key: cord-0113863-yf4kxxg2
authors: Sutar, Sagar T.; Kucche, Kishor D.
title: On Nonlinear Hybrid Fractional Differential Equations with Atangana-Baleanu-Caputo Derivative
date: 2020-07-21
journal: nan
DOI: nan
sha: 66ceef7e3367bc21a4fa2f142bcb4cd58bbc81a3
doc_id: 113863
cord_uid: yf4kxxg2

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.

Lakshmikantham and Vatsala [1, 2] developed the primary theory of fractional differential inequalities with Reimann-Liouville and Caputo fractional derivative. Authors utilized the investigated fractional inequalities and the comparison results to study the existence of local, extremal and global solutions to nonlinear fractional differential equations (FDEs). Dhage and Lakshmikantham [3] initiated the study of first order hybrid differential equations and investigated the basic results pertaining to existence and uniqueness of solution. Further, differential inequalities obtained in connection with hybrid FDEs utilized to examine comparison results and qualitative properties of solution. Adopting the similar approach of [3] , Zhao et al. in [4] extended the study of first order hybrid differential equations to hybrid FDEs involving Riemann-Liouville fractional derivative. Further, different class of Hybrid FDEs subject to various initial and boundary conditions have also been studied by several researchers [5, 6, 7, 8, 9] .

On the other hand, intending to eliminate the singular kernel in traditional fractional derivatives, Caputo and Fabrizio [10] presented a fractional derivative with the exponential kernel and Atangana-Baleanu [11] introduced a fractional derivative in the sense of Caputo with Mittag-Leffler function as its kernel, which notable as ABC-fractional derivative. The advantage of ABC-fractional derivative is that it is nonlocal and has a non-singular kernel.

Because of which it has numerous applications in demonstrating different problems that includes different diseases, such as, dengue fever outbreak [12] , tumor-immune surveillance mechanism [13] , the clinical implications of diabetes and tuberculosis coexistence [14] , the free motion of a coupled oscillator [15] , smoking models [16] and coronavirus [17] . For the fundamental development in the theory nonlinear ABC-FDEs, we refer the reader to the work of Jarad et al. [18] , Baleanu et al. [19] , Syam et al. [20] , Afshari et al. [21] , Shah et al. [22] and Ravichandran et al. [23, 24, 25, 26] .

Motivated by the works of [3, 4] and in continuation of a past work we have done in [27] , we develop the theory of nonlinear hybrid ABC-FDEs of the form

where,

, T > 0 and 0 < α < 1, (ii) ABC 0 D α τ denotes left ABC-fractional differential operator of order α with lower terminal 0,

The primary aim of the current study is to determine the equivalent fractional integral equation to ABC-hybrid FDEs (1.1)-(1.2) and explore the existence results. Further, we build up the theory of inequalities for ABC-hybrid FDEs and use it to examine the existence of a maximal and minimal solution and the comparison results.

The current paper is coordinated as follows. In section 2, we review essential definitions and results about ABC-fractional derivative. In section 3, we give equivalent fractional integral equations and derive existence result through it. In section 4, we acquire fractional differential inequalities for ABC-hybrid FDEs. Section 5 deals with the existence of maximal and minimal solutions of ABC-hybrid FDEs. In section 6, we determine comparison results relating to ABC-hybrid FDEs.

In this section, we recall the basic definitions and the results about ABC-fractional derivative which will be used later.

Definition 2.1 A function ω ∈ AC(J, R) is said to be solution of ABC-hybrid-FDEs (1.1)-

is absolutely continuous for each u ∈ R and ω satisfies 

Definition 2.3 [11] Let ω ∈ H 1 (0, T ) and α ∈ [0, 1], the left Atangana-Baleanu-Caputo fractional derivative of ω of order α is defined by

where B(α) > 0 is a normalization function satisfying B(0) = B(1) = 1 and E α is one parameter Mittag-Leffler function [28, 29] defined by

.

The associated fractional integral is defined by

is the Riemann-Liouville fractional integral [28, 29] of ω of order α.

Lemma 2.2 [20, 23] The equivalent fractional integral equation to the the ABC-FDEs

is given by

Definition 2.4 [31, 32, 33] The generalized Mittag-Leffler function E γ α,β (z) for the complex numbers α, β, γ with Re(α) > 0 is defined as

where (γ) k is the Pochhammer symbol given by (γ) 0 = 1, (γ) k = γ(γ + 1) · · · (γ + k − 1), k = 1, 2, · · · Note that, E 1 α,β (z) = E α,β (z) and E 1 α,1 (z) = E α (z).

Let 0 < α < 1 and β, σ, λ ∈ C (Re(β) > 0).

Lemma 2.4 [27] If m is any differentiable function on J such that ABC 0 D α τ m ∈ C(J) and there exists

Lemma 2.5 [3] Let S be a non-empty, closed convex and bounded subset of Banach algebra Ω and let F 1 : Ω → Ω and F 2 : S → Ω be two operators such that

then the operator F 1 ωF 2 ω = ω has a solution in S.

In the following Theorem, we derive an equivalent fractional integral equation to ABChybrid FDEs (1.1)-(1.2).

for each τ ∈ J. Then ω ∈ AC(J, R) is a solution of ABC-hybrid FDEs (1.1)-(1.2) if and only if ω is a solution of fractional integral equation 

This gives,

Using relation between fractional differential operators ABR 0 D α τ and ABC 0 D α τ given in Theorem 1 [11] , we obtain

Now putting τ = 0 in Eq.(3.2) and using the fact g(0, ω(0)) = 0, we obtain

For each τ ∈ J, consider the mapping h τ : R → R defined by,

By assumption h τ : R → R is increasing and hence it is injective. Using definition of h τ , Eq.(3.3) can be written as h 0 (ω(0)) = h 0 (ω 0 ).

Since h 0 is injective, we have ω(0) = ω 0 . This completes the proof of the Theorem. ✷

To prove existence results for solution of ABC-hybrid FDEs (1.1)-(1.2), we need following assumptions on f and g.

is increasing in R a.e. for each τ ∈ J.

(H2) The function g ∈ C is such that, |g(τ, ω(τ )| ≤ h(τ ), a.e. τ ∈ J, h ∈ C(J, R + ). 

Then Ω is Banach algebra with multiplication defined by

Define,

Consider the set, S = {ω ∈ Ω : ω ≤ R} .

One can verify that S is closed, convex and bounded subset of Banach algebra Ω. Consider the operators F 1 : Ω → Ω and F 1 : S → Ω defined by,

The equivalent fraction integral Eq.(3.1) to the ABC-hybrid FDEs (1.1)-(1.2) can be written in operator equation form given by

We prove that the operators F 1 and F 2 satisfies conditions of Lemma 2.5. The proof of the same have been given in following steps.

Step 1) F 1 is Lipschitz.

Using Lipschitz condition on f , for any ω, η ∈ Ω and τ ∈ J we obtain,

Step 2) F 2 is completely continuous.

We show that F 2 : S → Ω is a compact and continuous operator on S into Ω. First we show that F 2 is continuous on S. Let {ω n } be a sequence in S converging to a point ω ∈ S. Then by the Lebesgue dominated convergence theorem,

for all τ ∈ J. This shows that F 2 is a continuous operator on S. Using hypothesis (H2), for any ω ∈ S and τ ∈ J, we have

This gives,

, ω ∈ S, τ ∈ J, (3.8) which shows that F 2 is uniformly bounded on J. Next we prove that F 2 (S) is equicontinious set in Ω. Let any ω ∈ S and 0 ≤ τ 1 < τ 2 ≤ T . Then we have

Since g(τ, ω) is continuous on compact set J × [−R, R], it is uniformly continuous there and hence we have |g(τ 1 , ω(τ 1 )) − g(τ 2 , ω(τ 2 ))| → 0, as |τ 1 − τ 2 | → 0, for each ω ∈ S. (3.10)

Next using hypothesis (H2), we have

(3.11)

Therefore,

(3.12)

Therefore it follows from (3.9), (3.10) and (3.12) that

This proves F 2 (S) is equicontinious set in Ω. Since F 2 (S) is uniformly bounded and equicontinious set in Ω, by Ascoli-Arzela theorem F 2 is completely continuous.

Step 3) Let any η ∈ S. For ω ∈ Ω, consider the operator equation ω = F 2 ωF 2 η. Our aim is to prove that ω ∈ S.

Using hypothesis (H1) and condition (3.8), we have

This gives,

This proves ω ∈ S.

Step 4) The constants α and M of Lemma 2.5 corresponding to the operators F 1 and F 2 defined in equations (3.6) and (3.7) respectively are

By condition (3.4) , it follows that 

and satisfies the ABC-hybrid fractional differential inequalities,

where one of above inequality being is strict.

Then

Proof: Suppose that the conclusion of the theorem does not holds. Since v, w ∈ C(J) there exits τ 0 ∈ J such that v(τ 0 ) = w(τ 0 ) and v(τ ) < w(τ ) for all τ ∈ [0, τ 0 ).

Then v(τ 0 ) f (τ 0 , v(τ 0 )) = w(τ 0 ) f (τ 0 , w(τ 0 )) and using increasing property of the mapping ω → ω f (τ, ω)

Then m, ABC 0 D α τ m ∈ C(J). Further, τ 0 ∈ J is such that m(τ 0 ) = 0 and m(τ ) ≤ 0 for all τ ∈ [0, τ 0 ).

Since m satisfies all assumptions of Lemma 2.4, we get, ABC 0 D α τ m(τ 0 ) ≥ 0. This gives

Suppose that the inequality (i) is strict. Then we get

which is contradiction to v(τ 0 ) = w(τ 0 ). Therefore we must have v(τ ) < w(τ ), for all τ ∈ J.

This completes the proof of theorem. ✷ Theorem 4.2 Assume that the conditions of Theorem 4.1 holds with nonstrict inequalities (i) and (ii). Suppose that

Then

Proof: For any fix ǫ > 0, we define

This gives, for τ = 0

Again, using the Lipschitz condition on g and Eq.(4.2), we have

Using condition on L, from above inequality we obtain,

In the proof of Theorem 3.6 [27] it is produced that,

∈ C(J). Thus using the inequalities (ii), (4.3) and (4.4), for any τ ∈ J we have

Since v(0) < w ǫ (0), by application of Theorem 4.1 with w(τ ) = w ǫ (τ ), for each ǫ > 0 we have v(τ ) < w ǫ (τ ), τ ∈ J.

Taking limit as ǫ → 0, in the above inequality and utilizing equ. Given an arbitrary small real number ǫ > 0, consider the following ABC-hybrid FDEs

= g(τ, ω(τ )) + ǫ, a.e. τ ∈ J, (5.1)

where g ∈ C is such that g(0, ω 0 + ǫ) = 0. Proof: By hypothesis,

Then we can find ǫ 0 > 0 such that

Following simillar steps as in the proof of Theorem 3.2, one can complete the remaining part of the proof. Proof: Let {ǫ n } ∞ n=0 be a decreasing sequence of positive numbers converging to 0 where ǫ 0 is such that,

Using ǫ n ≤ ǫ 0 , n ∈ N ∪ {0}, it is easy to verify that

Due to above condition, by Theorem 5.1, for each n ∈ N ∪ {0}, ABC-hybrid FDEs

= g(τ, ω(τ )) + ǫ n , a.e. τ ∈ J, (5.3)

has a solution, say ω(τ, ǫ n ), hence we get

= g(τ, ω(τ, ǫ n )) + ǫ n > g(τ, ω(τ, ǫ n )) a.e. τ ∈ J, (5.5)

The equivalent integral equation of above ABC-hybrid FDEs is

Let u be any solution of ABC-hybrid FDEs (1.1)-(1.2), hence we get

= g(τ, u(τ )), a.e. τ ∈ J, (5.8)

Noting that, ω(0, ǫ n ) < u(0) for all n ∈ N ∪ {0}. Therefore using comparison Theorem 4.1, we have u(τ ) < ω(τ, ǫ n ), τ ∈ J, n ∈ N ∪ {0} . (5.10)

Let ω(τ, ǫ m ), ω(τ, ǫ n ) be the solutions of ABC-hybrid FDEs (5.3)-(5.4) corresponding to the m th , n th term of the sequence {ǫ n } ∞ n=0 , with m > n. Therefore we have,

≤ g(τ, ω(τ, ǫ m )) + ǫ n Applying Lemma 4.1 to the above set of inequalities, we get

This verifies that ω(τ, ǫ m ) decreasing sequence bounded bellow by any solution of ABChybrid FDEs (1.1)-(1.2). Therefore ω(τ ) = lim n→∞ ω(τ, ǫ n ) exists on J. We show that this converges is unoform on J. Therefore, it is enough to prove that the sequence {ω(τ, ǫ n )} is equicontinuous in C(J, R). Let τ 1 , τ 2 ∈ J with τ 1 < τ 2 be arbitrary. Then,

|f (τ 1 , ω(τ 1 , ǫ n ))g(τ 1 , ω(τ 1 , ǫ n )) − f (τ 2 , ω(τ 2 , ǫ n ))g(τ 2 , ω(τ 2 , ǫ n ))|

Since f, g are continuous on compact set J × [−R, R], they are uniformly continuous there. Hence, for each n ∈ N, |f (τ 1 , ω(τ 1 , ǫ n )) − f (τ 2 , ω(τ 2 , ǫ n ))| → 0, as |τ 1 − τ 2 | → 0 (5.12)

|f (τ 1 , ω(τ 1 , ǫ n ))g(τ 1 , ω(τ 1 , ǫ n )) − f (τ 2 , ω(τ 2 , ǫ n ))g(τ 2 , ω(τ 2 , ǫ n ))| → 0, as |τ 1 − τ 2 | → 0 (5.13)

. We find f (τ 1 , ω(τ 1 , ǫ n ))

(5.14)

Using inequalities (5.12), (5.13) and (5.14), we conclude from (5.11) that |ω(τ 1 , ǫ n ) − ω(τ 2 , ǫ n )| → 0, , as |τ 2 − τ 1 | → 0, This shows that {ω(τ, ǫ n )} converges uniformly to ω(τ ) as n → ∞. Hence taking limit as n → ∞ of equation Equ.(5.7), we get

Thus ω(τ ) is a solution of ABC-hybrid FDEs (1.1)-(1.2). Taking limit as n → ∞ of inequality (5.10), we get u(τ ) < ω(τ ), τ ∈ J. Hence ABC-hybrid FDEs (1.1)-(1.2) has a maximal solution. ✷

Theorem 6.1 Suppose the hypotheses (H1)-(H2) and condition (3.4) hold. Also assume that the function g satisfies the condition (4.1). If there exists a function u ∈ AC(J, R)

≤ g(τ, u(τ )), a.e. τ ∈ J, (6.1)

where r is maximal solution of the ABC-hybrid FDEs (1.1)-(1.2).

Proof: Let ǫ > 0 be arbitrary small. Then by Theorem 5.2, ω(τ, ǫ) is a solution of the ABC-hybrid FDEs (5.1)-(5.2). Therefore

= g(τ, ω(τ, ǫ)) + ǫ, a.e. τ ∈ J,

> g(τ, ω(τ, ǫ)), a.e. τ ∈ J, (6.3)

Therefore u is lower solution and ω(τ, ǫ) is upper solution of ABC 0 D α τ ω(τ ) f (τ, ω(τ )) = g(τ, ω(τ )).

Further, u(0) ≤ ω 0 < ω 0 + ǫ = ω(0, ǫ).

By applying Theorem 4.1, we obtain u(τ ) < ω(τ, ǫ), for all τ ∈ J,

In limiting case as ǫ → 0, we get

The proof of the following Theorem can be given in similar way as in the case of Theorem 6.1. 

≥ g(τ, v(τ )), a.e. τ ∈ J,

where ρ is minimal solution of the ABC-hybrid FDEs (1.1)-(1.2).

Using Theorem 6.1, we can prove the uniqueness result for ABC-hybrid FDEs (1.1)-(1.2). The detail of which is given in the following Theorem. , for all τ ∈ J; ω, η ∈ R.

If identically zero function is the only solution of the ABC-hybrid FDEs Define

, τ ∈ J.

We find by using linearity

η(τ ) f (τ, η(τ )) = g(τ, ω(τ )) − g(τ, η(τ )) ≤ |g(τ, ω(τ )) − g(τ, η(τ ))| ≤ G τ, ω f (τ, ω) − η f (τ, η) = G(τ, m(τ )), τ ∈ J. (6.9)

Again by using the fact that |f | ′ ≤ |f ′ |, we find ABC 0 D α τ m(τ ) =

Combining inequalities (6.9) and (6.10), we obtain ABC 0 D α τ m(τ ) ≤ G(τ, m(τ )), τ ∈ J. (6.11)

Using definition of m(τ ), we find

− ω 0 f (0, ω 0 ) = 0 (6.12)

From equations (6.11) and (6.12), using assumption, we get m(τ ) = 0, τ ∈ J. From which we can easily show that ω f (τ, ω) = η f (τ, η)

, τ ∈ J.

Hence ω = η. This proves the uniqueness of solution. ✷

Fractional integral inequalities and comparison results acquired in the present paper can utilized to analyze the various qualitative and quantitative properties of solutions for a different class of ABC-hybrid FDEs subject to enhanced initial and boundary conditions.

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