Variational inequalities for lattice-valued fuzzy relations with applications

Abstract

In this article, we study the notion of the variational inequalities for lattice-valued fuzzy relations. In this context, a variational inequality problem has been proposed that generalizes many results in the literature. The conditions for the existence of solutions of the proposed problem have been discussed. It has been shown that the proposed variational inequality problem is equivalent to a fixed point problem. This fixed point formulation allows us to present an iterative algorithm to approximate solution of the variational inequality problem. For applications, first the existence result for the solutions of an ℒ-fuzzy Caputo-Fabrizio fractional differential inclusion initial value problem involving a projection operator has been proved. Then the solutions of an obstacle boundary value variational inequality problem in function spaces has been obtained.

Keywords

fixed points variational inequalities iterative algorithm 46S40 47H10 54H25

1 Preliminaries

The central concept in numerous disciplines like engineering, operation research, game theory and economics is equilibrium. Matrix theory, optimization theory, complementarity theory and fixed point theory are the methodologies that have been applied for the formulation, qualitative analysis and computation of the equilibrium. The theory of variational inequalities has unified the above methodologies for studying equilibrium problems. The concept of variational inequalities was introduced by Hartman and Stampacchia as a tool for studying the theory of partial differential equations in infinite dimensional spaces in 1966 [14].

The revolution in finite dimensional theory was occurred, when Dafermos [9], reformulated the traffic network model of Smith [28] as a variational inequalities problem in 1980. This unveiled the theory to study further in finite dimensional spaces. The problems related to variational inequalities may include; environmental network problems, migration equilibrium problems, financial equilibrium problems, spatial price equilibrium problems, network problems and knowledge network problems [5 , 32].

Throughout this paper, let $H$ be a real Hilbert space and Ω be a closed and convex subset of $H$ . Denote by || · || and 〈 · , · 〉, the norm and inner product on $H$ . The dual of $H$ is denoted by $H^{*}$ . For $f \in H^{*}$ and $u \in H$ , 〈f, u〉 denotes the pairing. The elements will distinguish between the inner product and pairing.

The following problem was proposed by Stampacchia [14]:

Problem 1.1. Given a closed and convex subset Ω of a Hilbert space $H$ , and let $T : Ω \to H$ be a mapping. Then find u ∈ Ω such that $〈 Tu, v - u 〉 \geq 0, for all v \in Ω .$

Many generalizations have been proposed for the above Problem 1.1, for more details, see [4 , 29] and the references cited therein. One of the well-known generalizations of above Problem 1.1, was given by Fang and Peterson in 1982 [11]. Their proposed variational inequality problem was known as generalized variational inequality problem. Let $ℝ^{n}$ be an n-dimensional Euclidean space. Given a relation Γ on $ℝ^{n}$ and V be a subset of $ℝ^{n}$ , the generalized variational inequities problem is stated as follows:

Problem 1.2. Find all pairs (x, y) ∈ Γ such that

x ∈ V,

〈y, z - x〉≥0, for each z ∈ V .

This problem was a milestone in the theory of variational inequalities. After this the notion of fuzzy mappings were introduced by Heilpern [15], in 1981. Denote the set of all fuzzy sets on $H$ by $F (H)$ .

For α ∈ (0, 1] , the α-level set of any $T \in F (H)$ is denoted by [T] _α and defined as; $[T]_{α} = {z \in H : T (z) \geq α} .$ The variational problems for fuzzy mappings in the settings of topological vector spaces were introduced by Chang [7] in 1989. However the numerical algorithms for solving the variational inequalities problem for fuzzy mappings were presented by Noor in [23], as a generalization of the above both problems. This problem is stated as follows:

Problem 1.3. For a given fuzzy mapping $T : Ω \to F (H),$ find u ∈ Ω, such that w ∈ [Tu] _α and $〈 w, v - u 〉 \geq 0, for all v \in Ω .$ (1.1)

Let $(L, ⪯)$ be a complete distributive lattice with minimal element $0_{L}$ and maximal element $1_{L}$ . A generalization of fuzzy sets [34] was given by Goguen in [13], known as ℒ-fuzzy sets. It has been shown in [31], that intuitionistic fuzzy sets, and fuzzy sets are ℒ-fuzzy sets. Denote the family of all ℒ-fuzzy sets f on a Hilbert space $H$ by $F_{L} (H) : = {f : H \to L} .$ A mapping $T : Ω \to F_{L} (H)$ is called an ℒ-fuzzy mapping. The ℒ-fuzzy mappings were introduced by Rashid et. al. in [24], they used these mappings to find the coincidence points of β-admissible mappings in metric spaces. For more details we refer the readers for the articles [3 , 31], and the references cited therein. According to [13], we recall the following useful notions.

Let $T \in F_{L} (H),$ for $α_{L} \in L ∖ {0},$ the set $[T]_{α_{L}} = {u \in H : Tu ≽ α_{L}}$ is called an $α_{L}$ -level set of T .

A lattice-valued or an ℒ-fuzzy relation on X × Y is a function $ρ : X \times Y \to L .$ For $α_{L} \in L,$ the $α_{L}$ -level set of the ℒ-fuzzy relation ρ is $ρ^{α_{L}} = {(x, y) \in X \times Y : ρ (x, y) ≽ α_{L}} .$ That is, for each $α_{L} \in L,$ the set $ρ^{α_{L}}$ is a (binary) relation from X into Y, and denoted by $ρ^{α_{L}} : X ↫ Y .$

The set $ρ^{α_{L}}$ is called an $α_{L}$ -functional from X to Y if $ρ (x, y) ≽ α_{L}, ρ (x, z) ≽ α_{L}$ implies y = z for x ∈ X and y, z ∈ Y . Note that a mapping $(an α_{L} - mapping)$ is an $α_{L}$ -relation which is an $α_{L}$ -functional.

Remark 1.4. For $L = [0, 1]$ and $α_{L} = 1,$ we have

1-relation is a crisp binary relation from X to Y .

1-left(right) total relation is a just left (right) total relation as defined in [2].

1-functional is a functional in usual sense, and 1-mapping is a mapping as defined in [2].

For an ℒ-fuzzy, $α_{L}$ -relation $ρ^{α_{L}} : X ↫ Y$ , and S ⊂ X, define $\begin{matrix} ρ [S]_{α_{L}} : = {y \in Y : ρ (x, y) ≽ α_{L} forsome x \in S} \\ (Range of ρ^{α_{L}} for S) . \end{matrix}$ $dom (ρ^{α_{L}}) = {x \in X : ρ [{x}]_{α_{L}} \neq φ},$ $\begin{matrix} Range (ρ^{α_{L}}) = \\ {y \in Y : ρ (x, y) \geq α_{L} forsome x \in dom (ρ^{α_{L}})} . \end{matrix}$ For simplicity, use $ρ [x]_{α_{L}}$ for $ρ [{x}]_{α_{L}} .$ The class of $α_{L}$ -relations from X to Y is denoted by $R_{α_{L}} (X, Y)$ . Thus the collection $M_{α_{L}} (X, Y)$ of all $α_{L}$ -mappings from X to Y is a proper subcollection of $R_{λ} (X, Y)$ . Now let us propose our variational inequality problem which is called the variational inequality for Lattice-valued fuzzy relation, or an $L$ -fuzzy relation.

Problem 1.5. Given an ℒ-fuzzy relation $ρ : Ω \times H \to L,$ and $f \in H^{*},$ for $α_{L} \in L ∖ {0_{L}} .$ Find u ∈ Ω and $w \in H$ such that $ρ (u, w) \geq α_{L}$ and $〈 w, v - u 〉 \geq 〈 f, v - u 〉, for all v \in Ω .$ (1.2)

Now, let us define the following new concepts, to be used in our proofs.

Definition 1.6. An ℒ-fuzzy relation $ρ : Ω \times H \to L$ is called

$ρ^{α_{L}}$ -strongly monotone, if for all x, y ∈ Ω and $u, v \in H$ with $ρ (x, u) \geq α_{L}$ and $ρ (y, v) \geq α_{L}$ there exists δ > 0, such that $〈 u - v, x - y 〉 \geq δ ‖ x - y ‖^{2} .$

$ρ^{α_{L}}$ -Lipschitz continuous, if for all x, y ∈ Ω and $u, v \in H$ with $ρ (x, u) \geq α_{L}$ and $ρ (y, v) \geq α_{L}$ there exists 0 < β < 1, such that $‖ u - v ‖ \leq β ‖ x - y ‖ .$ For β = 1, the ℒ-fuzzy relation $ρ : Ω \times H \to L$ is called non-expansive.

Inverse $ρ^{α_{L}}$ -strongly monotone, if for all x, y ∈ Ω and $u, v \in H$ with $ρ (x, u) \geq α_{L}$ and $ρ (y, v) \geq α_{L}$ there exists γ > 0, such that $〈 u - v, x - y 〉 \geq γ ‖ u - v ‖^{2} .$

Define Λ, a canonical isomorphism from $H^{*}$ onto $H$ , such that for all $f \in H^{*}$ we have $〈 f, v 〉 = 〈 Λ f, v 〉 for all v \in Ω .$ (1.3) Where left hand side is pairing between the elements of $H^{*}$ and $H$ , while on the right hand side the inner product on the elements of $H$ is defined. Then $‖ Λ ‖_{H^{*}} = 1 = ‖ Λ^{- 1} ‖_{H}$ .

Definition 1.7. An ℒ-fuzzy relation $ρ : Ω \times H \to L$ is said to be closed valued, if the set ${(x, y) \in Ω \times H : ρ (x, y) \geq α_{L}}$ is a closed subset in $Ω \times H$ for each $α_{L} \in L .$

Remark 1.8. We know that every γ-inverse $ρ^{α_{L}}$ -strongly monotone and ℒ-fuzzy relation $ρ : Ω \times H \to L$ is $\frac{1}{γ}$ - $ρ^{α_{L}}$ -Lipschitz continuous.

In this paper our main purpose is to present a weaker version of above Problem 1.1-1.3, in which a weak relation, an ℒ-fuzzy relation is considered, an iterative sequence is constructed that converges to the solution of the problem 1.5.

2 Main Results

In this section, we present some useful lemmas, and basics results that constitute a base to prove our main existence results for proposed lattice-valued fuzzy variational inequalities Problem 1.5.

Lemma 2.1. Let $H$ be a Hilbert space and $ρ : Ω \times H \to L$ be a $ρ^{α_{L}}$ -Lipschitz closed valued continuous and ℒ-fuzzy relation. Then there exists u ∈ Ω such that $ρ (u, u) \geq α_{L}$ . This implies that the point u ∈ Ω is the $α_{L}$ - fixed point of the relation, i.e., $u \in ρ [u]_{α_{L}} .$

Proof. Let $u_{0}, u_{1} \in H$ be such that $ρ (u_{0}, u_{1}) \geq α_{L}$ , then there exists $u_{2} \in H$ such that $ρ (u_{1}, u_{2}) \geq α_{L}$ and using Definition 1.6 (ii) , we have $‖ u_{2} - u_{1} ‖ \leq β ‖ u_{1} - u_{0} ‖ for β \in (0, 1) .$ Since $ρ [u]_{α_{L}} \neg = φ$ for all $u \in H$ , and ρ is $ρ^{α_{L}}$ -Lipschitz continuous, then for $ρ (u_{1}, u_{2}) \geq α_{L}$ there exists $u_{3} \in H$ such that $ρ (u_{2}, u_{3}) \geq α_{L}$ such that $‖ u_{3} - u_{2} ‖ \leq β ‖ u_{2} - u_{1} ‖ \leq β^{2} ‖ u_{1} - u_{0} ‖ .$ Continuing in this way, we can find a sequence {u_n} in $H$ such that $ρ (u_{n}, u_{n + 1}) \geq α_{L}$ for all $n \in ℕ,$ and $‖ u_{n + 1} - u_{n} ‖ \leq β^{n} ‖ u_{1} - u_{0} ‖ for all n \geq 1 .$ Using above inequality, consoder $\begin{matrix} ‖ u_{n + p} - u_{n} ‖ \leq ‖ u_{n + p} - u_{n + p - 1} ‖ \\ + ‖ u_{n + p - 1} - u_{n + p - 2} ‖ + . . . + ‖ u_{n + 1} - u_{n} ‖ \\ \leq [β^{n + p - 1} + β^{n + p - 2} + . . . + β^{n}] ‖ u_{1} - u_{0} ‖ \\ \leq \frac{β^{n}}{1 - β} ‖ u_{1} - u_{0} ‖ for all n, p \geq 1 . \end{matrix}$ This shows that {u_n} is a Cauchy sequence in the Hilbert space $H$ , therefore $u_{n} \to u \in H$ . Continuity of ρ implies $ρ [u_{n}]_{α_{L}}$ converges to $ρ [u]_{α_{L}} .$ Since $u_{n + 1} \in ρ [u_{n}]_{α_{L}}$ for all n ≥ 1, previous statement implies $u_{n + 1} \in ρ [u]_{α_{L}} .$ This means u is a limit point of $ρ [u]_{α_{L}},$ but closedness of ρ implies $u \in ρ [u]_{α_{L}}$ or $ρ (u, u) \geq α_{L}$ . This proves the lemma.□

In the next lemma, we show that the proposed variational inequality problem (Problem 1.5) is equivalent to a fixed point problem, this gives a procedure to compute the solution by iterative scheme.

Lemma 2.2. Let Ω be a closed and convex subset of a Hilbert space $H .$ Then the point $(u, w) \in Ω \times H$ is a solution of the Problem 1.5 if and only if $u = {Pr}_{Ω} (u - k (w - Λ f)),$ for $ρ (u, w) \geq α_{L},$ where k > 0 is a constant and Pr _Ω is projection of $H$ onto Ω . That is, Problem 1.5 is equivalent to finding the fixed point of the mapping E : Ω → H, defined by $E (u) = {Pr}_{Ω} (u - k (w - Λ f)) .$

Proof. If the point $(u, w) \in Ω \times H$ is a solution of Problem 1.5, then $ρ (u, w) \geq α_{L}$ and $〈 w, v - u 〉 \geq 〈 Λ f, v - u 〉, for all v \in Ω .$ As for k > 0, we have 〈k (w - Λf) , v - u〉≥0, for all v ∈ Ω . Now for z ∈ Ω, { $\begin{matrix} ‖ z - (u - k (w - Λ f)) ‖^{2} \\ = ‖ z - u ‖^{2} + 2 〈 z - u, k (w - Λ f) 〉 + ‖ k (w - Λ f) ‖^{2} \\ \geq ‖ k (w - Λ f) ‖^{2} \\ = ‖ u - (u - k (w - Λ f)) ‖^{2} . \end{matrix}$ Thus $‖ z - (u - k (w - Λ f)) ‖ \geq ‖ u - (u - k (w - Λ f)) ‖,$ for all z ∈ Ω . Thus u = Pr _Ω (u - k (w - Λf)) .

Conversely, suppose that u = Pr _Ω (u - k (w - Λf)) . Then $‖ u - (u - k (w - Λ f)) ‖^{2} \leq ‖ z - (u - k (w - Λ f)) ‖^{2},$ for all z ∈ Ω . As for all z, u ∈ Ω, z_t = u + t (z - u) ∈ Ω for all t ∈ [0, 1] . Then $‖ u - (u - k (w - Λ f)) ‖^{2} \leq ‖ z_{t} - (u - k (w - Λ f)) ‖^{2} .$ Hence we have $\begin{matrix} ‖ k (w - Λ f) ‖^{2} \\ \leq ‖ u + t (z - u) - (u - k (w - Λ f)) ‖^{2} \\ = 〈 u + t (z - u) - (u - k (w - Λ f)), u + t (z - u) . \\ . - (u - k (w - Λ f)) 〉 \\ = 〈 t (z - u) + k (w - Λ f), t (z - u) + k (w - Λ f) 〉 \\ = t^{2} ‖ z - u ‖^{2} + 2 t 〈 k (w - Λ f), (z - u) 〉 \\ + ‖ k (w - Λ f) ‖^{2}, \end{matrix}$ which implies that $〈 k (w - Λ f), (z - u) 〉 \geq \frac{- t}{2} ‖ z - u ‖^{2},$ for all z ∈ Ω and t ∈ [0, 1] . In particular, for t = 0, we have $〈 w, z - u 〉 \geq 〈 Λ f, z - u 〉$ or from Definition 1.6 (iv) $〈 w, z - u 〉 \geq 〈 f, z - u 〉$ for all z ∈ Ω . This completes the proof.□

Lemma 2.3. [14] Let Ω be a closed and convex subset of $H$ . Then the projection operator Pr _Ω is nonexpansive and continuous.

Now, we are in a position to introduce and prove the main theorem.

Theorem 2.4. Let $(u, w) \in Ω \times H$ be a solution of the Problem 1.5 and $(u_{n}, w_{n}) \in Ω \times H$ satisfies u_n+1 = Pr _Ω (u_n - k (w_n - Λf)) and $ρ (u_{n}, w_{n}) \geq α_{L} .$ If an ℒ-fuzzy relation $ρ : Ω \times H \to L$ is closed valued $ρ^{α_{L}}$ -Lipschitz continuous and $ρ^{α_{L}}$ -strongly monotone with constants β and δ respectively, then u_n → u in Ω and w_n → w in $H$ for 0 < k < 2δ/β² .

Proof. By Lemma 2.2, Problem 1.5 can be written as follows: $E (u_{n}) : = u_{n + 1} = {Pr}_{Ω} (u_{n} - k (w_{n} - Λ f)) .$ Therefore $u_{n} - u = {Pr}_{Ω} (u_{n - 1} - k (w_{n - 1} - Λ f)) - {Pr}_{Ω} (u - k (w - Λ f)) .$ Since $ρ^{α_{L}}$ is $ρ^{α_{L}}$ -Lipschitz continuous, ||Λ||=1 and $ρ^{α_{L}}$ -strongly monotone, we have $\begin{matrix} ‖ u_{n} - u ‖^{2} & = & ‖ {Pr}_{Ω} [u_{n - 1} - k (w_{n - 1} - Λ f)] . \\ . - {Pr}_{Ω} [u - k (w - Λ f)] ‖^{2} \\ \leq & ‖ (u_{n - 1} - {kw}_{n - 1}) - (u - kw) ‖^{2} \\ = & ‖ u_{n - 1} - u ‖ + k^{2} ‖ w_{n - 1} - w ‖^{2} \\ - 2 k 〈 w_{n - 1} - w, u_{n - 1} - u 〉 \\ \leq & (1 + k^{2} β^{2} - 2 k δ) ‖ u_{n - 1} - u ‖^{2} . \end{matrix}$ (2.1) Therefore, $‖ u_{n} - u ‖ \leq θ ‖ u_{n - 1} - u ‖,$ where $θ = \sqrt{1 + k^{2} β^{2} - 2 k δ} < 1,$ as 0 < k < 2δ/β² . Note that E is a contraction so it has a fixed point, say u = E (u) . It also implies from (2.1) that w_n → w in $H$ . Since $ρ^{α_{L}}$ is closed valued, u ∈ Ω is such that $ρ (u, w) \geq α_{L}$ and $〈 w, v - u 〉 \geq 〈 Λ f, v - u 〉 = 〈 f, v - u 〉, for all v \in Ω .$ Hence the proof of our main theorem is completed.□

Theorem 2 allows us to write the following iterative algorithm to find the solution of the lattice-valued fuzzy variational inequality problem.

bfAlgorithm A. For any $(u_{0}, w_{0}) \in Ω \times H$ such that $ρ (u_{0}, w_{0}) \geq α_{L},$ the iterative scheme $u_{n + 1} = {Pr}_{Ω} [u_{n} - k (w_{n} - Λ f)], n = 0, 1, 2, 3 . . .$ converges to the unique solution of Problem 1.5, where 0 < k < 2δ/β², and $ρ (u_{n}, w_{n}) \geq α_{L} .$

Now, in order to study the proposed lattice-valued variational inequality problem in more abstract way in the sense of topological fixed point problem. Let us recall some notions that will be useful in this context.

Definition 2.5. [21] Let $F : Ω \to 2^{H}$ be a set-valued mapping. A mapping $f : Ω \to H$ is called a selection of F, if f (x) ∈ F (x) for all x ∈ Ω .

Lemma 2.6. [21] Let X and Y be two Banach spaces and F : X → 2^Y be a lower semi-continuous mapping with nonempty closed and convex values. Then T admits a continuous selection.

Theorem 2.7. Let $ρ : Ω \times H \to L$ be an ℒ-fuzzy lower semi-continuous relation such that, for each x ∈ Ω and $α_{L} \in L ∖ {0},$ the range set $ρ [x]_{α_{L}}$ is a closed and convex subset of $H$ . Then the solution of Problem 1.5, exists.

Proof. To prove this theorem, it is enough to show that (from Lemma 2), the mapping defined by $E (u) = {Pr}_{Ω} [u - k (w - Λ f)],$ for $w \in ρ [u]_{α_{L}},$ has a fixed point. As Pr _Ω is continuous by Lemma 2, and the set-valued mapping $F : Ω \to 2^{H},$ defined by $F (x) = ρ [x]_{α_{L}}$ is lower semi-continuous with closed and convex values. Therefore continuity of Pr _Ω, with this fact guarantees the fixed point of the mapping,

Pr _Ω ∘ (I - k (F (·) - Λf)) : = E, using Lemma 2, and hence the proof is completed.□

Lemma 2.8. Every ℒ-fuzzy mapping $T : X \to F_{L} (Y)$ is an ℒ-fuzzy relation.

Proof. Given an ℒ-fuzzy mapping $T : X \to F_{L} (Y) .$ For any $α_{L} \in L ∖ {0},$ define an ℒ-fuzzy relation $ρ : X \times Y \to L$ , by $ρ^{α_{L}} = {(x, y) \in X \times Y : T_{(x)} (y) ≽ α_{L}},$ then clearly for any x ∈ X, the range set of x under $ρ^{α_{L}}$ , $ρ [x]_{α_{L}}$ is considered as the $α_{L}$ -level set $[Tx]_{α_{L}},$ this is what we required.□

From above Lemma 2.8, we state our next theorem that describes the existence of the variational inequality problem for ℒ-fuzzy mappings.

Theorem 2.9. Let $(u, w) \in Ω \times H$ be a solution of the variational inequality problem for ℒ-fuzzy mappings, and $(u_{n}, w_{n}) \in Ω \times H$ satisfies u_n+1 = Pr _Ω [u_n - k (w_n - Λf)] , for $f \in H^{*}$ and $w_{n} \in [{Tu}_{n}]_{α_{L}} .$ If the mapping $T : Ω \to F_{L} (H)$ is closed valued $F_{α_{L}}$ -Lipschitz continuous and $F_{α_{L}}$ -strongly monotone with constants β and δ respectively, then u_n → u in Ω and w_n → w in $H$ , for 0 < k < 2δ/β² .

Clearly every ℒ-fuzzy mapping is a fuzzy mapping for $L = [0, 1],$ in this regard, the following corollary is a main theorem of [23] with the case when f ≡ 0. For the related theory about this corollary, we refer the readers to [23].

Corollary 2.10. [23] Let $(u, w) \in Ω \times H$ be a solution of the variational inequality problem for fuzzy mappings (Problem 1), and $(u_{n}, w_{n}) \in Ω \times H$ satisfies u_n+1 = Pr _Ω [u_n - kw_n] and w_n ∈ [Tu_n] _α . If the mapping $T : Ω \to F (H)$ is F_α-Lipschitz continuous and F_α-strongly monotone with constants β and δ respectively, then u_n → u in Ω and w_n → w in $H$ for 0 < k < 2δ/β² .

By Remark 1-(i), let us deduce the following corollary for existence of solution to the Problem 1 (Theorem 5.1 of [11]). Here γ denotes the range of the relation Γ, defined in Problem 1.

Corollary 2.11. [11] A pair (x, y) is a solution of generalized variational inequality problem (Problem 1.2) if and only if $x = {Pr}_{V} (x - y) and y \in γ (x) .$ Moreover, if γ is Lipschitz continuous and strongly monotone with constants β > 0 and δ > 0, respectively, then there exists a unique solution of the generalized variational inequality problem.

Finally, by Remark 1.4-(iii) with f = 0, we deduce the following corollary as a consequence result of Stampacchia given in [14].

Corollary 2.12. Let $H$ be a real Hilbert space and Ω be a closed convex subset of $H$ . Given k > 0, and let $T : Ω \to H$ be Lipschitz continuous and strongly monotone with constants β > 0 and δ > 0, respectively, then there exists a unique solution of the variational inequality Problem 1, for 0 < k < 2δ/β² .

3 Applications

In this section, first we present applications of $ρ^{α_{L}}$ -Lipschitz mappings and projection mappings for the solution of Caputo-Fabrizio fractional inclusion initial value problems and then obstacle type variational inequalities problem.

3.1 Application in Caputo-Fabrizio fractional inclusion involving projection operator

We recall some of basic notations will be useful in the sequel.

Definition 3.1. An ℒ-fuzzy subset u of $ℝ^{n}$ is called

normal, if there exists an $x \in ℝ^{n}$ such that $u (x) = 1_{L},$

ℒ-fuzzy convex, if for $x, y \in ℝ^{n}$ and $λ \in L,$ $u (λ x + (1 - λ) x) \geq min {u (x), u (y)},$

upper semi-continuous, if any $γ \in L,$ [u] _γ is closed set.

For an ℒ-fuzzy subset u of $ℝ^{n},$ the support of u is denoted and defined as; $[u]_{0} = \bar{({x \in ℝ^{n} : u (x) \geq 0})} .$ Let $E_{L}^{n}$ denotes the set of all those ℒ-fuzzy sets of $ℝ^{n},$ that are normal, fuzzy convex, upper semi-continuous and have compact support. It is well known that for each $u \in ℝ^{n},$ the closed level set $[u]_{γ_{L}}$ is compact and convex in $ℝ^{n}$ for each $γ_{L} \in L .$

Definition 3.2. [36] Let (X, ρ₁) and (Y, ρ₂) be two metric spaces and F : X → 2^Y ∖ {φ} be a set-valued mapping. Then F is said to be upper semicontinuous (u . s . c) on X, if for each x ∈ X and any open set V ⊂ Y with T (x) ⊂ V, there exists an open neighborhood U (x) of x such that T (U (x)) ⊂ V . F is called lower semi-continuous (l . s . c) on X, if for any point x ∈ X and any open set V ⊂ Y with T (x) ∩ V ¬ = ∅ , there exists an open neighborhood U (x) of x such that T (y)∩ V ¬ = ∅ for all y ∈ U (x) .

Definition 3.3. [20] An ℒ-fuzzy mapping $F : Ω \to E_{L}^{n},$ where $Ω \subset ℝ^{n},$ is called lower open, if F_(x) (w) is lower semicontinuous at each x ∈ Ω .

Let us start with a very useful proposition.

Proposition 3.4. [35] Let D be a paracompact Hausdorff topological space, Y be a topological vector space and N : D → 2^Y be a multivalued nonempty convex valued function. If N has open lower sections, that is, for any y ∈ Y, N^-1 (y) is open in D, then there exists a continuous selection f : D → Y such that f (x) ∈ N (x) for any x ∈ D .

Now we give a new definition of fractional derivative with exponential Kernel defined in [6].

Definition 3.5. [6] Let f ∈ H¹ (a, b) and α ∈ [0, 1] . Then the new definition of the left fractional derivative in the sense of Caputo and Fabrizio is $(_{a}^{CF} D^{α} f) (t) = \frac{B (α)}{1 - α} \overset{t}{\int_{a}} (\overset{´}{f} (x) exp [- α \frac{(t - x)}{1 - α}]) dx$ and the right fractional derivative is $(^{CF} D_{b}^{α} f) (t) = \frac{- B (α)}{1 - α} \overset{b}{\int_{t}} (\overset{´}{f} (x) exp [- α \frac{(x - t)}{1 - α}]) dx .$ The corresponding integrals are $(_{a}^{CF} I^{α} f) (t) = \frac{1 - α}{B (α)} f (t) + \frac{α}{B (α)} \overset{t}{\int_{a}} f (s) ds$ and $(^{CF} I_{b}^{α} f) (t) = \frac{1 - α}{B (α)} f (t) + \frac{α}{B (α)} \overset{b}{\int_{t}} f (s) ds$ respectively, where B (α) is a normalization function satisfying B (0) = B (1) =1 .

Lemma 3.6. [1] For 0 < α < 1, we have $(_{a}^{CF} I_{a}^{α CF} D^{α} f) (t) = f (t) - f (a)$ and $(^{CF} I_{b}^{α CF} D_{b}^{α} f) (t) = f (b) - f (t) .$

Problem 3.7. First we consider an ℒ-fuzzy fractional inclusion projection problem given as follows: $^{CF} D^{α} x (t) + g (x (t))$ $- {Pr}_{Ω} [g (x (t)) - λ M (x (t))] \in (F_{(t, x (t))})_{γ (x (t))},$ (3.1) for a.e t ∈ [0, T] with x (0) = x₀, where ^CFD^α is the Caputo-Fabrizio [6] fractional derivative of order α ∈ (0, 1) , $F : [0, T] \times ℝ^{n} \to E_{L}^{n}$ is an ℒ-fuzzy mapping, $g, M : ℝ^{n} \to ℝ^{n}$ are two mappings, $γ_{L} : ℝ^{n} \to L ∖ {0_{L}}$ is lower semi-continuous, Ω is closed convex subset of $ℝ^{n},$ λ > 0 is a constant and $P_{K} : ℝ^{n} \to Ω$ is a projection operator defined as; ${Pr}_{Ω} (x) = {w \in Ω : ‖ x - w ‖ = d (x, K)} .$

The similar problems are discussed in a different way in [33]. We prove our main theorem that describes the conditions for existence of the solutions of fractional inclusion projection problem (3.1) , which is stated as follows:.

Theorem 3.8. Let $F : [0, T] \to E_{L}^{n}$ be convex, lower open and $F^{γ_{L}}$ -Lipschitzean with constant β₁ and $γ : ℝ^{n} \to L$ be an upper semi-continuous mapping. Suppose that $g, M : ℝ^{n} \to ℝ^{n}$ are Lipschitz continuous mappings with positive constants β₂ and β₃ respectively. If $\frac{1 - α + α T}{B (α)} < \frac{1}{(2 β_{2} + λ β_{3} + β_{1})},$ then there exists a solution $x : [0, T] \to ℝ^{n}$ of the problem (3.1) that is a continuous function.

Proof. For each $(t, x) \in G = [0, T] \times ℝ^{n},$ define a multivalued mapping $\tilde{F} : G \to 2^{ℝ^{n}}$ as $\begin{matrix} \tilde{F} (t, x) & = & (F_{(t, x)})_{γ (x)} \\ = & {y \in ℝ^{n} : F_{(t, x)} (y) > γ (x)} . \end{matrix}$ Clearly for each (t, x) ∈ G, the set $\tilde{F} (t, x)$ is nonempty. For $y_{1}, y_{2} \in \tilde{F} (t, x)$ and η ∈ (0, 1) , $\begin{matrix} F_{(t, x)} (η x_{1} + (1 - η) x_{2}) \geq min {F_{(t, x)} (x_{1}), . \\ . F_{(t, x)} (x_{2})} > γ (x) \end{matrix}$ holds by convexity of F . Hence $η x_{1} + (1 - η) x_{2} \in \tilde{F} (t, x)$ , so $\tilde{F} (t, x)$ is convex on G . To prove $\tilde{F}$ has open lower sections, it is enough to prove that the set $({\tilde{F}}^{- 1} (y))^{c} = {(t, x) \in G : F_{(t, x)} (y) \leq γ (x)}$ is closed for each $y \in ℝ^{n} .$ Let (t_k, x_k) be a sequence in $({\tilde{F}}^{- 1} (y))^{c}$ such that (t_k, x_k) → (t, x) . As F is lower open and γ is upper semi-continuous, so we have $F_{(t, x)} (z) \leq lim inf F_{(t_{k}, x_{k})} (z) \leq lim sup F_{(t_{k}, x_{k})} (z) \leq γ (x),$ therefore $(t, x) \in ({\tilde{F}}^{- 1} (y))^{c} .$ Hence $\tilde{F}$ is convex valued mapping with lower sections. By using Proposition 3.4, there exists a continuous selection $f : G \to ℝ^{n}$ of $\tilde{F}$ such that for each (t, x) ∈ G, $f (t, x) \in \tilde{F} (t, x) .$ We get the following fractional initial value problem(IVP): $^{CF} D^{α} x (t) = {Pr}_{Ω} [g (x (t)) - λ M (x (t))] - g (x (t)) + f (t, x),$ (3.2) with x (0) = x₀ . For existence of this we define $\begin{matrix} T (x (t)) = x_{0} + \frac{1 - α}{B (α)} ({Pr}_{Ω} [g (x (t)) - λ M (x (t))] . \\ . - g (x (t)) + f (t, x)) \\ + \frac{α}{B (α)} \overset{t}{\int_{0}} ({Pr}_{Ω} [g (x (s)) - λ M (x (s))] . \\ . - g (x (s)) + f (s, x)) ds \end{matrix}$ with condition that Pr _Ω [g (x (t)) - λM (x (t))] - g (x (t)) + f (t, x) =0, at t = 0 .

To prove the existence of (3.2) , it is enough to prove that T has a unique fixed point. Consider for (t, x₁) , (t, x₂) ∈ G, $\begin{matrix} ‖ {Tx}_{1} - {Tx}_{2} ‖ \leq (\frac{1 - α}{B (α)} + \frac{α T}{B (α)}) \\ \times (‖ \begin{matrix} {Pr}_{Ω} [g (x_{1}) - λ M (x_{1})] - g (x_{1}) + f (t, x_{1}) \\ - ({Pr}_{Ω} [g (x_{2}) - λ M (x_{2})] - g (x_{2}) + f (t, x_{2})) \end{matrix} ‖) \\ \leq (\frac{1 - α}{B (α)} + \frac{α T}{B (α)}) \\ \times (\begin{matrix} ‖ {Pr}_{Ω} [g (x_{1}) - λ M (x_{1})] - {Pr}_{Ω} [g (x_{2}) - λ M (x_{2})] ‖ \\ + ‖ g (x_{1}) - g (x_{2}) ‖ + ‖ f (t, x_{1}) - f (t, x_{2}) ‖ \end{matrix}) . \end{matrix}$ Since Pr _Ω is nonexpansive, so we have $\begin{matrix} ‖ {Tx}_{1} - {Tx}_{2} ‖ \leq (\frac{1 - α}{B (α)} + \frac{α T}{B (α)}) \\ \times (\begin{matrix} 2 ‖ g (x_{1}) - g (x_{2}) ‖ + λ ‖ M (x_{1}) - M (x_{2}) ‖ \\ + ‖ f (t, x_{1}) - f (t, x_{2}) ‖ \end{matrix}) \end{matrix}$ Since F is Lipschitz, f is also with same constant β₁ . Hence, we have $\begin{matrix} ‖ {Tx}_{1} - {Tx}_{2} ‖ \leq (\frac{1 - α}{B (α)} + \frac{α T}{B (α)}) \\ (2 β_{2} + λ β_{3} + β_{1}) ‖ x_{1} - x_{2} ‖ . \end{matrix}$ Since $\frac{1 - α + α T}{B (α)} < \frac{1}{(2 β_{2} + λ β_{3} + β_{1})},$ T is a contraction and by Banach contraction principle, T has a unique fixed point. Consequently, the fractional IVP (3.2) has a unique solution. This completes the proof.□

In the following example we will apply above Theorem 3.8, to insure the existence of solution of given IVP.

Example 3.9. Let $L = {ϰ_{1}, ϰ_{2}, ϰ_{3}, ϰ_{4}}$ with ϰ₁ ⪯ ϰ₂ ⪯ ϰ₃ and ϰ₁ ⪯ ϰ₄ ⪯ ϰ₃, ϰ₂ and ϰ₃ are not comparable. Then $(L, ⪯)$ is a complete distributive lattice. Define $F : [0, 1] \times ℝ^{2} \to F_{L} (ℝ^{2})$ by $\begin{matrix} F_{(t, x)} (y) = \\ {\begin{matrix} ϰ_{1} if (t, x) \in [0, 1] \times ℝ^{2} and y = 0.1 x \\ ϰ_{2} if (t, x) \in [0, 1] \times ℝ^{2} and y = 0.2 x \\ ϰ_{3} if (t, x) \in [0, 1] \times ℝ^{2} and y = 0.3 x \\ ϰ_{4} if (t, x) \in [0, 1] \times ℝ^{2} and y \notin {0.1 x, 0.2 x, 0.3 x} \end{matrix} . \end{matrix}$ and consider the following Caputo-Fabrizio ℒ-fuzzy fractional differential inclusion initial value problem involving projection operator $\begin{matrix} ^{CF} D^{0.5} x (t) + g (x (t)) - {Pr}_{Ω} [g (x (t)) - λ M (x (t))] \\ \in (F_{(t, x (t))})_{γ (x (t))} \end{matrix}$ where $Ω = {(\begin{matrix} x_{1} \\ x_{2} \end{matrix}) : 0 \leq x_{1}, x_{2} \leq 1}$ $\begin{matrix} B (0.5) = 1, g (x (t)) = (\begin{matrix} 0.1 x_{1} \\ 0.01 x_{2} \end{matrix}), \\ M (x (t)) = (\begin{matrix} - 0.01 x_{1} \\ 0.02 x_{2} \end{matrix}), \end{matrix}$ and λ = 2 . To insure the existence we need to verify that $\frac{(2 β_{2} + λ β_{3} + β_{1}) (1 - α + α T)}{B (α)} < 1,$ where β₁, β₂, β₃ are the Lipschitzean constants of F, g and M respectively. Now for $y_{1}, y_{2} \in ℝ^{2}$ we have $‖ g (y_{1}) - g (y_{2}) ‖_{2} \leq 0.1 ‖ y_{1} - y_{2} ‖_{2},$ $‖ M (y_{1}) - M (y_{2}) ‖_{2} \leq 0.02 ‖ y_{1} - y_{2} ‖_{2},$ for $(t, x^{(1)}), (t, x^{(2)}) \in [0, 1] \times ℝ^{2}$ and $y^{(1)}, y^{(2)} \in ℝ^{2}$ with F_{(t,x⁽¹⁾)} (y⁽¹⁾) > ϰ₁, F_{(t,x⁽²⁾)} (y⁽²⁾) > ϰ₁ (for support of F, since ϰ₁ is zero of lattice ℒ) we have $‖ y^{(1)} - y^{(2)} ‖_{2} \leq 0.3 ‖ x^{(1)} - x^{(2)} ‖_{2} .$ From these inequalities we have β₁ = 0.3, β₂ = 0.1 and β₃ = 0.02 . It is important to note that for verification of initial condition, the condition on kernel for CF fractional derivative to be zero is satisfied since g (0) = M (0) =0, also we have $\frac{(2 β_{2} + λ β_{3} + β_{1}) (1 - α + α T)}{B (α)} = 0.54 < 1 .$ Therefore using Theorem 3.8, there exists a continuous solution of the given problem.

3.2 Obstacle type variational inequality problem

Now, we present an application of our main result, to obtain the solution of an obstacle type variational inequality problem:

For some closed bounded $Ω \subset ℝ^{n},$ let $H = H^{2} (Ω),$ f ∈ L² (Ω) and $K = {v \in H_{0}^{1} (Ω) : v (x) \geq 0, on Ω} .$ Then we know that K is a closed and convex subset in $H .$

Define an ℒ-fuzzy relation $ρ : K \times H \to L$ by $ρ (w, u) = χ_{K \times H} (w, u),$ where $χ_{K \times H}$ is the characteristic function on $K \times H$ defined in [24]. Then for $λ \in L ∖ {0_{L}},$ we want to find all $(w, u) \in K \times H$ with ρ (w, u) >0, satisfying $- k \nabla w (v - u) \geq f (v - u),$ (3.3) for all v ∈ K . u = 0 on ∂Ω, k > 1 .

For $u, v \in H$ , $〈 u, v 〉 = k \sum_{i = 1}^{n} \int_{Ω} \frac{\partial u}{\partial x_{i}} \frac{\partial v}{\partial x_{i}} dx .$ Now we consider

$- \int_{Ω} k \nabla wvdx = - k \sum_{i = 1}^{n} \int_{Ω} \frac{\partial^{2} w}{\partial x_{i}^{2}} vdx .$ Using Green’s Theorem, we have $- \int_{Ω} k \nabla wvdx = k \sum_{i = 1}^{n} \int_{Ω} \frac{\partial w}{\partial x_{i}} \frac{\partial v}{\partial x_{i}} dx - k \sum_{i = 1}^{n} \int_{\partial Ω} v \frac{\partial w}{\partial n} ds .$ Since $v \in H_{0}^{1} (Ω),$ $\sum_{i = 1}^{n} \int_{\partial Ω} v \frac{\partial w}{\partial n} ds = 0 .$

In a similar way, integrating (3.3), the above problem (3.3) reduced to $〈 kw, v - u 〉 \geq 〈 f, v - u 〉 for all v \in K .$ Clearly, the mapping Tw : = {kw} is Lipschitz continuous for k > 0 . Now for monotonicity, consider, $〈 k (w - u), w - u 〉 \geq k_{1} ‖ w - u ‖_{H_{0}^{1} (Ω)}^{2},$ for some k₁ > 0 (see Theorem 5.4.8 in [22]). Thanks to Theorem 2.4, in order to obtain the solution of problem (3.3).

Footnotes

Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea (2018R1D1A1B07045427). The paper has been prepared during the second author visit as a post doc researcher to the Department of Mathematical Science, UAEU, and would like to thanks the Department of Mathematical Science, UAEU for their support. The third author gratefully acknowledges with thanks the Department of Research Affairs at UAEU. This article is supported by the grant: UPAR(11) 2016, Fund No. 31S249(COS). And the third author gratefully acknowledges with thanks the Department of Research Affairs at UAEU. This article is supported by the grant: UPAR(11) 2016, Fund No. 31S249(COS).

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