PPF dependent fixed point theorems for ${\alpha }_c$-admissible rational type contractive mappings in Banach spaces

The fixed point theory in Banach spaces plays an important role and is useful in mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions (see [1–37]). In 1977, Bernfeld et al. [2] introduced the concept of PPF dependent fixed point or the fixed point with PPF dependence which is a fixed point for mappings that have different domains and ranges. They also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings. Very recently, some authors established the existence and uniqueness of PPF dependent fixed point for different types of contractive mappings and generalized some results of Bernfeld et al. [2] (see [1, 4, 12, 15, 20], and [33]).

In order to generalize the Banach contraction principle, Geraghty [9] proved the following theorem.

Theorem 1 (Geraghty [9])

then T has a unique fixed point.

Clearly, every constant function from I to E is a member of ${\mathcal{R}}_c$. It is easy to see that the class ${\mathcal{R}}_c$ is algebraically closed with respect to difference, i.e., $\varphi -\xi \in {\mathcal{R}}_c$ when $\varphi ,\xi \in {\mathcal{R}}_c$. Also the class ${\mathcal{R}}_c$ is topologically closed if it is closed with respect to the topology on $E_0$ generated by the norm ${\parallel \cdot \parallel }_{E_0}$.

Definition 1 ([2])

A mapping $\varphi \in E_0$ is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping $T:E_0\to E$ if $T\varphi =\varphi (c)$ for some $c\in I$.

Definition 2 ([2])

for all $\varphi ,\xi \in E_0$.

Samet in 2012 introduced the concepts of α-ψ-contractive and α-admissible mappings. Karapınar and Samet generalized these notions to obtain other fixed point results. Many authors generalized these notions to obtain fixed point results (see [18, 19, 21–23], and [32]).

Samet et al. [31], defined the notion of α-admissible mappings as follows:

Definition 3 ([31])

Definition 4 ([17])

Let $f:X\to X$ and $\alpha :X\times X\to [0,+\mathrm{\infty })$. We say that f is a triangular α-admissible mapping if

(T1) $\alpha (x,y)\ge 1$ implies $\alpha (fx,fy)\ge 1$, $x,y\in X$,

(T2) $\{\begin{array}{l}\alpha (x,z)\ge 1,\\ \alpha (z,y)\ge 1\end{array}$ implies $\alpha (x,y)\ge 1$, $x,y,z\in X$.

The concept of ${\alpha }_c$-admissible mapping was introduced by Agarwal et al. in 2013 (see [1]).

Definition 5 ([1])

Definition 6 ([4])

Let $c\in I$, $T:E_0\to E$, and $\alpha :E\times E\to [0,\mathrm{\infty })$. We say T is a triangular ${\alpha }_c$-admissible mapping if

(T1) $\alpha (\varphi (c),\xi (c))\ge 1$ implies $\alpha (T\varphi ,T\xi )\ge 1$,

(T2) $\alpha (\varphi (c),\mu (c))\ge 1$ and $\alpha (\mu (c),\xi (c))\ge 1$ implies $\alpha (\varphi (c),\xi (c))\ge 1$

for $\varphi ,\xi ,\mu \in E_0$.

Lemma 1 ([4])

Then T has a unique PPF dependent fixed point ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Moreover, for a fixed ${\varphi }_0\in {\mathcal{R}}_c$, if the sequence $\{{\varphi }_n\}$ of iterates of T is defined by $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$, then $\{{\varphi }_n\}$ converges to ${\varphi }^{\ast }\in {\mathcal{R}}_c$.

which is a contradiction. Hence, ${\varphi }^{\ast }={\phi }^{\ast }$. Then T has a unique PPF dependent fixed point in ${\mathcal{R}}_c$. □

Then T has a unique PPF dependent fixed point ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Moreover, for a fixed ${\varphi }_0\in {\mathcal{R}}_c$, if the sequence $\{{\varphi }_n\}$ of iterates of T is defined by $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$, then $\{{\varphi }_n\}$ converges to ${\varphi }^{\ast }\in {\mathcal{R}}_c$.