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Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics

GRONWALL-TYPE LEMMAS FOR AN INTEGRAL EQUATION WITH MODIFIED ARGUMENT MARIA DOBRI¸ TOIU Abstract. In this paper one gives some integral inequalities obtained as applications of the Picard operators technique and the Gronwall lemma for a nonlinear integral equation with modi…ed argument. Two examples are also given here. Key words and phrases: integral equation with modi…ed argument, Picard operator, integral inequality. 2000 Mathematics Subject Classi…cation: 45G10, 47H10. 1:Introduction We consider the nonlinear integral equation with modi…ed argument x(t) =

Z

b

K(t; s; x(s); x(g(s)); x(a); x(b))ds + f (t);

a

t 2 [a; b];

(1)

where K 2 C ([a; b] [a; b] B4 ; B), f 2 C([a; b]; B), g 2 C([a; b]; [a; b]) and (B; +; R; j j) is a Banach space. The existence and uniqueness and some properties of the solution of this integral equation have been studied by M. Dobri¸toiu in papers [8] and [9]. In this paper we will establish some integral inequalities obtained as applications of the Picard operators technique and the Gronwall lemma. Other integral equations of this type have been studied by M.Ambro in [1], M. Dobri¸toiu in [7], I.A. Rus in [18] and [19], M.A. S ¸erban in [20], Sz. András in [2] and [3]. Other integral inequalities have been studied by A. Buic¼ a in [5], D. Bainov and P. Simeonov in [4] and M. Zima in [21]. Also, we use in this paper the results obtained by I.A. Rus in [14], [15], [16] and [17], R. Precup in [12] and [13], Gh. Coman, I. Rus, G. Pavel, I.A.Rus in [6] and D. Guo, V. Lakshmikantham and X. Liu, in [11]. 1

Maria Dobri¸toiu - Gronwall-type lemmas for an integral equation

2.Notations and preliminaries Let X be a nonempty set, d a metric on X and A : X ! X an operator. In this paper we shall use the following notations and de…nition:

A

FA n+1

: = fx 2 Xj A(x) = xg - the …xed point set of A : = A An ; A0 := 1X ; A1 := A; n 2 N:

De…nition 1 Let (X; d) be a metric space. An operator A : X ! X is Picard operator if there exists x 2 X such that: (a) FA = fx g; (b) the sequence (An (x0 ))n2N converges to x , for all x0 2 X. In section 3 we will use the Picard operators technique and the Gronwall lemma, in order to obtain some integral inequalities for the integral equation (1), which are the properties of the solution of this integral equation. In this section we need the results below (see [5], [6], [8], [9], [14], [15], [16], [17], [21]). Theorem 1 (Contraction Principle). Let (X; d) be a complete metric space and A : X ! X an -contraction ( < 1). In these conditions we have: (i) FA = fx g; (ii) An (x0 ) ! x as n ! 1; n

(iii) d(x ; An (x0 )) Let

1

d(x0 ; A(x0 )) .

be an order relation on X.

Lemma 2 (Abstract Gronwall lemma) Let (X; d; ) be an ordered metric space and A : X ! X an operator. We suppose that: (i) A is Picard operator; (ii) the operator A is increasing. 2


If we denote with xA the unique …xed point of the operator A, then (a) x

A(x) ) x

xA ;

(b) x

A(x) ) x

xA .

In section 4, two examples are given, which are applications of the results obtained in the previous section. Here, we need the following results (see [6], [8], [14]). Theorem 3 Let Q 2 Mnn (R+ ) be a matrix. The following statements are equivalent: (i) Qk

! 0 as k ! 1;

(ii) The eigenvalues k = 1; n; (iii) The matrix I

k; k

= 1; n of the matrix Q verify the condition j k j < 1,

Q is non-singular and (I

Q)

1

= I + Q + : : : + Qn + : : : :

The following theorem contain the conditions of the existence and uniqueness of the solution of the system of integral equations Z b x(t) = K(t; s; x(s); x(g(s)); x(a); x(b))ds + f (t); t 2 [a; b]; (2) a

where K : [a; b] [a; b] Rm Rm Rm Rm ! Rm , f : [a; b] ! Rm and g : [a; b] ! [a; b]. Theorem 4 We suppose that (i) K 2 C ([a; b] [a; b] g 2 C([a; b]; [a; b];

Rm

Rm

Rm

Rm ; Rm ), f 2 C([a; b]; Rm ),

(ii) there exists a matrix Q 2 Mmm (R+ ) such that kK(t; s; u1 ; u2 ; u3 ; u4 ) Q (ku1

v1 k + ku2

K(t; s; v1 ; v2 ; v3 ; v4 )k

v2 k + ku3

for all t; s 2 [a; b] and ui ; vi 2 Rm ; i = 1; 4: 3

v3 k + ku4

v4 k)


(iii) [4(b

a)Q]n ! 0 as n ! 1 .

Then the system of integral equations (2) has a unique solution x 2 C([a; b]; Rm ), and this solution can be obtained using the method of successive approximations, starting at any element x0 2 C([a; b]; Rm ). 3.Main results Using the Picard operators technique for integral equations, we have obtained the following property of the solution of the integral equation (1). Theorem 5 We suppose that: (i) K 2 C([a; b]

[a; b]

B4 ; B), f 2 C([a; b]; B) and g 2 C([a; b]; [a; b]);

(ii) K(t; s; ; ; ; ) is increasing for all t; s 2 [a; b] and f ( ) is increasing for all t 2 [a; b]; (iii) there exists L > 0 such that kK(t; s; u1 ; u2 ; u3 ; u4 ) L (ku1

v1 k + ku2

K(t; s; v1 ; v2 ; v3 ; v4 )k

v2 k + ku3

v3 k + ku4

v4 k)

for all t; s 2 [a; b], ui ; vi 2 B, i = 1; 4; (iv) 4L(b

a) < 1.

Let x 2 C([a; b]; B) be the unique solution of the integral equation (1). In these conditions (a) if x 2 C([a; b]; B) is a low-solution of the integral equation (1), then x

x:

(b) if x 2 C([a; b]; B) is an upper-solution of the integral equation (1), then x . 4

x:


Proof. We consider the operator A : C([a; b]; B) ! C([a; b]; B), de…ned by Z b K(t; s; x(s); x(g(s)); x(a); x(b))ds + f (t); t 2 [a; b]: (3) A(x)(t) := a

From conditions (iii) and (iv) it results that the operator A is an contraction with the coe¢ cient = 4L (b a) and therefore the operator A is a Picard operator. In these conditions, by the Contraction Principle, it results that the integral equation (1) has in C([a; b]; B), a unique solution, which we denote by x . From condition (ii) it results that the operator A is increasing. The conditions of the abstract Gronwall lemma being satis…ed, it results the conclusions of the theorem, i.e. x

A(x) =) x

x

x

A(x) =) x

x:

The proof is complete. Remark 1 By substitution of the conditions (i), (iii) and (iv) with appropriate conditions, which assure the existence and uniqueness of the solution of the integral equation (1) in the following Banach spaces: R, Rm , l2 (R), L2 [a; b], the conclusions of the theorem 5 are true. Now, we consider the following system of integral equations (the case B = R ) Z b x(t) = K(t; s; x(s); x(g(s)); x(a); x(b))ds + f (t); (4) m

a

where K(t; s; x(s); x(g(s)); x(a); x(b)) = = (K1 (t; s; x(s); x(g(s)); x(a); x(b)); : : : ; Km (t; s; x(s); x(g(s)); x(a); x(b))), t; s 2 [a; b], f (t) = (f1 (t); f2 (t); : : : ; fm (t)), t 2 [a; b], and a g(s) b; s 2 [a; b]. We have the following lemma: 5


Lemma 6 We suppose that: (i) K 2 C ([a; b]

[a; b]

Rm

Rm

Rm

Rm ; Rm );

(ii) f 2 C([a; b]; Rm ); (iii) g 2 C([a; b]; [a; b]); (iv) there exists a matrix Q 2 Mmm (R+ ) such that 0 1 jK1 (t; s; u1 ; u2 ; u3 ; u4 ) K1 (t; s; v1 ; v2 ; v3 ; v4 )j @ A ::::::::: jKm (t; s; u1 ; u2 ; u3 ; u4 ) Km (t; s; v1 ; v2 ; v3 ; v4 )j 0 1 ju11 v11 j + ju21 v21 j + ju31 v31 j + ju41 v41 j A; ::::::::: Q@ ju1m v1m j + ju2m v2m j + ju3m v3m j + ju4m v4m j for all t; s 2 [a; b], ui ; vi 2 Rm , i = 1; 4 .

(v) [4(b

a)Q]n ! 0 as n ! 1 .

Let x 2 C([a; b]; Rm ) be the unique solution of the system of integral equations (4). Then (a) If x 2 C([a; b]; Rm ) is a low-solution of the system of the integral equations (4), then x x i.e. x(t)

Z

b

K(t; s; x(s); x(g(s)); x(a); x(b))ds + f (t); t 2 [a; b]:

a

(b) If x 2 C([a; b]; Rm ) is an upper-solution of the system of the integral equations (4), then x x i.e. x(t)

Z

a

b

K(t; s; x(s); x(g(s)); x(a); x(b))ds + f (t); t 2 [a; b]:

Two examples are given in the following section. 6


4.Examples Example 1. We consider the integral equation with modi…ed argument Z 1 t s s+1 1 1 13 x(t) = x(s) + x + x(0) + x(1) ds + t 7 5 2 7 5 14 0

17 ; 60

(5)

where

and

; x(0); x(1) = 7t x(s) + 5s x K t; s; x(s); x s+1 2 K 2 C ([0; 1] [0; 1] R4 ), 13 f (t) = 14 t 17 , f 2 C[0; 1], 60 s+1 g(s) = 2 , g 2 C([0; 1]; [0; 1]),

s+1 2

+ 17 x(0) + 15 x(1),

x 2 C[0; 1]. We attach to this integral equation the operator A : C[0; 1] ! C[0; 1], de…ned by Z 1 s+1 s 1 13 17 t 1 x(s) + x ; (6) A(x)(t) = + x(0) + x(1) ds + t 7 5 2 7 5 14 60 0 and one knows that the set of the solutions of the integral equation (5) coincides with the set of …xed points of the operator A, i.e. with FA . The function K satis…es a Lipschitz condition with respect to the arguments x(s) and x (0), with the constant 71 . Also, the function K satis…es a Lipschitz condition with respect to the arguments x s+1 and x(1), with the constant 2 1 24 . Now, it results that the operator A is a contraction with the coe¢ cient 35 5 and therefore A is a Picard operator. By Contraction Principle, it results that the integral equation (5) has a unique solution x 2 C[0; 1]. Since the function K(t; s; ; ; ; ) is increasing for all t; s 2 [0; 1] and the function f ( ) is increasing for all t 2 [0; 1], it results that the conditions of lemma 2 are satis…ed and therefore we have the following integral inequalities: - if x 2 C[0; 1] is a low-solution of the integral equation (5), then x x ; - if x 2 C[0; 1] is an upper-solution of the integral equation (5), then x x . Example 2. Now, we consider the system of integral equations (the case B = R2 ) ( R1 x1 (t) = 0 t+2 x (s) + 2t+1 x1 (s=2) + 51 x1 (0) + 51 x1 (1) ds + 2t + 1 15 R 115 t+21 ; (7) x2 (t) = 0 21 x2 (s) + 2t+1 x2 (s=2) + 17 x2 (0) + 17 x2 (1) ds + t 21 7


where

and

K 2 C ([0; 1] [0; 1] R2 R2 R2 R2 ; R2 ), K (t; s; x(s); x(s=2); x(0); x(1)) = (K1 (t; s; x(s); x(s=2); x(0); x(1)), K2 (t; s; x(s); x(s=2); x(0); x(1))), K1 (t; s; x(s); x(s=2); x(0); x(1)) = = t+2 x (s) + 2t+1 x1 (s=2) + 15 x1 (0) + 51 x1 (1) , 15 1 15 K2 (t; s; x(s); x(s=2); x(0); x(1)) = x (s) + 2t+1 x2 (s=2) + 17 x2 (0) + 71 x2 (1), = t+2 21 2 21 2 f 2 C([0; 1]; R ), f (t) = (f1 (t); f2 (t)), f1 (t) = 2t + 1, f2 (t) = t, g 2 C([0; 1]; [0; 1]), g(s) = s=2 ,

x 2 C([0; 1]; R2 ) . We consider the operator A : C([0; 1]; R2 ) ! C([0; 1]; R2 ), A(x)(t) = (A1 (x)(t); A2 (x)(t)), de…ned by ( R1 x (s) + 2t+1 x1 (s=2) + 51 x1 (0) + 15 x1 (1) ds + 2t + 1 A1 (x)(t) = 0 t+2 15 R 115 t+21 : x2 (s=2) + 71 x2 (0) + 17 x2 (1) ds + t A2 (x)(t) = 0 21 x2 (s) + 2t+1 21 (8) The set of the solutions of the system of integral equations (7) coincides with the set of …xed points of the operator A, i.e. with FA . The operator 4=5 0 A satis…es a Lipschitz condition with the matrix Q = ; Q 2 0 4=7 M22 (R+ ), and from the theorem 3, it results that the matrix Q converges to zero. Therefore the operator A is a contraction with this matrix Q. The conditions of the theorem 4 being satis…ed, it results that the system of integral equations (7) has a unique solution x 2 C([0; 1]; R2 ). Since the function K(t; s; ; ; ; ) is increasing for all t; s 2 [0; 1] and the function f ( ) is increasing for all t 2 [0; 1], it results that the conditions of lemma 6 are satis…ed and we have the following integral inequalities: - if x 2 C([0; 1]; R2 ) is a low-solution of the system of integral equations (7), then x x ; - if x 2 C([0; 1]; R2 ) is an upper-solution of the system of integral equations (7), then x x . Ref erences [1] M. Ambro, Aproximarea solu¸tiilor unei ecua¸tii integrale cu argument modi…cat, Studia Univ. Babe¸s-Bolyai, Mathematica, 2(1978), 26–32. 8


[2] Sz. András, Fiber '–contractions on generalized metric spaces and applications, Mathematica, 45(68)(2003):1, 3–8. [3] Sz. András, Ecua¸tii integrale Fredholm–Volterra, Editura Didactic¼ a ¸si Pedagogic¼ a, Bucure¸sti, 2005. [4] D. Bainov, P. Simeonov, Integral inequalities and applications, Kluwer, Dordrecht, 1992. [5] A. Buic¼ a, Gronwall-type nonlinear integral inequalities, Mathematica, 44(2002). [6] Gh. Coman, I. Rus, G. Pavel, I.A.Rus, Introducere în teoria ecua¸tiilor operatoriale, Editura Dacia, Cluj–Napoca, 1976. [7] M. Dobri¸toiu, An integral equation with modi…ed argument, Studia Univ. Babe¸s–Bolyai, Mathematica, vol. XLIX, 3(2004), 27–33. [8] M. Dobri¸toiu, Analysis of an integral equation with modi…ed argument, Studia Univ. Babe¸s-Bolyai, Mathematica, vol. 51, 1(2006), 81–94. [9] M. Dobri¸toiu, Properties of the solution of an integral equation with modi…ed argument, Carpathian Journal of Mathematics, 23(2007), No.1–2, 77–80 . [10] M. Dobri¸toiu, I.A. Rus, M.A. S ¸erban, An integral equation arising from infectious diseases, via Picard operators, Studia Univ. Babe¸s-Bolyai, Mathematica, to appear. [11] D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equation in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996. [12] R. Precup, Ecua¸tii integrale neliniare, Univ. Babe¸s-Bolyai Cluj–Napoca, 1993. [13] R. Precup, Methods in nonlinear integral equations, Dordrecht ; Boston : Kluwer Academic Publishers, 2002. [14] I.A. Rus, Principii ¸si aplica¸tii ale teoriei punctului …x, Editura Dacia, Cluj-Napoca, 1979. [15] I.A. Rus, Metrical …xed point theorems, University of Cluj-Napoca, 1979. [16] I.A. Rus, Teoreme de punct …x în spa¸tii Banach, Seminarul itinerant de ecau¸tii func¸tionale, aproximare ¸si convexitate, Univ. Babe¸s–Bolyai ClujNapoca, 1982, 327–332. [17] I.A. Rus, Ecua¸tii diferen¸tiale, ecua¸tii integrale ¸si sisteme dinamice, Casa de editur¼ a Transilvania Press, Cluj–Napoca, 1996. [18] I.A. Rus, Generalized contractions and applications, Cluj University Press, Cluj–Napoca, 2001. 9


[19] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae, 58(2003):1, 191–219. [20] M.A. S ¸erban, Application of …ber Picard operators to integral equations, Bul. S ¸tiin¸ti…c Univ. Baia–Mare, Seria B, Matematic¼ a–Informatic¼ a, vol. XVIII, 1(2002), 119–128. [21] M. Zima, The abstract Gronwall lemma for some nonlinear operators, Demonstratio Math., 31(1998), 325-332. Maria Dobri¸toiu Department of Mathematics University of Petro¸sani, Petro¸sani, Romania str. Universit¼ a¸tii, nr.20, Petro¸sani, 332006, jud.Hunedoara email: [email protected]

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