We consider generalized Orlicz–Morrey spaces MΦ,φ(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$M_{\Phi ,\varphi }({\mathbb {R}^n})$$\end{document} including their weak versions WMΦ,φ(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$WM_{\Phi ,\varphi }({\mathbb {R}^n})$$\end{document}. We find the sufficient conditions on the pairs (φ1,φ2)\documentclass[12pt]{minimal}
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\begin{document}$$(\varphi _{1},\varphi _{2})$$\end{document} and (Φ,Ψ)\documentclass[12pt]{minimal}
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\begin{document}$$(\Phi , \Psi )$$\end{document} which ensures the boundedness of the fractional maximal operator Mα\documentclass[12pt]{minimal}
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\begin{document}$$M_{\alpha }$$\end{document} from MΦ,φ1(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$M_{\Phi ,\varphi _1}({\mathbb {R}^n})$$\end{document} to MΨ,φ2(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$M_{\Psi ,\varphi _2}({\mathbb {R}^n})$$\end{document} and from MΦ,φ1(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$M_{\Phi ,\varphi _1}({\mathbb {R}^n})$$\end{document} to WMΨ,φ2(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$WM_{\Psi ,\varphi _2}({\mathbb {R}^n})$$\end{document}. As applications of those results, the boundedness of the commutators of the fractional maximal operator Mb,α\documentclass[12pt]{minimal}
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\begin{document}$$M_{b,\alpha }$$\end{document} with b∈BMO(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$b \in BMO({\mathbb {R}^n})$$\end{document} on the spaces MΦ,φ(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$M_{\Phi ,\varphi }({\mathbb {R}^n})$$\end{document} is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights φ(x,r)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi (x,r)$$\end{document}, which do not assume any assumption on monotonicity of φ(x,r)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi (x,r)$$\end{document} on r\documentclass[12pt]{minimal}
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\begin{document}$$r$$\end{document}.
