Author. It needs to be noted that the class of b-PX-478 manufacturer metric-like spaces
Author. It really should be noted that the class of b-metric-like spaces is larger that the class of metric-like spaces, considering that a b-metric-like is usually a FAUC 365 web metric like with s = 1. For some examples of metric-like and b-metric-like spaces (see [13,15,23,24]). The definitions of convergent and Cauchy sequences are formally the exact same in partial metric, metric-like, partial b-metric and b-metric-like spaces. For that reason we give only the definition of convergence and Cauchyness from the sequences in b-metric-like space. Definition 2. Ref. [1] Let x n be a sequence inside a b-metric-like space X, dbl , s 1 . (i) (ii) The sequence x n is said to be convergent to x if lim dbl ( x n , x ) = dbl ( x, x );nThe sequence x n is mentioned to be dbl -Cauchy in X, dbl , s 1 if and is finite. Ifn,mn,mlimdbl ( x n , x m ) existslimdbl ( x n , x m ) = 0, then x n is called 0 – dbl -Cauchy sequence.(iii)1 says that a b-metric-like space X, dbl , s 1 is dbl -complete (resp. 0 – dbl -complete) if for each and every dbl -Cauchy (resp. 0 – dbl -Cauchy) sequence x n in it there exists an x X such that lim dbl ( x n , x m ) = lim dbl ( x n , x ) = dbl ( x, x ).n,m nFractal Fract. 2021, five,3 of(iv)A mapping T : X, dbl , s 1 X, dbl , s 1 is named dbl -continuous when the sequence Tx n tends to Tx whenever the sequence x n X tends to x as n , that is definitely, if lim dbl ( x n , x ) = dbl ( x, x ) yields lim dbl Tx n , Tx = dbl Tx, Tx .n nHerein, we discuss initial some fixed points considerations for the case of b-metric-like spaces. Then we give a (s, q)-Jaggi-F- contraction fixed point theorem in 0 – dbl -complete b-metric-like space with no situations (F2) and (F3) utilizing the home of strictly increasing function defined on (0, ). Additionally, making use of this fixed point outcome we prove the existence of options for 1 type of Caputo fractional differential equation also as existence of solutions for one integral equation designed in mechanical engineering. 2. Fixed Point Remarks Let us commence this section with an essential remark for the case of b-metric-like spaces. Remark 1. Inside a b-metric-like space the limit of a sequence doesn’t need to be special in addition to a convergent sequence will not should be a dbl -Cauchy one particular. Nevertheless, when the sequence x n is often a 0 – dbl -Cauchy sequence inside the dbl -complete b-metric-like space X, dbl , s 1 , then the limit of such sequence is special. Indeed, in such case if x n x as n we get that dbl ( x, x ) = 0. Now, if x n x and x n y exactly where x = y, we receive that: 1 d ( x, y) dbl ( x, x n ) dbl ( x n , x ) dbl ( x, x ) dbl (y, y) = 0 0 = 0. s bl From (dbl 1) follows that x = y, that is a contradiction. We shall make use of the following outcome, the proof is similar to that inside the paper [25] (see also [26,27]). Lemma 1. Let x n be a sequence in b-metric-like space X, dbl , s 1 such that dbl ( x n , x n1 ) dbl ( x n-1 , x n )1 for some [0, s ) and for each and every n N. Then x n is often a 0 – dbl -Cauchy sequence.(two)(3)Remark 2. It really is worth noting that the previous Lemma holds in the setting of b-metric-like spaces for every [0, 1). For far more information see [26,28]. Definition three. Let T be a self-mapping on a b-metric-like space X, dbl , s 1 . Then the mapping T is stated to be generalized (s, q)-Jaggi F-contraction-type if there’s strictly increasing F : (0, ) (-, ) and 0 such that for all x, y X : dbl Tx, Ty 0 and dbl ( x, y) 0 yields F sq dbl Tx, TyA,B,C for all x, y X, exactly where Nbl ( x, y) = A bl A, B, C 0 having a B 2Cs 1 and q 1. d A,B,C F Nbl ( x, y) , (four)( x,Tx) bl (y,Ty)d.
