Vietnam J. Math. DOI 10.1007/s10013-015-0156-9

J H-Operator Pairs of Type (R) with Application

to Nonlinear Integral Equations Bahman Moeini1 · Abdolrahman Razani1

Received: 22 September 2013 / Accepted: 29 October 2014 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Abstract In this paper, a new class of noncommuting mappings as J H-operator pairs of type (R) are introduced and some examples are presented. Also, a common fixed point theorem for this kind of mappings is proved. Finally, as an application, the existence of a solution of nonlinear integral equations is proved. Keywords Common fixed point · J H-operator pair · J H-operator pair of type (R) · Nonlinear integral equation Mathematics Subject Classification (2010) 47H09

1 Introduction and Preliminaries Jungck [6] proved a common fixed point theorem for commuting mappings as generalizing the Banach’s contraction principle. Sessa [16] introduced weakly commuting mappings which was generalized by Jungck [7] as compatible mappings. Afterward, many authors studied about common fixed point theorems for noncommuting mappings and their applications (see [8–12, 17–21]). Hussain et al. [4] introduced a J H-operator class as a new class of noncommuting selfmappings that contains the occasionally weakly compatible. Sintunavarat et al. [22] introduced a generalized J H-operator class that contains the J Hoperator class. In 1922, Banach [2] used the contraction principle to establish the existence of a solution for an integral equation.

 Abdolrahman Razani

[email protected] Bahman Moeini moeini [email protected] 1

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

B. Moeini, A. Razani

The Hammerstein integral equation is as follows:  b x(t) = g(t) + k(t, τ )H (τ, x(τ ))dτ a

for all t ∈ I = [a, b] which is a Fredholm-type integral equation and the Volterra– Hammerstein integral equation is introduced as follows:  t k(t, τ )H (τ, x(τ ))dτ x(t) = g(t) + a

for all t ∈ I = [a, b]. Pathak et al. [13–15] proved a common fixed point theorem of various of compatible mappings and utilized this result to simultaneous Volterra–Hammerstein and simultaneous Volterra–Hammerstein nonlinear integral equations with infinite delay. Here, we introduce a new class of noncommuting mappings which contains the J Hoperator pairs as proper subclasses. Moreover, a common fixed point theorem is proved for a quadruple of selfmappings, satisfying certain generalized contraction. Finally, the result is applied to show the existence of solution of a pair of mixed type of a simultaneous Hammerstein-type integral equation and a pair of simultaneous Hammerstein-type integral equations with infinite delay. Our theorems extend and improve several known results. The set of fixed points of T is denoted by F (T ). The diameter of a set A is denoted by diam(A) = sup{d(x, y) : x, y ∈ A}. A point x ∈ X is a coincidence point (common fixed point) of f and S, if f x = Sx (respectively, x = f x = Sx). Let C(f, S), P C(f, S) denote the sets of all coincidence points, points of coincidence, respectively. Suppose (X, d) is a metric space and f, S are selfmappings on X, the pair (f, S) is called 1) commuting if and only if f Sx = Sf x for all x ∈ X; 2) weakly commuting if and only if f Sx = Sf x for some x ∈ X; 3) ([7]) compatible if and only if limn→∞ d(f Sxn , Sf xn ) = 0 whenever {xn } is a sequence in X such that limn→∞ f xn = limn→∞ Sxn = t for some t ∈ X; 4) ([22]) weakly compatible if and only if f Sx = Sf x for all x ∈ C(f, S); 5) ([22]) occasionally weakly compatible if and only if f Sx = Sf x for some x ∈ C(f, S); 6) ([4]) J H-operator if and only if d(w, x) ≤ diam(P C(f, S))

for some f x = Sx = w ∈ P C(f, S);

7) ([3]) subcompatible if and only if there exists a sequence {xn } in X such that limn→∞ f xn = limn→∞ Sxn = t, t ∈ X and which satisfy limn→∞ d(f Sxn , Sf xn ) = 0.

2 Class of J H-operator Pairs of Type (R) Definition 1 Let f and S be selfmappings on a metric space X. Then the pair (f, S) is said to be a J H-operator pair of type (R), if P C(f, S) = ∅ and lim d(xn, t) ≤ diam(P C(f, S))

n→∞

for some sequence {xn } in X with limn→∞ f xn = limn→∞ Sxn = t ∈ X. Proposition 1 If the pair (f, S) is J H-operator, then (f, S) is a J H-operator pair of type (R).

J H-Operator Pairs of Type (R) with Application to Equations

Proof By hypothesis, there exists a point t = fp = Sp ∈ P C(f, S) such that d(p, t) ≤ diam(P C(f, S)). Define a sequence {xn } in X by xn = p for n = 1, 2, . . . . Then limn→∞ f xn = limn→∞ Sxn = t and limn→∞ d(xn, t) ≤ diam(P C(f, S)). Therefore, (f, S) is a J H-operator pair of type (R). Example 1 Let f x = x 3 and Sx = 2 − x with X = R. So C(f, S) = {1}, P C(f, S) = {1} and diam(P C(f, S)) = 0. Define a sequence {xn } in X by xn = n+1 n for n = 1, 2, . . . . Then limn→∞ f xn = limn→∞ Sxn = t = 1. Since 0 = lim |xn − t| ≤ diam(P C(f, S)) = 0. n→∞

Therefore, the pair (f, S) is a J H-operator pair of type (R) and obviously this pair is also J H-operator. In the following, we shall show that the class of J H-operator pairs of Hussain et al. [4] is a proper subclass of J H-operator pairs of type (R). Remark 1 The inverse of Proposition 1 does not always hold. For example, suppose X = [0, 1] is a metric space with the usual metric and f, S : X → X are defined by: x if 0 < x ≤ 1, fx = 2 Sx = x 2 . 1 if x = 0, Then C(f, S) = { 12 } and P C(f, S) = { 14 }. Since f 12 = S 12 = 14 and     1 1 1 1 1 =d , =  −  > diam(P C(f, S)) = 0. 4 4 2 4 2 Thus, (f, S) is not a J H-operator pair. Now, we define a sequence {xn } by xn = (n = 1, 2, . . . ). We have

1 n+1 ,

lim f xn = lim Sxn = t = 0 ∈ X

n→∞

and

n→∞

   1   − 0 = 0 ≤ diam(P C(f, S)) = 0. lim d(xn , t) = lim  n→∞ n→∞ n + 1

Therefore, the pair (f, S) is a J H-operator pair of type (R). Remark 2 Suppose f and S are continuous selfmappings on a metric space (X, d). If the pair (f, S) is a J H-operator pair of type (R) and P C(f, S) is a singleton set, then f and S are subcompatible mappings. For this, there exists a sequence {xn } in X such that limn→∞ f xn = limn→∞ Sxn = t and which satisfy limn→∞ d(xn, t) ≤ diam(P C(f, S)) = 0, so limn→∞ xn = t and f t = St = t. Hence, limn→∞ f Sxn = limn→∞ Sf xn = t, that is f and S are subcompatible. Note that, the class of subcompatible mappings [3] and the class of J H-operator pairs of type (R) contain properly occasionally weakly compatible mappings but, these classes are two different concepts in general. Let X = [1, ∞) with the usual metric d. Define f and S as follows: ⎧ 3 ⎧ 2 if 1 < x < 2, x > 2, if 1 < x < 2, x > 2, ⎨x +x ⎨x +1 fx = 2 Sx = 2 if x = 1, if x = 1, ⎩ ⎩ 4 if x = 2, 4 if x = 2.

B. Moeini, A. Razani

Then, the pair (f, S) is a J H-operator pair of type (R) but these mappings are not subcompatible. Also, it can be shown that two selfmappings defined in Example 1.2 of [3] are subcompatible but not a J H-operator pair of type (R).

3 Some Common Fixed Point Theorems Here, we define a sequence as follows: Let f, g, S, and T be selfmappings of a metric space (X, d) such that f (X) ⊂ T (X)

and

g(X) ⊂ S(X),

(1)

and

d 2p (f x, gy) ≤ αd 2p (Sx, T y) + β max d q (Sx, f x) · d q (T y, gy), d r (Sx, gy) · d r (T y, f x),  1 s 1 t s t d (Sx, f x) · d (T y, f x), d (Sx, gy) · d (T y, gy) (2) 2 2

for each x, y ∈ X, where α, β > 0 and α +β < 1, 0 < p, q, q , r, r , s, s , t, t ≤ 1 are such that 2p = q +q = r +r = s +s = t +t . Then, for an arbitrary point x0 ∈ X, by (1), there exists a point x1 such that f x0 = T x1 and for this point x1 , we can choose a point x2 ∈ X such that gx1 = Sx2 and so on. Inductively, one can define a sequence {yn } in X, such that y2n = T x2n+1 = f x2n

and

y2n+1 = Sx2n+2 = gx2n+1

(3)

for n = 1, 2, . . . . Theorem 1 Let f, g, S, and T be selfmappings of a metric space (X, d) satisfying (2). If the pairs (f, S) and (g, T ) are J H-operators then f, g, S and T have a unique common fixed point in X. Proof By hypothesis, there exist points x, y ∈ X such that w = f x = Sx and z = gy = T y. Therefore, by (2), we have

d 2p (f x, gy) ≤ αd 2p (Sx, T y) + β max d q (Sx, f x) · d q (T y, gy), d r (Sx, gy) · d r (T y, f x),  1 s 1 d (Sx, f x) · d s (T y, f x), d t (Sx, gy) · d t (T y, gy) 2 2

= αd 2p (f x, gy) + β max 0, d r (f x, gy) · d r (gy, f x), 0, 0

= αd 2p (f x, gy) + βd r+r (f x, gy) = (α + β)d 2p (f x, gy) < d 2p (f x, gy),

a contradiction. Hence, f x = Sx = gy = T y. Moreover, if there is another point x ∈ X for which f x = Sx , then using (2), it follows that f x = Sx = gy = T y, in other words, f x = f x . Hence, w = f x = Sx is a unique point of coincidence of f and S. Then diam(P C(f, S)) = 0 and since (f, S) is a J H-operator pair, we have d(x, w) ≤ diam(P C(f, S)) = 0,

J H-Operator Pairs of Type (R) with Application to Equations

so, x = w is a unique common fixed point of f and S. Similarly, y = z is a unique common fixed point of g and T . On the other hand, since w = z, thus w is a unique common fixed point of f, g, S and T . Lemma 1 If dn = d(yn, yn+1 ), then limn→∞ dn = 0, where {yn } is defined by (3). Proof Inequality (2) implies

d 2p (y2n , y2n+1 ) ≤ αd 2p (y2n−1 , y2n ) + β max d q (y2n−1 , y2n ) · d q (y2n , y2n+1 ), 1 d r (y2n−1 , y2n+1 ) · d r (y2n , y2n ), d s (y2n−1 , y2n ) · d s (y2n , y2n ), 2  1 t d (y2n−1 , y2n+1 ) · d t (y2n , y2n+1 ) , 2

or



 t 1 t 2p 2p q q t d2n ≤ αd2n−1 + β max d2n−1 · d2n , 0, 0, d2n−1 · d2n + d2n 2

 1 t 2p q q t t t + d2n d2n ≤ αd2n−1 + β max d2n−1 · d2n , 0, 0, . d2n−1 d2n 2

If d2n > d2n−1 , then

 1  t+t 2p 2p q+q t+t d2n ≤ αd2n + β max d2n , 0, 0, , d2n + d2n 2



2p 2p 2p 2p 2p 2p 2p d2n ≤ αd2n + β max d2n , 0, 0, d2n ≤ αd2n + βd2n < d2n ,

or

which is a contradiction. Hence d2n ≤ d2n−1 .

(4)

Similarly, taking x = x2n+2 and y = x2n+1 in (2), we get

d 2p (y2n+2 , y2n+1 ) ≤ αd 2p (y2n+1 , y2n ) + β max d q (y2n+1 , y2n+2 ) · d q (y2n , y2n+1 ), 1 d r (y2n+1 , y2n+1 ) · d r (y2n , y2n+2 ), d s (y2n+1 , y2n+2 ) · d s (y2n , y2n+2 ), 2 1 t d (y2n+1 , y2n+1 ) · d t (y2n , y2n+1 ) , 2

or

   1 s 2p 2p q q s s ,0 d2n+1 ≤ αd2n + β max d2n+1 · d2n , 0, d2n+1 · d2n + d2n+1 2

  1 s 2p q q s s s ,0 . d2n+1 d2n + d2n+1 d2n+1 ≤ αd2n + β max d2n+1 · d2n , 0, 2 If d2n+1 > d2n , then

  1 s 2p 2p q q s s s + d2n+1 d2n+1 ,0 . d2n+1 ≤ αd2n+1 + β max d2n+1 · d2n+1 , 0, d2n+1 d2n+1 2 2p

2p

2p

2p

So, d2n+1 ≤ αd2n+1 + βd2n+1 < d2n+1 . This is a contradiction. Thus, d2n+1 ≤ d2n .

(5)

B. Moeini, A. Razani

From inequalities (4) and (5), dn+1 ≤ dn . Therefore, {dn } is a non-increasing sequence in R+ . Now, again by (2), we have 2p

d1

= d 2p (y2 , y1 ) = d 2p (f x2 , gx1 )

≤ αd 2p (Sx2 , T x1 ) + β max d q (Sx2 , f x2 ) · d q (T x1 , gx1 ), d r (Sx2 , gx1 ) · d r (T x1 , f x2 ),  1 s 1 d (Sx2 , f x2 ) · d s (T x1 , f x2 ), d t (Sx2 , gx1 ) · d t (T x1 , gx1 ) 2 2

2p q = αd (y1 , y0 ) + β max d (y1 , y2 ) · d q (y0 , y1 ), d r (y1 , y1 ) · d r (y0 , y2 ),  1 s 1 d (y1 , y2 ) · d s (y0 , y2 ), d t (y1 , y1 ) · d t (y0 , y1 ) 2 2

  1 s  s 2p q q s = αd0 + β max d1 · d0 , 0, d1 · d0 + d1 , 0 2

  1 2p q q ≤ αd0 + β max d0 · d0 , d0s · d0s + d0s 2 2p

= (α + β)d0 , 2p

2p

2p

2p

therefore d1 ≤ (α + β)d0 . In general, we have dn ≤ (α + β)n d0 , which implies that 2p 2p 2p limn→∞ dn ≤ limn→∞ (α + β)n d0 = 0. Thus, limn→∞ dn = 0. Lemma 2 The sequence {yn } defined by (3) is a Cauchy sequence. Proof Suppose the subsequence {y2n } is not a Cauchy sequence. Then there exists an ε > 0 such that for each even integer 2k, there exist even integers 2m(k) and 2n(k) with 2m(k) > 2n(k) such that d(y2m(k) , y2n(k) ) > ε. (6) For each even integer 2k, let 2m(k) be the least even integer exceeding 2n(k) satisfying (6), that is (7) d 2p (y2m(k)−1 , y2n(k) ) ≤ ε and d 2p (y2m(k) , y2n(k) ) > ε. Then ε < d 2p (y2m(k) , y2n(k) ) ≤ d 2p (y2m(k)−2 , y2n(k) ) + d 2p (y2m(k)−2 , y2m(k)−1 ) + d 2p (y2m(k)−1 , y2m(k) ) ≤ ε + d2m(k)−2 + d2m(k)−1 . Hence, from Lemma 1 and (7) lim d(y2n(k) , y2m(k) ) = ε.

k→∞

(8)

By the triangular inequality, we have     2p d (y2n(k) , y2m(k)−1 ) − d 2p (y2n(k) , y2m(k) ) ≤ d 2p (y2m(k)−1 , y2m(k) ), and     2p d (y2n(k)+1 , y2m(k)−1 ) − d 2p (y2n(k) , y2m(k) ) ≤ d 2p (y2m(k)−1 , y2m(k) ) + d 2p (y2n(k) , y2n(k)+1 ).

J H-Operator Pairs of Type (R) with Application to Equations

By Lemma 1 and (8), lim d(y2n(k) , y2m(k)−1 ) = ε

k→∞

and

lim d(y2n(k)+1 , y2m(k)−1 ) = ε.

k→∞

(9)

Now, using (2) to have     d 2p (y2n(k) , y2m(k) ) ≤ d 2p y2n(k) , y2n(k)+1 + d 2p y2n(k)+1 , y2m(k)     = d 2p y2n(k) , y2n(k)+1 + d 2p f x2m(k) , gx2n(k)+1     ≤ d 2p y2n(k) , y2n(k)+1 + αd 2p Sx2m(k) , T x2n(k)+1

    +β max d q Sx2m(k) , f x2m(k) d q T x2n(k)+1 , gx2n(k)+1 ,     1 d r Sx2m(k) , gx2n(k)+1 d r T x2n(k)+1 , f x2m(k) , d s (Sx2m(k) , 2   1 t  s f x2m(k) )d T x2n(k)+1 , f x2m(k) , d Sx2m(k) , gx2n(k)+1 2   t d T x2n(k)+1 , gx2n(k)+1     = d 2p y2n(k) , y2n(k)+1 + αd 2p y2m(k)−1 , y2n(k)

     +β max d q y2m(k)−1 , y2m(k) d q y2n(k) , y2n(k)+1 , d r y2m(k)−1 ,    1   y2n(k)+1 d r y2n(k) , y2m(k) , d s y2m(k)−1 , y2m(k) d s (y2n(k) , 2   1     y2m(k) , d t y2m(k)−1 , y2n(k)+1 d t y2n(k) , y2n(k)+1 . 2 Let k → ∞. By using Lemma 1, (8) and (9), we have

ε2p ≤ αε2p + β max 0, εr+r , 0, 0 = αε2p + βε2p < ε2p , which is a contradiction. Hence, {y2n } is a Cauchy sequence. Theorem 2 Let f, g, S, and T be selfmappings of a complete metric space (X, d) satisfying (1) and (2) and any one of the following: (i) (ii)

S is continuous and the pairs (f, S), (g, T ) are a J H-operator pair of type (R) and weakly compatible mappings, respectively, S and T are continuous and the pairs (f, S), (g, T ) are J H-operator pair of type (R).

Then f , S, g, and T have a unique common fixed point in X. Proof Suppose that condition (i) holds. First, we show that diam(P C(f, S)) = 0. By hypothesis P C(f, S) = ∅, hence there exists a point x in X that f x = Sx = w. From Lemma 2, {yn } is a Cauchy sequence in X. Since X is complete there exists a point z ∈ X such that limn→∞ yn = z and lim f x2n = lim T x2n+1 = lim Sx2n = lim gx2n+1 = z.

n→∞

n→∞

n→∞

n→∞

B. Moeini, A. Razani

Here, if z = w, from (2) we get

d 2p (f x, gx2n+1 ) ≤ αd 2p (Sx, T x2n+1 ) + β max d q (Sx, f x) · d q (T x2n+1 , gx2n+1 ), 1 d r (Sx, gx2n+1 ) · d r (T x2n+1 , f x), d s (Sx, f x) · d s (T x2n+1 , f x), 2  1 t d (Sx, gx2n+1 ) · d t (T x2n+1 , gx2n+1 ) . 2 Letting n → ∞, we have

d 2p (w, z) ≤ αd 2p (w, z) + β max 0, d r (w, z) · d r (w, z), 0, 0

= αd 2p (w, z) + βd r+r (w, z) < d 2p (w, z), a contradiction, therefore, w = z. Moreover, if there is another point x in X for which f x = Sx = w , then by using (2), we get w = z and w = w = z is a unique point of coincidence of f and S. So diam(P C(f, S)) = 0. Since the pair (f, S) is a J H-operator pair of type (R), there exists a sequence {tn } in X such that limn→∞ f tn = limn→∞ Stn = t ∈ X and limn→∞ d(tn , t) ≤ diam(P C(f, S)) = 0. Therefore, limn→∞ tn = t. Now, we show that t = z. If t = z, then by using (2) for x = tn and y = x2n+1 , we have

d 2p (f tn , gx2n+1 ) ≤ αd 2p (Stn , T x2n+1 ) + β max d q (Stn , f tn ) · d q (T x2n+1 , gx2n+1 ), 1 d r (Stn , gx2n+1 ) · d r (T x2n+1 , f tn ), d s (Stn , f tn ) · d s (T x2n+1 , f tn ), 2  1 t d (Stn , gx2n+1 ) · d t (T x2n+1 , gx2n+1 ) . 2

Letting n → ∞, we have

d 2p (t, z) ≤ αd 2p (t, z) + β max 0, d r (t, z) · d r (t, z), 0, 0

= αd 2p (t, z) + βd r+r (t, z) < d 2p (t, z), which is a contradiction. Hence t = z. Since S is continuous and limn→∞ Stn = t, so Sz = z. Putting x = z and y = x2n+1 , we have

d 2p (f z, gx2n+1 ) ≤ αd 2p (Sz, T x2n+1 ) + β max d q (Sz, f z) · d q (T x2n+1 , gx2n+1 ), 1 d r (Sz, gx2n+1 ) · d r (T x2n+1 , f z), d s (Sz, f z) · d s (T x2n+1 , f z), 2  1 t t d (Sz, gx2n+1 ) · d (T x2n+1 , gx2n+1 ) . 2 Letting n → ∞, we have d

2p

(f z, z) ≤ αd

2p

1 (z, z) + β max 0, 0, d s (z, f z) · d s (z, f z), 0 2

1 = β d s+s (z, f z) < d 2p (z, f z), 2



J H-Operator Pairs of Type (R) with Application to Equations

which is a contradiction. Thus f z = z. Since f (X) ⊂ T (X), there exists a point u ∈ X such that T u = f z = z. Using inequality (2), we have d 2p (z, gu) = d 2p (f z, gu)

≤ αd 2p (Sz, T u) + β max d q (Sz, f z) · d q (T u, gu), d r (Sz, gu) · d r (T u, f z),  1 s 1 d (Sz, f z) · d s (T u, f z), d t (Sz, gu) · d t (T u, gu) 2 2 

1 t+t (z, gu) < d 2p (z, gu), = β max 0, 0, 0, d 2

which is a contradiction. Hence gu = z. Since the pair (g, T ) is weakly compatible and gu = T u = z, we have gz = gT u = T gu = T z. Again from inequality (2), we have d 2p (z, gz) = d 2p (f z, gz)

≤ αd 2p (Sz, T z) + β max d q (Sz, f z) · d q (T z, gz),

 1 1 d r (Sz, gz) · d r (T z, f z), d s (Sz, f z) · d s (T z, f z), d t (Sz, gz) · d t (T z, gz) 2 2

 1 = αd 2p (z, gz) + β max 0, d r+r (z, gz), 0, d t+t (z, gu), 0 2 = αd 2p (z, gz) + βd 2p (z, gz) < d 2p (z, gz),

which means that gz = z. So, f z = Sz = gz = T z = z, that is z is a common fixed point of f , S, g, and T . The uniqueness of the common fixed point z follows from inequality (2). Similarly, we can also complete the proof when (ii) holds. This completes the proof. Corollary 1 Let f , g, S, and T be selfmappings of a complete metric space (X, d) satisfying (1) and d(f x, gy) ≤ αd(Sx, T y) (10) for all x, y ∈ X, where 0 < α < 1. If S is continuous and the pairs (f, S), (g, T ) are a J H-operator pair of type (R) and weakly compatible mappings, respectively. Then f , g, S and T have a unique common fixed point in X. Example 2 Suppose X = [0, 2] is a metric space with the X → X defined by: ⎧x ⎪ ⎨ 2 f x = gx = 1, Sx = 2 − x, T x = 1 ⎪ ⎩ 0

usual metric and f, g, S, T : if x ∈ [0, 1), if x = 1, if x ∈ (1, 2].

Then |f x − gy| = 0 ≤ α|Sx − T y| for all x, y ∈ X and S is continuous. Also   1 f (X) = {1} ⊆ T (X) = 0, ∪ {1}, g(X) = {1} ⊆ S(X) = [0, 2], 2

B. Moeini, A. Razani

C(f, S) = P C(f, S) = C(g, T ) = P C(g, T ) = {1}. To see that (f, S) is a J Hoperator pair of type (R), suppose {xn } is a sequence in [0, 2] defined by xn = n+1 n , then limn→∞ f xn = limn→∞ Sxn = t = 1 and   n + 1  − 1 = 0 ≤ diam(P C(f, S)) = 0, lim d(xn, t) = lim  n→∞ n→∞ n consequently, the pair (f, S) is a J H-operator pair of type (R). Now, since C(g, T ) = {1} and gT 1 = T g1 = 1, hence, the pair (g, T ) is weakly compatible. So all the conditions of Corollary 1 are satisfied and x = 1 is a unique common fixed point of f, g, S and T .

4 Application to Nonlinear Integral Equations Recently, Hussain and Taoudi [5] used the some Krasnosel’skii-type fixed point results to show the existence of weak solutions to the following Volterra integral equation in the space C([0, T ], X):  t x(t) = f (x(t)) + g(s, x(s))ds, t ∈ [0, T ]. 0

Agrawal et al. [1] utilized results of common fixed point theorems for a pair of nonlinear mappings to study the existence of nonnegative solutions of an implicit integral equation as follows in the space L1 [0, 1]:  1 p(t, x(t)) = ζ (t, s)f (s, x(s))ds, t ∈ [0, 1]. 0

In this section, we apply common fixed point methods to show the existence of a unique solution of systems of nonlinear integral equations in L [ a, ∞) under relaxed conditions. Also our results extend and generalize some known results. Let us consider the following pair of nonlinear integral equations:  ∞  ∞ x(t) = w(t) + μ m(t, s)ki (s, x(s))ds + λ n(t, s)hj (s, x(s))ds (11) a

a

for all t ∈ [ a, ∞) , where w ∈ L [ a, ∞) is known, m(t, s), n(t, s), ki (s, x(s)), hj (s, x(s)), i, j = 1, 2 and i = j are real or complex valued functions that are measurable both in t and s on [ a, ∞) and λ, μ are real or complex numbers. These functions satisfy the following conditions: ∞ (a) a supa≤s<∞ |m(t, s)|dt = M1 < +∞, ∞ (b) a supa≤s<∞ |n(t, s)|dt = M2 < +∞, (c) ki (s, x(s)) ∈ L[a, ∞) for all x ∈ L[a, ∞), and for i = 1, 2: |ki (s, x(s)) − ki (s, y(s))| > |x(s) − y(s)|

for all s ∈ [ a, ∞) , x, y ∈ L [ a, ∞) ,

also, there exists L1 > 0 such that for all s ∈ [ a, ∞) , |k1 (s, x(s)) − k2 (s, y(s))| ≤ L1 |x(s) − y(s)|

for all x, y ∈ L [ a, ∞) ,

(d) hi (s, x(s)) ∈ L[a, ∞) for all x ∈ L[a, ∞), and for i = 1, 2: |hi (s, x(s)) − hi (s, y(s))| > |x(s) − y(s)|

for all s ∈ [ a, ∞) , x, y ∈ L [ a, ∞) ,

also, there exists L2 > 0 such that for all s ∈ [ a, ∞) , |h1 (s, x(s)) − h2 (s, y(s))| ≤ L2 |x(s) − y(s)|

for all x, y ∈ L [ a, ∞) ,

J H-Operator Pairs of Type (R) with Application to Equations

for i = 1, 2, Ai is a nonempty set consisting of xpi ∈ L [ a, ∞) , such that  ∞  ∞ μ m(t, s)ki (s, xpi (s))ds = xpi (t) − w(t) − λ n(t, s)hi (s, xpi (s))ds = zpi ,

(e)

a

a

and for any xp2 ∈ A2 , this implies that  ∞  ∞   μ m(t, s)k (s, x (s))ds − w(t) − λ 2 p2  a

a

∞ a

  n(t, s)h2 (s, xp2 (s))ds  dt = 0,

for some sequence {xn (t)} in L[a, ∞), there exists τ1 (t) ∈ L[a, ∞) such that  ∞ lim μ m(t, s)k1 (s, xn (s))ds n→∞ a    ∞ = lim xn (t) − w(t) − λ n(t, s)h1 (s, xn (s))ds = τ1 (t),

(f)

n→∞

a

and which satisfies  ∞  ∞ |w(t)|dt + |λ|M2 |h1 (s, xp1 (s))|ds a a    ∞  ∞     − ≤ |μ| m(t, s)k (s, x (s))ds 1 q1    a

a

∞ a

  m(t, s)k1 (s, xr1 (s))ds  dt

for some xq1 , xr1 ∈ A1 . Theorem 3 With assumptions (a)–(f), if the following condition is also satisfied: ∞   ∞ (g) w(t) + λ a n(t, s)hi s, μ a m(s, τ )kj (τ, x(τ ))dτ = 0 i, j = 1, 2, i = j , then system (11) has a unique solution in L [ a, ∞) for each pair of real or complex numbers |μ|L1 M1 < 1. μ and λ with |λ|L2 M2 < 1 and 1−|λ|L 2 M2 Proof Define 



∞

f x(t) = μ

m(t, s)k1 (s, x(s))ds, a



m(t, s)k2 (s, x(s))ds, a

∞

Cx(t) = w(t)+λ

∞

gx(t) = μ

n(t, s)h1 (s, x(s))ds,



∞

Dx(t) = w(t)+λ

a

n(t, s)h2 (s, x(s))ds, a

Sx(t) = (I − C)x(t),

T x(t) = (I − D)x(t),

where I is the identity operator on L [ a, ∞) . Since  ∞  |f x(t)| ≤ |μ| |m(t, s)k1 (s, x(s))ds| ≤ |μ|supa≤s<∞ |m(t, s)| a

∞

|k1 (s, x(s))ds|,

a

thus by (a), (c)  ∞  |f x(t)|dt ≤ |μ| a

∞ a

 supa≤s<∞ |m(t, s)|dt

∞

|k1 (s, x(s))ds| < +∞,

a

hence f x ∈ L [ a, ∞) . Similarly gx ∈ L [ a, ∞) also. Now, by (b), (d) 

∞ a

 |Cx(t)|dt ≤ a

∞



∞

|w(t)|dt + |λ| a

 supa≤s<∞ |n(t, s)|dt

∞ a

|h1 (s, x(s))|ds < +∞.

B. Moeini, A. Razani

So Cx ∈ L [ a, ∞) . Similarly, Dx ∈ L [ a, ∞) also. Then f, g, C, D, S and T are operators from L [ a, ∞) into itself. On the other hand for all x, y ∈ L [ a, ∞)  ∞ f x − gy = |f x(t) − gy(t)|dt a   ∞  ∞  ∞    = m(t, s)k1 (s, x(s))ds − μ m(t, s)k2 (s, y(s))ds  dt μ a a a   ∞  ∞   μ  dt = m(t, s) [k (s, x(s))ds − k (s, y(s))] ds 1 2   a a  ∞  ∞ |μ|supa≤s<∞ |m(t, s)|dt |[k1 (s, x(s))ds − k2 (s, y(s))]ds ≤ a a  ∞ |x(s) − y(s)|ds ≤ |μ|M1 L1 a

≤ |μ|M1 L1 x − y .

(12)

Similarly, Cx − Dy ≤ |λ|M2 L2 x − y . Hence Sx − T y = (x − y) − (Cx − Dy) ≥ x − y − Cx − Dy ≥ x − y − |λ|M2 L2 x − y = (1 − |λ|M2 L2 ) x − y .

(13)

Thus, from (12) and (13), f x − gy ≤

|μ|M1 L1 Sx − T y . (1 − |λ|M2 L2 )

Now, we prove that f (L [ a, ∞) ) ⊆ T (L [ a, ∞) ). Let x(t) ∈ L [ a, ∞) be arbitrary, then we get T (f x(t)) = (I − D)f x(t) = f x(t) − Df x(t)  ∞ = f x(t) − w(t) − λ n(t, s)h2 (s, f x(s))ds a    ∞  ∞ = f x(t) − w(t) − λ n(t, s)h2 s, μ m(s, τ )k1 (τ, x(τ ))dτ ds a

a

= f x(t), by condition (g) of the theorem. Similarly, we can prove g(L[a, ∞)) ⊆ S(L[a, ∞)). From condition (f) for some sequence {xn (t)} in L[a, ∞), there exists τ1 (t) ∈ L[a, ∞) such that  ∞ m(t, s)k1 (s, xn (s))ds lim μ n→∞ a    ∞ = lim xn (t) − w(t) − λ n(t, s)h1 (s, xn (s))ds = τ1 (t), n→∞

and



a

 ∞ |w(t)|dt + |λ|M2 |h1 (s, xp1 (s))|ds a a    ∞  ∞     − ≤ |μ| m(t, s)k (s, x (s))ds 1 q1    ∞

a

a

a

∞

  m(t, s)k1 (s, xr1 (s))ds  dt (14)

J H-Operator Pairs of Type (R) with Application to Equations

for some xq1 , xr1 ∈ A1 . From (c) and (14), we have  lim xn − τ1 = lim

n→∞

∞

|xn (t) − τ1 (t)|dt    ∞ ∞   dt w(t) + λ n(t, s)h (s, x (s))ds lim 1 n   n→∞ a a  ∞  ∞ |w(t)|dt + |λ|M2 lim |h1 (s, xn (s))|ds n→∞ a a   ∞   ∞  ∞        m(t, s)k1 (s, xq1 (s))ds  −  m(t, s)k1 (s, xr1 (s))ds  dt |μ|  a a a   ∞  ∞  t    dt  |μ| m(t, s)k (s, x (s))ds − m(t, s)k (s, x (s))ds 1 q 1 r 1 1   a a a   ∞  ∞  ∞    dt μ m(t, s)k (s, x (s))ds − μ m(t, s)k (s, x (s))ds 1 q1 1 r1   a a a  ∞ |f xq1 (t) − f xr1 (t)|dt n→∞ a



= ≤ ≤ ≤ = =

a

= f xq1 − f xr1 = zq1 − zr1 ≤ diam(P C(f, S)).

So, the pair (f, S) is a J H-operator pair of type (R). Also, by condition (e) we can easily check that the pair (g, T ) is weakly compatible. Now, we shall show that the operator S is continuous. For any sequence {xn (t)} in L[a, ∞) that converges to x(t) ∈ L[a, ∞), we have Sxn − Sx = (I − C)xn (t) − (I − C)x(t) = xn (t) − x(t) + Cx(t) − Cxn (t)  ∞   ∞ |xn (t) − x(t)|dt + |λ| ≤ a

a

∞

 n(t, s)|h1 (s, xn (s)) − h1 (s, x(s))|ds dt.

a

By conditions (b), (c), we have Sxn − Sx ≤ xn (t) − x(t)  ∞ |h1 (s, xn (s)) − h2 (s, x(s)) + h2 (s, x(s)) − h1 (s, x(s))|ds +|λ|M2 a

≤ xn − x + |λ|M2 L2 ( xn − x + x − x ) → 0

as n → ∞.

This shows that S is continuous. Thus, all the conditions of Corollary 1 are satisfied. Therefore, there exists a unique common fixed point x ∈ L[a, ∞) such that x = f x = gx = T x = Sx, which proves the existence of a unique solution of system (11). Let us consider the following Hammerstein-type simultaneous equations with infinite delay:  ∞ n(t, s)hj (s, x(s))ds (15) x(t) = w(t) + ki (t, x(t)) + λ a

B. Moeini, A. Razani

for all t ∈ [a, ∞), where w ∈ L[a, ∞) is known, ki (t, x(t)), n(t, s), hj (s, x(s)), i, j = 1, 2 and i = j are real or complex valued functions that are measurable both in t and s on [a, ∞) and λ is a real or complex number, and assume the following conditions: ∞ (a’) a supa≤s<∞ |n(t, s)|dt = M1 < +∞, (b’) ki (s, x) are continuous in s and x, ki (s, x(s)) ∈ L[a, ∞) for all x ∈ L[a, ∞), and for i = 1, 2: |ki (s, x(s)) − ki (s, y(s))| > |x(s) − y(s)|

for all s ∈ [a, ∞), x, y ∈ L[a, ∞),

also, there exists L1 > 1 such that for all s ∈ [a, ∞), |k1 (s, x(s)) − k2 (s, y(s))| ≥ L1 |x(s) − y(s)| (c’)

for all x, y ∈ L[a, ∞),

hi (s, x(s)) ∈ L[a, ∞) for all x ∈ L[a, ∞), and for i = 1, 2: |hi (s, x(s)) − hi (s, y(s))| > |x(s) − y(s)|

for all s ∈ [a, ∞), x, y ∈ L[a, ∞),

also, there exists L2 > 0 such that for all s ∈ [a, ∞), |h1 (s, x(s)) − h2 (s, y(s))| ≤ L2 |x(s) − y(s)|

for all x, y ∈ L[a, ∞),

xp i

(d’) for i = 1, 2, Bi is a nonempty set consisting of ∈ L[a, ∞), such that  ∞ n(t, s)hi (s, xp i (s))ds = zp i , ki (t, xp i (t)) = xp i (t) − w(t) − λ a

and for any xp 2 ∈ B2 , this implies that  ∞ n(t, s)h2 (s, k2 (s, xp2 (s)))ds k2 (t, xp2 (t)) − w(t) − λ a  ∞ n(t, s)h2 (s, k2 (s, xp2 (s))ds), = k2 (t, xp2 (t) − w(t) − λ a

(e’)

for some sequence {xn (t)} in L[a, ∞), there exists τ1 (t) ∈ L[a, ∞) such that    ∞ lim k1 (t, xn (t)) = lim xn (t) − w(t) − λ n(t, s)h1 (s, xn (s))ds = τ1 (t), n→∞

n→∞

a

and which satisfies  ∞  |w(t)|dt + |λ|M2 lim

∞

n→∞ a

a

|h1 (s, xn (s))|ds ≤ xq 1 − xr 1

for some xq 1 , xr 1 ∈ B1 . Theorem 4 With assumptions (a’)–(e’), the integral equation (15) have a unique solution 2 M1 in L[a, ∞) for each real or complex number λ with 1+|λ|L < 1. L1 Proof Define



∞

f x(t) = x(t) − w(t) − λ gx(t) = x(t) − w(t) − λ

n(t, s)h1 (s, x(s))ds, a ∞ n(t, s)h2 (s, x(s))ds, a

Sx(t) = k1 (t, x(t)),

T x(t) = k2 (t, x(t)).

J H-Operator Pairs of Type (R) with Application to Equations

Thus  f x − gy = (x − y) + λ



∞

 n(t, s)(h2 (s, x(s)) − h1 (s, x(s))ds 

a

≤ x − y + |λ|M1 L2 x − y = (1 + |λ|M1 L2 ) x − y (1 + |λ|M1 L2 ) ≤ Sx − T y , L1 so f x − gy ≤ α Sx − T y , (0 < α < 1). From condition (e’) for some sequence {xn (t)} ∈ L[a, ∞), there exists τ1 ∈ L[a, ∞) such that    ∞ lim k1 (t, xn (t)) = lim Sxn = lim xn (t) − w(t) − λ n(t, s)h1 (s, xn (s))ds n→∞

n→∞

n→∞

a

= lim f xn = τ1 (t) n→∞

and



∞

 |w(t)|dt + |λ|M2 lim

n→∞ a

a

∞

|h1 (s, xn (s))|ds ≤ xq 1 − xr 1 .

(16)

From (b’) and (16), we have  lim xn1 − τ1 = lim

n→∞

n→∞ a



= lim

n→∞ a

∞

|xn (t) − τ1 (t)|dt    ∞ ∞  w(t) + λ  dt n(t, s)h (s, x (s))ds 1 n   a

 ∞ |w(t)|dt + |λ|M2 lim |h1 (s, xn (s))|ds n→∞ a a  ∞  ∞ |xq 1 (t) − xr 1 (t)|dt < |k1 (t, xq 1 (t)) − k1 (t, xr 1 (t))|dt ≤ xq 1 − xr 1 = a a  ∞ = |Sxq 1 (t) − Sxr 1 (t)|dt = Sxq 1 − Sxr 1 = zq 1 − zr 1 ≤ diam(P C(f, S)). 

≤

∞

a

Therefore, the pair (f, S) is a J H-operator pair of type (R). Also, by condition (d’), the pair (g, T ) is weakly compatible. Since f (L[a, ∞)) ⊂ T (L[a, ∞)) = L[a, ∞) and g(L[a, ∞)) ⊂ S(L[a, ∞)) = L[a, ∞), so by Corollary 1, there exists a unique common fixed point x ∈ L[a, ∞) such that x = f x = gx = Sx = T x. Hence, the simultaneous (15) have a unique solution in L[a, ∞). Acknowledgments The authors wish to express their sincere thanks to the referees for their careful reading, valuable comments, and effective suggestions on the improvement of the manuscript.

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