Multi-valued versions of Nadler, Banach, Branciari and Reich fixed point theorems in double controlled metric type spaces with applications

1 Department of Mathematics, University of Malakand, Chakdara Dir(L), KPK, Pakistan 2 Department of Mathematics and General Sciences, Price Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia 3 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 4 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan 5 Department of Mathematics, Islamia College Peshawar, Peshawar, KPK, Pakistan


Introduction
Numerous applications in engineering and scientific fields can be managed through Fredholm or Volterra integrals. A substantial amount of initial value and boundary value problems can be transformed to Fredholm or Volterra integral equations. Applications of which can be found in areas of mathematical modeling of physics and biology. For instance, one can observe the applications of these equations in biological sciences such as the heat transfer or heat radiation and biological species living together, the Volterra population growth model, scattering in quantum mechanics, kinetic not just stop here. Still, authors are finding ways to restrict or control the triangle inequalities and introducing some new metric type spaces. In recent times, Mlaiki et al. [35] introduced the concept of control metric spaces. Which is in fact, the extension of b-metric spaces and extended b-metric spaces. Abdeljawad et al. [49] modified the control metric space via two control functions which generalizes b-metrics, extended b-metrics and control metric spaces.
In this study, inspired theoretically from the above contribution, we establish some multi-valued fixed point theorems in the setting of double controlled metric spaces and then the established results are used to analyze the existence of solution to a Volterra integral inclusion and singular Fredholm integral inclusions of both types. The obtained results can be utilized in many of research problems seeking existence of solution to various kinds of integral equations. For example one can also use these results to obtain the of solution to Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < β < 2 [54].
Then ρ is a metric on X. The pair (X, ρ) is called a metric space.
Remark 2.4. From the above example it can be concluded that in general a b-metric is not continuous. It is obvious that for S = 1, every b-metric is a standard metric. Definition 2.5. Let (X, ρ) be a b-metric space, such that for all ζ, η ∈ X.
Definition 2.6. [51] Let X be a nonempty set. Define a distance function ρ : , be a function such that for all ζ, η, ν ∈ X, if ρ satisfies Then ρ is an extended b-metric on X. The pair (X, ρ) is called an extended b-metric space. And Note that the first two axioms in Definition (2.6) hold, trivially. For the real numbers x, y, with x = 0 or | x |≥ 1 and y = 0, or | y |≥ 1 we establish the following relation.
Recently, Mlaiki et al. [35] generalized the notion of b-metric space to control metric space.
Definition 2.8. [35] Let X be a nonempty set. Define a distance function ρ : X × X → [0, ∞). Let α : X 2 → [1, ∞), be a function such that for all ζ, η, ν ∈ X, if ρ satisfies Then ρ is a control metric on X. The pair (X, ρ) is called a control metric space. Thus the above metric is a control metric. However, it is not an extended b-metric as Thabet et al. [49] introduced double controlled metric type space.
Remark 2.11. A controlled metric type is in fact, a double controlled metric. However, the converse is not true as is verified by the examples given below. It is simple to show that the above metric is a double controlled metric. However, we can deduce that ρ is not an extended b metric as And define the functions α, β : It is obvious that ρ 1 and ρ 2 in Definition (2.10) hold. We claim that (ρ 3 ) is satisfied in Definition (2.10).
On the opposite side, we have So, the above double controlled metric is not a control metric. Now, we are going to explore the topological concepts of the double controlled type metric space. Consider the definitions follows immediately.
Definition 2.14. [49] Let (X, ρ) be a double controlled type metric space either by one or two functions (1) The sequence {ζ n } n∈N is convergent to some ζ ∈ X, if and only if for each > 0, ∃ an integer N such that ρ(ζ n , ζ) < for each n > N . It can be written mathematically, as lim It can be concluded that when τ is continuous at ζ in (X, ρ). Then ζ n → ζ implies that τζ n → τζ, as n → ∞.
The quest for the existence of fixed points for multi-valued mapping for complete metric spaces was originated by, Nadler, in 1979. He initiated the study of multi-valued version of Banach contraction mapping exercising the idea of Hausdorff metric.
Consider, some of the fundamental concepts from multi-valued fixed point theory that will help us analyzing the present study.

Definition 2.17. The Hausdorff metric H on CB(X) is defined as follow
The mapping H is a metric for CB(X) and is called the Hausdorff metric. It can be deduce that the metric H in fact depends on the metric of X and the two equivalent matrices for X can not generate equivalent Haudorff matrices for CB(X).
is the accumulation of all non-empty closed subsets of X. Consider, where ρ(ζ, A) = inf η∈A ρ(ζ, η). The pair (C(H), X) is known as generalized Hausdorff distance induced by d.
Definition 2.20. [44] Let (X, ρ 1 ) and (X, ρ 2 ) be metric spaces. A mapping F : X → CB(Y) is said to be a multi-valued Lipschitz mapping of X into Y iff ∀ ζ, η ∈ X we have, The constant c in (2.2) is called a Lipschitz constant.
Definition 2.21. If the Lipschitz constant c becomes less than 1 i.e., c < 1 in the case of (2.2) than F is called multi-valued contraction mapping.
Mathematically it can be expressed as lim Mathematically it can be expressed as lim Definition 2.24. An element ζ ∈ X is known to be a fixed point of multi-valued operator τ : X → N(X) if ζ ∈ τ(ζ). Mathematically, represented by Fix(τ) = ζ ∈ X : ζ ∈ τ(ζ) .

Multi-value fixed point results
In this section some fixed point results is proved in the setting of double controlled metric spaces. Our first result is Nadler fixed point theorem.
Then, τ has a fixed point.
For a positive constant u ∈ (0, 1), define the set H ζ u ⊂ X as, Theorem 3.4. Consider (X, ρ) be a complete a double controlled type metric space by two controlled functions α, β : X × X → [1, ∞). Let τ : X → C(X) be a multi-valued mapping. If ∃ a constant k ∈ (0, 1) such that for any ζ ∈ X there is η ∈ H ζ u , satisfying Moreover, consider that for each ζ ∈ X that lim n→∞ α(ζ, ζ n ) and lim n→∞ α(ζ n , ζ) exist and are finite. Then, τ has a fixed point provided that k < u and f is lower semi-continuous.
Let m, n be integers such that n < m then from Theorem (3.1) we get to the following.
The ratio test together with (3.6) implies that the limit of real number sequence {S n } exists and so, {S n } is a Cauchy sequence. In fact, the ratio test is applied to the term x i = i j=0 β(ζ j , ζ m ) α(ζ j , ζ i+1 ). Letting m, n → ∞ in (3.9), that is where L = k u now we know that k < u so we have L n → 0 as n → ∞. From which is deduced that the sequence {ζ n } ∞ n=1 is a Cauchy sequence. Since (X, ρ) is a complete double controlled metric space. So, the sequence {ζ n } ∞ n=1 converges to some point ζ 0 ∈ X. We claim that ζ is a fixed point of T .
As a matter of fact, from the given proof it is deduced that {ζ n } ∞ n=0 converges to ζ. While, on the other hand, f (ζ n ) is a decreasing sequence and hence, converges to 0. Since, f is lower semi-continuous, we get Therefore, f (ζ) = 0. Consequently, the closeness of τ(ζ) implies ζ ∈ τ(ζ).  The above double controlled metric is not a control metric, as can be seen.
It can be analyzed that (X, ρ) is a complete double controlled metric. Consider, the multi-value operator Now it is simple to deduce that f (ζ) = ρ(ζ, τ(ζ)) = 0.
We see that f (ζ) is continuous. In addition, there exists η ∈ H x 8 9 , such that the condition (3.5) is satisfied. Moreover, for every ζ 0 ∈ X the condition (3.6) is satisfied. Thus all the hypothesis of the Theorem (3.4) are fulfilled. Hence {a, b, c} are the fixed points of the τ for k = 2 3 . However, τ is not a contractive mapping in Nadler. For instance, which shows that Theorem (3.1) is the generalization of Nadler multi-valued fixed point theorem.
Now proceeding the work to extending Branciari integral fixed point theorem to double controlled type metric space. The theorem is proved in three steps.
Step 2. we now show that {ζ n } ∞ n=0 is a Cauchy sequence. From Theorem (3.1) we have (3.12) Step 3. The ratio test together with (3.11) implies that the limit of real number sequence {S n } exists and so, {S n } is a Cauchy sequence. In fact, the ratio test is applied to the term Letting m, n → ∞ in (3.12), that is lim m,n→∞ ρ(ζ m , ζ n ) = 0.

Applications to integral inclusions
In the current section existence theorems for Volterra type integral inclusion and singular Fredholm integral inclusions are obtained via multi-valued fixed point results.
The Volterra type integral inclusion can be expressed as where ϑ(α) is a continuous function on the given interval, Γ(α, τ) is the family of non-empty compact and convex sets on the interval, φ is the unknown solution belonging to the given inclusion, and a ≤ x ≤ b. Moreover, consider a multi-valued operator L : X → CB(X), defined by Then clearly the operator (4.2) is non-empty and closed. Furthermore, consider the double controlled metric ρ : X × X → [0, ∞) defined by ρ(α, β) = |α − β| and the controlled functions µ, ν : X × X → [1, ∞) defined by µ = α + 9β + 7 and ν = 6α + 2β + 4. Additionally, (4.2) be the operator from X → CB(X). The the integral (4.1) will have some solution provided that the following holds.
In general there arises two cases of singularity cases in Fredholm and Volterra integral equations [3,26,38] to be precise those cases are: (1) Dealing with the limit a → −∞ and b → ∞.
The general form of both the types are given below.
where ϑ(α) is a function continuous on [a, b], Γ(α, τ) is the family of non-empty compact and convex sets on the interval, φ is the unknown solution belonging to the given inclusion, and a ≤ x ≤ b in each case.
Proof. Since the given double controlled metric space is a complete space. Then, the theorem can be proved easily by following the same steps from Theorem (4.1). Then, by Theorem (3.10) the integral (4.4) possesses a solution.