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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