Generalization of Fixed Point Results via Iterative Process of F -Contraction

The aim of this paper to discuss generalized iterative process of F-contraction and establish new fixed point theorems in complete metric spaces. As an application of our results, we prove existence and uniqueness of functional equations and system of differential equations. Our results provide extension as well as substantial generalizations and improvements of several well known results in the existing comparable literature.


Introduction
In 2012, Wardowski [29] introduce a new type of contractions called F-contraction and proved new fixed point theorems concerning F-contraction. Afterwards Se-celean [28], proved fixed point theorems consisting of F-contractions by Iterated function systems. He generalized the Banach contraction principle in a different way than as it was done by different investigators.
Definition 1 [16] The set of all fixed points of T is denoted by ( ).
The set of all such points is denoted by ( ) , .

C f T
n ∀ ≥ these can be many f iterative sequence of T starting at 0 . x x Then clearly ( ) ( ) 1 .
We obtain a generalized iterative process is called a generalized dynamic process of f and T starting at 0 1.
x = So you can construct many f iterative sequences of T starting at x0 for different values.

Main Result
Throughout this section, we assume that the mapping F is right continuous. In the following we will consider only the dynamic processes ( ) n fx satisfying the following condition: For any in , , 0 , 0.  , ,          , , .
(say) and

( )
, , Thus all conditions of above Theorem 10 is satisfied and 0 is a fixed point of T. (1) Applications Decision space and a state space are two basic components of dynamic programming problem. State space is a set of states including initial states, action states and transitional states. So a state space is set of parameters representing different states. A decision space is the set of possible actions that can be taken to solve the problem. These general settings allow us to formulate many problems in mathematical optimization and computer programming. In particular the problem of dynamic programming related to multistage process reduces to the problem of solving functional equations where U and V are Banach spaces, W U ⊆ and D V ⊆ and , : , G G W D × × →   for more details on dynamic programming we refer to [6,7,8,9,23]. Suppose that W and D are the state and decision spaces respectively. We aim to give the existence and uniqueness of common and bounded solution of functional equations given in (2.6) and (2.7). Let ( )

(1) Application of system of integral equations:
Now we discuss an application of fixed point theorem we proved in the previous section in solving the system of Volterra type integral equations. Such system is given by the following equations:  Now we prove the following theorem to ensure the existence of solution of system of integral equations. For more details on such applications we refer the reader to [3,21].