Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives

Fuzzy fractional calculus has become a powerful tool with more accurate and successful results in modeling several complex and fuzzy physical phenomena in numerous seemingly diverse and widespread fields of science and engineering. Recently, the theory of fuzzy fractional differential equations was proposed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena. When a real physical phenomenon is modeled by a fractional initial value problem, we cannot usually be sure that the model is perfect. For example, the initial value of this problemmay not be known precisely. In order to get a perfect model under a precise initial condition, Agarwal et al., in [1], proposed the concept of fuzzy solutions for fractional differential equations with uncertainty. Arshad and Lupulescu, in [2], proved some results on the existence and uniqueness of solutions for fuzzy fractional differential equations. Later, Alikhani and Bahrami [3] proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. %e authors in [4, 5] discussed the concepts about generalized Hukuhara fractional Riemann–Liouville and Caputo differentiability of fuzzy-valued functions. %e equivalence between fuzzy fractional differential equation and fuzzy fractional integral equation was discussed in [6]. Ngo et al., in [7], proved the existence and uniqueness results of solutions for initial value problem under fuzzy Caputo–Katugampola fractional derivatives. For many basic works related to the nonlinear ordinary differential equations and the fuzzy fractional differential equations, we refer the readers to [8–13] and references therein. Motivated by the results mentioned above and by using the intuitionistic fuzzy sets theory introduced by Atanassov, in [14], we study the existence and uniqueness results for the following intuitionistic fuzzy local and nonlocal fractional boundary value problems:


Introduction
Fuzzy fractional calculus has become a powerful tool with more accurate and successful results in modeling several complex and fuzzy physical phenomena in numerous seemingly diverse and widespread fields of science and engineering. Recently, the theory of fuzzy fractional differential equations was proposed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena. When a real physical phenomenon is modeled by a fractional initial value problem, we cannot usually be sure that the model is perfect. For example, the initial value of this problem may not be known precisely. In order to get a perfect model under a precise initial condition, Agarwal et al., in [1], proposed the concept of fuzzy solutions for fractional differential equations with uncertainty. Arshad and Lupulescu, in [2], proved some results on the existence and uniqueness of solutions for fuzzy fractional differential equations. Later, Alikhani and Bahrami [3] proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. e authors in [4,5] discussed the concepts about generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions. e equivalence between fuzzy fractional differential equation and fuzzy fractional integral equation was discussed in [6]. Ngo et al., in [7], proved the existence and uniqueness results of solutions for initial value problem under fuzzy Caputo-Katugampola fractional derivatives. For many basic works related to the nonlinear ordinary differential equations and the fuzzy fractional differential equations, we refer the readers to [8][9][10][11][12][13] and references therein.
Motivated by the results mentioned above and by using the intuitionistic fuzzy sets theory introduced by Atanassov, in [14], we study the existence and uniqueness results for the following intuitionistic fuzzy local and nonlocal fractional boundary value problems: where c D α is the Caputo derivative of X(t) at order 0 < α < 1 and F: [0, T] × IF 1 ⟶ IF 1 and G: C([0, T], IF 1 ) ⟶ IF 1 are intuitionistic fuzzy continuous functions. e spaces IF 1 and C([0, T], IF 1 ) will be defined after. Our paper is organized as follows. Section 2 gives some basic definitions, lemmas, and theorems as preliminaries of intuitionistic fuzzy sets theory. e existence results for the intuitionistic fuzzy local and nonlocal fractional boundary value problems are given in Section 3 and Section 4. Illustrative example is presented in Section 5, followed by conclusion and future works in Section 6.

Preliminaries
Fuzzy set theory was introduced by Zadeh [15], and it is an extension of the classical crisp logic into a multivariate form. Atanassov generalizes this concept to intuitionistic fuzzy sets (IFSs) [14], and later, there has been much progress in the study of IFSs. As a special case of intuitionistic fuzzy sets, intuitionistic fuzzy numbers were introduced by Xu [16].
We denote by Definition 1 (see [17]). An element 〈u, v〉 ∈ IF(R) is called an intuitionistic fuzzy number if it satisfies the following conditions: (1) 〈u, v〉 is normal, i.e., there exists x 0 , x 1 ∈ R such that u(x 0 ) � 1 and v(x 1 ) � 1 (2) u is fuzzy convex and v is fuzzy concave (3) u is upper semicontinuous and v is lower semicontinuous We denote by IF 1 the collection of all intuitionistic fuzzy numbers.
Definition 4 (see [18]). Let then 〈w, z〉 is called Hukuhara difference of 〈u 1 , v 1 〉 and Definition 5 (see [18,19]). e generalized Hukuhara difference of two intuitionistic fuzzy number Journal of Mathematics Definition 6 (see [17]). Let f: We say that f is generalized Hukuhara differentiable at t 0 if there exists f ′ (t 0 ) ∈ IF 1 such that Definition 7 (see [17] is the set of all closed and bounded intervals of R. Definition 8 (see [17,19]). Let F: and we write Journal of Mathematics 3 en, we have the following result.
In the following sections, we will need some notations and definitions.
In this example, we calculate the intuitionistic fuzzy Caputo fractional derivative of the function 〈u, v〉(t). For this purpose, we start by giving the gH-derivative of 〈u, v〉(t) as follows: is implies that 〈u, v〉 ′ (t) � C. Since [C] α , us, Theorem 1 (Schaefer's fixed-point theorem (see [21])). Let P be a continuous and compact mapping of a Banach space X into itself such that the set is bounded; then, mapping P has a fixed point.

Intuitionistic Fuzzy Local Fractional Boundary Value Problems
Definition 12 (see [6]). An intuitionistic fuzzy function X: For the existence of solutions for problem (1), we need the following lemma.
Lemma 3 (see [6]). Let α ∈ ]0, 1[ and H: if and only if it is a solution of the following initial value problem: As a consequence of Lemma 3, we have the following result. A

d-monotone intuitionistic fuzzy function X(t) is a solution of the fractional integral equation,
if and only if is a solution of the fractional boundary value problem (1).

Theorem 2.
Assume that there exists a positive constant k such that In addition, if then problem (1) has a unique solution.
Proof. For this purpose, we transform problem (1) Let X, Y ∈ C([0, T], IF 1 ); then, we have Journal of Mathematics As a consequence of Banach fixed-point theorem, we can deduce that the operator P has a unique fixed point X which is the solution of problem (1). Proof. To show that problem (1) has at least one solution defined on [0, T], we use Schaefer's fixed-point theorem [22]. For this purpose, to prove that the operator P defined above has a fixed point, the proof of this theorem will be given in several steps.
Step 1: let us show that P is continuous.
Let (X n ) n ⊂ C([0, T], IF 1 ) such that X n converges to X in C([[0, T], IF 1 ) and t ∈ [0, T]; we have  application, an example is presented to illustrate the applicability of the obtained results.

Data Availability
e data used to support the findings of this study are included in the references within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.