Geodesic
Caristi's condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control, Tome 291 (2015), pp. 30-44

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We consider a lower bounded function on a complete metric space. For this function, we obtain conditions, including Caristi's conditions, under which this function attains its infimum. These results are applied to the study of the existence of a coincidence point of two mappings acting from one metric space to another. We consider both single-valued and set-valued mappings one of which is a covering mapping and the other is Lipschitz continuous. Special attention is paid to the study of a degenerate case that includes, in particular, generalized contraction mappings. 
@article{TM_2015_291_a2,
     author = {A. V. Arutyunov},
     title = {Caristi's condition and existence of a minimum of a lower bounded function in a metric space. {Applications} to the theory of coincidence points},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {30--44},
     publisher = {mathdoc},
     volume = {291},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2015_291_a2/}
}
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A. V. Arutyunov. Caristi's condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control, Tome 291 (2015), pp. 30-44. http://geodesic.mathdoc.fr/item/TM_2015_291_a2/