arXiv:1507.00188v1 [math.FA] 1 Jul 2015
EXISTENCE OF FIXED POINTS FOR CONDENSING OPERATORS UNDER AN INTEGRAL CONDITION VATAN KARAKAYA, NOUR EL HOUDA BOUZARA, KADRI DOGAN, AND YUNUS ATALAN Abstract. Our aim in this paper is to present results of existence of fixed points for continuous operators in Banach spaces using measure of noncompactness under an integral condition. This results are generalization of results given by A. Aghajania and M. Aliaskaria in [2] which are generalization of Darbo’s fixed point theorem. As application we use these results to solve an integral equations in Banach spaces.
1. INTRODUCTION AND PRELIMINARIES The Darbo’s fixed point theorem which guarantees the existence of fixed point for so called condensing mappings is very famous, since it generalizes two important theorems : The Banach principle theorem and The classical Schauder theorem. Over recent years, many authors presented works that give generalization of this theorem, see Aghajani et al. in [4], [2], [3], Samadi and Ghaemi in [11], [12] and others. Throughout this paper, X is assumed to be a Banach space and BC (R+ ) is the space of all real functions defined, bounded and continuous on R+ . The family of bounded subset, closure and closed convex hull of X are denoted by BX , X and ConvX, respectively. Definition 1. [6] Let X be a Banach space and BX the family of bounded subset of X. A map µ : BX → [0, ∞) is called measure of noncompactness defined on X if it satisfies the following: (1) µ (A) = 0 ⇔ A is a precompact set. (2) A ⊂ B ⇒ µ (A) 6 µ (B) . (3) µ (A) = µ A , ∀A ∈ BX . (4) µ (ConvA) = µ (A) . (5) µ (λA + (1 − λ) B) 6 λµ (A) + (1 − λ) µ (B) , for λ ∈ [0, 1] . (6) Let (An ) be a sequence of closed sets from BX such that An+1 ⊆ An , (n > 1) ∞ T and lim µ (An ) = 0, then the intersection set A∞ = An is nonempty n→∞
n=1
and A∞ is precompact.
Definition 2. A summable function is function for which the integral exists and is finite. 1991 Mathematics Subject Classification. 47H10, 47H08, 45Gxx. Key words and phrases. Measure of noncompactness, Fixed point, Integral condition, integral equation. 1
2 VATAN KARAKAYA, NOUR EL HOUDA BOUZARA, KADRI DOGAN, AND YUNUS ATALAN
Definition 3. Consider a function f : R → R and a point x0 ∈ R. The function f is said to be upper (resp. lower) semi-continuous at the point x0 if f (x0 ) > lim sup f (x) x→x0
(resp.f (x0 ) ≤ lim inf f (x). x→x0
Lemma 1. [4] Let ψ : R+ → R+ be a nondecreasing and upper semi-continuous function. Then, lim ψ n (t) = 0 for each t > 0 ⇔ ψ (t) < t for any t > 0. n→∞
Theorem 1 (Banach contraction theorem). [1] Let X be a Banach space and T : X → X be a contraction mapping on X, i.e. there is a nonnegative real number k < 1 such that kT x − T yk 6 k kx − yk ∀x, y ∈ X. Then the map T admits one and only one fixed point x∗ in X. Theorem 2 (Schauder theorem). [1] Let A be a nonempty, convex, compact subset of a Banach space X and suppose T : A → A is continuous. Then T has a fixed point. The famous Darbo’s theorem is as following Theorem 3 (Darbo’s theorem). [7] Let C be a nonempty closed, bounded and convex subset of X. If T : C → C is a continuous mapping µ (T A) 6 kµ (A) , k ∈ [0, 1) , then T has a fixed point. Theorem 4. [2] Let X be a Banach space and A be a nonempty, closed, bounded and convex subset of a Banach space X and let T : A → A be a continuous operator which satisfies the following inequality ! Z µ(T X) Z µ(X) ϕ (γ) dγ 6 Ψ ϕ (γ) dγ , 0
0
where µ is a measure of noncompactness, Ψ : R+ → R+ a nondecreasing function such that lim Ψn (t) = 0, ∀t > 0 and ϕ : [0, +∞[ → [0, +∞[ is integral mapn→∞
which is summable on each compact subset of [0, +∞[ and for each ǫ > 0, Rping ǫ ϕ (γ) dγ > 0. 0 Then T has at least one fixed point in X. 2. MAIN RESULTS Theorem 5. Let X be a Banach space and A be a nonempty, closed, bounded and convex subset of a Banach space X and let T : A → A be a continuous operator which satisfies the following inequality ! ! Z µ(T X) Z µ(X) (2.1) Φ ϕ (γ) dγ 6 Ψ ϕ (γ) dγ , 0
0
where µ is a measure of noncompactness and (i) Ψ : R+ → R+ is a nondecreasing and concave function such that lim Ψn (t) = n→∞
0, ∀t > 0.
EXISTENCE OF FIXED POINTS
3
(ii) Φ : R+ → R+ is a nondecreasing subadditive function such that Φ (t) > t and lim Φ (xn ) = 0 ⇔ lim xn = 0. n→∞
n→∞
(iii) ϕ : [0, +∞[ → [0, +∞[ is an integral mapping Rwhich is summable on each ǫ compact subset of [0, +∞[ and for each ǫ > 0, 0 ϕ (ω) dω > 0. Then T has at least one fixed point in X. ∞
Proof. Consider (An )n=0 , a closed and convex sequence of subset of X such that An+1 = Conv (T An ). We notice that A1 = Conv (T A0 ) ⊆ A0 and A2 = Conv (T A1 ) ⊆ A1 . By induction, we get ...An+1 ⊆ An ⊆ ... ⊆ A0 . In further, we have Z
µ(An+1 )
ϕ (γ) dγ
µ(Conv(T An ))
Z
=
0
ϕ (γ) dγ
0
Z
=
µ(T An )
ϕ (γ) dγ.
0
Using (2.1) , we get ! Z µ(An+1 ) Z (2.2) Φ ϕ (γ) dγ = Φ 0
µ(T An )
ϕ (γ) dγ 0
Moreover, using Φ (t) > t we get Z µ(An ) Z ϕ (γ) dγ 6 Φ 0
!
µ(An )
Z
6Ψ
ϕ (γ) dγ
0
µ(An )
ϕ (γ) dγ
0
!
,
and Z
µ(An )
ϕ (γ) dγ
=
µ(Conv(T An−1 ))
Z
ϕ (γ) dγ
0
0
=
Z
µ(T An−1 )
ϕ (γ) dγ.
0
Then, Ψ
Z
µ(An )
ϕ (γ) dγ
0
!
=
Z
Ψ
µ(T An−1 )
ϕ (γ) dγ
0
6
Z
Ψ Φ
!
µ(T An−1 )
ϕ (γ) dγ
0
6
Ψ
2
Z
µ(An−1 )
ϕ (γ) dγ 0
Repeating this process n times we get ! Z µ(An+1 ) Z n+1 Φ ϕ (γ) dγ 6 Ψ 0
0
!
µ(A0 )
ϕ (γ) dγ
!!
.
!
.
!
.
4 VATAN KARAKAYA, NOUR EL HOUDA BOUZARA, KADRI DOGAN, AND YUNUS ATALAN
In view of condition (i) , we get lim Ψn+1 n→∞
Z
lim Φ
n→∞
µ(An+1 )
R
µ(A0 ) ϕ (γ) dγ 0
ϕ (γ) dγ
0
!
= 0. Then,
= 0,
using (ii) we obtain lim
n→∞
It follows that
Z
µ(An+1 )
ϕ (γ) dγ = 0.
0
lim µ (An+1 ) = 0.
n→∞
Consequently, A∞ is compact and then T has at least one fixed point.
Remark 1. (i) Φ (x) = x, then Φ : R+ → R+ is nondecreasing additive (hence subadditive) function and lim Φ (xn ) = 0 ⇔ lim xn = 0. Thus, n→∞
n→∞
Φ (x) = x satisfies condition (ii) of theorem (5) and inequality (2.1) will be ! Z µ(T X) Z µ(X) ϕ (γ) dγ 6 Ψ ϕ (γ) dγ , 0
0
which is the condition given by Aghajani and Aliaskari in [2]. (ii) Φ (x) = x, and Ψ (x) = kx, where k ∈ [0, 1[ . Since lim Ψn (x) = lim k n x = n→∞
n→∞
0. Then, Ψ and Φ satisfy conditions (i) and (ii) of theorem (5) . Then, inequality (2.1) become Z µ(T X) Z µ(X) ϕ (γ) dγ 6 k ϕ (γ) dγ, 0
0
which is a generalization of the result given by Branciari in [9]. (iii) Φ (x) = x, Ψ (x) = kx and ϕ (x) = 1. These functions satisfy conditions (i) − (iii) of theorem (5) and instead of inequality (2.1) we obtain the following inequality µ (T X) 6 kµ (X) , which is the condition given by Darbo in his famous fixed point theorem (see [7]). 3. APPLICATIONS In this section we use Theorem 5 to study the resolvability of the following integral equation in the Banach space BC (R+ ) under more general hypothesis. Z t (3.1) x (t) = f t, g (t, s, x (s)) ds, x (t) , t ∈ R+ . 0
In what follow we formulate the assumptions under which equation (3.1) will be studied: (i) The function f : R+ × R+ × R+ → R+ is continuous and |f (t, x, 0)| ∈ BC(R+ ) for t ∈ R+ , and x ∈ R.
EXISTENCE OF FIXED POINTS
5
(ii) There exist a nondecreasing, concave and upper semi-continuous function ψ : R+ → R+ and a nondecreasing, upper semi-continuous and sub-additive function φ : R+ → R+ such that Φ (t) > t and lim φ (xn ) = 0 ⇔ lim xn = n→∞
n→∞
0, for which the function f : R+ × R+ × R+ → R+ satisfies the conditions ! ! Z |f (t,x,y1 )−f (t,x,y2 )| Z |y1 −y2 | Φ ϕ (γ) dγ 6 Ψ ϕ (γ) dγ , Φ
Z
|f (t,x1 ,y)−f (t,x1 ,y)|
ϕ (γ) dγ
!
6Ψ
Z
|x1 −x2 |
ϕ (γ) dγ
!
,
where ϕ : [0, ∞[→ [0, R ǫ∞[ is summable on every compact subset of [0, ∞[ and for every ǫ > 0, 0 ϕ (ω) dω > 0. (iii) The function g : R+ × R+ × R → R is continuous and there exist continuous functions a, b : R+ → R+ such that lim a (t) = 0, b ∈ L1 (R+ ) and t→∞
|g(t, s, x)| 6 a(t)b(s) for t, s ∈ R+ such that s 6 t and each x ∈ R. (iv) There exists at least one positive constant r0 such that the following inequality holds, Z r Z r ϕ (γ) dγ 6 ϕ (γ) dγ + M0 + M1 , 0
where M0 = sup
nR
|a(t)| 0
Rt 0
0
|b(s)|ds
o R sup|f (t,0,0)| ϕ (γ) dγ and M1 = Φ 0 ϕ (γ) dγ .
Theorem 6. Under the hypothesis (i) − (iv) the integral equation (3.1) has at least one solution in the space BC (R+ ).
Proof. Study the solvability of equation (3.1) is equivalent to study the existence of fixed points of the following operator Z t g (t, s, x (s)) ds, x (t) , t ∈ R+ . T x (t) = f t, 0
For that we need to verify that under assumptions (i) − (iv) the operator T satisfies the conditions of Theorem 5. First, let recall the following notions, the measure of noncompactness for a positive fixed t on BBC(R+ ) is given by the following µ (X) = ω 0 (X) + lim sup diamX (t) , t→∞
where, diamX (t) = sup {|x (t) − y (t)| : x, y ∈ X} , X (t) = {x (t) : x ∈ X} , and ω 0 (X) = lim ωL 0 (X) . L→∞
L ωL 0 (X) = lim ω (X, ǫ) , ǫ→0
ω L (X, ǫ) = sup ω L (x, ǫ) : x ∈ X ,
ω L (x, ǫ) = sup {|x (t) − x (s)| : t, s ∈ [0, L] , |t − s| 6 ǫ} , for L > 0.
6 VATAN KARAKAYA, NOUR EL HOUDA BOUZARA, KADRI DOGAN, AND YUNUS ATALAN
To show that T is self-mappings, that is, T map a ball Br0 into itself, let Z
|T x(t)|
Z
ϕ (γ) dγ 6 Φ 0
|T x(t)|
ϕ (γ) dγ
0
=Φ
!
Z |f (t,R t g(t,s,x(s))ds,x(t))| 0
ϕ (γ) dγ
0
!
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (t,R t g(t,s,x(s))ds,0)|+|f (t,R t g(t,s,x(s))ds,0)−f (t,0,0)|+|f (t,0,0)| 0 0 0
6Φ
ϕ (γ) dγ
0
6Φ +Φ
Z
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (t,R t g(t,s,x(s))ds,0)| 0 0
0 R |f (t, 0t g(t,s,x(s))ds,0)−f (t,0,0)|
ϕ (γ) dγ
0
!
+Φ
Z
ϕ (γ) dγ
!
!
|f (t,0,0)|
ϕ (γ) dγ
0
!
.
Using (ii) , we get Z
|T x(t)|
ϕ (ω) dω 6 Ψ
0
!
|x(t)|
Z
ϕ (γ) dγ +Ψ
0
Z |R t g(t,s,x(s))ds| 0
!
ϕ (γ) dγ +Φ
0
Z
sup|f (t,0,0)| t
ϕ (γ) dγ
0
In view of condition (iii) and Lemma 1, we obtain Z
|T x(t)|
ϕ (γ) dγ
6
0
Z
|x(t)|
ϕ (γ) dγ +
0
6
Z
Z
|a(t)|
Rt 0
|b(s)|ds
ϕ (γ) dγ + M1
0
|x(t)|
ϕ (γ) dγ + M0 + M1 .
0
Finally, the assumption (iv) guaranties the existence of a constant r0 such that T Br 0 ⊆ Br 0 . Now, let show that T satisfies Condition 2.1 of Theorem 5. Φ
Z
0
|T x(t)−T y(t)|
ϕ (γ) dγ
!
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (t,R t g(t,s,y(s))ds,y(t))| 0 0
6 Φ
ϕ (γ) dγ
!
ϕ (γ) dγ
!
0
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (t,R t g(t,s,y(s))ds,x(t))| 0 0
6 Φ
0
+Φ
Z |f (t,R t g(t,s,y(s))ds,x(t))−f (t,R t g(t,s,y(s))ds,y(t))| 0 0
ϕ (γ) dγ
0
Using assumption (ii) , we get (3.2) ! Z |T x(t)−T y(t)| Z Φ ϕ (γ) dγ 6 Ψ 0
0
|x(t)−y(t)|
!
ϕ (γ) dγ +Ψ
Z
0
Rt 0
!
|g(t,s,x(s))−g(t,s,y(s))|ds
ϕ (γ) dγ
.
!
!
EXISTENCE OF FIXED POINTS
Moreover, ! Z R t |g(t,s,x(s))−g(t,s,y(s))|ds 0 Ψ ϕ (γ) dγ
6
Ψ
0
L (where L is positve constant) t→∞ we have Z 2|a(t)| R t |b(s)|ds 0 ϕ (γ) dγ 6 ǫ, 0
where ǫ is an arbitrary positive number. Consequently, Inequality 3.2 will be ! Z |T x(t)−T y(t)| Z Φ ϕ (γ) dγ 6 Ψ 0
|x(t)−y(t)|
ϕ (γ) dγ
0
!
.
For t 6 L, we have Z t ωL (g, ǫ) = sup and kx − yk 6 ǫ . |g (t, s, x (s)) − g (t, s, y (s))| ds : t, s ∈ [0, L] , x, y ∈ B r0 1 0
Thus for t ∈ [0, L] , we obtain ! Z |T x(t)−T y(t)| Z Φ ϕ (γ) dγ 6 Ψ 0
|x(t)−y(t)|
ϕ (γ) dγ 0
!
+Ψ
Z
ωL 1 (g,ǫ)
ϕ (γ) dγ 0
!
.
Since g is continuous, it is uniformly continuous on [0, L] × [0, L] × [−r0 , r0 ]. Then, lim ωL 1 (g, ǫ) = 0.
ǫ→0
Consequently, Inequality 3.2 will be ! Z |T x(t)−T y(t)| Z Φ ϕ (γ) dγ 6 Ψ 0
ϕ (γ) dγ
!
.
ϕ (γ) dγ
!
,
|x(t)−y(t)|
0
Thus, for every t > 0 we have ! Z |T x(t)−T y(t)| Z Φ ϕ (γ) dγ 6 Ψ 0
|x(t)−y(t)|
0
hence, Φ
Z
0
lim sup Diam(T X(t)) t→∞
ϕ (γ) dγ
!
6Ψ
Z
0
lim sup Diam(X(t)) t→∞
ϕ (γ) dγ
!
.
8 VATAN KARAKAYA, NOUR EL HOUDA BOUZARA, KADRI DOGAN, AND YUNUS ATALAN
Now, let consider ! Z |T x(t)−T x(l)| Φ ϕ (γ) dγ
=
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (l,R l g(l,s,x(s))ds,x(l))| 0 0
Φ
0
ϕ (γ) dγ
0
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (l,R t g(t,s,x(s))ds,x(t))| 0 0
Φ
6
!
ϕ (γ) dγ
0
+Φ
Z |f (l,R t g(t,s,x(s))ds,x(t))−f (l,R l g(l,s,x(s))ds,x(t))| 0 0
!
ϕ (γ) dγ
0
+Φ
Z |f (l,R l g(l,s,x(s))ds,x(t))−f (l,R l g(l,s,x(s))ds,x(l))| 0 0
ϕ (γ) dγ
0
Z |f (t,R t g(t,s,x(s))ds,x(t))−f (l,R t g(t,s,x(s))ds,x(t))| 0 0
Φ
6
0
Z
+Ψ
Rt 0
|g(t,s,x(s))−g(l,s,x(s))|ds
ϕ (γ) dγ
0
!
+Ψ
Z
ϕ (γ) dγ
!
!
!
|x(t)−x(l)|
ϕ (γ) dγ 0
Putting, ω L (gx, ǫ) =
sup {|g (t, s, x (s)) − g (l, s, x (s))| : l, t, s ∈ [0, L] , x ∈ Br0 and |t − l| 6 ǫ} ,
L
sup {|f (t, x, y) − f (l, x, y)| : l, t ∈ [0, L] , x, y ∈ Br0 and |t − l| 6 ǫ} ,
L
sup {|T x (t) − T x (l)| : l, t ∈ [0, L] , x ∈ Br0 and |t − l| 6 ǫ} ,
ω (f x, ǫ) = ω (T x, ǫ) =
we get ! Z ωL (T x,ǫ) Z Φ ϕ (γ) dγ 6 Φ 0
ω L (f x,ǫ)
!
ϕ (γ) dγ +Ψ
0
Z
ω L (gx,ǫ)
!
ϕ (γ) dγ +Ψ
0
Z
ω L (x,ǫ)
ϕ (γ) dγ
0
We know that g is uniformly continuous on [0, L]×[0, L]×[−r0 , r0 ] and f is uniformly continuous on [0, L] × [−r0 , r0 ] × [−r0 , r0 ] , then we get ! ! Z ωL (T x,ǫ) Z ωL (x,ǫ) Φ ϕ (γ) dγ 6 Ψ ϕ (γ) dγ . 0
0
and then, Φ
Z
ω L (T X,ǫ)
ϕ (γ) dγ
0
!
6Ψ
Z
ω L (X,ǫ)
ϕ (γ) dγ
0
!
.
By taking ǫ → 0, L → ∞ and using the fact that Φ and Ψ are semicontinuous, we obtain ! ! Z ωL0 (T X) Z ωL0 (X) Φ ϕ (γ) dγ 6 Ψ ϕ (γ) dγ . 0
Finally, ! Z µ(T X) Φ ϕ (γ) dγ
0
6
Φ
0
Z
lim sup Diam(T X(t))
Z
lim sup Diam(X(t))
t→∞
ϕ (γ) dγ
0
6
Ψ
0
t→∞
ϕ (γ) dγ
!
!
+Φ
+Ψ
Z
ωL 0 (T X)
ϕ (γ) dγ
0
Z
0
ωL 0 (X)
ϕ (γ) dγ
!
!
!
.
!
.
EXISTENCE OF FIXED POINTS
using the fact that Ψ is concave, we get ! Z lim sup Diam(X(t)) Z t→∞ Ψ ϕ (γ) dγ + Ψ 0
= 6
=
Ψ Ψ Ψ
1 2
Z
lim sup Diam(X(t)) t→∞
2ϕ (γ) dγ
0
Z
!
ϕ (γ) dγ
lim sup Diam(X(t)) t→∞
1 2
+Ψ
2ϕ (γ) dγ +
0
Z
ωL 0 (X)
0
1 2 1 2
9
Z
ωL 0 (X)
2ϕ (γ) dγ
0
ωL 0 (X)
Z
2ϕ (γ) dγ
0
µ(T X)
2ϕ (γ) dγ 0
! !
!!
!
Finally, we obtain Φ
Z
µ(T X)
ϕ (γ) dγ
0
!
6Ψ
Z
µ(X)
ϕ (γ) dγ
0
!
,
and T has a fixed point.
Example 1. let the following integral equation Z t 1 −s2 e cos x (t) ds + ln (1 + x (t)) . (3.3) x (t) = sin t + ln 1 + 2 0 t +1 Considering that by putting f (t, x, y) = sin t + ln (1 + x) + ln (1 + y) and 2 1 e−s cos x, +1 we obtain an equation of the form (3.1) and it satisfies assumptions (i − v). Indeed, it is to see that assumption (i) is satisfied and by taking Φ (x) = kx for k ∈ [0, 1[, Ψ (x) = ln (1 + x) and ϕ (t) = 1, we get ! Z
g (t, s, x) =
t2
|f (t,x,y1 )−f (t,x,y2 )|
Φ
ϕ (γ) dγ
= k |f (t, x, y1 ) − kf (t, x, y2 )|
0
= = 6 6
1 − |y1 | k ln 1 − |y2 | 1 + (|y2 | − |y1 |) k ln 1 + 1 − |y2 | k ln (1 + (|y2 | − |y1 |)) ln (1 + |y2 − y1 |)
= Ψ (|y2 − y1 |) . Hence, Φ
Z
0
|f (t,x,y1 )−f (t,x,y2 )|
ϕ (γ) dγ
!
6Ψ
Z
0
|y1 −y2 |
ϕ (γ) dγ
!
.
10VATAN KARAKAYA, NOUR EL HOUDA BOUZARA, KADRI DOGAN, AND YUNUS ATALAN
The same way we prove that, Φ
Z
|f (t,x1 ,y)−f (t,x2 ,y)|
ϕ (γ) dγ
0
!
6Ψ
Z
|x1 −x2 |
ϕ (γ) dγ
0
!
.
Then, assumption (ii) holds. In further, |g (t, s, x)| = 6
1 −s2 sin x t2 + 1 e 1 −s2 t2 + 1 e , 2
then by taking a (t) = t21+1 and b (s) = e−s , it is easy to see that g, a and b satifies the conditions in (iii) . Finally, Z r Z r ϕ (γ) dγ 6 ϕ (γ) dγ + M0 + M1 , 0
0
R nR |a(t)| t |b(s)|ds
where M0 = sup 0 holds for every positive r.
0
ϕ (γ) dγ
o
and M1 = Φ
R
sup|f (t,0,0)| ϕ (γ) dγ 0
,
References [1] R. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2004. [2] A. Aghajani, M. Aliaskari, Generalization of Darbo’s fixed point theorem and application, Int. J. Nonlinear Anal. Appl., 2 (2011), 86-95. [3] A. Aghajani, R. Allahyari, M. Mursaleen. A generalization of Darbo’s theorem with application to the solvability of systems of integral equations, Comput. Math. Appl. 260 (2014), 68–77. [4] A. Aghajani, J. Bana´s and N, Sabzali, Some generalizations of Darbo fixed point theorem and applictions, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 345-358. [5] R.R. Akmerov, M.I. Kamenski, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhauser-Verlag, Basel, 1992. [6] J. Bana´s, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 21 (1980), 131–143. [7] J. Bana´s, K. Goebel, Measures of Noncompactness in Banach Spaces, in: Lecture Notes in Pure and Applied Mathematics, vol. 60, Dekker, New York, 1980. [8] J. Bana´s, Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Cent. Eur. J. Math. 10(6) (2012), 2003-2011. [9] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, IJMMS, 29:9 (2002), 531–536, PII. S0161171202007524. [10] K. Kuratowski, Sur les espaces complets, Fund. Math., 5 (1930), 301–309. [11] A. Samadi and M. B. Ghaemi, An Extension of Darbo’s Theorem and Its Application, Abstract and Applied Analysis, 2014 (2014), 1-11, Article ID 852324, doi:10.1155/2014/852324. [12] A. Samadi, M. B. Ghaemi, An Extension of Darbo Fixed Point Theorem and its Applications to Coupled Fixed Point and Integral Equations, Filomat, 28:4 (2014), 879–886, DOI 10.2298/FIL1404879S.
EXISTENCE OF FIXED POINTS
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Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yildiz Technical University, Istanbul, Turkey E-mail address:
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