Author. It needs to be noted that the class of b-Fmoc-Gly-Gly-OH manufacturer metric-like spaces
Author. It really should be noted that the class of b-metric-like spaces is bigger that the class of metric-like spaces, considering the fact that a b-metric-like is often a 3-Chloro-5-hydroxybenzoic acid Epigenetic Reader Domain 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 in the sequences in b-metric-like space. Definition 2. Ref. [1] Let x n be a sequence within a b-metric-like space X, dbl , s 1 . (i) (ii) The sequence x n is mentioned to become convergent to x if lim dbl ( x n , x ) = dbl ( x, x );nThe sequence x n is said 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 named 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 just about 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, 5,three 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 anytime the sequence x n X tends to x as n , that’s, 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 devoid of conditions (F2) and (F3) using the home of strictly rising function defined on (0, ). Furthermore, utilizing this fixed point result we prove the existence of solutions for a single type of Caputo fractional differential equation also as existence of solutions for 1 integral equation made in mechanical engineering. two. Fixed Point Remarks Let us begin this section with an essential remark for the case of b-metric-like spaces. Remark 1. Within a b-metric-like space the limit of a sequence doesn’t must be exclusive as well as a convergent sequence does not have to be a dbl -Cauchy one. Nevertheless, if the sequence x n is usually a 0 – dbl -Cauchy sequence inside the dbl -complete b-metric-like space X, dbl , s 1 , then the limit of such sequence is exclusive. 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 obtain 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 is a contradiction. We shall use 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 every n N. Then x n is a 0 – dbl -Cauchy sequence.(2)(3)Remark 2. It really is worth noting that the prior Lemma holds inside the setting of b-metric-like spaces for every [0, 1). For additional information 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 stated to become generalized (s, q)-Jaggi F-contraction-type if there is 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.