Jeong Sheok Ume

Monotone Generalized Nonlinear Contractions in Partially Ordered Metric Spaces

A concept of -monotone mapping is introduced, and some fixed and common fixed point theorems for -non-decreasing generalized nonlinear contractions in partially ordered complete metric spaces are proved. Presented theorems are generalizations of very recent fixed point theorems due to Agarwal et al. (2008).

IMPLICIT MANN FIXED POINT ITERATIONS FOR PSEUDO-CONTRACTIVE MAPPINGS

Abstract Let K be a compact convex subset of a real Hilbert space H and T : K → K a continuous hemi-contractive map. Let { a n } , { b n } and { c n } be real sequences in [0, 1] such that a n + b n + c n = 1 , and { u n } and { v n } be sequences in K . In this paper we prove that, if { b n } , { c n } and { v n } satisfy some appropriate conditions, then for arbitrary x 0 ∈ K , the sequence { x n } defined iteratively by x n = a n x n − 1 + b n T v n + c n u n ; n ≥ 1 , converges strongly to a fixed point of T .

Common fixed point theorems for families of weakly compatible maps

In this paper the existence and approximation of a unique common fixed point of two families of weakly compatible self-maps on a complete metric space are investigated. An example is presented to show that our results for the mappings considered satisfying non-linear contractive type conditions are genuine generalizations of the recent result for metric spaces [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439-448, Theorem 3.3] and many other known results.

Solvability of a Second Order Nonlinear Neutral Delay Difference Equation

This paper studies the second-order nonlinear neutral delay difference equation ?[?????(????

On Mann implicit iterations for strongly accretive and strongly pseudo-contractive mappings

Abstract For a Lipschitz strongly accretive map considered by Chidume in [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint No. IC2000098; C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantam on his 80th birthday, vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 2003, pp. 383–429], it is known that a classical Picard-type iteration process converges strongly to a zero of the operator. He also proved that the rate of convergence, in this case, is at least as fast as a geometric progression. In this paper we study the Mann implicit iteration sequence for strongly accretive and strongly pseudo-contractive mappings. We showed that this implicit scheme gives better convergence rate estimate. Presented results improve the corresponding results of [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint No. IC2000098; C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantam on his 80th birthday, vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 2003, pp. 383–429; L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Am. Math. Soc. 125 (2) (1997) 1363–1366; W.R. Sastry, G.V.R. Babu, Approximation of fixed points of strictly pseudo-contractive mappings on arbitrary closed, convex sets in a Banach space, Proc. Amer. Math. Soc. 128 (2000) 2907–2909; Y. Song, R. Chen, Viscosity approximative methods to Cesaro means for non-expansive mappings, Appl. Math. Comput. 186 (2) (2007) 1120–1128].

Existence Theorems for Generalized Distance on Complete Metric Spaces

We first introduce the new concept of a distance called -distance, which generalizes -distance, Tatarus distance, and -distance. Then we prove a new minimization theorem and a new fixed point theorem by using a -distance on a complete metric space. Our results extend and unify many known results due to Caristi, Ćirić, Ekeland, Kada-Suzuki-Takahashi, Kannan, Ume, and others.

Generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings

In this paper, we introduce the concept of generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings. By using the resolvent operator technique for maximal monotone mapping, we obtain that the generalized mixed quasivariational inclusions for fuzzy mappings are equivalent to the fuzzy fixed-point problems and the generalized mixed resolvent equations for fuzzy mappings, respectively. We suggest a number of iterative algorithms, establish the existence results of solutions for the generalized mixed quasivariational inclusions for fuzzy mappings and prove the convergence of the algorithms. The results presented in this paper are improvements of previously known results.

Completely generalized multivalued nonlinear quasi-variational inclusions.

We introduce and study a new class of completely generalized multivalued nonlinear quasivariational inclusions. Using the resolvent operator technique for maximal monotone mappings, we suggest two kinds of iterative algorithms for solving the completely generalized multivalued nonlinear quasi-variational inclusions. We establish both four existence theorems of solutions for the class of completely generalized multivalued nonlinear quasivariational inclusions involving strongly monotone, relaxed Lipschitz, and generalized pseudocontractive mappings, and obtain a few convergence results of iterative sequences generated by the algorithms. The results presented in this paper extend, improve, and unify a lot of results due to Adly, Huang, Jou-Yao, Kazmi, Noor, Noor-Al-Said, Noor-Noor, Noor-Noor-Rassias, Shim-Kang-Huang-Cho, Siddiqi-Ansari, Verma, Yao, and Zhang.

Some results on fixed point theorems for multivalued mappings in complete metric spaces

-distance, we first prove common fixed pointtheorems for multivalued mappings in complete metric spaces, then these theoremsare used to improve Ciri´ ´c’s fixed point theorem [ 1], Kada-Suzuki-Takahashi’s fixedpoint theorem [2], and Ume’s fixed point theorem [3].2. Preliminaries. Throughout, we denote by Nthe set of all positive integers andby Rthe set of all real numbers.Definition 2.1 (see [2]). Let

RESULTS ON COMMON FIXED POINTS

We establish common fixed point theorems related with families of self- mappings on metric spaces. Our results extend, improve, and unify the results due to Fisher (1977, 1978, 1979, 1981, 1984), Jungck (1988), and Ohta and Nikaido (1994).