• 대한수학회논문집
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• 제27권1호
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• pp.117-128
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• 2012

Using the concept of a g-monotone mapping we prove some common fixed point theorems for g-non-decreasing mappings which satisfy some generalized nonlinear contractions in partially ordered complete quasi b-metric spaces. The new theorems are generalizations of very recent fixed point theorems due to L. Ciric, N. Cakic, M. Rojovic, and J. S. Ume, [Monotone generalized nonlinear contractions in partailly ordered metric spaces, Fixed Point Theory Appl. (2008), article, ID-131294] and R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan [Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008), 1-8].

• 대한수학회지
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• 제55권4호
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• pp.849-875
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• 2018

The purpose of this paper is to present an operator method of regularization for the problem of finding a solution of a system of nonlinear ill-posed equations with a monotone hemicontinuous mapping and N inverse-strongly monotone mappings in Banach spaces. A regularization parameter choice is given and convergence rate of the regularized solutions is estimated. We also give the convergence and convergence rate for regularized solutions in connection with the finite-dimensional approximation. An iterative regularization method of zero order in a real Hilbert space and two examples of numerical expressions are also given to illustrate the effectiveness of the proposed methods.

• 대한수학회보
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• 제50권3호
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• pp.993-1005
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• 2013

In this paper, to include more generalized cases, the authors present a modified concept for the tripled and quadruple fixed point of the mixed monotone mappings. Also, they investigate the existence and uniqueness of fixed point of the ordered monotone operator with the Matkowski contractive conditions in the partial ordered metric spaces. As the direct consequences, the existence of coupled fixed point, tripled fixed point and quadruple fixed point are explored at the common framework and some previous results in [T. G. Bhaskar and V. Lakshmikan-tham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393; V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011), no. 15, 4889-4897; E. Karapinar and N. V. Luong, Quadruple fixed point theorems for nonlinear contractions, Computers and Mathematics with Applications (2012), doi:10.1016/j.camwa.2012.02061] are improved. Finally, some fixed point theorems are proved.

• 대한수학회논문집
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• 제25권4호
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• pp.583-607
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• 2010

The purpose of this paper is to prove strong convergence theorems for common fixed points of two families of weak relatively nonexpansive mappings and a family of equilibrium problems by a new monotone hybrid method in Banach spaces. Because the hybrid method presented in this paper is monotone, so that the method of the proof is different from the original one. We shall give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space $l^2$. Our results improve and extend the corresponding results announced in [W. Takahashi and K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008), Article ID 528476, 11 pages; doi:10.1155/2008/528476] and [Y. Su, Z. Wang, and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009), no. 11, 5616?5628] and some other papers.

• Kyungpook Mathematical Journal
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• 제53권3호
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• pp.307-318
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• 2013

In this paper, we introduce and study a new system of variational inclusions with B-monotone operators in Banach spaces. By using the proximal mapping associated with B-monotone operator, 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 for this system of variational inclusions. The results presented in this paper extend and improve some known results in the literature.

• 대한수학회논문집
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• 제26권4호
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• pp.685-693
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• 2011

General framework for proximal point algorithms based on the notion of (A, ${\eta}$)-maximal monotonicity (also referred to as (A, ${\eta}$)-monotonicity in literature) is developed. Linear convergence analysis for this class of algorithms to the context of solving a general class of nonlinear variational inclusion problems is successfully achieved along with some results on the generalized resolvent corresponding to (A, ${\eta}$)-monotonicity. The obtained results generalize and unify a wide range of investigations readily available in literature.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1453-1462
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• 2011

In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

• East Asian mathematical journal
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• 제18권1호
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• pp.95-109
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• 2002

In this paper, we study a class of variational inequalities, which is called the generalized set-valued mixed variational inequality. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert spaces, we have established an existence theorem of solutions for generalized set-valued mixed variational inequalities, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized set-valued mixed variational inequalities.

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.441-453
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• 2021

In this work, we define the concept of a mixed G-monotone mapping on a modular metric space endowed with a graph, and prove some fixed point theorems for this new class of mappings. Results of this paper extend coupled fixed point theorems from partially ordered metric spaces into the modular metric spaces endowed with a graph. An example is presented to illustrate the new result.

• Nonlinear Functional Analysis and Applications
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• 제26권4호
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• pp.839-853
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• 2021

Let 𝕽ⁿ be an Euclidean space and g : 𝕽ⁿ → 𝕽ⁿ be a monotone and continuous mapping. Suppose the convex constrained nonlinear monotone equation problem x ∈ 𝕮 s.t g(x) = 0 has a solution. In this paper, we construct an inertial-type algorithm based on the three-term derivative-free projection method (TTMDY) for convex constrained monotone nonlinear equations. Under some standard assumptions, we establish its global convergence to a solution of the convex constrained nonlinear monotone equation. Furthermore, the proposed algorithm converges much faster than the existing non-inertial algorithm (TTMDY) for convex constrained monotone equations.