existence results for nonlinear quadratic integral equations of fractional order in banach algebra

$: Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra$
RESEARCH PAPER

EXISTENCE RESULTS FOR NONLINEAR QUADRATICINTEGRAL EQUATIONS OF FRACTIONAL ORDER

IN BANACH ALGEBRA

Ahmed El-Sayed 1, Hind Hashem 1,2

Abstract

We present an existence theorem for at least one continuous solutionfor a nonlinear quadratic functional integral equation of fractional order.Also, a general quadratic integral of fractional order will be considered.

MSC 2010 : Primary 26A33; Secondary 45D05, 60G22, 33E30Key Words and Phrases : Banach algebra, fixed point theory, integral

equations, nonlinear operators

1. Introduction and Preliminaries

The theory of integral equations is rapidly developing with the helpof several tools of functional analysis, topology and fixed point theory. Inparticular, quadratic integral equations have many useful applications inproblems of the real world. For example, quadratic integral equations areoften applicable to theory of radiative transfer, kinetic theory of gases, the-ory of neutron transport, queuing theory and traffic theory. Many authorsstudied the existence of solutions for several classes of nonlinear quadraticintegral equations (see e.g. [2]-[6] and [17]-[18]). However, in most of thesepapers, the main results are realized with the help of the technique associ-ated with the measure of noncompactness.

c© 2013 Diogenes Co., Sofiapp. 816–826 , DOI: 10.2478/s13540-013-0051-6

EXISTENCE RESULTS FOR NONLINEAR QUADRATIC . . . 817

In recent years, many authors have focused on the existence of solutionof the equation

x = AxBx + Cx (1.1)

and obtained a lot of valuable results (see [7], [8], [10], [11], [12], [14], [15],[16], [19], and the references therein). These studies were mainly based onthe convexity and the closure of the bounded domain, the Schauder fixedpoint theorem in [19].

In this paper, we prove an existence theorem for the quadratic integralequation

x(t)=f(t, x(t))+g(t, x(t))∫ t

0

(t−s)α−1

Γ(α)u(s, x(s))ds, t∈J = [0, 1], α > 0.

(1.2)Let X = C(J, R) be the vector of all real-valued continuous functions onJ = [0, 1]. We equip the space X with the norm ||x|| = sup

t∈J|x(t)|.

Clearly, C(J, R) is a complete normed algebra with respect to thissupremum norm.

By a solution of the quadratic functional integral equation of fractionalorder (1.2) we mean a function x ∈ C(J, R) that satisfies Eq. (1.2).

In this section, we introduce some basic definitions and preliminaryfacts which we need in the sequel [1].

Definition 1.1. An algebra X is a vector space endowed with aninternal composition law noted by

(.) : X × X → X, (x, y) → x.y,

which is associative and bilinear.A normed algebra is an algebra endowed with a norm satisfying the

following property:For all x, y ∈ X we have

||x.y|| ≤ ||x||.||y||.A complete normed algebra is called a Banach algebra.

Definition 1.2. A mapping T : X → X is called totally boundedif T (S) is a totally bounded subset of X for any bounded subset S of X.Again a map T : X → X is completely continuous if it is continuous andtotally bounded on X. It is clearly that every compact operator is totallybounded, but the converse may not be true, however the two notions areequivalent on bounded subsets of a Banach space X.

818 A.M.A. El-Sayed, H.H.G. Hashem

Definition 1.3. Let X be a normed vector space. A mappingT : X → X is called D-Lipschitzian, if there exists a continuous andnondecreasing function φ such that

||Tx − Ty|| ≤ φD(||x − y||)for all x, y ∈ X where φ(0) = 0.

Sometimes, we call the function φD to be a D-function of the mapping Ton X. Obsviously, every Lipschitzian mapping is D-Lipschitzian. Further,if φ(r) < r, then T is called nonlinear contraction on X.

An important fixed point theorem that has been commonly used inthe theory of nonlinear integral equations is the generalization of Banachcontraction mapping principle proved in [9].

Recently B.C. Dhage in [15] proved a fixed point theorem involvingthree operators in a Banach algebra by blending the Banach fixed pointtheorem with that Shauder’s fixed point principle.

In [16], B.C. Dhage gave a proof of the fixed point theorem in [15] inthe case of Lipschtzian’s mapping. The following fixed point theorem wasproved in [1].

Theorem 1.1. Let S be a closed convex and bounded subset of aBanach algebra X and let A,C : X → X and B : S → X be threeoperators such that:

(i) A and C are D-Lipschitzian with a D-functions φA and φC respec-tively,

(ii) B is completely continuous, and,(iii) x = AxBy + Cx ⇒ x ∈ S, for all y ∈ S.Then the operator AB +C has a fixed point in S as soon as M φA(r)+

φC(r) < r, for r > 0 where M = ||B(S)||.

Remark 1.1. Since every Lipschitzian mapping is D -Lipschitzian,we obtain a theorem of Dhage in [13] as a corollary of Theorem 1.1.

2. Main Results

The goal of this section is to apply Theorem 1.1 to discuss the existenceof solutions to (1.1).Consider the following assumptions:

(i:) u : J ×R → R satisfies Caratheodory condition (i.e. measurablein t for all x ∈ R and continuous in x for almost all t ∈ J)


such that:

|u(t, x)| ≤ m(t) ∈ L1[0, 1] ∀ (t, x) ∈ J × R

and k = supt∈J

Iβ m(t) for any β ≤ α.

(ii:) f, g : J × R → R are continuous and bounded withK1 = sup

(t,x)∈J×R

|f(t, x)| K2 = sup(t,x)∈J×R

|g(t, x)| respectively.

(iii:) There exist two constants L1 and L2 satisfying

|f(t, x) − f(t, y)| ≤ L1 |x − y|and

|g(t, x) − g(t, y)| ≤ L2 |x − y|for all t ∈ J and x, y ∈ R.

Theorem 2.1. Let the assumptions (i:)-(iii:) be satisfied. Further-more, if (L1−K1)Γ(α−β+1)+L2k < K2 k, then the quadratic functionalintegral equation (1.2) has at least one solution in the space C(J, R).

P r o o f. Consider the mapping A,B and C on C(J, R) defined by:

(Ax)(t) = g(t, x(t))

(Bx)(t) =∫ t

0

(t − s)α−1

Γ(α)u(s, x(s)) ds

(Cx)(t) = f(t, x(t)).

Then the integral equations (1.2) can be written in the form:

Tx(t) = Cx(t) + Ax(t).Bx(t), (2.1)

and we shall show that A, B and C satisfy all the conditions of Theorem1.1.

Let us define a subset S of C(J, R) by

S := {x ∈ C(J, R), ||x|| ≤ r}.Obviously, S is nonempty, bounded, convex and closed subset of C(J, R).

For every x ∈ S we have

|(Tx)(t)| = |Cx(t) + Ax(t)Bx(t)| ≤ K1 + K2 Iα−β Iβ m(t)

≤ K1 + K2 k

∫ t

0

(t − s)α−β−1

Γ(α − β)ds ≤ K1 +

K2 k

Γ(α − β + 1)= r.

Then, Tx ∈ S and hence TS ⊂ S.


First, we begin by showing that C is Lipschitzian on S. To see this, letx, y ∈ S. So

|Cx(t) − Cy(t)| = |f(t, x(t)) − f(t, y(t))| ≤ L1|x(t) − y(t)|for all t ∈ J . Taking supremum over t

||Cx − Cy|| ≤ L1 ||x − y||for all x, y ∈ S. This shows that C is a Lipschitz mapping on S with theLipschitz constant L1.

In a similar way we can deduce that

||Ax − Ay|| ≤ L2 ||x − y||for all x, y ∈ S. This shows that A is a Lipschitz mapping on S with theLipschitz constant L2.

Secondly, we show that B is continuous and compact operator on S.First we show that B is continuous on S. To do this, let us fix arbitraryε > 0 and let {xn} be a sequence of points in S converging to a point x ∈ S.Then we get

|(Bxn)(t) − (Bx)(t)| ≤∫ t

0

(t − s)α−1

Γ(α)|u(s, xn(s)) − u(s, x(s))| ds

≤∫ t

0

(t − s)α−1

Γ(α)[|u(s, xn(s))| + |u(s, x(s))|] ds

≤ 2kΓ(α − β + 1)

≤ ε.

Thus|(Bxn)(t) − (Bx)(t)| → 0 as n → ∞.

Furthermore, let us assume that t ∈ J . Then, by the Lebesgue dominatedconvergence theorem, we obtain the estimate:

limn→∞(Bxn)(t) = lim

n→∞

∫ t

0

(t − s)α−1

Γ(α)u(s, xn(s)) ds

=∫ t

0

(t − s)α−1

Γ(α)u(s, x(s)) ds = (Bx)(t)

for all t ∈ J . Thus, Bxn → Bx as n → ∞ uniformly on R+ and hence B isa continuous operator on Sinto S. Now by (i:) and (ii:)

|Bxn(t)| ≤∫ t

0

(t − s)α−1

Γ(α)u(s, xn(s)) ds ≤ k

Γ(α − β + 1)

for all t ∈ J . Then ||Bxn(t)|| ≤ kΓ(α−β+1) = M for all n ∈ N . This shows

that {Bxn} is a uniformly bounded sequence in B(S).


Now, we proceed to show that it is also equicontinuous. Let t1, t2 ∈ J(without loss of generality assume that t1 < t2), then we have

|Bxn(t2) − Bxn(t1)|=

∣∣∣∣∫ t1

0

(t2 − s)α−1

Γ(α)u(s, xn(s)) ds +

∫ t2

t1

(t2 − s)α−1


−∫ t1

0

(t1 − s)α−1


∣∣∣∣≤

∣∣∣∣∫ t1

0

(t1 − s)α−1

Γ(α)u(s, xn(s)) ds +

∫ t2

t1

(t2 − s)α−1


−∫ t1

0

(t1 − s)α−1

Γ(α)u(s, x(s)) ds

∣∣∣∣ ≤∫ t2

t1

(t2 − s)α−1

Γ(α)|u(s, xn(s))| ds.

Therefore,

|Bxn(t2) − Bxn(t1)| ≤∫ t2

t1

(t2 − s)α−1

Γ(α)m(s) ds

≤ k(t2 − t1)α−β

Γ(α − β + 1).

Then, we obtain

|Bxn(t2) − Bxn(t1)| → 0 as t2 → t1.

As a consequence, |Bxn(t2) − Bxn(t1)| → 0 as t2 → t1. This shows that{Bxn} is an equicontinuous sequence in S. Now an application of theArzela-Ascoli theorem yields that {Bxn} has a uniformly convergent sub-sequence on the the compact subset J . Without loss of generality, call thesubsequence itself. We can easily show that {Bxn} is Cauchy sequence inS. Hence B(S) is relatively compact and consequently B is a continuousand compact operator on S.

Since all conditions of Theorem 1.1 are satisfied, then the operatorT = C + AB has a fixed point in S. �

3. General quadratic integral equation of fractional order

In this section, we study the general quadratic integral equation offractional order

x(t) =n∑

i=1

gi(t, x(t)).∫ t

0

(t − s)αi−1

Γ(αi)ui(s, x(s)) ds, t ∈ J, αi > 0, (3.2)

by applying the following fixed point theorem [20].


Theorem 3.1. Let n be a positive integer, and C be a nonempty,closed, convex and bounded subset of a Banach Algebra X. Assume thatthe operators Ai : X → X and Bi : C → X, i = 1, 2, ...., n, satisfy

(a) For each i ∈ {1, 2, ...., n}, Ai is D−Lipschitzian with a D−functionφi;

(b) For each i ∈ {1, 2, ...., n}, Bi is continuous and Bi(C) is precompact;

(c) For each y ∈ C, x =n∑

i=1Aix.Biy implies that x ∈ C.

Then, the operator equation x =n∑

i=1Aix.Bix has a solution provided

thatn∑

i=1

Miφi(r) < r, ∀r > 0,

where Mi = supx∈C

||Bix||, i = 1, 2, ..., n.

Eqn. (3.2) is investigated under the assumptions:(i∗) ui : J × R → R, i = 1, 2, ...., n satisfy Caratheodory condition

(i.e. measurable in t for all x ∈ R and continuous in x for almost allt ∈ J) such that:

|ui(t, x)| ≤ mi(t) ∈ L1[0, 1], i = 1, 2, ...., n ∀ (t, x) ∈ J × R

and ki = supt∈J

Iβi mi(t) for any βi ≤ αi, i = 1, 2, ...., n such that ki = 0 ∀ i,

(ii∗) gi : J × R → R, i = 1, 2, ...., n are continuous and boundedwith hi = sup

(t,x)∈J×R

|gi(t, x)|, i = 1, 2, ...., n,

(iii∗) There exist constants Li, i = 1, 2, ...., n satisfying

|gi(t, x) − gi(t, y)| ≤ Li |x − y|, i = 1, 2, ...., n

for all t ∈ J and x, y ∈ R.

Theorem 3.2. Let the assumptions (i∗)-(iii∗) be satisfied. Fur-

thermore, if

n∑i=1

Li ki

Γ(αi − βi + 1)<

n∑i=1

ki hi

Γ(αi − βi + 1), then the general qua-

dratic integral equation (3.2) has at least one solution in the space C(J, R).

P r o o f. Consider the mapping Ai and Bi on C(J, R) defined by:

(Aix)(t) = gi(t, x(t))

(Bix)(t) =∫ t

0

(t − s)αi−1

Γ(αi)ui(s, x(s)) ds.


Then the integral equation (3.2) can be written in the form:

Tx(t) =n∑

i=1

Aix(t).Bix(t) . (3.3)

We shall show that Ai and Bi satisfy all the conditions of Theorem 3.1.Let us define a subset C of C(J, R) by

C := {x ∈ C(J, R), ||x|| ≤ r}.Obviously, C is nonempty, bounded, convex and closed subset of C(J, R).

As done before in the proof of Theorem 2.1, we can get that for everyx ∈ C we have

|(Tx)(t)| ≤n∑

i=1

hiki

Γ(αi − βi + 1)= r.

Then, Tx ∈ C and hence TC ⊂ C.Easily, we can deduce that

||Aix − Aiy|| ≤ Li ||x − y||for all x, y ∈ C. This shows that Ai are Lipschitz mappings on C with theLipschitz constants Li. Also, we can prove that the operators Bi are contin-uous and compact operator on C for all t ∈ J and ||Bix(t)|| ≤ ki

Γ(αi−βi+1) =Mi for all x ∈ C.

Since all conditions of Theorem 3.1 are satisfied, then the operator

T =n∑

i=1Ai.Bi has a fixed point in C. �

4. Applications

As particular cases of Theorem 3.2 we can obtain theorems on the ex-istence of solutions belonging to the space C(J, R) for the following integralequations:

(i) Let n = 1, then we have

x(t) = g1(t, x(t)).∫ t

0

(t − s)α1−1

Γ(α1)u1(s, x(s)) ds, t ∈ J, α1 > 0.

(ii) Let n = 1 with g1(t, x(t)) = 1, then we have

x(t) =∫ t

0

(t − s)α1−1

Γ(α1)u1(s, x(s)) ds, t ∈ J, α1 > 0.

(iii) Let n = 2, then we have

x(t) = g1(t, x(t)).∫ t

0

(t − s)α1−1

Γ(α1)u1(s, x(s)) ds


+ g2(t, x(t)).∫ t

0

(t − s)α2−1

Γ(α2)u2(s, x(s)) ds,

where t ∈ J, α1, α2 > 0.(iv) Let n = 2, α1 → 1 and u1(t, x) = 1, then we have

x(t)= t g1(t, x(t)) + g2(t, x(t)).∫ t

0

(t−s)α2−1

Γ(α2)u2(s, x(s)) ds, t ∈ J, α2 > 0,

taking g(t, x(t)) = t g1(t, x(t)), we get

x(t)=g(t, x(t)) + g2(t, x(t)).∫ t

0

(t−s)α2−1

Γ(α2)u2(s, x(s)) ds, t ∈ J, α2 > 0.

This equation is studied in Section 2.

Example 4.1. Consider the following quadratic integral equation

x(t) =t2

2x(t) . I

12 [ t +

|x(t)|1 + 3|x(t)| ]

+ [√

t + 3 + |ln(x(t) + 1)| + 1] . I13 [

1 + 2t10

+|x(t)|

1 + |x(t)| ], t ∈ J.

Here g1(t, x(t)) = t2

2 x(t), g2(t, x(t)) =√

t + 3 + |ln(x(t) + 1)| + 1,

u1(t, x(t)) = t +|x(t)|

1 + 3|x(t)| and u2(t, x(t)) =1 + 2t

10+

|x(t)|1 + |x(t)| .

We can easily verify that g1, g2, u1 and u2 satisfy all the assumptions ofTheorem 3.2.

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1 Faculty of Science, Alexandria UniversityEl-Shatby, Aflaton St., Alexandria, EGYPT

e-mail: [email protected]

2 Faculty of Science, Qassim UniversityP. O. Box 6644, Buraidah – 81999, SAUDI ARABIA

e-mail: [email protected] Received: January 23, 2013

Please cite to this paper as published in:Fract. Calc. Appl. Anal., Vol. 16, No 4 (2013), pp. 816–826;DOI: 10.2478/s13540-013-0051-6

existence results for nonlinear quadratic integral equations of fractional order in banach algebra

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