Sentence examples for mappings were proved from inspiring English sources

Three fixed point theorems which extended the fixed point theory for Kannan's mappings were proved in [11].

In a separable metric space, random fixed point theorems for contraction mappings were proved by Hans̆ [8, 9], Hans̆ and S̆pac̆ek [10] and Mukherjee [11, 12].

In particular, some useful fixed point theorems for monotone mappings were proved by Zhang [19], Guo and Lakshmikantham [20], and Bhaskar and Lakshmikantham [21] under some weak assumptions.

Afterward, some fixed point results for generalized weak contraction mappings were proved by Choudhury et al. [2] by using some control function along with the notion of an altering distance function.

More recently, in Ghoncheh and Razani [5], the notion of C-condition introduced in [4] was generalized to the case of multi-valued version and some existence theorems of fixed point for these mappings were proved in a Ptolemy metric space.

In [12], some inequalities of Hermite-Hadamard type for differentiable convex mappings were proven using the following lemma.

Fixed point theorems for Perov-type α-quasi contraction mapping, Kannan-type contraction as well as Chatterjee-type contraction mappings are proved in a dislocated cone metric space over Banach algebra.

In addition, new results for cyclic mappings are proved.

Under suitable conditions some strong and weak convergence theorems for such mappings are proved.

Existence theorems for family of nonexpansive mappings are proved in [1 5] and actually many others.

Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved.