Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation

In this manuscript, we establish two Wardowski–Feng–Liu-type fixed point theorems for orbitally lower semicontinuous functions defined in orbitally complete b-metric spaces. The obtained results generalize and improve several existing theorems in the literature. Moreover, the findings are justified by suitable nontrivial examples. Further, we also discuss ordered version of the obtained results. Finally, an application is presented by using the concept of fractal involving a certain kind of fractal integral equations. An illustrative example is presented to substantiate the applicability of the obtained result in reducing the energy of an antenna.


Introduction
Through the whole of the last century, mathematicians engrossed themselves with touching up the underlying metric framework of the acclaimed Banach contraction principle. In their research articles, Bakhtin [3] and afterwards Czerwik [6][7][8] put forward another natural and impressive setting as an extension of metric spaces, b-metric spaces, to work with. Since then, plenty of fixed point, common fixed point and related results involving various classes of single-valued and multi-valued operators defined in such kind of spaces or in allied structures are published (see [1,2,5,10,11,[13][14][15][16][17][18]20]). Here we recall some basic definitions, notations and essential results, which will play crucial roles in this manuscript. Throughout this article, R + and R + 0 denote the set of all positive, respectively, nonnegative real numbers, and N stands for the set of positive integers. First of all, we recollect the definition of b-metric spaces. Definition 1. A b-metric on a nonempty set X is a function d b : X × X → R + 0 such that for a constant b 1 and all x, y, z ∈ X, the following conditions hold: Then (X, d b ) is called a b-metric space.
One may note that each metric space is a b-metric space considering b = 1. However, the converse does not hold. Furthermore, the topology on a b-metric space, the concept of Cauchy sequences, convergent sequences and the completeness of the setting are analogous to that of standard metric spaces. However, in general, a b-metric is not a continuous mapping in both variables (see, e.g., [13]).
On the other hand, Cosentino et al. in [5] introduced the set of functions TF b in the line of Wardowski's [19] approach to b-metric space as follows: 1 be a real number. TF b denotes the family of all functions F : R + → R with the following properties: (F1) F is strictly increasing; (F2) For each sequence {a n } of positive numbers, lim n→∞ a n = 0 if and only if lim n→∞ F (a n ) = −∞; (F3) For each sequence {a n } of positive numbers with lim n→∞ a n = 0, there exists k ∈ (0, 1) such that lim n→∞ a k n F (a n ) = 0; (F4) There exists τ ∈ R + such that for each sequence {a n } of positive numbers, if τ + F (ba n ) F (a n−1 ) for all n ∈ N, then τ + F (b n a n ) F (b n−1 a n−1 ) for all n ∈ N.
Definition 3. Let b 1 be a real number. TF * b denotes the family of all functions F : R + → R having the properties (F1)-(F4) and the following property: It is easy to see that the function F (x) = ln x or F (x) = x + ln x satisfies properties (F1)-(F5) for x > 0.
We recall the following result from [5]. In this result, H b denotes the b-Hausdorff-Pompeiu metric. Theorem 1. (See [5,Thm. 3.4].) Let (X, d b ) be a complete b-metric space, and let T : X → P cb (X). Assume that there exists a continuous from the right function F ∈ TF b and τ ∈ R + such that for all x, y ∈ X, T x = T y. Then T has a fixed point.
Moreover, Feng and Liu obtained the subsequent result (recall that a function f : X → R is said to be lower semicontinuous if for all sequences {x n } in X with lim n→∞ x n = x ∈ X, it satisfies f (x) lim inf n→∞ f (x n )).
Theorem 2. Let (X, d) be a metric space, T : X → CL(X), and let the function f : then T has a fixed point.
Against this background, we obtain fixed point results for multivalued mappings satisfying Wardowski-Feng-Liu-type conditions for orbitally lower semicontinuous functions in orbitally complete b-metric spaces. These results generalize, complement and unify the findings proposed in [2,12]. Besides, we illustrate a couple of examples to validate our obtained results, and also, we consider the ordered version of the attained results. Finally, we dish out an interesting application concerning our derived theorem and employing the notion of fractals to a certain type of fractal integral equations. An illustrative example is presented to show the applicability of the obtained result in reducing the energy of an antenna.

Main results
Let CL(X) denotes the family of nonempty closed subsets of X. Let T : X → CL(X) be a multivalued map, F ∈ TF b and η : Let T : X → X, and for some Cirić defined the orbit and orbital completeness in case of multivalued mappings in his paper [4]. In a similar fashion, we define these concepts in case of b-metric spaces. For a set X, we use the notation 2 X to denote the set of all subsets of X.
Definition 4. Let T : X → 2 X be a multivalued mapping on a b-metric space (X, d b ). An orbit for T at a point x 0 ∈ X is denoted by O(x 0 ) and is defined as a sequence {x n : x n ∈ T x n−1 }.
It is obvious that an orbitally complete b-metric space may not be complete. Now we present one of our main result.
Assume that the following conditions hold: Then T has a fixed point.
Proof. Suppose that T has no fixed points. Then for all Since T x ∈ CL(X) for every x ∈ X and F ∈ TF * b , then it is easy to prove that the set F x η is nonempty for every x ∈ X (proof will follow in the line of [12]). If x 0 ∈ X is any initial point, then there exists x 1 ∈ F x0 η such that and for x 1 ∈ X, there exists x 2 ∈ F x1 η satisfying Continuing this process, we get an iterative sequence {x n }, where x n+1 ∈ F xn η and for all n ∈ N ∪ {0}. It follows by (1) and property (F4) that for all n ∈ N. We verify that {x n } is a Cauchy sequence. Since x n+1 ∈ F xn η , then by the definition of F xn η we have which implies that From (2) and (3) we have i.e., Then using (5), the following holds: Let p n be the greatest number in {0, 1, . . . , n − 1} such that for all n ∈ N. In this case, { n } is a nondecreasing sequence. From (6) we get In a similar way, from (4) we can obtain Now consider the sequence {β( pn )}. We distinguish two cases. Case 1. For each n ∈ N, there is m > n such that β( pn ) > β( pm ). Then we obtain a subsequence { pn k } of { pn } with β( pn k ) > β( pn k+1 ) for all k. Since pn k → δ + , we deduce that lim inf for all k. Consequently, lim k→∞ F (b n k n k ) = −∞, and by (F2) lim k→∞ b n k n k = 0, which contradicts the fact that lim k→∞ n k > 0 as b > 1. By (7) the following holds for all n ∈ N: Passing to limit as n → ∞ in (9), we obtain lim n→∞ n(b n n ) k β( pn ) = 0.
Since ζ := lim inf n→∞ β( pn ) > 0, then there exists n 0 ∈ N such that β( pn ) > ζ/2 for all n = n 0 . Thus, for all n n 0 . Letting n → ∞ in (10), we have 0 lim n→∞ n(b n n ) k ζ/2 < lim n→∞ n(b n n ) k β( pn ) = 0, that is, From (11) there exits n 1 ∈ N such that n( n ) k 1 for all n n 1 . So, we have n 1/(b n n 1/k ) for all n n 1 . Now, the last limit implies that the series +∞ n=1 b n n is convergent, and hence, {x n } is a Cauchy sequence in X. Since X is a orbitally complete b-metric space, there exists z ∈ O(x 0 ) such that x n → z as n → ∞. On the other hand, from (8) and (F2) we have lim n→∞ d b (x n , T x n ) = 0. Since x → d b (x, T x) is orbitally lower semicontinuous, we have Therefore, z ∈ T z. Hence, T has a fixed point.
Next, we present an example in order to substantiate the above result.
For any x ∈ X, we have d b (x, T x) = ((3x − 1)/4) 2 . So the mapping x → d b (x, T x) is orbitally lower semicontinuous. Again, for any x ∈ X, we have y = (x + 1)/4 ∈ F x η , and for this y, we have Thus, we see that all the conditions of Theorem 3 hold true. So by that theorem T has a fixed point, and note that z = 1/3 is a fixed point of T .
Our second result is related to multivalued mappings T on the b-metric space X, where T x is compact for all x ∈ X. By taking into account Case 1, we can take η 0. Therefore, the proof of the following theorem is obvious.
Let us denote K(X) as the set of all nonempty compact subsets of X.
Theorem 4. Let (X, d b ) be an orbitally complete b-metric space, b 1, T : X → K(X) and F ∈ TF b . Assume that the following conditions hold: Then T has a fixed point.
Now we have the ordered version of the above results. To do this, we recall the definition of ordered b-metric space. (X, d b , ) is called an ordered b-metric space if d b is a b-metric space on X and (X, ) is a partially ordered set. Further, if (X, ) is a partially ordered set, then x, y ∈ X are called comparable if x y or y x holds. For If the condition (C) If {x n } ⊂ X is a nondecreasing sequence with x n → z in X, then x n z for all n holds, then T has a fixed point.
Proof. Following the line of proof of Theorem 3 and definition of F x η, ⊆ X, we can show that {x n } is a Cauchy sequence in (X, d b , ) with x n−1 x n for n ∈ N. From the orbital completeness of X there exists a z ∈ O(x 0 ) such that x n → z as n → ∞. By assumption (C), x n z for all n. Rest is followed from the proof of Theorem 3. Theorem 6. Let (X, d, ) be an ordered orbitally complete b-metric space with b 1, T : X → K(X) and F ∈ TF b . Assume that the following conditions hold: Then T has a fixed point provided (C) holds. Now we present an example to authenticate Theorem 5.
Example 2. Let us take X = N ∪ {0} and consider a relation on X by defining x y if and only if x divides y. Then it is easy to verify that (X, ) is a partially ordered set. Then (X, d b ) is a partially ordered b-metric space with b = 3. Now we define a mapping T : X → CL(X) by Let us choose F (α) = ln α for all α ∈ R + and τ (t) = 1/8, η(t) = 1/10 for all t ∈ (0, ∞). Then it is obvious that F ∈ TF b and τ (t) > η(t), lim inf s→t + τ (s) > lim inf s→t + η(s) for all t 0. Further, for x ∈ X, we have is orbitally lower semicontinuous. For x ∈ X, we have y = 4x ∈ F x η, , and for this y, we have Thus, we see that all conditions of Theorem 5 hold. So by the same theorem T has a fixed point. Indeed z = 0 is a fixed point of T .

Application
Fredholm equations stand up obviously in the scheme of signal processing, for instance, as the well-known spectral concentration issue. The operators convoluted are the similar as linear filters. They besides normally stand up in linear forward forming and inverse problems. In physics, the result of such integral equations permits for investigation spectra to be correlated to several fundamental disseminations, for example, the mass supply of polymers in a polymeric melts, or the supply of reduction times in the scheme. Moreover, Fredholm integral equations indicate in fluid mechanics issues connecting hydrodynamic connections near finite-sized elastic borders (see [9] for recent work). In antenna manufacturing, side lobes are the lobes (local maxima with local minimum energy) of the distant field radiation design of an antenna or other radioactivity foundation, which are not the chief lobe. The problem statement is that: is there a solution for which the side lobe energy is minimum? The answer of this question is to find a solution for Fredholm integral equation. In our discussion, we use a generalize fractal Fredholm integral equation based on the fractal integral [21] as follows: where Λ : J × J × R → R is defined and continuous on a fractal set of fractal dimension ℘ ∈ (0, 1), and σ : J → R is continuous function. Our aim is to show that (12) has a solution by using Theorem 3.
Theorem 7. Let X = C[J, R], and let Q the operator defined by where Λ : J × J × R → R is defined and continuous on a fractal set of fractal dimension ℘ ∈ (0, 1), and σ : J → R is continuous function. Moreover, assume the following conditions: (i) There occurs a continuous function γ : Then the integral equation (12) admits a solution.
Proof. We have to prove that the operator Q achieves all the assumptions of Theorem 3 in the single-valued type. Let χ ∈ X, then we obtain (Qχ)(t) − Q Qχ(t) Thus, we get the following inequality: Since the natural logarithm indicates to be in TF b , employing it on above inequality, we conclude that T + ln Γ 2 (℘ + 1) d Qχ, Q 2 χ ln d(χ, Qχ) .
Hence, in view of Theorem 3 (single-valued) with F (ψ) = ln ψ, the operator Q admits at least one fixed point, which is corresponding to the solution of Eq. (12). This completes the proof.