Fixed Points of Integral Type Multivalued Contractive Mappings with w-Distance

School of Mathematics, Dongbei University of Finance and Economics, Dalian, Liaoning 116025, China Department of General Studies, Dalian University of Finance and Economics, Dalian, Liaoning 116000, China Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China Department of General Studies, Dalian Neusoft University of Information, Dalian, Liaoning 116023, China Center for Studies of Marine Economy and Sustainable Development, Liaoning Normal University, Dalian, Liaoning 116029, China


Introduction and Preliminaries
In 2002, Branciari [1] generalized the famous Banach contraction principle and proved the following fixed point theorem for the contractive mapping of the integral type.
Theorem 1 (see [1]). Let T be a mapping from a complete metric space (X, d) into itself satisfying where c ∈ (0, 1) is a constant and φ: R + ⟶ R + is Lebesgue integrable, summable in each compact subset of R + and ε 0 ϕ(t)dt > 0 for each ε > 0. en, T has a unique fixed point a ∈ X and lim n⟶∞ T n x � a for each x ∈ X.
Using Hausdorff metric, Nadler [2] introduced the concept of multivalued contraction and proved a multivalued version of the well-known Banach contraction principle.
Theorem 2 (see [2]). Let (X, d) be a complete metric space and let T be a mapping from X into CB(X), where CB(X) is the family of all nonempty closed and bounded subsets of X. Assume that there exists c ∈ [0, 1) such that H(T(x), T(y)) ≤ cd(x, y), ∀x, y ∈ X. (2) en, T has a fixed point.
In the past decades, various fixed point theorems concerning multivalued contractive mappings have been proved. Especially, Feng and Liu [3] generalized eorem 2 and proved a few fixed point theorems for multivalued contractive mappings without Hausdorff metric.
Theorem 3 (see [3]). Let (X, d) be a complete metric space and T be a multivalued mapping from X into CL(X), where CL(X) is the family of all nonempty closed subsets of X. Assume that In 1996, Kada et al. [4] introduced the concept of w-distance in a metric space and proved several fixed point theorems for single-valued contractive mappings under w-distance. Some other fixed point results concerning w-distance can be found in [5][6][7][8][9]. In 2007, Guran [5] deduced the following fixed point theorem, which is a generalization of eorem 3.
Theorem 4 (see [5]). Let (X, d) be a complete metric space, T: X ⟶ CL(X) be a multivalued mapping, w: X × X ⟶ R + be a w-distance on X, and b ∈ (0, 1). Assume that w(y, T(y)) ≤ cw(x, y). (4) en, T has a fixed point in X. Motivated by the results in [1,[3][4][5], we prove two fixed point results for multivalued contractive mappings of integral type with respect to w-distance in complete metric spaces. e results presented in this paper improve eorems 2-4. Two examples with uncountably many points are included.
roughout this paper, we denote by N the set of positive integers, Definition 1 (see [4]). Let (X, d) be a metric space. A function w: X × X ⟶ R + is called a w-distance in X if it satisfies the following: Let X be a normed linear space with norm ‖·‖, α be a positive constant, and w: X × X ⟶ R + be defined by Proof. Let x, y, z ∈ X. It is clear that (w 2 ) holds and which yields (w 3 ). at is, w is a w-distance in X.
□ Example 2. Let X � R be endowed with the Euclidean metric d � |·| and w: X × X ⟶ R + be defined by where a is a constant in R. en, w is a w-distance in X.
Proof. Let x, y, z ∈ X. It is clear that (w 2 ) holds and which implies (w 1 ). For each ε > 0, put δ � (ε/2). If w(z, x) ≤ δ and w(z, y) ≤ δ, it is easy to see that which yields (w 3 ). at is, w is a w-distance in X.
Let (X, d) be a metric space. For any u ∈ X, D ⊆ X, T: X ⟶ CL(X), and w: A sequence x n n∈N 0 in X is called an orbit of T if x n ∈ T(x n− 1 ) for all n ∈ N. □ Definition 2. Let (X, d) be a metric space and T: X ⟶ CL(X) be a multivalued mapping. A function g: X ⟶ R + is said to be Lower semicontinuous in X if g(y) ≤ lim inf n⟶∞ g(y n ) for each y ∈ X and y n n∈N 0 ⊆ X with lim n⟶∞ y n � y (a 2 ) T-orbitally lower semicontinuous at z ∈ X if g(z) ≤ lim inf n⟶∞ g(x n ) for each orbit x n n∈N 0 of T with lim n⟶∞ x n � z (a 3 ) T-orbitally lower semicontinuous in X if it is T-orbitally lower semicontinuous at each z ∈ X Obviously, if g is lower semicontinuous in X, then g is T-orbitally lower semicontinuous in X.
e following lemmas play important roles in this paper.
Lemma 2 (see [4]). Let X be a metric space with metric d and let w be a w-distance in X. Let x n n∈N and (y n ) n∈N be sequences in X, let (α n ) n∈N and β n n∈N be sequences in R + converging to 0, and let x, y, z ∈ X, then the following hold: (a) If w(x n , y) ≤ α n and w(x n , z) ≤ β n for any n ∈ N, then y � z. In particular, if w(x, y) and w(x, z) � 0, then y � z. (b) If w(x n , y n ) ≤ α n and w(x n , z) ≤ β n for any n ∈ N, then y n n∈N converges to z. (c) If w(x n , x m ) ≤ α n for any n, m ∈ N with n > m, then x n n∈N is a Cauchy sequence. (d) If w(x, x n ) ≤ α n for any n ∈ N, then x n n∈N is a Cauchy sequence.
Lemma 3 (see [11]). Let (X, d) be a metric space, w be a w-distance on X, and D ∈ CL(X). Suppose that there exists u ∈ X such that w(u, u) � 0. en, w(u, D) � 0 if and only if u ∈ D.

Fixed Point Theorems
In this section, we establish fixed point theorems for multivalued contractive mappings (13) and (21), respectively. Theorem 5. Let (X, d) be a complete metric space, w be a w-distance in X, and T: X ⟶ CL(X) satisfy that for each where b and c are constants in (0, 1) with c < b and φ ∈ Φ 2 . en, For each x 0 ∈ X, there exists an orbit x n n∈N 0 of T such that lim n⟶∞ x n � u for some u ∈ X.
Proof. Now, we show (c 1 ). Let x 0 be an arbitrary point in X and a � (c/b). It follows from (13) that there exists Continuing this process, we choose easily a sequence x n n∈N 0 in X satisfying Next, we claim that x n n∈N 0 is a Cauchy sequence. It follows from (15) and φ ∈ Φ 2 that Letting n ⟶ ∞ in (16) and using Lemma 1, we infer that lim n⟶∞ w x n , x n+1 � 0, lim n⟶∞ f w x n � 0. (17)

Mathematical Problems in Engineering 3
Making use of (16), (w 1 ) and φ ∈ Φ 2 , we conclude that which together with Lemmas 1 and 2 yields that x n n∈N 0 is a Cauchy sequence. Completeness of X implies that there exists some u ∈ X such that lim n⟶∞ x n � u.
Finally, we show (c 2 ). Since f w : X ⟶ R + is T-orbitally lower semicontinuous at u, it follows from (17) that which means that f w (u) � 0. us, Lemma 3 and w(u, u) � 0 yield that u ∈ T(u). is completes the proof. □ Theorem 6. Let (X, d) be a complete metric space, w be a w-distance in X, and T: X ⟶ CL(X) satisfy that for each (x, y) ∈ X × T(x), and there exists z ∈ T(y) with where c is a constant in (0, 1) and φ ∈ Φ 2 . en, (c 1 ) and (c 2 ) hold.
Proof. Now, we show (c 1 ). Let (x 0 , x 1 ) be an arbitrary point in X × T(x 0 ). It follows from (21) that there exists Continuing this process, we construct a sequence x n n∈N 0 in X satisfying Next, we claim that x n n∈N 0 is a Cauchy sequence. It follows from (23) and φ ∈ Φ 2 that 0 ≤ w x n ,x n+1 Letting n ⟶ ∞ in (24) and using Lemma 1, we infer that lim n⟶∞ w x n , x n+1 � 0, which together with (23) implies that Combining (25) and (26), we conclude that e rest of the proof is similar to that of eorem 5 and is omitted. is completes the proof.
It is clear that w(u, u) � 0, φ ∈ Φ 2 , and is continuous in X. It follows that f w (x) is lower semicontinuous in X. In order to verify (13), for each x ∈ X, we have to consider the following cases.