Generalizations of Darbo’s fixed point theorem for new condensing operators with application to a functional integral equation

Abstract: In this paper, we provide some generalizations of the Darbo’s fixed point theorem associated with the measure of noncompactness and present some results on the existence of the coupled fixed point theorems for a special class of operators in a Banach space. To acquire this result, we define α-ψ and β-ψ condensing operators and using them we propose new fixed point results. Our results generalize and extend some comparable results from the literature. Additionally, as an application, we apply the obtained fixed point theorems to study the nonlinear functional integral equations.


Introduction and preliminaries
The study of nonlinear integral equations, nowadays, is a subject of interest for many researchers in nonlinear functional analysis. Integral equations arise in many practical problems including potential theory and other physics-related problems. On the other hand, fixed point theory is one of the most effective and fruitful tool used in nonlinear analysis to solve functional integral equations. It's concerned with the conditions for the existence of one or more fixed points of a mapping T from a topological space X into itself. Brouwer [1] established a fixed point result what has become the well-known Brouwer's fixed point theorem for finite dimensional spaces. While in 1922, Banach [2] introduced his celebrated Banach contraction principle for complete metric spaces which guarantee the existence and uniqueness of fixed point. Afterwards, in 1930, Schauder [3] extended the Brouwer's fixed point theorem to infinite dimensional spaces using the condition of compactness. There are many developments in fixed point theory in various directions, one among them is single-valued mappings (see [4][5][6][7][8][9][10] and references therein). Furthermore, Kuratowski [11] in 1930, opened up a new direction of research with the introduction of the concept of a measure of noncompactness, which gives the degree of noncompactness for bounded sets. The measure of noncompactness can also be used in the study of single-valued and multivalued mappings, especially in metric and topological fixed point theory. The measure of noncompactness combining with some algebraic arguments is beneficial for studying mathematical formulations, especially solving the existence of solutions of some nonlinear problems under certain situations.

Main results
Definition 2.1. (α-ψ condensing operator) Let E be a Banach space and let T : E → E be a given operator. We say that T is an α-ψ condensing operator if there exist two functions α : E × E → [0, +∞) and ψ ∈ Ψ such that α(x, Tx)µ(TX) ≤ ψ(µ(X)), for any bounded subset X of E and x ∈ X with µ an arbitrary measure of noncompactness.
Theorem 2.2. Let C be a nonempty, bounded, closed and convex subset of a Banach space E and there exists α : E × E → [0, +∞) such that T : C → C is a continuous, α-admissible and α-ψ condensing operator satisfying the following: (i) there exist closed and convex X 0 ⊆ C and x 0 ∈ X 0 such that where µ be an arbitrary measure of noncompactness and ψ ∈ Ψ . Then T has at least one fixed point in the set C.
If there exists an integer N ≥ 0 such that µ(X N ) = 0, then X N is a relatively compact set and also TX N ⊆ X N . Thus, Theorem 1.3, implies that T has fixed point. Next, we assume that µ(Xn) ≠ 0 for any n ≥ 0. From equation (2.1) we have α(x 0 , x 1 ) = α(x 0 , Tx 0 ) ≥ 1, and also T is a α-admissible operator implies that α(x 1 , x 2 ) ≥ 1.
Recursively, we get the following inequality Furthermore, from our assumptions and equation (2.2), we have continuing in this manner, we reach the following inequality Thus, equation (2.3) implies that µ(Xn) → 0 as n → ∞. Since the sequence {Xn} is nested so from Definition 1.1 (axiom MNC6), we deduce that the set X∞ = ⋂︀ ∞ n=1 Xn is nonempty, closed and convex subset of the set X 0 . On the other hand, µ(X∞) ≤ µ(Xn), ∀ n ∈ N implies that µ(X∞) = 0. Hence we get that X∞ is a member of the kerµ, which implies X∞ is compact. Moreover, we have X∞ ⊂ Xn and T(Xn) ⊂ Xn for all n ∈ N. Therefore, T : X∞ → X∞ is well defined. For any bounded A ⊂ X∞, we have T(A) ⊂ X∞ and T(A) is a compact subset of X∞, implies that T is compact operator. Therefore, Theorem 1.3, completes the proof. Remark 2.3. In Theorem 2.2, we get Darbo's theorem if we take α(x, y) = 1 and ψ(t) = kt for all t ≥ 0 and for k ∈ [0, 1). Now using the above theorem, we prove the following corollary which belongs to the classical metric fixed point theory.
Corollary 2.4. Let C be a nonempty, bounded, closed and convex subset of a Banach space E and there exists α : E × E → [0, +∞) such that T : C → C is a continuous and α-admissible operator satisfying the following: (ii) there exist closed and convex X 0 ⊆ C and x 0 ∈ X 0 such that where ψ ∈ Ψ . Then T has at least one fixed point in C.
where diamX = sup{‖x − y‖ : x, y ∈ X} stands for the diameter of X. It is easy to see that µ is a measure of noncompactness in a space E in the sense of Definition 1. Proof. Let F = {x ∈ C : Tx = x} be the set of all fixed points of T and µ(F) ≠ 0, then by α-ψ condensing operator of T we have which is a contradiction from above inequality since T(F) = F. This implies that F is a relatively compact set. Now taking into account any convergent sequence {xn} ⊂ F and xn → x * , we have x * ∈ C, because C is closed. The continuity of T implies that xn = Txn → Tx * and Tx * = x * , which means that x * ∈ F, i.e. F is a compact set.
Example 2.6. The operator T : BC(R+) → BC(R+) defined by and let BC(R+) denote the space of all real-valued bounded and continuous functions on R+. First, we observe that Theorem 1.4, cannot be applied in the case when ‖x‖, ‖y‖ > 1 and we obtain and by taking supremum value on both sides we have µ(TX) = 2µ(X). Now, we define the mapping α : Clearly, T is a α-ψ condensing operator with ψ(t) = t 2 for t ≥ 0, and µ(X) = diamX.
In order to prove the next results, we need the following definitions.
We say that T is β-admissible operator if for every X ∈ 2 E , we have β(X) ≥ 1 ⇒ β(co TX) ≥ 1. We say that T is β-ψ condensing operator if there exist two functions β : 2 E → [0, +∞) and ψ ∈ Ψ such that for any bounded subset X of E with µ an arbitrary measure of noncompactness.
Theorem 2.11. Let C be a nonempty, bounded, closed and convex subset of a Banach space E and there exists β : 2 E → [0, +∞) such that T : C → C is a continuous, β-admissible and β-ψ condensing operator satisfying the following: (i) there exist closed and convex X 0 ⊆ C such that

7)
where µ be an arbitrary measure of noncompactness and ψ ∈ Ψ . Then T has at least one fixed point in C.
Proof. Similarly, as in the proof of Theorem 2.2, we define the following sequence: Xn = co(TX n−1 ).
If there exists an integer N 0 ≥ 0 such that µ(X N0 ) = 0, implies X N0 is relatively compact and also TX N0 ⊆ X N0 . Thus, Theorem 1.3, implies that T has fixed point. Moreover, we assume that µ(Xn) ≠ 0 for all n ≥ 0. For T to be a β-admissible operator and from equation (2.7), we obtain β(X 1 ) = β(co TX 0 ) ≥ 1. Recursively, we obtain the following inequality continuing in this manner, we reach at the following inequality Equation (2.9) implies that µ(Xn) → 0 as n → ∞. Since the sequence {Xn} is nested and in view of Definition 1.1(axiom MNC6), we deduce that the set X∞ = ⋂︀ ∞ n=1 Xn is nonempty, closed and convex subset of the set X 0 . Hence, we get that X∞ is a member of the kerµ and T maps X∞ into itself and taking into account Theorem 1.3, gives the desired result.
Remark 2.12. From Theorem 2.11, we get Darbo's theorem if we take β(X) = 1 and ψ(t) = kt for all t ≥ 0 and for some k ∈ [0, 1). Now let us pay attention to the following corollary from the above theorem which belongs to the classical metric fixed point theory.
Corollary 2.13. Let C be a nonempty, bounded, closed and convex subset of a Banach space E and there exists β : 2 E → [0, +∞) such that T : C → C is a continuous and β-admissible operator satisfying the following: (i) for any X ∈ M C and x, y ∈ X, we have (ii) there exist closed and convex X 0 ⊆ C such that where ψ ∈ Ψ . Then T has at least one fixed point in the set C.
Proof. Let µ : M E → R+ be defined as: where diamX = sup{‖x − y‖ : x, y ∈ X} stands for the diameter of X. Clearly, from Definition 1.1, µ is a measure of noncompactness in the space E. Further, from (2.10) and ψ to be nondecreasing we have So from Theorem 2.11, we get desired result.
Proposition 2.14. If β(X) ≥ 1 for all X ∈ 2 E having T(X) = X, then the set of all fixed points of T in Theorem 2.11, is a compact set.
Proof. Let F = {x ∈ C : Tx = x} be the set of all fixed points of T and µ(F) ≠ 0, from the assumption of which is a contradiction from above inequality since T(F) = F. This implies that F is a relatively compact set. Now taking into account any convergent sequence {xn} ⊂ F and xn → x * , we have x * ∈ C, because C is closed. The continuity of T implies that xn = Txn → Tx * and Tx * = x * , which means that x * ∈ F, i.e. F is a compact set.

Example 2.15. The operator T : BC(R+) → BC(R+) defined by
At first we take a look at that Theorem 1.4, cannot be carried out in the case while ‖x‖, ‖y‖ > 1 and we obtain Through taking supremum value on both sides we have However, if we define β : 2 BC(R+) → [0, +∞) by Certainly T is an β-ψ condensing operator with ψ(t) = t 2 for t ≥ 0, and µ(X) = diamX.

Coupled fixed point theorems
In this section, we prove some coupled fixed point theorems using α-admissible and β-admissible operators.
Additionally, as a result of Lemma 2.17, we present the following examples.
Theorem 2.21. Let C be a nonempty, bounded, closed and convex subset of a Banach space E and : E 2 ×E 2 → [0, +∞), let T : C × C → C be continuous and also fulfilling the following conditions:

12)
for any (x 1 , y 1 ) ∈ X 1 × X 2 ; and where µ be an arbitrary measure of noncompactness on E and ψ ∈ Ψ . Then T has at least one coupled fixed point in C × C.
Proof. To prove this theorem, we need to define G : From Example 2.19, we take µ * is a measure of noncompactness on E 2 as follows: where X 1 and X 2 are the natural projections of X on E. Further, we define α : X 2 × X 2 → [0, +∞) as follows: We need to show that G satisfies all the conditions of Theorem 2.2, then G has a fixed point in C × C, which is the coupled fixed point of the operator T. We list the following conditions which we want to meet for the desired result: • G is continuous and α-admissible operator; )︀ , for (x 1 , y 1 ) ∈ X; • There exists closed and convex subset A 0 ⊆ C × C, and (x 0 , y 0 ) ∈ A 0 such that We can easily see from equation ( Eventually, from equations (2.16) and (2.17) we have (i) let X 1 × X 2 ⊆ C × C and T(x 1 , y 1 ) = x 2 , T(y 1 , x 1 ) = y 2 such that

23)
where µ be an arbitrary measure of noncompactness on E and ψ ∈ Ψ . Then T has at least one coupled fixed point in C × C.

Theorem 2.23. Let C be a nonempty, bounded, closed and convex subset of a Banach space E and : 2 E1×E2 →
[0, +∞), let T : C × C → C be continuous and satisfying the following conditions: (i) for any X 1 × X 2 ⊆ C × C, we have where µ be an arbitrary measure of noncompactness and ψ ∈ Ψ . Then T has at least one coupled fixed point in C × C.
Proof. Firstly, we define G : also take µ * is measure of noncompactness on E 2 as follows: where X 1 and X 2 are the natural projections of X on E. Further, we define β : 2 E1×E2 → [0, +∞) in the following way: where X 1 and X 2 are the natural projections of X on E. To get required result we need to show that G satisfies all the conditions of Theorem 2.11, which are following: • G is continuous and β-admissible operator; • β(X)µ * (G(X)) ≤ ψ (︀ µ * (X) )︀ ; • There exists closed and convex subset A 0 ⊆ C × C such that GA 0 ⊆ A 0 and β(A 0 ) ≥ 1.
Clearly G is continuous and also whenever β(X 1 × X 2 ) ≥ 1, we have β (︀ co G(X 1 × X 2 ) )︀ ≥ 1, which shows that G is β-admissible. From our hypothesis we have From equations (2.28), (2.29) and (2.30) we obtain Finally, from equations (2.27) and (2.28), we get the following inequality In the end, from equation ( (i) for any X 1 × X 2 ⊆ C × C, we have where µ be an arbitrary measure of noncompactness and ψ ∈ Ψ . Then T has at least one coupled fixed point in C × C.

A functional integral equation
In this section, we are going to present an application of Theorem 2.11, a study of existence of solution for an integral equation defined on the Banach spaces BC(R+), which includes all continuous real-valued and bounded functions on R+ and equipped with the norm, i.e.
The measure of noncompactness on BC(R+) [21,[30][31][32] for a positive fixed t on M BC(R+) is defined as follows: . Before defining the ω 0 (X), we first need to define the modulus of continuity for any x ∈ X and ϵ > 0. The modulus of the continuity of x on the interval [0, T] denoted by ω T (x, ϵ), i.e. where As an application of the Theorem 2.11, we are going to have a look at the existence of the solution for the following integral equation: For this cause, we assume the following conditions: i) the function A : R+ → R+ is continuous and bounded with M 1 = sup{|A(t)| : t ∈ R+}; ii) ξ , η, β : R+ → R+ are continuous functions and ξ (t) → ∞ as t → ∞; iii) the function φ is continuous and there exist α, δ > 0, such that for any t 1 , t 2 ∈ R+ and moreover, φ(0) = 0; iv) the functions h : R+ × R → R and f : R+ × R × R → R are continuous and ψ ∈ Ψ , and there exists nondecreasing continuous function θ : R+ → R with θ(0) = 0. Also, there exists ζ : R 2 → R with ζ (x 1 , y 1 ) ≥ 0 such that and for all x 1 , y 1 ∈ R for any t ≥ 0; v) the functions defined by t → |h(t, 0)| and t → |f (t, 0, 0)| are bounded on R+; i.e. where M 4 is a positive constant defined by the following equalities uniformly with respect to x, u ∈ BC(R+); vii) for ζ (x(t), y(t)) ≥ 0, for all x, y ∈ X ⊆ BC(R+) and for any t ∈ R+, implies that ζ (u(t), v(t)) ≥ 0, for all u, v ∈ coT(X) and for any t ∈ R+. Moreover, ζ (x 0 (t), y 0 (t)) ≥ 0 for all x 0 , y 0 ∈ Br 0 (ball of radius of r 0 in BC(R+)), for any t ∈ R+.
Theorem 2.25. Suppose that (i)-(vii) holds; then the system of integral equation has at least one solution in the space BC(R+).
Proof. Let T : BC(R+) → BC(R+) be an operator defined by Moreover, the space BC(R+) is equipped the following norm: We can easily show that the solution of equation (2.44) in BC(R+) is equivalent to the fixed point of T. Obviously Tx is continuous function for any x ∈ BC(R+). Furthermore, using the triangular inequality and ζ (x(t), 0) ≥ 0 for t ∈ R+, and additionally by means of our assumptions we obtain (2.47) So, from above equation and making use of equations (2.41) and (2.42), we have Thus, T is well-defined and we obtain T(Br 0 ) ⊂Br 0 . Further, we prove that the mapping T :Br 0 →Br 0 is continuous. Let x, u ∈Br 0 such that ζ (x(t), u(t)) ≥ 0 for t ∈ R+ and for ϵ > 0, ‖x − u‖B r 0 < ϵ 2 , then we have (2.49) Now using the equation (2.43) there exists T > 0 such that if t > T, Then we have for any x, u ∈ BC(R+). Now we consider the following two cases: Case 1. If t > T, then from equations (2.49) and (2.50) we get Finally, from equations (2.51) and (2.52), we conclude that T is a continuous function fromBr 0 intoBr 0 . Now we show that the map T satisfies all the conditions of Theorem 2.11. To do this, for an arbitrary T > 0 and ϵ > 0, assume that X 1 are arbitrary nonempty subsets ofBr 0 and t 1 , t 2 ∈ [0, T], with |t 1 − t 2 | ≤ ϵ. Without loss of generality let β(t 1 ) ≤ β(t 2 ), and ζ (x(t 1 ), x(t 2 )) ≥ 0, for any arbitrary x ∈ X 1 , we have (2.54) Now we make the following substitutions }︁ , }︁ .

Conclusion
Taking into account its interesting applications, looking for newly fixed point theorems concerning the new setup of contractive type conditions has acquired considerable attention over the last few decades. In this regard, the main purpose of this paper is to provide new ideas of α-ψ and β-ψ condensing operators and make use of them to establish a new fixed point and coupled fixed point theorems. An application to a solution of the functional integral equation is illustrated to the usability of the obtained fixed point results.