Composition operators and multiplication operators between two Orlicz function spaces are investigated. First, necessary and sufficient conditions for their continuity are presented in several forms. It is shown that, in general, the Radon-Nikodým derivative $\frac{d(\mu\circ\tau^{-1})}{d\mu}(s)$ need not belong to $L^{\infty}(\Omega)$ to guarantee the continuity of the composition operator $c_{\tau}x(t)=x(\tau(t))$ from $L^{\Phi}(\Omega)$ into $L^{\Psi}(\Omega)$ . Next, the problem of compactness of these operators is considered. We apply a compactness criterion in Orlicz spaces which involves compactness with respect to the topology of local convergence in measure and equi-absolute continuity in norm of all the elements of the set under consideration. In connection with this, we state some sufficient conditions for equi-absolute continuity of the composition operator $c_{\tau}$ and the multiplication operator $M_{w}$ from one Orlicz space into another. Also the problem of necessary conditions is discussed.

Maluta’s coefficient of Musielak-Orlicz sequence spaces equipped with the Orlicz norm is calculated. A sufficient condition for the Schur property of these spaces is given.

In this paper, we present a series of fixed point results for Mann’s iteration scheme in the framework of $G_b$-metric spaces. First, we introduce the concept of convex $G_b$-metric space by means of a convex structure and Mann’s iteration algorithm is extended to this space. Furthermore, using Mann’s iteration scheme, we prove some fixed point results for several mappings satisfying various suitable conditions on complete convex $G_b$-metric spaces. Some examples supporting our main results are also presented. We also discuss the well-posedness of the fixed point problems and the P property for given mappings. Moreover, as an application, we apply our main result to prove the existence of the solutions to integral equations.

In this paper, we will use the convex modular $$\rho ^{*}(f)$$ρ ∗ ( f ) to investigate $$\Vert f\Vert _{\Psi ,q}^{*}$$‖ f ‖ Ψ , q ∗ on $$(L_{\Phi })^{*}$$( L Φ ) ∗ defined by the formula $$\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))$$‖ f ‖ Ψ , q ∗ = inf k > 0 1 k s q ( ρ ∗ ( k f ) ) , which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$‖ f ‖ Ψ , q ∗ are discussed, the interval for dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$‖ f ‖ Ψ , q ∗ attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of $$L_{\Phi ,p}~(1\le p\le \infty )$$L Φ , p ( 1 ≤ p ≤ ∞ ) is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.