Fixed-Point Theorems for α -Admissible Mappings with w -Distance and Applications to Nonlinear Integral Equations

Two ﬁxed-point theorems for α -admissible mappings satisfying contractive inequality of integral type with w -distance in complete metric spaces are proved. Our results extend and improve a few existing results in the literature. As applications, we use the ﬁxed-point theorems obtained in this paper to establish solvability of nonlinear integral equations. Examples are included.


Introduction and Preliminaries
In 2002, Branciari [1] discussed the existence and uniqueness of fixed points for mappings satisfying contractive condition of integral type, which is a generalization of the Banach contraction principle in metric spaces.
Theorem 1 (see [1]). Let f be a mapping from a complete metric space (X, d) into itself satisfying for all x, y ∈ X, where c ∈ (0, 1) is a constant and ϕ ∈ Φ 2 . en, f has a unique fixed point a ∈ X and lim n⟶∞ f n x � a for each x ∈ X.
In 2012, Samet et al. [2] introduced the concept of α-ψ-contractive type mappings and established some fixedpoint theorems for these mappings in complete metric spaces.
In 1996, Kada et al. [3] introduced the notion of w-distance and elaborated with the help of examples, that the concept of w-distance is general than that of metric on a nonempty set. Several researchers [4][5][6][7] used this notion to obtain some fixed point results not only in usual metric spaces but also in other spaces.
Definition 2 (see [3]). Let (X, d) be a metric space. A function p: X × X ⟶ R + is called a w-distance in X if it satisfies the following: In 2016, Lakzian et al. [8] introduced the concept of generalized (α − ψ − p)-contractive mappings and proved the following fixed point results for such mappings, which generalize eorems 2 and 3.
Theorem 4 (see [8]). Let p be a w-distance on a complete metric space (X, d), and let f: X ⟶ X be an (α − ψ− p)-contractive mapping, that is, where ψ ∈ Φ 1 . Assume that either f is continuous or, for any sequence x n n∈N in X if α(x n , x n+1 ) ≥ 1 for all n ∈ N and x n ⟶ x ∈ X as n ⟶ ∞, then α(x n , x) ≥ 1 for all n ∈ N.
Theorem 5 (see [8]). Let p be a w-distance on a complete metric space (X, d), and let f: X ⟶ X satisfy (4), (b 1 ), (b 2 ), and the following Then, there exists a point u ∈ X such that fu � u.
Motivated by the results in [1][2][3][4][5][6][7][8][9][10], in this paper we prove the existence and uniqueness of fixed points for α-admissible mappings satisfying contractive inequality of integral type via w-distance in complete metric spaces, which are used to study solvability of nonlinear Fredholm and Volterra integral equations. Our results generalize eorems 1-5 and two examples are given.
Throughout this paper, we denote by N the set of positive integers, Lemma 1 (see [10]). Let ϕ ∈ Φ 2 and r n n∈N be a nonnegative sequence.

Fixed-Point Theorems
In this section, we prove the existence and uniqueness of fixed points for α-admissible mapping (6) with w-distance. Theorem 6. Let p be a w-distance in a complete metric space (X, d) and let f: X ⟶ X satisfy that if one of the following conditions holds: (2) f has a unique fixed point u ∈ X if for any fixed points x, y ∈ X of f, there exists a point z ∈ X such that α(z, x) ≥ 1 and α(z, y) ≥ 1.
Proof. Firstly, we show (1). Define a sequence x n n∈N in X by x n+1 � fx n , ∀n ∈ N 0 , where x 0 satisfies (c 2 ). Assume that there exists some n 0 ∈ N with x n 0 � x n 0 − 1 . Put u � x n 0 − 1 . Clearly, u is a fixed point of f and u � lim n⟶∞ f n x 0 . Now, we assume that x n ≠ x n− 1 for all n ∈ N. It follows from (c 1 ) and (c 2 ) that 2 Mathematical Problems in Engineering It is easy to see that In view of (6) and (8) and which implies that which together with (ψ, ϕ) ∈ Φ 1 × Φ 3 and Lemma 1 yields that Let ε > 0 and δ be defined by ( In the following, we claim that x n n∈N is a Cauchy sequence. Making use of (6), (9), (12), (p 1 ), and (ψ, ϕ) ∈ Φ 1 × Φ 3 , we infer that which implies that p x n , x m < δ, ∀m, n ∈ N with m > n ≥ n 0 .
It follows from (14) that which together with (p 3 ) gives that at is, x n n∈N is a Cauchy sequence. By completeness of X, there exists a point u ∈ X such that lim n⟶∞ x n � u. (17) Assume that (c 3 ) holds. Using (17) and (c 3 ), we gain that Assume that (c 4 ) holds. By virtue of (8), (17), and (c 4 ), we deduce that Similar to the proofs of (12)-(14), we know that for each ε 1 > 0 there exists n 1 ∈ N satisfying which together with (p 2 ) and (17) yields that that is, In light of (6), (19), and (ψ, which together with Lemma 1 implies that lim n⟶∞ p x n+1 , fu � 0. In view of (11), (24) and, (p 1 ), we infer that 0 ≤ p x n , fu ≤ p x n , x n+1 + p x n+1 , fu ⟶ 0 as n ⟶ ∞, that is, Let ε 2 > 0. It follows from (p 3 ) that there exists δ 1 > 0 such that p(u, v) ≤ δ 1 and p(u, z) ≤ δ 1 imply d(v, z) < ε 2 . Combining (22) and (26), we know that there exists n 2 ∈ N such that p(x n , u) ≤ δ 1 and p(x n , fu) ≤ δ 1 for all n ≥ n 2 . Hence, d(u, fu) ≤ ε 2 . Letting ε 2 ⟶ 0 + , we have Next, we show that p(u, u) � 0 if α(u, u) ≥ 1. Suppose that p(u, u) > 0. In view of (6) and (ψ, ϕ) ∈ Φ 1 × Φ 3 , we infer that Mathematical Problems in Engineering 3 which is ridiculous. Hence, p(u, u) � 0. Secondly, we show (2). Suppose that x and y are two fixed points of f in X. It follows that there exists a point z ∈ X satisfying In light of (29) and (c 1 ), we get that which together with (6) and which gives that Similarly, we conclude that Making use of (32), (33), and Lemma 1, we get that which together with the proof of (27) yields similarly that x � y.
at is, f has a unique fixed point in X. is completes the proof. □ Theorem 7. Let p be a w-distance in a complete metric space (X, d), and let f: X ⟶ X satisfy (6), (c 1 ), (c 2 ), and the following (c 5 ) inf p(x, y) + p(x, fx): x ∈ X } > 0 for each y ∈ X with y ≠ fy. en, (1) and (2) hold.
Proof. Following the proof of eorem 6, we deduce that x n n∈N is a Cauchy sequence. By completeness of (X, d), there exists a point u ∈ X that satisfies (17) holds. Suppose that u ≠ fu. In light of (11), (17), and (c 5 ), we conclude that which is impossible. Hence, u � fu. e rest of the proof is similar to that of eorem 6 and is omitted. is completes the proof. □ Remark 1. eorem 6 generalizes and improves eorems 1-4. e following example manifests that eorem 6 extends substantially eorem 1.
It is clear that p is a w-distance in X and (ψ, ϕ) ∈ Φ 1 × Φ 3 . Let x, y ∈ X. In order to verify (6), we have to consider the following cases.
Finally, let x n n∈N be a sequence in X such that α(x n , x n+1 ) ≥ 1 for all n ∈ N and x n ⟶ x ∈ X as n ⟶ ∞. It is easy to see that x n ∈ [0, (1/2)] ∪ 2/3 { } for all n ∈ N. Since x n n∈N is a sequence in the closed subset [0, (1/2)] ∪ 2/3 { } in the metric space (X, d), it follows that the point x belongs to [0, (1/2)] ∪ 2/3 { }. erefore, α(x n , x) � 1 for all n ∈ N. Hence, the conditions of eorem 6 are satisfied. It follows from eorem 6 that f has a fixed point 0 ∈ X.
However, we cannot invoke eorem 1 to show that the mapping f has a fixed point in X. Suppose that the conditions of eorem 1 are satisfied. Clearly, y * � (2/3).
which is impossible. us, eorem 1 is not applicable in proving the existence of fixed points for the mapping f in X.
Hence, the conditions of eorem 7 are satisfied. It follows from eorem 7 that f has a fixed point 0 ∈ X.
Mathematical Problems in Engineering 5