Session 35. Topological fixed point theory and related topics

The concept of b-generalized pseudodistances and best proximity points for set-valued contractions of Nadler type in b-metric spaces

Robert Plebaniak, University of Łódź, Poland
In this talk we study, in \(b\)-metric space, the concept of \(b\) -generalized pseudodistance (introduced in [1]) which is an extension of \(b\)-metric. Next, inspired by the ideas of S. B. Nadler [2] and A. Abkar and M. Gabeleh [3], we define a new set-valued non-self-mapping contraction of Nadler type with respect to this \(b\)-generalized pseudodistance, which is a generalization of Nadler's contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for \(T:A\rightarrow 2^{B}\). A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error \(\inf \{d(x,y):y\in T(x)\}\), and hence the existence of a consummate approximate solution to the equation \(T(x)=x\). In other words, the best proximity points theorem achieves a global optimal minimum of the map \( x\rightarrow \inf \{d(x;y):y\in T(x)\}\) by stipulating an approximate solution \(x\) of the point equation \(T(x)=x\) to satisfy the condition that \( \inf \{d(x;y):\) \(y\in T(x)\}\) \(=dist(A;B)\). The examples which illustrate the main result given. The talk includes also the comparison of our results with those in the literature.
References
  1. R. Plebaniak, New generalized pseudodistance and coincidence point theorem in a \(b\)-metric space , Fixed Point Theory and Applications 2013, 2013:270 doi:10.1186/1687-1812-2013-270.
  2. S. B. Nadler, Multi-valued contraction mappings , Pacific J. Math. 30, 1969, 475-488.
  3. A. Abkar, M. Gabeleh, The existence of best proximity points for multivalued non-self-mappings , Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Mathematicas, Volume 107, Issue 2, 2013, 319-325.
  4. R. Plebaniak, On best proximity points for set-valued contractions of Nadler type with respect to \(b\)-generalized pseudodistances in \(b\)-metric spaces , Fixed Point Theory and Applications, 2014, 2014:39 doi:10.1186/1687-1812-2014-39.
Print version