Nonlinear Functional Analysis and Applications
• /
• v.26 no.3
• /
• pp.451-474
• /
• 2021

In this paper, we investigate a hybrid algorithm for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators which is also a common fixed point problem for finite family of relatively quasi-nonexpansive mappings and split feasibility problem in uniformly convex real Banach spaces which are also uniformly smooth. The iterative algorithm employed in this paper is design in such a way that it does not require prior knowledge of operator norm. We prove a strong convergence result for approximating the solutions of the aforementioned problems and give applications of our main result to minimization problem and convexly constrained linear inverse problem.

• Communications of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.421-436
• /
• 2024

In this article, we propose a shrinking projection algorithm for solving a finite family of generalized equilibrium problem which is also a fixed point of a nonexpansive mapping in the setting of Hadamard manifolds. Under some mild conditions, we prove that the sequence generated by the proposed algorithm converges to a common solution of a finite family of generalized equilibrium problem and fixed point problem of a nonexpansive mapping. Lastly, we present some numerical examples to illustrate the performance of our iterative method. Our results extends and improve many related results on generalized equilibrium problem from linear spaces to Hadamard manifolds. The result discuss in this article extends and complements many related results in the literature.

• Nonlinear Functional Analysis and Applications
• /
• v.28 no.1
• /
• pp.175-203
• /
• 2023

In this paper, we study an iterative algorithm that is based on inertial proximal and contraction methods embellished with relaxation technique, for finding common solution of monotone variational inclusion, and fixed point problems of pseudocontractive mapping in real Hilbert spaces. We establish a strong convergence result of the proposed iterative method based on prediction stepsize conditions, and under some standard assumptions on the algorithm parameters. Finally, some special cases of general problem are given as applications. Our results improve and generalized some well-known and related results in literature.

• Journal of applied mathematics & informatics
• /
• v.28 no.1_2
• /
• pp.13-30
• /
• 2010

Let C be a nonempty closed convex subset of a real Hilbert space H. Consider the following iterative algorithm given by $x_0\;{\in}\;C$ arbitrarily chosen, $x_{n+1}\;=\;{\alpha}_n{\gamma}f(W_nx_n)+{\beta}_nx_n+((1-{\beta}_n)I-{\alpha}_nA)W_nP_C(I-s_nB)x_n$, ${\forall}_n\;{\geq}\;0$, where $\gamma$ > 0, B : C $\rightarrow$ H is a $\beta$-inverse-strongly monotone mapping, f is a contraction of H into itself with a coefficient $\alpha$ (0 < $\alpha$ < 1), $P_C$ is a projection of H onto C, A is a strongly positive linear bounded operator on H and $W_n$ is the W-mapping generated by a finite family of nonexpansive mappings $T_1$, $T_2$, ${\ldots}$, $T_N$ and {$\lambda_{n,1}$}, {$\lambda_{n,2}$}, ${\ldots}$, {$\lambda_{n,N}$}. Nonexpansivity of each $T_i$ ensures the nonexpansivity of $W_n$. We prove that the sequence {$x_n$} generated by the above iterative algorithm converges strongly to a common fixed point $q\;{\in}\;F$ := $\bigcap^N_{i=1}F(T_i)\;\bigcap\;VI(C,\;B)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)q,\;p\;-\;q{\rangle}\;{\leq}\;0$ for all $p\;{\in}\;F$. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.

• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.655-664
• /
• 1997

The existence and behavior of a bounded solution for a perturbed nonlinear differential equation of the type $$ (DE) x'(t) + Ax(t) \ni G(x(t)), t \in [0, \infty) $$ is considered. First, we consider the existence of a bounded solution with more simple assumptions using the concept of "the method of lines". Then we devote to study its behavior using recent results of almost non-expansive curve which is developed by Djafari Rouhani.i Rouhani.