On a functional-integral equation with deviating arguments

Applied Mathematics and Computation 246 (2014) 64–70

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On a functional-integral equation with deviating arguments Mohamed Abdalla Darwish a,⇑, Bothayna S. Kashkari a, Kishin Sadarangani b a b

Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira, Baja, 35017 Las Palmas de Gran Canaria, Spain

a r t i c l e

i n f o

Keywords: Functional integral equation Nonnegative and nonincreasing solutions Deviating arguments Measure of noncompactness

a b s t r a c t We study the solvability of a functional-integral equation with deviating arguments, where our investigations take place in the space of Lebesgue integrable functions on an unbounded interval. In this space, we show that our functional-integral equation has at least one nonnegative and nonincreasing solution. The proof of our main result is based on a suitable combination of the technique associated with measures of noncompactness (in both the weak and the strong sense) and the Darbo fixed point. In the end, we conclude an example to illustrate our abstract results. Ó 2014 Published by Elsevier Inc.

1. Introduction The nonlinear integral equations occur in solving several problems arising in engineering, physics and economics. In the theory of nonlinear analysis, one of the most frequently investigated nonlinear integral equations is the Hammerstein integral equation (cf. [2,8,13,19,20]),

xðtÞ ¼ f ðtÞ þ

Z

kðt; sÞgðs; xðsÞÞds;

t 2 I;

I

where I  R is an interval (bounded or unbounded). In this paper we study the problem of existence of nonnegative and nonincreasing solutions for the functional integral equations of Hammerstein type with deviating arguments

 Z xðtÞ ¼ f t; k

1

 kðt; sÞgðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞds ;

ð1:1Þ

0

t 2 Rþ ¼ ½0; 1Þ and k  0. Throughout k : Rþ  Rþ ! Rþ ; g : Rþ  Rn ! Rþ and /i : Rþ ! Rþ are functions which satisfy special hypotheses (i ¼ 1; 2; . . . ; n). Also, special cases for Eq. (1.1) were investigated in connection with some applications of such a kind of problems in economics, engineering and physics. Also, Special cases for Eq. (1.1) appear in problems considered in the theory of partial differential equations [6,9–11,16]. The aim of this paper is to prove the existence of nonincreasing solutions of Eq. (1.1) in the space of Lebesgue integrable functions on an unbounded interval. Our proof depends on a suitable combination of the Darbo fixed point theorem and the technique associated with both measures of weak noncompactness and measures of noncompactness in the strong sense.

⇑ Corresponding author. E-mail addresses: [email protected] (M.A. Darwish), [email protected] (B.S. Kashkari), [email protected] (K. Sadarangani). http://dx.doi.org/10.1016/j.amc.2014.08.020 0096-3003/Ó 2014 Published by Elsevier Inc.

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2. Notation and auxiliary facts This section is devoted to collecting some definitions and results which will be needed further on. Let L1 ðXÞ denote the space of Lebesgue integrable functions on the measurable set X with the standard norm

kyk ¼

Z

jyðtÞjdt:

X

Let us assume that I  R is a given interval, bounded or unbounded. A function f ðt; xÞ ¼ f : I  R ! R satisfies the Carathéodory conditions if it is measurable in t for any x 2 R and continuous in x for almost all t 2 I. Then, to every function x ¼ xðtÞ which is measurable on the interval I, we may assign the function ðFxÞðtÞ ¼ f ðt; xðtÞÞ; t 2 I. The function Fx is measurable and the operator F defined in such a way is called the superposition operator generated by the function f, see [1] and references therein. The necessary and sufficient condition guaranteeing that the superposition operator F is a self-continuous map present in the following theorem. The case when I is a bounded interval was proved by Krasnosel’skii [17], while the case when I is an unbounded interval was proved by Appell and Zabrejko [1]. Theorem 2.1. The superposition operator F generated by the function f maps the space L1 ðIÞ continuously into itself if and only if

jf ðt; xÞj 6 aðtÞ þ bjxj for all t 2 I and all x 2 R, where a 2 L1 ðIÞ and b  0 is a constant. Next, we recall some basic facts concerning measures of noncompactness, [3,5]. Let us assume that E is an infinite dimensional Banach space with norm k:k and zero element h. Denote by ME the family of all nonempty and bounded subsets of E and by NE ; NW E its subfamilies consisting of all relatively compact and relatively weakly compact sets, respectively. For a subset X of R, the symbols X; X W stand for the closure and the weak closure of a set X, respectively. The symbol coX will denote the convex closed hull (with respect to the norm topology) of a set X. We denote by Bðx; rÞ the ball centered at x and of radius r. We write Br instead of Bðh; rÞ. Definition 2.2. A mapping conditions:

l : ME ! Rþ is said to be a measure of noncompactness in E if it satisfies the following

(1) The family kerl ¼ fX 2 ME : lðXÞ ¼ 0g is nonempty and kerl  NE . (2) X  Y ) lðXÞ 6 lðYÞ. (3) lðXÞ ¼ lðcoXÞ ¼ lðXÞ. (4) lðkX þ ð1  kÞYÞ 6 k lðXÞ þ ð1  kÞ lðYÞ for 0 6 k 6 1. (5) If X n 2 ME , X n ¼ X n ; X nþ1  X n for n ¼ 1; 2; 3; . . . and if limn!1 lðX n Þ ¼ 0 then \1 n¼1 X n – /. The family kerl described above is called the kernel of the measure of noncompactness

l.

Definition 2.3. A mapping l : ME ! Rþ is said to be a measure of weak noncompactness in E if it satisfies the following conditions: (2)–(4) of Definition 2.2 and the following two conditions: 0

(1) The family kerl ¼ fX 2 ME : lðXÞ ¼ 0g is nonempty and kerl  NW E . 0 1 (5) If X n 2 ME , X n ¼ X W n ; X nþ1  X n for n ¼ 1; 2; 3; . . . and if limn!1 lðX n Þ ¼ 0 then \n¼1 X n – /. Definition 2.4. [3] Let X be a bounded subset of E. The Hausdorff measure of noncompactness

v is defined by

vðXÞ ¼ inffe > 0 : there is a finite subset of Y of E such thatX  Y þ Be g: The first important and convenient measure of weak noncompactness was defined by De Blasi [12] in the following way

bðXÞ ¼ inffe > 0 : there is a weakly compact subset Y of E such that X  Y þ Be g: The measures of noncompactness v and b have some interesting properties. They play a significant role in nonlinear analysis and find many applications (cf. [3,5,12,18]). We will make use of the following fixed point theorem due to Darbo [14]. To quote this theorem, we need the following definition. Definition 2.5. Let C be a nonempty subset of a Banach space E and let P : C ! E be a continuous operator which transforms bounded sets into bounded ones. We say that P satisfies the Darbo condition (with constant j  0) with respect to a measure of noncompactness l if for any bounded subset X of C we have

lðPXÞ 6 jlðXÞ: If P satisfies the Darbo condition with

j < 1 then it is called a contraction operator with respect to l.

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Theorem 2.6 [14]. Let Q be a nonempty, bounded, closed and convex subset of the space E and let

P:Q !Q be a continuous mapping which is a contraction with respect to the measure of noncompactness point in the set Q.

l. Then P has at least one fixed

Remark 2.7 [15]. Theorem 2.6 remains valid if l is a measure of weak noncompactness and if we assume that P is a weakly continuous operator, i.e., mapping weakly convergent sequences into itself. Also, we recall a theorem concerning the compactness in measure of a subset X of L1 ðIÞ, see [4]. Theorem 2.8. Let X be a bounded subset of L1 ðIÞ consisting of all functions which are a.e. nondecreasing (or nonincreasing) on the interval I. Then X is compact in measure. We recall the formula for a measure of weak noncompactness in L1 ðRþ Þ which appears in [7]. Let us fix a bounded subset X of L1 ðRþ Þ and define

  Z cðXÞ ¼ lim sup sup jxðtÞjdt : X  Rþ ; e!0

x2X

 measðXÞ 6 e

X

and

 Z dðXÞ ¼ lim sup T!1

1

jxðtÞjdt : x 2 X

 :

T

Put

cðXÞ ¼ cðXÞ þ dðXÞ:

ð2:1Þ

Then we have the following results, see [7] and references therein. Theorem 2.9. The function c is a measure of weak noncompactness in the space L1 ðRþ Þ such that

bðXÞ 6 cðXÞ  2bðXÞ; where b denotes the De Blasi measure of noncompactness. Moreover, cðBL1 ðRþ Þ Þ ¼ 2. Theorem 2.10. Let X be a nonempty, bounded and compact in measure subset of the space L1 ðRþ Þ. Then

vðXÞ 6 cðXÞ  2vðXÞ: Theorem 2.11. Let X be a nonempty, bounded and compact in measure subset of the space L1 ðRþ Þ. If P : X ! L1 ðRþ Þ is a continuous map then it is weakly sequentially continuous on X. By combining all the above established facts (Theorems 2.8, 2.10, 2.11 and Theorem 2.6) we can deduce the following result. Theorem 2.12. Let Q be a nonempty, bounded, closed and convex and compact in measure subset of the space L1 ðRþ Þ and let

P:Q !Q be a continuous mapping which is a contraction with respect to the measure of weak noncompactness c. Then P has at least one fixed point in the set Q.

3. Main results In this section we will study Eq. (1.1) assuming that the following hypotheses are satisfied: ðh1 Þ The function f ðt; xÞ ¼ f : Rþ  R ! R satisfies the Carathéodory conditions, and there exist a function a1 2 L1 ðRþ Þ and a constant b1 > 0 such that

jf ðt; xÞj 6 a1 ðtÞ þ b1 jxj for t 2 Rþ and for x 2 R. Moreover, f : Rþ  Rþ ! Rþ and f ðt; xÞ is nonincreasing with respect to t and nondecreasing with respect to x on Rþ . ðh2 Þ The function kðt; sÞ ¼ k : Rþ  Rþ ! R satisfies Carathéodory conditions and there exist two functions p : Rþ ! Rþ and q : Rþ ! Rþ , where p continuous and integrable over Rþ and q is bounded with jkðt; sÞj 6 pðtÞqðsÞ, for t; s 2 Rþ . Moreover, k is nonincreasing with respect to t.

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ðh3 Þ The function gðt; x1 ; x2 ; . . . ; xn Þ ¼ g : Rþ  Rn ! Rþ satisfies the Carathéodory conditions, and there exist a function a2 2 L1 ðRþ Þ and a constant b2 > 0 such that

gðt; x1 ; x2 ; . . . ; xn Þ 6 a2 ðtÞ þ b2

n X

jxi j

i¼1

for t 2 Rþ and for x 2 R. ðh4 Þ The function /i : Rþ ! Rþ ; i ¼ 1; 2; . . . ; n, is increasing and absolutely continuous. Moreover, there is a constant M i > 0; i ¼ 1; 2; . . . ; n, such that /0i ðtÞ  M i for almost all t  0 and /0i is bounded on Rþ by K i ; i ¼ 1; 2; . . . ; n. P 1 n 1 ðh5 Þ kb1 b2 kKk < M, where M ¼ . i¼1 M i

Remark 3.1. In ðh1 Þ and ðh2 Þ, when we say that the function uðt; xÞ ¼ u : Rþ  R ! R satisfies the Carathéodory conditions, we mean that the function t ! uðt; xÞ is measurable for any x 2 R and the function x ! uðt; xÞ is continuous for almost all t 2 Rþ . Now, we are in a position to state and prove our main result. Theorem 3.2. Let the hypotheses ðh1 Þ  ðh5 Þ be satisfied. Then Eq. (1.1) has at least one solution x 2 L1 ðRþ Þ which is a.e. nonnegative and nonincreasing on Rþ . For the proof of Theorem 3.2, we will use the following lemmas. Lemma 3.3. Suppose that / : Rþ ! Rþ is an increasing function such that /ðtÞ  M for almost t  0, where M > 0. Then the operator M / defined on L1 ðRþ Þ by ðM / xÞðtÞ ¼ xð/ðtÞÞ for any t 2 Rþ maps the space L1 ðRþ Þ continuously into itself. Proof. In fact, for x 2 L1 ðRþ Þ

Z

kM / xk ¼

1

jðM/ xÞðtÞj dt 6

0

Z 0

1

jxð/ðtÞÞj

/0 ðtÞ 1 dt ¼ M M

Z

1

jxðuÞj du ¼

0

1 kxk: M

Since M / is linear, we obtain the desired result. h Lemma 3.4. Under hypothesis ðh2 Þ the linear Fredholm integral operator

ðKxÞðtÞ ¼

Z

1

kðt; sÞxðsÞds

0

transforms the space L1 ðRþ Þ continuously into itself. Proof. In fact, for x 2 L1 ðRþ Þ

  Z 1 Z 1 Z 1 Z 1 Z 1 Z 1



jðKxÞðtÞj dt ¼ kðt; sÞxðsÞ ds

dt 6 jkðt; sÞj jxðsÞj ds dt 6 pðtÞqðsÞ jxðsÞj ds dt

0 0 0 0 0 0 0 Z 1  Z 1  Z 1 Z 1 Z 1 ¼ pðtÞ qðsÞ jxðsÞj ds dt 6 pðtÞR jxðsÞj ds dt 6 pðtÞRkxk dt ¼ Rkxk kpk;

kKxk ¼

Z

1

0

0

0

0

0

where R ¼ supfqðtÞ : t 2 Rþ g. Therefore, since K is a linear operator, K transforms L1 ðRþ Þ continuously into itself.

h

Now, we are ready to give the proof of Theorem 3.2. Proof of Theorem 3.2. For a given x 2 L1 ðRþ Þ, we consider the operator F defined by

 Z ðF xÞðtÞ ¼ f t; k

1

 kðt; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds :

0

Notice that the operator F is given by the following composition of operators n Y L1 ðRþ Þ ð/1 ; /2 ; . . . ; /n Þ L1 ðRþ Þ Hg L1 ðRþ Þ kK L1 ðRþ Þ Hf L1 ðRþ Þ; ! ! i¼1 ! !

where ð/1 ; /2 ; . . . ; /n Þ is defined by x ! ðM/1 x; M /2 x; . . . ; M /n xÞ which, in virtue of Lemma 3.3, is continuous, Hf and Hg are the superposition operators generated by the functions f and g which are continuous in virtue of hypotheses ðh1 Þ and ðh3 Þ and Theorem 2.1 and kK is continuous by Lemma 3.4. Therefore, F transforms L1 ðRþ Þ continuously into itself. Moreover, taking into account our hypotheses, we have

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 Z 1 



f t; k

dt kðt; sÞ gðs; xð/ ðsÞÞ; xð/ ðsÞÞ; . . . ; xð/ ðsÞÞÞ ds 1 2 n



0 0

Z 1

 Z 1



a1 ðtÞ þ kb1

kðt; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds

dt 6

kF xk ¼

Z

1

0

6 ka1 k þ kb1

Z

1

Z

0

0

1

kðt; sÞ a2 ðsÞ ds dt þ kb1 b2

Z

0

1

0

n Z X 6 ka1 k þ kb1 kKkka2 k þ kb1 b2 kKk i¼1

Z

1

kðt; sÞ

n X jxð/i ðsÞÞj ds dt

0

i¼1

1

jxð/i ðtÞÞj dt 6 ka1 k þ kb1 kKkka2 k þ kb1 b2 kKk

0

Z 1 n X 1 jxð/i ðtÞÞj/0i ðtÞ dt M i 0 i¼1

Z 1 n X 1 ¼ ka1 k þ kb1 kKkka2 k þ kb1 b2 kKk jxðtÞj dt ¼ ka1 k þ kb1 kKkka2 k þ kb1 b2 kKkM 1 kxk; M i 0 i¼1

where kKk denotes the norm of the linear Fredholm integral operator appearing in Lemma 3.4 and M 1 ¼

Pn

1 i¼1 M i .

From the last estimate we deduce that the operator F transforms the ball Br into itself for r ¼ ðka1 k þ kb1 kKkka2 kÞ =ð1  kb1 b2 kKkM 1 Þ, thanks to hypothesis ðh5 Þ. In what follows let us define D to be the subset of the ball Br consisting of all functions which are a.e. nonnegative and nonincreasing on Rþ . Then the set D is nonempty, bounded, closed and convex, see [4]. Moreover, D is compact in measure, thanks to hypotheses ðh1 Þ and ðh2 Þ. Next, we will prove that F maps the set D into itself. In fact, in virtue of hypothesis ðh1 Þ, it is clear that if x  0 then F x  0. On the other hand, for x 2 D and t 1 6 t2 and, taking into account that f ðt; xÞ is nonincreasing with respect to t and nondecreasing with respect to x (hypothesis ðh1 Þ) and kðt; sÞ is nonincreasing with respect to t (hypothesis ðh2 Þ) we have

  Z 1 ðF xÞðt2 Þ ¼ f t2 ; k kðt2 ; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds 0   Z 1 6 f t1 ; k kðt 2 ; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds 0   Z 1 6 f t1 ; k kðt 1 ; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds ¼ ðF xÞðt1 Þ: 0

This proves that F maps the set D into itself. We now show that the operator F is a contraction with respect to the measure of weak compactness c. For this purpose take a nonempty subset X of D and fix e > 0. Further, let us take a nonempty subset X of Rþ such that X is measurable and measðXÞ 6 e. Then for an arbitrary x 2 X and in view of our hypotheses we have

Z

jðF xÞðtÞj dt 6

Z 

X

X

Z 6

Z

a1 ðtÞ þ kb1

0

a1 ðtÞ dt þ kb1

X

Z 6 X

Z 6 X

¼

Z X

Z

1



kðt; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds

dt

Z Z X

1

kðt; sÞ a2 ðsÞ ds dt þ kb1 b2

Z Z

0

X

a1 ðtÞ dt þ kb1 kKa2 kL1 ðXÞ þ kb1 b2

1

kðt; sÞ

0

n X jxð/i ðsÞÞj ds dt i¼1

n X

kKxð/i ÞkL1 ðXÞ

i¼1

a1 ðtÞ dt þ b1 kkKkX a1 ðtÞ dt þ kb1 kKkX

Z X

Z X

n X kxð/i ÞkL1 ðXÞ i¼1

a2 ðtÞ dt þ kb1 b2 kKkX

n Z X i¼1

jxð/i ðtÞÞj dt

X

Z n X 1 jxð/i ðtÞÞj/0i ðtÞ dt Mi X X X i¼1 Z Z Z n X 1 6 a1 ðtÞ dt þ kb1 kKk a2 ðtÞ dt þ kb1 b2 kKk jxðv Þj dv M i / i ðXÞ X X i¼1  Z  Z Z n X 1 6 a1 ðtÞ dt þ kb1 kKk a2 ðtÞ dt þ kb1 b2 kKk  sup sup jxðtÞj dt : X  Rþ ; measðXÞ 6 Hi e ; Mi X X X i¼1 6

a1 ðtÞ dt þ kb1 kKkX

where

Hi ¼ supf/0i ðtÞ : t 2 Rþ g:

Z

a2 ðtÞ dt þ kb1 b2 kKkX

a2 ðtÞ dt þ kb1 b2 kKkX

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Now, taking into account the fact that

 Z  ¼ 0; lim sup jai ðtÞj dt : X  Rþ ; measðXÞ 6 e e!0

i ¼ 1; 2

X

and keeping in mind the definition of cðXÞ, it follows that

cðF XÞ 6 jcðXÞ;

ð3:1Þ 1

Pn

1

1 i¼1 Mi .

where j ¼ kb1 b2 kKkM , where M ¼ Obviously, in view of hypothesis ðh5 Þ we have that j < 1. Next, let us fix an arbitrary number T > 0. Then, taking into account our hypotheses, for an arbitrary function x 2 X we have

Z

1

jðF xÞðtÞj dt 6

T

Z

1

T

Z

Z 

a1 ðtÞ þ kb1

1

a1 ðtÞ dt þ kb1

6 T

Z T

1

kðt; sÞ a2 ðsÞ ds dt þ kb1 b2

Z

0

1

Z

T

1

n X kðt; sÞ jxð/i ðsÞÞj ds dt

0

i¼1

n X a1 ðtÞ dt þ kb1 kKa2 kL1 ð½T;1ÞÞ þ kb1 b2 kKxð/i ÞkL1 ð½T;1ÞÞ

1

a1 ðtÞ dt þ kb1 kKk½T;1Þ

T

Z

Z

a1 ðtÞ dt þ b1 kkKk½T;1Þ

T

¼

1

1

6 Z

0

Z



kðt; sÞ gðs; xð/1 ðsÞÞ; xð/2 ðsÞÞ; . . . ; xð/n ðsÞÞÞ ds

dt

T 1

6 Z

1

i¼1

Z

1

a2 ðtÞ dt þ kb1 b2 kKk½T;1Þ

T

Z

i¼1

1

a2 ðtÞ dt þ kb1 b2 kKk½T;1Þ

T

n Z X i¼1

1

jxð/i ðtÞÞj dt T

Z 1 n X 1 jxð/i ðtÞÞj/0i ðtÞ dt M i T T T i¼1 Z 1 Z 1 Z 1 n X 1 6 a1 ðtÞ dt þ kb1 kKk a2 ðtÞ dt þ kb1 b2 kKk jxðv Þj dv ; M i /i ðTÞ T T i¼1

6

1

a1 ðtÞ dt þ kb1 kKk½T;1Þ

Z

n X kxð/i ÞkL1 ð½T;1ÞÞ

1

a2 ðtÞ dt þ kb1 b2 kKk½T;1Þ

where kKk½T;1Þ denotes the norm of the linear Fredholm integral operator mapping L1 ð½T; 1ÞÞ into itself. Since /i is absolutely continuous

/i ðTÞ  /i ð0Þ ¼

Z 0

T

/0i ðtÞ dt 

Z

T

Mi dt ¼ Mi T 0

and, consequently, /i ðTÞ ! 1 as T ! 1. Therefore, from the last estimate we obtain

dðF XÞ 6 jdðXÞ;

ð3:2Þ 1

due to the fact that dðYÞ ¼ 0 for any singleton Y of L ðRþ Þ. From (3.1) and (3.2) and the definition of the measure of noncompactness c given by formula (2.1), we obtain

cðF XÞ 6 jcðXÞ:

ð3:3Þ

Now, an application of Theorem 2.12 implies that the operator F has at least one fixed point in D. This completes the proof. Now, we present an example to illustrating our results.

h

Example 3.5. Consider the following integral equation

xðtÞ ¼

1 1 þ arctan k 1 þ t2 1 þ t

Z 0

1

ets es þ

n X

! !  2 ln 1 þ xði sÞ ds ;

t 2 Rþ :

ð3:4Þ

i¼1

1 1 ts Notice that Eq. (3.4) is a particular case of Eq. (1.1), where f ðt; xÞ ¼ 1þt ; gðt; x1 ; x2 ; . . . ; xn Þ ¼ et 2 þ 1þt arctan x, kðt; sÞ ¼ e P 2 þ ni¼1 ln ð1 þ jxi jÞ and /i ðtÞ ¼ i t, for i ¼ 1; 2; . . . ; n. Next, we will check that Eq. (3.4) satisfies hypotheses of Theorem 3.2. Notice that f ðt; xÞ satisfies the Carathéodory conditions since f is continuous on Rþ  Rþ . Also, we have



jf ðt; xÞj ¼





1 1

6 1 þ 1 j arctan xj 6 1 þ jxj; þ arctan x

2 1 þ t 1þt 1 þ t2 1 þ t 1 þ t2

1 where we have used the fact that j arctan xj 6 jxj for any x 2 R. Therefore, in our case, a1 ðtÞ ¼ 1þt 2 and b1 ¼ 1. It is easily seen

that a1 2 L1 ðRþ Þ and ka1 k ¼ p2 . Moreover, it is clear that f applies Rþ  Rþ into Rþ and f is nonincreasing with respect to t and 1 nondecreasing with respect to x on Rþ since ðarctan xÞ0 ¼ 1þx 2 > 0. This proves that hypothesis ðh1 Þ of Theorem 3.2 is satisfied.

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M.A. Darwish et al. / Applied Mathematics and Computation 246 (2014) 64–70

In connection with hypothesis ðh2 Þ, notice that, since kðt; sÞ ¼ ets is continuous on Rþ  Rþ ; k satisfies the Carathéodory conditions. Moreover, jkðt; sÞj ¼ jets j ¼ et es for any s; t 2 Rþ . Therefore, pðtÞ ¼ et and qðsÞ ¼ es . It is easily seen that p is integrable over Rþ with kpk ¼ 1 and q is bounded with supfqðsÞ : s 2 Rþ g ¼ 1. Finally, it is clear that K is nonincreasing with respect to t and, consequently, hypothesis ðh2 Þ of Theorem 3.2 is satisfied. P Notice that, since gðt; x1 ; x2 ; . . . ; xn Þ ¼ et þ ni¼1 ln ð1 þ jxi jÞ; g transforms continuously Rþ  Rn into Rþ and, therefore, it satisfies the Carathéodory conditions. Moreover,

jgðt; x1 ; x2 ; . . . ; xn Þj ¼ et þ

n X

ln ð1 þ jxi jÞ 6 et þ

i¼1

n X

jxi j;

i¼1

for any t 2 Rþ and x1 ; x2 ; . . . ; xn 2 R, where we have used the fact that lnð1 þ jxjÞ 6 jxj for any x 2 R. Therefore, a2 ðtÞ ¼ et and b2 ¼ 1. Since a2 2 L1 ðRþ Þ, hypothesis ðh3 Þ of Theorem 3.2 is satisfied. 2 2 In our case, /i ðtÞ ¼ i t, for i ¼ 1; 2; . . . ; n which are increasing and absolutely continuous functions with /0i ðtÞ ¼ i > 0. 2

2

Therefore, Mi ¼ i > 0 and, moreover, /0i ðtÞ 6 i for t > 0. This proves that hypothesis ðh4 Þ of Theorem 3.2 is satisfied. P 1 n 1 Finally, since it is easily seen that kKk 6 1, if k 6 then Eq. (3.4) has at least one solution x 2 L1 ðRþ Þ which is a.e. i¼1 i2 nonnegative and nonincreasing on Rþ by Theorem 3.2. Acknowledgment This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (381/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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