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 $$ M^T(x,y):=\max \{ \varrho(x,y),D_{\varrho}(x,T(x)),D_{\varrho}(y,T(y)), 2^{-1} [ D_{\varrho}(x,T(y))+D_{\varrho}(y,T(x)) ]\}. $$