Solving Integral Equations by Means of Fixed Point Theory

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung, Taiwan Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey Department of Mathematics and Computer Sciences, Universitatea Transilvania Brasov, Brasov, Romania Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.B. 80203, Jeddah 21589, Saudi Arabia Department of Statistics and Operations Research, University of Granada, Granada, Spain


Introduction
Nowadays, nonlinear analysis is one of the most active branches of mathematics. Its applications to real-life contexts have attained great success. Physics, engineering, chemistry, biology, and economy are some of the scientific areas that have benefited the most from the techniques developed in nonlinear analysis. In this context, fixed point theory has played an important role in the development of new methodologies for the determination of solutions of certain equations of several types, such as matrix equations, integral equations, and differential equations.
In principle, the elements used by fixed point theory are few and very simple to handle a nonlinear operator for which we want to find its possible fixed points, a real metric that endows the underlying space with a complete character, and an inequality (called the contractivity condition) that is strong enough to ensure the existence of fixed points. With these three ingredients, it is possible to propose good fixed point theorems, as has been done for the last seventy years (see, for instance, Boyd and Wong [1], Caristi [2], Chatterjea [3], Hardy and Rogers [4], Kannan [5,6], Ćirić [7], Geragthy [8], Meir and Keeler [9], Samet et al. [10], Khojasteh et al. [11], Kutbi et al. [12], and Jleli and Samet [13]).
Based on these three initial tools, the possibilities that this field of study has shown have been practically endless. On the one hand, researchers have worked with increasingly abstract metric spaces. In some of these cases, the object associated with the distance between two points has not been a single real number but much more general abstract objects. On the other hand, the operators involved in these studies have been increasingly general, including the possibility of studying multidimensional fixed points (see [14]). Finally, the contractivity condition is the part that has received the most attention within the field of fixed point theory.
In recent times, major efforts have been done in order to introduce as weak as possible contractivity conditions. For instance, it is usual to find auxiliary functions that help to consider extremely weak inequalities. Having this aim in mind, we would like to highlight here two possible extensions.
(i) On the one hand, although the first contractivity conditions only considered a small quantity of terms, after the appearance of the Ćirić theorem [7], the current versions involve more and more terms in their developments. This is the case, for instance, of Karapnar's interpolative-type contractions [15], but many other results can be cited in this line of research (see [16,17]) (ii) On the other hand, in general, notice that the good and reasonable properties that an operator T : M ⟶ M satisfies are usually inherited by the selfcomposition T 2 = T ∘ T, but it is possible that T 2 enjoys those good properties without T doing it. This is the case, for instance, of continuity: it is possible for T 2 to be continuous without T being continuous. In this sense, some results (like Istrăţescu's fixed point theorem; see [18,19]) employing T 2 are more general than their corresponding ones with T One of the powerful applications of fixed point theory can be found in the context of integral equations, whose recent numerical treatments have made great scientific advances in this field (see, for instance, collocation methods [20], operational matrix methods [21][22][23], Galerkin methods [24,25], and Krylov subspace methods [26]).
In this paper, we introduce a new family of contractive mappings that we call hybrid-interpolative Reich-Istrăţescutype contractions because they are inspired by the previous classes of contractive operators. The main advantage of this new family of contractive mappings is that they allow us to present, at the same time, contractivity conditions that involve a large number of terms, including some with the selfcomposition T 2 of the operator, and which are placed either adding or multiplying to the other terms. Furthermore, we introduce some fixed point results that confirm that this kind of operators is appropriate in this field of study. Finally, we illustrate the utility of the novel theorems by introducing a novel application in the setting of integral equations.
This work is organized as follows. Section 2 is dedicated to presenting some notations, preliminaries, and related results in the field of fixed point theory. In Section 3, we describe a complete study about the behavior of the Picard sequences that we will handle in the following sections. The main results of this paper can be found in the Section 4, and direct consequences are placed in Section 5. The application of the main statements is developed in Section 6. Finally, some conclusions and prospect works are discussed in Section 7.

Background on Fixed Point Theory
In this work, we denote by ℝ and ℕ = f0, 1, 2, ⋯g the set of all real numbers and the set of nonnegative integers, respectively. Let ðM, dÞ be a metric space and let T : M ⟶ M be a mapping. A point u ∈ M is a fixed point of T if Tu = u. We will denote by * Fix T ðMÞ the set of all fixed points of T in M.
Given n ∈ ℕ, the mapping T n = T ∘ T ∘ ⋯ ðnÞ ∘ T : M ⟶ M is the n-th iterate of T (as convention, we agree that T 0 is the identity mapping on M). Given z 0 ∈ M, the sequence fz n g n∈ℕ defined by z n = T n z 0 for all n ∈ ℕ is the Picard sequence of T based on z 0 . Such sequence can recursively be defined as z n+1 = Tz n for all n ∈ ℕ. A mapping T is called a Picard operator if each Picard sequence of such operator converges to one of its fixed points. A binary relation on the set M is a nonempty subset R of the Cartesian product M × M. We will write zRt when two points z, t ∈ M verify ðu, tÞ ∈ R.
is known as an α-orbital admissible mapping.
One of the first results in fixed point theory in which the contractivity condition was stated in terms of its selfcomposition T 2 = T ∘ T rather than in terms of T was due to Istrăţescu (see [18,19]).
Theorem 2 (Istrăţescu [18,19]). Given a complete metric space ðM, dÞ, every continuous map T : M ⟶ M is a Picard operator provided that there exist a, c ∈ ð0, 1Þ such that a + c < 1 and for all z, t ∈ M.
Notice that the good properties (like continuity) of an operator T are usually inherited by T 2 , but it is possible that T 2 enjoys those good properties without T doing it. In this sense, some results employing T 2 are more general that their corresponding ones with T. Some generalizations of this result in different abstract metric spaces (b-metric spaces, ordered metric spaces, cone metric spaces, etc.) were presented in recent papers (see [27][28][29]).
In other lines of research, inspired by Kannan's theorem [5,6], Karapnar introduced in [15] a family of contractions in which the distances of the right-hand side of the contractivity condition are multiplying instead of adding up. Also, notice that his contractivity condition must only be verified by pairs of points in the metric space that are not fixed points of the considered nonlinear operator, which avoids any kind of indetermination of the involved powers. Journal of Function Spaces Theorem 3 (Karapnar [15]). Let ðM, dÞ be a complete metric space and let T : M ⟶ M be a mapping such that there exist constants k ∈ 0, 1Þ and λ ∈ ð0, 1Þ satisfying for all z, t ∈ M \ * Fix T ðMÞ. Then, T has a unique fixed point in M.
In the previous result, as z and t are not fixed points of T , then dðz, TzÞ > 0 and dðt, TtÞ > 0. Furthermore, as λ > 0 and 1 − λ > 0, then the expressions dðz, TzÞ λ and d ðt, TtÞ 1−λ are well defined. However, in the main results that we will introduce later, we will employ expressions such as which we would like to explain for the sake of clarity: on the one hand, (4) means the product a 4 · ½dðTz, TtÞ λ , where a 4 and λ are nonnegative real numbers (notice that the exponent λ only affects the distance dðTz, TtÞ, and we avoid to write the brackets); on the other hand, the power d ðTz, TtÞ λ is not well defined when the base and the exponent take the value 0 at the same time. However, for our purposes, we must advise the reader that, when the base and the exponent are 0 at the same time, we will use the convention 0 0 = 1 .

Study of the Behavior of Some Picard Sequences
In this section, we describe the behavior of some sequences that will be of importance in the proofs of the main results of this work.
Proof. For n = 0 in (5), and for n = 1 in (5), This means that inequalities (6) hold for n = 1. Suppose that (6) holds for some n ∈ ℕ, that is, Therefore, This completes the induction.☐ Lemma 5. Let fz n g n∈ℕ be a sequence on a metric space ðM, dÞ. Suppose that there is c ∈ ½0, 1Þ such that Then, fz n g n∈ℕ is a Cauchy sequence in ðM, dÞ.
Proof. Let us consider the sequence fr n g defined by r n = dð z n , z n+1 Þ for all n ∈ ℕ. By the hypothesis, this sequence verifies (5). Then, Proposition 4 guarantees that where Δ = max fr 0 , r 1 g. In particular, If c = 0 or Δ = 0, then fz n g n≥2 is a constant sequence, so it is a Cauchy sequence. Suppose that c > 0 and Δ > 0. In order to prove that the sequence fz n g n∈ℕ is a Cauchy sequence in ðM, dÞ, let ε > 0 be arbitrary. Since ε/ð2ΔÞ > 0 and 0 < c < 1, there is a natural number n 0 > 1 such that In particular,

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Let n, m ∈ ℕ such that m > n ≥ 2n 0 . Let p be a natural number such that p ≥ n 0 + 1 and 2p ≥ m. Therefore, by (14), This proves that fz n g n∈ℕ is a Cauchy sequence in ðM, dÞ .☐ Remark 6. Taking into account that, in general, the notation Δ α n is not well defined because the number Δ ðα n Þ is distinct to ðΔ α Þ n , we clarify that, in the next statement, we use the convention: Proposition 7. Given c ∈ ½0,∞Þ and α ∈ ð0, 1Þ, let fr n g ⊂ ½0, ∞Þ be a sequence such that Let Δ = max fr 0 , r 1 , 1g. Then, Therefore, Proof. If c = 0, the announced properties are trivial. Suppose that c > 0. Since α ∈ ð0, 1Þ, we know that Therefore, as Δ ≥ 1, then Using n = 0 in (19), and if n = 1 in (19), using (23), The previous two inequalities mean that (20) holds for n = 1. Suppose that (20) is fulfilled for some n ∈ ℕ, and we are going to prove it for n + 1. Indeed, and using (23), which completes the induction. Then, (20) holds. Taking into account that fα n g n∈ℕ ⟶ 0, we know that fΔ α n g n∈ℕ ⟶ Δ 0 = 1. On the other hand, Hence, In order to check how different the conditions (19), where α ∈ ð0, 1Þ, and (5), where α = 1, are, let us consider the following example.☐ 4 Journal of Function Spaces Example 8. Let fr n g n∈ℕ ⊂ ð0,∞Þ be the sequence defined by where c = α = 0.5. Then, it can be easily proven by induction that fr n g is the constant sequence given by r n = 0.25 for all n ∈ ℕ. Indeed, if r n = r n+1 = 0.25 for some n ∈ ℕ, then r n+2 = c · max r n , r n+1 f g α = 0:5 · max 0:25,0:25 f g 0:5 = 0:25: As a consequence, fr n g ⟶ 0:25, but it does not converge to zero, as in Proposition 4.

Corollary 9.
Let fz n g n∈ℕ be a sequence on a metric space ð M, dÞ. Suppose that there are c ∈ ½0, 1Þ and α, β ∈ ½0, 1 verify- Then, fz n g n∈ℕ is a Cauchy sequence in ðM, dÞ.
Proof. Notice that, for all n ∈ ℕ, Then, Lemma 5 is applicable.☐

Fixed Point Theorems for Hybrid-Interpolative Reich-Istrăţescu-Type Contractions
In this section, we introduce the novel class of contractive mappings based on Reich and Istrăţescu's approaches.
Definition 10. Let ðM, dÞ be a metric space and let α : M × M ⟶ ½0,∞Þ be a function. A mapping T : M ⟶ M is a hybrid-interpolative Reich-Istrăţescu-type contraction in the case that for some λ ∈ 0,∞Þ, there exist a constant k ∈ 0 , 1Þ and six numbers a 1 , a 2 , a 3 , a 4 , where Remark 11.
(1) As we have commented in the second section, when λ > 0, the expression dðz, tÞ λ is well defined even if the base is zero. However, although z ≠ t in (36), in the case λ = 0, it is possible that we can find the indetermination 0 0 in the expression of I λ ðz, tÞ. In such a case, we will use the convention 0 0 = 1 to avoid such indetermination. In other words, if some exponent in the expression of I λ ðz, tÞ is zero, then its correspondent power will take the value 1. Notice that in this case, it is impossible that all exponents are zero because ∑ 5 i=1 a i + δ = 1 (2) We can believe that the cases are equivalent because if ∑ 5 i=1 a i + δ < 1, then we can replace δ by δ ′ ∈ 0, 1Þ such that ∑ 5 i=1 a i + δ ′ = 1, and the mapping T is also a hybrid-interpolative Reich-Istrăţescu-type contraction by considering the new parameters a 1 , a 2 , a 3 , a 4 , a 5 , δ′ ≥ 0. However, as we will show later, when λ = 0, we cannot permit the sum of the exponents to be less than 1 because, in 5 Journal of Function Spaces such case, the sequence of distances between a term and its consecutive might not converge to zero (so it is not Cauchy) (3) Although the definition of I λ ðz, tÞ, for λ > 0, is very different to the definition of I 0 ðz, tÞ (λ = 0) because the first case uses additions and the second case involves products, there is a particular case in which both algebraic expressions lead to the same contractivity condition. It corresponds to the choice: In this case, if λ > 0, and if λ = 0, Notice that this case corresponds to the Banach contractivity condition particularized to T 2 instead of T: which appears when αðz, tÞ = 1 for all z, t ∈ M. Other similar cases will be discussed in Remark 23.
The first main theorem of this work is the following one.
Theorem 12. Let ðM, dÞ be a complete metric space. A continuous hybrid-interpolative Reich-Istrăţescu-type contraction T : M ⟶ M has at least a fixed point provided that the mapping T is α-orbital admissible and there exists z 0 ∈ M such that αðz 0 , Tz 0 Þ ≥ 1.
Proof. From the hypothesis, we know that there exists z 0 ∈ M such that αðz 0 , T 0 Þ ≥ 1. Since T is α-orbital admissible, αðT z 0 , T 2 z 0 Þ ≥ 1, and by an inductive reasoning, we get that αð T n z 0 , T n+1 z 0 Þ ≤ 1 for any n ∈ ℕ. Starting from this point z 0 ∈ M, we define the sequence fz n g in M as follows: If there is some n ∈ ℕ satisfying that z n = z n+1 , then z n is a fixed point of T, and the proof finishes here. On the contrary case, suppose that z n is distinct to z n+1 for all n ∈ ℕ.☐ We will divide the proof into two cases, namely, λ > 0 and λ = 0. In both cases, we prove that the sequence fz n g is Cauchy.
Case A. For the first case, λ > 0, given any n ∈ ℕ, taking in (35) z = z n and t = z n+1 , we have Using the power of λ, Therefore, which means that, for all n ∈ ℕ, or, equivalently, Let us denote Journal of Function Spaces Clearly, This proves that there is c ∈ 0, 1Þ such that Lemma 5 concludes that fz n g n∈ℕ is a Cauchy sequence in ðM, dÞ.
In both cases (λ > 0 and λ = 0), we have demonstrated that fz n g n∈ℕ is a Cauchy sequence in ðM, dÞ. As it is complete, then there exists a point u ∈ M such that fz n g ⟶ u as n ⟶ ∞. Moreover, due to the continuity of the mapping T, we conclude that Tu = u; that is, u is a fixed point of T.

Remark 14.
Notice that in the previous proof we have shown, without using the continuity of the mapping T; that is, in both cases (λ > 0 and λ = 0), the Picard sequence fz n g n∈ℕ is Cauchy in the metric space ðM, dÞ. Using its completeness, it follows that there exists a point u ∈ M such that f z n g ⟶ u as n ⟶ ∞. Then, we can use this argument in the next results because the continuity of the mapping T is only used in the last part of the proof.
When λ = 0 and ∑ 5 i=1 a i + δ < 1, the statement given by Theorem 12 is false. For instance, if a 1 = a 2 = a 3 = a 4 = a 5 = δ = 0, the contractive condition (36) does not provide any kind of control on dðT 2 z, T 2 tÞ because the value of I 0 ðz, t Þ is always 1 (all exponents are zero). However, even if all constants a 1 , a 2 , a 3 , a 4 , a 5 , and δ are strictly positive, the operator T could be fixed point free, as we show in the next example.  following table: ð61Þ As a consequence, if α is given by αðz, tÞ = 1 for all z, t ∈ M, then the mapping T satisfies for all distinct z, t ∈ M. Therefore, T is a hybrid-interpolative Reich-Istrăţescu-type contraction. Furthermore, ðM, dÞ is complete, T is α-orbital admissible, and there exists z 0 ∈ M such that αðz 0 , Tz 0 Þ ≥ 1. However, T is fixed point free.
Remark 16. The reason why Theorem 12 could fail when λ = 0 and ∑ 5 i=1 a i + δ < 1 is the following one: a sequence fz n g n∈ℕ satisfying could be non-Cauchy when α + β < 1. For instance, let fz n g n∈ℕ be the sequence defined by z n = 0:25n for all n ∈ ℕ: If M = fz n : n ∈ ℕg ⊂ ½0,∞Þ is endowed with the Euclidean distance, then Therefore, if we take then, for all n ∈ ℕ, However, the sequence fz n g n∈ℕ positively diverges, so it is not Cauchy.
If the operator T is continuous on M, then the composition T 2 = T ∘ T also is. However, the mapping T 2 could be continuous even if T is not continuous. In this case, we can replace the continuity of the mapping T with a weaker condition, namely, the continuity of T 2 , whose set of fixed points is nonempty, as it is shown in the next statement. Proof. Let fz n = T n z 0 g n∈ℕ be the Picard sequence of T whose initial point is z 0 . In Remark 14, we commented that this sequence has a limit u ∈ M. Since the mapping T 2 is continuous, then Thereby, T 2 u = u; that is, u is a fixed point of T 2 . As a result, the mapping T 2 has at least one fixed point; that is, the set * Fix T 2 ðMÞ is nonempty. Furthermore, T 3 u = Tu. In order to check that u is also a fixed point of T, suppose, by contradiction, that u ≠ Tu. In this case, Tu is not a fixed point either because, in such a case, Tu = T 2 u = u, which is false. Then, u, Tu ∈ M \ * Fix T ðMÞ.☐ Case A. If λ > 0, then Since αðu, TuÞ ≥ 1, then from (35), which is a contradiction. Then, Tu = u, so u is a fixed point of T.

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Case B. If λ = 0, then Proof. Theorem 17 guarantees that the set of fixed points of T is nonempty. Suppose that T has two distinct fixed points u, v ∈ * Fix T ðMÞ.☐ Case A. If λ > 0, replacing such points in (35), we obtain Therefore, which is impossible. Then, T cannot have two distinct fixed points. Case B. If λ = 0, then If a 2 > 0 or a 3 > 0 or a 5 > 0 or δ > 0, then I 0 ðu, vÞ = 0, so the contractivity condition (35) leads to which is false because u and v are distinct points. On the contrary, if a 2 = a 3 = a 5 = δ = 0, we agreed that 0 a 2 = 0 a 3 = 0 a 5 = 0 δ = 1, so The argument shown in (73) proves that this case is also impossible, so the mapping T cannot have two distinct fixed points in any case.
Also, a particular result holds for the case λ = 0; more exactly, we can remove the continuity conditions of T or T 2 .
Theorem 19. Let ðM, dÞ be a complete metric space, and let T : M ⟶ M be a hybrid-interpolative Reich-Istrăţescu-type contraction for λ = 0 such that a 1 > 0 or a 2 > 0 or a 5 > 0. If we suppose that (i) T is α-orbital admissible (ii) there exists z 0 ∈ M such that αðz 0 , Tz 0 Þ ≥ 1 (iii) for any sequence fz n g n∈ℕ in M such that αðz n , z n+1 Þ ≥ 1 for n ∈ ℕ and fz n g ⟶ u as n ⟶ ∞, we have that αðz n , uÞ ≥ 1 for all n ∈ ℕ then T has a fixed point.
Proof. Following the proof of Theorem 12, we considered the Picard sequence z n = T n z 0 for all n ∈ ℕ. If this sequence contains a fixed point, the proof is finished. On the contrary case, we have shown that it is a Cauchy sequence on ðM, d Þ, so it converges to a point u ∈ M. To prove that u is a fixed point of T, suppose, by contradiction, that u ≠ Tu. Without loss of generality, we can assume that fz n g n∈ℕ satisfies z n ≠ z m for all n, m ∈ ℕ such that n ≠ m. In this case, there is n 0 ∈ ℕ such that z n and u are distinct and they are not fixed points of T for all n ≥ n 0 . Let us check that u is a fixed point of T 2 . Indeed, for all n ≥ n 0 , Since a 1 > 0 or a 2 > 0 or a 5 > 0, letting n ⟶ ∞ in (77), we find out that dðu, T 2 uÞ = 0; that is, u is a fixed point of T 2 . Now, following the lines from Theorem 17, we obtain a contradiction that proves that u is also a fixed point of T.☐ Its self-composition is given by Clearly, T and T 2 are not continuous mappings. Next, let us show that T is a hybrid-interpolative Reich-Istrăţescutype contraction w.r.t. the function α : M × M ⟶ ½0,∞Þ defined as follows: Let us take k = 3 4 , a 1 = a 4 = 1 2 , a 2 = a 3 = a 5 = δ = 0: ð81Þ (b) For z = −1 and t = 1, we have (c) For all other cases, Hence, from Theorem 19, we conclude that T has a fixed point.

Consequences
In the field of fixed point theory, it is commonly accepted that a contractivity condition is all the more general if it has more possibilities of being particularized, giving rise to versions of already known theorems. Therefore, in order to show the power of the main introduced results, in this section we are going to illustrate several contexts in which they can be applied.
The first important framework appears when the mapping α : M × M ⟶ ½0,∞Þ constantly takes the value 1; that is, αðz, tÞ = 1 for each z, t ∈ M. In this case, the hypotheses about the α-orbital admissibility and the existence of a point z 0 ∈ M such that αðz 0 , Tz 0 Þ ≥ 1 are trivial.

Corollary 21.
Let ðM, dÞ be a complete metric space, and let T : M ⟶ M be a mapping such that (i) either T or T 2 is continuous (ii) for some λ ∈ 0,∞Þ, there exist a constant k ∈ 0, 1Þ and six numbers a 1 , a 2 , a 3 , a 4 , a 5 , δ ∈ ½0, 1 such that, for all distinct z, t ∈ M \ * Fix T ðMÞ, where I λ ðz, tÞ is defined by (36) and Then, T has a fixed point.

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In addition to this, for the case λ > 0, if we suppose that the contractivity condition (85) holds for all distinct points u, v ∈ M, then T has a unique fixed point.
Proof. It follows from Theorems 12, 17, and 18 applied to the case in which αðz, tÞ = 1 for each z, t ∈ M.☐ The following result follows by choosing in the set of constants fa 1 , a 2 , a 3 , a 4 , a 5 , δg one of them as 1 and the other ones as 0. Notice that six corollaries are being summarized into only one. (ii) there exists a constant k ∈ 0, 1Þ such that at least one of the following conditions is fulfilled for all distinct z, t ∈ M \ * Fix T ðMÞ: Then, T has a fixed point.
Proof. This result corresponds to the case λ > 0 in Theorem 12 (T continuous) or Theorem 17 (T 2 continuous) when the contractivity condition (35) is considered by using the following respective choices for constants: Then, T has a fixed point.
Particular cases are especially interesting, like in the following case, in which it is not necessary to assume the continuity of the mapping T.
Corollary 25 (Istrăţescu [18,19]). Let ðM, dÞ be a complete metric space, and let T : M ⟶ M be a continuous mapping such that there exist a, b ∈ ð0, 1Þ with a + b < 1 satisfying d T 2 z, T 2 t À Á ≤ a · d z, t ð Þ+ b · d Tz, Tt ð Þfor all z, t ∈ M: ð90Þ Then, T has a unique fixed point.
Proof. Let us consider the choices λ = 1 and αðz, tÞ = 1 for each z, t ∈ M. Let k = a + b ∈ ð0, 1Þ, and let a 1 = a k , a 4 = b k , a 2 = a 3 = a 5 = δ = 0: Then, for each distinct z, t ∈ M, which means that the contractivity condition α z, t ð Þd T 2 z, T 2 t À Á ≤ k · I 1 z, t ð Þfor all distinct z, t ∈ M \ * Fix T M ð Þ holds because of (90). Under this framework, the proof of Theorem 12 (using any initial point z 0 ∈ M) guarantees that the Picard sequence fz n = T n z 0 g n∈ℕ converges to a point 11 Journal of Function Spaces u 0 ∈ M. Since T is continuous, then fz n+1 = Tz n g n∈ℕ converges to Tu 0 , so Tu 0 = u 0 .
Furthermore, T has a unique fixed point because if u 0 and v 0 were two distinct fixed points of T, then which is impossible. In the following result, we employ a binary relation for controlling the pairs of points that must satisfy the contractivity condition. Let R be a binary relation on the set M. A mapping T : M ⟶ M is R-orbital admissible if TzRTt for all z, t ∈ M such that zRt.☐ Corollary 26. Let ðM, dÞ be a complete metric space endowed with a binary relation R, and let T : M ⟶ M be a continuous mapping. Assume that for some λ ∈ 0,∞Þ, there exist a constant k ∈ 0, 1Þ and six numbers a 1 , a 2 , a 3 , a 4 , a 5 , δ ≥ 0 satisfying (37) such that, for all distinct z, t ∈ M \ * Fix T ðMÞ such that zRt, If T is R-orbital admissible and there is z 0 ∈ M such that z 0 RTz 0 , then T has at least one fixed point.
Proof. Let us consider the function α R : M × M ⟶ ½0,∞Þ defined by Then, T is α R -orbital admissible and there is z 0 ∈ M such that α R ðz 0 , Tz 0 Þ = 1. The contractivity condition (95) is equivalent to (35) under the assumptions (37). Hence, Theorem 12 is applicable.☐ In the previous corollary, when M is endowed with a binary relation R, we can replace the completeness of the metric space by the weaker version: the metric space ðM, d Þ is R-increasingly complete if each d-Cauchy sequence fz n g n∈ℕ ⊆ M satisfying that z n Rz n+1 for all n ∈ ℕ is d-convergent to a point of M. In this case, Corollary 26 can be stated as follows.