It is possible to remove the heavy condition of continuity of the self-mapping f in Theorem 2.5.
We obtained sufficient conditions for the existence of a common fixed point of the mappings F, H in the metric space X endowed with a graph G such that the set of vertices of G, V ( G ) = X and the set of edges of G, E ( G ) ⊆ X × X. MSC 47H10, 47H04, 47H07, 54C60, 54H25.
The assumption of continuity of one of the mappings f or g in Theorem 4 can be replaced by another condition, which is often used in similar situations.
To obtain results about coincidence points or common fixed points of mappings f and g, in the case when f X ⊂ gX, usually the technique of so-called Jungck sequence y n = fx n = gxn+1was used (see, [1, 2]).
where m, n ∈ ℕ, A + 2B + 4D < 1. Remark 2 Similarly as in Remark 1, we note that if the cone P is normal, then continuity of mappings f and g in Theorem 3 can be replaced by conditions.
A pair of self mappings f and g in a metric space ((X, d)) are said to be weakly compatible if for all (tin X) the equality (ft=gt) implies (fgt=gft).
Then every increasing mapping G : HL loc ( [ a, b ), E ) → [ w −, w + ] has the smallest and greatest fixed points, and they are increasing with respect to G. Let us impose the following hypotheses on the mappings f and D in problem (5.1).
In the present section we are interested in mappings (f : X rightarrow X) satisfying pbigl(f y),f(x bigr) leq varphi bigl(p y,x bigr) (3.1) or pbigl(f y),f(x bigr) leq varphi bigl(m_{f} y,x bigr) (3.2) for m_{f} y,x) = maxbigl{ p y,x),pbigl(f y),ybigr),p bigl(f(x),xbigr bigr}, (3.3) where ((X,p)) is a d-metric space, and φ is a comparison function.
Example 1.3 Consider the set-valued mappings F and G defined in Example 1.2.
Hence, the mappings F and g are compatible in X.
