# Common Fixed Point Theorems for Contractive Mappings of Integral Type inG-Metric Spaces and Applications

Authors: Fengrong Zhang; Xiangshuai Zhang; Yan Hao
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Two common fixed point theorems for weakly compatible mappings satisfying contractive conditions of integral type inG-metric spaces are demonstrated. The results obtained in this paper generalize and differ from a few results in the literature and are used to prove the existence and uniqueness of common bounded and continuous solutions for certain functional equations and nonlinear Volterra integral equations. A nontrivial example is included.


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## Body

## 1. Introduction

The Banach fixed point theorem which was first presented by Banach in 1922 is a significant result in fixed point theory. Because of its importance in proving the existence of solutions for functional equations, nonlinear Volterra integral equations and nonlinear integro-differential equations, this result has been extended in many different directions (see, e.g., [1–22] and the references cited therein). In particular, Rhoades [12] and Branciari [4] generalized the Banach fixed point theorem and gave the following fixed point theorems, respectively.Theorem 1 (see [12]).
Letf be a mapping from a complete metric space X,d into itself satisfying
(1)dfx,fy≤dx,y−φdx,y,∀x,y∈X,where φ∈Φ4. Then, f has a unique fixed point in X.Theorem 2 (see [4]).
LetX,d be a complete metric space and f:X→X be a mapping satisfying
(2)∫0dfx,fyφtdt≤c∫0dx,yφtdt,∀x,y∈X,where φ∈Φ1 and c∈0,1 is a constant. Then, f has a unique fixed point a∈X such that limn→∞fnx=a for each x∈X.In 2013, Gupta and Mani [21] obtained the existence and uniqueness of a fixed point for contractive mappings of an integral type in complete metric spaces by using iterative approximations. In 2007, Kumar et al. [6] proved a common fixed point theorem for a pair of compatible mappings satisfying a contractive inequality of integral type, which improves Theorem 2.Theorem 3 (see [6]).
LetX,d be a complete metric space and f,g:X→X be compatible mappings such that
(3)fX⊆gX,giscontinuous,∫0dfx,fyφtdt≤c∫0dgx,gyφtdt,∀x,y∈X,
whereφ∈Φ1 and c∈0,1 is a constant. Then, f and g have a unique common fixed point in X.In 2006, Mustafa and Sims [9] introduced a new concept of generalized metric space called G-metric space. From then on, lots of research works have been carried out on generalizing contractive conditions for different contractive mappings satisfying various known properties in G-metric spaces [1–3, 5, 10, 11, 13, 15, 19, 20]. In 2018, Gupta et al. [19] proved some fixed point theorems for the functions satisfying ϕ-contraction and mixed g-monotone property in G-metric spaces. In 2015, Gupta and Deep [20] gave a few common fixed point theorems using the property E.A. in the setting of G-metric and fuzzy metric spaces by taking a set of three conditions for self-mappings. In 2011, Aydi [1] proved a fixed point theorem for mappings satisfying a ψ,ϕ-weakly contractive condition in G-metric spaces.Theorem 4 (see [1]).
LetX,G be a complete G-metric space and f be a mapping from X into itself satisfying
(4)ψGfx,fy,fz≤ψGx,y,z−ϕGx,y,z,∀x,y,z∈X,where ψ,ϕ∈Φ2. Then, f has a unique fixed point u∈X and f is G-continuous at u.In 2012, Aydi [2] obtained the following common fixed point theorem for a pair of mappings involving a contractive condition of integral type in G-metric spaces.Theorem 5 (see [2]).
LetX,G be a G-metric space and f,g be two mappings from X into itself such that
(5)∫0Gfx,fy,fzφtdt≤α∫0Ggx,gy,gzφtdt,∀x,y,z∈X,where φ∈Φ1 and α∈0,1 is a constant. If fX⊆gX and gX is a complete subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.The objective of this paper is both to introduce two new classes of contractive mappings of integral type in the setting ofG-metric spaces and to prove the existence and uniqueness of points of coincidence and common fixed points for these mappings. Our results extend Theorem 5, are different from Theorem 4, and are used to show solvability of the functional equations arising in dynamic programming and nonlinear Volterra integral equations. A nontrivial example is given.


## 2. Preliminaries

Throughout this paper,ℕ denotes the set of all positive integers, ℕ0=0∪ℕ, ℝ+=0,+∞, and ℝ=−∞,+∞. Put
(6)Φ1=∫0ϵφtdtφ∣φ:R+→R+is Lebesgue integrable and summable on each compact subset ofR+and∫0ϵφtdt>0,∀ϵ>0,Φ2=φ∣φ:R+→R+is a continuous and nondecreasing function such thatφt=0if and only ift=0,Φ3=φ∣φ:R+→R+satisfies thatliminfn→∞φan>0↔liminfn→∞an>0for eachann∈N⊂R+,Φ4=φ∣φ∈Φ2andlimt→+∞φt=+∞,Φ5=φ∣φ∈Φ2andφt1+t2≤φt1+φt2,∀t1,t2∈R+.Definition 6 (see [9]).
LetX be a nonempty set and G:X×X×X→ℝ+ be a function satisfying the following properties:
(G1) Gx,y,z=0 if x=y=z,
(G2) 0<Gx,x,y for all x,y∈X with x≠y,
(G3) Gx,x,y≤Gx,y,z for all x,y,z∈X with y≠z,
(G4) Gx,y,z=Gx,z,y=Gy,z,x=⋯, symmetry in all three variables,
(G5) Gx,y,z≤Gx,a,a+Ga,y,z for all x,y,z,a∈X.The functionG is called a G-metric in X, and the pair X,G is called a G-metric space.Definition 7 (see [9]).
LetX,G be a G-metric space and xnn∈ℕ be a sequence of points of X. The sequence xnn∈ℕ is G-convergent to x∈X if
(7)limn,m→∞Gx,xn,xm=0,that is, for any ε>0, there exists N∈ℕ such that
(8)Gx,xn,xm<ε,∀m,n≥N.The pointx is called the limit of the sequence xnn∈ℕ and writexn→x or limn→∞xn=x.Lemma 8 (see [9]).
LetX,G be a G-metric space. Then, the following statements are equivalent:
(1)
xnn∈ℕ is G-convergent to x,(2)
Gxn,xn,x→0 as n→∞,(3)
Gxn,x,x→0 as n→∞,(4)
Gxn,xm,x→0 as n,m→∞.Definition 9 (see [9]).
LetX,G be a G-metric space. A sequence xnn∈ℕ in X is called G-Cauchy if for every ε>0, there is N∈ℕ such that
(9)Gxl,xn,xm<ε,∀l,n,m≥N,that is, Gxl,xn,xm→0asl,n,m→∞.Lemma 10 (see [9]).
LetX,G be a G-metric space. Then, the following statements are equivalent:
(1)
xnn∈ℕ is G-Cauchy,(2)
For everyε>0, there exists N∈ℕ such that Gxn,xn,xm<ε for all n,m≥N.Definition 11 (see [9]).
AG-metric space X,G is called G-complete if every G-Cauchy sequence is G-convergent in X,G.Lemma 12 (see [9]).
LetX,G be a G-metric space. Then, the function Gx,y,z is jointly continuous in all three of its variables.Lemma 13 (see [9]).
LetX,G be a G-metric space. Then,
(10)Gx,y,y≤2Gy,x,x,∀x,y∈X,Gx,y,z−Gx,y,a≤maxGa,z,z,Gz,a,a,∀x,y,z,a∈X.Definition 14 (see [14]).
LetS and T be self-mappings of a nonempty set X.
(1)
A pointx∈X is said to be a fixed point of T if Tx=x.(2)
A pointx∈X is said to be a coincidence point of S and T if Tx=Sx and w=Sx=Tx is said to be a point of coincidence of S and T.(3)
A pointx∈X is said to be a common fixed point of S and T if x=Tx=Sx.Definition 15.
A pair of self-mappingsf and g in a G-metric space X,G are said to be weakly compatible if for any x∈X, the equality fx=gx gives that fgx=gfx.Lemma 16 (see [14]).
LetX be a nonempty set and f,g:X→X be weakly compatible mappings. If f and g have a unique point of coincidence w∈X, then w is the unique common fixed point of f and g.Lemma 17 (see [7]).
Letφ∈Φ1 and rnn∈ℕ be a nonnegative sequence with limn→∞rn=a. Then,
(11)limn→∞∫0rnφtdt=∫0aφtdt.Lemma 18 (see [8]).
Letφ∈Φ3. Then, φt>0 if and only if t>0.


## 3. Main Results

Now, we study the existence and uniqueness of points of coincidence and common fixed points for contractive mappings (12) and (51) below in G-metric spaces, respectively.Theorem 19.
LetX,G be a G-metric space, f and g:X→X be two mappings satisfying
(12)ψ∫0Gfx,fy,fzφtdt≤ψ∫0M1x,y,zφtdt−ϕ∫0M1x,y,zφtdt,∀x,y,z∈X,where φ,ψ,ϕ∈Φ1×Φ2×Φ3 and
(13)M1x,y,z=maxGgx,gy,gz,1+Ggx,gx,fxGgy,gy,fz2+Ggx,fx,fy,1+Ggx,gx,fxGgz,gz,fy2+Ggx,fx,fz,1+Ggy,gy,fyGgz,gz,fx2+Ggy,fy,fz,1+Ggy,gy,fyGgx,gx,fz21+Ggy,fx,fy,1+Ggz,gz,fzGgy,gy,fx21+Ggz,fy,fz,1+Ggz,gz,fzGgx,gx,fy21+Ggz,fx,fz,1+Ggx,gy,gzGfx,fy,fz1+Ggz,fy,fy+Ggx,gy,gz,1+Ggx,gy,gzGfx,fy,fz1+Ggy,fx,fx+Ggx,gy,gz,1+Ggx,gy,gzGfx,fy,fz1+Ggx,fz,fz+Ggx,gy,gz.IffX⊆gX and gX is a complete subset of X, then f and g have a unique point of coincidence in X. Furthermore, if f and g are weakly compatible mappings, then f and g have a unique common fixed point in X.Proof.
Letx0 be an arbitrary point in X. Since fX⊆gX, it follows that there exists a sequence xnn∈ℕ0 in X satisfying
(14)fxn=gxn+1,∀n∈ℕ0.
PutGn=Gfxn,fxn+1,fxn+2 for all n∈ℕ0. Assume that fxn0=fxn0+1 for some n0∈ℕ0. It is clear that gxn0+1=fxn0+1, that is, fxn0+1 is a point of coincidence of f and g. Assume that fxn≠fxn+1 for all n∈ℕ0. Clearly, Gn>0 for all n∈ℕ0. By virtue of (G3)–(G5) and (14), we observe that
(15)1+Ggxn+1,gxn+1,fxn+1Ggxn,gxn,fxn+221+Ggxn+1,fxn,fxn+1=1+Gfxn,fxn,fxn+1Gfxn−1,fxn−1,fxn+221+Gfxn,fxn,fxn+1=Gfxn−1,fxn−1,fxn+22=Gfxn+2,fxn−1,fxn−12≤Gfxn+2,fxn,fxn+Gfxn,fxn−1,fxn−12=Gfxn,fxn,fxn+2+Gfxn−1,fxn−1,fxn2≤Gfxn,fxn+2,fxn+1+Gfxn−1,fxn,fxn+12=Gn−1+Gn2,∀n∈ℕ.
Set(16)m1xn,xn+1,xn+2=max1+Ggxn,gxn,fxnGgxn+1,gxn+1,fxn+22+Ggxn,fxn,fxn+1,1+Ggxn,gxn,fxnGgxn+2,gxn+2,fxn+12+Ggxn,fxn,fxn+2,1+Ggxn+1,gxn+1,fxn+1Ggxn+2,gxn+2,fxn2+Ggxn+1,fxn+1,fxn+2,1+Ggxn+1,gxn+1,fxn+1Ggxn,gxn,fxn+221+Ggxn+1,fxn,fxn+1,1+Ggxn+2,gxn+2,fxn+2Ggxn+1,gxn+1,fxn21+Ggxn+2,fxn+1,fxn+2,1+Ggxn+2,gxn+2,fxn+2Ggxn,gxn,fxn+121+Ggxn+2,fxn,fxn+2,∀n∈ℕ,which together with (15) yields that
(17)m1xn,xn+1,xn+2=max1+Gfxn−1,fxn−1,fxnGfxn,fxn,fxn+22+Gfxn−1,fxn,fxn+1,1+Gfxn−1,fxn−1,fxnGfxn+1,fxn+1,fxn+12+Gfxn−1,fxn,fxn+2,1+Gfxn,fxn,fxn+1Gfxn+1,fxn+1,fxn2+Gfxn,fxn+1,fxn+2,1+Gfxn,fxn,fxn+1Gfxn−1,fxn−1,fxn+221+Gfxn,fxn,fn+1,1+Gfxn+1,fxn+1,fxn+2Gfxn,fxn,fxn21+Gfxn+1,fxn+1,fxn+2,1+Gfxn+1,fxn+1,fxn+2Gfxn−1,fxn−1,fxn+121+Gfxn+1,fxn,fxn+2≤max1+Gn−1Gn2+Gn−1,0,1+GnGn−12+Gn,Gn−1+Gn2,0,1+2GnGn−121+Gn≤maxGn−1,Gn,∀n∈ℕ.
In light of (G1), (G3), (G5), and (13)–(17), we get that
(18)M1xn,xn+1,xn+2=maxGgxn,gxn+1,gxn+2,m1xn,xn+1,xn+2,1+Ggxn,gxn+1,gxn+2Gfxn,fxn+1,fxn+21+Ggxn+2,fxn+1,fxn+1+Ggxn,gxn+1,gxn+2,1+Ggxn,gxn+1,gxn+2Gfxn,fxn+1,fxn+21+Ggxn+1,fxn,fxn+Ggxn,gxn+1,gxn+2,1+Ggxn,gxn+1,gxn+2Gfxn,fxn+1,fxn+21+Ggxn,fxn+2,fxn+2+Ggxn,gxn+1,gxn+2=maxGfxn−1,fxn,fxn+1,m1xn,xn+1,xn+2,1+Gfxn−1,fxn,fxn+1Gfxn,fxn+1,fxn+21+Gfxn+1,fxn+1,fxn+1+Gfxn−1,fxn,fxn+1,1+Gfxn−1,fxn,fxn+1Gfxn,fxn+1,fxn+21+Gfxn,fxn,fxn+Gfxn−1,fxn,fxn+1,1+Gfxn−1,fxn,fxn+1Gfxn,fxn+1,fxn+21+Gfxn−1,fxn+2,fxn+2+Gfxn−1,fxn,fxn+1=maxGn−1,m1xn,xn+1,xn+2,Gn,Gn,1+Gn−1Gn1+Gfxn−1,fxn+2,fxn+2+Gn−1=maxGn−1,Gn,∀n∈ℕ.
Now we assert thatGn≤Gn−1, ∀n∈ℕ. Suppose that there exists some n0∈ℕ satisfying Gn0>Gn0−1. It follows from (12), φ,ψ,ϕ∈Φ1×Φ2×Φ3 and Lemma 18, we infer that
(19)ψ∫0Gn0φtdt=ψ∫0Gfxn0,fxn0+1,fxn0+2φtdt≤ψ∫0M1xn0,xn0+1,xn0+2φtdt−ϕ∫0M1xn0,xn0+1,xn0+2φtdt=ψ∫0Gn0φtdt−ϕ∫0Gn0φtdt<ψ∫0Gn0φtdt,which is a contradiction. Therefore, Gn≤Gn−1 for all n∈ℕ and
(20)M1xn,xn+1,xn+2=Gn−1,∀n∈ℕ.
It is apparent that the sequenceGnn∈ℕ0 is nonincreasing and bounded, which implies that there exists r with
(21)limn→∞Gn=r≥0.
Now, we demonstrate thatr=0. Suppose that r>0. On account of (12), (20), and (21), φ,ψ,ϕ∈Φ1×Φ2×Φ3 and Lemma 17, we deduce that
(22)ψ∫0rφtdt=limsupn→∞ψ∫0Gnφtdt=limsupn→∞ψ∫0Gfxn,fxn+1,fxn+2φtdt≤limsupn→∞ψ∫0M1xn,xn+1,xn+2φtdt−ϕ∫0M1xn,xn+1,xn+2φtdt=limsupn→∞ψ∫0Gn−1φtdt−ϕ∫0Gn−1φtdt≤limsupn→∞ψ∫0Gn−1φtdt−liminfn→∞ϕ∫0Gn−1φtdt=ψ∫0rφtdt−liminfn→∞ϕ∫0Gn−1φtdt<ψ∫0rφtdt,which is impossible. Thus, r=0. That is,
(23)limn→∞Gn=0.
It follows from (G3), (G4), and (23) that
(24)0≤Gfxn−1,fxn−1,fxn≤Gfxn−1,fxn,fxn+1→0asn→∞,0≤Gfxn−1,fxn,fxn=Gfxn,fxn,fxn−1≤Gfxn,fxn−1,fxn−2→0asn→∞,which yield that
(25)limn→∞Gfxn−1,fxn−1,fxn=limn→∞Gfxn−1,fxn,fxn=0.
Next, we verify thatfxnn∈ℕ0 is a G-Cauchy sequence. Suppose that fxnn∈ℕ0 is not a G-Cauchy sequence. It follows from Lemma 10 that there exist a constant ε>0 and two subsequences fxmkk∈ℕ and fxnkk∈ℕ of fxnn∈ℕ0 such that nk is minimal in the sense that
(26)k<mk<nk<mk+1andGfxmk,fxmk,fxnk>ε,∀k∈ℕ,which means that Gfxmk,fxmk,fxnk−1≤ε for all k∈ℕ. By means of (G3)–(G5) and Lemma 13, we deduce that
(27)ε<Gfxmk,fxmk,fxnk=Gfxnk,fxmk,fxmk≤Gfxnk,fxnk−1,fxnk−1+Gfxnk−1,fxmk,fxmk≤ε+Gfxnk,fxnk−1,fxnk−1≤ε+Gnk−1,∀k∈ℕ,(28)Gfxnk,fxmk−1,fxmk−Gfxmk,fxmk,fxnk≤maxGfxmk−1,fxmk,fxmk,Gfxmk,fxmk−1,fxmk−1≤2Gfxmk−1,fxmk,fxmk+1=2Gmk−1,∀k∈ℕ,(29)Gfxmk−1,fxmk−1,fxnk−Gfxnk,fxmk−1,fxmk≤maxGfxmk−1,fxmk,fxmk,Gfxmk,fxmk−1,fxmk−1≤2Gfxmk−1,fxmk,fxmk+1=2Gmk−1,∀k∈ℕ,(30)∣Gfxmk−1,fxmk−1,fxnk−1−Gfxmk−1,fxmk−1,fxnk∣≤maxGfxnk−1,fxnk,fxnk,Gfxnk,fxnk−1,fxnk−1≤2Gfxnk−1,fxnk,fxnk+1=2Gnk−1,∀k∈ℕ,(31)∣Gfxmk,fxmk,fxnk−1−Gfxmk,fxmk,fxnk∣≤maxGfxnk,fxnk−1,fxnk−1,Gfxnk−1,fxnk,fxnk≤2Gfxnk−1,fxnk,fxnk+1=2Gnk−1,∀k∈ℕ(32)Gfxnk−1,fxmk−1,fxmk−Gfxmk,fxmk,fxnk−1≤maxGfxmk,fxmk−1,fxmk−1,Gfxmk−1,fxmk,fxmk≤2Gfxmk−1,fxmk,fxmk+1=2Gmk−1,∀k∈ℕ.
Lettingk→∞ in (27)–(32) and using (23) and (25), we obtain that
(33)ε=limk→∞Gfxmk,fxmk,fxnk=limk→∞Gfxnk,fxmk−1,fxmk=limk→∞Gfxmk−1,fxmk−1,fxnk=limk→∞Gfxmk−1,fxmk−1,fxnk−1=limk→∞Gfxmk,fxmk,fxnk−1=limk→∞Gfxnk−1,fxmk−1,fxmk.
In view of (G3)–(G5) and Lemma 13, we infer that
(34)Gfxnk−1,fxnk−1,fxmk=Gfxmk,fxnk−1,fxnk−1≤Gfxmk,fxnk,fxnk+Gfxnk,fxnk−1,fxnk−1≤Gfxmk,fxmk−1,fxnk+Gfxnk+1,fxnk,fxnk−1=Gfxmk,fxmk−1,fxnk+Gnk−1,∀k∈N,(35)Gfxmk,fxmk−1,fxnk=Gfxnk,fxmk−1,fxmk≤Gfxnk,fxmk,fxmk+Gfxmk,fxmk−1,fxmk≤Gfxnk,fxnk−1,fxmk+Gfxmk,fxmk−1,fxmk≤Gfxnk,fxnk−1,fxnk−1+Gfxnk−1,fxnk−1,fxmk+Gfxmk,fxmk−1,fxmk≤Gnk−1+2Gmk−1+Gfxnk−1,fxnk−1,fxmk,∀k∈ℕ.
Takingk→∞ in (34) and (35) and utilizing (23) and (33), we conclude that
(36)ε=limk→∞Gfxnk−1,fxnk−1,fxmk.
On the basis of (G3)–(G5) and Lemma13, we arrive at
(37)Gfxnk−1,fxmk,fxnk=Gfxnk,fxnk−1,fxmk≤Gfxnk,fxnk−1,fxnk−1+Gfxnk−1,fxnk−1,fxmk≤Gnk−1+Gfxnk−1,fxnk−1,fxmk,∀k∈ℕ,(38)Gfxnk−1,fxnk−1,fxmk≤Gfxnk−1,fxnk,fxnk+Gfxnk,fxnk−1,fxmk≤2Gfxnk,fxnk−1,fxnk−1+Gfxnk−1,fxmk,fxnk≤2Gnk−1+Gfxnk−1,fxmk,fxnk,∀k∈ℕ.
Lettingk→∞ in (37) and (38) and using (23) and (36), we deduce that
(39)ε=limk→∞Gfxnk−1,fxmk,fxnk.
On account of (G3)–(G5) and Lemma 13, we receive that
(40)Gfxnk,fxnk,fxmk−1=Gfxmk−1,fxnk,fxnk≤Gfxmk−1,fxnk−1,fxnk−1+Gfxnk−1,fxnk,fxnk≤Gfxnk−1,fxmk−1,fxmk+2Gnk−1,∀k∈ℕ,(41)Gfxnk−1,fxmk−1,fxmk≤Gfxnk−1,fxmk−1,fxmk−1+Gfxmk−1,fxmk−1,fxmk≤Gfxnk−1,fxnk,fxmk−1+Gmk−1≤Gfxnk−1,fxnk,fxnk+Gfxnk,fxnk,fxmk−1+Gmk−1≤2Gnk−1+Gfxnk,fxnk,fxmk−1+Gmk−1,∀k∈ℕ.
Takingk→∞ in (40) and (41) and using (23) and (33), we conclude that
(42)ε=limk→∞Gfxnk,fxnk,fxmk−1.
Making use of (12)–(14), (23), (25), (33), (36), (39), (42), φ,ψ,ϕ∈Φ1×Φ2×Φ3, and Lemma 12, we obtain that
(43)limk→∞M1xmk,xmk,xnk=limk→∞maxGgxmk,gxmk,gxnk,1+Ggxmk,gxmk,fxmkGgxmk,gxmk,fxnk2+Ggxmk,fxmk,fxmk,1+Ggxmk,gxmk,fxmkGgxnk,gxnk,fxmk2+Ggxmk,fxmk,fxnk,1+Ggxmk,gxmk,fxmkGgxnk,gxnk,fxmk2+Ggxmk,fxmk,fxnk,1+Ggxmk,gxmk,fxmkGgxmk,gxmk,fxnk21+Ggxmk,fxmk,fxmk,1+Ggxnk,gxnk,fxnkGgxmk,gxmk,fxmk21+Ggxnk,fxmk,fxnk,1+Ggxnk,gxnk,fxnkGgxmk,gxmk,fxmk21+Ggxnk,fxmk,fxnk,1+Ggxmk,gxmk,gxnkGfxmk,fxmk,fxnk1+Ggxnk,fxmk,fxmk+Ggxmk,gxmk,gxnk,1+Ggxmk,gxmk,gxnkGfxmk,fxmk,fxnk1+Ggxmk,fxmk,fxmk+Ggxmk,gxmk,gxnk,1+Ggxmk,gxmk,gxnkGfxmk,fxmk,fxnk1+Ggxmk,fxnk,fxnk+Ggxmk,gxmk,gxnk=limk→∞maxGfxmk−1,fxmk−1,fxnk−1,1+Gfxmk−1,fxmk−1,fxmkGfxmk−1,fxmk−1,fxnk2+Gfxmk−1,fxmk,fxmk,1+Gfxmk−1,fxmk−1,fxmkGfxnk−1,fxnk−1,fxmk2+Gfxmk−1,fxmk,fxnk,1+Gfxmk−1,fxmk−1,fxmkGfxnk−1,fxnk−1,fxmk2+Gfxmk−1,fxmk,fxnk,1+Gfxmk−1,fxmk−1,fxmkGfxmk−1,fxmk−1,fxnk21+Gfxmk−1,fxmk,fxmk,1+Gfxnk−1,fxnk−1,fxnkGfxmk−1,fxmk−1,fxmk21+Gfxnk−1,fxmk,fxnk,1+Gfxnk−1,fxnk−1,fxnkGfxmk−1,fxmk−1,fxmk21+Gfxnk−1,fxmk,fxnk,1+Gfxmk−1,fxmk−1,fxnk−1Gfxmk,fxmk,fxnk1+Gfxnk−1,fxmk,fxmk+Gfxmk−1,fxmk−1,fxnk−1,1+Gfxmk−1,fxmk−1,fxnk−1Gfxmk,fxmk,fxnk1+Gfxmk−1,fxmk,fxmk+Gfxmk−1,fxmk−1,fxnk−1,1+Gfxmk−1,fxmk−1,fxnk−1Gfxmk,fxmk,fxnk1+Gfxmk−1,fxnk,fxnk+Gfxmk−1,fxmk−1,fxnk−1=maxε,ε2,ε2+ε,ε2+ε,ε2,0,0,1+εε1+2ε,ε,1+εε1+2ε=ε.
Making use of (12), (33), (43), φ,ψ,ϕ∈Φ1×Φ2×Φ3, and Lemma 17, we conclude that
(44)ψ∫0εφtdt=limsupk→∞ψ∫0Gfxmk,fxmk,fxnkφtdt≤limsupk→∞ψ∫0M1xmk,xmk,xnkφtdt−ϕ∫0M1xmk,xmk,xnkφtdt≤limsupk→∞ψ∫0M1xmk,xmk,xnkφtdt−liminfk→∞ϕ∫0M1xmk,xmk,xnkφtdt=ψ∫0εφtdt−liminfk→∞ϕ∫0M1xmk,xmk,xnkφtdt<ψ∫0εφtdt,which is ridiculous. Thus, fxnn∈ℕ0 is a G-Cauchy sequence. Since gX is complete, it follows that there exists w∈gX such that
(45)limn→∞fxn=w.
In light of Lemma8 and w∈gX, there exists a∈X satisfying ga=w and
(46)limn→∞Gfxn,fxn,ga=limn→∞Gfxn,ga,ga=0.
Next, we provega=fa. Suppose that ga≠fa. In view of (12), (13), (25), (46), φ,ψ,ϕ∈Φ1×Φ2×Φ3, and Lemmas 12 and 17, we obtain that
(47)limn→∞M1xn,xn,a=limn→∞maxGgxn,gxn,ga,1+Ggxn,gxn,fxnGgxn,gxn,fa2+Ggxn,fxn,fxn,1+Ggxn,gxn,fxnGga,ga,fxn2+Ggxn,fxn,fa,1+Ggxn,gxn,fxnGga,ga,fxn2+Ggxn,fxn,fa,1+Ggxn,gxn,fxnGgxn,gxn,fa21+Ggxn,fxn,fxn,1+Gga,ga,faGgxn,gxn,fxn21+Gga,fxn,fa,1+Gga,ga,faGgxn,gxn,fxn21+Gga,fxn,fa,1+Ggxn,gxn,gaGfxn,fxn,fa1+Gga,fxn,fxn+Ggxn,gxn,ga,1+Ggxn,gxn,gaGfxn,fxn,fa1+Ggxn,fxn,fxn+Ggxn,gxn,ga,1+Ggxn,gxn,gaGfxn,fxn,fa1+Ggxn,fa,fa+Ggxn,gxn,ga=limn→∞maxGfxn−1,fxn−1,ga,1+Gfxn−1,fxn−1,fxnGfxn−1,fxn−1,fa2+Gfxn−1,fxn,fxn,1+Gfxn−1,fxn−1,fxnGga,ga,fxn2+Gfxn−1,fxn,fa,1+Gfxn−1,fxn−1,fxnGga,ga,fxn2+Gfxn−1,fxn,fa,1+Gfxn−1,fxn−1,fxnGfxn−1,fxn−1,fa21+Gfxn−1,fxn,fxn,1+Gga,ga,faGfxn−1,fxn−1,fxn21+Gga,fxn,fa,1+Gga,ga,faGfxn−1,fxn−1,fxn21+Gga,fxn,fa,1+Gfxn−1,fxn−1,gaGfxn,fxn,fa1+Gga,fxn,fxn+Gfxn−1,fxn−1,ga,1+Gfxn−1,fxn−1,gaGfxn,fxn,fa1+Gfxn−1,fxn,fxn+Gfxn−1,fxn−1,ga,1+Gfxn−1,fxn−1,gaGfxn,fxn,fa1+Gfxn−1,fa,fa+Gfxn−1,fxn−1,ga=max0,Gga,ga,fa2,0,0,Gga,ga,fa2,0,0,Gga,ga,fa,Gga,ga,fa,Gga,ga,fa1+Gga,fa,fa=Gga,ga,fa(48)ψ∫0Gga,ga,faφtdt=limsupn→∞ψ∫0Gfxn,fxn,faφtdt≤limsupn→∞ψ∫0M1xn,xn,aφtdt−ϕ∫0M1xn,xn,aφtdt≤limsupn→∞ψ∫0M1xn,xn,aφtdt−liminfn→∞ϕ∫0M1xn,xn,aφtdt=ψ∫0Gga,ga,faφtdt−liminfn→∞ϕ∫0M1xn,xn,aφtdt<ψ∫0Gga,ga,faφtdt,which is absurd. Consequently, w=ga=fa, that is, w is a point of coincidence of f and g.
Lastly, we certify thatf and g have a unique point of coincidence in X. Assume that there exists b∈X with fb=gb≠fa. In terms of (13), (G2), and Lemma 13, we receive that
(49)M1a,a,b=maxGga,ga,gb,1+Gga,ga,faGga,ga,fb2+Gga,fa,fa,1+Gga,ga,faGgb,gb,fa2+Gga,fa,fb,1+Gga,ga,faGgb,gb,fa2+Gga,fa,fb,1+Gga,ga,faGga,ga,fb21+Gga,fa,fa,1+Ggb,gb,fbGga,ga,fa21+Ggb,fa,fb,1+Ggb,gb,fbGga,ga,fa21+Ggb,fa,fb,1+Gga,ga,gbGfa,fa,fb1+Ggb,fa,fa+Gga,ga,gb,1+Gga,ga,gbGfa,fa,fb1+Gga,fa,fa+Gga,ga,gb,1+Gga,ga,gbGfa,fa,fb1+Gga,fb,fb+Gga,ga,gb=maxGfa,fa,fb,Gfa,fa,fb2,Gfb,fb,fa2+Gfa,fa,fb,Gfb,fb,fa2+Gfa,fa,fb,Gfa,fa,fb2,0,0,1+Gfa,fa,fbGfa,fa,fb1+2Gfa,fa,fb,Gfa,fa,fb,1+Gfa,fa,fbGfa,fa,fb1+Gfa,fb,fb+Gfa,fa,fb=Gfa,fa,fb>0.
According to (12), (49), φ,ψ,ϕ∈Φ1×Φ2×Φ3, and Lemma 18, we gain that
(50)ψ∫0Gfa,fa,fbφtdt≤ψ∫0M1a,a,bφtdt−ϕ∫0M1a,a,bφtdt=ψ∫0Gfa,fa,fbφtdt−ϕ∫0Gfa,fa,fbφtdt<ψ∫0Gfa,fa,fbφtdt,which is contradictive. Therefore, f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible mappings, by Lemma 16, we know that f and g have a unique common fixed point in X. This completes the proof.
Similar to the argument of Theorem19, we derive the following result and omit its proof.Theorem 20.
LetX,G be a G-metric space, f and g:X→X be two mappings satisfying
(51)ψ∫0Gfx,fy,fzφtdt≤ψ∫0M2x,y,zφtdt−ϕ∫0M2x,y,zφtdt,∀x,y,z∈X,where φ,ψ,ϕ∈Φ1×Φ2×Φ3 and
(52)M2x,y,z=maxGgx,gy,gz,1+Ggx,fx,fxGgz,fy,fy2+Ggx,gy,fx,1+Ggy,fy,fyGgz,fx,fx2+Ggx,gy,fy,1+Ggz,fz,fzGgx,fy,fy2+Ggy,gz,fz,1+Ggx,fx,fxGgy,fz,fz21+Ggx,gz,fx,1+Ggy,fy,fyGgx,fz,fz21+Ggy,gz,fy,1+Ggz,fz,fzGgy,fx,fx21+Ggx,gz,fz,1+Ggx,gy,gzGfx,fy,fz1+Ggz,gz,fy+Ggx,gy,gz,1+Ggx,gy,gzGfx,fy,fz1+Ggy,gy,fx+Ggx,gy,gz,1+Ggx,gy,gzGfx,fy,fz1+Ggx,gx,fz+Ggx,gy,gz.IffX⊆gX and gX is a complete subset of X, then f and g have a unique point of coincidence in X. Furthermore, if f and g are weakly compatible mappings, then f and g have a unique common fixed point in X.Remark 21.
In caseψt=t,ϕt=1−λt,∀t∈R+ and λ∈0,1 is a constant, then Theorems 19 and 20 reduce to results, which include Theorem 5 as a special case. The following example shows that Theorems 19 and 20 generalize substantially Theorem 5 and differ from Theorem 4.Example 22.
LetX=0,2. Define f,g:X→X, φ,ψ,ϕ,η:ℝ+→ℝ+, and G:X×X×X→R+ by
(53)fx=0,∀x∈0,1,16,∀x∈1,2,gx=x,∀x∈0,1,76,∀x∈1,2,φt=2t,ψt=3t,ϕt=t,ηt=2t,∀t∈ℝ+,Gx,y,z=x−y+y−z+z−x,∀x,y,z∈X.Clearly,X,G is a G-metric space, φ,ψ,ϕ∈Φ1×Φ2×Φ3, f and g are weakly compatible mappings, fX⊆gX and gX is complete, ηt=ψt−ϕt,∀t∈ℝ+ and η is nondecreasing in ℝ+. Let x,y,z∈X. In order to verify (12) and (51), we consider the following seven cases:Case 1.
x,y,z∈0,1 or x,y,z∈1,2. It follows that
(54)ψ∫0Gfx,fy,fzφtdt=0≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Case 2.
x,y∈0,1 and z∈1,2. It follows that
(55)M1x,y,z≥1+Ggy,gy,fyGgz,gz,fx2+Ggy,fy,fz=1+Gy,y,0G7/6,7/6,02+Gy,0,1/6=7/31+2y2+y+1/6+y−1/6≥1,M2x,y,z≥1+Ggy,fy,fyGgz,fx,fx2+Ggx,gy,fy=1+Gy,0,0G7/6,0,02+Gx,y,0=7/31+2y2+x+y+∣x−y∣≥712.
It is easy to see that(56)Mix,y,z≥712,ψ∫0Gfx,fy,fzφtdt=3∫01/32tdt=13<2·49144=η∫07/12φtdt≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Case 3.
x,z∈0,1 and y∈1,2. It follows that
(57)M1x,y,z≥1+Ggz,gz,fzGgy,gy,fx21+Ggz,fy,fz=1+Gz,z,0G7/6,7/6,021+Gz,1/6,0=7/31+2z21+1/6+z+z−1/6≥78,M2x,y,z≥max1+Ggx,fx,fxGgy,fz,fz21+Ggx,gz,fx,1+Ggz,fz,fzGgy,fx,fx21+Ggx,gz,fz=max1+Gx,0,0G7/6,0,021+Gx,z,0,1+Gz,0,0G7/6,0,021+Gx,z,0=max7/31+2x21+x+z+∣x−z∣,7/31+2z21+x+z+∣x−z∣≥76.
It is obvious that(58)Mix,y,z≥78,ψ∫0Gfx,fy,fzφtdt=3∫01/32tdt=13<2·4964=η∫07/8φtdt≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Case 4.
y,z∈0,1 and x∈1,2. It follows that
(59)M1x,y,z≥1+Ggy,gy,fyGgx,gx,fz21+Ggy,fx,fy=1+Gy,y,0G7/6,7/6,021+Gy,1/6,0=7/31+2y21+y+1/6+y−1/6≥78,M2x,y,z≥1+Ggz,fz,fzGgx,fy,fy2+Ggy,gz,fz=1+Gz,0,0G7/6,0,02+Gy,z,0=7/31+2z2+y+z+∣y−z∣≥712.
It is apparent that(60)Mix,y,z≥712,ψ∫0Gfx,fy,fzφtdt=3∫01/32tdt=13<2·49144=η∫07/12φtdt≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Case 5.
x∈0,1 and y,z∈1,2. It follows that
(61)M1x,y,z≥1+Ggz,gz,fzGgy,gy,fx21+Ggz,fy,fz=1+G7/6,7/6,1/6G7/6,7/6,021+G7/6,1/6,1/6=7/31+221+2=76,M2x,y,z≥1+Ggy,fy,fyGgz,fx,fx2+Ggx,gy,fy=1+G7/6,1/6,1/6G7/6,0,02+Gx,7/6,1/6=7/31+22+1+∣x−7/6∣+∣x−1/6∣≥2113.
It is distinct that(62)Mix,y,z≥76,ψ∫0Gfx,fy,fzφtdt=3∫01/32tdt=13<2·4936=η∫07/6φtdt≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Case 6.
y∈0,1 and x,z∈1,2. It follows that
(63)M1x,y,z≥1+Ggy,gy,fyGgx,gx,fz21+Ggy,fx,fy=1+Gy,y,0G7/6,7/6,1/621+Gy,1/6,0=21+2y21+y+1/6+y−1/6≥34,M2x,y,z≥1+Ggy,fy,fyGgz,fx,fx2+Ggx,gy,fy=1+Gy,0,0G7/6,1/6,1/62+G7/6,y,0=21+2y2+y+7/6+∣y−7/6∣≥613.
It is clear that(64)Mix,y,z≥613,ψ∫0Gfx,fy,fzφtdt=3∫01/32tdt=13<2·36169=η∫06/13φtdt≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Case 7.
z∈0,1 and x,y∈1,2. It follows that
(65)M1x,y,z≥1+Ggy,gy,fyGgx,gx,fz21+Ggy,fx,fy=1+G7/6,7/6,1/6G7/6,7/6,021+G7/6,1/6,1/6=7/31+221+2=76,M2x,y,z≥1+Ggy,fy,fyGgx,fz,fz21+Ggy,gz,fy=1+G7/6,1/6,1/6G7/6,0,021+G7/6,z,1/6=7/31+221+1+∣z−7/6∣+∣z−1/6∣≥2120.It is easy to obtain that(66)Mix,y,z≥2120,ψ∫0Gfx,fy,fzφtdt=3∫01/32tdt=13<2·441400=η∫021/20φtdt≤η∫0Mix,y,zφtdt=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀i∈1,2.Therefore, (12) and (51) hold. That is, the conditions of Theorems 19 and 20 are fulfilled. It follows from each of Theorems 19 and 20 that f and g have a unique common fixed point in X.However, Theorem4 cannot be applied to testify the existence of fixed points of the mapping f in X. Suppose that there exists ψ,ϕ∈Φ2 satisfying the conditions of Theorem 4. In virtue of (4), we infer that
(67)ψ13=ψGf1,f1,f76≤ψG1,1,76−ϕG1,1,76=ψ13−ϕ13<ψ13,which is a contradiction.Now we claim that Theorem5 is useless in proving the existence of common fixed points of the mappings f and g in X. Suppose that there exist α∈0,1 and φ∈Φ1 satisfying the conditions of Theorem 5. Taking advantage of (5), we receive that
(68)∫01/3φtdt=∫0Gf1,f2,f2φtdt≤α∫0Gg1,g2,g2φtdt=α∫01/3φtdt<∫01/3φtdt,which is impossible.


## 4. Applications

In this section, we study the existence and uniqueness of common solutions for the below functional equations (72) and nonlinear Volterra integral equations (94) by using the results obtained in Section 3.LetU and V denote two Banach spaces; S⊆U and D⊆V signify the state and decision spaces, respectively. BS indicates the Banach space of all bounded functions in S with norm
(69)h=suphx:x∈S,∀h∈BS.DefineG:BS3→ℝ+ by
(70)Gx,y,z=maxx−y,y−z,z−x,∀x,y,z∈BS.It is clear thatBS,G is a complete G-metric space.Consider the functional equations arising in dynamic programming:(71)fx=supy∈Dux,y+Hx,y,fax,y,∀x∈S,(72)gx=supy∈Dvx,y+Lx,y,gbx,y,∀x∈S,whereu,v:S×D→ℝ, a,b:S×D→S, and H,L:S×D×ℝ→ℝ are mappings. Put
(73)fhx=supy∈Dux,y+Hx,y,hax,y,∀x,h∈S×BS,(74)ghx=supy∈Dvx,y+Lx,y,hbx,y,∀x,h∈S×BS.Theorem 23.
Letu,v:S×D→ℝ, a,b:S×D→S, and H,L:S×D×ℝ→ℝ satisfy that
(C1) u,v,H, and L are bounded;
(C2) fgh=gfh for each h∈BS with fh=gh;
(C3) fBS⊆gBS and gBS is complete;
(C4) for each i∈1,2,3,x∈S,y∈D, and h1,h2,h3∈BS(75)ψ∫0∣Hx,y,hiax,y−Hx,y,hi+1ax,y∣φtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt,where φ,ψ,ϕ∈Φ1×Φ5×Φ3,h4=h1 and
(76)M1∗h1,h2,h3=maxGgh1,gh2,gh3,1+Ggh1,gh1,fh1Ggh2,gh2,fh32+Ggh1,fh1,fh2,1+Ggh1,gh1,fh1Ggh3,gh3,fh22+Ggh1,fh1,fh3,1+Ggh2,gh2,fh2Ggh3,gh3,fh12+Ggh2,fh2,fh3,1+Ggh2,gh2,fh2Ggh1,gh1,fh321+Ggh2,fh1,fh2,1+Ggh3,gh3,fh3Ggh2,gh2,fh121+Ggh3,fh2,fh3,1+Ggh3,gh3,fh3Ggh1,gh1,fh221+Ggh3,fh1,fh3,1+Ggh1,gh2,gh3Gfh1,fh2,fh31+Ggh3,fh2,fh2+Ggh1,gh2,gh3,1+Ggh1,gh2,gh3Gfh1,fh2,fh31+Ggh2,fh1,fh1+Ggh1,gh2,gh3,1+Ggh1,gh2,gh3Gfh1,fh2,fh31+Ggh1,fh3,fh3+Ggh1,gh2,gh3.Then, the functional equations (72) have a unique common bounded solution h∗∈BS.Proof.
By virtue of (C1) and (73), we obtain that fh and gh are bounded for each h∈BS, which yields that f and g are self mappings in BS. It follows from φ∈Φ1 that for each ε>0 there exists δ>0 such that
(77)∫Cφtdt<ε,∀C⊂ℝ+withmC≤δ,where m denotes the Lebesgue measure. Put x∈S and h1,h2,h3∈BS. (73) means that there exist y1,y2∈D such that
(78)fh1x<ux,y1+Hx,y1,h1ax,y1+δ,(79)fh2x<ux,y2+Hx,y2,h2ax,y2+δ,(80)fh1x≥ux,y2+Hx,y2,h1ax,y2,(81)fh2x≥ux,y1+Hx,y1,h2ax,y1.
In terms of (78) and (81), we gain that
(82)fh1x−fh2x<Hx,y1,h1ax,y1−Hx,y1,h2ax,y1+δ≤∣Hx,y1,h1ax,y1−Hx,y1,h2ax,y1∣+δ.
On account of (79) and (80), we derive that
(83)fh2x−fh1x<Hx,y2,h2ax,y2−Hx,y2,h1ax,y2+δ≤Hx,y2,h2ax,y2−Hx,y2,h1ax,y2+δ.
In light of (82) and (83), we get that
(84)fh1x−fh2x<maxT1,T2+δ,where
(85)T1=Hx,y1,h1ax,y1−Hx,y1,h2ax,y1,T2=Hx,y2,h2ax,y2−Hx,y2,h1ax,y2.
It follows from (75), (84), and φ,ψ,ϕ∈Φ1×Φ5×Φ3 that
(86)ψ∫0fh1x−fh2xφtdt≤ψ∫0maxT1,T2+δφtdt=maxψ∫0T1+δφtdt,ψ∫0T2+δφtdt=maxψ∫0T1φtdt+∫T1T1+δφtdt,ψ∫0T2φtdt+∫T2T2+δφtdt≤maxψ∫0T1φtdt+ψ∫T1T1+δφtdt,ψ∫0T2φtdt+ψ∫T2T2+δφtdt≤maxψ∫0T1φtdt,ψ∫0T2φtdt+maxψ∫T1T1+δφtdt,ψ∫T2T2+δφtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt+ψε.
Takingε→0+ in the above inequalities and using ψ∈Φ5 and (70), we infer that
(87)ψ∫0∣fh1x−fh2x∣φtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt,ψ∫0∥fh1−fh2∥φtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt.
Similarly, we deduce that(88)ψ∫0∥fh2−fh3∥φtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt,ψ∫0∥fh3−fh1∥φtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt.
It follows that(89)ψ∫0Gfh1,fh2,fh3φtdt=ψ∫0max∥fh1−fh2∥,∥fh2−fh3∥,∥fh3−fh1∥φtdt=maxψ∫0∥fh1−fh2∥φtdt,ψ∫0∥fh2−fh3∥φtdt,ψ∫0∥fh3−fh1∥φtdt≤ψ∫0M1∗φtdt−ϕ∫0M1∗φtdt,∀h1,h2,h3∈BS.Consequently, the conditions of Theorem19 are satisfied. It follows from Theorem 19 that f and g have a unique common fixed point h∗∈BS, that is, the functional equations (72) have a unique common bounded solution h∗∈BS. This completes the proof.As in the proof of Theorem23, we obtain similarly the following result and omit its proof.Theorem 24.
Letu,v:S×D→ℝ, a,b:S×D→S, and H,L:S×D×ℝ→ℝ satisfy (C1)–(C3) and (C5) for each i∈1,2,3,x∈S,y∈D and h1,h2,h3∈BS(90)ψ∫0∣Hx,y,hiax,y−Hx,y,hi+1ax,y∣φtdt≤ψ∫0M2∗φtdt−ϕ∫0M2∗φtdt,where φ,ψ,ϕ∈Φ1×Φ5×Φ3,h4=h1 and
(91)M2∗h1,h2,h3=maxGgh1,gh2,gh3,1+Ggh1,fh1,fh1Ggh3,fh2,fh22+Ggh1,gh2,fh1,1+Ggh2,fh2,fh2Ggh3,fh1,fh12+Ggh1,gh2,fh2,1+Ggh3,fh3,fh3Ggh1,fh2,fh22+Ggh2,gh3,fh3,1+Ggh1,fh1,fh1Ggh2,fh3,fh321+Ggh1,gh3,fh1,1+Ggh2,fh2,fh2Ggh1,fh3,fh321+Ggh2,gh3,fh2,1+Ggh3,fh3,fh3Ggh2,fh1,fh121+Ggh1,gh3,fh3,1+Ggh1,gh2,gh3Gfh1,fh2,fh31+Ggh3,gh3,fh2+Ggh1,gh2,gh3,1+Ggh1,gh2,gh3Gfh1,fh2,fh31+Ggh2,gh2,fh1+Ggh1,gh2,gh3,1+Ggh1,gh2,gh3Gfh1,fh2,fh31+Ggh1,gh1,fh3+Ggh1,gh2,gh3.Then, the functional equations (72) have a unique common bounded solution h∗∈BS.LetC0,T,ℝ denote the Banach space of all continuous functions in 0,T with norm
(92)x=sup∣xt∣:t∈0,T,∀x∈C0,T,R.PutX=C0,T,ℝ and define G:X×X×X→ℝ+ by
(93)Gx,y,z=x−y+y−z+z−x,∀x,y,z∈X.It is obvious thatX,G is a complete G-metric space.Consider the nonlinear Volterra integral equations:(94)xt=p1t+∫0tK1t,s,xsds,∀t∈0,T,yt=p2t+∫0tK2t,s,ysds,∀t∈0,T,where T>0 is a constant, p1,p2:0,T→ℝ and K1,K2:0,T2×ℝ→ℝ are given functions.Put(95)fxt=p1t+∫0tK1t,s,xsds,∀t,x∈0,T×X,(96)gxt=p2t+∫0tK2t,s,xsds,∀t,x∈0,T×X.Theorem 25.
LetK1,K2:0,T2×ℝ→ℝ and p1,p2:0,T→R satisfy that
(d1) K1,K2,p1, and p2 are continuous;
(d2) fgx=gfx for each x∈X with fx=gx;
(d3) fX⊆gX and gX is complete;
(d4) there exists a continuous function Q:0,T×0,T→ℝ+ such that
(97)K1t,s,xs−K1t,s,ys≤Qt,sgxs−gys,∀t,s∈0,T,x,y∈X.
(d5)supt∈0,T∫0tQt,sds≤1/2.Then, the nonlinear Volterra integral equations (94) have a unique common continuous solution in X.Proof.
Defineφ,ψ,ϕ:ℝ+→ℝ+ by
(98)φt=1,ψt=2t,ϕt=t,∀t∈R+.
Clearly,φ,ψ,ϕ∈Φ1×Φ2×Φ3. On account of (d1) and (95), we deduce that for each x∈X, fx and gx are continuous functions in 0,T, that is, f and g are mappings from X into itself. By means of (93), (95), (97), and (d5), we obtain that
(99)ψ∫0Gfx,fy,fzφtdt=2∫0Gfx,fy,fzφtdt=2Gfx,fy,fz=2fx−fy+fy−fz+fz−fx≤2supt∈0,T∫0tK1t,s,xs−K1t,s,ysds+supt∈0,T∫0tK1t,s,ys−K1t,s,zsds+supt∈0,T∫0tK1t,s,zs−K1t,s,xsds≤2supt∈0,T∫0tQt,sgxs−gysds+supt∈0,T∫0tQt,sgys−gzsds+supt∈0,T∫0tQt,sgzs−gxsds≤2supt∈0,T∫0tQt,sdsgx−gy+gy−gz+gz−gx≤Ggx,gy,gz≤Mix,y,z=ψ∫0Mix,y,zφtdt−ϕ∫0Mix,y,zφtdt,∀x,y,z∈X,i∈1,2,where M1x,y,z and M2x,y,z are defined by (13) and (52), respectively. That is, the conditions of Theorems 19 and 20 are fulfilled. Therefore, each of Theorems 19 and 20 guarantees that f and g have a unique common fixed point x∈X, which is a unique common continuous solution of the nonlinear Volterra integral equations (94) in X. This completes the proof.



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*Source: 6619964-2021-01-31.xml*