International Journal of Analysis and Applications Volume 19, Number 1 (2021), 138-152 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-138 SOME RESULTS OF RATIONAL CONTRACTION MAPPING VIA EXTENDED CF -SIMULATION FUNCTION IN METRIC-LIKE SPACE WITH APPLICATION HABES ALSAMIR∗ Finance and Banking Department, College of Business Administration, Dar Aluloom University, riyadh, Saudi Arabia ∗Corresponding author: habes@dau.edu.sa; h.alsamer@gmail.com Abstract. In this paper, we introduce a new contraction via CF -simulation function and prove the existence and the uniqueness of our mapping defined on a metric-like space. Our work generalizes and extends some theorems in the literature. An example and application of second type of Fredholm integral equation are given. 1. Introduction Many problems in mathematics and other sciences such as physics, chemistry, computer science and engineering resolved by using fixed point theory. The Banach contraction mapping principle [1] is one of the essential results in fixed point theory. Thus, a huge number of mathematical researchers generalized and extended it in a lot of spaces that appeared after 1922. One of the most spaces introduced in this decade is metric-like space that was presented by Amini-Harandi [11] in 2012. After that, a lot of researchers proved (common) fixed point results by using different types of contractive conditions in the setting of metric-like spaces, for example see( [2], [3], [6]- [10]). Definition 1.1. [11] Let χ is a nonempty set. A function σ : χ × χ → [0,∞) is said to be a metric like space (or dislocated metric) on χ if for any α,ν,ξ ∈ χ, the following conditions hold: Received November 13th, 2020; accepted December 10th, 2020; published January 7th, 2021. 2010 Mathematics Subject Classification. 54H25, 47H10. Key words and phrases. An extended CF -simulation function, fixed point, metric-like spaces. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 138 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-138 Int. J. Anal. Appl. 19 (1) (2021) 139 (σ1) σ(α,ξ) = 0 ⇒ α = ξ, (σ2) σ(α,ξ) = σ(ξ,α), (σ3) σ(α,ξ) ≤ σ(α,w) + σ(w,ξ). The pair (χ,σ) is called a metric-like space. Let (χ,σ) be a metric-like space. A sequence {αn} in χ, if and only if lim n→∞ σ(αn,α) = σ(α,α) A sequence {αn} is called σ-Cauchy if the limit limn,m→∞σ(αn,αm) exists and is finite. The metric-like space (χ,σ) is called complete if for each σ-Cauchy sequence {αn}, there is some α ∈ χ such that lim n→∞ σ(αn,α) = σ(α,α) = lim n,m→∞ σ(αn,αm). Lemma 1.1. [12] Let (χ,σ) be a metric-like space. Let {αn} be a sequence in χ such that αn → u where α ∈ χ and σ(α,α) = 0. Then, for all ξ ∈ χ, we have limn→∞σ(αn,ξ) = σ(α,ξ). Definition 1.2. [24] Let χ be a nonempty set. A function ß : χ×χ → [0,∞) is a partial metric if for all α,ξ,w ∈ χ, the following conditions are satisfied: (1) α = ξ ⇔ ß(α,α) = ß(α,ξ) = ß(ξ,ξ), (2) ß(α,α) ≤ ß(α,ξ), (3) ß(α,ξ) = ß(ξ,ξ), (4) ß(α,ξ) ≤ ß(α,w) + ß(w,ξ) − ß(w,w). In this case, the pair (χ, ß) is called a partial metric space. It is known that each partial metric is a metric-like, but the converse is not true in general. Example 1.1. Let χ = {0, 1} and σ : χ×χ → [0,∞) defined by σ(0, 0) = 2, σ(u,v) = 1 if (α,ξ) 6= (0, 0) Then, pair (χ,σ) is a metric-like space. Note that σ is not a partial metric on χ because σ(0, 0) � σ(1, 0). Remark 1.1. Let χ = {0, 1}, and σ(α,ξ) = 1 for each α,ξ ∈ χ and αn = 1 for each n ∈ N. Then it is easy to see that αn → 0 and αn → 1 and so in metric-like spaces the limit of a convergent sequence is not necessarily unique. Definition 1.3. [27] A function ζ : [0,∞) × [0,∞) → R is called an extended simulation function if ζ satisfies the following conditions: (ζ1) ζ(α,ξ) < α− ξ for all α,ξ > 0, Int. J. Anal. Appl. 19 (1) (2021) 140 (ζ2) if {αn} and {ξn} are sequences in (0,∞) such that limn→∞αn = limn→∞ξn = ` ∈ (0,∞) > 0, and αn > l, n ∈ N, then lim n→∞ sup ζ(αn,ξn) < 0. (ζ2)let {αn} be a sequences in (0,∞) such that lim n→∞ αn = ` ∈ [0,∞) > 0, ζ(αn, l) ≥ 0, n ∈ N, then l = 0. Many researchers have used the above notation to prove some fixed and common fixed point results, see for example ( [13], [23]). In 2014, Ansari [26] introduced the concept of C-class functions as follows: Definition 1.4. [26] A mapping F : [0,∞)2 → R is called a C -class function if for any α,ξ ∈ [0,∞), the following conditions hold: (i) F(α,ξ) ≤ α, (ii) F(α,ξ) = α implies that either α = 0 or ξ = 0. As examples of C -class functions, we state: (1) F(α,ξ) = α− ξ for all α,α ∈ [0,∞); (2) F(α,ξ) = lα for all α,ξ ∈ [0,∞) where 0 CF ⇒ α > ξ, (Fii) F(ξ,ξ) ≤ CF for all ξ ∈ [0,∞). The following example of C -class functions that have property CF (1) F1(α,ξ) = α 1+ξ , CF = 1, 2. (2) F2(α,ξ) = α− ξ, CF = r, r ∈ [0,∞). Liu [5] linked between a C-class function and CF -simulation function and presented it as the following: Definition 1.6. [5] A mapping ζ : R+ × R+ → R is CF -simulation function if satisfying the following conditions: (ζi) ζ(0, 0) = 0 Int. J. Anal. Appl. 19 (1) (2021) 141 (ζii) ζ(α,ξ) < F(α,ξ), where α,ξ > 0, with property CF (ζiii) if {αn},{ξn} are sequences in (0,∞) such that lim n→∞ αn = lim n→∞ ξn > 0, and αn < ξn, then lim sup n→∞ ζ(αn,ξn) < CF , Example 1.2. [5] Let ζ : R+ ×R+ → R be a function defined by ζ(α,ξ) = mF(α,ξ), where α,ξ ∈ [0,∞) and m ∈ R be such that m < 1 and for each α,ξ ∈ [0,∞). Considering CF = 1,ζ is a CF -simulation function. Choosing F(α,ξ) = α 1+ξ , we get ζ(α,ξ) = mα 1+ξ is also a CF -simulation function with CF = 1. Chanda et. al. [25] brought the concept of CF -extended simulation function as the following: Definition 1.7. [25] A mapping ζ : R+ ×R+ → R an extended CF -simulation function if satisfying the following conditions: (ζ1) ζ(α,ξ) < F(α,ξ), where α,ξ > 0, with property CF (ζ2) if {αn},{ξn} are sequences in (0,∞) such that lim n→∞ αn = lim n→∞ ξn = l, where l ∈ (0,∞) and ξn > l for all n ∈ N, then lim sup n→∞ ζ(αn,ξn) < CF , (ζ3) if {αn} be a sequence (0,∞), such that lim n→∞ αn = l ∈ [0,∞), ζ(αn, l) ≥ CF ⇒ l = 0. Example 1.3. [25] Let ζ : R+ ×R+ → R be a function defined by ζ(α,ξ) = 3 4 α− ξ, where α,ξ ∈ [0,∞). Considering F(α,ξ) = α− ξ with CF = 1, for all α,ξ ∈ [0,∞), we assured that (ζ1)is proved. Now if {αn},{ξn} are sequences in (0,∞) such that lim n→∞ αn = lim n→∞ ξn = l > 0 and ξn > l for all n ∈ N, we obtain lim sup n→∞ ζ(αn,ξn) = lim sup n→∞ [ 3 4 αn − ξn] = −l 4 < CF = 1. Int. J. Anal. Appl. 19 (1) (2021) 142 Thus ζ(α,ξ) = 3 4 α− ξ meets (ζ2). Now, we check for (ζ3). We choose a sequence {ξn} in (0,∞) with lim n→∞ αn = l ≥ 0 for each n ∈ N such that ζ(αn, l) ≥ CF = 1 = 3 4 l−αn ≥ 1 ⇒ αn ≤ 3 4 l− 1. Letting n →∞, we have l ≤ 3 4 l− 1 ⇒ 1 4 l ≤ −1 ⇒ l = −4 which is a contradiction to l ≥ 0. Hence ζ(α,ξ) = 3 4 α− ξ satisfies all conditions od Definition 1.3 and so is an extended CF -simulation function. A functional ϕ : [0,∞) → [0,∞) is lower semicontinuous at a point α0 ∈ χ if (1) ϕ(α0) ≤ lim infα→α0 ϕ(α), (2) ϕ(α) = 0 ⇔ α = 0, Lemma 1.2. Let (χ,σ) be a metric like space and let {wn} be a sequence in χ such that limn→∞σ(αn,αn+1) = 0. If limn,m→∞σ(αn,αm) 6= 0, then there exist � > 0 and two sequences {nl} and {ml} of positive integers with nl > ml > l such that following three sequences σ(α2nl,α2ml ), σ(α2nl−1,α2ml ), and σ(α2nl,α2ml+1) converge to � + when l →∞. In this article, motivated by the idea of an extended CF -simulation function due to Chanda et al 1.3, we prove the existence and the uniqueness of a common fixed point for two mappings satisfying a contraction which involve a lower semicontinuous function is established. An example and application are given to support the obtained work. 2. Main Result Theorem 2.1. Assume that p,q : χ → χ are two self-maps on a complete metric-like space (χ,σ). Suppose that there exist an extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.1) ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),m(α,ξ)) ≥ CF Int. J. Anal. Appl. 19 (1) (2021) 143 for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ) + ϕ(α) + ϕ(ξ),σ(α,pα) + ϕ(α) + ϕ(pα),σ(ξ,qξ) + ϕ(ξ) + ϕ(qξ), σ(α,qξ) + ϕ(α) + ϕ(qξ) + σ(pα,ξ) + ϕ(pα) + ϕ(ξ) 4 }.(2.2) Then, (p,q) has a common fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. Proof. Let α0 ∈ χ, and define a sequence {αn} by α2n+1 = pα2n and α2n+2 = qα2n+1 for all n ≥ 0. If α2n = α2n+1 for some n, then the proof is done. Therefore, if α2n 6= α2n+1 and σ(α2n,α2n+1) = 0, then by (σ1), which is a discrepancy. Applying (2.1), we obtain CF ≤ ζ(σ(pα2n,qα2n+1) + ϕ(pα2n) + qϕ(α2n+1),m(α2n,α2n+1)) = ζ(σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2),m(α2n,α2n+1)).(2.3) By applying (ζ2) in (2.3), we obtain CF < F(m(α2n,α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2)), which implies (2.4) σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2) < m(α2n,α2n+1) where m(α2n,α2n+1) = max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n,pα2n) + ϕ(α2n) + ϕ(pα2n),σ(α2n+1,qα2n+1) +ϕ(α2n+1) + ϕ(qα2n+1), 1 4 (σ(α2n,qα2n+1) + ϕ(α2n) + ϕ(qα2n+1) + σ(pα2n,α2n+1) + ϕ(pα2n) +ϕ(α2n+1))} = max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) +ϕ(α2n+1) + ϕ(α2n+2),σ(α2n+1,α2n+2) + ϕ(uα2n+1) + ϕ(α2n+2), 1 4 (σ(α2n,α2n+2) + ϕ(α2n) +ϕ(α2n+2) + σ(α2n+1,α2n+1) + ϕ(α2n+1) + ϕ(α2n+1))} ≤ max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2), 1 4 (σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1) + σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2))} = max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2)}.(2.5) Int. J. Anal. Appl. 19 (1) (2021) 144 Thus, from (2.4), we get σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2) < max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2)}.(2.6) By a similar process, one can also get the following σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1) < max{σ(α2n−1,α2n) + ϕ(α2n−1) + ϕ(α2n),σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1)}.(2.7) Therefore, from (2.6) and (2.7), (2.8) σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1) < max{σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn),σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)}, for all n ∈ N. Necessarily, we obtain (2.9) max{σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn),σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)} = σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn), for all n ∈ N. Consequently, for all n ∈ N, we have σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1) < σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn) Therefore, we find that {σ(αn,αn+1) + +ϕ(αn) + ϕ(αn+1)} is a decreasing sequence. So, there exists l ≥ 0 such that lim n→∞ (σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)) = l. Assume that l > 0. Then, we deal with {αn} and {ξn} with same limit where αn = σ(pαn,pαn+1) > 0 and αn = σ(qαn,qαn+1) > 0 for all n ∈ N and αn > l for all n ∈ N. Lastly we get from condition (ζ2), CF ≤ ζ(σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1),σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn) < CF which is a contradiction. Then, we conclude that l = 0 and lim n→∞ (σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)) = 0, Int. J. Anal. Appl. 19 (1) (2021) 145 which implies (2.10) lim n→∞ σ(αn,αn+1) = 0, and (2.11) lim n→∞ ϕ(αn) = 0. Now, we will prove that {αn} is Cauchy sequence. After that, we will prove lim n→∞ σ(αn,αm) = 0. Assume that lim n→∞ σ(αn,αm) 6= 0. By contradiction.Thus, that is l = 0. There exists � > 0 and two sequences {αny} and {αmy} of {αn} with ny > my ≥ l such that for every y with the (smallest number satisfying the condition below) (2.12) σ(αny,αmy ) ≥ �. and (2.13) σ(αny−1,αmy−1) < �. By using (2.12) and (2.13) and the triangular inequality, we get � ≤ σ(αny,αmy ) ≥ σ(αny,αmy−1) + σ(αmy−1,αmy ) < σ(αmy−1,αmy ) + �. By (??) (2.14) lim y→∞ σ(αny,αmy ) = lim y→∞ σ(αny−1,αmy−1) = �. We also have (2.15) σ(αny,αmy−1) −σ(αny,αny−1) −σ(αmy,αmy−1) ≤ σ(αny−1,αmy ), and (2.16) σ(αny−1,αmy ) ≤ σ(αny−1,αny ) + σ(αny,αmy ). Letting y →∞ in (2.15) and (2.16) and by using (2.10) and (2.14), we obtain (2.17) lim y→∞ σ(αny−1,αmy ) = �. Again, using the triangular inequality, we have (2.18) | σ(αny−1,αmy ) −σ(αny−1,αmy−1) | σ(αmy−1,αmy ). Int. J. Anal. Appl. 19 (1) (2021) 146 Letting y →∞ in (2.18)and by using (2.17), we get (2.19) lim y→∞ σ(αny−1,αmy−1) = �. From (2.34), we have m(αny−1,αmy−1) = max{σ(αny−1,αmy−1) + ϕ(αny−1) + ϕ(αmy−1),σ(αny−1,pαny−1) + ϕ(αmy−1) +ϕ(qαmy−1), 1 4 (σ(αny−1,qαmy−1) + ϕ(αny−1) + ϕ(qαmy−1) + σ(pαny−1,αmy−1) +ϕ(pαny−1) + ϕ(αmy−1))} = max{σ(αny−1,αmy−1) + ϕ(αny−1) + ϕ(αmy−1),σ(αny−1,αny ) + ϕ(αn−1) + ϕ(αn), σ(αmy−1,αmy ) + ϕ(αmy−1) + ϕ(αmy ), 1 4 (σ(αny−1,αmy ) + ϕ(αny−1) + ϕ(αmy ) + σ(αny,αmy−1) + ϕ(αny ) + ϕ(αmy−1))}.(2.20) Letting y →∞ in (2.20)and by (2.10),(2.11),(2.14),(2.17) and (2.19), it follows that (2.21) lim y→∞ σ(αny,αmy ) = lim y→∞ m(αny−1,αmy−1) = �. Applying (theta2), we get CF ≤ ζ(σ(αny,αmy ) + ϕ(αn) + ϕ(αm),m(αny−1,αmy−1)) < CF which is a contradiction. Hence αn is a Cauchy sequence and hence limn→∞αn = k ∈ χ exists because χ is complete. Since ϕ is lower semicontinuous, ϕ(k) ≤ lim inf n→∞ ϕ(αn) ≤ lim n→∞ ϕ(αn), which implies (2.22) ϕ(k) = 0. We claim that k is a common fixed point of p and q. Put α = αn and ξ = k in (2.33) for all n, and we obtain (2.23) ζ(σ(pαn,qk) + ϕ(pαn) + ϕ(qk),m(αn,k)) ≥ CF m(αn,k) = max{σ(αn,k) + ϕ(αn) + ϕ(k),σ(αn,pun) + ϕ(αn) + ϕ(pαn),σ(k,qk) + ϕ(k) + ϕ(qk), 1 4 (σ(αn,qk) + ϕ(αn) + ϕ(qk) + σ(pαn,k) + ϕ(pαn) + ϕ(k))} = max{σ(αn,k) + ϕ(αn) + ϕ(k),σ(αn,un+1) + ϕ(αn) + ϕ(αn+1),σ(k,qk) + ϕ(k) + ϕ(qk), 1 4 (σ(αn,qk) + ϕ(αn) + ϕ(qk) + σ(αn+1,k) + ϕ(αn+1) + ϕ(k))}. Int. J. Anal. Appl. 19 (1) (2021) 147 Let n →∞ in (2.23) and using (2.22), we have CF ≤ ζ(σ(k,qk) + ϕ(qk),σ(k,qk) + ϕ(qk)) < F(σ(k,qk) + ϕ(qk),σ(k,qk) + ϕ(qk))(2.24) ⇒ (2.25) σ(k,qk) + ϕ(qk) < σ(k,qk) + ϕ(qk) which is absurd. Hence σ(k,qk) + ϕ(qk) = 0, and hence (2.26) k = qk and ϕ(qk) = 0. Similarly, when we take α = αn and ξ = k in (2.33) for all n we get (2.27) k = pk and ϕ(pk) = 0. Equations (2.26) and (2.27) show that k is a common fixed point of p and q. To prove the uniqueness of the common fixed point, we suppose that h is another fixed point of p and q. We argue by contradiction. Assume that there exists h 6= k(so σ(h,k) > 0.) such that (2.28) ζ(σ(ph,qk) + ϕ(ph) + ϕ(qk),m(h,k)) ≥ CF , where m(h,k) = max{σ(h,k) + ϕ(h) + ϕ(k),σ(h,ph) + ϕ(h) + ϕ(ph),σ(k,qk) + ϕ(k) + ϕ(qk), σ(h,qk) + ϕ(h) + ϕ(qk) + σ(ph,k) + ϕ(ph) + ϕ(k) 4 } = σ(h,qk).(2.29) Hence from (2.30), we obtain CF ≤ ζ(σ(h,k),σ(h,k)) < F(σ(h,k),σ(h,k)) < CF ,(2.30) which is absurd and hence h = k. � We will use the same manner in 2.1 to obtain the following result. Theorem 2.2. Assume that p,q : χ → χ are two self-maps on a complete partial metric space (χ,σ). Suppose that there exists a extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.31) ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),m(α,ξ)) ≥ CF Int. J. Anal. Appl. 19 (1) (2021) 148 for all u,v ∈ χ, where m(α,ξ) = max{dpar(α,ξ) + ϕ(α) + ϕ(v),dpar(α,pα) + ϕ(α) + ϕ(pα),dpar(ξ,qξ) + ϕ(ξ) + ϕ(qξ), dpar(α,qξ) + ϕ(α) + ϕ(qξ) + dpar(pα,ξ) + ϕ(pα) + ϕ(ξ) 2 }.(2.32) Then, (p,q) has a common fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. If we put q = p in 2.1, we have the following Corollary Corollary 2.1. Assume that p : χ → χ be self-map on a complete metric-like space (χ,σ). Suppose that there exists a extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.33) ζ(σ(pα,pξ) + ϕ(pα) + ϕ(pξ),m(α,ξ)) ≥ CF for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ) + ϕ(α) + ϕ(ξ),σ(α,pα) + ϕ(α) + ϕ(pα),σ(ξ,pξ) + ϕ(ξ) + ϕ(pξ), σ(α,pξ) + ϕ(α) + ϕ(pξ) + σ(pα,ξ) + ϕ(pα) + ϕ(ξ) 4 }.(2.34) Then, p has a unique fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. Corollary 2.2. Assume that p,q : χ → χ are two self-maps on a complete metric-like space (χ,σ). Suppose that there exists a extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.35) ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),σ(α,ξ) + ϕ(α) + ϕ(ξ)) ≥ CF for all α,ξ ∈ χ. Then, (p,q) has a unique common fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. Corollary 2.3. Assume that p : χ → χ be self-map on a complete metric-like space (χ,σ). Suppose that there exists a extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.36) ζ(σ(pα,pξ) + ϕ(pα) + ϕ(pξ),σ(α,ξ) + ϕ(α) + ϕ(ξ)) ≥ CF for all α,ξ ∈ χ. Then, p has a unique fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. If we take ϕ(t) = 0 in 2.1 and 2.2, we obtain the following two corollaries. Corollary 2.4. Assume that p,q : χ → χ are two self-maps on a complete metric-like space (χ,σ). Suppose that there exists a extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.37) ζ(σ(pα,qξ),m(α,ξ)) ≥ CF for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ),σ(α,pα),σ(ξ,qξ), σ(α,qξ) + σ(pα,ξ) 4 }.(2.38) Int. J. Anal. Appl. 19 (1) (2021) 149 Then, (p,q) has a unique common fixed point z ∈ χ such that σ(z,z) = 0. Corollary 2.5. Assume that p : χ → χ be self-map on a complete metric-like space (χ,σ). Suppose that there exists a extended CF−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.39) ζ(σ(pα,qξ),m(α,ξ)) ≥ CF for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ),σ(α,pα),σ(ξ,qξ), σ(α,qξ) + σ(pα,ξ) 4 }.(2.40) Then, p has a unique fixed point z ∈ χ such that σ(z,z) = 0. Example 2.1. Let χ = [0, 1] be equipped with the metric-like mapping σ(α,ξ) = α2 + ξ2 for all α,ξ ∈ χ. Let p,q : χ → χ be defined by pα =   α2 α+1 if 0 ∈ [0, 1], α2, otherwise. , and qα =   α3 α+1 if 0 ∈ [0, 1], α3, otherwise. . We also consider ζ(s,t) = 1 3 s− t for all s,t ≥ 0, CF = 0 and ϕ(t) = t for all α ∈ χ. Note that (χ,σ) is a complete metric-like space. Without loss of generality we assume that α,ξ ∈ χ, σ(pα,qξ) + ϕ(pα) + ϕ(qξ) = σ( α2 α + 1 , ξ3 ξ + 1 + ϕ( α2 α + 1 ) + ϕ( ξ3 ξ + 1 ) = ( α2 α + 1 )2) + ( ξ3 ξ + 1 )2)3 + ϕ( α2 ξ + 1 ) + ϕ( α2 α + 1 ) ≤ 1 6 (α2 + ξ3) + 1 3 (α + ξ) ≤ 1 3 (α2 + ξ3) + α + ξ) = 1 3 (σ(α,ξ) + ϕ(α) + ϕ(ξ)) ≤ 1 3 m(α,ξ). It follows that ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),m(α,ξ)) = 1 3 m(α,ξ) − [σ(pα,qξ) + ϕ(pα) + ϕ(qξ)] ≥ 0. Then Theorem 2.1 is applicable to (p,q) and ϕ on (χ,σ). Moreover, α = 0 is a common fixed point of (p,q). Int. J. Anal. Appl. 19 (1) (2021) 150 3. Application In this part, we will apply Corollary 2.3 to study the existence and uniqueness of solutions of second type of Fredholm integral equation: (3.1) α(ϑ) = ∫  0 π(ϑ,κ)$(κ,θ(κ))dκ α(ϑ) = ∫  0 π(ϑ,κ)$(κ,τ(κ))dκ. for all (ϑ,κ) ∈ [0, ]2. Let T = C([0, ],R) is the set of real continuous functions on [0, ] for  > 0, defined by σ(α,ξ) =‖ α− ξ ‖∞= sup t∈ |α(t) − ξ(t)| for all α,ξ ∈ T. Then (T,σ) is a complete metric-like space. We consider the operators pα(ϑ) = ∫  0 π(ϑ,κ)$(κ,θ(κ))dκ, qξ(ϑ) = ∫  0 π(ϑ,κ)$(κ,τ(κ))dκ, Theorem 3.1. Assume that Equation (3.1) with the following axioms: (1) π : [0, ] × [0, ] → [0,∞) is a continuous function, (2) $ : [0, ] ×R → R where $(κ,.) is monotone nondecreasing mapping for all κ ∈ [0, ], (3) supϑ,κ∈[0,] ∫  0 π(ϑ,κ)dκ ≤ 1, (4) for every δ ∈ (0, 1) such that for all (ϑ,κ) ∈ [0, ]2 and θ,τ ∈ R, ‖ $(κ,θ(κ)) −$(κ,τ(κ)) ‖≤ δ ‖ α(t) − ξ(t) ‖, Then, the system (3.1)has a unique solution. Proof. For α,ξ ∈ T and from (3) and (4), for all ϑ and κ, we have σ(pα(ϑ),qξ(ϑ)) = | pα(ϑ) −qξ(ϑ) |(3.2) = | ∫  0 π(ϑ,κ)$(κ,θ(κ))dκ− ∫  0 π(ϑ,κ)$(κ,τ(κ))dκ | ≤ ∫  0 π(ϑ,κ) ‖ $(κ,θ(κ)) −$(κ,τ(κ)) ‖ dκ ≤ ∫  0 π(ϑ,κ)δ ‖ α(ϑ) − ξ(ϑ) ‖∞ dκ ≤ π(ϑ,κ)δ ‖ α(ϑ) − ξ(ϑ) ‖∞ ≤ δσ(α,ξ) ≤ δm(α,ξ).(3.3) Int. J. Anal. Appl. 19 (1) (2021) 151 Let (ζ1) and ζ(α,ξ) = δα− ξ for all α,ζ ∈ [0,∞),Cg = 0. Now (3.4) σ(pα(ϑ),qξ(ϑ)) < δm(α,ξ). Then, from (3.2), we obtain ζ(σ(pα,qξ),m(α,ξ)) ≥ Cf. Applying Corollary (3.1), we obtain that (p,q) has a unique common fixed point in C([0, 1]), say x. 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Ansari, L. Kanta Dey, B. Damjanovic̀. On Non-Linear Contractions via Extended CF-Simulation Functions. Filomat 32(10) (2018), 3731–3750 [26] A.H. Ansari. Note on φ − ψ-contractive type mappings and related fixed point. In: The 2nd Regional Conference on Mathematics and Applications, Payame Noor University, pp. 377–380, 2014. [27] A.F. Roldán-López-de-Hierro, B. Samet. ϕ-admissibility results via extended simulation functions. J. Fixed Point Theory Appl. 19(3) (2017), 1997-2015. https://www.researchgate.net/publication/332396635_Fixed_point_results_for_new_contraction_involving_C-class_functions_in_partail_metric_spaces https://www.researchgate.net/publication/332396635_Fixed_point_results_for_new_contraction_involving_C-class_functions_in_partail_metric_spaces 1. Introduction 2. Main Result 3. Application References