Abstract
In this paper, a generalization of Nadler's fixed point theorem is presented for H + -type k-multi- valued weak contractive mappings. We consider a nonconvex Hammerstein type integral inclusion and prove an existence theorem by using an H + -type multi-valued weak contractive mapping.
Highlights
In 1969, Nadler [16] proved a fixed point theorem for the set-valued contractions, which is of fundamental importance in nonlinear analysis
Inspired from the fixed point result of Nadler [16], the fixed point theory of set-valued contraction was further developed in different directions by many authors, in particular, by Reich [20, 21], Mizoguchi and Takahashi [15], Ciric [3], Kaneko [9], Lim [13], Lami Dozo [14], Feng and Liu [5], Klim and Wardowski [10], Suzuki [22], Pathak and Shahzad [17, 18] and many others
Using the same argument as used for the set valued map G(·), we deduce that G1(·) is measurable with nonempty closed values
Summary
In 1969, Nadler [16] proved a fixed point theorem for the set-valued contractions, which is of fundamental importance in nonlinear analysis. Inspired from the fixed point result of Nadler [16], the fixed point theory of set-valued contraction was further developed in different directions by many authors, in particular, by Reich [20, 21], Mizoguchi and Takahashi [15], Ciric [3], Kaneko [9], Lim [13], Lami Dozo [14], Feng and Liu [5], Klim and Wardowski [10], Suzuki [22], Pathak and Shahzad [17, 18] and many others. For other applications of the same result see, for example, [4] [6], [7], [8], [12] and [19]
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