EXISTENCE RESULTS FOR SINGLE AND MULTIVALUED MAPPINGS IN METRIC SPACES WITH APPLICATIONS

. We present some new existence results for single and multivalued mappings in metric spaces on very general settings. Some illustrative examples are presented to validate our theorems. Finally, we discuss an application to the Volterra-type integral inclusions


INTRODUCTION
The Banach contraction principle (BCP) is one of the most simple, elegant and classical tool in non-linear analysis with numerous extensions and generalizations. Some of the notable early generalizations and extensions of the BCP can be found in [3,5,6,8,12,17,19,26]. For more details, one may refer to Rhoades [27]. Some recent extensions and generalizations of the BCP can be found in [10,13,22,25,29,33]. In 1969, Boyd and Wong [3], presented an important generalization of the BCP by replacing the contraction constant with a real valued control function.
In 2018, Bisht [2] claimed that the Lemma 1.3 is incorrect. He presented a counter example for his claim. He obtained the following theorem.
In 2008, Suzuki [33] introduced a new class of contraction mappings where the contraction condition to be hold only on certain elements of the underlying space. He presented a remarkable generalization of the BCP which also characterises completeness of the metric space.
Theorem 1.6. [33]. Suppose (E, ρ) is a complete metric space and f : E → E a self-mapping such that for all x, y ∈ E and h ∈ [0, 1), where θ : [0, 1) → 1 2 , 1 is a non-decreasing function such that Then f has a unique fixed point in E.
Motivated by [3], [25], [29], [33] and others, we obtain some new fixed point results for single and multivalued mappings. The article is organized as follows. The section 1 is introductory.
In section 2, we obtain a generalization of the BCP, Theorems 1. In section 4, we discuss an application to Volterra-type integral inclusion problems.

FIXED POINT RESULTS FOR SINGLE-VALUED MAPPINGS
Now onwards, N denotes the set of all natural numbers, R the set of all real numbers, and ϕ : The orbit of f at some u 0 ∈ E is defined as for some u ∈ E converges in E then E is said to be f -orbitally or orbitally complete [5].
The following lemma will be used to prove our theorems.
Lemma 2.1. Assume that (E, ρ) be a metric space. Let (v n ) be a non-Cauchy sequence in E such that lim n→∞ ρ(v n , v n+1 ) = 0. Then there exist an ξ > 0 and two sequences (m(k)) and (n(k)) of positive integers such that: Proof. Proof is trivial. Therefore we omit it.
Next, we present our main result of this section.
Theorem 2.2. Suppose (E, ρ) is a metric space. Let f : E → E a mapping such that for some v 0 in E, .
If E is forbitally complete then the sequence of iterations ( f n (v 0 )) is Cauchy in E and converges to Thus ρ n+1 ≤ ϕ(ρ n ) < ρ n . So, the sequences (ρ n ) and (ϕ(ρ n )) are bounded below and monotone decreasing. This implies that lim n→∞ ρ n and lim n→∞ ϕ(ρ n ) exist.
Suppose lim n→∞ ρ n = ρ > 0 and ρ n = ρ + ξ n with ξ n > 0. Since for all t > 0, lim sup s→t + ϕ(s) < t for (t n ) with t n ↓ ρ + , we have lim sup t n →ρ + ϕ(t n ) < ρ. Hence, we get a contradiction. Thus lim n→∞ ρ n = 0. This shows that f is asymptotically regular at point v 0 of E. Using Lemma 2.1, it is easy to show the sequence (v n ) is Cauchy. Since E is f -orbitally complete therefore there exists z ∈ E such that v n → z as n → ∞. Also, the sequence of iterations Now, for all n ∈ N, we show that Assuming by contradiction, we suppose that for all n ∈ N. Then, by triangle inequality we have This is a contradiction. Thus, the inequality (2.3) is true for all n ∈ N.
In the first case, since which is a contradiction unless f (z) = z. Similarly, in the other case, we can deduce that f (z) = z. Uniqueness of fixed point follows easily.
Define the mapping f : E → E by Hence Thus all the hypothesis of Theorem 2.2 are verified and f has a fixed point at (12, 5.5).
However for x = (0, 0) and y = (0, 10), If E is f -orbitally complete then the sequence of iterations ( f n (v 0 )) is Cauchy in E and converges to the unique fixed point of f in O(v 0 , f ).
If we replace N(x, y) with M(x, y) in Theorem 2.2, then we get the following result.
If E is f -orbitally complete then the sequence of iterations ( f n (v 0 )) is Cauchy in E and con- The following result is a consequence of Corollary 2.5 which generalizes the Theorem 1.5 without using the assumption of orbitally continuity on f .
If E is f -orbitally complete then the sequence of iterations ( f n (v 0 )) is Cauchy in E and converges to the unique fixed point of f in O(v 0 , f ).
If we take N(x, y) = ρ(x, y) in Theorem 2.2, then we get following extension of Theorem 1.4 and 1.6.
If E is f -orbitally complete then the sequence of iterations ( f n (v 0 )) is Cauchy in E and converges to the unique fixed point of f in O(v 0 , f ).
Next, we get a generalized version of Matkowski's fixed point result [15].
Let (E, ρ) be a metric space and let f : E → E be a mapping such that for some v 0 ∈ E, If E is f -orbitally complete then the sequence of iterations ( f n (v 0 )) is Cauchy in E and con- Similarly, we get a generalized version of Boyd and Wong's [3] result as follows: Corollary 2.9. Let (E, ρ) be a metric space and let f : E → E be a mapping such that for some v 0 ∈ E, where ψ is defined in Theorem 1.1. If E is f -orbitally complete then the sequence of iterations

FIXED POINT RESULTS FOR MULTIVALUED MAPPINGS
In this section, we discuss fixed point results for multivalued mappings on very general settings. First, we recall some notations, definitions and results from [5], [19] and [20]. Let (E, ρ) Then H(A, B) ≤ ϑ .
Definition 3.2. [28,30]. A mapping F : E → CB(E) is called asymptotically regular at v 0 ∈ E, If F is asymptotically regular at each point v of E then we called F is asymptotically regular on E.
Then it is easily seen that for each u ∈ E and (u n ) ∈ E such that u n ∈ F(u n−1 ), ρ(u n , u n+1 ) → 0.
Thus F is asymptotic regular on E.
In the above definition, if we take F = f as a single-valued self mapping on E then we get the following definition of asymptotic regularity [4].
and if equality (3.1) is true for all u ∈ E, then the mapping f is called asymptotically regular on

E.
A mapping f is asymptotically regular at its fixed points but the converse is not always true.
The following example illustrates this fact. Then for any u ∈ E, we have f n (u) = log[e u + n] and lim n→∞ ρ( f n (u), f n+1 (u)) = 0.
However, the mapping f is fixed point free.  Hicks and Rhoades [9] introduced the following notion of orbital lower semi-continuity.  [11]). Remark 3.11. The condition of F-orbitally continuity is more general than the condition of orbital continuity and k-continuity for k ≥ 1 (see [ for all x, y ∈ E, where D(x, y) = ρ(x, y) + µ[ρ(x, F(x)) + ρ(y, F(y))] and µ ≥ 0. Then, there exists a sequence of iterations (v n ) ∈ E starting from v 0 , converges to a point z ∈ E. If g : E → R defined by g(x) = ρ(x, F(x)) for all x ∈ E, is lower semi-continuous at the point z then z is a fixed point of F.

Such a choice is permissible because
Continuing in the same manner, we get a sequence Since F is asymptotically regular at v 0 then Then by Lemma 2.1, there exist ξ > 0 and two positive sequences n(k), m(k) with k ≤ m(k) < n(k) such that
Making k → +∞ and using Lemma 2.1, we have We also note that lim k→+∞ D(v m(k) , v n(k) ) = ξ . Then by lim sup s→t + ϕ(s) < t for all t > 0, we get a contradiction unless ξ = 0. Thus, (v n ) is a Cauchy sequence. Now the E is complete there exists z ∈ E such that v n → z as n → ∞. If g is lower semi-continuous at the point z then we have The compactness of F(z) implies z ∈ F(z).
The following example illustrates the validity of our Theorem 3.12.
Example 3.13. Let E = [0, 1] and ρ(x, y) = |x − y| for x, y ∈ E be a usual metric on E. Define Then F is asymptotic regular on E and the mapping , if x = 1 n , n ∈ N, , Similarly, when x = 1 n and y = 1 m , we find that H(F(x), F(y)) = 1 Corollary 3.14. Let (E, ρ) be a complete metric space and f : E → E is an asymptotically regular at some point v 0 ∈ E such that for all x, y ∈ E, where L(x, y) is defined in Theorem 1.2. Then the sequence of iterations (v n ) ∈ E starting from v 0 , converges to a point z ∈ E. If g(x) = ρ(x, f (x)) for all x ∈ E is lower semi-continuous at the point z then z is a fixed point of f .
The following result is an extension of Theorem 2.2 in setting of multivalued mappings.
Theorem 3.15. Let (E, ρ) be a metric space and let F : E → CB(E) be a multivalued asymptotically regular mapping at some point v 0 ∈ E such that where D(x, y) is defined in Theorem 3.12.
If E is f -orbitally complete then any sequence (v n ) ⊆ O(v 0 , F) is convergent to a point z ∈ O(v 0 , F). If g(x) = ρ(x, F(x)) for all x ∈ E, is F-orbitally lower semi-continuous at the point z then z is a fixed point of F.
We claim that (v n ) is a Cauchy sequence. If not then by Lemma 2.1, there exist ξ > 0 and two sequences m(k), n(k) with k ≤ m(k) < n(k) such that Furthermore, by (3.6) and (3.7), we have Then, by triangle inequality and contraction (3.5), we have Letting k → +∞ and from Lemma 2.1, we get , v n(k) )).
Also, we note that lim k→+∞ D(v m(k) , v n(k) ) = ξ and by lim sup Taking ϕ(t) = t 2 , we get δ (F(x), F(y)) ≤ ϕ(ρ(x, f (x))). Hence F satisfies all the assumptions of Theorem 3.15 and F has a fixed point in E. Here F has two fixed points at x = 0 and 1 in E.
where L(x, y) is defined in Theorem 1.2. If E is f -orbitally complete then the sequence of iterations ( f n (v 0 )) is convergent to a point z ∈ E. If g(x) = ρ(x, f (x)) for all x ∈ E is f -orbitally lower semi-continuous at the point z then z is a fixed point of f . One can easily verify that at x = 8, f is asymptotically regular and O(6, f ) = E. Also, for all x, y ∈ E ρ( f (x), f (y)) ≤ 7 13 L(x, y).

APPLICATION TO VOLTERRA-TYPE INTEGRAL INCLUSION
In this section, we study the existence result for integral inclusion of Volterra-type. Consider |υ(t) − ω(t)|. A selection for F is a continuous mapping f : E → E such that f (υ) ∈ F(υ) ( see [18]) and
Now, under the above assumptions, we have the following theorem.
Proof. Since the F is asymptotic regular and g is lower semi-continuous mapping on E. Now, we will prove that F satisfies condition (3.2) of Theorem 3.12. Let υ ∈ E and υ(t) ∈ F(υ).
Thus, all the assumptions of Theorem 3.12 are verified and hence the inclusion problem (4.1) has a solution in E.