Gregus Type Research Articles

STRONG CONVERGENCE THEOREM FOR UNIFORMLY L-LIPSCHITZIAN MAPPING OF GREGUS TYPE IN BANACH SPACES

In this paper, we introduced a new mapping called Uniformly L-Lipschitzian mapping of Gregus type, and used the Mann iterative scheme to approximate the fixed point. A Strong convergence result for the sequence generated by the scheme is shown in real Banach space. Our result generalized and unifybmany recent results in this area of research. In addition, using Java(jdk1.8.0_101), we give a numericalbexample to support our claim.

Fixed point theorems of generalized Gregus type in quasi-metric spaces for two pairs of mappings satisfying common coincidence range property

The purpose of this paper is to prove some general fixed point theorems for two pairs of mappings satisfying implicit relations of generalized Gregus type in quasi-metric spaces without the notion of sequence and inequality. As applications we obtain new results for mappings satisfying contractive / extensive conditions of integral type and in G-metric spaces.

COMMON FIXED POINT FOR TANGENTIAL MAPS OF GREGUS TYPE ON FUZZY METRIC SPACES

In this paper, we define extend the results of many others. We prove common fixed point theorems on tangential property for a Gregus type on pair of fuzzy metric spaces. We also deal on some coupled coincidence and common fixed point theorems.

Some common fixed point theorems for Gregus type mappings

In this paper, sufficient conditions for the existence of common fixed points for a compatible pair of self maps of Gregustype in the framework of convex metric spaces have been obtained. Also, established the existence of common fixed points for a pair of compatible mappings of type (B) and consequently for compatible mappings of type (A). The proved results generalize and extend some of the well known results of the literature.

An integral type fixed point theorem for multi-valued mappings employing strongly tangential property

Sintunavarat and Kumam (W. Sintunavarat, P. Kumam, Gregus-type common fixed point theorems for tangential multi-valued mappings of integral type in metric spaces, Int. J. Math. Math. Sci. 2011 12 (Article ID 923458)) extended the tangential property to hybrid pair of mappings which generalizes the idea of tangential property due to Pathak and Shahzad (H.K. Pathak, N. Shahzad, Gregus type fixed point results for tangential mappings satisfying contractive conditions of integral type, Bull. Belg. Math. Soc. Simon Stevin 16(2) (2009) 277–288). In the present paper, we introduce the notion of strong tangential property and utilize the same to prove an integral type metrical common fixed point theorem for non-self mappings. An illustrative example is also furnished to support our main result. Our results are corrected, improved and generalized versions of a multitude of relevant common fixed point theorems of the existing literature.

Some Integral Type Fixed-Point Theorems and an Application to Systems of Functional Equations

In this paper, we prove a new common fixed point theorem for four self mappings by using the notions of compatibility and subsequential continuity (alternate subcompatibility and reciprocal continuity) in metric spaces satisfying a general contractive condition of integral type. We give some examples to support the useability of our main result. Also, we obtain some fixed point theorems of Gregus type for four mappings satisfying a strict general contractive condition of integral type in metric spaces. We conclude the paper with an application of our main result to solvability of systems of functional equations.

COMMON FIXED POINT THEOREM AND INVARIANT APPROXIMATION IN COMPLETE LINEAR METRIC SPACES

A common fixed point result of Gregus type for subcompatible mappings defined on a complete linear metric space is obtained. The considered underlying space is generalized from Banach space to complete linear metric spaces, which include Banach space and complete metrizable locally convex spaces. Invariant approximation results have also been determined as its application.

Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type

In this article, we define a tangential property which can be used not only for single-valued mappings but also for multi-valued mappings, and used it in the prove for the existence of a common fixed point theorems of Gregus type for four mappings satisfying a strict general contractive condition of integral type in metric spaces. Our theorems generalize and unify main results of Pathak and Shahzad (Bull. Belg. Math. Soc. Simon Stevin 16, 277-288, 2009) and several known fixed point results.

Common fixed points and invariant approximation for Gregus type contraction mappings

Some new common fixed point theorems for Gregus type contraction mappings have been obtained in convex metric spaces. As applications, invariant approximation results for these types of mappings are obtained. The proved results generalize, unify and extend some of the known results of M.A. Al-Thagafi (Int. J. Math. Sci. 18:613–616, 1995; J. Approx. Theory 85:318–323, 1996), M.A. Al-Thagafi and N. Shahzad (Nonlinear Anal. 64:2778–2786, 2006), L. Ciric (Publ. Inst. Math. 49:174–178, 1991; Arch. Math. (BRNO) 29:145–152, 1993), M.L. Diviccaro, B. Fisher, S. Sessa (Publ. Math. (Debr.) 34:83–89, 1987), B. Fisher and S. Sessa (Int. J. Math. Math. 9:23–28, 1986), M. Gregus (Boll. Un. Mat. Ital. (5) 7-A:193–198, 1980), L. Habiniak (J. Approx. Theory 56:241–244, 1989), N. Hussain, B.E. Rhoades and G. Jungck (Numer. Func. Anal. Optim. 28:1139–1151, 2007), G. Jungck (Int. J. Math. Math. Sci. 13:497–500, 1990), G. Jungck and S. Sessa (Math. Jpn. 42:249–252, 1995), R.N. Mukherjee and V. Verma (Math. Jpn. 33:745–749, 1988), T.D. Narang and S. Chandok (Ukr. Math. J. 62:1367–1376, 2010), S.A. Sahab, M.S. Khan and S. Sessa (J. Approx. Theory 55:349–351, 1988), N. Shahzad (J. Math. Anal. Appl. 257:39–45, 2001; Rad. Math. 10:77–83, 2001; Int. J. Math. Game Theory Algebra 13:157–159, 2003), S.P. Singh (J. Approx. Theory 25:89–90, 1979), A. Smoluk (Mat. Stosow. 17:17–22, 1981), P.V. Subrahmanyam (J. Approx. Theory 20:165–172, 1977) and of few others.

Common fixed point theorems for uniformly subcompatible mappings satisfying more general condition

We prove common fixed point theorems for uniformly subcompatible mappings satisfying a more generalized ciric type condition and a condition more general than gregus type condition in a locally convex domain. As an application, we have also established best approximation result. Our results extend recent results existing in the literature.

Common fixed point theorems for uniformly subcompatible mappings satisfying more general condition

We prove common fixed point theorems for uniformly subcompatible mappings satisfying a more generalized ciric type condition and a condition more general than gregus type condition in a locally convex domain. As an application, we have also established best approximation result. Our results extend recent results existing in the literature.

Gregus‐Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces

The concept of tangential for single‐valued mappings is extended to multivalued mappings and used to prove the existence of a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. Consequently, several known fixed point results generalized and improved the corresponding recent result of Pathak and Shahzad (2009) and many authors.

A Gregus type common fixed point theorem in normed spaces with application

n this paper, we introduce the notion of $\phi$-weakly compatible mapping for a pair of mappings. A fixed point theorem for two pairs of $\phi$-weakly compatible mappings satisfying a rational type contraction in a normed space is also established. Subsequently we use our result to find existence of solutions of variational inequalities.

Gregus type fixed point results for tangential mappings satisfying contractive conditions of integral type

The notion of pair-wise tangential mappings, which is a generalization of mappings satisfying (E.A) property, is introduced and used to prove a common fixed point theorem of Gregus type for a quadruple of self mappings of a metric space satisfying a strict general contractive condition of integral type. Our main result generalizes a recent result of A. Djoudi, A. Alioche [Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, J. Math. Anal. Appl. 329 (2007), 31-45].

Common fixed point theorems of Gregus type for weakly compatible mappings satisfying generalized contractive conditions

We prove a common fixed point theorem of Gregus type for four mappings satisfying a generalized contractive condition in metric spaces using the concept of weak compatibility which generalizes theorems of [I. Altun, D. Turkoglu, B.E. Rhoades, Fixed points of weakly compatible mappings satisfying a general contractive condition of integral type, Fixed Point Theory Appl. 2007 (2007), article ID 17301; A. Djoudi, L. Nisse, Gregus type fixed points for weakly compatible mappings, Bull. Belg. Math. Soc. 10 (2003) 369–378; A. Djoudi, A. Aliouche, Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, J. Math. Anal. Appl. 329 (1) (2007) 31–45; P. Vijayaraju, B.E. Rhoades, R. Mohanraj, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 15 (2005) 2359–2364; X. Zhang, Common fixed point theorems for some new generalized contractive type mappings, J. Math. Anal. Appl. 333 (2) (2007) 780–786]. We prove also a common fixed point theorem which generalizes Theorem 3.5 of [H.K. Pathak, M.S. Khan, T. Rakesh, A common fixed point theorem and its application to nonlinear integral equations, Comput. Math. Appl. 53 (2007) 961–971] and common fixed point theorems of Gregus type using a strict generalized contractive condition, a property (E.A) and a common property (E.A).

Common Fixed Point and Invariant Approximation Results for Gregus Type I-Contractions

A fixed point theorem of Ciric, Diviccaro et al., Fisher and Sessa, Gregus, Jungck, and Mukherjee and Verma is generalized to weakly compatible maps. As applications, common fixed point and approximation results for Gregus type I-contractions are obtained. Our results unify and generalize various known results to the more general classes of noncommuting mappings.

Common fixed points in best approximation for Banach operator pairs with Ćirić type I-contractions

The common fixed point theorems, similar to those of Ćirić [Lj.B. Ćirić, On a common fixed point theorem of a Gregus type, Publ. Inst. Math. (Beograd) (N.S.) 49 (1991) 174–178; Lj.B. Ćirić, On Diviccaro, Fisher and Sessa open questions, Arch. Math. (Brno) 29 (1993) 145–152; Lj.B. Ćirić, On a generalization of Gregus fixed point theorem, Czechoslovak Math. J. 50 (2000) 449–458], Fisher and Sessa [B. Fisher, S. Sessa, On a fixed point theorem of Gregus, Internat. J. Math. Math. Sci. 9 (1986) 23–28], Jungck [G. Jungck, On a fixed point theorem of Fisher and Sessa, Internat. J. Math. Math. Sci. 13 (1990) 497–500] and Mukherjee and Verma [R.N. Mukherjee, V. Verma, A note on fixed point theorem of Gregus, Math. Japon. 33 (1988) 745–749], are proved for a Banach operator pair. As applications, common fixed point and approximation results for Banach operator pair satisfying Ćirić type contractive conditions are obtained without the assumption of linearity or affinity of either T or I. Our results unify and generalize various known results to a more general class of noncommuting mappings.

On common fixed point and approximation results of Gregus type

Fixed point theorems of Ciric [3], Fisher and Sessa [4], Gregus [5], Jungck [10] and Mukherjee and Verma [17] are generalized to a locally convex space. As applications, common fixed point and invariant app roximation results for subcompatible maps are obtained. Our results unify and generalize various known results to a more general class of noncommuting mappings.

Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type

We prove a common fixed point theorem of Gregus type for four mappings satisfying a contractive condition of integral type in metric spaces using the concept of weak compatibility which generalizes Theorem 2 of [A. Djoudi, L. Nisse, Gregus type fixed points for weakly compatible mappings, Bull. Belg. Math. Soc. 10 (2003) 369–378] and other papers. We prove also common fixed point theorems of Gregus type using a strict contractive condition of integral type, a property (E.A) and a common property (E.A) introduced by [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188] and [W. Liu, J. Wu, Z. Li, Common fixed points of single-valued and multi-valued maps, Int. J. Math. Math. Sci. 19 (2005) 3045–3055], respectively.

Common fixed point theorems of gregus type in convex metric spaces

In this paper, we prove common fixed point theorems of Gregus type for three mappings in convex metric spaces. We extend and generalize some well known results by many authors.