• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.255-267
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• 2007

In this work, by using Xu's inequality, Nalder's results, the notion of $(A,\;{\eta})-accretive$ mappings and the new resolvent operator technique associated with $(A,\;{\eta})-accretive$ mappings due to Lan et al., we study the existence of solutions for a new class of $(A,\;{\eta})-accretive$ variational inclusion problems with non-accretive set-valued mappings and the convergence of the iterative sequences generated by the algorithms in Banach spaces. Our results are new and extend, improve and unify the corresponding results in this field.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.785-827
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• 2023

In this paper, under some new appropriate conditions imposed on the parameter and mappings involved in the resolvent operator associated with a P-accretive mapping, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. This paper is also concerned with the construction of a new iterative algorithm using the resolvent operator technique and Nadler's technique for solving a new system of generalized multi-valued resolvent equations in a Banach space setting. The convergence analysis of the sequences generated by our proposed iterative algorithm under some appropriate conditions is studied. The final section deals with the investigation and analysis of the notion of H(·, ·)-co-accretive mapping which has been recently introduced and studied in the literature. We verify that under the conditions considered in the literature, every H(·, ·)-co-accretive mapping is actually P-accretive and is not a new one. In the meanwhile, some important comments on H(·, ·)-co-accretive mappings and the results related to them appeared in the literature are pointed out.

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.243-267
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• 2013

This paper is dedicated to study a new class of general nonlinear random A-maximal $m$-relaxed ${\eta}$-accretive (so called (A, ${\eta}$)-accretive [49]) equations with random relaxed cocoercive mappings and random fuzzy mappings in $q$-uniformly smooth Banach spaces. By utilizing the resolvent operator technique for A-maximal $m$-relaxed ${\eta}$-accretive mappings due to Lan et al. and Chang's lemma [13], some new iterative algorithms with mixed errors for finding the approximate solutions of the aforesaid class of nonlinear random equations are constructed. The convergence analysis of the proposed iterative algorithms under some suitable conditions are also studied.

• East Asian mathematical journal
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• v.27 no.1
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• pp.11-21
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• 2011

In this paper, we try to extend the viscosity approximation technique to find a particular common zero of a finite family of accretive mappings in a Banach space which is strictly convex reflexive and has a weakly sequentially continuous duality mapping. The explicit viscosity approximation scheme is proposed and its strong convergence to a solution of a variational inequality is proved.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.313-325
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• 2010

General models for the relaxed proximal point algorithm using the notion of relative maximal accretiveness (RMA) are developed, and then the convergence analysis for these models in the context of solving a general class of nonlinear inclusion problems differs significantly than that of Rockafellar (1976), where the local Lipschitz continuity at zero is adopted instead. Moreover, our approach not only generalizes convergence results to real Banach space settings, but also provides a suitable alternative to other problems arising from other related fields.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.337-348
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• 2004

In this paper, we suggest and analyze Noor (three-step) iterative scheme for solving nonlinear strongly accretive operator equation Tχ = f. The results obtained in this paper represent an extension as well as refinement of previous known results.

• East Asian mathematical journal
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• v.17 no.2
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• pp.349-365
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• 2001

Let X be a real Banach space and $A:X{\rightarrow}2^X$ be an $\alpha$-strongly accretive operator. It is proved that if the duality mapping J of X satisfies Condition (I) with additional conditions, then the Ishikawa and Mann iteration processes with errors converge strongly to the unique solution of operator equation $z{\in}Ax$. In addition, the convergence of the Ishikawa and Mann iteration processes with errors for $\alpha$-strongly pseudo-contractive operators is given.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.935-950
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• 2012

In this paper, we introduce and study a new system of generalized set-valued quasi-variational-like inclusions with (A, ${\eta}$)-accretive mapping in Banach spaces. By using the resolvent operator associated with (A, ${\eta}$)-accretive mappings, we construct a new iterative algorithm for approximating the solution of this system of variational inclusions. We also prove the existence of solutions and the convergence of the sequences generated by the algorithm in Banach spaces. The results presented in this paper extend and improve some known results in the literature.

• Korean Journal of Mathematics
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• v.18 no.4
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• pp.441-464
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• 2010

In this paper, we introduce an iterative method for finding common elements of the set of solutions to a general system of variational inequalities for inverse-strongly accretive mappings and of the set of fixed points of strict pseudo-contractions in a real Banach space. The results presented in this paper mainly improve and extend the corresponding results announced by many others.

• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.477-485
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• 1997

Let E be a smooth Banach space. Suppose T:$E \rightarrow E$ is a strongly accretive map. It is proved that each of the two well known fixed point iteration methods (the Mann and ishikawa iteration methods), under suitable conditions converges strongly to a solution of the equation $T_x=f$.