Author. It ought to be noted that the class of b-metric-like spaces
Author. It should be noted that the class of b-metric-like spaces is larger that the class of metric-like spaces, because a b-metric-like is 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 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 from the sequences in b-metric-like space. Definition 2. Ref. [1] Let x n be a sequence Aztreonam web within a b-metric-like space X, dbl , s 1 . (i) (ii) The sequence x n is stated to become 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 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, 5,3 of(iv)A mapping T : X, dbl , s 1 X, dbl , s 1 is known as dbl -continuous in the event 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 go over initially 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 out conditions (F2) and (F3) working with the house of strictly rising function defined on (0, ). Furthermore, using this fixed point outcome we prove the existence of options for a single variety of Caputo fractional differential equation at the same time as existence of solutions for one particular integral equation developed 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 does not need to be distinctive in addition to a convergent sequence doesn’t need to be a dbl -Cauchy 1. Nonetheless, when the sequence x n is often a 0 – dbl -Cauchy sequence within the dbl -complete b-metric-like space X, dbl , s 1 , then the limit of such sequence is one of a kind. 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 exactly where x = y, we acquire 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 make use of the following result, 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 each and every n N. Then x n is a 0 – dbl -Cauchy sequence.(two)(three)Remark 2. It really is worth noting that the previous Lemma holds within the setting of b-metric-like spaces for each [0, 1). For much more information see [26,28]. Definition three. Let T be a Fmoc-Gly-Gly-OH Purity & Documentation 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 is certainly 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 having a B 2Cs 1 and q 1. d A,B,C F Nbl ( x, y) , (4)( x,Tx) bl (y,Ty)d.