Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 35 https://internationalpubls.com Outcomes for The Analysis of The Existence of Fuzzy Fractional Differential Equations P.Dhanalakshmi1, S.Padmavathi2 1Department of Applied Mathematics, Bharathiar University, Coimbatore, India. viga_dhanasekar@yahoo.co.in 2Department of Mathematics, Vellalar College for Women, Erode, India. padmavathi201@gmail.com Article History: Received: 23-05-2024 Revised: 26-06-2024 Accepted: 18-07-2024 Abstract: In this article, a Cauchy problem for a fuzzy q-fractional differential equation of order α has been considered.The results for generalized Hukuhara q-differentiablity of a fuzzy function are established. This work has led to the study of existence and uniqueness of the fuzzy function with Caputo Hukuhara q-differentiability and generalized Banach fixed point theorem. A crucial qualitative property of the differential equation, which is continuous dependence on initial conditions and the functions involved is analysed. An illustrative example is given which ensures the result. Keywords: Fuzzy Fractional Differential Equations, Fuzzy q-Fractional Differential Equation, Existence of Solution, Fixed point theorem, Continuous dependence. 1 Introduction The generalized form of differential equations is termed as fractional differential equations (FDE), which emerged as an application of fractional calculus. FDE engages as an important tool to perform many physical phenomena which includes viscoelasticity, control theory of dynamical systems, optics and signal processing etc. FDE attracted many scientists and mathematicians. The existence theory of fractional differential equations of nonlinear type finds its consistent study by many researchers. In recent years, Cauchy problem for nonlinear FDE has become the most interesting field to study the existence, uniqueness, long time behavior etc. Fixed point theory has created extensive interest in the researchers to utilize it for the existence of solution of the fractional differential equation. Fractional derivative is derived mostly by two operators Riemann-Liouville and Caputo operators. But the Caputo operator has advantages for initial value problems. The core attention towards the analysis of existence and uniqueness of the solution of the FDEs is prolific to study. In particular, the study of the existence of solution to Fuzzy Fractional Differential Equation is quite interesting. In addition, fixed point theory is an inevitable tool to study the existence and uniqueness of some mathematical models, which has been studied since many decades. Research is progressing extensively to study the Cauchy problems, with the utilization of fixed point theory approach. Further, the study of uncertainty becomes the most interesting study in these days. Also the dynamics itself is uncertain due to its dependence on time. Basically, the proposed ideas are a generalisation of the theory and solution of fuzzy differential equations . However, the authors considered fuzzy fractional differential equations under the Riemann-Liouville H-derivative[4]. Again, it requires a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 36 https://internationalpubls.com quantity of fractional H-derivative of an unknown solution at the fuzzy initial point. Also, the q- calculus appears to be the connection between mathematics and physics. It has several applications in the areas like quantum theory, hypo geometric functions and electronics[1, 3, 8, 9]. Z.Noeiaghdam et.al and Obaidat et.al [6, 7] study the existence of q-fractional differential equations with uncertainty. The existence of solution of a fractional q-integro differential equation with q-nonlocal condition has been examined by Ibrahim et.al [5]. Z.Noeiaghdam et.al analysed the fuzzy q-derivative and fuzzy q- fractional derivative in Caputo sense and used generalized Hukuhara difference. Fuzzy q-fractional differential equation solving creates strenuous attention to the researchers [8]. Motivated by those, this work studies the existence of integro differential equations 𝐶𝐷𝛼(𝜙(𝜏)) = 𝜆𝜙(𝜏) + 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏)), (1.1) 𝜙(𝜏0) = 𝜙0 (1.2) where 𝒢𝜙(𝜏) = ∫ 𝜏 𝜏0 𝒦(𝜏, 𝑠)𝜙(𝑠)𝑑𝑠, 𝒦 ∈ 𝐶[𝒟, ℝ+], 𝑤ℎ𝑒𝑟𝑒 𝒟 = {(𝜏, 𝑠) ∈ ℝ2: 0 ≤ 𝑠 ≤ 𝜏 ≤ 𝕋}, 𝒮𝜙(𝜏) = ∫ 𝜏 𝜏0 ℋ(𝜏, 𝑠)𝜙(𝑠)𝑑𝑠, ℋ ∈ 𝐶[𝒟0, ℝ+], 𝑤ℎ𝑒𝑟𝑒 𝒟0 = {(𝜏, 𝑠) ∈ ℝ2: 0 ≤ 𝜏, 𝑠 ≤ 𝕋} where 0 < 𝛼 ≤ 1,𝜏 ∈ [𝜏0, 𝕋] 𝜆 is a constant and 𝑓: 𝕋𝑞 × ℝ𝔽 × ℝ𝔽 × ℝ𝔽 → ℝ𝔽 are continuous and 𝑓(𝜏, 𝜙(𝜏), 𝐺𝜙(𝜏), 𝑆𝜙(𝜏)) satisfies the following condition (𝐻1)‖𝑓(𝜏, 𝜙, 𝒢𝜙, 𝒮𝜙) − 𝑓(𝜏, 𝜓, 𝒢𝜓, 𝒮𝜓)‖ ≤ ℒ[‖𝜙 − 𝜓‖ + ‖𝒢𝑢 − 𝒢𝑣‖ + ‖𝒮𝑢 − 𝒮𝑣‖] with 0 < ℒ < 1. 2 Preliminaries 2.1 Basic Fuzzy Concepts Definition 2.1 [2] Let ℝ𝔽 be the set of all fuzzy valued functions. Let 𝜙: ℝ𝔽 → [0,1] be satisfying the following conditions. 1. 𝜙 is upper semi-continuous on ℝ 2. 𝜙 is fuzzy convex 3. 𝜙 is normal 4. Closure of {𝑡 ∈ ℝ𝔽|𝜙(𝑡) > 0} is compact. Let ℝ𝔽 be the space of above said fuzzy numbers. Also, for 0 < 𝑟 ≤ 1, denote 𝜙(𝑟) = {𝑡 ∈ ℝ𝑛|𝜙(𝑡) > 𝑟}, which is known as the 𝑟-level set, which is closed for all 𝑟 ∈ [0,1]. In ℝ𝔽, for arbitrary 𝜙, 𝜓 ∈ ℝ𝔽 and a scalar 𝑘, define the following binary operations, namely, addition and scalar multiplication, respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 37 https://internationalpubls.com Addition: (𝜙 ⊕ 𝜓)(𝑟) = 𝜙(𝑟) + 𝜓(𝑟) Scalar multiplication: (𝑘 ⊙ 𝜙)(𝑟) = (𝑘𝜙(𝑟), 𝑘𝜙(𝑟)), 𝑘 ≥ 0 (𝑘 ⊙ 𝜙)(𝑟) = (𝑘𝜙(𝑟), 𝑘𝜙(𝑟)), 𝑘 ≤ 0 2.2 Hukuhara Difference We utilise Hukuhara difference from [?], for 𝜙, 𝜓 ∈ ℝ𝔽, as follows. i.e., 𝜙 = 𝜓 + 𝜔, if 𝑤 ∈ ℝ𝔽. Then 𝜔 is called as the Hukuhara difference of 𝜙 and 𝜓. Also, the generalized Hukuhara difference, denoted as 𝑔𝐻 in short, is also defined as, 𝜙 ⊖ 𝑔𝐻 𝜓 = 𝜔 ⇔ { (𝑖)𝜙 = 𝜓 + 𝜔 (𝑖𝑖)𝜙 = 𝜓 + (−1)𝜔 It is obvious that (𝑖) and (𝑖𝑖) are true, if and only if 𝜔 is a crisp number. 2.3 q Calculus The usual derivative of a function ′𝑓(𝜏)′ is defined as 𝑙𝑖𝑚⏟ 𝜏→𝜏0 𝑓(𝜏)−𝑓(𝜏0) 𝜏−𝜏0 . Now we can study 𝔮-derivative of 𝑓. For that, we use the following definitions from [?]. Let 𝕋𝔮 be the time scale, for 0 < 𝔮 < 1. 𝕋𝔮 = {𝔮𝑛: 𝑛 ∈ ℤ} ∪ {0} 𝕋𝔮 𝑛 = {𝔮𝛼+𝑛: 𝑛 ∈ ℤ} ∪ {0} Consider an arbitrary function 𝑓: 𝕋𝔮 → ℝ. Its 𝔮-differential is 𝑑𝔮𝑓(𝜏) = 𝑓(𝜏) − 𝑓(𝔮𝜏) Then the 𝔮-derivative of 𝑓 is given by, 𝐷𝔮𝑓(𝜏) = 𝑑𝔮𝑓(𝜏) 𝑑𝔮𝜏 = 𝑓(𝜏)−𝑓(𝔮𝜏) (1−𝔮)𝜏 , 𝜏 ∈ 𝕋𝔮 − {0} The q-gamma function denoted by Γ𝑞(. ) can be defined as, Γ𝔮(𝛼) = (1−𝔮)𝔮 𝛼−1 (1−𝑞)𝛼−1 , 𝛼 ∈ ℝ − {0} ∪ ℤ, 0 < 𝔮 < 1, which is satisfied in the following relation Γ𝔮(𝛼 + 1) = (1−𝔮)𝛼 (1−𝔮) Γ𝔮(𝛼) Γ𝔮(𝛼) = 1, 𝛼 > 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 38 https://internationalpubls.com Definition 2.2 [6] Let for 𝛼 > 0, 𝑓(𝜏) is a 𝔮-integrable function, the fractional 𝔮-integral of order 𝛼 is defined by, 𝔮𝐼𝑎 𝛼𝑓(𝜏) = 1 Γ𝔮(𝛼) ∫ 𝜏 𝑎 (𝜏 − 𝔮𝑠)𝔮 𝛼−1𝑓(𝑠)𝑑𝔮𝑠 Definition 2.3 [6] For an arbitrary fuzzy valued function 𝑓: 𝕋𝑞 → ℝ𝔽, the 𝔮-differential by 𝑔𝐻 difference is 𝑔𝐻𝑑𝔮𝑓(𝜏) = 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝔮𝜏). If 𝑔𝐻𝑑𝔮𝑓(𝜏) = 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝔮𝜏) exists, the fuzzy generalized Hukuhara 𝔮-derivative of 𝑓 is defined by, 𝐹𝐷𝔮𝑓(𝜏) = 𝑔𝐻𝑑𝔮𝑓(𝜏) 𝑑𝔮𝜏 = 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝔮𝜏) (1 − 𝔮)𝜏 , 𝜏 ∈ 𝕋𝔮 − {0} Definition 2.4 [6] Let 𝑓: 𝕋𝑞 → ℝ𝔽 be a 𝔮-integrable function. The fuzzy 𝔮-fractional integral of order 𝛼 for 𝑓 is defined by, 𝐹𝐼𝑎 𝛼𝑓(𝜏) = 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝑎 (𝜏 − 𝔮𝑠)𝔮 𝛼−1 ⊙ 𝑓(𝑠)𝑑𝔮𝑠 Definition 2.5 [6] Let ∀𝑚, 𝔮 𝐹𝐷𝑚𝑓 be continuous and 𝔮-integrable functions in 𝕋𝔮. The fuzzy Caputo q-fractional derivative of order 𝛼 of fuzzy valued function is defined, 𝔮 𝐹𝐶𝐷𝛼𝑓(𝑡) =𝑎 𝐹 𝐼𝑎 𝑚−𝛼(𝐹𝐷𝔮 𝑚𝑓)(𝜏) = 1 Γ𝔮(𝑚 − 𝛼) ⊙ ∫ 𝜏 𝑎 (𝜏 − 𝔮𝑠)𝔮 𝑚−𝛼−1 ⊙𝔮 𝐹 𝐷𝑚𝑓(𝑠)𝑑𝔮𝑠 where 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ ℕ, 𝑡 > 𝑎. Definition 2.6 [6] For 0 < 𝛼 ≤ 1, the fuzzy Caputo 𝔮-fractional 𝑔𝐻 derivative will be, 𝔮 𝐹𝐶𝐷𝛼𝑓(𝜏) =𝑎 𝐹 𝐼𝑎 1−𝛼(𝐹𝐷𝔮𝑓)(𝜏) = 1 Γ𝔮(1 − 𝛼) ⊙ ∫ 𝜏 𝑎 (𝜏 − 𝔮𝑠)𝔮 −𝛼 ⊙𝔮 𝐹 𝐷𝑓(𝑠)𝑑𝔮 for 𝜏 > 𝑎. Definition 2.7 [7] For a fuzzy valued function 𝑓(𝜏), where 𝜏 ∈ 𝑇𝑞, the Caputo 𝔮-integral of order 𝛼 is 𝔮𝐼𝑎 𝛼𝑓(𝜏) = 1 Γ𝔮(𝛼) ∫ 𝜏 𝑎 (𝜏 − 𝔮𝑠)𝔮 𝛼−1𝑓(𝑠)𝑑𝔮𝑠 Definition 2.8 [6] For a fuzzy valued function 𝑓(𝜏), where 𝜏 ∈ 𝑇𝑞, 𝔮𝐼𝑎 𝛼(𝔮 𝐶𝐷𝛼)(𝑓(𝜏)) = 𝑓(𝜏) − ∑ 𝑚−1 𝑘=0 (𝜏 − 𝑎)𝔮 𝑘 Γ𝔮(𝑘 + 1) 𝐷𝔮 𝑘𝑓(𝑎) where 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ ℕ and for the special case 0 < 𝛼 ≤ 1, 𝔮𝐼𝑎 𝛼(𝔮 𝐶𝐷)(𝑓(𝜏)) = 𝑓(𝜏) − 𝑓(𝑎) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 39 https://internationalpubls.com Definition 2.9 [6] Let 𝑓: 𝕋𝑞 → ℝ𝔽 be Caputo differentiable at 𝜙0 ∈ 𝕋𝔮. We say that 𝑓 is Caputo [(𝑖)−𝑔𝐻] differentiable at 𝜙0, if (𝑖) 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜙0; 𝑟) = [ 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜙0; 𝑟), 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜙0; 𝑟)] and 𝑓 is Caputo [(𝑖𝑖)−𝑔𝐻] differentiable at 𝜙0, if (𝑖𝑖) 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜙0; 𝑟) = [ 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜙0; 𝑟), 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜙0; 𝑟)], for 0 ≤ 𝑟 ≤ 1 Definition 2.10 The function 𝐸𝛼(𝛼 > 0) defined by 𝐸𝛼(𝑧) = ∑ ∞ 𝑘=0 𝑧𝑘 Γ(𝑘𝛼 + 1) is called Mittag-Leffler function of order 𝛼. Lemma 2.1 Suppose that 𝑓: 𝕋𝔮 → ℝ𝔽 be a fuzzy valued function and 𝑓 is 𝑔𝐻 𝔮-differentiable and 𝔮-integrable in 𝕋𝔮, then for 0 < 𝛼 ≤ 1, then 𝔮 𝐹𝐼𝑎 𝛼(𝔮 𝐹𝐶𝐷𝛼𝑓(𝜏)) = 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝑎) Lemma 2.2 Let 𝑣: [𝜏0, 𝑇] → [0, +∞) be a real function and 𝑤(. ) is a non negative, locally integrable function on [𝜏0, 𝑇]. Assume that there is a constant 𝛼 such that 0 < 𝛼 ≤ 1. 𝑣(𝜏) ≤ 𝑤(𝜏) + 𝑎 ∫ 𝜏 0 (𝜏 − 𝑠)−𝛼𝑣(𝑠)𝑑𝑠. Then, there exists a constant 𝐾 = 𝐾(𝛼) such that 𝑣(𝜏) ≤ 𝑤(𝜏) + 𝐾𝑎 ∫ 𝜏 0 (𝜏 − 𝑠)−𝛼𝑤(𝑠)𝑑𝑠 for every 𝜏 ∈ [𝜏0, 𝑇]. We consider here the generalized Banach fixed point theorem, which is used to prove the existence results. Theorem 2.3 Let 𝑈 be a nonempty closed subset of a Banach space 𝐵, and let 𝛼𝑛 ≥ 0, 𝑛 ∈ ℕ ∪ {0}, be a sequence such that ∑∞ 𝑛=0 𝛼𝑛 converges. Moreover, let the mapping 𝐹: 𝑈 → 𝑈 satisfy the inequality ‖𝐹𝑛𝑢 − 𝐹𝑛𝑣‖ ≤ 𝛼𝑛‖𝑢 − 𝑣‖ ∀ 𝑛 ∈ ℕ ∪ {0}, and for every 𝑢, 𝑣 ∈ 𝑈. Then 𝐹 has a uniquely defined fixed point 𝑢∗.Furthermore, the sequence {𝐹𝑛𝑢0}𝑛=1 ∞ converges to the fixed point 𝑢∗ for every 𝑢0 ∈ 𝑈. 3 Results on Fuzzy q-fractional derivative by 𝒈𝑯 difference Let 𝑋 be a nonempty closed subset of a ℝ. Consider an arbitrary fuzzy valued function ℎ: 𝑋 → ℝ𝐹 , where ℝ is the set of all real numbers, and [ℎ(𝜏)]𝑟 = [ℎ(𝜏; 𝑟), ℎ(𝜏; 𝑟)] is called as 𝑟 -cut or parametric form of the fuzzy valued function 𝑓. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 40 https://internationalpubls.com The results on Fuzzy 𝔮 -fractional derivative by 𝑔𝐻 difference of the fuzzy function 𝑓(𝜏, 𝑢(𝜏), 𝒢𝑢(𝜏), 𝒮𝑢(𝜏)) has been studied by Z.Noeiaghdam et.al in [?]. For an arbitrary fuzzy valued function 𝑓: 𝕋𝔮 → ℝ𝔽, q-differential by 𝑔𝐻 difference is 𝑑𝑞,𝑔𝐻 = 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝔮𝜏) if 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝔮𝜏) exists. Then the fuzzy generalized Hukuhara 𝔮 derivative of 𝑓 is defined by 𝐷𝐹,𝑞𝑓(𝜏) = 𝑑𝑞,𝑔𝐻 𝑑𝑞𝜏 = 𝑓(𝜏) ⊖ 𝑔𝐻 𝑓(𝔮𝜏) (1 − 𝔮)𝜏 , 𝜏 ∈ 𝕋𝔮 − 0 Lemma 3.1 The function 𝑓 is 𝑔𝐻 differentiable if and only if, 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏); 𝑟) and 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏)) are differentiable with respect to 𝜏 for all 𝑟 ∈ [0,1] and 𝑓′ 𝑔𝐻 (𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏); 𝑟) = [min (𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏); 𝑟), 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏); 𝑟)) , max (𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏); 𝑟), 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏); 𝑟))] Theorem 3.2 Let 𝑓: 𝕋𝔮 × ℝ𝔽 × ℝ𝔽 × ℝ𝔽 → ℝ𝔽 be a fuzzy valued function on 𝕁. (a) If 𝑓 is [(𝑖)− 𝑔𝐻] differentiable at 𝜙0 ∈ 𝕋𝔮, then 𝑓 is Caputo [(𝑖)− 𝑔𝐻] differentiable at 𝜙0 (b) If 𝑓 is [(𝑖𝑖)− 𝑔𝐻] differentiable at 𝜙0 ∈ 𝕋𝔮, then 𝑓 is Caputo [(𝑖𝑖)− 𝑔𝐻] differentiable at 𝜙0 Proof. C 𝐷𝛼(𝑓(𝜏0, 𝜙(𝜏0), 𝒢𝜙(𝜏0), 𝒮𝜙(𝜏0)); 𝑟) = 𝐼1−𝛼(𝑓′)(𝜏0, 𝜙(𝜏0), 𝒢𝜙(𝜏0), 𝒮𝜙(𝜏0)); 𝑟) = [ 1 Γ(1−𝛼) ∫ 𝑓′(𝜏0,𝜙(𝜏0),𝒢𝜙(𝜏0),𝒮𝜙(𝜏0));𝑟) (𝜏0−𝑠)𝛼 𝑑𝑠, 1 Γ(1−𝛼) ∫ 𝑓′(𝜏0,𝜙(𝜏0),𝒢𝜙(𝜏0),𝒮𝜙(𝜏0));𝑟) (𝜏0−𝑠)𝛼 𝑑𝑠] = [ 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜏0, 𝜙(𝜏0), 𝒢𝜙(𝜏0), 𝒮𝜙(𝜏0)), 𝑔𝐻 𝐶 𝐷𝛼𝑓(𝜏0, 𝜙(𝜏0), 𝒢 𝜙(𝜏0), 𝒮𝜙(𝜏0))] , 0 ≤ 𝑟 ≤ 1 Therefore, 𝑓 is [(𝑖)− 𝑔𝐻] Caputo differentiable. Similarly we can prove that 𝑓 is [(𝑖𝑖)− 𝑔𝐻] Caputo differentiable. 4 Existence result for fuzzy 𝖖-fractional differential equations Theorem 4.1 If the function 𝑓: 𝕋𝔮 × ℝ𝔽 × ℝ𝔽 × ℝ𝔽 → ℝ𝔽 is a continuous fuzzy valued function, then the given equation (1.1) with the initial condition (1.2) is equivalent to the following integral equation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 41 https://internationalpubls.com 𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0 = 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))𝑑𝔮𝑠 (4.1) Proof. Let 𝜙(𝜏) be a solution of (1.1). We claim that 𝜙(𝜏) is also a solution of (4.1). To assure that, let us consider, 𝑟(𝜏) = 𝜆𝜙(𝜏) + 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏)). Then we have, 𝑟(𝜏) =𝐶 𝐷𝛼𝜙(𝜏). Assume that here 𝜙(𝜏) is a monotone function for 𝜏 ∈ 𝕋𝔮. Using Lemma (2.1), we have 𝐼𝛼(𝐶𝐷𝛼)(𝜙(𝜏)) = 𝜙(𝜏) ⊖ 𝑔𝐻 𝜙(𝜏0) = 𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0 Hence 𝐼𝛼𝑟(𝜏) = 𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0 i.e., 𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0 = 𝜆 Γ𝑞(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))𝑑𝔮𝑠 The necessary condition is satisfied. Next, let us consider that 𝜙(𝜏) is a monotone function for 𝜏 ∈ 𝕋𝔮 , such that (4.1) is satisfied. Operating fuzzy Caputo 𝑔𝐻 derivative of order 𝛼, by employing Definition (2.5), on both the sides of equation (4.1), we get 𝐶𝐷𝛼(𝜙(𝜏)) = 𝜆𝜙(𝜏) + 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏)) for 𝜏 ∈ 𝕋𝔮. Now, this part is sufficient to prove the theorem. Theorem 4.2 Let 𝕋𝔮 ≥ 0 and let 𝜖 ≥ 0 be a constant such that 0 ∈ [𝜙0 − 𝜖, 𝜙0 + 𝜖]. Assume that 𝑓: 𝕋𝔮 × [𝜙0 − 𝜖, 𝜙0 + 𝜖] × [𝜙0 − 𝜖, 𝜙0 + 𝜖] × [𝜙0 − 𝜖, 𝜙0 + 𝜖] → [𝜙0 − 𝜖, 𝜙0 + 𝜖] satisfies the condition (𝐻1). 𝜅 = 𝑚𝑖𝑛 {𝕋𝔮, [ Γ𝔮(𝛼 + 1)𝜖 (𝜖 + ‖𝜙0‖)(𝜆 + 𝐿(1 + 𝐾∗ + 𝐻∗) + 𝑀 ] 1 𝛼 }, where 𝑀 = 𝑆𝑢𝑝𝜏∈[𝜏0,𝑇]|𝑓(𝜏, 0,0,0)| . Then the cauchy problem for (1.1) has a unique solution 𝑢: [0, 𝜅] → ℝ𝔽. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 42 https://internationalpubls.com Proof. Define the set 𝔹 = {𝜙 ∈ 𝐶([𝜏0, 𝜅], ℝ𝔽): 𝜙(𝜏0) = 𝜙0, ‖𝜙 − 𝜙0‖ ≤ 𝜖}. Since 𝜙0 ∈ 𝔹, 𝔹 is nonempty. Also, 𝔹 is a closed, bounded and convex subset of Banach space 𝐶([0, 𝜅], ℝ𝔽). On 𝔹, define an operator 𝒦 by 𝒦𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0 = 𝜆 Γ𝑞(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝑢(𝑠), 𝒢𝑢(𝑠), 𝒮𝑢(𝑠))𝑑𝔮𝑠 Now, it needs to prove that 𝒦 maps 𝔹 to itself. Consider any 𝜙 ∈ 𝔹 and 𝜏 ∈ [0, 𝜅]. ‖𝒦𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0‖ ≤ 𝜆 Γ𝑞(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙(𝑠)‖𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))‖𝑑𝔮𝑠 ≤ 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1(‖𝜙(𝑠) − 𝜙0‖ + ‖𝜙0‖)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1(‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝑢(𝑠), 𝒮𝑢(𝑠)) − 𝑓(𝑠, 0,0,0)‖)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 0,0,0)‖𝑑𝔮𝑠 By the assumption (𝐻1) and the definition of 𝔹 for any 𝜏 ∈ [0, 𝜅], ‖𝜙(𝜏)‖ ≤ ‖𝜙(𝜏) − 𝜙0‖ + ‖𝜙0‖ ≤ 𝜖 + ‖𝜙0‖ (4.2) Consider, 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1(‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠)) − 𝑓(𝑠, 0,0,0)‖)𝑑𝔮𝑠 ≤ 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1ℒ[‖𝜙‖ + ‖𝒢𝜙‖ + ‖𝒮𝜙‖]𝑑𝔮𝑠 ℒ Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙‖(1 + 𝐾∗ + 𝐻∗)𝑑𝔮𝑠 where 𝑚𝑎𝑥‖𝒢𝜙‖ = 𝐾∗ and 𝑚𝑎𝑥‖𝒮𝜙‖ = 𝐻∗ Therefore, ‖𝒦𝜙(𝜏) ⊖ 𝑔𝐻 𝜙0‖ ≤ 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1(𝜖 + ‖𝜙0‖)𝑑𝔮𝑠 + ℒ Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1(𝜖 + ‖𝜙0‖) + 𝑀 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑑𝔮𝑠 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 43 https://internationalpubls.com = (𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀 Γ𝔮(𝛼) Γ𝔮(𝛼) Γ𝔮(𝛼+1) (𝜏 − 𝜏0)𝛼 ≤ (𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀 Γ𝔮(𝛼) Γ𝔮(𝛼) Γ𝔮(𝛼+1) (𝜅)𝛼 = 𝜖 Hence 𝒦 maps 𝔹 to itself. Next we consider that, for 0 ≤ 𝜏1 ≤ 𝜏2 ≤ 𝜅, ‖𝒦𝜙(𝜏1) − 𝒦𝜙(𝜏2)‖ ≤ 𝜆 Γ𝔮(𝛼) ⊙ ‖∫ 𝜏1 𝜏0 (𝜏1 − 𝔮𝑠)𝛼−1𝜙(𝑠) 𝑑𝔮𝑠 − ∫ 𝜏2 𝜏0 (𝜏2 − 𝔮𝑠)𝛼−1𝜙(𝑠) 𝑑𝔮𝑠‖ + 1 Γ𝔮(𝛼) ⊙ ‖∫ 𝜏1 𝜏0 (𝜏1 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝑢(𝑠), 𝒮𝑢(𝑠)) 𝑑𝔮𝑠 − ∫ 𝜏2 𝜏0 (𝜏2 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠)) 𝑑𝔮𝑠‖ Consider 𝜆 Γ𝔮(𝛼) ⊙ ‖∫ 𝜏1 𝜏0 (𝜏1 − 𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠 − ∫ 𝜏2 𝜏0 (𝜏2 − 𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠‖ ≤ 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏1 𝜏0 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)‖𝜙(𝑠)‖𝑑𝔮 + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏0 ((𝜏2 − 𝔮𝑠)𝛼−1)‖𝜙(𝑠)‖𝑑𝔮𝑠 ≤ 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏1 𝜏0 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)(‖𝜙(𝑠) − 𝜙0‖ + ‖𝜙0‖)𝑑𝔮𝑠 + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏0 ((𝜏2 − 𝔮𝑠)𝛼−1)(‖𝜙(𝑠) − 𝜙0‖ + ‖𝜙0‖)𝑑𝔮𝑠 ≤ 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏1 𝜏0 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)(𝜖 + ‖𝜙0‖)𝑑𝔮𝑠 + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏0 ((𝜏2 − 𝔮𝑠)𝛼−1)(𝜖 + ‖𝜙0‖)𝑑𝔮𝑠 ≤ 𝜆(𝜖+‖𝜙0‖) Γ𝔮(𝛼) Consider 1 Γ𝔮(𝛼) ⊙ ‖ ∫ 𝜏1 𝜏0 (𝜏1 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))𝑑𝔮𝑠 − ∫ 𝜏2 𝜏0 (𝜏2 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))𝑑𝔮𝑠 ‖ ≤ 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏1 𝜏0 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠)) − 𝑓(𝑠, 0,0,0)‖𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏1 𝜏0 (𝜏2 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 0,0,0)‖𝑑𝔮𝑠 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 44 https://internationalpubls.com + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏1 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠)) − 𝑓(𝑠, 0,0,0)‖𝑑𝔮𝑠 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏1 (𝜏2 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 0,0,0)‖𝑑𝔮𝑠 ≤ ℒ(1+𝐾∗+𝐻∗) Γ𝔮(𝛼) ⊙ ∫ 𝜏1 𝜏0 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)(𝜖 + ‖𝜙0‖)𝑑𝔮𝑠 + 𝑀 Γ𝑞(𝛼) ⊙ ∫ 𝜏1 𝜏0 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)𝑑𝔮𝑠 + ℒ(1+𝐾∗+𝐻∗) Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏1 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)(𝜖 + ‖𝜙0‖)𝑑𝔮𝑠 + 𝑀 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏1 ((𝜏1 − 𝔮𝑠)𝛼−1 − (𝜏2 − 𝔮𝑠)𝛼−1)𝑑𝔮𝑠 Then ‖𝒦𝜙(𝜏1) − 𝒦𝜙(𝜏2)‖ ≤ (𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀 Γ𝑞(𝛼) ⊙ ∫ 𝜏1 𝜏0 ( (𝜏1 − 𝔮𝑠)𝛼−1 −(𝜏2 − 𝔮𝑠)𝛼−1) 𝑑𝔮𝑠 + (𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀 Γ𝔮(𝛼) ⊙ ∫ 𝜏2 𝜏1 (𝜏2 − 𝔮𝑠)𝛼−1𝑑𝔮𝑠 = (𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀 Γ𝑞(𝛼) ⊙ (2(𝜏2 − 𝜏1)𝛼 + 𝜏1 𝛼 − 𝜏2 𝛼) Hence 𝒦𝜙 is continuous. Hence for any 𝜙 ∈ 𝔹, we have 𝒦𝜙 ∈ 𝐶([0, 𝜅], ℝ𝔽) i.e. 𝒦𝜙(𝜏0) = 𝜙0 and ‖𝒦𝜙 − 𝜙0‖ ≤ 𝜖. This concludes that 𝒦𝜙 ∈ 𝔹 whenever 𝑢 ∈ 𝔹. i.e. 𝒦𝜙 maps 𝔹 into itself. The next step is to prove that, for every 𝑛 ∈ ℕ ∪ {0}, and every 𝜙, 𝜓 ∈ 𝔹, we have ‖𝒦𝑛𝜙 − 𝒦𝑛𝜓‖ ≤ [𝜆+ℒ(1+𝐾∗+𝐻∗)]𝑛 Γ𝑞(𝑛𝛼+1) ‖𝜙 − 𝜓‖, 𝜏 ∈ [0, 𝜅] (4.3) This can be seen by induction. For 𝑛 = 0, the inequality (4.4) is trivially true. We assume that (4.4) is true for 𝑛 = 𝑚 − 1 and prove it for 𝑛 = 𝑚. By using definition of operator 𝒦, we have ‖𝒦𝑚𝜙(𝜏) − 𝒦𝑚𝜓(𝜏)‖ ≤ 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝑠)𝛼−1‖𝒦𝑚−1𝜙(𝑠) − 𝒦𝑚−1𝜓(𝑠)‖𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝑠)𝛼−1 ‖ 𝑓(𝑠, 𝒦𝑚−1𝜙(𝑠), 𝒦𝑚−1𝒢𝜙(𝑠), 𝒦𝑚−1𝒮𝜙(𝑠)) −𝑓(𝑠, 𝒦𝑚−1𝜓(𝑠), 𝒦𝑚−1𝒢𝜓(𝑠), 𝒦𝑚−1𝒮𝜓(𝑠)) ‖ 𝑑𝑞𝑠 For 𝑛 = 𝑚 − 1, we get ‖𝒦𝑚𝜙(𝜏) − 𝒦𝑚𝜓(𝜏)‖ ≤ 𝜆 Γ𝑞(𝛼) (𝜆+ℒ(1+𝐾∗+𝐻∗))𝑚−1 Γ𝑞((𝑚−1)𝛼+1) ‖𝜙 − 𝜓‖ ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝑠)𝛼−1𝑠𝑚𝛼−𝛼𝑑𝔮𝑠 + ℒ(1+𝐾∗+𝐻∗) Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝑠)𝛼−1‖𝒦𝑚−1𝜙(𝑠) − 𝒦𝑚−1𝜓(𝑠)‖𝑑𝔮𝑠 ≤ (𝜆+ℒ) Γ𝔮(𝛼) (𝜆+ℒ(1+𝐾∗+𝐻∗))𝑚−1 Γ𝔮((𝑚−1)𝛼+1) ‖𝜙 − 𝜓‖ ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝑠)𝛼−1𝑠𝑚𝛼−𝛼𝑑𝔮𝑠 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 45 https://internationalpubls.com = [(𝜆+ℒ(1+𝐾∗+𝐻∗))𝜏𝛼]𝑚 Γ𝔮(𝑚𝛼+1) ‖𝜙 − 𝜓‖ which is our desired inequality (4.4). Hence we have ‖𝒦𝑛𝜙 − 𝒦𝑛𝜓‖ ≤ [(𝜆+ℒ(1+𝐾∗+𝐻∗))𝜅𝛼]𝑛 Γ𝔮(𝑚𝛼)+1 ‖𝜙 − 𝜓‖ By definition(2.10), we have ∑∞ 𝑛=0 [(𝜆+ℒ(1+𝐾∗+𝐻∗))𝜅𝛼]𝑛 Γ𝑞(𝑛𝛼+1) = 𝐸(𝜆 + ℒ(1 + 𝐾∗ + 𝐻∗))𝜅𝛼 We have proved that the operator 𝒦 satisfies all the conditions of theorem (2.3) and hence ℒ has a unique fixed point 𝜙: [0, 𝜅] → ℝ𝔽 which is the solution of (1.1). 5 Continuous Dependence and Uniqueness of Solutions Theorem 5.1 Suppose that the function 𝑓: 𝕋𝑞 × ℝ𝔽 × ℝ𝔽 × ℝ𝔽 → ℝ𝔽. Let 𝜙1(𝜏) and 𝜙2(𝑡) be the solutions of equation, 𝐶𝐷𝛼𝜙(𝜏) = 𝜆𝜙(𝜏) + 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏)), 𝜏 ∈ 𝕋𝑞 (5.1) corresponding to 𝜙1(0) = 𝜙0 and 𝜙2(0) = 𝜙0 ∗ respectively. Then ‖𝜙1 − 𝜙2‖ ≤ {1 + 𝐾(𝜆+ℒ(1+𝐾∗+𝐻∗)) Γ𝔮(𝛼) 𝑇𝛼} ‖𝜙0 − 𝜙0 ∗‖, 𝜏 ∈ 𝕋𝑞 Let 𝜙1(𝜏) and 𝜙2(𝜏) be the solutions of equation (5.1) corresponding to 𝜙1(0) = 𝜙0 and 𝜙2(0) = 𝜙0 ∗ respectively. 𝐶𝐷𝛼𝜙1(𝜏) = 𝜆𝜙1(𝜏) + 𝑓(𝜏, 𝜙1(𝜏), 𝒢𝜙1(𝜏), 𝒮𝜙1(𝜏)), 𝜙1(0) = 𝜙0, 𝐶𝐷𝛼𝜙2(𝜏) = 𝜆𝜙2(𝜏) + 𝑓(𝜏, 𝜙2(𝜏), 𝒢𝜙2(𝜏), 𝒮𝜙2(𝜏)), 𝜏2(0) = 𝜙0 ∗ This implies that, 𝜙1(𝜏) = 𝜙0 + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 0 (𝜏 − 𝔮𝑠)𝛼−1𝜙1(𝑠)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 0 (𝜏 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙1(𝑠), 𝒢𝜙1(𝑠), 𝒮𝜙1(𝑠))𝑑𝔮𝑠 and 𝜙2(𝜏) = 𝜙0 ∗ + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 0 (𝜏 − 𝔮𝑠)𝛼−1𝜙2(𝑠)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 0 (𝜏 − 𝑠)𝛼−1𝑓(𝑠, 𝜙2(𝑠), 𝒢𝜙2(𝑠), 𝒮𝜙2(𝑠))𝑑𝔮𝑠 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 46 https://internationalpubls.com Using the hypothesis (𝐻1), for any 𝜏 ∈ [𝜏0, 𝑇], we obtain ‖𝜙1(𝜏) − 𝜙2(𝜏)‖ ≤ ‖𝜙0 − 𝜙0 ∗‖ + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙1(𝑠) − 𝜙2(𝑠)‖𝑑𝔮𝑠 + 1 Γ(𝛼) ⊙ ∫ 𝜏 0 (𝜏 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 𝜙1(𝑠), 𝒢𝜙1(𝑠), 𝒮𝜙1(𝑠)) − 𝑓(𝑠, 𝜙2(𝑠), 𝒢𝜙2(𝑠), 𝒮𝑢2(𝑠))‖𝑑𝔮𝑠 ≤ ‖𝜙0 − 𝜙0 ∗‖ + 𝐾(𝜆 + ℒ(1 + 𝐾∗ + 𝐻∗)) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙0 − 𝜙0 ∗‖𝑑𝔮𝑠 = ‖𝜙0 − 𝜙0 ∗‖ {1 + 𝐾(𝜆+ℒ(1+𝐾∗+𝐻)∗)) Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝑠)𝛼−1𝑑𝔮𝑠} ≤ ‖𝜙0 − 𝜙0 ∗‖ {1 + 𝐾(𝜆+ℒ(1+𝐾∗+𝐻)∗)) Γ𝔮(𝛼) 𝑇𝛼}, 𝜏 ∈ 𝕋𝑞. This proves the uniqueness of the solution of (1.1). 6 Continuous Dependence on Functions involved and Parameters Consider (1.1) and 𝐶𝐷𝛼𝜒(𝜏) = 𝜆𝜒(𝜏) + 𝑓∗(𝜏, 𝜒(𝜏), 𝒢𝜒(𝜏), 𝒮𝜒(𝜏)) (6.1) 𝜒(0) = 𝜒0, where 𝑓: 𝕋𝑞 × ℝ𝔽 × ℝ𝔽 × ℝ𝔽 → ℝ𝔽 Theorem 6.1 Suppose that 𝑓 in (1.1) satisfies the hypothesis (𝐻1). Let 𝜒(𝑡) be a solution of (6.1) and suppose that ‖𝑓(𝜏, 𝜒(𝜏), 𝒢𝜒(𝜏), 𝒮𝜒(𝜏)) − 𝑓∗(𝜏, 𝜒(𝜏), 𝒢𝜒(𝑡), 𝒮𝜒(𝑡))‖ ≤ 𝜖, 𝜏 ∈ 𝕋𝑞 and ‖𝜙0 − 𝜓0‖ < 𝛿 where 𝜖, 𝛿 > 0 are arbitrary small constants. Then the solution of 𝜙(𝜏) of (1.1) depends continuously on the functions involved therein. Proof. Let 𝜙(𝜏) and 𝜓(𝜏) be solutions of (1.1) and (6.1) respectively. 𝐶𝐷𝛼𝜙(𝜏) = 𝜆𝜙(𝜏) + 𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝑡)), 𝜙(0) = 𝜙0 𝐶𝐷𝛼𝜓(𝜏) = 𝜆𝜓(𝜏) + 𝑓∗(𝜏, 𝜓(𝜏), 𝒢𝜓(𝜏), 𝒮𝜓(𝜏)), 𝜓(0) = 𝜓0 This implies that, 𝜙(𝜏) = 𝜙0 + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))𝑑𝔮𝑠 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 47 https://internationalpubls.com and 𝜓(𝑡) = 𝜓0 + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝜓(𝑠)𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑓∗(𝑠, 𝜓(𝑠), 𝒢𝜓(𝑠), 𝒮𝜓(𝑠))𝑑𝔮𝑠 ‖𝜙(𝜏) − 𝜓(𝜏)‖ ≤ ‖𝜙0 − 𝜓0‖ + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙(𝑠) − 𝜓(𝑠)‖𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠)) − 𝑓∗(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠))‖𝑑𝔮𝑠 ≤ ‖𝜙0 − 𝜓0‖ + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙(𝑠) − 𝜓(𝑠)‖𝑑𝔮𝑠 + 1 Γ𝑞(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 𝜙(𝑠), 𝒢𝜙(𝑠), 𝒮𝜙(𝑠)) − 𝑓(𝑠, 𝜓(𝑠), 𝒢𝜓(𝑠), 𝒮𝜓(𝑠))‖𝑑𝔮𝑠 + 1 Γ𝑞(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝑓(𝑠, 𝜓(𝑠), 𝒢𝜓(𝑠), 𝒮𝜓(𝑠)) − 𝑓∗(𝑠, 𝜓(𝑠), 𝒢𝜓(𝑠), 𝒮𝜓(𝑠))‖𝑑𝔮𝑠 ≤ ‖𝜙0 − 𝜓0‖ + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙(𝑠) − 𝜓(𝑠)‖𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1ℒ[‖𝜙 − 𝜓‖ + ‖𝒢𝜙 − 𝒢𝜓‖ + ‖𝒮𝜙 − 𝒮𝜓‖]𝑑𝔮𝑠 + 𝜖 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑑𝔮𝑠 ≤ ‖𝜙0 − 𝜓0‖ + 𝜆 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙 − 𝜓‖𝑑𝔮𝑠 + 1 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1ℒ(1 + 𝐾∗ + 𝐻∗)‖𝜙 − 𝜓‖𝑑𝔮𝑠 + 𝜖 Γ𝔮(𝛼) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1𝑑𝔮𝑠 ≤ 𝛿 + 𝜖 Γ𝔮(𝛼+1) 𝜏𝛼 + ( 𝜆+ℒ(1+𝐾∗+𝐻∗) Γ𝔮(𝛼) ) ⊙ ∫ 𝜏 𝜏0 (𝜏 − 𝔮𝑠)𝛼−1‖𝜙 − 𝜓‖𝑑𝔮𝑠 By Lemma (2.2), ‖𝜙(𝑡) − 𝜓(𝑡)‖ ≤ 𝛿 + 𝜖 Γ𝔮(𝛼+1) 𝜏𝛼 + 𝐾(𝜆+ℒ(1+𝐾∗+𝐻∗))𝛿 Γ𝔮(𝛼+1) 𝜏𝛼 + 𝐾 𝜆+𝐿(1+𝐾∗+𝐻∗) Γ𝑞(2𝛼+1) 𝜏2𝛼 , 𝜏 ∈ 𝕋𝑞. From the above inequality it comes to know that the solution 𝜙(𝜏) depends continuously on the functions involved in the given FDE (1.1). If 𝜖 = 0, the continuous dependence of solutions of the inequality is on the initial conditions. Here it is noted that as 𝜖, 𝛿 > 0 were arbitrary, by taking 𝜖, 𝛿 > 0, we have 𝜙 → 𝜓, where 𝜙: 𝕋𝑞 → ℝ𝔽 and 𝜓: 𝕋𝑞 → ℝ𝔽 are the solutions of (1.1) and (6.1) respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 48 https://internationalpubls.com 7 Illustrative example The incorporation of an illustrative example can serve as a powerful tool to amplify the overall impact of the presented idea in the article. To accomplish the same, we provide an example to illustrate the execution of our main result. We consider 𝐶𝐷𝛼[𝜙(𝜏)] = 𝜏+1 11 𝜙(𝜏) + 𝜏2 2 + ∫ 1 0 1 (𝑠+𝜏+2)2 𝜙(𝑠)𝑑𝑠 + ∫ 1 0 𝜏 𝑠+1 𝜙(𝑠)𝑑𝑠, 𝜏 ∈ [𝜏0, 𝑇] (7.1) 𝜙(0) = 0 Define 𝑓(𝜏, 𝜙, 𝒢𝜙, 𝒮𝜙) = 𝜏+1 11 𝑢(𝜏) + 𝜏2 2 + ∫ 1 0 1 (𝑠+𝜏+2)2 𝜙(𝑠)𝑑𝑠 + ∫ 1 0 𝜏 𝑠+1 𝜙(𝑠)𝑑𝑠 Clearly the function 𝑓 and 𝑔 continuous, since ‖𝑓(𝜏, 𝜙(𝜏), 𝒢𝜙(𝜏), 𝒮𝜙(𝜏)) − 𝑓(𝜏, 𝜓(𝜏), 𝒢𝜓(𝜏), 𝒮𝜓(𝜏))‖ ≤ 169 264 ‖𝜙 − 𝜓‖ Here by (1.2), we get ℒ = 169 264 ≤ 1 . Therefore by theorem(3.2) this problem has a unique solution. References [1] R. P. Agarwal. Certain fractional q-integrals and q-derivatives. Mathematical Proceedings of the Cambridge Philosophical Society, 66(2):365–370, 1969. [2] Tofigh Allahviranloo, Atefeh Armand, and Zienab Gouyandeh.Fuzzy frac- tional differential equations under generalized fuzzy caputo derivative. Jour- nal of Intelligent& Fuzzy Systems, 26(3):1481–1490, 2014. [3] Jehad Alzabut, Mohamed Houas, and Mohamed I. Abbas. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential caputo fractional q-derivatives. 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