We define the new measure arising from the gauge function and denote it by G α ( F, a, b ).
where W = ( W 1, …, W m ), B ( x, y ) is the gauge function, and W α ∈ A, then X is called a partial Noether operator corresponding to L, and X is the α th prolongation of the generalized operator (2.7).
Corollary 3 Let ( X, p ) be a partial metric space such that the induced weightable T 0 -qpm d p is complete, let φ : [ 0, ∞ ) → [ 0, ∞ ) be a Bianchini-Grandolfi gauge function and let T : X → Cl ( d p ) s ( X ) be a multivalued map. If one of the following two conditions is satisfied, then T has a fixed point.
Corollary 1 Let ( X, d ) be a complete T 0 -qpm space, q a Q-function on ( X, d ), φ : [ 0, ∞ ) → [ 0, ∞ ) a Bianchini-Grandolfi gauge function and T X → ClCl d s ( X ) a multivalued map such that for each x, y ∈ X and u ∈ T x, there is v ∈ T y satisfying q ( u, v ) ≤ φ ( q ( x, y ) ). Then T has a fixed point. If we take φ ( t ) = r t where r ∈ [ 0, 1 ) we get one of the main results in [2].
Such functions are called gauge functions or Minkowski functionals, and are well studied in convex analysis and functional analysis.
The purpose of this paper is to establish new fixed point results formulti-valued mappings satisfying an -contractive condition, and viaBianchini-Grandolfi gauge functions, on α-complete metric spaces.Our results unify, generalize, and complement various results from theliterature.
In this paper we prove the existence of a fixed point for multivalued maps satisfying a contraction condition in terms of Q-functions, and via Bianchini-Grandolfi gauge functions, for complete T 0 -quasipseudometric spaces.
In the literature, such mappings are called in two different ways: (c -comparison func -comparisone sources (see, e.g., [11]), and Bianchini-Grandolfunctionsfunctinnsome sourcesherseese.g.e.g., [11]14]).
Moreover, Proinov [5] established two general theorems for equivalence between the Meir-Keeler type contractive conditions (1.2) and the contractive conditions involving gauge functions (1.1).
These functions are known in the literature as Bianchini-Grandolfi gauge functions in some sources (see e.g. [12 14]) and as ( c ) -comparison functions in some other sources (see e.g. [15]).
