# Solving Integral Equations by Means of Fixed Point Theory

Authors: E. Karapinar; A. Fulga; N. Shahzad; A. F. Roldán López de Hierro
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

One of the most interesting tasks in mathematics is, undoubtedly, to solve any kind of equations. Naturally, this problem has occupied the minds of mathematicians since the dawn of algebra. There are hundreds of techniques for solving many classes of equations, facing the problem of finding solutions and studying whether such solutions are unique or multiple. One of the recent methodologies that is having great success in this field of study is the fixed point theory. Its iterative procedures are applicable to a great variety of contexts in which other algorithms fail. In this paper, we study a very general class of integral equations by means of a novel family of contractions in the setting of metric spaces. The main advantage of this family is the fact that its general contractivity condition can be particularized in a wide range of ways, depending on many parameters. Furthermore, such a contractivity condition involves many distinct terms that can be either adding or multiplying between them. In addition to this, the main contractivity condition makes use of the self-composition of the operator, whose associated theorems used to be more general than the corresponding ones by only using such mapping. In this setting, we demonstrate some fixed point theorems that guarantee the existence and, in some cases, the uniqueness, of fixed points that can be interpreted as solutions of the mentioned integral equations.


---

## Body

## 1. 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 thecontractivity 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 operatorT:M⟶M satisfies are usually inherited by the self-composition T2=T∘T, but it is possible that T2 enjoys those good properties without T doing it. This is the case, for instance, of continuity: it is possible for T2 to be continuous without T being continuous. In this sense, some results (like Istrăţescu’s fixed point theorem; see [18, 19]) employing T2 are more general than their corresponding ones with TOne 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 [21–23], Galerkin methods [24, 25], and Krylov subspace methods [26]).In this paper, we introduce a new family of contractive mappings that we callhybrid-interpolative Reich-Istrăţescu-type 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 self-composition T2 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. Section2 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.


## 2. Background on Fixed Point Theory

In this work, we denote byℝ and ℕ=0,1,2,⋯ the set of all real numbers and the set of nonnegative integers, respectively. Let M,d 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 ∗FixTM the set of all fixed points of T in M.Givenn∈ℕ, the mapping Tn=T∘T∘⋯n∘T:M⟶M is the n-th iterate of T (as convention, we agree that T0 is the identity mapping on M). Given z0∈M, the sequence znn∈ℕ defined by zn=Tnz0 for all n∈ℕ is the Picard sequence of T based on z0. Such sequence can recursively be defined as zn+1=Tzn 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 setM is a nonempty subset R of the Cartesian product M×M. We will write zRt when two points z,t∈M verify u,t∈R.Definition 1.
Letα:M×M⟶0,∞ be a function. A mapping T:M⟶M such that
(1)ifz∈M,αz,Tz≥1⇒αTz,T2z≥1is 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 self-compositionT2=T∘T rather than in terms of T was due to Istrăţescu (see [18, 19]).Theorem 2 (Istrăţescu [18, 19]).
Given a complete metric spaceM,d, every continuous map T:M⟶M is a Picard operator provided that there exist a,c∈0,1 such that a+c<1 and
(2)dT2z,T2t≤adTz,Tt+cdz,t,for all z,t∈M.Notice that the good properties (like continuity) of an operatorT are usually inherited by T2, but it is possible that T2 enjoys those good properties without T doing it. In this sense, some results employing T2 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 [27–29]).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.Theorem 3 (Karapnar [15]).
LetM,d be a complete metric space and let T:M⟶M be a mapping such that there exist constants k∈0,1 and λ∈0,1 satisfying
(3)dTz,Tt≤k·dz,Tzλ·dt,Tt1−λ,for all z,t∈M\∗FixTM. Then, T has a unique fixed point in M.In the previous result, asz and t are not fixed points of T, then dz,Tz>0 and dt,Tt>0. Furthermore, as λ>0 and 1−λ>0, then the expressions dz,Tzλ and dt,Tt1−λ are well defined. However, in the main results that we will introduce later, we will employ expressions such as
(4)a4dTz,Ttλ,which we would like to explain for the sake of clarity: on the one hand, (4) means the product a4·dTz,Ttλ, where a4 and λ are nonnegative real numbers (notice that the exponent λ only affects the distance dTz,Tt, and we avoid to write the brackets); on the other hand, the power dTz,Ttλ 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 00=1.


## 3. 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.Proposition 4.
Givenc∈0,∞, let rn⊂0,∞ be a sequence such that
(5)rn+2≤c·maxrn,rn+1foralln∈ℕ.
LetΔ=maxr0,r1. Then,
(6)r2n≤cnΔ,r2n+1≤cnΔforalln≥1,which is equivalent to
(7)rn+2≤c∗En/2·Δforalln∈ℕ.
In particular, ifc∈0,1, then rn⟶0.Proof.
Forn=0 in (5),
(8)r2≤c⋅maxr0,r1=cΔand for n=1 in (5),
(9)r3≤c·maxr1,r2≤c·maxr1,c·maxr0,r1=c·maxr1,cr0,cr1≤c·maxr1,cr0≤c·maxr1,r0=cΔ.
This means that inequalities (6) hold for n=1. Suppose that (6) holds for some n∈ℕ, that is,
(10)r2n≤cnΔ,r2n+1≤cnΔ.
Therefore,(11)r2n+2≤c·maxr2n,r2n+1≤c·maxcnΔ,cnΔ=cn+1Δ,r2n+3≤c·maxr2n+1,r2n+2≤c·maxcnΔ,cn+1Δ=cn+1Δ.
This completes the induction.☐Lemma 5.
Letznn∈ℕ be a sequence on a metric space M,d. Suppose that there is c∈0,1 such that
(12)dzn+2,zn+3≤c·maxdzn,zn+1,dzn+1,zn+2foralln∈ℕ.
Then,znn∈ℕ is a Cauchy sequence in M,d.Proof.
Let us consider the sequencern defined by rn=dzn,zn+1 for all n∈ℕ. By the hypothesis, this sequence verifies (5). Then, Proposition 4 guarantees that
(13)dz2n,z2n+1=r2n≤cnΔ,dz2n+1,z2n+2=r2n+1≤cnΔforalln≥1,where Δ=maxr0,r1. In particular,
(14)dz2n,z2n+1+dz2n+1,z2n+2≤2cnΔforalln≥1.
Ifc=0 or Δ=0, then znn≥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 znn∈ℕ is a Cauchy sequence in M,d, let ε>0 be arbitrary. Since ε/2Δ>0 and 0<c<1, there is a natural number n0>1 such that
(15)∑k=n0+∞ck<ε2Δ.
In particular,(16)2Δ∑k=n0pck<2Δ∑k=n0+∞ck<εforallp∈ℕsuchthatp≥n0.
Letn,m∈ℕ such that m>n≥2n0. Let p be a natural number such that p≥n0+1 and 2p≥m. Therefore, by (14),
(17)dzn,zm≤∑j=nm−1dzj,zj+1≤∑j=2n02p−1dzj,zj+1=∑k=n0p−1dz2k,z2k+1+dz2k+1,z2k+2≤∑k=n0p−12ckΔ≤2Δ∑k=n0pck<2Δ∑k=n0+∞ck<ε.
This proves thatznn∈ℕ is a Cauchy sequence in M,d.☐Remark 6.
Taking into account that, in general, the notationΔαn is not well defined because the number Δαn is distinct to Δαn, we clarify that, in the next statement, we use the convention:
(18)Δαn=Δαn.Proposition 7.
Givenc∈0,∞ and α∈0,1, let rn⊂0,∞ be a sequence such that
(19)rn+2≤c·maxrn,rn+1αforalln∈ℕ.
LetΔ=maxr0,r1,1. Then,
(20)r2n≤c1+α+α2+⋯+αn−1Δαn,r2n+1≤c1+α+α2+⋯+αn−1Δαnforalln≥1.
Therefore,(21)limsupn⟶+∞rn≤c1/1−α.Proof.
Ifc=0, the announced properties are trivial. Suppose that c>0. Since α∈0,1, we know that
(22)αm+1<αm<αm−1<⋯<α2<αforallm∈ℕ,m≥1.
Therefore, asΔ≥1, then
(23)Δαm+1<Δαm<Δαm−1<⋯<Δα2<Δαforallm∈ℕ,m≥1.
Usingn=0 in (19),
(24)r2≤c·maxr0,r1α≤cΔα,and if n=1 in (19), using (23),
(25)r3≤c·maxr1,r2α≤c·maxΔ,cΔαα=c·maxΔα,cαΔα2≤c·maxΔα,cαΔα=cΔα·max1,cα=cΔα.
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,
(26)r2n+2≤c·maxr2n,r2n+1α≤c·maxc1+α+α2+⋯+αn−1Δαn,c1+α+α2+⋯+αn−1Δαnα=c·c1+α+α2+⋯+αn−1Δαnα=c·cα+α2+⋯+αn−1+αnΔαn+1=c1+α+α2+⋯+αn−1+αnΔαn+1,and using (23),
(27)r2n+3≤c·maxr2n+1,r2n+2α≤c·maxc1+α+α2+⋯+αn−1Δαn,c1+α+α2+⋯+αn−1+αnΔαn+1α=c·maxcα+α2+⋯+αn−1+αnΔαn+1,cα+α2+⋯+αn−1+αn+αn+1Δαn+2≤c·maxcα+α2+⋯+αn−1+αnΔαn+1,cα+α2+⋯+αn−1+αn+αn+1Δαn+1=cΔαn+1·maxcα+α2+⋯+αn−1+αn,cα+α2+⋯+αn−1+αn+αn+1=cΔαn+1·cα+α2+⋯+αn−1+αn=c1+α+α2+⋯+αn−1+αnΔαn+1,which completes the induction. Then, (20) holds.
Taking into account thatαnn∈ℕ⟶0, we know that Δαnn∈ℕ⟶Δ0=1. On the other hand,
(28)limn⟶+∞1+α+α2+⋯+αn−1+αn=∑k=0+∞αk=11−α,then
(29)limn⟶+∞c1+α+α2+⋯+αn−1+αn=c1/1−α.
Hence,(30)limsupn⟶+∞rn≤c1/1−α.
In order to check how different the conditions (19), where α∈0,1, and (5), where α=1, are, let us consider the following example.☐Example 8.
Letrnn∈ℕ⊂0,∞ be the sequence defined by
(31)r1=r2=0.25,rn+2=c·maxrn,rn+1αforalln∈ℕ,where c=α=0.5. Then, it can be easily proven by induction that rn is the constant sequence given by rn=0.25 for all n∈ℕ. Indeed, if rn=rn+1=0.25 for some n∈ℕ, then
(32)rn+2=c·maxrn,rn+1α=0.5·max0.25,0.250.5=0.25.
As a consequence,rn⟶0.25, but it does not converge to zero, as in Proposition 4.Corollary 9.
Letznn∈ℕ be a sequence on a metric space M,d. Suppose that there are c∈0,1 and α,β∈0,1 verifying α+β=1 such that
(33)dzn+2,zn+3≤c·dzn,zn+1α·dzn+1,zn+2βforalln∈ℕ.
Then,znn∈ℕ is a Cauchy sequence in M,d.Proof.
Notice that, for alln∈ℕ,
(34)dzn+2,zn+3≤c·dzn,zn+1α·dzn+1,zn+2β≤c·maxdzn,zn+1,dzn+1,zn+2α·maxdzn,zn+1,dzn+1,zn+2β=c·maxdzn,zn+1,dzn+1,zn+2α+β=c·maxdzn,zn+1,dzn+1,zn+2.
Then, Lemma5 is applicable.☐


## 4. 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.
LetM,d be a metric space and let α:M×M⟶0,∞ be a function. A mapping T:M⟶M is a hybrid-interpolative Reich-Istrăţescu-type contraction in the case that for some λ∈0,∞, there exist a constant k∈0,1 and six numbers a1,a2,a3,a4,a5,δ≥0 such that, for all distinct z,t∈M\∗FixTM,
(35)αz,tdT2z,T2t≤k·Iλz,t,where
(36)Iλz,t=a1dz,tλ+a2dz,Tzλ+a3dt,Ttλ+a4dTz,Ttλ+a5dTz,T2zλ+δdTt,T2tλ1/λ,forλ>0,dz,ta1·dz,Tza2·dt,Tta3·dTz,Tta4·dTz,T2za5·dTt,T2tδ,forλ>0,(37)∑i=15ai+δ≤1,ifλ>0,∑i=15ai+δ=1,ifλ=0.Remark 11.
(1)
As we have commented in the second section, whenλ>0, the expression dz,tλ 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 00 in the expression of Iλz,t. In such a case, we will use the convention 00=1 to avoid such indetermination. In other words, if some exponent in the expression of Iλz,t 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 ∑i=15ai+δ=1(2)
We can believe that the cases(38)∑i=15ai+δ≤1,∑i=15ai+δ=1are equivalent because if ∑i=15ai+δ<1, then we can replace δ by δ′∈0,1 such that ∑i=15ai+δ′=1, and the mapping T is also a hybrid-interpolative Reich-Istrăţescu-type contraction by considering the new parameters a1,a2,a3,a4,a5,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 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 ofIλz,t, for λ>0, is very different to the definition of I0z,t (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:
(39)a1=1,a2=a3=a4=a5=δ=0.
In this case, ifλ>0,
(40)Iλz,t=1·dz,tλ1/λ=dz,t,and if λ=0,
(41)I0z,t=dz,t1·dz,Tz0·dt,Tt0·dTz,Tt0·dTz,T2z0·dTt,T2t0=dz,t·1·1·1·1·1=dz,t.
Notice that this case corresponds to the Banach contractivity condition particularized toT2 instead of T:
(42)αz,tdT2z,T2t≤k·dz,t,which appears when αz,t=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.
LetM,d 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 z0∈M such that αz0,Tz0≥1.Proof.
From the hypothesis, we know that there existsz0∈M such that αz0,T0≥1. Since T is α-orbital admissible, αTz0,T2z0≥1, and by an inductive reasoning, we get that αTnz0,Tn+1z0≤1 for any n∈ℕ. Starting from this point z0∈M, we define the sequence zn in M as follows:
(43)z1=Tz0,z2=Tz1=T2z0,⋯,zn=Tzn−1=Tnz0.
If there is somen∈ℕ satisfying that zn=zn+1, then zn is a fixed point of T, and the proof finishes here. On the contrary case, suppose that zn is distinct to zn+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 zn is Cauchy.Case A. For the first case,λ>0, given any n∈ℕ, taking in (35) z=zn and t=zn+1, we have
(44)dzn+2,zn+3≤αzn+2,zn+3dT2zn,T2zn+1≤k·Iλzn,zn+1=k·a1dzn,zn+1λ+a2dzn,Tznλ+a3dzn+1,Tzn+1λ+a4dTzn,Tzn+1λ+a5dTzn,T2znλ+δdTzn+1,T2zn+1λ1/λ=k·a1dzn,zn+1λ+a2dzn,zn+1λ+a3dzn+1,zn+2λ+a4dzn+1,zn+2λ+a5dzn+1,zn+2λ+δdzn+2,zn+3λ1/λ=k·a1+a2dzn,zn+1λ+a3+a4+a5dzn+1,zn+2λ+δdzn+2,zn+3λ1/λ≤k·a1+a2+a3+a4+a5maxdzn,zn+1λ,dzn+1,zn+2λ+δdzn+2,zn+3λ1/λ.Using the power ofλ,
(45)dzn+2,zn+3λ≤kλa1+a2+a3+a4+a5maxdzn,zn+1λ,dzn+1,zn+2λ+kλδdzn+2,zn+3λ.Therefore,(46)1−kλδdzn+2,zn+3λ≤kλa1+a2+a3+a4+a5maxdzn,zn+1λ,dzn+1,zn+2λ≤kλ1−δmaxdzn,zn+1,dzn+1,zn+2λ,which means that, for all n∈ℕ,
(47)dzn+2,zn+3λ≤kλ1−δ1−kλδ·maxdzn,zn+1,dzn+1,zn+2λ,or, equivalently,
(48)dzn+2,zn+3≤kλ1−δ1−kλδ1/λ·maxdzn,zn+1,dzn+1,zn+2.Let us denote(49)c=kλ1−δ1−kλδ1/λ.Clearly,(50)c=kλ1−δ1−kλδ1/λ=kλ−kλδ1−kλδ1/λ<1.This proves that there isc∈0,1 such that
(51)dzn+2,zn+3≤cmaxdzn,zn+1,dzn+1,zn+2foralln∈ℕ.Lemma5 concludes that znn∈ℕ is a Cauchy sequence in M,d.Case B. For the caseλ=0, given n∈ℕ, taking z=zn and t=zn+1, we have
(52)Iλzn,zn+1=dzn,zn+1a1·dzn,Tzna2·dzn+1,Tzn+1a3·dTzn,Tzn+1a4·dTzn,T2zna5·dTzn+1,T2zn+1δ=dzn,zn+1a1·dzn,zn+1a2·dzn+1,zn+2a3·dzn+1,zn+2a4·dzn+1,zn+2a5·dzn+2,zn+3δ=dzn,zn+1a1+a2·dzn+1,zn+2a3+a4+a5·dzn+2,zn+3δ.Hence, the contractivity condition (35) means that
(53)dzn+2,zn+3≤αzn+2,zn+3dzn+2,zn+3=αzn+2,zn+3dT2zn,T2zn+1≤k·Iλzn,zn+1=k·dzn,zn+1a1+a2·dzn+1,zn+2a3+a4+a5·dzn+2,zn+3δ.Ifδ=1, then a1=a2=a3=a4=a5=0, and the previous inequality would be
(54)0<dzn+2,zn+3≤k·dzn+2,zn+3,which is false. Then, necessarily δ<1, so ∑i=15ai=1−δ>0, which means that the numbers
(55)α=a1+a21−δ,β=a3+a4+a51−δsatisfy α+β=1. As a result, property (53) is equivalent to
(56)dzn+2,zn+3≤k·dzn,zn+1a1+a2·dzn+1,zn+2a3+a4+a5·dzn+2,zn+3δ⇔dzn+2,zn+31−δ≤k·dzn,zn+1a1+a2·dzn+1,zn+2a3+a4+a5⇔dzn+2,zn+3≤k1/1−δ·dzn,zn+1a1+a2/1−δ·dzn+1,zn+2a3+a4+a5/1−δ⇔dzn+2,zn+3≤k1/1−δ·dzn,zn+1α·dzn+1,zn+2β,for all n∈ℕ. Taking into account that k∈0,1,
(57)0<1−δ≤1⇒1≤11−δ⇒k1/1−δ≤k<1.Since0<k1/1−δ<1 and α+β=1, Corollary 9 finally concludes that znn∈ℕ is a Cauchy sequence in M,d.In both cases (λ>0 and λ=0), we have demonstrated that znn∈ℕ is a Cauchy sequence in M,d. As it is complete, then there exists a point u∈M such that zn⟶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 mappingT; that is, in both cases (λ>0 and λ=0), the Picard sequence znn∈ℕ is Cauchy in the metric space M,d. Using its completeness, it follows that there exists a point u∈M such that zn⟶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.Whenλ=0 and ∑i=15ai+δ<1, the statement given by Theorem 12 is false. For instance, if a1=a2=a3=a4=a5=δ=0, the contractive condition (36) does not provide any kind of control on dT2z,T2t because the value of I0z,t is always 1 (all exponents are zero). However, even if all constants a1, a2, a3, a4, a5, and δ are strictly positive, the operator T could be fixed point free, as we show in the next example.Example 15.
LetM=0,0.001,0.002,0.003,0.004 be endowed with the Euclidean distance d. Let T:M⟶M be defined by
(58)Tz=z+0.001,ifz<0.004,0,ifz=0.004.
Clearly,dz,t≤0.004, so
(59)dT2z,T2t≤0.004forallz,t∈M.
If we define(60)k=0.95,a1=a2=a3=a4=a5=δ=0.05,the values of k·I0z,tare given in the following table:
(61)7667499.eq-inline.001
As a consequence, ifα is given by αz,t=1 for all z,t∈M, then the mapping T satisfies
(62)dT2z,T2t≤0.004<0.1<k·I0z,t,for all distinct z,t∈M. Therefore, T is a hybrid-interpolative Reich-Istrăţescu-type contraction. Furthermore, M,d is complete, T is α-orbital admissible, and there exists z0∈M such that αz0,Tz0≥1. However, T is fixed point free.Remark 16.
The reason why Theorem12 could fail when λ=0 and ∑i=15ai+δ<1 is the following one: a sequence znn∈ℕ satisfying
(63)dzn+2,zn+3≤k·dzn,zn+1α·dzn+1,zn+2βforalln∈ℕ,could be non-Cauchy when α+β<1. For instance, let znn∈ℕ be the sequence defined by
(64)zn=0.25nforalln∈ℕ.
IfM=zn:n∈ℕ⊂0,∞ is endowed with the Euclidean distance, then
(65)dzn,zn+1=zn+1−zn=0.25n+1−0.25n=0.25foralln∈ℕ.
Therefore, if we take(66)k=δ=12,a1=a2=116,a3=a4=a5=124,then, for all n∈ℕ,
(67)k·dzn,zn+1a1+a2/1−δ·dzn+1,zn+2a3+a4+a5/1−δ=0.5·0.250.25·0.250.25=0.25=dzn+2,zn+3.
However, the sequenceznn∈ℕ positively diverges, so it is not Cauchy.If the operatorT is continuous on M, then the composition T2=T∘T also is. However, the mapping T2 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 T2, whose set of fixed points is nonempty, as it is shown in the next statement.Theorem 17.
LetM,d be a complete metric space and T:M⟶M be a hybrid-interpolative Reich-Istrăţescu-type contraction such that T2 is continuous. If, in addition,
(1)
T is α-orbital admissible(2)
there existsz0∈M such that αz0,Tz0≥1(3)
αu,Tu≥1 for u∈∗FixT2Mthen T has a fixed point.Proof.
Letzn=Tnz0n∈ℕ be the Picard sequence of T whose initial point is z0. In Remark 14, we commented that this sequence has a limit u∈M. Since the mapping T2 is continuous, then
(68)du,T2u=limn⟶∞dzn+2,T2u=limn⟶∞dT2zn,T2u=0.
Thereby,T2u=u; that is, u is a fixed point of T2. As a result, the mapping T2 has at least one fixed point; that is, the set ∗FixT2M is nonempty. Furthermore, T3u=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=T2u=u, which is false. Then, u,Tu∈M\∗FixTM.☐Case A. Ifλ>0, then
(69)Iλu,Tu=a1du,Tuλ+a2du,Tuλ+a3dTu,T2uλ+a4dTu,T2uλ+a5dTu,T2uλ+δdT2u,T3uλ1/λ=a1du,Tuλ+a2du,Tuλ+a3dTu,uλ+a4dTu,uλ+a5dTu,uλ+δdu,Tuλ1/λ=a1+a2+a3+a4+a5+δdu,Tuλ1/λ≤du,Tuλ1/λ=du,Tu.Sinceαu,Tu≥1, then from (35),
(70)0<du,Tu≤αu,Tudu,Tu=αu,TudT2u,T3u≤k·Iλu,Tu≤k·du,Tu,which is a contradiction. Then, Tu=u, so u is a fixed point of T.Case B. Ifλ=0, then
(71)I0u,Tu=du,Tua1·du,Tua2·dTu,T2ua3·dTu,T2ua4·dTu,T2ua5·dT2u,T3uδ=du,Tua1·du,Tua2·dTu,ua3·dTu,ua4·dTu,ua5·du,Tuδ=du,Tua1+a2+a3+a4+a5+δ=du,Tu,and the argument of the previous case can be repeated here.For the caseλ>0, by adding a supplementary condition, we can show the uniqueness of the fixed point.Theorem 18.
Under the hypotheses of Theorem17, if we suppose that the contractivity condition (35) holds for all distinct points u,v∈M and also that αu,v≥1 for every u,v∈∗FixTM, then T has a unique fixed point.Proof.
Theorem17 guarantees that the set of fixed points of T is nonempty. Suppose that T has two distinct fixed points u,v∈∗FixTM.☐Case A. Ifλ>0, replacing such points in (35), we obtain
(72)Iλu,v=a1du,vλ+a2du,Tuλ+a3dv,Tvλ+a4dTu,Tvλ+a5dTu,T2uλ+δdTv,T2vλ1/λ=a1du,vλ+a4du,vλ1/λ=a1+a41/λdu,v≤du,v.Therefore,(73)du,v≤αu,vdu,v=αu,vdT2u,T2v≤k·Iλu,v≤k·du,v<du,v,which is impossible. Then, T cannot have two distinct fixed points.Case B. Ifλ=0, then
(74)I0u,v=du,va1·du,Tua2·dv,Tva3·dTu,Tva4·dTu,T2ua5·dTv,T2vδ=du,va1·0a2·0a3·du,va4·0a5·0δ.Ifa2>0 or a3>0 or a5>0 or δ>0, then I0u,v=0, so the contractivity condition (35) leads to
(75)du,v≤αu,vdu,v=αu,vdT2u,T2v≤k·Iλu,v=0,which is false because u and v are distinct points. On the contrary, if a2=a3=a5=δ=0, we agreed that 0a2=0a3=0a5=0δ=1, so
(76)a1+a4=∑i=15ai+δ=1,I0u,v=du,va1+a4=du,v.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 caseλ=0; more exactly, we can remove the continuity conditions of T or T2.Theorem 19.
LetM,d be a complete metric space, and let T:M⟶M be a hybrid-interpolative Reich-Istrăţescu-type contraction for λ=0 such that a1>0 or a2>0 or a5>0. If we suppose that
(i)
T is α-orbital admissible(ii)
there existsz0∈M such that αz0,Tz0≥1(iii)
for any sequenceznn∈ℕ in M such that αzn,zn+1≥1 for n∈ℕ and zn⟶u as n⟶∞, we have that αzn,u≥1 for all n∈ℕthen T has a fixed point.Proof.
Following the proof of Theorem12, we considered the Picard sequence zn=Tnz0 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 M,d, 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 thatznn∈ℕ satisfies zn≠zm for all n,m∈ℕ such that n≠m. In this case, there is n0∈ℕ such that zn and u are distinct and they are not fixed points of T for all n≥n0. Let us check that u is a fixed point of T2. Indeed, for all n≥n0,
(77)dzn+2,T2u≤αzn,udT2zn,T2u≤k·I0zn,u=k·dzn,ua1·dzn,Tzna2·du,Tua3·dTzn,Tua4·dTzn,T2zna5·dTu,T2uδ=k·dzn,ua1·dzn,zn+1a2·du,Tua3·dzn+1,Tua4·dzn+1,zn+2a5·dTu,T2uδ.
Sincea1>0 or a2>0 or a5>0, letting n⟶∞ in (77), we find out that du,T2u=0; that is, u is a fixed point of T2. Now, following the lines from Theorem 17, we obtain a contradiction that proves that u is also a fixed point of T.☐Let us show an example in which we can deduce the existence of a fixed point from Theorem19 even when T and T2 are not continuous at the same time.Example 20.
LetM,d be the complete metric space where M=−1,1, and d:M×M⟶0,∞ is the metric dz,t=z−t for all z,t∈M. Let T:M⟶M be the mapping defined for each z∈M by
(78)Tz=1−z2,ifz∈−1,0,z22,ifz∈0,1.
Its self-composition is given by(79)T2z=1−z22,ifz∈−1,0,z48,ifz∈0,1.
Clearly,T and T2 are not continuous mappings. Next, let us show that T is a hybrid-interpolative Reich-Istrăţescu-type contraction w.r.t. the function α:M×M⟶0,∞ defined as follows:
(80)αz,t=32,ifz,t∈0,1,1,ifz=−1,t=1,0,otherwise.
Let us take(81)k=34,a1=a4=12,a2=a3=a5=δ=0.
Assumptions (i), (ii), and (iii) of Theorem19 are satisfied. Furthermore, the contractivity condition can be checked as follows:
(a)
For everyz,t∈0,1, we have
(82)αz,tdT2z,T2t=32z48−t48=3z2−t2z2+t216≤38z2−t2=38·z2−t2·z2−t2=38·z+t·z−t·2·z2−t22=38·2·z+t·z−t·z22−t22≤38·2·2·z−t1/2·z22−t221/2=34·dz,t1/2·dTz,Tt1/2,(b)
Forz=−1 and t=1, we have
(83)α−1,1dT2−1,T21=18<34=34·d−1,11/2·dT−1,T11/2.(c)
For all other cases,(84)αz,tdT2z,T2t=0≤34·dz,t1/2·dTz,Tt1/2.
Hence, from Theorem19, we conclude that T has a fixed point.


## 5. 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 mappingα:M×M⟶0,∞ constantly takes the value 1; that is, αz,t=1 for each z,t∈M. In this case, the hypotheses about the α-orbital admissibility and the existence of a point z0∈M such that αz0,Tz0≥1 are trivial.Corollary 21.
LetM,d be a complete metric space, and let T:M⟶M be a mapping such that
(i)
eitherT or T2 is continuous(ii)
for someλ∈0,∞, there exist a constant k∈0,1 and six numbers a1,a2,a3,a4,a5,δ∈0,1 such that, for all distinct z,t∈M\∗FixTM,
(85)dT2z,T2t≤k·Iλz,t,where Iλz,t is defined by (36) and
(86)∑i=15ai+δ≤1,ifλ>0,∑i=15ai+δ=1,ifλ=0.Then,T has a fixed point.In addition to this, for the caseλ>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 Theorems12, 17, and 18 applied to the case in which αz,t=1 for each z,t∈M.☐The following result follows by choosing in the set of constantsa1,a2,a3,a4,a5,δ one of them as 1 and the other ones as 0. Notice that six corollaries are being summarized into only one.Corollary 22.
LetM,d be a complete metric space, let α:M×M⟶0,∞ be a function, and let T:M⟶M be a mapping such that
(i)
eitherT or T2 is continuous, T is α-orbital admissible, and there exists z0∈M such that αz0,Tz0≥1(ii)
there exists a constantk∈0,1 such that at least one of the following conditions is fulfilled for all distinct z,t∈M\∗FixTM:
(87)αz,tdT2z,T2t≤k·dz,t,αz,tdT2z,T2t≤k·dz,Tz,αz,tdT2z,T2t≤k·dt,Tt,αz,tdT2z,T2t≤k·dTz,Tt,αz,tdT2z,T2t≤k·dTz,T2z,αz,tdT2z,T2t≤k·dTt,T2t.Then,T has a fixed point.Proof.
This result corresponds to the caseλ>0 in Theorem 12 (T continuous) or Theorem 17 (T2 continuous) when the contractivity condition (35) is considered by using the following respective choices for constants:
(88)a1=1,a2=a3=a4=a5=δ=0;a2=1,a1=a3=a4=a5=δ=0,a3=1,a1=a2=a4=a5=δ=0,a4=1,a1=a2=a3=a5=δ=0,a5=1,a1=a2=a3=a4=δ=0,δ=1,a1=a2=a3=a4=a5=0.☐Remark 23.
Notice that we would have obtained the same six contractivity conditions (87) for the case λ=0 under the six cases given by (88).The following result follows by usingαz,t=1 for each z,t∈M in the previous statement.Corollary 24.
LetM,d be a complete metric space, and let T:M⟶M be a mapping such that
(i)
eitherT or T2 is continuous(ii)
there exists a constantk∈0,1 such that at least one of the following conditions is fulfilled for all distinct z,t∈M\∗FixTM:
(89)dT2z,T2t≤k·dz,t,dT2z,T2t≤k·dz,Tz,dT2z,T2t≤k·dt,Tt,dT2z,T2t≤k·dTz,Tt,dT2z,T2t≤k·dTz,T2z,dT2z,T2t≤k·dTt,T2t.
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 mappingT.Corollary 25 (Istrăţescu [18, 19]).
LetM,d be a complete metric space, and let T:M⟶M be a continuous mapping such that there exist a,b∈0,1 with a+b<1 satisfying
(90)dT2z,T2t≤a·dz,t+b·dTz,Ttforallz,t∈M.
Then,T has a unique fixed point.Proof.
Let us consider the choicesλ=1 and αz,t=1 for each z,t∈M. Let k=a+b∈0,1, and let
(91)a1=ak,a4=bk,a2=a3=a5=δ=0.
Then, for each distinctz,t∈M,
(92)k·I1z,t=k·akdz,t+bkdTz,Tt=a·dz,t+b·dTz,Tt,which means that the contractivity condition
(93)αz,tdT2z,T2t≤k·I1z,tforalldistinctz,t∈M\∗FixTMholds because of (90). Under this framework, the proof of Theorem 12 (using any initial point z0∈M) guarantees that the Picard sequence zn=Tnz0n∈ℕ converges to a point u0∈M. Since T is continuous, then zn+1=Tznn∈ℕ converges to Tu0, so Tu0=u0.
Furthermore,T has a unique fixed point because if u0 and v0 were two distinct fixed points of T, then
(94)du0,v0=dT2u0,T2v0≤a·du0,v0+b·dTu0,Tv0=a+bdu0,v0<du0,v0,which is impossible.
In the following result, we employ a binary relation for controlling the pairs of points that must satisfy the contractivity condition. LetR 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.
LetM,d 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,∞, there exist a constant k∈0,1 and six numbers a1,a2,a3,a4,a5,δ≥0 satisfying (37) such that, for all distinct z,t∈M\∗FixTM such that zRt,
(95)dT2z,T2t≤k·Iλz,t.
IfT is R-orbital admissible and there is z0∈M such that z0RTz0, then T has at least one fixed point.Proof.
Let us consider the functionαR:M×M⟶0,∞ defined by
(96)αRz,t=1,ifzRt,0,otherwise.
Then,T is αR-orbital admissible and there is z0∈M such that αRz0,Tz0=1. The contractivity condition (95) is equivalent to (35) under the assumptions (37). Hence, Theorem 12 is applicable.☐In the previous corollary, whenM is endowed with a binary relation R, we can replace the completeness of the metric space by the weaker version: the metric space M,d is R-increasingly complete if each d-Cauchy sequence znn∈ℕ⊆M satisfying that znRzn+1 for all n∈ℕ is d-convergent to a point of M. In this case, Corollary 26 can be stated as follows.Corollary 27.
LetM,d be a R-increasingly complete metric space w.r.t. a binary relation R on M, and let T:M⟶M be a continuous mapping. Assume that for some λ∈0,∞, there exist a constant k∈0,1 and six numbers a1,a2,a3,a4,a5,δ≥0 satisfying (37) such that, for all distinct z,t∈M\∗FixTM such that zRt,
(97)dT2z,T2t≤k·Iλz,t.
IfT is R-orbital admissible and there is z0∈M such that z0RTz0, then T has at least one fixed point.Proof.
Using the functionαR defined in (96) and following the lines of the proof of Theorem 12, we get that the Picard sequence zn=Tnz0n∈ℕ is a Cauchy sequence in M,d such that znRTzn for all n∈ℕ, that is, znRzn+1 for all n∈ℕ. Since M,d is an R-increasingly complete metric space, such sequence is convergent to a point u0∈M. Then, the rest of the proof of Theorem 12 can be repeated.☐The following consequence corresponds to the choiceαz,t=1 for each z,t∈M in Theorem 19. Notice that here the operator T has not been continuous.Corollary 28.
LetM,d be a complete metric space, and let T:M⟶M be a mapping for which there exist a constant k∈0,1 and six numbers a1,a2,a3,a4,a5,δ≥0 such that, for all distinct z,t∈M\∗FixTM,
(98)dT2z,T2t≤k·dz,ta1·dz,Tza2·dt,Tta3·dTz,Tta4·dTz,T2za5·dTt,T2tδ.Suppose that∑i=15ai+δ=1 and a1+a2+a5>0. Then, T has at least one fixed point.Furthermore, if the contractivity condition (98) holds for all distinct points u,v∈M and also αu,v≥1 for every u,v∈∗FixTM, then T has a unique fixed point.Proof.
This result follows from the caseλ=0 in Theorem 19 by using the function αz,t=1 for each z,t∈M. Notice that the condition a1+a2+a5>0 means that at least one of the constants a1, a2, or a5 is strictly positive.
To prove the uniqueness, suppose thatu,v∈∗FixTM are two distinct fixed points of T. Then,
(99)I0u,v=du,va1·du,Tua2·dv,Tva3·dTu,Tva4·dTu,T2ua5·dTv,T2vδ=du,va1·0a2·0a3·du,va4·0a5·0δ.
Ifa2>0 or a3>0 or a5>0 or δ>0, we can deduce that I0u,v=0, so the contractivity condition (98) leads to
(100)du,v=dT2u,T2v≤k·Iλu,v=0,which is false because u and v are distinct points. On the contrary, if a2=a3=a5=δ=0, we agreed that 0a2=0a3=0a5=0δ=1, so
(101)a1+a4=∑i=15ai+δ=1,I0u,v=du,va1+a4=du,v.
Therefore,(102)du,v=dT2u,T2v≤k·I0u,v=k·du,v<du,v,which is impossible. Then, T cannot have two distinct fixed points.☐


## 6. An Application to Integral Equations

LetCτ0,Θ,0,1 be the space of all continuous functions with value on 0,1 where Θ>0 and for all z,t∈Cτ0,Θ,0,1:
(103)dz,t=supτ∈τ0,Θzτ−tτ.Obviously, the pairM,d, where M=Cτ0,Θ,0,1, forms a complete metric space.A familyCτ:τ∈ℝ of bounded linear operators is said to be a strongly continuous cosine family if and only if
(1)
C0=I (here I is the identity operator)(2)
for eachz∈M, Cτz is strongly continuous in τ on ℝ(3)
Cτ+υ+Cτ−υ=2CτCυ for all τ,υ∈ℝFor a strongly continuous cosine familyCτ:τ∈ℝ in M, we consider the following:
(1)
The associated sine familySτ:τ∈ℝ where Sτz=∫0τCυzdυ, for τ∈ℝ and z∈M(2)
The infinitesimal generatorA:M⟶M defined by Az=d2/dτ2Cτzτ=0, for z∈DA, where DA=z∈M∣C·z∈C2ℝ,MLet alsoM≥1 and N>0 be two constants such that Ct≤M and St≤N for any τ∈τ0,Θ.We consider the equation(104)z″τ=Azτ+ξτ,zτ,∫τ0τζτ,υ,zυdυ,0<τ0<Θ,(105)zτ0=z0,z′τ0=w0,z0,w0∈0,1,where ξ:τ0,Θ×0,1×ℝ⟶ℝ and ζ:τ0,Θ×τ0,Θ×0,1⟶ℝ are continuous functions.A functionz∈M defined by
(106)zτ=Cτ−τ0z0+Sτ−τ0w0+∫τ0τSτ−υξυ,zυ,∫τ0υζυ,s,zsdsdυ,τ∈τ0,Θis said to be the mild solution of equation (104).Theorem 29.
In this context, if(a)
the mappingT:M⟶M defined for τ∈τ0,Θ by
(107)Tzτ=Cτ−τ0z0+Sτ−τ0w0+∫τ0τSτ−υξυ,zυ,∫τ0υζυ,s,zsdsdυ,is continuous(b)
forτ,υ∈τ0,Θ and z1,z2,t1,t2∈M, there exist some constants k1>0,k2>1 such that
(108)ξτ,z1τ,t1τ−ξτ,z2τ,t2τ≤k1z1τ−z2τ+t1τ−t2τ,ζτ,υ,z1υ−ζτ,υ,z2υ≤k2z1υ−z2υ1/2.(c)
Nk1Θ2Nk11+k2Θ+k2<1 and Nk11+k2Θ>1then equation (104) has a unique mild solution on τ0,Θ.Proof.
We notice that the mild solution of equation (104) is precisely a fixed point of the mapping T. For this reason, we check if the assumption of Theorem 12 holds. Considering the function α:M×M⟶0,∞ defined as αz,t=1 for any z,t∈M, we have that T is α-orbital admissible and there exists z0∈M such that αz0,Tz0=1. Moreover, for z,t∈M, we have
(109)dTz,Tt=supτ∈τ0,ΘTzτ−Ttτ=supτ∈τ0,Θ∫τ0τSτ−υξυ,zυ,∫τ0υζυ,s,zsdsdυ−−∫τ0τSτ−υξυ,tυ,∫τ0υζυ,s,tsdsdυ≤supτ∈τ0,Θ∫τ0τSτ−υξυ,zυ,∫τ0υζυ,s,zsds−ξυ,tυ,∫τ0υζυ,s,tsdsdυ≤Nsupτ∈τ0,Θ∫τ0τk1∣zυ−tυ∣+∣∫τ0υζυ,s,zsds−∫τ0υζυ,s,tsds∣dυ≤N∫τ0τk1supτ∈τ0,Θ∣zυ−tυ∣+∫τ0υk2supτ∈τ0,Θzs−ts1/2dsdυ=Nk1dz,tτ−τ0+∫τ0τk2dz,t1/2υ−τ0dυ=Nk1dz,tτ−τ0+k2dz,t1/2τ−τ022.
Analogically,(110)dT2z,T2t=supτ∈τ0,ΘT2zτ−T2tτ=supτ∈τ0,ΘTTzτ−TTtτ≤supτ∈τ0,Θ∫τ0τSτ−υξυ,Tzυ,∫τ0υζυ,s,Tzsds−−ξυ,Ttυ,∫τ0υζυ,s,Ttsdsdυ≤Nsupτ∈τ0,Θ∫τ0τk1Tzυ−Ttυ+∫τ0υζυ,s,Tzsds−∫τ0υζυ,s,Ttsdsdυ≤N∫τ0τk1supτ∈τ0,Θ\∣Tzυ−Ttυ∣+∫τ0υk2supτ∈τ0,Θzs−ts1/2dsdυ=Nk1dTz,Ttτ−τ0+∫τ0τk2dTz,Tt1/2υ−τ0dυ=Nk1dTz,Ttτ−τ0+k2dTz,Tt1/2τ−τ022,and taking (109) into account, we have
(111)dT2z,T2t≤Nk1Nk1dz,tτ−τ0+k2dz,t1/2τ−τ022τ−τ0+k2dTz,Tt1/2τ−τ022≤Nk1Nk1dz,t1/2Θ2+k2Θ32+k2dTz,Tt1/2Θ22=Nk1Θ21+k2Θ2·dz,t1/2+Nk1k2Θ22·dTz,Tt1/2≤Nk1Θ21+k2Θ·dz,t1/2+Nk1k2Θ2·dTz,Tt1/2.
If we denotek=Nk1Θ2Nk11+k2Θ+k2<1, choosing a1=Nk1Θ21+k2Θ/k<1, a4=Nk1Θ2k2/k<1, a2=a3=a5=δ=0, and λ=1/2, we have that the relation (35) holds. Therefore, T is a hybrid-interpolative Reich-Istrăţescu-type contraction, and from Theorem 12, it follows that the mapping T has a fixed point or, equivalent, that equation (104) has a mild solution.☐


## 7. Conclusions and Future Lines of Research

In this paper, we have introduced a great family of contractions that we have called hybrid-interpolative Reich-Istrăţescu-type contractions in the setting of metric spaces. The main advantage of this family is the fact that its general contractivity condition can be particularized in several ways, depending on many parameters (λ,k,a1,a2,a3,a4,a5,δ). Furthermore, such contractivity condition involves many distinct terms that can be either adding or multiplying between them (it depends on λ). In addition to this, we must highlight that the main contractivity condition makes use of the self-composition T2=T∘T, whose associated theorems used to be more general than the corresponding results by only using T.For the study of the behavior of the Picard sequence that is used in the main results, an analysis of the convergence of certain sequences of nonnegative real terms that naturally appear in the proofs that we have introduced has been carried out. If the reader pays attention, some of these sequences repeatedly appear in the field of the fixed point theory. For example, the condition “dzn+1,zn+2≤λdzn,zn+1 for all n∈ℕ” appears in many demonstrations in this field of study. Therefore, the results of Section 3 have been of great help in the development of the main results.In this setting, we have demonstrated some fixed point theorems that guarantee the existence and, in some cases, the uniqueness, of fixed points. To illustrate the power of the introduced results, we have shown several consequences of the main statements. We have also remarked how the proposed results are optimal with respect to the involved parameters; that is, when a result includes the hypothesis about∑i=15ai+δ<1, we have described an example in which the equality in the previous strict inequality leads to the invalidity of the corresponding result.Finally, we have taken advantage of the main contents of the manuscript in order to introduce a novel application in which, under very general conditions, we are able to deduce the existence of solution of a kind of integral equations.Prospective work must be done in this line of research. On the one hand, further studies should be done on how to reformulate the contractivity condition in order to be able to develop fixed point theory in more general abstract metric spaces (for instance,b-metric spaces or fuzzy metric spaces). On the other hand, we propose studying the possible existence of fixed points for an operator T such that
(112)αz,tdT2z,T2t≤k·maxdz,t,dz,Tz,dt,Tt,dTz,Tt,dTz,T2z,dTt,T2t,for all distinct points z,t∈Ω, where Ω is an appropriate subset of a metric space M,d.



---

*Source: 7667499-2022-02-10.xml*