Introduction

Numerous mathematical concepts have been presented to describe or characterize many phenomena in the real world. Differential equation, as one of the most valuable concepts, has been widely used in constructing mathematical models, which can be regarded as an essential tool toward the comprehension of nature. (See, for instance, [21, 23, 27] for applications tied to models from mathematical biology and physics.) On account of the significance of the theory and application of differential equation, it attracts scholars interest extensively to study the qualitative theories of systems constructed by differential equation. Especially, the theory of solvability is one of the fundamental and important topics in practice, such as the existence of the solutions for differential equation; we can refer to [18, 22, 30] for the recent developments.

The existence results about nonlinear systems utilize differential equations of integer order in the majority. As is known to us, fractional dynamic is widely applied in various fields of science, like physics, engineering, biology and modeling infectious diseases, neural networks, as well as anomalous diffusion [17, 26, 32]. For instance, scholars [25] simulated the outbreak of influenza A(H1N1) using a fractional-order SEIR model. Re- cently, it has a rapid development [1, 2, 19, 24, 25, 28].

One of the methods for solving the existence of solutions of nonlinear systems is relying on fixed point theorems. If the fixed point theory involves the mixed hypotheses of algebra, topology, and geometry, then it is called hybrid fixed point theory. Hybrid fixed point theorem can do well in solving existence of hybrid differential equations, which is also useful for the theory of nonlinear quadratic differential equations; see in literature [6, 20]. Recently, many researchers like Dhage [9, 11], Lakshmikantham [7], Bashiri [5] give several hybrid fixed point theorems to help perfecting the theories. Achievements have been made in this area.

In 2011, Zhao [31] studied fractional quadratic differential equations involving Rie- mann–Liouville differential operators of order

The existence theorem for fractional quadratic differential equations was proved under mixed Lipschitz and Caratheodory conditions.

Recently, Dhage [8] introduced a notion of partially condensing mappings in a par- tially ordered normed linear space and proved some hybrid fixed point theorems under certain mixed conditions of algebra, analysis, and topology. The author combined the compactness with some algebraic arguments to prove some hybrid fixed point theorems for the mappings in ordered spaces and apply the results to obtain the solutions of integral equations under some mixed compactness and monotonic conditions, which was the main contribution of the works; see Dhage [10, 12–15] for recent development.

In 2015, Dhage [16] considered the periodic boundary value problem of first-order nonlinear quadratic ordinary differential equations

forThey applied iteration principle embodied in hybrid fixed point theorem in partially ordered normed linear spaces to obtain the existence as well as approximations of the positive solutions for this system.

In 2020, Ardjouni [3] investigated the nonlinear third-order hybrid differential equa- tion

By employing Dhage iteration principle in a partially ordered normed linear space they proved the existence and approximation of solutions of the initial value problems of this system.

The previous works were mostly based on differential equations of integer order. We intend to study a class of hybrid fractional differential equations with impulses in a partially ordered linear algebra, which contains linear perturbations and quadratic per- turbations in the meantime. Our results not only rely on iteration principles and related hybrid fixed point theorems, which have been proposed, but also extend some known theories to fractional-order systems. Furthermore, the conclusions may be subject to the influence of impulses to some extent. These advantages inspire us to continue current works.

In this paper, we discuss the following hybrid fractional (in the sense of Caputo) differential equation with impulses, which is given by

where denotes the Caputo fractional derivative of order

The plan of this paper is as follows. In Section 2, we recall some necessary notions, definitions, and lemmas that used in the rest. In Section 3, we apply hybrid fixed point theorem to solve the initial value problems of (1) and give rigorous proof in the meantime. In Section 4, an example is given to illustrate our conclusions.

Preliminaries

In this section, we recall some basic and essential definitions of fractional calculus. Mean- while, we introduce some definitions, lemmas, as well as relative theories in linear par- tially ordered spaces for better obtaining our main results.

Definition 1. (See [26].) The Riemann–Liouville fractional integral of order of a function is given by

where is the Gamma function, provided the right side is pointwise defined on

Definition 2. (See [26].) The Caputo fractional derivative of order for a function [0.1] is defined by.

Lemma 1. (See [26].) Let Then the solution of the fractional differential equation

is

Unless otherwise specified, we assume thatdenotes a partially orderedBanach space with an order relation4and the norm Two elements and in are said to be comparable if either the relation or holds. A nonempty subset of is called a chain or totally ordered if all the elements of are comparable. The set is said to be regular if each sequence is nondecreasing (resp. nonincreasing) in such that as , then (resp.) for all. The detailscan be found in [8] and the reference therein

Definition 3. (See [16].) A mapping is called isotone or nondecreasing if it preserves the order relation that is, if implies for all

Definition 4. (See [16].) A mapping is called partially continuous at a point if for, there exists a such that whenever is comparable to a and is called partially continuous on if it is partially continuous at every point of . It is clear that if is partially continuous on , then it is continuous on every chain contained in

Definition 5. (See [16].) A mapping is called partially bounded if () is bounded for every chain in . is called uniformly partially bounded if () is bounded for every chain in by a unique constant. is called bounded if () is a bounded subset of

Definition 6. (See [4].) The set S is said to be quasi-equicontinuous in if for any , there exists a such that if, and

Lemma 2 [Compactness criterion]. (See [4].) The set is relatively compact if and only if

(i) S is bounded,

(ii) S is quasi-equicontinuous in

Definition 7. (See [16].) A mapping is called partially compact if () is a relatively compact subset of S for all totally ordered sets or chains in .

Definition 8. (See [13,14].) The order relation and the metricdon a nonempty set are said to be-compatible if is a monotone sequence, that is, monotone nondecreasingor monotone nondecreasing sequence in and if subsequence convergestox∗implies that the original sequence converges to x^∗. Similarly, given a partiallyordered normed linear space , the order relation and the norm are said tobe-compatible if and the metricddefined through the norm are -compatible.A subset E of is called a Janhavi set if the order relation and the metricdor the norm are -compatible in it. In particular, if , then is called a Janhavi metric spaceor Janhavi Banach space

Definition 9. (See [16].) A upper semicontinuous and nondecreasing function is called a-function, provided . Let be a partially ordered normed linear space. A mapping is called partially nonlinear-Lipschitz if there exists a-functionsuch that for all comparable elements,, thenT is called a partiallyLipschitz with a Lipschitz constant k

We recall that a nonempty closed and convex subset of the Banach algebra is called a cone if

(i)

(ii) and

(iii) where . is a zero element of ..

The cone in S is called positive if

(i) , where “ o ” is a multiplicative composition in; see Dhage [10] and the reference therein.

Now we assume that denotes a positive cone.

Lemma 3. (See [16].) If are such that and, then

Lemma 4. (See [14].) Every ordered Banach space () is regular.

Lemma 5. (See [14].) Every partially compact subset E of an ordered Banach space (

Lemma 6. (See [9].) Let be a regular partially ordered complete normed linear algebra, and let every compact chain C in S be Janhavi set. Let and be three nondecreasing operators such that:

(i) and are partially bounded and partially nonlinear-Lipschitz with-func- tions and, respectively;

(ii) B is partially continuous and compact;

(iii) , where is a chain in and

(iv) there exists an element such that x

Then the operator equation has a solution x^∗, and the sequence xn of successive iterations defined by= 0, 1, . . . , converges monotonically to x^∗.

We consider the space () defined on with norm , which is given by

Clearly ()is a Banach space with the supremum norm. We define the order cone in () by

which is a positive cone in (), and define order relation in () by

Clearly, ( (),) is a regular ordered Banach space with respect to the above norm and order relation in (), and every compact chain in () is Janhavi set in view of the Lemmas 4 and 5. For the sake of convenience, we denote (0, 0), g(0, 0) := g₀ in the sequel.

Lemma 7.(See [29].) Let (t)[0, 1] be a solution of the following impulsive hybrid fractional differential equation:

Then it satisfies the following impulsive hybrid fractional integral equation:

Definition 10. Let (t)∈ () satisfy

Then the function . is called a mild solution of IVPs (2).

Definition 11. Let ∈ () satisfy

Then the function u is called a mild lower solution of IVPs (2).

Main results

In this section, we attempt to obtain the existence and approximation for a mild solution of IVPs for impulsive hybrid fractional differential equation (1).

In order to obtain our main results, hypotheses (H1)–(H5) are given by:

(H1) There exist-positive constants M and N such that

and

(H2) There exist-functions and such that

wheret

(H3) is nondecreasing in for each, and there exists a constant such that

where

(H4) I_k()/g(t, x) is nondecreasing in for each , and there exists a con- stant H such that

whe

(H5) IVPs (1) has a mild lower solution ().

Theorem 1. Under assumptions (H1)–(H5) and further, we assume that

where we denote

Then IVPs (1) has a mild solution x∗, and the sequence of successive approxima- tions defined by

converges monotonically to x*(4)

Proof. At first, we define S = (). Then by Lemma 5 we have that every compact chain in possesses the compatibility property with respect to the norm, and the order relation so that every compact chain is a Janhavi set in

By Lemma 7 a mild solution of impulsive hybrid fractional differential equation (1) is a solution for impulsive hybrid integral equation

Define operators

In this way, the impulsive hybrid fractional integral equation (5) can be written as the operator equation

For the sake of showing our main results, we prove that and satisfy the conditions of Lemma 6. This is achieved in the following steps.

Step 1., and are nondecreasing operators on

Let be such that. Then by hypothesis (H2) we obtain

for all . By the definition of the order relation in S we get This shows that A and C are nondecreasing operators. Similarly,

It is shown that the operator B is also nondecreasing. Thus, and are nondecreasing positive operators.

Step 2. We note that A, C are partially bounded and partially-Lipschitz operator on

Let x ∈ S be arbitrary. The by (H1)

for all. Taking supremum over t, we obtain So, A and C are bounded. This further implies that A and C are partially bounded on S

Let and Then by (H2) we have

Taking the supremum over ., we obtain

Therefore,A and C are partially nonlinear -Lipschitz on. Assumption (i) of Lemma 6is satisfied.

Step 3. We show thatBis a partially continuous operator onS

At first, let be a sequence in a chain C of S, which satisfies

Then by Lebesgue dominated convergence theorem

This indicates that converges monotonically to pointwise on

Next, we prove that is a quasi-equicontinuous sequence of function in .Let

Note that , then we have

So

as t′ → t′′ for all, which shows that the convergence → is uniform, and hence, B is partially continuous on S.

Step 4. B is a partially compact operator on S.

Let C be an arbitrary chain in S. We need to prove that (C) is a uniformly bounded set in S. Let y (C) be any element. Then there is an element x C such thatNow by hypotheses (H3), (H4) we have

where. Taking supremum over t, we obtain kyk = kBxk 6 L for

Next, we prove that B(C) is a quasi-equicontinuous set in S. Let be arbitrary, k = 0, 1, . . . , m. Likewise, we discuss above in Step 3:

when t′→t′′. Hence,B(C)is a quasi-equicontinuous subset ofS. Now, B(C)is a uni-formly bounded and quasi-equicontinuous set of functions in S. B y applying Lemma 2 itis compact. Consequently,Bis a partially compact operator onSinto itself

Therefore, assumption (ii) of Lemma 6 is satisfied.

tep 5. There is a u ∈ S such that satisfies. IVPs (1) has a mildlower solutionu. Then by application of Lemma 7 and Definition 11 we have

Then from the definitions of A, B, and C we can deduce that

Hence,, which meets (iii) in Lemma 6.

Step 6. and satisfy .It is clear thatψandφare D- functions of operatorsAandCfrom condition (H2),L denotessup {‖B(C)‖: C is a chain inS}from the proof given in Step 4. Then, com-bining with condition (3), we have

Assumption (iv) of Lemma 6 is satisfied

In a word, all the conditions of Lemma 6 are satisfied. Thus, we can conclude that the operator equation has a solution x ∗ . Consequently, IVPs (1) has a mild solution x ∗ defined on Furthermore, the sequence of successive approximations defined by (4) converges monotonically to x ^∗ . This completes the proof

Example

Given the interval J = [0, 1] and the points t1 = 1/4, t2 = 1/2, t3 = 3/4, we consider the IVPs of the following impulsive hybrid fractional differential equation:

where functions f, g, h are defined as follo

for arbitrary. Through calculation, we hav

Take= 3/40, H = 1/4 in Theorem 1. Then we calculate that L = 1.338, moreover,r > 0. Therefore, all the conditions of Theorem 1 are satisfied. Then we can conclude that system (6) has a mild solution x∗, and xn of successive approximation is defined by where x1 = u = 0, converges monotonically to x∗.

where

Acknowledgments

The authors sincerely thank the reviewers for their valuable sugges- tions and useful comments that have led to the present improved version of the original manuscript

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Notes