# Wardowski’s Contraction and Fixed Point Technique for Solving Systems of Functional and Integral Equations

Authors: Hasanen A. Hammad; Monica-Felicia Bota; Liliana Guran
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this manuscript, some tripled fixed point results are presented in the framework of complete metric spaces. Furthermore, Wardowski’s contraction was mainly applied to discuss some theoretical results with and without a directed graph under suitable assertions. Moreover, some consequences and supportive examples are derived to strengthen the main results. In the last part of the paper, the obtained theoretical results are used to find a unique solution to a system of functional and integral equations.


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## Body

## 1. Introduction

Mathematics is one of the most important ways to understand things that happen around us. Mathematics has been divided into branches, and with its help, one can analyze other sciences. Integral and differential equations are very important tools that can be used to build patterns in order to understand the models that happen around us. The fixed point theory also plays a crucial role in integral and differential equations.A commonly used tool that has a major role in nonlinear analysis is the fixed point technique, which was given by the well-known scientist Banach. The famous “Banach Contraction Principle” [1] can be announced as follows.Theorem 1.
Assume thatΘ,ϖ is a complete metric space (CMS) and Q is a self-mapping defined on it, such that for all l1,l2∈Θ and ϱ∈0,1, the following inequality holds:
(1)ϖQl1,Ql2≤ϱϖl1,l2.
Then, there exists a unique fixed point (FP) ofQ and the sequence Qnl°1n∈ℕ converges to it, for all l°1∈Θ.There are more generalizations of the inequality (1) either by replacing the contraction condition or by using more general spaces. For more results, see [2–4].We construct the present paper as follows: in Section1, we recall the background of our work; in Section 2, we give essential results, which are useful for understanding the aim of the paper; in Section 3, we discuss the existence of tripled fixed point (TFP) results via π-contraction mappings in CMS with and without a directed graph (DG); in Section 4, we prove the existence of a solution of different types of tripled systems of functional integral equations; and in Section 5, illustrative examples are given to support our study.


## 2. Preliminaries

In 2012, a new type of contraction was given by Wardowski, called Wardowski’s contraction orπ-contraction (see [5]). He generalized the condition in Banach’s theorem and stated the following definition.Definition 2 (see [5]).
Assume thatΘ,ϖ is a metric space and Q is a self-mapping defined on it. We say Q is π-contraction, if there is π∈Ϝ and ℓ∈0,+∞ such that
(2)ϖQl1,Ql2>0impliesℓ+πϖQl1,Ql2≤πϖl1,l2∀l1,l2∈Θ,where z is the family of all functions π:0,+∞⟶R such that the conditions below hold:
πi: for each l1,l2∈ℝ+, if l1<l2, then πl1<πl2; i.e., π is strictly increasing.
πii: limn⟶∞ln1=0 if and only if limn⟶∞πln1=−∞, where ln1n∈N is a sequence of positive numbers.
πiii: liml1⟶0+l1μπl1=0 for each μ∈0,1.
By the inequality (2), the same author introduced some examples of various contractions as follows: for all l1,l2∈Θ with υ>0 and Ql1≠Ql2,
(i)
π1υ=lnυ, ϖQl1,Ql2/ϖl1,l2≤e−ℓ(ii)
π2υ=lnυ+υ, ϖQl1,Ql2eϖQl1,Ql2≤ϖl1,l2eϖl1,l2−ℓ(iii)
π3υ=−1/υ, ϖQl1,Ql21+ℓϖl1,l22≤ϖl1,l2(iv)
π4υ=lnυ2+υ, ϖQl1,Ql21+ϖQl1,Ql2≤e−ℓϖl1,l21+ϖl1,l2where all functions πn:n=1,2,3,4∈Ϝ..Remark 3.
The inequality (2) implies that Q is a contractive mapping, that is,
(3)ϖQl1,Ql2<ϖl1,l2,for all l1,l2∈Θ such that Ql1≠Ql2. Hence, every π-contraction is continuous.Remark 4 (see [6]).
Letπυ=−1/υζ, where ζ>1 and υ>0. Then, π∈Ϝ.Wardowski states his theorem as follows.Theorem 5 (see [6]).
Assume that the mappingQ satisfies the contraction condition (2) on a CMS Θ,ϖ. Then, there is a unique fixed point of Q and Qnl°1n∈ℕ converges to the fixed point for all l°1∈Θ..For two mappings, this theorem has been generalized by Isik [7] as follows.Lemma 6 (see [7]).
Suppose thatΘ,ϖ is a CMS and Q and R are self-mappings defined on it. If there is ℓ>0 and π∈Ϝ such that
(4)ℓ+πϖQl1,Rl2≤πϖl1,l2,for all l1,l2∈Θ and minϖQl1,Rl2,ϖl1,l2>0, then there exists a unique common fixed point of Q and R.A number of papers related toπ-contraction and related fixed point theorems in the setting of various spaces were published. See, for example, [8–10].In the paper [11], the concept of the coupled fixed point (CFP) was presented and studied. In partially ordered metric spaces and abstract spaces, some main results in this direction have been considered. See [12, 13].Definition 7.
Assume thatΘ≠∅ and Q,R:Θ×Θ⟶Θ are given mappings; then, the pair l1,l2∈Θ×Θ is called
(i)
CFP ofQ if Ql1,l2=l1 and Ql2,l1=l2(ii)
a common CFP ofQ and R, if Ql1,l2=Rl1,l2=l1 and Ql2,l1=Rl2,l1=l2Using the generalized notion of CFP, Berinde and Borcut [14] defined the notion of a tripled fixed point (TFP) for self-mappings and established some interesting consequences in partially ordered metric spaces. Many other research results were given in this direction, for different spaces and different types of mappings. For additional results, see [4, 15–17].Definition 8.
Assume thatΘ≠∅ and Q,R:Θ3⟶ΘwhereΘ3=Θ×Θ×Θ are given mappings; then, the pair l1,l2,l3∈Θ3 is called a TFP of Q if Ql1,l2,l3=l1, Ql2,l3,l1=l2, and Ql3,l1,l2=l3..
Here, the symbolΩ refers to the set of all TFPs of the mapping Q, that is,
(5)Ω=l1,l2,l3∈Θ3:Ql1,l2,l3=l1,Ql2,l3,l1=l2,andQl3,l1,l2=l3.In [18], Jachymski used the following notations.Assume thatΘ,ϖ is a MS and Y is the diagonal of the Cartesian product Θ×Θ. Consider ℶ=△ℶ,∇ℶ a directed graph (DG), where △ℶ is the set of vertices that coincides with Θ and ∇ℶ is the set of edges that contains all loops, i.e., ∇ℶ⊇Y.The two definitions below were introduced by Chaobankoh and Charoensawa [19].Definition 9 (see [19]).
A mappingQ:Θ3⟶Θ is called edge-preserving if
(6)l1,u1,l2,u2,l3,u3∈∇ℶ,implies
(7)Ql1,l2,l3,Qu1,u2,u3,Ql2,l3,l1,Qu2,u3,u1,Ql3,l1,l2,Qu3,u1,u2∈∇ℶ.Definition 10 (see [19]).
A mappingQ:Θ3⟶Θ is called ℶ-continuous for each l1,l2,l3∈Θ3 and for any sequence mjj∈ℕ of positive integers with
(8)Qlmj1,lmj2,lmj3⟶l1,Qlmj2,lmj3,lmj1⟶l2,Qlmj3,lmj1,lmj2⟶l3,as j⟶∞, and
(9)Qlmj1,lmj2,lmj3,umj+11,umj+12,umj+13,Qlmj2,lmj3,lmj1,umj+12,umj+13,umj+11,Qlmj3,lmj1,lmj2,umj+13,umj+11,umj+12∈∇ℶ.
Then, forj⟶∞, we have
(10)QQlmj1,lmj2,lmj3,Qlmj2,lmj3,lmj1,Qlmj3,lmj1,lmj2⟶Ql1,l2,l3,QQlmj2,lmj3,lmj1,Qlmj3,lmj1,lmj2,Qlmj1,lmj2,lmj3⟶Ql2,l3,l1,QQlmj3,lmj1,lmj2,Qlmj1,lmj2,lmj3,Qlmj2,lmj3,lmj1⟶Ql3,l1,l2.Definition 11 (see [18]).
LetΘ,ϖ be a CMS and ℶ be a directed graph. A triple Θ,ϖ,ℶ has the property K if for any sequence lmm∈ℕ⊂Θ with limn⟶∞lm=l and lm,lm+1∈∇ℶ, for n∈ℕ, we get lm,l∈∇ℶ.


## 3. Tripled Fixed Point Technique

Let us start this section by giving the following lemma, which is useful in theproof of the main result.Lemma 12.
LetΘ,ϖ be a CMS and Θ3 be a Cartesian product. Define a distance ϖmax by
(11)ϖmaxl1,l2,l3,u1,u2,u3=maxϖl1,u1,ϖl2,u2,ϖl3,u3.
Then,Θ3,ϖmax is also CMS.Proof.
The proof of the lemma is obvious.Furthermore, let us give the first main theorem of this section.Theorem 13.
Assume thatΘ,ϖ is a CMS and Q,R:Θ3⟶Θ are continuous mappings. If there is ℓ>0 and π∈Ϝ such that ϖl1,l2,l3,u1,u2,u3>0 implies
(12)ℓ+πϖQl1,l2,l3,Ru1,u2,u3≤πmaxϖl1,u1,ϖl2,u2,ϖl3,u3,for each l1,l2,l3,u1,u2,u3∈Θ3, then Q and R have a unique common TFP.Proof.
Define the mappingsM∗,H∗:Θ3⟶Θ3 by
(13)M∗l1,l2,l3=Ql1,l2,l3,Ql2,l3,l1,Ql3,l1,l2,H∗l1,l2,l3=Rl1,l2,l3,Rl2,l3,l1,Rl3,l1,l2.
Next, for a CMSΘ3 (see Lemma 12), we shall show that M∗ and H∗ justify the inequality (4). For l1,l2,l3,u1,u2,u3∈Θ3, let
(14)ϖmaxM∗l1,l2,l3,H∗u1,u2,u3=ϖmaxQl1,l2,l3,Ql2,l3,l1,Ql3,l1,l2,Ru1,u2,u3,Ru2,u3,u1,Ru3,u1,u2=maxϖQl1,l2,l3,Ru1,u2,u3,ϖQl2,l3,l1,Ru2,u3,u1,ϖQl3,l1,l2,Ru3,u1,u2>0.
Here, if we put(15)DQR=maxϖQl1,l2,l3,Ru1,u2,u3,ϖQl2,l3,l1,Ru2,u3,u1,ϖQl3,l1,l2,Ru3,u1,u2,then three cases will be discussed for ϖQl1,l2,l3,Ru1,u2,u3>0 as follows:
★i: if DQR=ϖQl1,l2,l3,Ru1,u2,u3, then, by relation (12), we obtain
(16)ℓ+πϖmaxM∗l1,l2,l3,H∗u1,u2,u3=ℓ+πϖQl1,l2,l3,Ru1,u2,u3≤πmaxϖl1,l2,l3,ϖu1,u2,u3=πmaxϖl1,u1,ϖl2,u2,ϖl3,u3.
★ii: if DQR=ϖQl2,l3,l1,Ru2,u3,u1, then, by (12), we have
(17)ℓ+πϖmaxM∗l1,l2,l3,H∗u1,u2,u3=ℓ+πϖQl2,l3,l1,Ru2,u3,u1≤πmaxϖl2,l3,l1,ϖu2,u3,u1=πmaxϖl2,u2,ϖl1,u1≤πmaxϖl1,u1,ϖl2,u2,ϖl3,u3.
★iii: if DQR=ϖQl3,l1,l2,Ru3,u1,u2, it follows from (12) that
(18)ℓ+πϖmaxM∗l1,l2,l3,H∗u1,u2,u3=ℓ+πϖQl3,l1,l2,Ru3,u1,u2≤πmaxϖl3,l1,l2,ϖu3,u1,u2=πmaxϖl3,u3,ϖl2,u2,ϖl1,u1.
The above cases prove that the condition (4) is fulfilled. Then, M∗ and H∗ have a unique common FP l′1,l′2,l′3∈Θ3. This means
(19)l′1,l′2,l′3=M∗l′1,l′2,l′3=Ql′1,l′2,l′3,Ql′2,l′3,l′1,Ql′3,l′1,l′2,l′1,l′2,l′3=H∗l′1,l′2,l′3=Rl′1,l′2,l′3,Rl′2,l′3,l′1,Rl′3,l′1,l′2.
Hence,(20)Ql′1,l′2,l′3=Rl′1,l′2,l′3=l′1,Ql′2,l′3,l′1=Rl′2,l′3,l′1=l′2,Ql′3,l′1,l′2=Rl′3,l′1,l′2=l′3.
Therefore,l′1,l′2,l′3 is a common TFP of Q and R.
The uniqueness follows immediately from the definition ofM∗ and H∗.A pivotal result follows below by lettingQ=R in Theorem 13.Corollary 14.
Assume thatΘ,ϖ is a CMS and Q:Θ3⟶Θ is a continuous mapping. If there is ℓ>0 and π∈Ϝ such that ϖl1,l2,l3,u1,u2,u3>0 implies
(21)ℓ+πϖQl1,l2,l3,Qu1,u2,u3≤πmaxϖl1,u1,ϖl2,u2,ϖl3,u3,for all l1,l2,l3,u1,u2,u3∈Θ3, then Q has a unique TFP.Now, we will discuss the existence and uniqueness of a TFP in a CMS with a directed graph.Following the paper [19], we define the set Θ3Q by
(22)Θ3Q=l1,l2,l3∈Θ3:l1,Ql1,l2,l3,l2,Ql2,l3,l1,l3,Ql3,l1,l2∈∇ℶ.Proposition 15.
LetQ:Θ3⟶Θ be an edge-preserving mapping; then, for all n∈ℕ,
†1: l1,u1,l2,u2,l3,u3∈∇ℶ⇒Qnl1,l2,l3,Qnu1,u2,u3,Qnl2,l3,l1,Qnu2,u3,u1,Qnl3,l1,l2,Qnu3,u1,u2∈∇ℶ.
†2: l1,l2,l3∈Θ3Q⇒Qnl1,l2,l3,Qn+1l1,l2,l3,Qnl2,l3,l1,Qn+1l2,l3,l1,Qnl3,l1,l2,Qn+1l3,l1,l2∈∇ℶ.
†3: l1,l2,l3∈Θ3Q⇒Qnl1,l2,l3,Qnl2,l3,l1,Qnl3,l1,l2∈Θ3Q.Proof.
†1: consider l1,u1,l2,u2,l3,u3∈∇ℶ. Because R is a preserving mapping, we get Ql1,l2,l3,Qu1,u2,u3∈∇ℶ. Using the same property, we can write Q2l1,l2,l3,Q2u1,u2,u3∈∇ℶ. It follows that, by induction, Qnl1,l2,l3,Qnu1,u2,u3∈∇ℶ. In the same manner, we can prove Qnl2,l3,l1,Qnu2,u3,u1∈∇ℶ and Qnl3,l1,l2,Qnu3,u1,u2∈∇ℶ.
†2: assume that
(23)l1,l2,l3∈Θ3Q:l1,Ql1,l2,l3,l2,Ql2,l3,l1,l3,Ql3,l1,l2∈∇ℶ.
By†1, we get
(24)Qnl1,l2,l3,Qn+1l1,l2,l3=Qnl1,l2,l3,QnQl1,l2,l3,Ql2,l3,l1,Ql3,l1,l2∈∇ℶ.
Similarly, one can show thatQnl2,l3,l1,Qn+1l2,l3,l1∈∇ℶ and Qnl3,l1,l2,Qn+1l3,l1,l2∈∇ℶ.†3: from †2, we get
(25)Qnl1,l2,l3,QQnl1,l2,l3,Qnl2,l3,l1,Qnl3,l1,l2=Qnl1,l2,l3,Qn+1l1,l2,l3∈∇ℶ,which is equivalent to Qnl1,l2,l3,Qnl2,l3,l1,Qnl3,l1,l2∈Θ3Q.Definition 16.
We sayQ:Θ3⟶Θ is a πℶ-rational contraction mapping πℶ‐RCM if
♡i: Q is edge-preserving.
♡ii: there is a positive ℓ>0 such that
(26)ℓ+πϖQl1,l2,l3,Qu1,u2,u3≤πϖl1,u1+ϖl2,u2+ϖl3,u33,for all l1,u1,l2,u2,l3,u3∈∇ℶ, with ϖQl1,l2,l3,Qu1,u2,u3>0.Lemma 17.
Assume thatΘ,ϖ is a MS and Q:Θ3⟶Θ is a πℶ‐RCM with a DG ℶ. Then, for each l1,u1,l2,u2,l3,u3∈∇ℶ, we have
(27)πϖQl1,l2,l3,Qu1,u2,u3≤πϖl1,u1+ϖl2,u2+ϖl3,u33−nℓ.Proof.
Letl1,u1,l2,u2,l3,u3∈∇ℶ. Because Q is edge-preserving, we have
(28)Ql1,l2,l3,Qu1,u2,u3∈∇ℶ.
It follows from Proposition15†1 that Qnl1,l2,l3,Qnu1,u2,u3∈∇ℶ. Because Q is a πℶ‐RCM, one can obtain
(29)πϖQ2l1,l2,l3,Q2u1,u2,u3=πϖQQl1,l2,l3,Ql2,l3,l1,Ql3,l1,l2,QQu1,u2,u3,Qu2,u3,u1,Qu3,u1,u2≤πϖQl1,l2,l3,Qu1,u2,u3+ϖQl2,l3,l1,Qu2,u3,u1+ϖQl3,l1,l2,Qu3,u1,u23−ℓ≤πϖl1,u1+ϖl2,u2+ϖl3,u33−2ℓ.
Moreover, we have the same result ifQnl2,l3,l1,Qnu2,u3,u1∈∇ℶ or Qnl3,l1,l2,Qnu3,u1,u2∈∇ℶ. Therefore, the conclusion follows using mathematical induction.Lemma 18.
LetQ:Θ3⟶Θ be a πℶ‐RCM on a CMS Θ,ϖ with a DG ℶ. Then, for each l1,l2,l3∈Θ3Q, there is l′1,l′2,l′3∈Θ3 such that Qnl1,l2,l3n∈ℕ⟶l′1, Qnl2,l3,l1n∈ℕ⟶l′2, and Qnl3,l1,l2n∈ℕ⟶l′3, as .Proof.
Suppose thatl1,l2,l3∈Θ3Q; then,
(30)l1,Ql1,l2,l3,l2,Ql2,l3,l1,l3,Ql3,l1,l2∈∇ℶ.
Setu1=Ql1,l2,l3, u2=Ql2,l3,l1, and u3=Ql3,l1,l3 in the contractive condition of Lemma 17 and put
(31)Il1,l1,l2=ϖl1,Ql1,l2,l3+ϖl2,Ql2,l3,l1+ϖl3,Ql3,l1,l23.
Then, we have(32)πϖQnl1,l2,l3,QnQl1,l2,l3,Ql2,l3,l1,Ql3,l1,l2≤πIl1,l1,l2−nℓ,,or equivalently,
(33)πϖQnl1,l2,l3,Qn+1l1,l2,l3≤πIl1,l1,l2−nℓ.
Asn⟶∞ in (33), we can write
(34)limn⟶∞πϖQnl1,l2,l3,Qn+1l1,l2,l3=−∞.
Applying conditionπii, we have
(35)limn⟶∞ϖQnl1,l2,l3,Qn+1l1,l2,l3=0.
Using the same steps, we can write(36)limn⟶∞ϖQnl2,l3,l1,Qn+1l2,l3,l1=0,limn⟶∞ϖQnl3,l1,l2,Qn+1l3,l1,l2=0.
From conditionπiii to (35), there exists μ∈0,1 such that
(37)limn⟶∞ϖQnl1,l2,l3,Qn+1l1,l2,l3μϖQnl1,l2,l3,Qn+1l1,l2,l3=0.
For alln∈ℕ, the inequality (33) yields
(38)ϖQnl1,l2,l3,Qn+1l1,l2,l3μ×πϖQnl1,l2,l3,Qn+1l1,l2,l3−πIl1,l1,l2≤−nϖQnl1,l2,l3,Qn+1l1,l2,l3μℓ≤0.
Take into account (35) and (37), and taking n⟶∞ in (38), we get
(39)limn⟶∞ϖQnl1,l2,l3,Qn+1l1,l2,l3μ=0.
By (39), there is n°∈ℕ, such that nϖQnl1,l2,l3,Qn+1l1,l2,l3μ≤1, for all n≥n°, or
(40)ϖQnl1,l2,l3,Qn+1l1,l2,l3≤1n1/μ,foralln≥n°.
Using (40), for m>n≥n°, we get
(41)ϖQnl1,l2,l3,Qml1,l2,l3≤ϖQnl1,l2,l3,Qn+1l1,l2,l3+⋯+ϖQm−1l1,l2,l3,Qml1,l2,l3≤∑n≥n°∞1n1/μ.
The convergence series∑n≥n°∞1/n1/μ leads to limn,m⟶∞ϖQnl1,l2,l3,Qml1,l2,l3=0. Moreover, we can write
(42)limn,m⟶∞ϖQnl2,l3,l1,Qml2,l3,l1=0,limn,m⟶∞ϖQnl3,l1,l2,Qml3,l1,l2=0.
This implies thatQnl1,l2,l3n∈ℕ, Qnl2,l3,l1n∈ℕ, and Qnl3,l1,l2n∈ℕ are Cauchy sequences in Θ. The completeness of Θ,ϖ tells us that there is l′1,l′2,l′3∈Θ3 such that Qnl1,l2,l3n∈ℕ⟶l′1, Qnl2,l3,l1n∈ℕ⟶l′2, and Qnl3,l1,l2n∈ℕ⟶l′3, as n⟶∞. Then, the conclusion follows.Theorem 19.
Assume thatQ:Θ3⟶Θ is a πℶ‐RCM on a CMS Θ,ϖ with a DG ℶ. Let
(a)
Q be ℶ-continuous(b)
the tripleΘ,ϖ,ℶ satisfy the property K and π be continuous
Then,Ω≠∅ if and only if Θ3Q≠∅.Proof.
LetΩ≠∅; then, there is l′1,l′2,l′3∈Ω so that l′1,Ql′1,l′2,l′3=l′1,l′1∈Y⊂∇ℶ, l′2,Ql′2,l′3,l′1=l′2,l′2∈Y⊂∇ℶ, and l′3,Ql′3,l′1,l′2=l′3,l′3∈Y⊂∇ℶ. So, l′1,Ql′1,l′2,l′3,l′2,Ql′2,l′3,l′1,l′3,Ql′3,l′1,l′2∈∇ℶ; this yields Θ3Q≠∅.
Conversely, suppose thatΘ3Q≠∅; this means that there is l1,l2,l3∈Θ3Q such that
(43)l1,Ql1,l2,l3,l2,Ql2,l3,l1,l3,Ql3,l1,l2∈∇ℶ.
Considering a positive integer sequencenii∈ℕ, by Proposition 15†2, we obtain
(44)Qnil1,l2,l3,Qni+1l1,l2,l3∈∇ℶ.
Applying Lemma18 to (44), there are l′1,l′2,l′3∈Θ such that
(45)limi⟶∞Qnil1,l2,l3=l′1,limi⟶∞Qnil2,l3,l1=l′2,limi⟶∞Qnil3,l1,l2=l′3.(a)
LetQ be ℶ-continuous; then, we get(46)QQnil1,l2,l3,Qnil2,l3,l1,Qnil3,l1,l2⟶Ql′1,l′2,l′3,asi⟶∞.
From triangle inequality, it follows(47)ϖQl′1,l′2,l′3,l′1≤ϖQl′1,l′2,l′3,Qni+1l1,l2,l3+ϖQni+1l1,l2,l3,l′1.
The continuity ofQ and (45) leads to ϖQl′1,l′2,l′3,l′1=0, i.e., Ql′1,l′2,l′3=l′1. Similarly, one can show that Ql′2,l′3,l′1=l′2 and Ql′3,l′1,l′2=l′3. Hence, a triple l′1,l′2,l′3 is a TFP of Q and Ω≠0.
(b)
If a tripleΘ,ϖ,ℶ satisfies the property K, then we get(48)ϖQnl1,l2,l3,l′1∈∇ℶ.
Again, by the triangle inequality, we have(49)ϖQl′1,l′2,l′3,l′1≤ϖQl′1,l′2,l′3,Qn+1l1,l2,l3+ϖQn+1l1,l2,l3,l′1≤ϖQl′1,l′2,l′3,QQnl1,l2,l3,Qnl2,l3,l1,Qnl3,l1,l2+ϖQn+1l1,l2,l3,l′1.
Using mappingπ yields
(50)πϖQl′1,l′2,l′3,l′1−ϖQn+1l1,l2,l3,l′1≤πϖQl′1,l′2,l′3,QQnl1,l2,l3,Qnl2,l3,l1,Qnl3,l1,l2≤πϖt′1,Qnl1,l2,l3+ϖl′2,Qnl2,l3,l1+ϖl′3,Qnl3,l1,l23−τ.
Asn⟶∞ in (50), we obtain that πϖQl′1,l′2,l′3,l′1≤−∞, that is, ϖQl′1,l′2,l′3,l′1=0, i.e., Ql′1,l′2,l′3=l′1. Similarly, one can prove that Ql′2,l′3,l′1=l′2 and Ql′3,l′1,l′2=l′3. So l′1,l′2,l′3∈Ω.


## 4. Applications

The fixed point theory is a very important tool in nonlinear analysis, due to its applications in various domains (see [20, 21]).Before stating the main results of this section, we need the following lemma.Lemma 20 (see [22]).
Assume thatφℓζ:0,∞⟶0,∞ is a function defined by
(51)φℓζe=e1+ℓeζζ,for ζ>1 and ℓ>0. Then,
(i)
φℓζe is strictly increasing(ii)
φℓζ0=0 and φℓζe is a concave function(iii)
fore,r∈0,∞, φℓζr−φℓζe≤φℓζr−e


### 4.1. System of Tripled Functional Equations

The fixed point technique contributes to the study of dynamic programming, which is considered an essential tool in optimization problems such as the study of dynamic economic models. This technique has been studied by many researchers to give a unique solution to a system of functional equations via suitable contraction conditions in various spaces. For more results, we refer to Bhakta and Mitra [23], Liu [24], Pathak et al. [25], Zhang [26], and Bellman and Lee [27].Consider a system of tripled functional equations below:(52)zl1=supl2∈Dcl1,l2+Jl1,l2,zol1,l2,bol1,l2,aol1,l2,bl1=supl2∈Dcl1,l2+Jl1,l2,bol1,l2,aol1,l2,zol1,l2,al1=supl2∈Dcl1,l2+Jl1,l2,aol1,l2,zol1,l2,bol1,l2,where S and D are state and decision spaces, respectively, l1∈S, o:S×D⟶S, c:S×D⟶ℝ, and J:S×D×ℝ3⟶ℝ.We denote the set of all bounded real-valued functions on a nonempty setS, by AS. Define
(53)ν=supl1∈Sνl1,for any ν∈AS. Moreover, on AS, define a distance as follows:
(54)ϖr,u=supl1∈Srl1−ul1,for all r,u∈AS. Clearly, the pair AS,ϖ is a CMS.Problem (52) will be considered via the two hypotheses below:‡i: the functions c:S×D⟶ℝ and J:S×D×ℝ3⟶ℝ are bounded.‡ii: for all l1∈S, l2∈D, and z,b,e,z∗,b∗,e∗∈ℝ, for ζ>1 and ℓ>0, we have
(55)Jl1,l2,z,b,a−Jl1,l2,z∗,b∗,a∗≤maxz−z∗,b−b∗,a−a∗1+ℓmaxz−z∗,b−b∗,a−a∗ζζ.Theorem 21.
Using the hypotheses‡i and ‡ii on AS×AS, the problem (52) has a unique bounded common solution.Proof.
On the spaceAS, let us define an operator Q as follows:
(56)Qz,b,al1=supl2∈Dcl1,l2+Jl1,l2,zol1,l2,bol1,l2,aol1,l2,for each z,b,a∈AS and l1∈S. The boundedness of the functions c and J assures that the mapping Q is well defined.
Suppose thatz,b,a,z∗,b∗,a∗∈AS×AS, and take
(57)χz∗,b∗,a∗z,b,a=maxzol1,l2−z∗ol1,l2,bol1,l2−b∗ol1,l2,aol1,l2−a∗ol1,l2,ϑz∗,b∗,a∗z,b,a=maxzol1,l2−z∗ol1,l2,bol1,l2−b∗ol1,l2,aol1,l2−a∗ol1,l2.
Then, by hypothesis‡ii, we have
(58)ϖQz,b,a,Qz∗,b∗,a∗=supl1∈SQz,b,al1−Qz∗,b∗,a∗l1=supl1∈Ssupl2∈Dcl1,l2+Jl1,l2,zol1,l2,bol1,l2,aol1,l2−supl2∈Dcl1,l2+Jl1,l2,z∗ol1,l2,b∗ol1,l2,a∗ol1,l2=supl1∈Ssupl2∈DJl1,l2,zol1,l2,bol1,l2,aol1,l2−Jl1,l2,z∗ol1,l2,b∗ol1,l2,a∗ol1,l2≤supl1∈Ssupl2∈Dχz∗,b∗,a∗z,b,a1+ℓχz∗,b∗,a∗z,b,aζζ≤supl1∈Sϑz∗,b∗,a∗z,b,a1+ℓϑz∗,b∗,a∗z,b,aζζ≤maxϖz,z∗,ϖb,b∗,ϖa,a∗1+ℓmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζζ,where the nondecreasing character of φℓζ was used (Lemma 20). Then,
(59)ϖQz,b,e,Qz∗,b∗,e∗≤maxϖz,z∗,ϖb,b∗,ϖe,e∗1+ℓmaxϖz,z∗,ϖb,b∗,ϖe,e∗ζζ.
Takingζ on both sides, we have
(60)ϖQz,b,e,Qz∗,b∗,a∗ζ≤maxϖz,z∗,ϖb,b∗,ϖa,a∗ζ1+ℓmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζ,or equivalently,
(61)1+ℓmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζ≤1ϖQz,b,a,Qz∗,b∗,a∗ζ,yields
(62)1maxϖz,z∗,ϖb,b∗,ϖa,a∗ζ+ℓ≤1ϖQz,b,a,Qz∗,b∗,a∗ζ, and this leads to
(63)ℓ−1ϖQz,b,a,Qz∗,b∗,a∗ζ≤−1maxϖz,z∗,ϖb,b∗,ϖa,a∗ζ.
This confirms that the inequality (21) of Corollary 14 holds with πυ=−1/υζ∈Ϝ (Remark 4). Then, it follows that the operator Q has a unique TFP. At the same time, it is a unique bounded solution of the problem (52) on AS×AS.


### 4.2. Tripled System of the First Type of Integral Equations

In this subsection, the theoretical results of Corollary14 will be applied to discuss the existence and uniqueness of a solution of an integral equation tripled system. Let us consider the following system:
(64)l1e=ke+∫01Ye,r,l1r,l2r,l3rdr,l2e=ke+∫01Ye,r,l2r,l3r,l1rdr,l3e=ke+∫01Ye,r,l3r,l1r,l2rdr,where ke is defined for all e∈0,1.ConsiderC0,1, the set of all real continuous functions defined on 0,1, and together with the distance defined above, we can notice that C0,1,ϖ is a CMS.Now, we discuss the problem (64) according to the assumptions below:♠1: k:0,1⟶ℝ is a continuous function.♠2: Y:0,1×0,1×ℝ3⟶ℝ is a continuous function verifying
(65)Ye,r,l1r,l2r,l3r−Ye,r,l′1r,l′2r,l′3r≤maxl1−l′1,l2−l′2,l3−l′31+ℓmaxl1−l′1,l2−l′2,l3−l′3ζζ,for each e,r∈0,1 and l1,l2,l3,l′1,l′2,l′3∈ℝ, and ℓ>0 and ζ>1.Furthermore, let us present the main theorem of this subsection.Theorem 22.
There is a unique solution of system (64) l′1,l′2,l′3∈C0,13, as long as the conditions ♠1 and ♠2 are satisfied.Proof.
Define a mappingQ on C0,1 as follows:
(66)Ql1,l2,l3e=ke+∫01Ye,r,l1r,l2r,l3rdr,for all l1,l2,l3∈C0,1. In virtue of ♠1 and ♠2, we conclude that Ql1,l2,l3∈C0,1 for each l1,l2,l3∈C0,1. Thus, we can write
(67)Q:C0,13⟶C0,1.
LetϖQl1,l2,l3,Ql′1,l′2,l′3>0; then, for e∈0,1, we get
(68)Ql1,l2,l3e−Ql′1,l′2,l′3t=∫01Ye,r,l1r,l2r,l3rdr−∫01Ye,r,l′1r,l′2r,l′3rd≤∫01Ye,r,l1r,l2r,l3r−Ye,r,l′1r,l′2r,l′3rdr≤∫01maxl1r−l′1r,l2r−l′2r,l3r−l′3r1+ℓmaxl1r−l′1r,l2r−l′2r,l3r−l′3rζζdr≤∫01maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3r1+ℓmaxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζζdr=maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3r1+ℓmaxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζζ,
where the nondecreasing characters ofφℓζ were used (Lemma 20). Thus,
(69)ϖQl1,l2,l3,Ql′1,l′2,l′3≤maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3r1+ℓmaxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζζ.
By the same approach used at the inequalities (60)–(62), we get
(70)ℓ−1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ≤−1maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζ.
Hence, the hypotheses of Corollary14 are fulfilled on πυ=−1/υζ∈Ϝ (Remark 4). There is a unique TFP of the mapping Q. In other words, there is l′1,l′2,l′3∈C0,13 such that
(71)l′1e=Γl°′1,l°′2,l°′3e=ke+∫01Ye,r,l°′1r,l°′2r,l°′3rdr,l′2e=Γl°′2,l°′3,l°′1e=ke+∫01Ye,r,l°′2r,l°′3r,l°′1rdr,l′3e=Γl°′3,l°′1,l°′2e=ke+∫01Ye,r,l°′3r,l°′1r,l°′2rdr.


### 4.3. Tripled System of the Second Type of Integral Equations

Let us consider the following type of system of integral equations:(72)l1e=∫0MWe,rXe,r,l1r,l2r,l3rdr,l2e=∫0MWe,rXe,r,l2r,l3r,l1rdr,l3e=∫0MWe,rXe,r,l3r,l1r,l2rdr,where e,r∈0,M with M>0.This subsection is devoted to discussing the influence of the theoretical results of a DG for solving this new type of system of integral equations.LetΠ=C0,M,ℝn endowed with l1=max0≤e≤Ml1e for all l1∈Π. Moreover, define a partial order on a graph ℶ as follows, for all l1,l2,l3,l′1,l′2,l′3∈Π and e∈0,M,
(73)l1≤l′1⇔l1e≤l′1e,l2≤l′2⇔l2e≤l′2e,l3≤l′3⇔l3e≤l′3e.Thus,Π,. is a CMS equipped with a directed graph ℶ.LetΠ,.,ℶ be a triple with the property K and
(74)Π3=Π×Π×ΠQ=l1,l2,l3∈Π3:l1≤Ql1,l2,l3,l2≤Ql2,l3,l1,andl3≤Ql3,l1,l2.We can state the main theorem.Theorem 23.
There is at least one solution of the problem (72), if the assumptions below are fulfilled:
▶1: the functions X:0,M×0,M×ℝn×ℝn×ℝn⟶ℝn and W:0,M×0,M⟶ℝn are continuous such that
(75)∫0MWe,rdr≤Mℓ,for all e,r∈0,M and ℓ>0.
▶2: for all l1,l2,l3,l′1,l′2,l′3∈ℝn with l1≤l′1, l2≤l′2, and l3≤l′3, we have
(76)Xe,r,l1r,l2r,l3r≤Xe,r,l′1r,l′2r,l′3r,for all e,r∈0,M.
▶3: there are ℓ>0 and ζ>1 so that
(77)Xe,r,l1r,l2r,l3r−Xe,r,l′1r,l′2r,l′3r≤ℓM1/3maxl1−l′1,l2−l′2,l3−l′31+ℓ1/3maxl1−l′1,l2−l′2,l3−l′3ζζ,for any e,r∈0,M.
▶4: there is l°1,l°2,l°3∈Π3 such that
(78)l°1e=∫0MWe,rXe,r,l°1r,l°2r,l°3rdr,l°2e=∫0MWe,rXe,r,l°2r,l°3r,l°1rdr,l°3e=∫0MWe,rXe,r,l°3r,l°1r,l°2rdr,where e∈0,M.Proof.
Let the mappingQ:Π3⟶Π defined by
(79)Ql1,l2,l3e=∫0MWe,rXe,r,l1r,l2r,l3rdr,e∈0,M.
Next, we show thatQ is ℶ-edge-preserving. Let l1,l2,l3,l′1,l′2,l′3∈Π with l1≤l′1, l2≤l′2, and l3≤l′3. Then, we get
(80)Ql1,l2,l3e=∫0MWe,rXe,r,l1r,l2r,l3rdr≤∫0MWe,rXe,r,l′1r,l′2r,l′3rdr=Ql′1,l′2,t′3e.
Using the same steps, we can writeQl2,l3,l1e≤Ql′2,l′3,l′1,e and Ql3,l1,l2e≤Ql′3,l′1,l′2e, for all e∈0,M.
Next, from▶4, it follows
(81)Π3Q=l1,l2,l3∈Π3:l1≤Ql1,l2,l3,l2≤Ql2,l3,l1,andl3≤Ql3,l1,l2≠∅.
Ultimately,(82)Ql1,l2,l3e−Ql′1,l′2,l′3e≤∫0MWe,rXe,r,l1r,l2r,l3r−Xe,r,l′1r,l′2r,l′3rdr≤∫0MWe,rℓM1/3maxl1−l′1,l2−l′2,l3−l′31+ℓ1/3maxl1−l′1,l2−l′2,l3−l′3ζζdr≤1/3maxl1−l′1,l2−l′2,l3−l′31+ℓ1/3maxl1−l′1,l2−l′2,l3−l′3ζζ≤1/3l1−l′1+l2−l′2+l3−l′31+ℓ1/3l1−l′1+l2−l′2+l3−l′3∣ζζ=ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/31+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζζ,where the nondecreasing characters of φℓζ were used (Lemma 20). Thus,
(83)ϖQl1,l2,l3,Ql′1,l′2,l′3≤ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/31+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζζ.
Takingζ on both sides, we get
(84)ϖQl1,l2,l3,Ql′1,l′2,l′3ζ≤ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ1+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ,or equivalently,
(85)1+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ≤1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ,or
(86)1ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ+ℓ≤1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ.
This leads to(87)ℓ−1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ≤−1ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ.
Hence,Q is πℶ‐RCM with πυ=−1/υζ∈Ϝ (Remark 4). So, it follows from Theorem 19 that the mapping Q has a TFP, which is a solution of the problem (72).


## 4.1. System of Tripled Functional Equations

The fixed point technique contributes to the study of dynamic programming, which is considered an essential tool in optimization problems such as the study of dynamic economic models. This technique has been studied by many researchers to give a unique solution to a system of functional equations via suitable contraction conditions in various spaces. For more results, we refer to Bhakta and Mitra [23], Liu [24], Pathak et al. [25], Zhang [26], and Bellman and Lee [27].Consider a system of tripled functional equations below:(52)zl1=supl2∈Dcl1,l2+Jl1,l2,zol1,l2,bol1,l2,aol1,l2,bl1=supl2∈Dcl1,l2+Jl1,l2,bol1,l2,aol1,l2,zol1,l2,al1=supl2∈Dcl1,l2+Jl1,l2,aol1,l2,zol1,l2,bol1,l2,where S and D are state and decision spaces, respectively, l1∈S, o:S×D⟶S, c:S×D⟶ℝ, and J:S×D×ℝ3⟶ℝ.We denote the set of all bounded real-valued functions on a nonempty setS, by AS. Define
(53)ν=supl1∈Sνl1,for any ν∈AS. Moreover, on AS, define a distance as follows:
(54)ϖr,u=supl1∈Srl1−ul1,for all r,u∈AS. Clearly, the pair AS,ϖ is a CMS.Problem (52) will be considered via the two hypotheses below:‡i: the functions c:S×D⟶ℝ and J:S×D×ℝ3⟶ℝ are bounded.‡ii: for all l1∈S, l2∈D, and z,b,e,z∗,b∗,e∗∈ℝ, for ζ>1 and ℓ>0, we have
(55)Jl1,l2,z,b,a−Jl1,l2,z∗,b∗,a∗≤maxz−z∗,b−b∗,a−a∗1+ℓmaxz−z∗,b−b∗,a−a∗ζζ.Theorem 21.
Using the hypotheses‡i and ‡ii on AS×AS, the problem (52) has a unique bounded common solution.Proof.
On the spaceAS, let us define an operator Q as follows:
(56)Qz,b,al1=supl2∈Dcl1,l2+Jl1,l2,zol1,l2,bol1,l2,aol1,l2,for each z,b,a∈AS and l1∈S. The boundedness of the functions c and J assures that the mapping Q is well defined.
Suppose thatz,b,a,z∗,b∗,a∗∈AS×AS, and take
(57)χz∗,b∗,a∗z,b,a=maxzol1,l2−z∗ol1,l2,bol1,l2−b∗ol1,l2,aol1,l2−a∗ol1,l2,ϑz∗,b∗,a∗z,b,a=maxzol1,l2−z∗ol1,l2,bol1,l2−b∗ol1,l2,aol1,l2−a∗ol1,l2.
Then, by hypothesis‡ii, we have
(58)ϖQz,b,a,Qz∗,b∗,a∗=supl1∈SQz,b,al1−Qz∗,b∗,a∗l1=supl1∈Ssupl2∈Dcl1,l2+Jl1,l2,zol1,l2,bol1,l2,aol1,l2−supl2∈Dcl1,l2+Jl1,l2,z∗ol1,l2,b∗ol1,l2,a∗ol1,l2=supl1∈Ssupl2∈DJl1,l2,zol1,l2,bol1,l2,aol1,l2−Jl1,l2,z∗ol1,l2,b∗ol1,l2,a∗ol1,l2≤supl1∈Ssupl2∈Dχz∗,b∗,a∗z,b,a1+ℓχz∗,b∗,a∗z,b,aζζ≤supl1∈Sϑz∗,b∗,a∗z,b,a1+ℓϑz∗,b∗,a∗z,b,aζζ≤maxϖz,z∗,ϖb,b∗,ϖa,a∗1+ℓmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζζ,where the nondecreasing character of φℓζ was used (Lemma 20). Then,
(59)ϖQz,b,e,Qz∗,b∗,e∗≤maxϖz,z∗,ϖb,b∗,ϖe,e∗1+ℓmaxϖz,z∗,ϖb,b∗,ϖe,e∗ζζ.
Takingζ on both sides, we have
(60)ϖQz,b,e,Qz∗,b∗,a∗ζ≤maxϖz,z∗,ϖb,b∗,ϖa,a∗ζ1+ℓmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζ,or equivalently,
(61)1+ℓmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζmaxϖz,z∗,ϖb,b∗,ϖa,a∗ζ≤1ϖQz,b,a,Qz∗,b∗,a∗ζ,yields
(62)1maxϖz,z∗,ϖb,b∗,ϖa,a∗ζ+ℓ≤1ϖQz,b,a,Qz∗,b∗,a∗ζ, and this leads to
(63)ℓ−1ϖQz,b,a,Qz∗,b∗,a∗ζ≤−1maxϖz,z∗,ϖb,b∗,ϖa,a∗ζ.
This confirms that the inequality (21) of Corollary 14 holds with πυ=−1/υζ∈Ϝ (Remark 4). Then, it follows that the operator Q has a unique TFP. At the same time, it is a unique bounded solution of the problem (52) on AS×AS.


## 4.2. Tripled System of the First Type of Integral Equations

In this subsection, the theoretical results of Corollary14 will be applied to discuss the existence and uniqueness of a solution of an integral equation tripled system. Let us consider the following system:
(64)l1e=ke+∫01Ye,r,l1r,l2r,l3rdr,l2e=ke+∫01Ye,r,l2r,l3r,l1rdr,l3e=ke+∫01Ye,r,l3r,l1r,l2rdr,where ke is defined for all e∈0,1.ConsiderC0,1, the set of all real continuous functions defined on 0,1, and together with the distance defined above, we can notice that C0,1,ϖ is a CMS.Now, we discuss the problem (64) according to the assumptions below:♠1: k:0,1⟶ℝ is a continuous function.♠2: Y:0,1×0,1×ℝ3⟶ℝ is a continuous function verifying
(65)Ye,r,l1r,l2r,l3r−Ye,r,l′1r,l′2r,l′3r≤maxl1−l′1,l2−l′2,l3−l′31+ℓmaxl1−l′1,l2−l′2,l3−l′3ζζ,for each e,r∈0,1 and l1,l2,l3,l′1,l′2,l′3∈ℝ, and ℓ>0 and ζ>1.Furthermore, let us present the main theorem of this subsection.Theorem 22.
There is a unique solution of system (64) l′1,l′2,l′3∈C0,13, as long as the conditions ♠1 and ♠2 are satisfied.Proof.
Define a mappingQ on C0,1 as follows:
(66)Ql1,l2,l3e=ke+∫01Ye,r,l1r,l2r,l3rdr,for all l1,l2,l3∈C0,1. In virtue of ♠1 and ♠2, we conclude that Ql1,l2,l3∈C0,1 for each l1,l2,l3∈C0,1. Thus, we can write
(67)Q:C0,13⟶C0,1.
LetϖQl1,l2,l3,Ql′1,l′2,l′3>0; then, for e∈0,1, we get
(68)Ql1,l2,l3e−Ql′1,l′2,l′3t=∫01Ye,r,l1r,l2r,l3rdr−∫01Ye,r,l′1r,l′2r,l′3rd≤∫01Ye,r,l1r,l2r,l3r−Ye,r,l′1r,l′2r,l′3rdr≤∫01maxl1r−l′1r,l2r−l′2r,l3r−l′3r1+ℓmaxl1r−l′1r,l2r−l′2r,l3r−l′3rζζdr≤∫01maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3r1+ℓmaxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζζdr=maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3r1+ℓmaxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζζ,
where the nondecreasing characters ofφℓζ were used (Lemma 20). Thus,
(69)ϖQl1,l2,l3,Ql′1,l′2,l′3≤maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3r1+ℓmaxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζζ.
By the same approach used at the inequalities (60)–(62), we get
(70)ℓ−1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ≤−1maxϖl1r,l′1r,ϖl2r,l′2r,ϖl3r,l′3rζ.
Hence, the hypotheses of Corollary14 are fulfilled on πυ=−1/υζ∈Ϝ (Remark 4). There is a unique TFP of the mapping Q. In other words, there is l′1,l′2,l′3∈C0,13 such that
(71)l′1e=Γl°′1,l°′2,l°′3e=ke+∫01Ye,r,l°′1r,l°′2r,l°′3rdr,l′2e=Γl°′2,l°′3,l°′1e=ke+∫01Ye,r,l°′2r,l°′3r,l°′1rdr,l′3e=Γl°′3,l°′1,l°′2e=ke+∫01Ye,r,l°′3r,l°′1r,l°′2rdr.


## 4.3. Tripled System of the Second Type of Integral Equations

Let us consider the following type of system of integral equations:(72)l1e=∫0MWe,rXe,r,l1r,l2r,l3rdr,l2e=∫0MWe,rXe,r,l2r,l3r,l1rdr,l3e=∫0MWe,rXe,r,l3r,l1r,l2rdr,where e,r∈0,M with M>0.This subsection is devoted to discussing the influence of the theoretical results of a DG for solving this new type of system of integral equations.LetΠ=C0,M,ℝn endowed with l1=max0≤e≤Ml1e for all l1∈Π. Moreover, define a partial order on a graph ℶ as follows, for all l1,l2,l3,l′1,l′2,l′3∈Π and e∈0,M,
(73)l1≤l′1⇔l1e≤l′1e,l2≤l′2⇔l2e≤l′2e,l3≤l′3⇔l3e≤l′3e.Thus,Π,. is a CMS equipped with a directed graph ℶ.LetΠ,.,ℶ be a triple with the property K and
(74)Π3=Π×Π×ΠQ=l1,l2,l3∈Π3:l1≤Ql1,l2,l3,l2≤Ql2,l3,l1,andl3≤Ql3,l1,l2.We can state the main theorem.Theorem 23.
There is at least one solution of the problem (72), if the assumptions below are fulfilled:
▶1: the functions X:0,M×0,M×ℝn×ℝn×ℝn⟶ℝn and W:0,M×0,M⟶ℝn are continuous such that
(75)∫0MWe,rdr≤Mℓ,for all e,r∈0,M and ℓ>0.
▶2: for all l1,l2,l3,l′1,l′2,l′3∈ℝn with l1≤l′1, l2≤l′2, and l3≤l′3, we have
(76)Xe,r,l1r,l2r,l3r≤Xe,r,l′1r,l′2r,l′3r,for all e,r∈0,M.
▶3: there are ℓ>0 and ζ>1 so that
(77)Xe,r,l1r,l2r,l3r−Xe,r,l′1r,l′2r,l′3r≤ℓM1/3maxl1−l′1,l2−l′2,l3−l′31+ℓ1/3maxl1−l′1,l2−l′2,l3−l′3ζζ,for any e,r∈0,M.
▶4: there is l°1,l°2,l°3∈Π3 such that
(78)l°1e=∫0MWe,rXe,r,l°1r,l°2r,l°3rdr,l°2e=∫0MWe,rXe,r,l°2r,l°3r,l°1rdr,l°3e=∫0MWe,rXe,r,l°3r,l°1r,l°2rdr,where e∈0,M.Proof.
Let the mappingQ:Π3⟶Π defined by
(79)Ql1,l2,l3e=∫0MWe,rXe,r,l1r,l2r,l3rdr,e∈0,M.
Next, we show thatQ is ℶ-edge-preserving. Let l1,l2,l3,l′1,l′2,l′3∈Π with l1≤l′1, l2≤l′2, and l3≤l′3. Then, we get
(80)Ql1,l2,l3e=∫0MWe,rXe,r,l1r,l2r,l3rdr≤∫0MWe,rXe,r,l′1r,l′2r,l′3rdr=Ql′1,l′2,t′3e.
Using the same steps, we can writeQl2,l3,l1e≤Ql′2,l′3,l′1,e and Ql3,l1,l2e≤Ql′3,l′1,l′2e, for all e∈0,M.
Next, from▶4, it follows
(81)Π3Q=l1,l2,l3∈Π3:l1≤Ql1,l2,l3,l2≤Ql2,l3,l1,andl3≤Ql3,l1,l2≠∅.
Ultimately,(82)Ql1,l2,l3e−Ql′1,l′2,l′3e≤∫0MWe,rXe,r,l1r,l2r,l3r−Xe,r,l′1r,l′2r,l′3rdr≤∫0MWe,rℓM1/3maxl1−l′1,l2−l′2,l3−l′31+ℓ1/3maxl1−l′1,l2−l′2,l3−l′3ζζdr≤1/3maxl1−l′1,l2−l′2,l3−l′31+ℓ1/3maxl1−l′1,l2−l′2,l3−l′3ζζ≤1/3l1−l′1+l2−l′2+l3−l′31+ℓ1/3l1−l′1+l2−l′2+l3−l′3∣ζζ=ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/31+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζζ,where the nondecreasing characters of φℓζ were used (Lemma 20). Thus,
(83)ϖQl1,l2,l3,Ql′1,l′2,l′3≤ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/31+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζζ.
Takingζ on both sides, we get
(84)ϖQl1,l2,l3,Ql′1,l′2,l′3ζ≤ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ1+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ,or equivalently,
(85)1+ℓϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ≤1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ,or
(86)1ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ+ℓ≤1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ.
This leads to(87)ℓ−1ϖQl1,l2,l3,Ql′1,l′2,l′3ζ≤−1ϖl1,l′1+ϖl2,l′2+ϖl3,l′3/3ζ.
Hence,Q is πℶ‐RCM with πυ=−1/υζ∈Ϝ (Remark 4). So, it follows from Theorem 19 that the mapping Q has a TFP, which is a solution of the problem (72).


## 5. Examples

In this section, some important examples satisfying theoretical consequences are presented, with the role to strengthen our results.Example 1.
Assume thatΘ=0,∞ and ϖl1,l2=l1−l2. Clearly, Θ,ϖ is a CMS. Define Q,R:Θ3⟶Θ by
(88)Ql1,l2,l3=l1−4l2+l35,l1+l3≥4l2,Rl1,l2,l3=l1−l2+l35,l1+l3≥l2,for all l1,l2,l3∈Θ.Moreover, from the definition,Q and R are continuous. Let π:ℝ+⟶ℝ be a function defined by πυ=lnυ for υ>0. To verify the inequality (12) of Theorem 13, we consider the following cases:•1: if l1+l3≥4l2 and l′1+l′3≥l′2, we can write Ql1,l2,l3=l1−4l2+l3/5 and Rl′1,l′2,l′3=l′1−l′2+l′3/5; then,
(89)ϖQl1,l2,l3,Rl′1,l′2,l′3=l1−4l2+l35−l′1−l′2+l′35=l1−l′15+l′2−4l25+l3−l′35≤l1−l′15+l′2−4l25+l3−l′35≤l1−l′15+l2−l′25+l3−l′35≤35maxl1−l′1,l2−l′2,l3−l′3=35maxϖl1,l′1,ϖl2,l′2,ϖl3,l′3.Takingπ into account, we can write
(90)lnϖQl1,l2,l3,Rl′1,l′2,l′3≤ln35maxϖl1,l′1,ϖl2,l′2,ϖl3,l′3=ln35+lnmaxϖl1,l′1,ϖl2,l′2,ϖl3,l′3,or
(91)ln53+lnϖQl1,l2,l3,Rl′1,l′2,l′3≤lnmaxϖl1,l′1,ϖl2,l′2,ϖl3,l′3,which leads to
(92)ℓ+πϖQl1,l2,l3,Rl′1,l′2,l′3≤πmaxϖl1,l′1,ϖl2,l′2,ϖl3,l′3.•2: if l1+l3≥4l2 and l′1+l′3<l′2, we have Ql1,l2,l3=l1−4l2+l3/5 and Rl′1,l′2,l′3=0; then,
(93)ϖQl1,l2,l3,Rl′1,l′2,l′3=l1−4l2+l35−0≤l1−l′1+l′2−l2+l3−l′3+l′1+l′3−l′25≤l1−l′15+l′2−4l25+l3−l′35,since for all i,j,k∈Θ, i+j+k≤3maxi,j,k, one can get
(94)ϖQl1,l2,l3,Rl′1,l′2,l′3≤35maxl1−l′15+l′2−4l25+l3−l′35≤35maxϖl1,l′1,ϖl2,l′2,ϖl3,l′3,and by the same manner of •1, we get (92).•3: if l1+l3<4l2 and l′1+l′3≥l′2, then Ql1,l2,l3=0 and Rl′1,l′2,l′3=l′1−l′2+l′3/5. Hence, by the same method of •2, we obtain (92).•4: if l1+l3<4l2 and l′1+l′3<l′2, we get Ql1,l2,l3=0 and Rl′1,l′2,l′3=0; it is trivial.It follows from•1−•4 that the inequality (12) of Theorem 13 with ℓ=ln5/3>0 is verified.Then,0,0,0∈Θ3 is a unique common TFP of Q and R.Example 2.
We consider the following tripled system of functional equations:(95)zl1=supl2∈ℝarctanl1+5l2+11+l12+11+el2+13zo1+8zo33+13bo1+5bo33+13ao1+4ao33,bl1=supl2∈ℝarctanl1+5l2+11+l12+11+el2+13bo1+8bo33+13ao1+5ao33+13zo1+4zo33al1=supl2∈ℝarctanl1+5l2+11+l12+11+el2+13ao1+8ao33+13zo1+5zo33+13bo1+4bo33,,for l1∈0,1.It is clear that the system (95) is a special form of system (52) with S=0,1 and D=ℝ. The condition ‡i of Theorem 21 is clear. For ‡ii, we can write
(96)Jl1,l2,zol1,l2,bol1,l2,aol1,l2−Jl1,l2,z∗ol1,l2,b∗ol1,l2,a∗ol1,l2≤13zo1+8zo33−z∗o1+8z∗o33+13bo1+5bo33−b∗o1+5b∗o33+13ao1+4ao33−a∗o1+4a∗o33=13φ83zo−φ83z∗o+13φ53bo−φ53b∗o+13φ43ao−φ53a∗o≤13φ83zo−z∗o+13φ53bo−b∗o+13φ43ao−a∗o≤13φ83zo−z∗o+13φ53bo−b∗o+13φ43ao−a∗o≤13φ83maxz−z∗,b−b∗,a−a∗+13φ53maxz−z∗,b−b∗,a−a∗+13φ43maxz−z∗,b−b∗,a−a∗≤3×13φ43maxz−z∗,b−b∗,a−a∗=maxz−z∗,b−b∗,a−a∗1+4maxz−z∗,b−b∗,a−a∗33,where Lemma20 is used. Hence, ‡ii is satisfied with ℓ=4 and ζ=3. According to Theorem 21, the system (95) has a unique solution in AS×AS.Example 3.
Suppose the following tripled system of integral equations:(97)l1w=ew+∫01w2+r1+r+13l1r1+10l1r55+13l2r1+7l2r55+13l3r1+6l3r55dr,l2w=ew+∫01w2+r1+r+13l2r1+10l2r55+13l3r1+7l3r55+13l1r1+6l1r55dr,l3w=ew+∫01w2+r1+r+13l3r1+10l3r55+13l1r1+7l1r55+13l2r1+6l2r55dr,for w∈0,1.Again, system (97) is a special case of system (64), where kw=ew.It is obvious that the condition♠1 of Theorem 22 holds. For the condition ♠2, we get
(98)Yw,r,l1r,l2r,l3r−Yw,r,l′1r,l′2r,l′3r≤13l1r1+10l1r55−l′1r1+10l′1r55+13l2r1+7l2r55−l′2r1+7l′2r55+13l3r1+6l3r55−l′3r1+6l′3r55=13φ105l1r−φ105l′1r+13φ75l2r−φ75l′2r+13φ65l3r−φ75l′3r≤13φ105l1r−l′1r+13φ75l2r−l′2r+13φ65l3r−l′3r≤13φ105l1r−l′1r+13φ75l2r−l′2r+13φ65l3r−l′3r≤12φ105maxl1−l′1,l2−l′2,l3−l′3+12φ75maxl1−l′1,l2−l′2,l3−l′3+13φ65maxl1−l′1,l2−l′2,l3−l′3≤3×13φ65maxl1−l′1,l2−l′2,l3−l′3=maxl1−l′1,l2−l′2,l3−l′31+6maxl1−l′1,l2−l′2,l3−l′355,where Lemma 20 was used. Hence, ♠2 holds with ℓ=6 and ζ=5. According to Corollary 14, system (97) has a unique solution l′1,l′2,l′3∈C0,13.


## 6. Conclusions

The present paper is dedicated to the study of the existence and uniqueness of tripled fixed points in a CMS with and without a directed graph. Common tripled fixed point results are given too. Moreover, some applications of the main results in solving different types of tripled equation systems are presented. Then, using our main results, we study the existence and uniqueness of a solution of some systems of tripled functional and integral equations used in the study of dynamic programming. To sustain our results, the last part of the paper is dedicated to some illustrative examples. Our results come to improve some results from the related literature and give new directions in the study of economic phenomena, using the tripled fixed point technique.



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