### Fixed point results for generalized $\varphi$-contraction on a set with two metrics

#### Abstract

The aim of this paper is to present fixed point theorems for multivalued operators $ T\colon X \to P(X)$, on a nonempty

set $X$ with two metrics $d$ and $\varrho$, satisfying the following generalized $\varphi$-contraction condition:

$$

H_{\varrho}(T(x),T(y))\leq \varphi(M^T(x,y)),\quad

\text{for every } x,y \in X,

$$

where

$$

\multline

M^T(x,y):=\max \{ \varrho(x,y),D_{\varrho}(x,T(x)),D_{\varrho}(y,T(y)),\\

2^{-1} [ D_{\varrho}(x,T(y))+D_{\varrho}(y,T(x)) ]\}.

\endmultline

$$

set $X$ with two metrics $d$ and $\varrho$, satisfying the following generalized $\varphi$-contraction condition:

$$

H_{\varrho}(T(x),T(y))\leq \varphi(M^T(x,y)),\quad

\text{for every } x,y \in X,

$$

where

$$

\multline

M^T(x,y):=\max \{ \varrho(x,y),D_{\varrho}(x,T(x)),D_{\varrho}(y,T(y)),\\

2^{-1} [ D_{\varrho}(x,T(y))+D_{\varrho}(y,T(x)) ]\}.

\endmultline

$$

#### Keywords

Set with two metrics; multivalued operator; fixed point; homotopy result; data dependence

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