The aim of this paper is to obtain some new common fixed point theorems for weakly compatible mappings in symmetric spaces satisfying an implicit function. Some illustrative examples to highlight the realized improvements are furnished. Our results generalize and extend some recent results contained in Ali and Imdad [Sarajevo J. Math. 4(17)(2008), 269-285] to symmetric spaces and consequently a host of metrical common fixed theorems are generalized and improved. We state an integral type fixed point theorem in symmetric space. In the process, we also derive a fixed point result on common fixed point in probabilistic symmetric spaces.

In this paper, we prove some recent even coincidence theorems due to Imdad et al. (Bull Math Anal Appl 5(4): 19-39, 2013) using a method of reduction from the respective coincidence theorems for mappings with one variable in ordered complete metric spaces. Our technique of proof is different, slightly simpler, shorter and more effective than the ones used in Imdad et al.

In this paper, we prove n-tupled fixed point theorems (for even n) for mappings satisfying Meir-Keeler type contractive condition besides enjoying mixed monotone property in ordered partial metric spaces. As applications, some results of integral type are also derived. Our results generalize the corresponding results of Erduran and Imdad (J. Nonlinear Anal. Appl. 2012Appl. :jnaa-00169, 2012.

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Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].