On multivalued maps for \(\varphi \)-contractions involving orbits with application. (English) Zbl 1485.54044
Summary: In [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 8, 2361–2369 (2007; Zbl 1130.54021)], P. D. Proinov established the existence of fixed point theorems regarding as a generalization of the Banach contraction principle (BCP) of self mapping under an influence of gauge function (GF). In this paper, we develop some existence results on \(\varphi \)-contraction for multivalued maps via \(b \)-Bianchini-Grandolfi gauge function (B-GGF) in class of \(b \)-metric spaces and consequently assure the existence results in the module of simulation function as well \(\alpha \)-admissible mapping. An extensive set of nontrivial example is given to justify our claim. At the end, we give an application to prove the existence behavior for the system of integral inclusion.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54C60 | Set-valued maps in general topology |
| 54E40 | Special maps on metric spaces |
Keywords:
fixed point; \(b\)-Bianchini-Grandolfi gauge function; simulation function; \(b\)-metric spaceCitations:
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