CUBO, A Mathematical Journal

Vol.22, N◦03, (325–350). December 2020
http://dx.doi.org/10.4067/S0719-06462020000300325

Received: 20 May, 2020 | Accepted: 22 October, 2020

Existence and Attractivity Theorems for Nonlinear Hybrid
Fractional Integrodifferential Equations with Anticipation

and Retardation

Bapurao C. Dhage

Kasubai, Gurukul Colony, Thodga Road,

Ahmedpur-413 515, Dist. Latur,

Maharashtra, India.

bcdhage@gmail.com

ABSTRACT

In this paper, we establish the existence and a global attractivity results for a nonlinear

mixed quadratic and linearly perturbed hybrid fractional integrodifferential equation

of second type involving the Caputo fractional derivative on unbounded intervals of

real line with the mixed arguments of anticipations and retardation. The hybrid fixed

point theorem of Dhage is used in the analysis of our nonlinear fractional integrod-

ifferential problem. A positivity result is also obtained under certain usual natural

conditions. Our hypotheses and claims have also been explained with the help of a

natural realization.

RESUMEN

En este art́ıculo, se establecen resultados de existencia y de atractividad global para una

ecuación no lineal cuadrática mixta e h́ıbrida fraccionaria integrodiferencial linealmente

perturbada de segundo tipo involucrando la derivada fraccional de Caputo en intervalos

no acotados de la recta real con argumentos mixtos de anticipación y retardo. El

teorema de punto fijo h́ıbrido de Dhage es usado en el análisis de nuestro problema no

lineal fraccionario integrodiferencial. También se obtiene un resultado de positividad

bajo ciertas condiciones naturales usuales. Nuestras hipótesis y afirmaciones también

se explican con la ayuda de una realización natural.

Keywords and Phrases: Hybrid fractional integrodifferential equation, Dhage fixed point the-

orem, Existence theorem, Attractivity of solutions, Asymptotic stability.

2020 AMS Mathematics Subject Classification: 34K10, 47H10.

c©2020 by the author. This open access article is licensed under a Creative
Commons Attribution-NonCommercial 4.0 International License.

http://dx.doi.org/10.4067/S0719-06462020000300325


326 Bapurao C. Dhage CUBO
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1 Introduction

Let t0 ∈ R be a fixed real number and let J∞ = [t0,∞) be a closed but unbounded interval in

R. Let CRB(J∞) denote the class of pulling functions a : J∞ → (0,∞) satisfying the following

properties:

(i) a is continuous, and

(ii) lim
t→∞

a(t) = ∞.

The notion of the pulling function is introduced in Dhage [15, 17] and Dhage et al. [21]. There

do exist functions a : J∞ → (0,∞) satisfying the above two conditions. In fact, if a1(t) = |t| + 1,

a2(t) = e
|t|, then a1,a2 ∈ CRB(J∞). Again, the class of continuous and strictly monotone functions

a : J∞ → (0,∞) going to ∞ satisfy the above criteria. Note that if a ∈ CRB(J∞), then the

reciprocal function a : J∞ → R+ defined by a(t) =
1

a(t)
is continuous and lim

t→∞
a(t) = 0. It has

been shown in Dhage [16, 18, 19, 20] and Dhage et. al [21] that the pulling functions are useful in

proving different asymptotic characterizations of the solutions of nonlinear differential and integral

equations. In this paper we employ the pulling functions for characterizing the solutions of a

nonlinear hybrid fractional differential equation when the value of independent variable is large.

It is now well-known that several anomalous real world problems in sciences and engineering

are adequately modelled on fractional differential equations (see Hilfer [25] and Kilbas et. al [27]).

Sometimes one may be interested in the behaviour of the anomalous dynamic system in the long

duration of time which depend upon both past history as well as the future data of the process

in question. In such cases, we take help of fractional differential equations with retardatory and

anticipatory arguments on the unbounded intervals of real line. Motivated by this reason, in this

paper we discuss asymptotic behaviour of a nonlinear hybrid fractional integrodifferential equation

with retardation and anticipation on the unbounded intervals via hybrid fixed point theory of

Dhage [8, 9, 16].

We need the following fundamental definitions from fractional calculus (see Podlubny [28],

Kilbas et al. [27] and references therein) in what follows.

Definition 1.1. If J∞ = [t0,∞) be an interval of the real line R for some t0 ∈ R with t0 ≥ 0,

then for any x ∈ L1(J∞,R), the Riemann-Liouville fractional integral of fractional order q > 0 is

defined as

I
q
t0
x(t) =

1

Γ(q)

∫ t

t0

x(s)

(t − s)1−q
ds, t ∈ J∞,

provided the right hand side is pointwise defined on (t0,∞), where Γ is the Euler’s gamma function

defined by Γ(q) =

∫ ∞

0

e−ttq−1 dt.



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 327

Definition 1.2. If x ∈ Cn(J∞,R), then the Caputo fractional derivative
CD

q
t0
x of x of fractional

order q is defined as

CD
q
t0
x(t) =

1

Γ(n − q)

∫ t

t0

(t − s)n−q−1x(n)(s)ds, t ∈ J∞,

where n − 1 < q ≤ n, n = [q] + 1, [q] denotes the integer part of the real number q, and Γ is the

Euler’s gamma function. Here Cn(J∞,R) denotes the space of real valued functions x(t) which are

n times continuously differentiable on J∞.

Given a pulling function a ∈ CRB(J∞)
⋂

C1(J∞,R), we consider the following nonlinear

hybrid fractional integrofractional differential equation (in short HFRIGDE) involving the Caputo

fractional derivative,

CD
q
t0













a(t)x(t) −

m
∑

j=1

Iαjhj(t,x(t),x(η(t)))

f(t,x(t),x(θ(t)))













= g(t,x(t),x(γ(t))), t ∈ J∞,

x(t0) = x0,































(1.1)

where CD
q
t0

is the Caputo fractional derivative of fractional order 0 < q ≤ 1, Iαj are the Riemann-

Louville fractional integration of fractional order αj ≥ 0 for j = 1, . . . ,m, f : J∞×R×R → R\{0},

hj : J∞ ×R×R → R are continuous, g : J∞ ×R×R → R is Carathéodory and η,θ,γ : J∞ → J∞

are the continuous functions such that η and θ are anticipatory and γ is retardatory, that is, η(t) ≥ t,

θ(t) ≥ t and γ(t) ≤ t for all t ∈ J∞ with η(t0) = t0 = θ(t0).

Definition 1.3. By a solution for the hybrid fractional differential equation (1.1) we mean a

function x ∈ C1(J∞,R) such that

(i) the map (x,y,z) 7→
a(t)x − Σmj=1I

αjhj(t,x,z)

f(t,x,y)
is well defined for each t ∈ J∞,

(ii) the map t 7→
a(t)x(t) − Σmj=1I

αjhj(t,x(t),x(θ(t)))

f(t,x(t),x(θ(t)))
= z(t) is differentiable on J∞ and z

′ ∈

C(J∞,R), and

(iii) x satisfies the equations in (1.1) on J∞,

where C1(J∞,R) is the space of continuous real-valued functions defined on J∞ whose first deriva-

tive x′ exists and x′ ∈ C(J∞,R).

As the functions θ and γ in the HFRIGDE (1.1) are respectively anticipatory and retardatory,

the arguments in the problem (1.1) are deviating over the unbounded interval J∞. Therefore,

the behaviour of the dynamic system modelled on the HFRIGDE (1.1) depends upon both back



328 Bapurao C. Dhage CUBO
22, 3 (2020)

history as well as future data. As a result the existence analysis of the HFRIGDE (1.1) involves

both anticipation and retardation information of the state variable. In a nutshell, the HFRIGDE

(1.1) is a nonlinear problem with anticipation and retardation.

The HFRIGDE (1.1) is a mixed linear and quadratic perturbation of second type obtained

by multiplying the unknown function under Caputo derivative with a scalar function a together

with a subtraction of the term containing unknown function and dividing by a nonlinearity f. The

classification of the different types of perturbations of a differential equation is given in Dhage [6].

When hj ≡ h on J∞ × R × R, the HFRIGDE (1.1) reduces to the nonlinear ordinary quadratic

Caputo fractional differential equation,

CD
q
t0

[

a(t)x(t) − h(t,x(t),x(η(t)))

f(t,x(t),x(θ(t)))

]

= g(t,x(t),x(γ(t))), t ∈ J∞,

x(t0) = x0,











(1.2)

which again, when hj ≡ 0, includes the class of the nonlinear quadratic Caputo fractional differ-

ential equations

CD
q
t0

[

a(t)x(t)

f(t,x(t),x(θ(t)))

]

= g(t,x(t),x(γ(t))), t ∈ J∞,

x(t0) = x0,











(1.3)

as a special case. The HFRIGDE (1.2) is new to the literature whereas the HFRIGDE (1.3) is

studied in Dhage [18] for existence and attractivity of the solutions on unbounded interval J∞.

When f(t,x,y) = 1 and g(t,x,y) = g(t,x) for all (t,x,y) ∈ J∞ × R × R, we obtain the following

Caputo fractional differential equation,

CD
q
t0
[a(t)x(t)] = g(t,x(t)), t ∈ J∞,

x(t0) = x0 ∈ R.

}

(1.4)

The equation (1.4) is studied in Dhage [17] for existence, uniqueness and asymptotic attrac-

tivity and stability of solutions via classical fixed point theory.

We note that when q = 1, the hybrid fractional differential equations (in short HFRDEs)

(1.2), (1.3) and (1.4) reduce to the ordinary nonlinear hybrid differential equations,

d

dt

[

a(t)x(t) − h(t,x(t),x(η(t)))

f(t,x(t),x(θ(t)))

]

= g(t,x(t),x(γ(t))), t ∈ J∞,

x(t0) = x0,











, (1.5)

d

dt

[

a(t)x(t)

f(t,x(t),x(θ(t)))

]

= g(t,x(t),x(γ(t))), t ∈ J∞,

x(t0) = x0,











(1.6)



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 329

and
d

dt
[a(t)x(t)] = g(t,x(t)), t ∈ J∞,

x(t0) = x0 ∈ R,







(1.7)

which are discussed in Dhage [17], Dhage [18] and [15] respectively. The hybrid differential equation

(1.7) also includes the nonlinear differential equation treated in Burton and Furumochi [4] as the

special case. Therefore the existence and attractivity results of this paper include the similar results

for the ordinary nonlinear hybrid classical and fractional differential equations (1.2) through (1.7)

as special cases.

Now we state a couple of well-known results fractional calculus which are helpful in transform-

ing the Caputo fractional differential equations into Riemann-Louville fractional integral equations

and vice versa.

Lemma 1.1 (Kilbas et al. [27]). Suppose that x ∈ Cn(J,R) and q ∈ (n − 1,n), n ∈ N. Then, the

general solution of the fractional differential equation

cD
q
t0
x(t) = 0

is given by

x(t) = c0 + c1(t − t0) + c2(t − t0)
2 + · · · + cn−1(t − t0)

n−1

for all t ∈ J, where ci, i = 0,1, . . . ,n − 1 are constants and C
n(J,R) is the space of n times

continuously differentiable real-valued functions defined on J = [a,b].

Lemma 1.2. (Kilbas et al. [27, page 96]) Let x ∈ Cn(J,R) and q > 0. Then, we have

I
q
t0

(

CD
q
t0
x(t)

)

= x(t) −

n−1
∑

k=0

x(k)(t0)

k !
(t − t0)

k = x(t) +

n−1
∑

k=0

ck(t − t0)
k

for all t ∈ J = [a,b], where n − 1 < q ≤ n, n = [q] + 1 and c0, . . . ,cn−1 are constants.

The converse of the above lemma is not true. It is mentioned in Kilbas et al. [27, page 95]

that if q > 0 and x ∈ C(J,R), then CD
q
t0

(

I
q
t0
x(t)

)

= x(t) for all t ∈ J = [a,b], however it has been

proved recently in Cohen and Salem [1, 2] that it is not true for any continuous function on J.

Remark 1.1. The conclusion of the above Lemmas 1.1 and 1.2 also remains true if we replace the

function spaces Cn([a,b],R) and C([a,b],R) with the function spaces BCn(J∞,R) and BC(J∞,R)

respectively.

2 Auxiliary Results

Let X be a non-empty set and let T : X → X. An invariant point under T in X is called a

fixed point of T , that is, the fixed points are the solutions of the functional equation T x = x. Any



330 Bapurao C. Dhage CUBO
22, 3 (2020)

statement asserting the existence of fixed point of the mapping T is called a fixed point theorem

for the mapping T in X. The fixed point theorems are obtained by imposing the conditions on T

or on X or on both T and X. By experience, better the mapping T or X, we have better fixed

point principles. As we go on adding richer structure to the non-empty set X, we derive richer

fixed point theorems useful for applications to different areas of mathematics and particularly

to nonlinear differential and integral equations. Below we give some fixed point theorems useful

in establishing the attractivity and ultimate positivity of the solutions for HFRIGDE (1.1) on

unbounded intervals. Before stating these results we give some preliminaries.

Definition 2.1 (Dhage [8, 9, 10]). An upper semi-continuous and nondecreasing function ψ :

R+ → R+ satisfying ψ(0) = 0 is called a D-function on R+. Let X be an infinite dimensional

Banach space with the norm ‖ · ‖. A mapping T : X → X is called D-Lipschitz if there is a

D-function ψT : R+ → R+ satisfying

‖T x − T y‖ ≤ ψT (‖x − y‖) (2.1)

for all x,y ∈ X.

If ψT (r) = k r, k > 0, then T is called Lipschitz with the Lipschitz constant k. In particular, if

k < 1, then T is called a contraction on X with the contraction constant k. Further, if ψT (r) < r

for r > 0, then T is called nonlinear D-contraction and the function ψT is called D-function

of T on X. There do exist D-functions and the commonly used D-functions are ψT (r) = k r and

φ(r) =
r

1 + r
, etc. (see Banas and Dhage [3] and the references therein).

Definition 2.2. An operator T on a Banach space X into itself is called totally bounded if for any

bounded subset S of X, T (S) is a relatively compact subset of X. If T is continuous and totally

bounded, then it is called completely continuous on X.

The operator theoretic technique is a powerful method often times used in the analysis of

different types of nonlinear equations. Our essential tool used in the chapter is the following hybrid

fixed point theorem of Dhage [9, 16] for a quadratic operator equation involving three operators in

a Banach algebra X which uses arguments from analysis and topology. See also Dhage [6, 7, 9, 16]

and Dhage and O’Regan [22] for some related results and applications.

Theorem 2.1 (Dhage fixed point theorem [9, 16]). Let S be a non-empty, closed convex and

bounded subset of the Banach algebra X and let A,C : X → X and B : S → X be three

operators such that

(a) A and C are D-Lipschitz with D-functions ψA and ψC respectively,

(b) B is completely continuous,



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 331

(c) MB ψA(r) + ψC(r) < r, where MB = ‖B(S)‖ = sup{‖Bx‖ : x ∈ S}, and

(d) x = AxBy + Cx =⇒ x ∈ S for all y ∈ S.

Then the operator equation AxBx + Cx = x has a solution in S.

The above hybrid fixed point theorem of Dhage is a fifth important operator theoretic tech-

nique or tool that used in the subject of nonlinear analysis in line with Banach, Schauder, Kras-

noselskii and Dhage (see [23],[5]). The nonlinear alternatives related to Dhage fixed point theorem,

Theorem 2.1 on the lines of Leray-Schauder and Schafer are also available in the literature (see

Dhage [7, 8, 9, 10] and references therein), however the present version is more convenient to apply

in the theory of nonlinear hybrid differential equations. A collection of a good number of applicable

fixed point theorems may be found in the monographs of Granas and Dugundji [23], Deimling [5],

Dhage [16] and the references therein. In the following section we give different types of char-

acterizations of the solutions for nonlinear fractional integrodifferential equations on unbounded

intervals of the real line.

3 Characterizations of Solutions

We seek solutions of the HFRIGDE (1.1) in the function space BC(J∞,R) of continuous and

bounded real-valued functions defined on J∞. Define a standard supremum norm ‖ · ‖ and a

multiplication “ · ” in BC(J∞,R) by

‖x‖ = sup
t∈J∞

|x(t)|

and

(x · y)(t) = (xy)(t) = x(t)y(t), t ∈ J∞.

Clearly, BC(J∞,R) becomes a Banach algebra w.r.t. the above norm and the multiplication.

Let A,B,C : BC(J∞,R) → BC(J∞,R) be three continuous operators and consider the following

operator equation in the Banach algebra BC(J∞,R),

Ax(t) Bx(t) + Cx(t) = x(t) (3.1)

for all t ∈ J∞. Below we give different characterizations of the solutions for the operator equation

(3.1) in the function space BC(J∞,R).

Definition 3.1. We say that solutions of the operator equation (3.1) are locally attractive if

there exists a closed ball Br(x0) in the space BC(J∞,R) for some x0 ∈ BC(J∞,R) such that for

arbitrary solutions x = x(t) and y = y(t) of equation (3.1) belonging to Br(x0) we have that

lim
t→∞

(x(t) − y(t)) = 0. (3.2)



332 Bapurao C. Dhage CUBO
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In the case when the limit (3.2) is uniform with respect to the set Br(x0), i.e., when for each

ε > 0 there exists T > 0 such that

|x(t) − y(t)| ≤ ǫ (3.3)

for all x,y ∈ Br(x0) being solutions of ((3.1) and for t ≥ T, we will say that solutions of equation

(3.1) are uniformly locally attractive on J∞.

Definition 3.2. A solution x = x(t) of equation (3.1) is said to be globally attractive if (3.2)

holds for each solution y = y(t) of (3.1) in BC(J∞,R). In other words, we may say that solu-

tions of the equation (3.1) are globally attractive if for arbitrary solutions x(t) and y(t) of (3.1)

in BC(J∞,R), the condition (3.2) is satisfied. In the case when the condition (3.2) is satisfied

uniformly with respect to the space BC(J∞,R), i.e., if for every ǫ > 0 there exists T > 0 such that

the inequality (3.2) is satisfied for all x,y ∈ BC(J∞,R) being the solutions of (3.1) and for t ≥ T,

we will say that solutions of the equation (3.1) are uniformly globally attractive on J∞.

Remark 3.1. Let us mention that the details of the global attractivity of solutions may be found

in a recent paper of Hu and Yan [26] while the concepts of uniform local and global attractivity (in

the above sense) may be found in Banas and Dhage [3], Dhage [10, 12, 13] and references therein.

Now we introduce the new concept of local and global ultimate positivity of the solutions for

the operator equation (3.1) in the space BC(J∞,R).

Definition 3.3 (Dhage [11]). A solution x of the equation (3.1) is called locally ultimately

positive if there exists a closed ball Br(x0) in the space BC(J∞,R) for some x0 ∈ BC(J∞,R)

such that x ∈ Br(0) and

lim
t→∞

[

|x(t)| − x(t)
]

= 0. (3.4)

In the case when the limit (3.4) is uniform with respect to the solution set of the operator

equation (3.1) in BC(J∞,R), i.e., when for each ε > 0 there exists T > 0 such that

| |x(t)| − x(t)| ≤ ǫ (3.5)

for all x being solutions of (3.1) in BC(J∞,R) and for t ≥ T, we will say that solutions of equation

(3.1) are uniformly locally ultimately positive on J∞.

Definition 3.4 (Dhage [13]). A solution x ∈ BC(J∞,R) of the equation (3.1) is called globally

ultimately positive if (3.4) is satisfied. In the case when the limit (3.5) is uniform with respect

to the solution set of the operator equation (3.1) in BC(J∞,R), i.e., when for each ε > 0 there

exists T > 0 such that (3.5) is satisfied for all x being solutions of (3.1) in in BC(J∞,R) and for

t ≥ T, we will say that solutions of equation (3.1) are uniformly globally ultimately positive

on J∞.



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 333

Finally, we have the the following characterization of the asymptotic stability of the solution

of the equation (3.1) on J∞.

Definition 3.5. A solution of the equation (3.1) is called asymptotically stable to t-axis or zero

if limt→ x(t) = 0. Again, x is called uniformly asymptotically stable to zero if for ǫ > 0 there

exists a real number T ≥ t0 such that |x(t)| ≤ ǫ for all t ≥ T.

Remark 3.2. We note that global attractivity implies the local attractivity and uniform global

attractivity implies the uniform local attractivity of the solutions for the operator equation (3.1)

on J∞. Similarly, global ultimate positivity implies local ultimate positivity of the solutions for

the operator equation (3.1) on an unbounded interval J∞. However, the converse of the above two

statements may not be true.

4 Attractivity and Positivity Results

Now, in this section, we discuss the attractivity results for the ordinary hybrid functional fractional

integrodifferential equation (1.1) on J∞. We need the following definition in the sequel.

Definition 4.1. A function β : J∞ × R × R → R is called Carathéodory if

(i) the map t 7→ β(t,x,y) is measurable for each x,y ∈ R, and

(ii) the map (x,y) 7→ β(t,x,y) is jointly continuous for each t ∈ J∞.

The following lemma is often used in the study of nonlinear differential equations (see Granas

et al. [24] and references therein).

Lemma 4.1 (Carathéodory). Let β : J∞ × R× R → R be a Carathéodory function. Then the map

(t,x,y) 7→ β(t,x,y) is jointly measurable. In particular the map t 7→ β(t,x(t),y(t)) is measurable

on J∞ for all x,y ∈ C(J∞,R).

We need the following hypotheses in the sequel.

(A1) The function f is continuous and there exists a function ℓ ∈ BC(J∞,R+) and a constant

K > 0 such that

∣

∣f(t,x1,x2) − f(t,x1,x2)
∣

∣ ≤
ℓ(t) max{|x1 − x2|, |x2 − y2|}

K + max{|x1 − x2|, |x2 − y2|}

for all t ∈ J∞ and x1,x2,y1,y2 ∈ R. Moreover, sup
t∈J∞

ℓ(t) = L.

(A2) The function t 7→ |f(t,0,0)| is bounded with bound F .



334 Bapurao C. Dhage CUBO
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(B1) The function g is Carathéodory and bounded on J∞ × R × R with bound Mg.

(C1) The functions hj are continuous and there exist a functions ℓj ∈ BC(J∞,R+) and a constants

Kj > 0 such that

∣

∣hj(t,x1,x2) − hj(t,x1,x2)
∣

∣ ≤
ℓj(t) max{|x1 − x2|, |x2 − y2|}

Kj + max{|x1 − x2|, |x2 − y2|}

for all t ∈ J∞ and x1,x2,y1,y2 ∈ R, where j = 1, . . . ,m. Moreover, sup
t∈J∞

ℓj(t) = Lj.

(C2) The function t 7→ |hj(t,0,0)| is bounded with bound Hj.

(D1) The pulling function a satisfies lim
t→∞

a(t)tq = 0 = lim
t→∞

a(t)tαj for each j = 1, . . . ,m.

Remark 4.1. If a ∈ CRB(J∞), then a ∈ BC(J∞,R+) and so the number ‖a‖ = supt∈J∞ a(t)

exists. Again, since the hypothesis (D1) holds, the function w : R+ → R+ defined by the expression

w(t) = a(t)tq is continuous on J∞ and satisfies the relation lim
t→∞

w(t) = 0. So the number W =

supt≥t0 w(t) exists. Similarly, the function wj : R+ → R+ defined by the expression wj(t) = a(t)t
αj

is continuous on J∞ and satisfies the relation lim
t→∞

wj(t) = 0 for each j = 1, . . . ,m. Hence, the

number Wj = supt≥t0 wj(t) exists for each j = 1, . . . ,m.

The following lemma is useful in the sequel.

Lemma 4.2. If for any function h ∈ L1(J∞,R), the function x ∈ BC(J∞,R) is a solution of the

HFRIGDE

CD
q
t0

[

a(t)x(t) −
∑m

j=1 I
αj hj(t,x(t),x(η(t))

f(t,x(t),x(θ(t)))

]

= h(t), t ∈ J∞, (4.1)

and

x(0) = x0, (4.2)

then x satisfies the hybrid fractional integral equation (in short HFRIE)

x(t) =
[

f(t,x(t),x(θ(t)))
]

(

c0 a(t) +
a(t)

Γq

∫ t

t0

(t − s)q−1h(s)ds

)

+ a(t)

m
∑

j=1

Iαjhj(t,x(t),x(η(t))

(4.3)

for all t ∈ J∞, where c0 =
a(t0)x0

f(t0,x0,x0)
and x0 6= 0.

Proof. Let h ∈ L1(J∞,R). Assume first that x is a solution of the HFRIGDE (4.1)-(4.2) defined

on J∞ and x0 6= 0. We apply the Riemann-Liouville fractional integration I
q
t0

of fractional order

q from t0 to t on both sides of the HFRIGDE (4.1). Then, by an application Lemma 1.2, the

HFRIGDE (4.1)-(4.2) is transformed into the HFRIE (4.3) on J∞.



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 335

Definition 4.2. A solution x ∈ BC(J∞,R) of the FRIE (4.3) is called a mild solution of the

HFRIGDE (4.1)-(4.2) defined on J∞.

In the following we shall deal with the mild solution of the HFRIGDE (1.1) on unbounded

interval J∞ of the real line R. Our main existence and global attractivity result is as follows.

Theorem 4.1. Assume that the hypotheses (A1) - (A2), (B1), (C1) - (C2) and (D1) hold. Further,

assume that

(m + 1)· max

{

L
(
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

,
L1 W1
Γ(α1)

, . . . ,
Lm Wm
Γ(αm)

}

≤ min{K,K1, . . . ,Km}.

(4.4)

Then the HFRIGDE (1.1) has a mild solution and mild solutions are uniformly globally attractive

defined on J∞.

Proof. Now, using Lemma 4.2, it can be shown that the mild solution x of the HFRIGDE (1.1) is

equivalent to the nonlinear hybrid fractional integral equation (in short HFRIE)

x(t) =
[

f(t,x(t),x(θ(t)))
]

(

c0 a(t) +
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

)

+ a(t)

m
∑

j=1

Iαjhj(t,x(t),x(η(t))

(4.5)

for all t ∈ J∞, where c0 =
a(t0)x0

f(t0,x0,x0)
. Set X = BC(J∞,R) and define a closed ball Br(0) in X

centered at origin of radius r given by

r =
(

L + F
)

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

+

m
∑

j=1

Lj + Hj
Γ(αj)

Wj.

Define three operators A and C on X and B on Br(0) by

Ax(t) = f(t,x(t),x(θ(t))), t ∈ J∞, (4.6)

Bx(t) = c0a(t) +
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds, t ∈ J∞ (4.7)

and

Cx(t) = a(t)

m
∑

j=1

Iαj hj(t,x(t),x(η(t)), t ∈ J∞. (4.8)

Then the HFRIE (4.5) is transformed into the operator equation as

Ax(t) Bx(t) + Cx(t) = x(t), t ∈ J∞. (4.9)



336 Bapurao C. Dhage CUBO
22, 3 (2020)

We show that the operators A, B and C satisfy all the conditions of Theorem 2.1 on BC(J∞,R).

First we we show that the operators A, B and C define the mappings A,C : X → X and

B : Br(0) → X. Let x ∈ X be arbitrary. Obviously, Ax is a continuous function on J∞. We

show that Ax is bounded on J∞. Thus, if t ∈ J∞, then we obtain:

|Ax(t)| = |f(t,x(t),x(θ(t)))|

≤ |f(t,x(t),x(θ(t))) − f(t,0,0)| + |f(t,0,0)|

≤ ℓ(t)
max{|x(t)|, |x(θ(t))|}

K + max{|x(t)|, |x(θ(t))|}
+ F

≤ L + F.

Therefore, taking the supremum over t,

‖Ax‖ ≤ L + F = N.

Thus Ax is continuous and bounded on J∞. As a result Ax ∈ X. Again, we have

∣

∣Cx(t)
∣

∣ ≤

∣

∣

∣

∣

∣

∣

a(t)

m
∑

j=1

Iαjhj(t,x(t),x(η(t))) − a(t)

m
∑

j=1

Iαj hj(t,0,0)

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

a(t)
m
∑

j=1

Iαj hj(t,0,0)

∣

∣

∣

∣

∣

∣

≤ a(t)
m
∑

j=1

Iαj
∣

∣h(t,x(t),x(η(t))) − h(t,0,0)
∣

∣ + a(t)
m
∑

j=1

Iαj
∣

∣hj(t,0,0)
∣

∣

≤ a(t)

m
∑

j=1

Iαj
ℓj(t) max{|x(t)| , |x(η(t))|}

Kj + max{|x(t)| , |x(η(t))|}
+ a(t)

m
∑

j=1

IαjHj

≤ a(t)

m
∑

j=1

Iαj
ℓj(t) ‖x‖

Kj + ‖x‖
+ a(t)

m
∑

j=1

IαjHj

≤ a(t)

m
∑

j=1

IαjLj + a(t)

m
∑

j=1

IαjHj

≤

m
∑

j=1

Lj
Γ(αj)

Wj +

m
∑

j=1

Hj
Γ(αj)

Wj

≤

m
∑

j=1

Lj + Hj
Γ(αj)

Wj

for all t ∈ t∞. Taking the supremum over t as t → ∞, we obtain

‖Cx‖ ≤

m
∑

j=1

Lj + Hj
Γ(αj)

Wj.

As a result (Cx) is continuous and bounded on J∞. Hence, Cx ∈ X. Similarly, it can be shown

that Bx ∈ X and in particular, A,C : X → X and B : Br(0) → X. We show that A is a Lipschitz



CUBO
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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 337

on X. Let x,y ∈ X be arbitrary. Then, by hypothesis (A1),

‖Ax − Ay‖ = sup
t∈J∞

|Ax(t) − Ay(t)|

≤ sup
t∈J∞

ℓ(t)
max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|}

K + max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|}

≤
L‖x − y‖

K + ‖x − y‖

= ψA(‖x − y‖)

for all x,y ∈ X. This shows that A is a D-Lipschitz on X with D-function ψA(r) =
Lr

K + r
.

Similarly, by hypothesis (C1). we have

‖Cx − Cy‖ = sup
t∈J∞

|Cx(t) − Cy(t)|

≤ sup
t∈J∞

a(t)

m
∑

j=1

Iαj
∣

∣hj(t,x(t),x(η(t))) − hj(t,y(t),y(η(t))
∣

∣

≤ sup
t∈J∞

a(t)

m
∑

j=1

Iαj
ℓj(t) max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|}

Kj + max{|x(t) − y(t)|, |x(θ(t)) − y(θ(t))|}

≤ sup
t∈J∞

a(t)

m
∑

j=1

Iαj
Lj ‖x − y‖

Kj + ‖x − y‖

≤

m
∑

j=1

Lj Wj
Γ(αj)

‖x − y‖

Kj + ‖x − y‖

≤

m
∑

j=1

Wj
Γ(αj)

·
Lj ‖x − y‖

Kj + ‖x − y‖

≤ m ·
max

{

L1 W1
Γ(α1)

, . . . , Lm Wm
Γ(αm)

}

‖x − y‖

min{K1, . . . ,Km} + ‖x − y‖

This shows that C is a D-Lipschitz on X with D-function ψC(r) given by

ψC(r) = m ·
max

{

L1 W1
Γ(α1)

, . . . , Lm Wm
Γ(αm)

}

r

min{K1, . . . ,Km} + r
.

Next, we shows that B is a completely continuous operator on Br(0). First, we show that B

is continuous on Br(0). To do this, let us fix arbitrarily ǫ > 0 and let {xn} be a sequence of points



338 Bapurao C. Dhage CUBO
22, 3 (2020)

in Br(0) converging to a point x ∈ Br(0). Then we get:

|(Bxn)(t) − (Bx)(t)|

≤
a(t)

Γq

∫ t

t0

(t − s)q−1|g(s,xn(s),xn(γ(s))) − g(s,x(s),x(γ(s)))|ds

≤
a(t)

Γq

∫ t

t0

(t − s)q−1[|g(s,xn(s),xn(γ(s)))| + |g(s,x(s),x(γ(s)))|]ds

≤ 2Mg
a(t)

Γq

∫ t

t0

(t − s)q−1 ds

=
2Mg
Γq

· w(t),

where, w(t) = a(t)tq.

Hence, by virtue of hypothesis (D1), we infer that there exists a T > 0 such that w(t) ≤ ǫ for

t ≥ T . Thus, for t ≥ T , from the estimate (3.3) we derive that

|(Bxn)(t) − (Bx)(t)| ≤
2Mg
Γq

ǫ as n → ∞.

Furthermore, let us assume that t ∈ [t0,T ]. Then, by dominated convergence theorem, we

obtain the estimate:

lim
n→∞

Bxn(t) = lim
n→∞

[

c0a(t) +
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,xn(s),xn(γ(s)))ds

]

= c0a(t) +
a(t)

Γq

∫ t

t0

(t − s)q−1
[

lim
n→∞

g(s,xn(s),xn(γ(s)))
]

ds

= c0a(t) +
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

= Bx(t)

for all t ∈ [t0,T ]. Moreover, it can be shown as below that {Bxn} is an equicontinuous sequence

of functions in X. Now, following the arguments similar to that given in Granas et al. [23], it is

proved that B is a a continuous operator on Br(0).

Next, we show that B is a compact operator on Br(0). To finish, it is enough to show that

every sequence {Bxn} in B(Br(0)) has a Cauchy subsequence. Now, proceeding with the earlier

arguments it is proved that

‖Bxn‖ ≤ |c0| ‖a‖ +
MfW

Γq
= r

for all n ∈ N. This shows that {Bxn} is a uniformly bounded sequence in B(Br(0)).

Next, we show that {Bxn} is also a equicontinuous sequence in B(Br(0)). Let ǫ > 0 be given.

Since limt→∞ w(t) = 0, there is a real number T1 > t0 ≥ 0 such that |w(t)| <
ǫ

8Mf/Γ(q)
for all



CUBO
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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 339

t ≥ T1. Similarly, since lim
t→∞

a(t) = 0, for above ǫ > 0, there is a real number T2 > t0 ≥ 0 such

that |a(t)| <
ǫ

8|c0|
for all t ≥ T2. Thus, if T = max{T1,T2}, then

|w(t)| <
ǫ

8Mf/Γ(q)
and |a(t)| <

ǫ

8|c0|
(4.10)

for all t ≥ T . Let t,τ ∈ J∞ be arbitrary. If t,τ ∈ [t0,T ], then we have

|Bxn(t) − Bxn(τ)|

≤ |c0| |a(t) − a(τ)|

+

∣

∣

∣

∣

a(t)

Γq

∫ t

t0

(t − s)q−1f(s,x(s))ds −
a(τ)

Γq

∫ τ

t0

(τ − s)q−1f(s,x(s))ds

∣

∣

∣

∣

≤ |c0| |a(t) − a(τ)|

+

∣

∣

∣

∣

a(t)

Γq

∫ t

t0

(t − s)q−1f(s,x(s))ds −
a(τ)

Γq

∫ t

t0

(τ − s)q−1f(s,x(s))ds

∣

∣

∣

∣

+

∣

∣

∣

∣

a(τ)

Γq

∫ t

t0

(τ − s)q−1f(s,x(s))ds −
a(τ)

Γq

∫ τ

t0

(τ − s)q−1f(s,x(s))ds

∣

∣

∣

∣

≤ |c0| |a(t) − a(τ)|

+
Mf
Γq

∫ t

t0

∣

∣a(t)(t − s)q−1 − a(τ)(τ − s)q−1
∣

∣ ds

+
Mf
Γq

∣

∣

∣

∣

∫ t

τ

∣

∣

∣
a(τ)(τ − s)q−1

∣

∣

∣
ds

∣

∣

∣

∣

≤ |c0| |a(t) − a(τ)|

+
Mf
Γq

∫ T

t0

∣

∣a(t)(t − s)q−1 − a(τ)(τ − s)q−1
∣

∣ ds

+
Mf ‖a‖

Γq
|(τ − t)q| .

Since the functions t 7→ a(t) and t 7→ a(t)(t − s)q−1 are continuous on compact [t0,T ], they

are uniformly continuous there. Therefore, by the uniform continuity, for above ǫ we have the real

numbers δ1 > 0 and δ2 > 0 depending only on ǫ such that

|t − τ| < δ1 =⇒ |a(t) − a(τ)| <
ǫ

9|c0|

and

|t − τ| < δ2 =⇒
∣

∣a(t)(t − s)q−1 − a(τ)(τ − s)q−1
∣

∣ <
ǫ

9MfT
/

Γq
.

Similarly, choose the real number δ3 =

(

ǫ

9Mf ‖a‖
/

Γ(q)

)1/q

> 0 so that

|t − τ| < δ3 =⇒ |(t − τ)
q| <

ǫ

9Mf‖a‖
/

Γ(q)
.



340 Bapurao C. Dhage CUBO
22, 3 (2020)

Let δ4 = min{δ1,δ2,δ3}. Then

|t − τ| < δ4 =⇒ |Bxn(t) − Bxn(τ)| <
ǫ

3

for all n ∈ N. Again, if t,τ > T , then we have a δ5 > 0 depending only on ǫ such that

|Bxn(t) − Bxn(τ)|

≤ |c0| |a(t) − a(τ)| +
a(t)

Γq

∣

∣

∣

∣

∫ t

t0

(t − s)q−1f(s,xn(s))ds

∣

∣

∣

∣

+
a(τ)

Γq

∣

∣

∣

∣

∫ τ

t0

(τ − s)q−1f(s,xn(s))ds

∣

∣

∣

∣

≤
∣

∣c0
[

|a(t)| + |a(τ)|
]

+
Mf
Γ(q)

[

w(t) + w(τ)
]

<
ǫ

2
< ǫ

for all n ∈ N whenever |t − τ| < δ5. Similarly, if t,τ ∈ R+ with t < T < τ, then we have

|Bxn(t) − Bxn(τ)| ≤ |Bxn(t) − Bxn(T)| + |Bxn(T) − Bxn(τ)|.

Take δ = min{δ4,δ5} > 0 depending only on ǫ. Therefore, from the above obtained estimates,

it follows that

|Bxn(t) − Bxn(T)| <
ǫ

2
and |Bxn(T) − Bxn(τ)| <

ǫ

2

for all n ∈ N whenever |t − τ| < δ. As a result, |Bxn(t) − Bxn(τ)| < ǫ for all t,τ ∈ J∞ and for all

n ∈ N whenever |t − τ| < δ. This shows that {Bxn} is a equicontinuous sequence in X. Now an

application of Arzelà-Ascoli theorem yields that {Bxn} has a uniformly convergent subsequence

on the compact subset [t0,T ] of J∞. Without loss of generality, call the subsequence to be the

sequence itself. We show that {Bxn} is Cauchy in X. Now |Bxn(t) − Bx(t)| → 0 as n → ∞ for all

t ∈ [t0,T ]. Then for given ǫ > 0 there exists an n0 ∈ N such that

sup
t0≤t≤T

a(t)

Γq

∫ t

t0

(t − s)q−1
∣

∣f(s,xm(s)) − f(s,xn(s))
∣

∣ ds <
ǫ

2

for all m,n ≥ n0. Therefore, if m,n ≥ n0, then we have

‖Bxm − Bxn‖

= sup
t0≤t<∞

∣

∣

∣

∣

a(t)

Γq

∫ t

t0

(t − s)q−1
∣

∣f(s,xm(s)) − f(s,xn(s))
∣

∣ ds

∣

∣

∣

∣

≤ sup
t0≤t≤T

∣

∣

∣

∣

a(t)

Γq

∫ t

t0

(t − s)q−1
∣

∣f(s,xm(s)) − f(s,xn(s))
∣

∣ ds

∣

∣

∣

∣

+ sup
t≥T

∣

∣

∣

∣

a(t)

Γq

∫ t

t0

(t − s)q−1
[

∣

∣f(s,xm(s))
∣

∣ +
∣

∣f(s,xn(s))
∣

∣

]

ds

∣

∣

∣

∣

< ǫ.



CUBO
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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 341

This shows that {Bxn} ⊂ B(Br(0)) ⊂ X is Cauchy. Since X is complete, {Bxn} converges to

a point in X. As B(Br(0)) is closed, we have that {Bxn} converges to a point in B(Br(0)). Hence

B(Br(0)) is relatively compact and consequently B is a continuous and compact operator on Br(0)

into itself.

Next, we estimate the value of the constant MB of the hypothesis (c) of the Theorem 2.1. By

definition of MB, one has

‖B(Br(0))‖ = sup{‖Bx‖ : x ∈ Br(0)}

= sup

{

sup
t∈J∞

|Bx(t)| : x ∈ Br(0)

}

≤ sup
x∈Br(0)

{

sup
t∈J∞

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
|a(t)|

+
1

Γq
· sup
t∈J∞

|a(t)|

∫ t

t0

(t − s)q−1|g(s,x(s),x(γ(s)))|ds

}

≤
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

Mg
Γq

· sup
t∈J∞

a(t)

∫ t

t0

(t − s)q−1 ds

≤
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

Mg
Γq

· sup
t∈J∞

a(t)tq

≤
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq
= MB.

Thus,

‖Bx‖ ≤
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq
= MB

for all x ∈ Br(0). Hence, we have

MBψA(r) + ψC(r)

≤

L

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

r

K + r

+ m ·
max

{

L1 W1
Γ(α1)

, . . . , Lm Wm
Γ(αm)

}

r

min{K1, . . . ,Km} + r

≤ (m + 1) ·
max

{

L
(
∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣ ‖a‖ +
MgW
Γq

)

, L1 W1
Γ(α1)

, . . . , Lm Wm
Γ(αm)

}

r

min{K,K1, . . . ,Km} + r

< r

for r > 0, because

(m + 1)· max

{

L
(
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

,
L1 W1
Γ(α1)

, . . . ,
Lm Wm
Γ(αm)

}

≤ min{K,K1, . . . ,Km}

Therefore, hypothesis (c) of Theorem 2.1 is satisfied.



342 Bapurao C. Dhage CUBO
22, 3 (2020)

Next, let y ∈ Br(0) be arbitrary and let x = AxBy + Cx. Then,

|x(t)| ≤ |Ax(t)| |By(t)| + |Cx(t)|

≤ ‖Ax‖ ‖By‖ + ‖Cx‖

≤ ‖A(X)‖ ‖B(Br(0))‖ + ‖C(X)‖

≤
(

L + F
)

MB +
m
∑

j=1

Lj + Hj
Γ(αj)

Wj

≤
(

L + F
)

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

+

m
∑

j=1

Lj + Hj
Γ(αj)

Wj

for all t ∈ J∞. Therefore, we have:

‖x‖ ≤
(

L + F
)

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

+

m
∑

j=1

Lj + Hj
Γ(αj)

Wj = r.

This shows that x ∈ Br(0) and hypothesis (c) of Theorem 2.1 is satisfied. Now we apply

Theorem 2.1 to the operator equation AxBx + Cx = x to yield that the HFRIGDE (1.1 ) has a

mild solution on J∞. Moreover, the mild solutions of the HFRIGDE (1.1) are in Br(0). Hence,

mild solutions are global in nature.

Finally, let x,y ∈ Br(0) be any two mild solutions of the HFRIGDE (1.1) on J∞. Then, from



CUBO
22, 3 (2020)

Existence and Attractivity Theorems for Nonlinear Hybrid . . . 343

(4.5) we obtain

|x(t) − y(t)| ≤

∣

∣

∣

∣

∣

[

f(t,x(t),x(θ(t)))
]

×

×

(

a(t0)x0a(t)

f(t0,x0,x0)
+
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

)

−
[

f(t,y(t),y(θ(t)))
]

×

×

(

a(t0)x0a(t)

f(t0,x0,x0)
+
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,y(s),y(γ(s)))ds

)

∣

∣

∣

∣

∣

+ sup
t∈J∞

a(t)

m
∑

j=1

Iαj
∣

∣hj(t,x(t),x(η(t))) − hj(t,y(t),y(η(t))
∣

∣

≤

∣

∣

∣

∣

∣

[

f(t,x(t),x(θ(t))) − f(t,y(t),y(θ(t)))
]

×

(

a(t0)x0a(t)

f(t0,x0,x0)
+
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

)

∣

∣

∣

∣

∣

+
∣

∣ f(t,y(t),y(θ(t)))
∣

∣ ×

×

∣

∣

∣

∣

(

a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds − g(s,y(s),y(γ(s)))ds

)
∣

∣

∣

∣

+ sup
t∈J∞

a(t)
m
∑

j=1

Iαj
Lj ‖x − y‖

Kj + ‖x − y‖

≤
∣

∣f(t,x(t),x(θ(t))) − f(t,y(t),y(θ(t)))
∣

∣ ×

×

∣

∣

∣

∣

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
|a(t)| +

MgW

Γq
w(t)

)
∣

∣

∣

∣

+ 2
[

|f(t,x(t),x(θ(t))) − f(t,0,0)| + |f(t,0,0)|
]MgW

Γq
w(t)

+

m
∑

j=1

wj(t)

Γ(αj)
·
Lj ‖x − y‖

Kj + ‖x − y‖

≤ ℓ(t)
|x(t) − y(t)|

K + |x(t) − y(t)|

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

+
2 MgW

Γq

[

ℓ(t) max
{

|x(t)| , |x(θ(t))|
}

K + max
{

|x(t)| , |x(θ(t))|
} + F

]

w(t)

+

m
∑

j=1

Lj wj(t)

Γ(αj)

≤

L

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

|x(t) − y(t)|

K + |x(t) − y(t)|

+
2 MgW

Γq
(L + F)w(t) +

m
∑

j=1

Lj wj(t)

Γ(αj)
. (4.11)



344 Bapurao C. Dhage CUBO
22, 3 (2020)

Taking the limit superior as t → ∞ in the above inequality (4.11) yields, limt→∞ |x(t)−y(t)| =

0. Therefore, there is a real number T > 0 such that |x(t) − y(t)| < ǫ for all t ≥ T . Consequently,

the mild solutions of HFRIGDE (1.1) are uniformly globally attractive on J∞. This completes the

proof.

Remark 4.2. The conclusion of Theorem 4.1 also remains true under if we replace the hypotheses

(A1), (A2), (C1) and (C2) with the following modified conditions:

(A′1) The function f is continuous and there exists a D-function ψf ∈ D such that

∣

∣f(t,x1,x2) − f(t,y1,y2)
∣

∣ ≤ ψf
(

max{|x1 − y1|, |x2 − y2|}
)

for all t ∈ J∞ and x1,x2,y1,y2 ∈ R.

(A′2) The function f is bounded on J∞ × R × R with bound Mf.

(C′1) The functions h
′
js are continuous and there exist D-functions ψhj ∈ D such that

∣

∣hj(t,x1,x2) − hj(t,y1,y2)
∣

∣ ≤ ψhj
(

max{|x1 − y1|, |x2 − y2|}
)

for all t ∈ J∞ and x1,x2,y1,y2 ∈ R, where j = 1, . . . ,m.

(C′2) The functions hj are bounded on J∞ × R × R with bound Mhj .

Theorem 4.2. Assume that the hypotheses (A′1) - (A
′
2), (B1), (C

′
1) - (C

′
2) and (D1) hold. Fur-

thermore, assume that

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
‖a‖ +

MgW

Γq

)

ψf (r) +

m
∑

j=1

Wj
Γ(αj)

ψhj (r) < r, r > 0. (4.12)

Then the HFRIGDE (1.1) has a mild solution and mild solutions are uniformly globally attractive

defined on J∞.

Proof. The proof is similar to Theorem 4.1 and hence we omit the details.

Theorem 4.3. Assume that the hypotheses (A1) - (A2), (B1), (C1) - (C2) and (D1) hold. Then

the HFRIGDE (1.1) has a mild solution and mild solutions are uniformly globally attractive and

ultimately positive defined on J∞.

Proof. By Theorem 4.1, the HFRIGDE (1.1) has a global mild solution in the closed ball Br(0),

where the radius r is given as in the proof of Theorem 4.1, and the mild solutions are uniformly

globally attractive on J∞. We know that for any x,y ∈ R, one has the inequality,

|x| |y| = |xy| ≥ xy,



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22, 3 (2020)

Existence and Attractivity Theorems for Nonlinear Hybrid . . . 345

and therefore,
∣

∣|xy| − (xy)
∣

∣ ≤ |x|
∣

∣|y| − y
∣

∣ +
∣

∣|x| − x
∣

∣ |y| (4.13)

for all x,y ∈ R. Now, for any mild solution x of the HFRIGDE (1.1) in Br(0), one has

∣

∣|x(t)| − x(t)
∣

∣

=

∣

∣

∣

∣

∣

∣

∣
f(t,x(t),x(θ(t)))

∣

∣

∣
×

×
∣

∣

∣

(

a(t0)x0a(t)

f(t0,x0,x0)
+
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

)

∣

∣

∣

−
[

f(t,x(t),x(θ(t)))
]

×

×

(

a(t0)x0a(t)

f(t0,x0,x0)
+
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

)
∣

∣

∣

∣

+

∣

∣

∣

∣

m
∑

j=1

Iαj
∣

∣hj(t,x(t),x(η(t)))
∣

∣ −

m
∑

j=1

Iαj hj(t,x(t),x(η(t)))

∣

∣

∣

∣

≤

∣

∣

∣

∣

∣

∣

∣

∣
f(t,x(t),x(θ(t)))

∣

∣

∣

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
−

a(t0)x0
f(t0,x0,x0)

)

a(t)

∣

∣

∣

∣

∣

+
∣

∣

∣
f(t,x(t),x(θ(t)))

∣

∣

∣
×

×

∣

∣

∣

∣

∣

∣

∣

∣

∣

a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

∣

∣

∣

∣

−
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

∣

∣

∣

∣

∣

+
∣

∣

∣
|f(t,x(t),x(θ(t)))

∣

∣ − f(t,x(t),x(θ(t)))
∣

∣

∣

×

∣

∣

∣

∣

a(t0)x0a(t)

f(t0,x0,x0)
+
a(t)

Γq

∫ t

t0

(t − s)q−1g(s,x(s),x(γ(s)))ds

∣

∣

∣

∣

+

m
∑

j=1

Iαj
∣

∣

∣

∣

∣hj(t,x(t),x(η(t)))
∣

∣ − hj(t,x(t),x(η(t)))
∣

∣

∣

≤

[

4(L + F)
∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣

]

a(t) +

[

4(L + F)
Mg
Γq

]

w(t)

+ 2

m
∑

j=1

Hj
Γ(αj)

wj(t) (4.14)

for all t ∈ J∞.

Taking the limit superior as t → ∞ in the above inequality (4.14), we obtain the estimate that

lim
t→∞

∣

∣|x(t)| − x(t)
∣

∣ = 0. Therefore, there is a real number T > 0 such that
∣

∣ |x(t)| − x(t)
∣

∣ ≤ ǫ for

all t ≥ T . Hence, mild solutions of the HFRIGDE (1.1) are uniformly globally attractive as well

as ultimately positive defined on J∞. This completes the proof.



346 Bapurao C. Dhage CUBO
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Theorem 4.4. Assume that the hypotheses (A1) - (A2) and (B1) hold. Then the HFRDE (1.1) has

a mild solution and mild solutions are uniformly globally attractive, uniformly ultimately positive

and uniformly asymptotically stable to zero defined on J∞.

Proof. By Theorems 4.1 and 4.2, the HFRIGDE (1.1) has a global mild solution in the closed

ball Br(0), where the radius r is given as in the proof of Theorem 4.1, and the mild solutions are

uniformly globally attractive and uniformly ultimately positive on J∞. Now, for any mild solution

x ∈ Br(0), we have from (4.10),

|x(t)| ≤ (L + F)

(

∣

∣

∣

a(t0)x0
f(t0,x0,x0)

∣

∣

∣
a(t) +

Mg
Γq

w(t)

)

+
m
∑

j=1

Lj + Hj
Γ(αj)

wj(t).

Taking the limit superior as t → ∞ in the above inequality yields that limt→∞ |x(t)| = 0.

Therefore, for ǫ > 0 there exists a real number T ≥ t0 such that |x(t)| < ǫ whenever t ≥ T .

Consequently, the mild solution x is a uniformly asymptotically stable to zero defined on J∞. This

completes the proof.

Example 4.1 Let J∞ = R+ = [0,∞) ⊂ R. Given a pulling function a(t) = e
t ∈ CRB(R+),

consider the following nonlinear hybrid fractional Caputo differential equation with the mixed

arguments of anticipation and retardation,

CD
q
0









etx(t) −
t

t2 + 1
I3/2

(

|x(t)| + |x(3t)|

4 + |x(t)| + |x(3t)|

)

1 +
1

t2 + 1

(

|x(t)| + |x(2t)|

2 + |x(t)| + |x(2t)|

)









=
e−t log

(

1 + |x(t)| + |x(t/2)|
)

2 + |x(t)| + |x(t/2)|
, t ∈ R+,

x(0) = 0,



























(4.15)

for all t ∈ R+, where
CD

q
0 is the Caputo fractional derivative of fractional order 0 < q ≤ 1.

Here, a(t) = et, θ(t) = 2t, η(t) = 3t, γ(t) =
t

2
for t ∈ R+ and hence θ(0) = 0 = η(0). Next,

α = 3/2 and the functions f : R+ × R × R → R+ \ {0} and g,h : R+ × R × R → R are defined by

f(t,x,y) = 1 +
1

t2 + 1

[

|x| + |y|

2 + |x| + |y|

]

,

h(t,x,y) =
t

t2 + 1

[

|x| + |x|

4 + |x| + |x|

]

and

g(t,x,y) =
e−t log(|x| + |y|)

1 + |x| + |y|
.

Clearly, the function f is continuous and bounded real function on R+ × R × R with bound

Mf = 2 and in particular, F = 1. Now, it can be shown as in Banas and Dhage [3] that the function

f satisfies the hypothesis (A1) with ℓ(t) =
1

t2 + 1
and K = 1. So we have L = 1. Furthermore,



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 347

the function h is also continuous and bounded on J∞ × R × R with bound Mh = 1. Next, the

function h satisfies the hypothesis (C1) with the function ℓh(t) =
t

t2 + 1
so that we have Lh =

1

2
and Kh = 4. Again, the function g is continuous and bounded on J∞ × R × R and therefore,

satisfies the hypotheses (B1) with Mg = 1. Next, we have

lim
t→∞

w(t) = lim
t→∞

e−ttq = 0 = lim
t→∞

e−tt3/2 = lim
t→∞

wh(t)

and so the hypothesis (D1) is satisfied. Now, ‖a‖ = supt∈R+ e
−t = 1,W = supt∈R+ e

−t tq = 1 and

Wh = 1. Finally, it is verified that the the functions a, f, g and h satisfy the condition (4.4)

of Theorem 4.1. Consequently, the HFRIGDE (4.15) has a mild solution and mild solutions are

globally uniformly attractive, uniformly ultimately positive and uniformly asymptotically stable

to zero defined on R+. In particular, the HFRIGDE

CD
2/3
0









etx(t) −
t

t2 + 1
I3/2

(

|x(t)| + |x(3t)|

4 + |x(t)| + |x(3t)|

)

1 +
1

t2 + 1

(

|x(t)| + |x(2t)|

2 + |x(t)| + |x(2t)|

)









=
e−t log

(

1 + |x(t)| + |x(t/2)|
)

2 + |x(t)| + |x(t/2)|
, t ∈ R+,

x(0) = 0,



























has a mild solution and mild solutions are globally uniformly attractive, uniformly ultimately

positive and uniformly asymptotically stable to zero defined on R+.

Remark 4.3. Finally, we remark that the ideas of this paper may be extended with appropriate

modifications to a more general hybrid fractional integrdifferential equation with Caputo fractional

derivative,

CD
q
t0













a(t)x(t) −

m
∑

j=1

Iαjhj(t,x(t),x(η1(t)), . . . ,x(ηn))

f(t,x(θ1(t)), · · · ,x(θn(t)))













= g(t,x(γ1(t)), · · · ,x(γn(t))), t ∈ J∞,

x(t0) = x0 ∈ R,















































(4.16)

where CD
q
t0

is the Caputo fractional derivative of fractional order 0 < q ≤ 1, Γ is a Euler’s gamma

function, f : J∞ × R × ...(n times) × R → R \ {0}, g,hj : J∞ × R × ...(n times) × R → R are

continuous and θi,γi : J∞ → J∞ are continuous functions which are respectively anticipatory and

retardatory, that is, θi(t) ≥ t and γi(t) ≤ t for all t ∈ J∞ with θi(t0) = t0 = ηi(t0) for i = 1, . . . ,n.

Remark 4.4. If g is assumed to be continuous function on J∞ × R × R, then the attractivity

and existence results for the HFRIGDE (1.1) may be obtained via another approach of using

measure of noncompactness. In that case we need to construct a handy tool for the measure of

noncompactness which is not the case with the present approach in the qualitative study of such

nonlinear fractional integrodifferential equations. See the details of this procedure that appears in

Banas and Dhage [3], Hu and Yan [26], Dhage [11, 14] and the references therein.



348 Bapurao C. Dhage CUBO
22, 3 (2020)

5 The Conclusion

From the foregoing discussion, it is clear that the pulling functions and the hybrid fixed point

theorems are very much useful for proving the existence theorems as well as characterizing the

mild solutions of different types of nonlinear fractional integrodifferential equations on unbounded

intervals of the real line when the nonlinearity is not necessarily continuous. The choices of the

pulling function and the fixed point theorem depends upon the situations and the circumstances

of the nonlinearities involved in the nonlinear problem. The clever selection of the fixed point

theorems yields very powerful existence results as well as different characterizations of the nonlinear

fractional differential equations. In this article, we have been able to prove in Theorems 4.1, 4.2,

4.3 and 4.4 the existence as well as global attractivity, ultimate positivity and asymptotic stability

of the mild solutions for a quadratic type of nonlinear hybrid fractional differential equation (1.1)

on the unbounded interval J∞ = [t0,∞) of right half of the real line R+, however, other nonlinear

fractional integrodifferential equations can be treated in the similar way for these and some other

characterizations such as monotonic global attractivity, monotonic asymptotic attractivity and

monotonic ultimate positivity etc. of the mild solutions on unbounded intervals of the real line. It

is known that several real world phenomena in physics and chemistry such as growth and decay

of the radioactive elements continue for a very long period of time and the existence results of

the type proved in this paper may be applicable for the situation to understand the behavior of

the process after a sufficient lapse of time. In a forthcoming paper, it is proposed to discuss the

global asymptotic and monotonic attractivity of the mild solutions for nonlinear hybrid fractional

integrodifferential equations involving three nonlinearities via classical and applicable hybrid fixed

point theory.

Acknowledgement The author is thankful to the referees for giving some suggestions for

the improvement of this paper.



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Existence and Attractivity Theorems for Nonlinear Hybrid . . . 349

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	Introduction
	 Auxiliary Results
	Characterizations of Solutions
	 Attractivity and Positivity Results
	The Conclusion