Author. It ought to be noted that the class of b-metric-like spaces
Author. It ought to be noted that the class of b-metric-like spaces is larger that the class of metric-like spaces, since a b-metric-like is really a 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 MRTX-1719 Purity & Documentation Cauchy sequences are formally exactly the 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 on the sequences in b-metric-like space. Definition two. Ref. [1] Let x n be a sequence in a b-metric-like space X, dbl , s 1 . (i) (ii) The sequence x n is said to PX-478 site become convergent to x if lim dbl ( x n , x ) = dbl ( x, x );nThe sequence x n is stated to become 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 known as 0 – dbl -Cauchy sequence.(iii)One particular says that a b-metric-like space X, dbl , s 1 is dbl -complete (resp. 0 – dbl -complete) if for every single 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 called dbl -continuous in the event the sequence Tx n tends to Tx anytime 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 1st 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 without circumstances (F2) and (F3) applying the home of strictly escalating function defined on (0, ). In addition, applying this fixed point outcome we prove the existence of solutions for a single form of Caputo fractional differential equation as well as existence of options for one integral equation developed in mechanical engineering. two. Fixed Point Remarks Let us get started this section with a vital remark for the case of b-metric-like spaces. Remark 1. In a b-metric-like space the limit of a sequence doesn’t ought to be unique and a convergent sequence will not really need to be a dbl -Cauchy one. Nonetheless, if 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 exceptional. Certainly, 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 where x = y, we get 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, which can be a contradiction. We shall use the following outcome, the proof is equivalent 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 every single n N. Then x n is a 0 – dbl -Cauchy sequence.(two)(3)Remark 2. It is actually worth noting that the earlier Lemma holds in the setting of b-metric-like spaces for every single [0, 1). For far more facts see [26,28]. Definition 3. Let T be a self-mapping on a b-metric-like space X, dbl , s 1 . Then the mapping T is mentioned to become generalized (s, q)-Jaggi F-contraction-type if there’s strictly escalating 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, where Nbl ( x, y) = A bl A, B, C 0 with a B 2Cs 1 and q 1. d A,B,C F Nbl ( x, y) , (4)( x,Tx) bl (y,Ty)d.