Terminal Value Problem for Implicit Katugampola Fractional Differential Equations in b-Metric Spaces

Laboratory of Mathematics, University of Saïda–Dr. Moulay Tahar, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89, Sidi Bel-Abbes 22000, Algeria Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam Department of Mathematics, Cankaya University, 06790 Etimesgut, Ankara, Turkey Department of Medical Research, China Medical University Hospital, China Medical University, 40402 Taichung, Taiwan


Introduction and Preliminaries
An interesting extension and unification of fractional derivatives of the type Caputo and the type Caputo-Hadamard is called Katugampola fractional derivative that has been introduced by Katugampola [1,2]. Some fundamental properties of this operator are presented in [3,4]. Several results of implicit fractional differential equations have been recently provided (see [4][5][6][7][8][9][10][11][12][13][14] and the references therein). A new class of mixed monotone operators with concavity and applications to fractional differential equations has been considered in [15]. In [16], the authors presented some existence and uniqueness results for a class of terminal value problem for differential equations with Hilfer-Katugampola fractional derivative.
On the other side, a novel extension of b-metric was suggested by Czerwik [17,18]. Although the b-metric standard looks very similar to the metric definition, it has a quite different structure and properties. For example, in the b -metric topology framework, an open (closed) set is not open (closed). Additionally, the b-metric function is not continuous. These weaknesses make this new structure more interesting (see [19][20][21][22][23][24][25][26][27][28]).
Throughout the paper, any mentioned set is nonempty. We consider the following type of terminal value problems of Katugampola implicit differential equations of noninteger orders: with T > 0 and the function κ : Here, ρ D r 0 + is the Katugampola fractional derivative of order r ∈ ð0, 1.

Journal of Function Spaces
Definition 6 [29,30]. Let (S, d, c) with c ≥ 1 be a b.m.s and α : S × Sℝ * + . We say that S is α − regular if for any sequence fν n g n∈ℕ in S such that x n ⟶ x as n ⟶ ∞ and αðν n , ν n+1 Þ ≥ 1 for each n; there exists a subsequence fν n ðκÞg κ∈ℕ of fν n gn with αðν nðκÞ , xÞ ≥ 1 for all k.
Theorem 2 [29,30]. We presume that a self-operator T over a complete b.m.s. (S, d, c) with c ≥ 1 forms a generalized α − φ − Geraghty contraction. Furthermore, Then T possesses a fixed point. Furthermore, if (iii) for all fixed points μ, ν of T, either αðμ, νÞ ≥ 1 or αð ν, μÞ ≥ 1, then the found fixed point is unique This manuscript launches the study of Katugampola implicit fractional differential equations on b.m.s.
On account of Lemma 4, we deduce that solutions of (1) are the fixed points of N.