Fixed Points and Stability for Integral-Type Multivalued Contractive Mappings

The existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces are proved, and the stability of fixed point sets relative to these multivalued contractive mappings is established. The results obtained in this article generalize and improve some known results in the literature. An illustrative example is given.


Introduction
The famous Banach fixed point theorem has both various extensions and valuable applications in a mass of differential equations, difference equations, functional equations, matrix equations, and integral equations ( ). In 2002, Branciari [3] obtained an interesting integral-type fixed point theorem for the contractive mapping of integral type, which is an integral version of the Banach contraction mapping.
Theorem 1 (see [3]). Let f be a mapping from a complete metric space ðX, ρÞ into itself satisfying where c ∈ ð0, 1Þ is a constant and p : ½0,+∞Þ ⟶ ½0,+∞Þ is Lebesgue integrable, summable on each compact subset of ½0 , +∞Þ and Ð ε 0 pðsÞds > 0 for each ε > 0. Then, f has a unique fixed point u ∈ X such that lim n⟶∞ f n x = u for each x ∈ X.
In 1969, Nadler [15] gave a multivalued analog of the Banach fixed point theorem by using the Hausdorff metric and introducing the multivalued contraction mapping, that is, he presented a nice fixed point theorem for the multivalued contraction mapping.
Theorem 2 (see [10]). Let ðX, ρÞ be a complete metric space and T : X ⟶ CBðXÞ be a multivalued contraction mapping, that is, there exists a constant r ∈ ½0, 1Þ satisfying H Tx, Ty ð Þ≤ rρ x, y ð Þ,∀x, y ∈ X: Then, T has a fixed point in X.
Czerwik [5] and Gordji et al. [8] extended Theorem 2 and proved fixed point theorems for some multivalued contractive mappings, which include (2) as special cases. The researchers [4,11,13,[22][23][24] gained fixed point theorems for several multivalued contractive mappings and studied also the stability of fixed point sets with respect to the multivalued contractive mappings. Lim [11] established the stability of fixed point sets associated with the multivalued contraction mappings in Theorem 2. Choudhury et al. [24] proved that a uniformly convergent sequence of α * − ψ multivalued contractions has stable fixed point sets.
By combining the ideas of Nadler, Branciari, and Lim, in this article, we study the existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces and present stability of fixed point sets relative to a sequence of integral-type multivalued contractive mappings. Our results generalize and unify a few results in [5,8,11,15]. An example is also presented to illustrate the efficiency of our results.

Preliminaries
Throughout this paper, ℕ denotes the set of all positive integers, ℝ + = ½0,+∞Þ, ℕ 0 = f0g ∪ ℕ and Assume that ðX, ρÞ is a metric space, CLðXÞ stands for the family of all nonempty closed subsets of X, and CBðXÞ denotes the family of all nonempty closed bounded subsets of X. For C, D ∈ CLðXÞ and T, fT i g i∈N 0 : X ⟶ CLðXÞ, define A sequence fx n g n∈ℕ 0 in X is called an orbit of T at x 0 if x n+1 ∈ Tx n for each n ∈ ℕ 0 . Lemma 3 (see [12]). Let p ∈ Φ 1 and fr n g n∈ℕ be a nonnegative sequence and lim n⟶∞ r n = a. Then Lemma 4 (see [12]). Let p ∈ Φ 1 and fr n g n∈N be a nonnegative sequence. Then It follows from [13] the following.

Lemma 5.
Assume that ðX, ρÞ is a metric space and C, D ∈ CLðXÞ. Then, for any r > 1 and a ∈ C, there exists b ∈ D such that Lemma 6 (see [13]). Assume that ðX, ρÞ is a metric space. Then Lemma 7. Assume that ðX, ρÞ is a metric space, C ∈ CLðXÞ, and fx n g n∈ℕ ⊂ X converges to a ∈ X. Then

Journal of Function Spaces
Proof. It follows from Lemma 6 that that is This completes the proof.☐ Proof. (see (12)). Let sup C = c. It follows that and there exist a sequence fa n g n∈ℕ in C satisfying lim n⟶∞ a n = c: Thus, (14) and which yields that On account of (15) and Lemma 3, we infer that Clearly, (12) follows from (17) and (18). The proof of (13) is similar to that of (12) and is omitted. This completes the proof.☐

Fixed Point Theorems and an Example
Now, we investigate the existence and iterative approximations of fixed points for the integral-type multivalued contractive mappings (19) and (42), respectively. Theorem 9. Assume that ðX, ρÞ is a complete metric space and T : X ⟶ CLðXÞ satisfies that where q is a constant in ð0, 1Þ and p ∈ Φ 2 . Then, for each x 0 ∈ X, there exists an orbit fx n g n∈ℕ 0 of T at x 0 such that it converges to some fixed point a ∈ X of T and Proof. For any x 0 in X and x 1 in Tx 0 , Lemma 5 guarantees that Note that which together with (19), (21), and p ∈ Φ 2 yields. and Lemma 5 reveals that

Journal of Function Spaces
Notice that which together with (19), (25), and p ∈ Φ 2 infers and Making use of (24) and (28), we deduce Continuing the process, we obtain an order fx n g n∈N 0 of T at x 0 satisfying Thus, (30), q ∈ ð0, 1Þ, and p ∈ Φ 2 mean By (31) and p ∈ Φ 2 , we obtain It is clear that (33), q ∈ ð0, 1Þ, p ∈ Φ 2 , and Lemma 8 guarantee Hence, fx n g n∈N 0 is a Cauchy sequence. Completeness of ðX, ρÞ means that there exists a point a in X with Letting m ⟶ ∞ in (33) and using (37) and Lemma 3, we arrive at that is, (20) holds.
where q is a constant in ð0, 1Þ and p ∈ Φ 2 . Then, for each x 0 ∈ X, there exists an orbit fx n g n∈N 0 of T at x 0 such that it converges to some fixed point a ∈ X of T and (20) holds.