Nenad Cakić

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 .

Modified approach to generalized Stirling numbers via differential operators

Abstract In this paper we give a modified approach to the generalized Stirling numbers of the second kind S r , s ( n , k ) and S r , s ( k ) . These numbers were firstly defined by Carlitz and recently studied extensively by Blasiak, Penson and Solomon. This approach depends on the previous results obtained by Carlitz, Toscano and Cakic. We show that Blasiak’s results can be investigated from Carlitz and Cakic’s results. Some interesting combinatorial identities are obtained.

A note on tvs-cone metric fixed point theory

Abstract In this paper, the concept of set-valued contraction of Nadler type in the setting of tvs-cone spaces was introduced and a fixed point theorem in the setting of tvs-cone spaces with respect to a solid cone was proved. Obtained results extend and generalize the main results of [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific Journal of Mathematics 30 (1969) 475–488] and [D. Wardowski, On set-valued contractions of Nadler type in cone metric spaces, Applied Mathematics Letters 24 (2011) 275–278]. Two examples are given to illustrate the usability of the results.

Generalized higher order Stirling numbers

In this paper new explicit expressions for both kinds of Comtet numbers and some interesting special cases are derived. Moreover, we define and study the generalized multiparameter non-central Stirling numbers and generalized Comtet numbers via differential operators. Furthermore, recurrence relations and new explicit formulas for those numbers are obtained. Finally some interesting special cases, new combinatorial identities and a connection between these numbers and some interesting polynomials are deduced.

Coincidence and fixed point theorems for generalized (ψ,φ)-weak nonlinear contraction in ordered K-metric spaces

In this paper, we establish coincidence and fixed point theorems for a pair of mappings satisfying a new generalized (@j,@f)-weak nonlinear contraction type condition in ordered K-metric spaces. The presented results generalize and extend the very recent results of Choudhury and Metiya [B.S. Choudhury and N. Metiya, The point of coincidence and common fixed point for a pair of mappings in cone metric spaces, Computers and Mathematics with Applications, 60 (2010) 1686-1695].

On common fixed point theorems for non-self hybrid mappings in convex metric spaces

Abstract In this paper, we introduce the concept of a contractive type non-self mappings for a two pair of multi-valued and single-valued mappings in metric spaces and prove some results on coincidence and common fixed points in complete convex metric spaces. Our theorems generalize and improve the theorems of Imdad and Khan [M. Imdad, L. Khan, Some common fixed point theorems for a family of mappings in metrically convex spaces, Nonlinear Anal. – Theor. 67 (9) (2007) 2717–2726; M. Imdad, L. Khan, Fixed point theorems for a family of hybrid pairs of mappings in metrically convex spaces, Fixed Point Theory Appl. 2005 (3) (2005) 281–294] and of Assad [N.A. Assad, Fixed point theorems for set valued transformations on compact sets, Boll. Un. Math. Ital. 4 (1973) 1–7] and several others theorems. To accomplish that, we shall use slightly improved version of methods of proofs, usually used in proving fixed point theorems for non-self mappings. An example is constructed to show that our results are genuine generalizations of the above mentioned results.

Strong convergence theorems for the implicit iteration process for a finite family of hemicontractive mappings in Banach space

Abstract The purpose of this work is to study the following implicit iteration scheme x n = α n x n − 1 + ( 1 − α n ) T n x n , n ≥ 1 , where T n = T n m o d N , and to prove several strongly convergent theorems of the iteration for a finite family of hemicontractive mappings in Banach space. Our results extend a recent result of Haiyun Zhou [Haiyun Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal. 68 (2008) 2977–2983] and Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], and we have proved that the sequence { x n } converges strongly to a common fixed point of a finite family of hemicontractive mappings { T i } i = 1 N .

q-Analogues of multiparameter non-central Stirling and generalized harmonic numbers

Abstract In this paper we derive q -analogues of the multiparameter non-central Stirling numbers of the first and second kind, introduced by El-Desouky. Moreover, recurrence relations, explicit formulas and a connection between these numbers and generalized q -harmonic numbers are obtained. Furthermore, some important special cases and new combinatorial identities are given. Finally, algorithms of these numbers and matrix representation using Maple are derived.

The Hyperspherical Functions of a Derivative

We present the results of theoretical research of the generalized hypersherical functionHS� by generalizing two known functions related to the sphere hypersurface and hypervolume and the recurrent relation between them. By introducing two-dimensional degrees of freedom k and nand the third, radius r� , we develop the derivative functions for all three arguments and the possibilities of their use. The symbolical evolution, numerical experiment, and graphical presentation of functions are realized using the Mathcad Professional and Mathematica softwares.