Fuzzy measure of non-compactness with applications in fractional anti-periodic boundary value problems involving nonsingular kernel

Abstract

In this article we introduce the notion of fuzzy measure of noncompactness. We define the fuzzy condensing and fuzzy k-set contractions using fuzzy measure of noncompactness. An extension of Stanislaw Szulfa’s fixed point theorem for a self-operator on a closed bounded and convex subset of a Banach space has been proved. Using the defined multivalued k-set contractions, generalizations of Kakutani-Fan and Krasnoselskii type theorems have been proved. For applications the existence result for solutions of a fractional Caputo-Fabrizio anti-periodic boundary value problem has been proved. We give some examples to validate our results.

Keywords

fuzzy measure of noncompactness Szulfa’s theorem K-Fan theorem multivalued fuzzy condensing operators Krasnoselskii’s theorem fixed points

1 Introduction

In this article, unless otherwise stated, let E be a Banach space over a field F and Γ_n (E) , Γ_cl (E) , Γ_cv (E), Γ_prc (E) , Γ_cp (E) , Γ_b (E) be the set of all nonempty, nonempty closed, nonempty convex, nonempty precompact, nonempty compact and nonempty bounded subsets of E respectively. However one can combine any of these sub-notations, for example Γ_cl,b (E) the set of all nonempty closed and bounded subsets of E. A function T : E → Γ_n (E) is called a multivalued operator and a point u ∈ E is called a fixed point of T, if u ∈ Tu, let us denote by Fix_T, the set of all fixed points of the operator T . For any A ⊂ E, we denote $T (A) = \underset{u \in A}{\cup} Tu$ .

Now we recall some definitions and results which will be useful to understand and prove our main results. Our first lemma is a Schauder’s fixed point theorem, which is one of most celebrated results in the theory of operator analysis.

1.1 Lemma [1]

Let Ω be a closed and convex subset of a normed space E. Then every compact and continuous mapping T : Ω → Ω has at least one fixed point.

We recall some generalizations of above result in the settings of multivalued operators.

1.2 Definition [1]

Let E and $\overset{´}{E}$ be two Banach spaces, a multivalued operator $T : E \to Γ_{n} (\overset{´}{E})$ is said to be an upper semi-continuous (resp, lower semi-continuous/continuous) if the set {u∈ E : Tu ⊂ A } (resp, {u∈ E : Tu ∩ A ¬ = ϕ } / {u∈ E : ∪ Tu = A }) is an open set in E for each open subset A of $\overset{´}{E} .$

The lower and upper semi-continuity of multivalued operators is very important, and is used in many important results to ensure the existence of fixed points. The following is the famous result of Kakutani-Fan [2], to find the fixed points of a upper semi-continuous multivalued operator.

1.3 Theorem

Let Ω be a compact subset of a Banach space E and let T : Ω → Γ_cv,cp (E) be an upper semi-continuous multivalued operator. Then T has a fixed point.

1.4 Definition [1]

A multivalued operator $T : E \to Γ_{cp} (\overset{´}{E})$ is called compact if $\bar{T (E)}$ is compact subset of $\overset{´}{E} .$ T is called totally bounded if for any bounded subset B of E, the set T (B) is totally bounded subset of $\overset{´}{E} .$

1.5 Remark [1]

It is well known that every compact multivalued operator is totally bounded and converse may not true.

1.6 Definition [1]

The multivalued operator $T : E \to Γ_{cp} (\overset{´}{E})$ is completely continuous if it is upper semi-continuous and totally bounded.

1.7 Definition [1]

The graph of a multivalued operator $T : E \to Γ_{n} (\overset{´}{E})$ is defined as; $G r (T) = {(u, v) \in E \times \overset{´}{E} : v \in Tu},$ and the graph is closed if every sequence (u_n, v_n) in $G r (T),$ such that $(u_{n}, v_{n}) \to (u, v) \in G r (T) .$ T is closed if its graph is closed.

1.8 Lemma [1]

A multivalued operator $T : E \to Γ_{cl} (\overset{´}{E})$ is upper semi-continuous if and only if it is closed and compact.

The following is the well known generalization of above Theorem 1.3 given by Bohnentblust-Karlin in which the domain of the operator is relaxed and is given as follows:

1.9 Theorem [3]

Let Ω be a closed and convex subset of a Banach space E and let T : Ω → Γ_cp,cv (E) be an upper semicontinuous compact operator. Then T has a fixed point.

The following theorem is more general case of the above theorem and is stated as follows.

1.10 Theorem [4]

Let Ω be a closed convex and bounded subset of a Banach space E and let T : Ω → Γ_cp (E) be compact and closed multivalued operator. Then T has a fixed point.

One can further weaken the compactness of the operator by using the notion of condensing operators. Condensing operators play an important role in the existence theory of linear and nonlinear operators. Kuratowski [5] defines a ball measure of noncompactness α, and the Hausdorff measure of noncompactness β was defined by Sadovskii [6]. For a bounded subset B of a given Banach space E, $α (B) = inf {λ > 0 : B \subset \underset{i = 0}{\overset{n}{\cup}} S_{i}, δ (S_{i}) \leq λ},$ where δ denotes the diameter of a set, and $β (B) = {λ > 0 : B \subset \underset{i = 0}{\overset{n}{\cup}} B (u_{i}, λ) for some u_{i} \in E},$ where B (u_i, λ) are the open balls with centered at u_i of radius λ .

Recently in the followings [7, 8], a generalized measure of noncompactness is defined in this way.

1.11 Definition

A function μ : Γ_b (E) → [0, ∞) is called a measure of noncompactness if for A, B ∈ Γ_b (E) , the following conditions hold;

(MNC1) μ (A) = 0, implies A is precompact,

(MNC2) $μ (\bar{A}) = μ (A),$ where $\bar{A}$ denotes the closure of A,

(MNC3) μ (Conv (A)) = μ (A) , where Conv (A) denotes the convex hull of A,

(MNC4) μ is nondecreasing, i.e., if A ⊂ B then μ (A) ≤ μ (B) ,

(MNC5) if {A_n} is a sequence in Γ_b (E) such that $lim_{n \to \infty} μ (A_{n}) = 0,$ implies $A_{\infty} = \underset{n = 1}{\overset{\infty}{\cap}} \bar{A_{n}}$ is nonempty.

Moreover μ is called sublinear if

(MNC6) for λ ∈ F, and

(MNC7) μ (A + B) ≤ μ (A) + μ (B) for A, B ∈ Γ_cl,b (E) .

The ball measure of noncompactness α, and the Hausdorff measure of noncompactness β are special case of μ .

1.12 Definition [5]

A multivalued operator T : E → Γ_cl,b (E) is called a μ-condensing, if for any B ∈ Γ_b (E) , the set T (B) ∈ Γ_b (E) and $μ (T (B)) < μ (B)$ for μ (B) > 0.

1.13 Definition [5]

A multivalued operator T : E → Γ_cl,b (E) is called a μ-k-set contraction, if for any B ∈ Γ_b (E) , the set T (B) ∈ Γ_b (E) and $μ (T (B)) \leq k \cdot μ (B),$ for some k ∈ (0, 1).

Clearly above two definitions will remained valid if we replace μ by α or β defined above.

The following is the generalized version of Bohnentblust-Karlin theorem which weakens the condition of compactness on operator.

1.14 Theorem [9]

Let Ω be a closed convex and bounded subset of a Banach space E. Let T : Ω → Γ_cl,cv (E) be β-condensing and upper semi-continuous operator. Then T has a fixed point.

In the next section we will generalize the above theorem using generalized type of condensing operators.

2 Fixed Point results via fuzzy measure of noncompactness

In this section we introduce the concept of fuzzy-measure of noncompactness, fuzzy condensing and fuzzy k-set contractions. We generalize M $\ddot{o}$ nch [10], Darbo [11], Kakutani [12] and Bohnentblust-Karlin [13] fixed point theorems, indeed many results in the literature.

The following definition will play a crucial role in all coming results.

2.1 Definition

A fuzzy set μ_F : Γ_b (E) × [0, ∞) → [0, 1] is called fuzzy measure of noncompactness (FMNC) if for A, B ∈ Γ_b (E) and t > 0, the following conditions are satisfied;

(1) μ_F (A, 0) = 0,

(2) The family {A∈ Γ_b (E) : μ_F (A, t) = 1 } is nonempty and contained in Γ_prc (E) ,

(3) μ_F (A, t) ≥ μ_F (B, t) for A ⊆ B, and t > 0,

(4) $μ_{F} (A, t) = μ_{F} (\bar{A}, t) = μ_{F} (conv (A), t)$ for any A ∈ Γ_b (E) ,

(5) μ_F (A, ·) : [0, ∞) → [0, 1] is left continuous and $lim_{t \to \infty} μ_{F} (A, t) = 1 .$

(6) for any decreasing sequence {A_n} in Γ_b (E) such that $lim_{n \to \infty} μ_{F} (A_{n}, t) = 1,$ implies $A_{\infty} = \underset{n = 1}{\overset{\infty}{\cap}} \bar{A_{n}}$ is nonempty.

μ_F is called fuzzy sublinear if

(7) for k ∈ F,

(8) μ_F (A + B, t + s) ≥ μ_F (A, t) ∗ μ_F (B, s) , where ∗ is continuous t-norm.

2.2 Remark

For a given measure of noncompactness α,(resp, β or μ), and for t > 0, we define $μ_{F} (Ω, t) = \frac{t}{t + α (Ω)},$ then μ_F is fuzzy measure of noncompactness, called standard fuzzy measure of noncompactness induced by α, (resp, β or μ).

2.3 Examples

Let $g : ℝ^{+} \to ℝ^{+}$ be an increasing and continuous function, for δ > 0, define a function $μ_{F} (Ω, t) = \frac{g (t)}{g (t) + δ \cdot α (Ω)},$ then μ_F is a FMNC with ∗ as a usual multiplication · . The standard fuzzy measure of noncompactness is a special case.

Define with the same function g $μ_{F} (Ω, t) = e^{- \frac{α (Ω)}{g (t)}},$ then μ_F is a FMNC with ∗ as a usual multiplication · .

Define with the same function g $μ_{F} (Ω, t) = 1 - \frac{α (Ω)}{g (t)},$ then μ_F is a FMNC with ∗ as Lukasieviczt-norm $L$ .

2.4 Remark

If (X, N, ∗) is a fuzzy Banach space [14], then μ_F : Γ_b,cv (X) × [0, ∞) → [0, 1] , (defined by any (1) - (3) in above examples) is a fuzzy measure of noncompactness and in this whole fuzzy structure we do not prefer to use crisp measure of noncompactness. In this environment the existence theory of fuzzy differential and integral equations is still not explored.

The concept of fuzzy measure of noncompactness is different from the concept of fuzzy measures, we define fuzzy measure and fuzzy measure space then explain how both concept are different.

2.5 Definition ([15] Chapter 6)

Let X be an arbitrary set and $B$ be Borel field of X . A set function $σ : B \to [0, 1]$ is called a fuzzy measure (FM) if the following properties are satisfied.

(FM1) σ (ϕ) = 0 and σ (X) = 1,

(FM2) for $A, B \in B$ and A ⊂ B, then σ (A) ≤ σ (B) ,

(FM3) if {F_n } _n≥1 is a monotone sequence (in the sense of inclusion) in $B,$ then $lim_{n \to \infty} σ (F_{n}) = σ (lim_{n \to \infty} F_{n}) .$

2.6 Remark

Note that if we assume μ_F (Ω, ·) is independent of t (just to behave like fuzzy measure). Since ϕ being a finite set is compact set then μ_F (ϕ, ·) = 1 but clearly from (FM1) σ (ϕ) = 0, so we cannot develop any useful relationship between FMNC and FM.

Now we present our first results regarding the fixed point of a continuous self operator on a closed bounded and convex subset of a Banach space. This theorem generalize the results of [16] and [17]. This result has a unique importance because it will be very useful in the existence theory of linear and nonlinear operators.

2.7 Theorem

Let Ω be a bounded closed and convex subset of a Banach space such that 0 ∈ Ω and T be a continuous self mapping on Ω. If $\begin{matrix} V & = & Conv (T (V)) or V = T (V) \cup {0} \\ \Rightarrow & μ_{F} (V, t) = 1, for t > 0, \end{matrix}$ holds for every subset V of Ω, then T has a fixed point.

Proof. Let w₀ : =0 and w_n+1 : = T (w_n) for n = 0, 1, 2, 3, . . . . Let W = { w_n : n = 0, 1, 2, 3, . . . } , then clearly W = T (W)∪ { 0 } and using hypothesis of theorem we have μ_F (W, t) = 1, therefore W is relatively compact in Ω . Let K be the set of all limit points of W, then we have K = T (K) (see [18] page 34, 7(iii)). Let $Γ = {Y \subseteq Ω : K \subset Y and ConvT (Y) \subset Y} .$ Clearly Ω ∈ Γ so Γ is nonempty. Let $U = \underset{Y \in Γ}{\cap} Y$ , as K ⊂ U so U is nonempty and K = T (K) ⊂ ConvT (K) ⊂ ConvT (U) which implies $K \subset ConvT (U) .$ (*) Also since ConvT (U) ⊂ ConvT (Y) ⊂ Y for all Y ∈ Γ, therefore $ConvT (U) \subset \underset{Y \in Γ}{\cap} Y = U,$ (**) which implies U ∈ Γ. Now as Conv (Conv (T (U))) ⊂ ConvT (U) , and K ⊂ ConvT (U) by (∗) , therefore ConvT (U) ∈ Γ . Also as ConvT (U) ⊂ U by (∗∗) and obviously U ⊂ ConvT (U) ∈ Γ (U being intersection of all elements of Γ), this implies U = Conv (T (U)). Hence by assumption in statement of theorem μ_F (U, t) = 1, so U is relatively compact and hence $\bar{U}$ is compact subset of Ω . Applying Schauder’s fixed point theorem on mapping $T |_{\bar{U}}$ (i.e, $T : \bar{U} \to Ω)$ , we conclude that T has a fixed point.■

We define fuzzy μ_F-condensing and fuzzy μ_F-k-set contractions by means of fuzzy measure of noncompactness.

2.8 Definition

A mapping T : Ω → Γ_cl,cv,b (E) is called a fuzzy μ_F-condensing if there exists a fuzzy measure of noncompactness μ_F such that $μ_{F} (T (Ω), t) > μ_{F} (Ω, t)$ for each closed convex and bounded subset Ω of Banach space E .

2.9 Definition

A mapping T : Ω → Γ_cl,cv,b (E) is called a fuzzy μ_F-k-set contraction if there exists some k ∈ (0, 1) and a fuzzy measure of noncompactness μ_F such that $μ_{F} (T (Ω), kt) \geq μ_{F} (Ω, t)$ for each closed convex and bounded subset Ω of Banach space E .

The next result is regarding the fixed point of a given multivalued fuzzy μ_F-k-set contraction using fuzzy measure of noncompactness. This result generalize the well known results of Darbo [11], and [5].

2.10 Theorem

Every continuous fuzzy μ_F-k-set contraction on a closed, convex and bounded subset Ω of E has a fixed point and Fix_T ∈Γ_prc (E) .

Proof. For any Ω₀ ∈ Γ_cl,cv,b (E) define the sequence $Ω = Ω_{0}$ and $Ω_{n + 1} = \bar{Conv (T (Ω_{n}))}, n = 0, 1, 2, 3, . . . .$ As Ω₀ is closed and convex, then $Ω_{1} = \bar{Conv (T (Ω_{0}))} \subset Ω_{0}$ and by induction we have $. . . Ω_{n + 1} \subset Ω_{n} . . . Ω_{2} \subset Ω_{1} \subset Ω_{0},$ which implies that {Ω_n} is a nonincreasing sequence of closed, convex and bounded subsets of E . Now consider for t > 0, $\begin{matrix} μ_{F} (Ω_{n + 1}, t) & = & μ_{F} (\bar{Conv (T (Ω_{n}))}, t) \\ = & μ_{F} (T (Ω_{n}), t) \\ \geq & μ_{F} (Ω_{n}, \frac{t}{k}) \\ = & μ_{F} (\bar{Conv (T (Ω_{n - 1}))}, \frac{t}{k}) \\ \geq & μ_{F} (Ω_{n - 1}, \frac{t}{k^{2}}) \\ : \\ \geq & μ_{F} (Ω_{0}, \frac{t}{k^{n}}) \to 1 \end{matrix}$ as n → ∞ , using (5) , in Definition 2.1, we conclude that $lim_{n \to \infty} μ_{F} (Ω_{n + 1}, t) = 1,$ which implies $Ω_{\infty} = \underset{n = 1}{\overset{\infty}{\cap}} \bar{Ω_{n}}$ is nonempty. Since Ω_∞ ⊆ Ω_n for all $n \in ℕ,$ which implies (using (3) , in Definition 2.1), μ_F (Ω_∞, t)≥ $lim_{n \to \infty} μ_{F} (Ω_{n + 1}, t) = 1 .$ Which implies $\bar{Ω_{\infty}} = Ω_{\infty}$ (by construction Ω_∞ is closed set), is compact and convex. Therefore by Lemma 1.8, the operator T : Ω_∞ → Γ_cv,cp (Ω_∞) is upper semi-continuous, and application of Kakutani theorem implies T has a fixed point.

Finally to prove Fix_T ∈Γ_prc (E) . Consider for k ∈ (0, 1), t > 0, and using the fact that Fix_T is invariant under T $\begin{matrix} μ_{F} ({Fix}_{T}, t) & = & μ_{F} (T^{n} ({Fix}_{T}), t), \\ \geq & μ_{F} ({Fix}_{T}, \frac{t}{k^{n}}) \to 1, as n \to \infty . \end{matrix}$ Which implies $μ_{F} ({Fix}_{T}, t) = 1 .$ This implies Fix_T is precompact, this proves the theorem.■

2.11 Remark

From above Remark 2.2, we conclude that every α-condensing/k-set contraction (resp, β or μ-condensing/k-set contraction) operator is a fuzzy μ_F-condensing/μ_F-k-set contraction operator.

From above definitions and results we have the following corollaries.

2.12 Corollary

Every continuous μ-k-set (β-k-set, α-k-set), contraction on a closed, convex and bounded subset Ω of E has a fixed point.

3 Krasnoselskii-type results for sum of two multivalued operators

In this section we use Theorem 2.10, to prove a multivalued version of Krasnoselskii-type fixed point theorem, first we recall some more results from literature.

3.1 Definition

A multivalued mapping T : E → Γ_cl,cv (E) is said to be multivalued k-contraction if there exists k ∈ (0, 1) such that $H (Tu, Tv) \leq k \cdot d (u, v)$ for all u, v ∈ E, where H is the Hausdorff Pompeiu distance [19].

3.2 Remark

In [4], it has been proved that every multivalued k-contraction T is a α-k-set contraction and hence a μ_F-k-set contraction.

We prove our main result of this section stated as follows;

3.3 Theorem

Let Ω be a closed, convex and bounded subset of a Banach space E and μ_F be a fuzzy sublinear, fuzzy measure of noncompactness. Let S, T : Ω → Γ_cl,cv (Ω) be multivalued operators such that,

(a) S is closed and μ_F- $\frac{k}{2}$ -set contraction for some k ∈ (0, 1),

(b) T is compact and closed,

Then there exist a solution of operator inclusion u ∈ Su + Tu and the solution set Fix_S+T ∈ Γ_pre (E) .

Proof. Define ψ : Ω → Γ_cl,cv (Ω) by $ψ (u) = Su + Tu$ for each u ∈ Ω . As S is closed and T is compact, therefore their sum ψ is also closed. Also the sum is Minkowski therefore ψ is convex. Therefore ψ : Ω → Γ_cl,cv (Ω) is well defined. Also for any nonempty subset B of Ω, we have $ψ (B) \subseteq Ω$ which implies ψ (B) is bounded as Ω is bounded, and by definition $ψ (B) \subseteq S (B) + T (B) .$ Now using fuzzy sublinearity of μ_F, for some k ∈ (0, 1) , and t > 0, consider $\begin{matrix} μ_{F} (ψ (B), kt) \\ \geq & μ_{F} (S (B) + T (B), \frac{k}{2} t + \frac{k}{2} t) \\ \geq & μ_{F} (S (B), \frac{k}{2} t) * μ_{F} (T (B), \frac{k}{2} t), \end{matrix}$ since T is compact therefore $μ_{F} (T (B), \frac{k}{2} t) = 1,$ and using the fact that S is μ_F- $\frac{k}{2}$ -set contraction, we have $\begin{matrix} μ_{F} (ψ (B), kt) & \geq & μ_{F} (S (B), \frac{k}{2} t) * 1 \\ = & μ_{F} (S (B), \frac{k}{2} t) \\ \geq & μ_{F} (B, t) . \end{matrix}$ Hence ψ satisfies all the conditions of Theorem 2.10, therefore there exists a point z ∈ ψ (z) , the required solution of the operator inclusion z ∈ Sz + Tz . Next we claim $μ_{F} ({Fix}_{S + T}, t) = 1 .$ On contrary, suppose that for t > 0, $μ_{F} ({Fix}_{S + T}, t) < 1,$ then for some k ∈ (0, 1) , as Fix_S+T is invariant under ψ, so μ_F (Fix_S+T, t) = μ_F (ψⁿ (Fix_S+T) , t), consider $\begin{matrix} 1 & > & μ_{F} ({Fix}_{S + T}, t) = μ_{F} (ψ^{n} ({Fix}_{S + T}), t) \\ \geq & μ_{F} ({Fix}_{S + T}, \frac{t}{k^{n}}) \to 1, as n \to \infty, \end{matrix}$ gives 1 > 1, a contradiction, therefore $μ_{F} ({Fix}_{S + T}, t) = 1 .$ This proves the theorem.■

The following corollaries are due to Remark 2.11.

3.4 Corollary [9]

Let Ω be a closed, convex and bounded subset of a Banach space E and μ be a sublinear measure of noncompactness. Let S, T : Ω → Γ_cl,cv (Ω) be multivalued operators such that,

(a) S is closed and μ-k-set contraction (resp, α or β-k-set contraction),

(b) T is compact and closed,

Then there exist a solution of operator inclusion u ∈ Su + Tu and the solution set S_S+T ∈ Γ_cp (E) .

The next corollary is due to Remark 3.2.

3.5 Corollary [4]

Let Ω be a closed, convex and bounded subset of a Banach space E and S, T : Ω → Γ_cl,cv (Ω) be multivalued operators such that,

(a) S is closed and multivalued k-contraction, for some k ∈ (0, 1) ,

(b) T is compact and closed,

Then there exist a solution of operator inclusion u ∈ Su + Tu and the solution set S_S+T ∈ Γ_cp (E) .

4 Applications in fractional differential equations

In this section, we prove an existence result for Caputo Fabrizio fractional anti-periodic boundary value problem of order 1 < α ≤ 2 . We use our Theorem 2.7, to find the sufficient conditions for existence of solution of said Caputo Fabrizio fractional BVP. First we recall some definitions and results which will be useful for main results of this section. For more applications of fixed point theory in the existence theory of integral and differential equations, we refer the readers to the following articles [20 –23].

4.1 Definition [25]

Let f ∈ H¹ (a, b) , and α ∈ [0, 1] , then the definition of the new 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 $(^{CF} D_{b}^{α} f) (t) = \frac{- B (α)}{1 - α} \overset{b}{\int_{t}} (\overset{´}{f} (x) exp [- α \frac{(x - t)}{1 - α}]) dx .$ Where $\overset{´}{f}$ is an ordinary derivative of f . 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 .

4.2 Lemma [25]

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

The higher order fractional derivative and integrals are given in the following definition.

4.3 Definition [25]

Let n < α ≤ n + 1 and f be such that fⁿ ∈ H¹ (a, b) . Set β = α - n . Then β ∈ (0, 1] and we define $(_{a}^{CF} D^{α} f) (t) = (_{a}^{CF} D^{β} f^{n}) (t),$ the associated fractional integral is $(_{a}^{CF} I^{α} f) (t) = (_{a} I^{n}_{a}^{CF} I^{β} f) (t) .$

4.4 Definition [25]

Let n < α ≤ n + 1 and f be such that fⁿ ∈ H¹ (a, b) . Set β = α - n . Then β ∈ (0, 1] and we define $(^{CF} D_{b}^{α} f) (t) = (^{CF} D_{b}^{β} {(- 1)}^{n} f^{n}) (t),$ the associated fractional integral is $(^{CF} I_{b}^{α} f) (t) = (I_{b}^{n}^{CF} I_{b}^{β} f) (t) .$ The next proposition will be crucial for our main results.

4.5 Proposition [25]

For y (t) defined on [a, b] and α ∈ (n, n + 1] for some $n \in ℕ,$ we have $(_{a}^{CF} I^{α}_a^{CF} D^{α} y) (t) = y (t) - \underset{k = 0}{\overset{n}{Σ}} \frac{y^{k} (a)}{k!} {(t - a)}^{k}$ and $(^{CF} I_{b}^{α CF} D_{b}^{α} y) (t) = y (t) - \underset{k = 0}{\overset{n}{Σ}} \frac{{(- 1)}^{k} y^{k} (b)}{k!} {(b - t)}^{k} .$

Let H be the space of all continuous functions from J = [0, T] into E, that is H : = C [J, E] .

4.6 Definition [26]

A function f : J × H → H is said to be Carathéodory if

(a) t → f (t, v) is measurable for each v ∈ H,

(b) v → f (t, v) is continuous for almost all t ∈ J .

4.7 Lemma [26]

Let Ω be closed convex and bounded subset of the Banach space H, G be a continuous function on J × J and f be a Carathéodory on J × H . If there exists $q \in L^{1} (J, ℝ^{+})$ and for each bounded subset β of E, we have $lim_{k \to 0^{+}} (f (J_{t, h} \times β)) \leq q (t) α (β),$ where J_t,h = J ∩ [t - h, t] . If U is equicontinuous subset of Ω, then $\begin{matrix} α ({\int_{J} G (s, t) f (s, v (s)) ds : v \in U}) \\ \leq \int_{J} G ∥ (s, t) ∥ q (s) α (U (s)) ds . \end{matrix}$

In this section we discuss the conditions for the existence of the following Caputo Fabrizio fractional anti-periodic boundary value problem (CFFBVP). $(^{CF} D_{b}^{α} y) (t) = f (t, y (t)), 1 < α \leq 2, 0 < t < T,$ (4.1) with $y (0) = - y (T) and \overset{´}{y} (0) = - \overset{´}{y} (T),$ (4.2) where $f : [0, T] \times ℝ \to ℝ$ is continuous and y (t) ∈ H¹ (0, T) .

4.8 Lemma

Let $K : [0, T] \to ℝ$ be continuous. A function y (t) ∈ H¹ (0, T) is a solution of the following anti-periodic boundary value problem $(^{CF} D_{b}^{α} y) (t) = K (t), 1 < α \leq 2, 0 < t < T,$ (4.3) with boundary conditions (4.2) if and only if y (t) is the solution of the following integral equation $y (t) = g (t) + \overset{T}{\int_{0}} G (t, s) K (s) ds,$ where, $G (t, s) = {\begin{matrix} \frac{(T - 2 t) (α - 1) - 2 (2 - α)}{4 B (α - 1)} \\ - \frac{(α - 1)}{2 B (α - 1)} (T - s) for 0 \leq s \leq t \leq T \\ \frac{(T - 2 t) (α - 1) + (2 - α) 2}{4 B (α - 1)} \\ + \frac{(α - 1) [2 t - (s + T)]}{2 B (α - 1)} for 0 \leq t \leq s \leq T \end{matrix},$ and $g (t) = \frac{(T - 2 t) (2 - α)}{4 B (α - 1)} K (T) .$

Proof. We have $(^{CF} D_{b}^{α} y) (t) = K (t), 1 < α \leq 2, 0 < t < T .$ Applying above Proposition 4.5, we have $\begin{matrix} y (t) & = & c_{1} + c_{2} t + \frac{(2 - α)}{B (α - 1)} \overset{t}{\int_{0}} K (s) ds \\ + \frac{(α - 1)}{B (α - 1)} \overset{t}{\int_{0}} (t - s) K (s) ds . \end{matrix}$ Differentiating both sides, we have $\overset{´}{y} (t) = c_{2} + \frac{(2 - α)}{B (α - 1)} K (t) + \frac{(α - 1)}{B (α - 1)} \overset{t}{\int_{0}} K (s) ds$ Using $\overset{´}{y} (0) = - \overset{´}{y} (T),$ with assumption given in [25], i.e., K (0) = 0, we have $c_{2} = - (c_{2} + \frac{(2 - α)}{B (α - 1)} K (T) + \frac{(α - 1)}{B (α - 1)} \overset{T}{\int_{0}} K (s) ds),$ which implies $c_{2} = - \frac{1}{2} (\frac{(2 - α)}{B (α - 1)} K (T) + \frac{(α - 1)}{B (α - 1)} \overset{T}{\int_{0}} K (s) ds) .$ Now by using y (0) = - y (T) , we have $c_{1} = - (\begin{matrix} c_{1} + c_{2} T + \frac{(2 - α)}{B (α - 1)} \overset{T}{\int_{0}} K (s) ds \\ + \frac{(α - 1)}{B (α - 1)} \overset{T}{\int_{0}} (T - s) K (s) ds \end{matrix}),$ which gives $\begin{matrix} c_{1} & = & - c_{2} \frac{T}{2} - \frac{(2 - α)}{2 B (α - 1)} \overset{T}{\int_{0}} K (s) ds \\ - \frac{(α - 1)}{2 B (α - 1)} \overset{T}{\int_{0}} (T - s) K (s) ds . \end{matrix}$ Putting the value of c₂ in above equation we have $\begin{matrix} c_{1} & = & \frac{T (2 - α)}{4 B (α - 1)} K (T) + \frac{T (α - 1)}{4 B (α - 1)} \overset{T}{\int_{0}} K (s) ds \\ - \frac{(2 - α)}{2 B (α - 1)} \overset{T}{\int_{0}} K (s) ds - \frac{(α - 1)}{2 B (α - 1)} \overset{T}{\int_{0}} (T - s) K (s) ds . \end{matrix}$ Thus the solution is $\begin{matrix} y (t) & = & \frac{(T - 2 t) (2 - α)}{4 B (α - 1)} K (T) \\ + \overset{t}{\int_{0}} \frac{(T - 2 t) (α - 1) + (2 - α) 2}{4 B (α - 1)} K (s) ds \\ + \overset{t}{\int_{0}} \frac{(α - 1) [2 t - (s + T)]}{2 B (α - 1)} K (s) ds \\ + \overset{T}{\int_{t}} \frac{(T - 2 t) (α - 1) - 2 (2 - α)}{4 B (α - 1)} K (s) ds \\ - \overset{T}{\int_{t}} \frac{(α - 1)}{2 B (α - 1)} (T - s) K (s) ds . \end{matrix}$ Let $G (t, s) = {\begin{matrix} \frac{(T - 2 t) (α - 1) - 2 (2 - α)}{4 B (α - 1)} \\ - \frac{(α - 1)}{2 B (α - 1)} (T - s) for 0 \leq s \leq t \leq T \\ \frac{(T - 2 t) (α - 1) + (2 - α) 2}{4 B (α - 1)} \\ + \frac{(α - 1) [2 t - (s + T)]}{2 B (α - 1)} for 0 \leq t \leq s \leq T \end{matrix},$ hence, we have $y (t) = g (t) + \overset{T}{\int_{0}} G (t, s) K (s) ds .$ This proves the lemma.■

Clearly G (t, s) is continuous on J × J and denote $G^{*} = sup_{(t, s) \in J \times J} ‖ G (t, s) ‖ .$ .As g is continuous on J therefore there exists $g^{*} = sup_{t \in J} ‖ g (t) ‖ .$

To prove the existence result of the Caputo Fabrizio fractional differential equation (4.1) with boundary conditions (4.2) , we assume the following conditions;

(C1) f : J × E → E is a Caratheodory function.

(C2) There exists $q \in L^{1} (J, ℝ^{+}),$ such that $‖ f (t, u) ‖ \leq q (t) ‖ u ‖,$ for t ∈ J and u ∈ E .

(C3) For each t ∈ J and each bounded subset β of E, we have $lim_{k \to 0^{+}} (f (J_{t, h} \times β)) \leq q (t) α (β),$ where J_t,h = J ∩ [t - h, t] .

4.9 Theorem

If the conditions (C1) - (C3) are satisfied and $G^{*} \underset{o}{\int^{T}} q (s) ds < 1$ holds, then the CFFBVP (4.1)-(4.2) has a solution.

Proof. Define an operator Q : H → H by $Q (u (t)) = g (t) + \underset{o}{\int^{T}} G (t, s) f (s, u (s)) ds .$ Clearly the solution of CFFBVP (4.1) - (4.2) exists if and only if Q has a fixed point.

Set $r \geq \frac{g^{*}}{1 - G^{*} \underset{o}{\int^{T}} q (s) ds},$ and define the set $B_{r} = {v \in H : {∥ v ∥}_{\infty} \leq r} .$ The set B_r is closed convex and bounded. We show that the operator Q satisfies all conditions of Theorem 2.7.

First we show that Q is continuous. For this let (v_n) be a sequence in H such that v_n → v in H . Now for each t ∈ J, consider $\begin{matrix} ‖ Q (v_{n}) - Q (v) ‖ \\ = ‖ \underset{o}{\int^{T}} G (t, s) f (s, v_{n} (s)) ds - \underset{o}{\int^{T}} G (t, s) f (s, v (s)) ds ‖ \\ = ‖ \underset{o}{\int^{T}} G (t, s) [f (s, v_{n} (s)) - f (s, v (s))] ds ‖ \\ \leq \underset{o}{\int^{T}} ‖ G (t, s) ‖ ‖ f (s, v_{n} (s)) - f (s, v (s)) ‖ ds \\ \leq G^{*} \underset{o}{\int^{T}} ‖ f (s, v_{n} (s)) - f (s, v (s)) ‖ ds . \end{matrix}$ Since f is a Caratheodory function, then by Lebesgue dominated convergence theorem [26] $‖ f (s, v_{n} (s)) - f (s, v (s)) ‖ \to 0, as \to \infty .$ Therefore Q is continuous.

Now we show that Q maps B_r into B_r . For each v ∈ B_r and t ∈ J, we have $\begin{matrix} ‖ Q (v (t)) ‖ \leq ‖ g (t) ‖ + \underset{o}{\int^{T}} ‖ G (t, s) ‖ ‖ f (s, v (s)) ‖ ds \\ \leq ‖ g (t) ‖ + \underset{o}{\int^{T}} ‖ G (t, s) ‖ q (s) ‖ v (s) ‖ ds \\ \leq g^{*} + G^{*} r \underset{o}{\int^{T}} q (s) ds \\ \leq r \end{matrix}$ which was required.

Next to show that Q (B_r) is bounded and equicontinuous. From above step Q (B_r) is bounded. Now for t₁, t₂ ∈ J with t₁ < t₂ and v ∈ B_r, consider $\begin{matrix} ‖ Q (v (t_{2})) - Q (v (t_{1})) ‖ \\ \leq ‖ g (t_{1}) - g (t_{2}) ‖ \\ + ‖ \underset{o}{\int^{T}} [G (t_{2}, s) - G (t_{1}, s)] f (s, v (s)) ds ‖ \\ \leq ‖ g (t_{1}) - g (t_{2}) ‖ \\ + r ‖ \underset{o}{\int^{T}} q (s) [G (t_{2}, s) - G (t_{1}, s)] ds ‖ . \end{matrix}$ As t₁ → t₂ the right hand side of the above inequality approaches to zero.

Let U be a subset of B_r such that $U \subset \bar{Conv} (Q (U) \cup {0}) .$ U is equicontinuous and bounded therefore the function u (t) : = α (U (t)) is continuous on J . Using Remark 2.2, we have $μ_{F} (U, τ) = \frac{τ}{τ + α (U)},$ for some τ > 0 . Now consider $\begin{matrix} u (t) & = & α (U (t)) \\ \leq & α (Q (U) (t) \cup {0}) \end{matrix}$ which implies $‖ u (t) ‖_{\infty} \leq ‖ u (t) ‖_{\infty} G^{*} \underset{o}{\int^{T}} q (s) ds$ since $G^{*} \underset{o}{\int^{T}} q (s) ds < 1,$ so we have u (t) = 0 for all t ∈ J, so u = 0, therefore α (U (t)) = 0 for all t ∈ J, hence $μ_{F} (U, τ) = 1,$ therefore by Theorem 2.10, U is relatively compact in H . By virtue of Arzila Ascoli’s theorem U is relatively compact in B_r . Hence by Theorem 2.7, Q has a fixed point, this proves the result.■

We need to mention here that FMNC can be used to prove existence theorems for fuzzy differential and integral equations. To end up this discussion we recall the following results and notations.

Let E ⁿ be the space of all fuzzy numbers, a fuzzy number is a fuzzy set $u : ℝ^{n} \to [0, 1],$ which is normal, fuzzy convex, upper semi-continuous and have compact support [27, 28].

4.10 Remark

From Theorem 2.1 of [29], E ⁿ can be embedded as a closed convex cone in Banach space X . The embedding j : E ⁿ → X is an isometry and an isomorphism, moreover if u : [0, a] → E ⁿ is integrable then j (u) is strongly measureable and Bochner integrable and $j (\underset{o}{\int^{t}} u (s) ds) = \underset{o}{\int^{t}} j (u) (s) ds for all t \in [0, a] .$

The following two lemmas are consequence of above remark.

4.11 Lemma [30]

Let X be a separable Banach space and let $A = {x_{n} : n \in ℕ}$ be an integrally bounded sequence of measurable functions from [0, a] into X . Then the function $t \to β ({x_{n} : n \in ℕ})$ is measurable and $β (\underset{o}{\int^{t}} A (s) ds) \leq \underset{o}{\int^{t}} β (A (s)) ds, t \in [0, a] .$ Where β is a Hausdorff measure of noncompactness defined in Section 1.

Finally we have the following lemma.

4.12 Lemma [31]

Let $A = {u_{n} : n \in ℕ}$ be an integrable fuzzy functions from [0, a] into E ⁿ . Then the function $t \to β ({u_{n} (t) : n \in ℕ})$ is measurable and $β (\underset{o}{\int^{t}} A (s) ds) \leq 2 \underset{o}{\int^{t}} β (A (s)) ds, t \in [0, a] .$

4.13 Remark

The above Lemma 4.12 is used in [27 , 33], to find the existence of fuzzy integral and differential equations, where β is crisp Hausdorff measure of noncompactness, from here we conclude that we can use FMNC to find the existence of solutions of fuzzy integral equations. However it is worthy to mention here that if we assume fuzzy Banach spaces instead of Banach spaces then we should not prefer to use crisp MNC but FMNC will be more useful. This case is still open to study for researchers in which many notions will be required to define and develop the whole theory.

Footnotes

Acknowledgments

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

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