Fixed Point Theorems for Contractions of Rational Type with PPF Dependence in Banach Spaces

Abstract

We prove the existence of the PPF dependent fixed point in the Razumikhin class for contractions of rational type in Banach spaces, by using a general class of pairs of functions. Our result has as particular cases a great number of interesting consequences which extend and generalize some results appearing in the literature.

1. Introduction

Banach’s contraction principle is one of the pivotal results of analysis. Its significance lies in its vast applicability to a great number of branches of mathematics and other sciences, for example, theory of existence of solutions for nonlinear differential, integral, and functional equations, variational inequalities, and optimization and approximation theory.

Generalizations of the contractive mapping theorem have been a heavily investigated branch of research. In particular, this principle was extended in [1], where the domain of the nonlinear operator involved is 𝒞([𝑎,𝑏],𝐸) and the range is 𝐸, where 𝐸 is a Banach space. This result is known as the contraction theorem for operator with PPF (past, present, and future) dependence. The PPF fixed point theorems are useful for proving the existence of solutions for nonlinear functional-differential and integral equations which may depend upon the past history, present data, and future considerations. Some papers about fixed point theorems with PPF dependence have appeared in the literature (see, e.g., [1–5]).

On the other hand, Dass and Gupta in [6] and Jaggi in [7] were the pioneers in proving fixed point theorems using contractive conditions involving rational expressions. In [4], the authors present a fixed point theorem for contractions of rational type with PPF dependence.

The purpose of this paper is to present a fixed point theorem for generalized contractions of rational type with PPF dependence which has, as particular cases, interesting consequences. Particularly, our result extends the one appearing in [4].

2. Preliminaries

Throughout this paper, 𝐸 will denote a Banach space with norm ‖⋅‖_𝐸 and 𝐸₀=𝒞([𝑎,𝑏],𝐸) will denote the space of the continuous 𝐸-valued functions defined on [𝑎,𝑏] and equipped with the norm ‖⋅‖_𝐸0 given by 𝜙_𝐸0=sup_{[]𝑡∈𝑎,𝑏}𝜙(𝑡)_𝐸for𝜙∈𝐸₀. (1)

Let 𝑇∶𝐸₀→𝐸 be a mapping. A point 𝜙∈𝐸₀ is said to be a PPF dependence fixed point of  𝑇 or a fixed point with PPF dependence of  𝑇 if  𝑇𝜙=𝜙(𝑐), for some 𝑐∈[𝑎,𝑏].

For a fixed 𝑐∈[𝑎,𝑏], the Razumikhin class 𝑅_𝑐 is defined by 𝑅_𝑐=𝜙∈𝐸₀∶𝜙_𝐸0=𝜙(𝑐)_𝐸. (2)

Remark 1. Notice that, for 𝑥∈𝐸 fixed, the function 𝜙_𝑥, defined by 𝜙_𝑥(𝑡)=𝑥for𝑡∈[𝑎,𝑏], (3) satisfies 𝜙_𝑥∈𝐸₀, ‖𝜙_𝑥‖_𝐸0=‖𝜙_𝑥(𝑐)‖_𝐸=‖𝑥‖ for any 𝑐∈[𝑎,𝑏] and, therefore, 𝜙_𝑥∈𝑅_𝑐 for any 𝑐∈[𝑎,𝑏]. Consequently, 𝑅_𝑐≠0/ for any 𝑐∈[𝑎,𝑏].

We say that the class 𝑅_𝑐 is algebraically closed with respect to difference if for any  𝜙,𝜉∈𝑅_𝑐 we have  𝜙−𝜉∈𝑅_𝑐. Similarly, we say that the class 𝑅_𝑐 is topologically closed if it is closed with respect to the topology on 𝐸₀ induced by the norm ‖⋅‖_𝐸0.

The Razumikhin class plays an important role in the existence of PPF fixed point.

The first result about the existence of PPF fixed point appears in [1] and it is presented in the following theorem.

Theorem 2 (see [1]). Suppose that 𝑇∶𝐸₀→𝐸 is a mapping such that there exists 𝛼∈[0,1) satisfying 𝑇𝜙−𝑇𝜉_𝐸≤𝛼𝜙−𝜉_𝐸0 (4) for any  𝜙,𝜉∈𝐸₀. If  𝑅_𝑐 is topologically closed and algebraically closed with respect to difference for some 𝑐∈[𝑎,𝑏], then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Recently, in [4] the authors proved the following PPF dependent fixed point theorem for rational type contraction mappings.

Theorem 3 (see [4]). Let 𝑇∶𝐸₀→𝐸 be a mapping satisfying 𝑇𝜙−𝑇𝜉_𝐸≤𝛼𝜙−𝜉_𝐸0+𝛽𝜙(𝑐)−𝑇𝜙_𝐸𝜉(𝑐)−𝑇𝜉_𝐸1+𝑇𝜙−𝑇𝜉_𝐸 (5) for any 𝜙,𝜉∈𝐸₀ and where 𝛼,𝛽∈[0,1) with 𝛼+𝛽<1 and 𝑐∈[𝑎,𝑏]. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

The main purpose of this paper is, by using a class of pairs of functions satisfying certain assumptions, to present new PPF dependent fixed point theorems for contractions of rational type. Particularly, our result generalizes the main result of  [4] (Theorem 3).

3. Main Results

We start this section presenting the following class of pairs of functions 𝔉. A pair of functions (𝜑,𝜙) is said to belong to the class 𝔉 if they satisfy the following conditions:(i)𝜑,𝜙∶[0,∞)→[0,∞);(ii)for 𝑡,𝑠∈[0,∞), if 𝜑(𝑡)≤𝜙(𝑠), then 𝑡≤𝑠;(iii)for (𝑡_𝑛) and (𝑠_𝑛) sequences in [0,∞) such that lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝑠_𝑛=𝑎, (6) if 𝜑(𝑡_𝑛)≤𝜙(𝑠_𝑛) for any 𝑛∈ℕ, then 𝑎=0.

Remark 4. Notice that if (𝜑,𝜙)∈𝔉 and 𝜑(𝑡)≤𝜙(𝑡), then 𝑡=0, since we can take 𝑡_𝑛=𝑠_𝑛=𝑡 for any 𝑛∈ℕ and by (iii) we deduce 𝑡=0.

In the sequel, we present some interesting examples of pairs of functions belonging to the class 𝔉 which will be very important in our study.

Example 5. Let 𝜑∶[0,∞)→[0,∞) be a continuous and increasing function such that 𝜑(𝑡)=0 if and only if 𝑡=0 (these functions are known in the literature as altering distance functions).
Let 𝜙∶[0,∞)→[0,∞) be a nondecreasing function such that 𝜙(𝑡)=0 if and only if  𝑡=0 and suppose that 𝜙≤𝜑.
Then the pair (𝜑,𝜑−𝜙)∈𝔉.
In fact, it is clear that (𝜑,𝜑−𝜙) satisfy (i).
To prove (ii), suppose that 𝑡,𝑠∈[0,∞) and 𝜑(𝑡)≤(𝜑−𝜙)(𝑠). Then, from 𝜑(𝑡)≤𝜑(𝑠)−𝜙(𝑠)≤𝜑(𝑠) (7) and taking into account the increasing character of 𝜑, we can deduce that 𝑡≤𝑠.
In order to prove (iii), we suppose that 𝜑𝑡_𝑛≤𝜑𝑠_𝑛−𝜙𝑠_𝑛≤𝜑𝑠_𝑛forany𝑛∈ℕ, (8) where 𝑡_𝑛,𝑠_𝑛∈[0,∞) and lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝑠_𝑛=𝑎. (9) Taking 𝑛→∞ in (8), we infer that lim_𝑛→∞𝜙(𝑠_𝑛)=0.
Let us suppose that 𝑎>0. Since lim_𝑛→∞𝑠_𝑛=𝑎>0, we can find 𝜀>0 and a subsequence (𝑠_𝑛𝑘) of (𝑠_𝑛) such that 𝑠_𝑛𝑘>𝜀 for any 𝑘∈ℕ. As 𝜙 is nondecreasing, we have 𝜙(𝑠_𝑛𝑘)>𝜙(𝜀) for any 𝑘∈ℕ and, consequently, lim_𝑘→∞𝜙(𝑠_𝑛𝑘)≥𝜙(𝜀). This contradicts the fact that lim_𝑘→∞𝜙(𝑠_𝑛𝑘)=0. Therefore, 𝑎=0.
This proves that (𝜑,𝜑−𝜙)∈𝔉.

An interesting particular case is when 𝜑 is the identity mapping, 𝜑=1_[0,∞), and 𝜙∶[0,∞)→[0,∞) is a nondecreasing function such that 𝜙(𝑡)=0 if and only if  𝑡=0 and 𝜙(𝑡)≤𝑡 for any 𝑡∈[0,∞).

Example 6. Let 𝑆 be the class of functions defined by 𝑆=𝛼∶[0,∞)⟶[0,1)∶𝛼𝑡_𝑛⟶1⟹𝑡_𝑛⟶0. (10) Let us consider the pairs of functions (1_[0,∞),𝛼1_[0,∞)), where 𝛼∈𝑆 and 𝛼1_[0,∞) is defined by (𝛼1_[0,∞))(𝑡)=𝛼(𝑡)𝑡, for 𝑡∈[0,∞).
Then (1_[0,∞),𝛼1_[0,∞))∈𝔉.
It is clear that the pairs (1_[0,∞),𝛼1_[0,∞)) with 𝛼∈𝑆 satisfy (i).
To prove (ii), from 1_[0,∞)(𝑡)≤𝛼1_[0,∞)(𝑠)for𝑡,𝑠∈[0,∞) (11) we infer, since 𝛼∶[0,∞)→[0,1), that 𝑡≤𝛼(𝑠)𝑠<𝑠 (12) and, consequently, (1_[0,∞),𝛼1_[0,∞)) satisfies (ii).
In order to prove (iii), we suppose that 1_[0,∞)𝑡_𝑛=𝑡_𝑛≤𝛼1_[0,∞)𝑠_𝑛=𝛼𝑠_𝑛𝑠_𝑛forany𝑛∈ℕ, (13) where 𝑡_𝑛,𝑠_𝑛∈[0,∞) and lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝑠_𝑛=𝑎.
Let us suppose that 𝑎>0.
Since lim_𝑛→∞𝑠_𝑛=𝑎>0, we can find a subsequence (𝑠_𝑛𝑘) such that 𝑠_𝑛𝑘>0 for any 𝑘∈ℕ. Now, as 𝑡_𝑛≤𝛼𝑠_𝑛𝑠_𝑛≤𝑠_𝑛forany𝑛∈ℕ, (14) in particular, we have 𝑡_𝑛𝑘≤𝛼𝑠_𝑛𝑘𝑠_𝑛𝑘≤𝑠_𝑛𝑘forany𝑘∈ℕ, (15) and, since 𝑠_𝑛𝑘>0 for any 𝑘∈ℕ, 𝑡_𝑛𝑘𝑠_𝑛𝑘≤𝛼𝑠_𝑛𝑘≤1. (16) Taking 𝑘→∞ in the last inequality, we obtain lim_𝑘→∞𝛼𝑠_𝑛𝑘=1. (17) Finally, since 𝛼∈𝑆, we infer that lim_𝑘→∞𝑠_𝑛𝑘=0 and this contradicts the fact that lim_𝑛→∞𝑠_𝑛=𝑎>0.
Therefore, 𝑎=0.
This proves that (1_[0,∞),𝛼1_[0,∞))∈𝔉 for 𝛼∈𝑆.

Remark 7. Suppose that 𝑔∶[0,∞)→[0,∞) is an increasing function and (𝜑,𝜙)∈𝔉. Then it is easily seen that the pair (𝑔∘𝜑,𝑔∘𝜙)∈𝔉.

Now, we are ready to present our main result.

Theorem 8. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exists a pair of functions (𝜑,𝜙)∈𝔉 and 𝑐∈[𝑎,𝑏] such that 𝜑𝑇𝜉−𝑇𝜂_𝐸≤max𝜙𝜉−𝜂_𝐸0,𝜙𝜉(𝑐)−𝑇𝜉_𝐸⋅𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (18) for any 𝜉,𝜂∈𝐸₀. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Proof. Let 𝜉₀ be an arbitrary function in 𝑅_𝑐 (whose existence is guaranteed by Remark 1). Since 𝜉₀∈𝑅_𝑐⊂𝐸₀, put 𝑥₁=𝑇𝜉₀∈𝐸.
Again, by Remark 1, we can find 𝜉₁∈𝑅_𝑐 such that 𝑇𝜉₀=𝑥₁=𝜉₁(𝑐). (19) Since 𝜉₁∈𝑅_𝑐⊂𝐸₀, put 𝑥₂=𝑇𝜉₁∈𝐸. Using the same argument, we can find 𝜉₂∈𝑅_𝑐 such that 𝑇𝜉₁=𝑥₂=𝜉₂(𝑐). (20) Repeating this process, we can obtain a sequence (𝜉_𝑛) in 𝑅_𝑐 such that 𝑇𝜉_𝑛−1=𝜉_𝑛(𝑐)forany𝑛∈ℕ. (21) Since 𝑅_𝑐 is algebraically closed with respect to difference, we have 𝜉_𝑝−𝜉_𝑞𝐸0=𝜉_𝑝(𝑐)−𝜉_𝑞(𝑐)_𝐸 (22) for any 𝑝,𝑞∈ℕ.
First, we will prove that lim_𝑛→∞‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0=0. In fact, taking into account (21) and (22), we get 𝜉_𝑛+1−𝜉_𝑛𝐸0=𝜉_𝑛+1(𝑐)−𝜉_𝑛(𝑐)_𝐸=𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸forany𝑛∈ℕ (23) and, therefore, applying the contractive condition, we have 𝜑𝜉_𝑛+1−𝜉_𝑛𝐸0=𝜑𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸≤max𝜙𝜉_𝑛−𝜉_{𝑛−1𝐸0},𝜙𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸⋅𝜉_𝑛−1(𝑐)−𝑇𝜉_𝑛−1𝐸1+𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸. (24)
Let us suppose that there exists 𝑛₀∈ℕ such that ‖𝜉_𝑛0+1−𝜉_𝑛0‖_𝐸0=0.
In this case, 𝜉_𝑛0+1=𝜉_𝑛0 and, consequently, 𝜉_𝑛0+1(𝑐)=𝜉_𝑛0(𝑐).
By (21), we have 𝑇𝜉_𝑛0=𝜉_𝑛0+1(𝑐)=𝜉_𝑛0(𝑐) (25) and 𝜉_𝑛0 would be the PPF dependent fixed point.
In the sequel we suppose that ‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0≠0 for any 𝑛∈ℕ.
Now, we can distinguish two cases.
Case  1. Consider max𝜙𝜉_𝑛−𝜉_{𝑛−1𝐸0},𝜙𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸⋅𝜉_𝑛−1(𝑐)−𝑇𝜉_𝑛−1𝐸1+𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸=𝜙𝜉_𝑛−𝜉_{𝑛−1𝐸0}. (26) In this case, from (24), we infer 𝜑𝜉_𝑛+1−𝜉_𝑛𝐸0≤𝜙𝜉_𝑛−𝜉_{𝑛−1𝐸0}, (27) and, since (𝜑,𝜙)∈𝔉, we deduce 𝜉_𝑛+1−𝜉_𝑛𝐸0≤𝜉_𝑛−𝜉_{𝑛−1𝐸0}. (28)Case  2. Consider max𝜙𝜉_𝑛−𝜉_{𝑛−1𝐸0},𝜙𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸⋅𝜉_𝑛−1(𝑐)−𝑇𝜉_𝑛−1𝐸1+𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸𝜉=𝜙^𝑛(𝑐)−𝑇𝜉_𝑛𝐸⋅𝜉_𝑛−1(𝑐)−𝑇𝜉_𝑛−1𝐸1+𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸. (29) In this case, from (24) and since (𝜑,𝜙)∈𝔉, we infer 𝜉_𝑛+1−𝜉_𝑛𝐸0≤𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸⋅𝜉_𝑛−1(𝑐)−𝑇𝜉_𝑛−1𝐸1+𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸. (30) By (21) and (22), we have 𝜉_𝑛+1−𝜉_𝑛𝐸0≤𝜉_𝑛(𝑐)−𝜉_𝑛+1(𝑐)_𝐸⋅𝜉_𝑛−1(𝑐)−𝜉_𝑛(𝑐)_𝐸1+𝜉_𝑛+1(𝑐)−𝜉_𝑛(𝑐)_𝐸=𝜉_𝑛+1−𝜉_𝑛𝐸0⋅𝜉_𝑛−𝜉_{𝑛−1𝐸0}1+𝜉_𝑛+1−𝜉_𝑛𝐸0. (31) Since ‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0≠0, from the last inequality, it follows that 𝜉1≤^𝑛−𝜉_{𝑛−1𝐸0}1+𝜉_𝑛+1−𝜉_𝑛𝐸0 (32) and, therefore, 𝜉_𝑛+1−𝜉_𝑛𝐸0<1+𝜉_𝑛+1−𝜉_𝑛𝐸0≤𝜉_𝑛−𝜉_{𝑛−1𝐸0}. (33) In both cases, we obtain that inequality (28) is satisfied and, consequently, the sequence (‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0) is a decreasing sequence of nonnegative real numbers.
Put 𝑟=lim_𝑛→∞‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0, where 𝑟≥0, and denote {(,{(.𝐴=𝑛∈ℕ:𝑛satisfies26)}𝐵=𝑛∈ℕ:𝑛satisfies29)} (34) We remark the following.(1)If  Card𝐴=∞, then from (24) we can find infinitely many natural numbers 𝑛 satisfying inequality (27) and, since lim_𝑛→∞‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0=lim_𝑛→∞‖𝜉_𝑛−𝜉_𝑛−1‖_𝐸0=𝑟 and (𝜑,𝜙)∈𝔉, we deduce that 𝑟=0.(2)If  Card𝐵=∞, then from (24) we can find infinitely many 𝑛∈ℕ such that 𝜑𝜉_𝑛+1−𝜉_𝑛𝐸0𝜉≤𝜙^𝑛(𝑐)−𝑇𝜉_𝑛𝐸⋅𝜉_𝑛−1(𝑐)−𝑇𝜉_𝑛−1𝐸1+𝑇𝜉_𝑛−𝑇𝜉_𝑛−1𝐸. (35) Since (𝜑,𝜙)∈𝔉 and using a similar argument to the one used in Case 2, we obtain 𝜉_𝑛+1−𝜉_𝑛𝐸0≤𝜉_𝑛+1−𝜉_𝑛𝐸0⋅𝜉_𝑛−𝜉_{𝑛−1𝐸0}1+𝜉_𝑛+1−𝜉_𝑛𝐸0 (36) for infinitely many 𝑛∈ℕ. Taking 𝑛→∞ in the last inequality and taking into account that lim_𝑛→∞‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0=𝑟 we deduce 𝑟𝑟≤²,1+𝑟 (37) and, consequently, 𝑟≤0. Since 𝑟≥0, we obtain 𝑟=0.
Therefore, lim_𝑛→∞𝜉_𝑛+1−𝜉_𝑛𝐸0=0. (38)
Next, we will prove that (𝜉_𝑛) is a Cauchy sequence in 𝐸₀. In contrary case, since lim_𝑛→∞‖𝜉_𝑛+1−𝜉_𝑛‖_𝐸0=0, by Lemma 2.1 of  [8], we can find 𝜀>0 and subsequences (𝜉_𝑛(𝑘)) and (𝜉_𝑚(𝑘)) of (𝜉_𝑛) satisfying (i)𝑛(𝑘)(𝑘)()𝜉>𝑚≥𝑘for𝑘>0,ii𝜀≤_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0},𝜉_{𝑛(𝑘)−1}−𝜉_{𝑚(𝑘)𝐸0}()<𝜀for𝑘>0,iiilim_𝑘→∞𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0}=lim_𝑘→∞𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)+1𝐸0}=lim_𝑘→∞𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)𝐸0}=lim_𝑘→∞𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}=𝜀. (39) Since 𝜉_𝑛(𝑘)+1,𝜉_𝑚(𝑘)+1∈𝑅_𝑐 for any 𝑘∈ℕ, from (21) and (22), we have 𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}=𝜉_𝑛(𝑘)+1(𝑐)−𝜉_𝑚(𝑘)+1(𝑐)_𝐸=𝑇𝜉_𝑛(𝑘)−𝑇𝜉_{𝑚(𝑘)𝐸} (40) for any 𝑘∈ℕ.
Using the contractive condition and (21) and (22), we obtain 𝜑𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}=𝜑𝑇𝜉_𝑛(𝑘)−𝑇𝜉_{𝑚(𝑘)𝐸}≤max𝜙𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0},𝜙𝜉_𝑛(𝑘)(𝑐)−𝑇𝜉_{𝑛(𝑘)𝐸}⋅𝜉_𝑚(𝑘)(𝑐)−𝑇𝜉_{𝑚(𝑘)𝐸}1+𝑇𝜉_𝑛(𝑘)−𝑇𝜉_{𝑚(𝑘)𝐸}=max𝜙𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0},𝜙𝜉_𝑛(𝑘)(𝑐)−𝜉_𝑛(𝑘)+1(𝑐)_𝐸⋅𝜉_𝑚(𝑘)(𝑐)−𝜉_𝑚(𝑘)+1(𝑐)_𝐸×1+𝜉_𝑛(𝑘)+1(𝑐)−𝜉_𝑚(𝑘)+1(𝑐)_𝐸⁻¹=max𝜙𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0},𝜙𝜉_𝑛(𝑘)+1−𝜉_{𝑛(𝑘)𝐸0}⋅𝜉_𝑚(𝑘)+1−𝜉_{𝑚(𝑘)𝐸0}×1+𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}⁻¹ (41) for any 𝑘∈ℕ.
Let us put 𝜑𝜉𝐶=𝑘∈ℕ∶_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}≤𝜙𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0},𝜑𝜉𝐷=𝑘∈ℕ∶_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}𝜉≤𝜙^𝑛(𝑘)+1−𝜉_{𝑛(𝑘)𝐸0}⋅𝜉_𝑚(𝑘)+1−𝜉_{𝑚(𝑘)𝐸0}1+𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}. (42) By (41) we have  Card𝐶=∞ or  Card𝐷=∞.
Let us suppose that  Card𝐶=∞. Then there exist infinitely many 𝑘∈ℕ such that 𝜑𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}≤𝜙𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0} (43) and since (𝜑,𝜙)∈𝔉 and lim_𝑘→∞𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}=lim_𝑘→∞𝜉_𝑛(𝑘)−𝜉_{𝑚(𝑘)𝐸0}=𝜀 (44) we infer from (39) that 𝜀=0. This is a contradiction.
Let us suppose that  Card𝐷=∞. In this case, we can find infinitely many 𝑘∈ℕ such that 𝜑𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}𝜉≤𝜙^𝑛(𝑘)+1−𝜉_{𝑛(𝑘)𝐸0}⋅𝜉_𝑚(𝑘)+1−𝜉_{𝑚(𝑘)𝐸0}1+𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0} (45) and since (𝜑,𝜙)∈𝔉, we infer 𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}≤𝜉_𝑛(𝑘)+1−𝜉_{𝑛(𝑘)𝐸0}⋅𝜉_𝑚(𝑘)+1−𝜉_{𝑚(𝑘)𝐸0}1+𝜉_𝑛(𝑘)+1−𝜉_{𝑚(𝑘)+1𝐸0}. (46) Taking  𝑘→∞ and in view of (38) and (39), it follows that 𝜀≤0 and this is a contradiction.
Therefore, since in both possibilities  Card𝐶=∞  and  Card𝐷=∞ we obtain a contradiction, we deduce that (𝜉_𝑛) is a Cauchy sequence in 𝐸₀.
Since 𝐸₀ is a Banach space, we can find 𝜉^∗∈𝐸₀ such that lim_𝑛→∞𝜉_𝑛=𝜉^∗. As 𝜉_𝑛∈𝑅_𝑐 and 𝑅_𝑐 is topologically closed, we have 𝜉^∗∈𝑅_𝑐.
Next, we will prove that 𝜉^∗ is a PPF dependent fixed point of 𝑇. In fact, by the contractive condition, we obtain 𝜑𝑇𝜉^∗−𝑇𝜉_𝑛𝐸≤max𝜙𝜉^∗−𝜉_𝑛𝐸0,𝜙𝜉^∗(𝑐)−𝑇𝜉^∗_𝐸⋅𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸1+𝑇𝜉^∗−𝑇𝜉_𝑛𝐸 (47) for any 𝑛∈ℕ.
We can distinguish two cases again.(1)There exist infinitely many 𝑛∈ℕ such that 𝜑𝑇𝜉^∗−𝑇𝜉_𝑛𝐸≤𝜙𝜉^∗−𝜉_𝑛𝐸0. (48) In this case, since (𝜑,𝜙)∈𝔉, we obtain 𝑇𝜉^∗−𝑇𝜉_𝑛𝐸≤𝜉^∗−𝜉_𝑛𝐸0 (49) for infinitely many 𝑛∈ℕ. Since lim_𝑛→∞𝜉_𝑛=𝜉^∗, taking 𝑛→∞ in the last inequality, we obtain lim_𝑛→∞𝑇𝜉_𝑛=𝑇𝜉^∗, (50) where, to simplify our considerations, we will denote the subsequence by the same symbol (𝑇𝜉_𝑛). By (21), 𝑇𝜉^∗=lim_𝑛→∞𝑇𝜉_𝑛=lim_𝑛→∞𝜉_𝑛+1(𝑐). (51) 𝜉_𝑛→𝜉^∗ in 𝐸₀; this means that sup_{𝑡∈[𝑎,𝑏]}𝜉_𝑛(𝑡)−𝜉^∗(𝑡)_𝐸⟶0 (52) and, consequently, lim_𝑛→∞𝜉_𝑛+1(𝑐)=𝜉^∗(𝑐). From this last result and from (51) we deduce that 𝑇𝜉^∗=𝜉^∗(𝑐) (53) and, therefore, 𝜉^∗ is a PPF dependent fixed point of 𝑇 in 𝑅_𝑐.(2) There exist infinitely many 𝑛∈ℕ such that 𝜑𝑇𝜉^∗−𝑇𝜉_𝑛𝐸𝜉≤𝜙^∗(𝑐)−𝑇𝜉^∗_𝐸⋅𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸1+𝑇𝜉^∗−𝑇𝜉_𝑛𝐸. (54) To simplify our considerations, we will denote the subsequence by the same symbol (𝑇𝜉_𝑛). Since (𝜑,𝜙)∈𝔉, we infer 𝑇𝜉^∗−𝑇𝜉_𝑛𝐸≤𝜉^∗(𝑐)−𝑇𝜉^∗_𝐸⋅𝜉_𝑛(𝑐)−𝑇𝜉_𝑛𝐸1+𝑇𝜉^∗−𝑇𝜉_𝑛𝐸 (55) for any 𝑛∈ℕ. Using (21), we have that 𝑇𝜉_𝑛=𝜉_𝑛+1(𝑐) and, therefore, 𝑇𝜉^∗−𝑇𝜉_𝑛𝐸≤𝜉^∗(𝑐)−𝑇𝜉^∗_𝐸⋅𝜉_𝑛(𝑐)−𝜉_𝑛+1(𝑐)_𝐸1+𝑇𝜉^∗−𝑇𝜉_𝑛𝐸 (56) for any 𝑛∈ℕ. Taking 𝑛→∞ and by (38) since lim_𝑛→∞‖𝜉_𝑛(𝑐)−𝜉_𝑛+1(𝑐)‖_𝐸=0, we infer (50). From the above case, we deduce that 𝜉^∗ is a PPF dependent fixed point of 𝑇 in 𝑅_𝑐. Therefore, we have proved that in both cases 𝜉^∗ is a PPF dependent fixed point of 𝑇 in 𝑅_𝑐.
Finally, we will prove the uniqueness of PPF dependent fixed point of 𝑇 in 𝑅_𝑐.
Suppose that 𝜙^∗ is another PPF dependent fixed point of 𝑇 in 𝑅_𝑐. Then, since 𝜉^∗, 𝜙^∗∈𝑅_𝑐 and 𝑅_𝑐 is algebraically closed, we obtain 𝜉^∗−𝜙^∗_𝐸0=𝜉^∗(𝑐)−𝜙^∗(𝑐)_𝐸. (57) As 𝜉^∗(𝑐)=𝑇𝜉^∗ and 𝜙^∗(𝑐)=𝑇𝜙^∗ because 𝜉^∗ and 𝜙^∗ are PPF dependent fixed points of 𝑇, we infer 𝜉^∗−𝜙^∗_𝐸0=𝜉^∗(𝑐)−𝜙^∗(𝑐)_𝐸=𝑇𝜉^∗−𝑇𝜙^∗_𝐸. (58) Consequently, using the contractive condition, we get 𝜑𝜉^∗−𝜙^∗_𝐸0=𝜑𝑇𝜉^∗−𝑇𝜙^∗_𝐸≤max𝜙𝜉^∗−𝜙^∗_𝐸0,𝜙𝜉^∗(𝑐)−𝑇𝜉^∗_𝐸⋅𝜙^∗(𝑐)−𝑇𝜙^∗_𝐸1+𝑇𝜉^∗−𝑇𝜙^∗_𝐸=max𝜙𝜉^∗−𝜙^∗_𝐸0,𝜙(0). (59) We can distinguish two cases.(i)Consider max{𝜙(‖𝜉^∗−𝜙^∗‖_𝐸0),𝜙(0)}=𝜙(‖𝜉^∗−𝜙^∗‖_𝐸0). In this case, from (59) we have 𝜑𝜉^∗−𝜙^∗_𝐸0≤𝜙𝜉^∗−𝜙^∗_𝐸0. (60) Now, since (𝜑,𝜙)∈𝔉 and using Remark 4, we get ‖𝜉^∗−𝜙^∗‖_𝐸0=0 and, therefore, 𝜉^∗=𝜙^∗.(ii)Consider max{𝜙(‖𝜉^∗−𝜙^∗‖_𝐸0),𝜙(0)}=𝜙(0). From (59) we obtain 𝜑𝜉^∗−𝜙^∗_𝐸0≤𝜙(0), (61) and, since (𝜑,𝜙)∈𝔉, we infer that ‖𝜉^∗−𝜙^∗‖_𝐸0≤0. Therefore, ‖𝜉^∗−𝜙^∗‖_𝐸0=0 or, equivalently, 𝜉^∗=𝜙^∗.
This result finishes the proof.

By Theorem 8, we obtain the following corollaries.

Corollary 9. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exists a pair of functions (𝜑,𝜙)∈𝔉 and 𝑐∈[𝑎,𝑏] satisfying 𝜑𝑇𝜉−𝑇𝜂_𝐸≤𝜙𝜉−𝜂_𝐸0 (62) for any 𝜉,𝜂∈𝐸₀. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Corollary 10. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exists a pair of functions (𝜑,𝜙)∈𝔉 and 𝑐∈[𝑎,𝑏] satisfying 𝜑𝑇𝜉−𝑇𝜂_𝐸≤𝜙𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (63) for any 𝜉,𝜂∈𝐸₀. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

The main result of [4] is Theorem 3. Notice that the contractive condition appearing in this theorem 𝑇𝜉−𝑇𝜂_𝐸≤𝛼𝜉−𝜂_𝐸0+𝛽𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (64) for any 𝜉,𝜂∈𝐸₀, where 𝛼,𝛽∈[0,1) with 𝛼+𝛽<1 and 𝑐∈[𝑎,𝑏], implies that 𝑇𝜉−𝑇𝜂_𝐸≤𝛼+𝛽max𝜉−𝜂_𝐸0,𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸≤max𝛼+𝛽𝜉−𝜂_𝐸0,𝛼+𝛽𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (65) for any 𝜉,𝜂∈𝐸₀. This condition is a particular case of the contractive condition appearing in Theorem 8 with the pair of functions (𝜑,𝜙)∈𝔉 given by 𝜑=1_[0,∞) and 𝜙=(𝛼+𝛽)1_[0,∞). Therefore, Theorem 3 is a particular case of the following corollary.

Corollary 11. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exist real numbers 𝛼,𝛽∈[0,1) with 𝛼+𝛽<1 and 𝑐∈[𝑎,𝑏] satisfying 𝑇𝜉−𝑇𝜂_𝐸≤max𝛼+𝛽𝜉−𝜂_𝐸0,𝛼+𝛽𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (66) for any 𝜉,𝜂∈𝐸₀. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Taking into account Example 5, we have the following corollary.

Corollary 12. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exist two functions 𝜑,𝜙∶[0,∞)→[0,∞) and 𝑐∈[𝑎,𝑏] such that 𝜑𝑇𝜉−𝑇𝜂_𝐸≤max𝜑𝜉−𝜂_𝐸0−𝜙𝜉−𝜂_𝐸0,𝜑𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸−𝜙𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (67) for any 𝜉,𝜂∈𝐸₀, where 𝜑 is a continuous and increasing function satisfying 𝜑(𝑡)=0 if and only if 𝑡=0, and 𝜙 is a nondecreasing function such that 𝜙(𝑡)=0 if and only if 𝑡=0, and 𝜙≤𝜑.
If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Corollary 12 has the following consequences.

Corollary 13. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exist two functions 𝜑,𝜙∶[0,∞)→[0,∞) and 𝑐∈[𝑎,𝑏] such that 𝜑𝑇𝜉−𝑇𝜂_𝐸≤𝜑𝜉−𝜂_𝐸0−𝜙𝜉−𝜂_𝐸0 (68) for any 𝜉,𝜂∈𝐸₀, where 𝜑 is an increasing function and 𝜙 is a nondecreasing function and they satisfy 𝜑(𝑡)=𝜙(𝑡)=0 if and only if 𝑡=0, and 𝜑 is continuous with 𝜙≤𝜑. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Corollary 13 can be considered as the version, in the context of PPF dependent fixed point theorems, of the following result about fixed point theorems which appears in [9].

Theorem 14 (see [9]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 a mapping satisfying 𝜑𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦−𝜙𝑑𝑥,𝑦 (69) for 𝑥,𝑦∈𝑋, where 𝜑 and 𝜙 satisfy the same conditions as in Corollary 13. Then 𝑇 has a unique fixed point.

Corollary 15. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exist two functions 𝜑,𝜙∶[0,∞)→[0,∞) satisfying the same conditions as in Corollary 13 and 𝑐∈[𝑎,𝑏] such that 𝜑𝑇𝜉−𝑇𝜂_𝐸≤𝜑𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸−𝜙𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (70) for any 𝜉,𝜂∈𝐸₀. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Taking into account Example 6, we have the following corollary.

Corollary 16. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exist 𝛼∈𝑆 (see Example 6) and 𝑐∈[𝑎,𝑏] satisfying 𝑇𝜉−𝑇𝜂_𝐸≤max𝛼𝜉−𝜂_𝐸0𝜉−𝜂_𝐸0,𝛼𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸×𝜉(𝑐)−𝑇𝜉_𝐸𝜂(𝑐)−𝑇𝜂_𝐸1+𝑇𝜉−𝑇𝜂_𝐸 (71) for any 𝜉,𝜂∈𝐸₀. If 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

A consequence of Corollary 16 is the following result.

Corollary 17. Let 𝑇∶𝐸₀→𝐸 be a mapping such that there exists 𝛼∈𝑆 satisfying 𝑇𝜉−𝑇𝜂_𝐸≤𝛼𝜉−𝜂_𝐸𝜉−𝜂_𝐸0 (72) for any 𝜉,𝜂∈𝐸₀.
If 𝑐∈[𝑎,𝑏] such that 𝑅_𝑐 is topologically closed and algebraically closed with respect to difference, then 𝑇 has a unique PPF dependent fixed point in 𝑅_𝑐.

Corollary 17 is the version, in the context of PPF dependent fixed point theorems, of the following result about fixed point theorems appearing in [10].

Theorem 18. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 a mapping satisfying 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑦𝑑𝑥,𝑦 (73) for any 𝑥,𝑦∈𝑋, where 𝛼∈𝑆. Then 𝑇 has a unique fixed point.

Conflict of Interests

The authors declare that there is no conflict of interests in the submitted paper.