# Common Fixed Points of(α,η) − (θ,ϝ) Rational Contractions with Applications

Authors: Mujahid Abbas; Hira Iqbal; Manuel De La Sen
Journal: Journal of Mathematics
 (2019)

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## Abstract

We present the notion of set valued(α,η)-(θ,ϝ) rational contraction mappings and then some common fixed point results of such mappings in the setting of metric spaces are established. Some examples are presented to support the concepts introduced and the results proved in this paper. These results unify, extend, and refine various results in the literature. Some fixed point results for both single and multivalued (θ,ϝ) rational contractions are also obtained in the framework of a space endowed with partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.


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## Body

## 1. Introduction and Preliminaries

Let(X,d) be a metric space. A mapping T:X→X is called a contraction if there exists a constant c∈[0,1) such that, for any x,y∈X, we have (1)dTx,Ty≤cdx,y.The widely known Banach contraction theorem [1] states that a contraction mapping on a complete metric space X has a unique fixed point; that is, there exists a point x in X such that x=Tx.In the last few decades, several authors have extended and generalized this principle in various directions.Jleli and Samet [2] presented a new type of contractive mapping, namely θ-contraction mapping and established an interesting fixed point theorem for such mappings in a generalized metric space. The concept of generalized metric spaces was introduced by Branciari [3], where the triangle inequality is replaced by the inequality d(w,z)≤d(w,x)+d(x,y)+d(y,z) for all pairwise distinct points w,x,y,z∈X.Jleli and Samet [2] considered the set Θ of real valued functions θ:(0,∞)→(1,∞) which satisfy the following conditions:θ1
θ is nondecreasing;θ2
for each sequence{cn}⊂(0,∞), limn→∞⁡θ(cn)=1 if and only if limn→∞cn=0+;θ3
there existm∈(0,1) and l∈(0,∞] such that limc→0+θ(c)-1/cm=l.Example 1.
Defineθi:(0,∞)→(1,∞) where i=1,2 by (2)θ1t=etandθ2t=etet.Then θ1,θ2∈Θ.A mappingT:X→X on a metric space (X,d) is called a θ-contraction if for any x,y∈X and θ∈Θ, we have (3)θdTx,Ty≤θdx,ycwhenever d(Tx,Ty)>0 and 0≤c<1.Theorem 2 (see [2]).
Let(X,d) be a complete generalized metric space and T:X→X. If there exist θ∈Θ and 0≤c<1 such that (4)θdTx,Ty≤θdx,ycholds for any x,y∈X whenever d(Tx,Ty)≠0. Then T has a fixed point.Ahmadet al. [4] modified the class Θ of mappings as follows: (5)Ω=θ:0,∞→1,∞satisfyθ1,θ2andθ3′where (θ3′)θ:(0,∞)→(1,∞) is continuous.Example 3.
Defineθi:(0,∞)→(1,∞) for i=1,2,3 by (6)θ1t=et,θ2t=etandθ3t=etet.Then, θ1,θ2,θ3∈Ω.Authors in [4] considered the following result of Jleli and Samet [2] with the function θ∈Ω instead of θ∈Θ:Theorem 4.
Let(X,d) be a complete metric space and T:X→X a θ-contraction, where θ∈Ω. Then T has a unique fixed point x~∈X and for any x0∈X, the sequence {Tnx0} converges to x~.Note that the Banach contraction theorem immediately follows from the above theorem.LetX be a nonempty set endowed with a metric d. Let P(X) be the set of all nonempty subsets of X, K(X) denotes the set of all nonempty compact subsets of X, and CB(X) denotes the set all nonempty closed and bounded subsets. For A,B∈CB(X) and x∈X, define distance of a point x from the set A by (7)dx,A=infx∈A⁡dx,A.A mapping H:CB(X)×CB(X)→R+ defined by (8)HA,B=maxsupa∈A⁡da,B,supb∈B⁡dA,b is called the generalized Pompeiu-Hausdorff distance induced by d.LetT:X→CB(X). A point x∈X is called a fixed point of T if x∈Tx.Nadler [5] obtained the following multivalued version of Banach contraction principle.Theorem 5.
Let(X,d) be a complete metric space. If T:X→CB(X) satisfies (9)HTx,Ty≤cdx,yfor any x,y∈X and c∈(0,1), then T has a fixed point.Afterwards, many researchers have obtained fixed point results for multivalued mappings satisfying certain generalized contractive conditions. Hançeret al. [6] introduced multivalued θ- contraction mappings as follows.Let(X,d) be a metric space, θ∈Θ, and T:X→P(X). Then, T is called a multivalued θ-contraction if, for any x,y∈X, (10)θHTx,Ty≤θdx,ycholds whenever H(Tx,Ty)>0 where 0≤c<1.They established the following fixed point results for multivaluedθ-contraction mappings on complete metric spaces.Theorem 6.
Let(X,d) be a complete metric space and T:X→K(X) a multivalued θ-contraction. Then T has a fixed point.For further results in this direction, we refer to [7–10].Another variation of contraction mapping that can be found in literature isα-contraction mapping.Aslet al. [11] initiated the concept of α∗-admissibility in case of multivalued mappings, whereas Mohammadiet al. [12] presented the notion of α-admissibility in case of multifunctions.Karapinaret al. [13] presented the idea of a triangular α-admissible mapping.Definition 7.
Letα,η:[0,∞)×[0,∞)→R+. A pair (α,η) is called triangular if for any x,y,z∈X, α(x,y)≥η(x,y) and α(y,z)≥η(y,z) imply that α(x,z)≥η(x,z).Recently, Abbaset al.[14] proposed a concept of α -closed mappings for set valued mappings. We present the following generalization of the definition.Definition 8.
LetT,S:X→P(X) and α,η:X×X→R+. We say that a pair (T,S) is triangular (α,η)-closed if the pair (α,η) is triangular and for any x,y∈X with α(x,y)≥η(x,y) we have α(u,v)≥η(u,v) for all u∈Tx and v∈Su.IfS=T then a mapping T which is triangular (α,η) closed is referred to simply as a T triangular (α,η)-closed mapping.Example 9.
LetX=[0,1]. Let T,S:X→P(X) be defined by (11)Tx=1,x.andSx=0,x2Define the mappings α,η:X×X→[0,∞) by (12)αx,y=ex-y,wheneverx≥y0,wheneverx<yand (13)ηx,y=x2forallx∈0,1.It is obvious that the pair (T,S) is triangular (α,η)-closed.The following lemma is crucial in our results.Lemma 10.
LetT,S:X→P(X). Suppose that the pairs (T,S) and (S,T) are a triangular (α,η)-closed. Assume that there exists x0∈X with α(x0,x1)≥η(x0,x1) where x1∈Tx0. Define sequences x2n+1∈Tx2n and x2n+2∈Sx2n+1, then α(xm,xn)≥η(xm,xn) for all m,n∈N with m>n.Proof.
By assumption, there existx0∈X and x1∈Tx0 such that α(x0,x1)≥η(x0,x1). Since (T,S) is (α,η)-closed and α(x0,x1)≥η(x0,x1) we obtain α(x1,x2)≥η(x1,x2) where x1∈Tx0 and x2∈Sx1. As (S,T) is (α,η)-closed, we have x3 in Tx2 such that α(x2,x3)≥η(x2,x3). Continuing this way, we have sequences x2n+1∈Tx2n and x2n+2∈Sx2n+1 with α(xn,xn+1)≥η(xn,xn+1) and α(xn+1,xn+2)≥η(xn+1,xn+2) for all n∈N.
Since the pair(α,η) is triangular we obtain that (14)αxn,xn+2≥ηxn,xn+2.Thus by induction, we have (15)αxn,xm≥ηxn,xm.for any m,n∈N with m>n.Parvanehet al. [15] introduced the concept of α-HΘ-contraction with respect to a family of functions H and obtained some θ-contraction fixed point results in metric and ordered metric spaces.They introduced the following family of functions:Letϝ denote the set of functions F:R4→R+ satisfying condition (A∗):For alls1,s2,s3,s4∈R+ with s1.s2.s3.s4=0 there exists c∈[0,1) such that (16)Fs1,s2,s3,s4=cFollowing are some examples of such functions [15].Example 11.
F ( s 1 , s 2 , s 3 , s 4 ) = L  m i n ( s 1 , s 2 , s 3 , s 4 ) + c, where L∈R+ and c∈[0,1).Example 12.
F ( s 1 , s 2 , s 3 , s 4 ) = c e L  m i n ( s 1 , s 2 , s 3 , s 4 ), where L∈R+ and c∈[0,1).The following definition which is a generalization ofα-continuity [16] is needed in the sequel.Definition 13.
Let(X,d) be a metric space, α,η:X×X→R+ and T,S:X→CB(X). A pair (T,S) is (α,η)-continuous at the point x∈X if, for any sequence {xn} in X, limn→∞d(xn,x)=0 and α(xn,xn+1)≥η(xn,xn+1) for all n∈N implies that limn→∞H(Txn,Sx)=0. We say that pair (T,S) is (α,η)-continuous on (CB(X),H) if the pair (T,S) is (α,η)-continuous on each x∈X.In this paper, we introduce multivalued(α,η)-(θ,ϝ) rational contraction pair of multivalued mappings and prove the existence of common fixed points of the pair in a metric space. We also obtain some fixed point results for both single and multivalued (θ,ϝ) rational contraction mappings in a space endowed with a partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.


## 2. Common Fixed Point Results

Throughout this section we assume that(X,d) is a metric space and θ∈Ω where θ:(0,∞)→(1,∞) satisfies (θ1), (θ2) and (θ′3). Let ϝ be a family of continuous and nondecreasing functions where F:R4→R+ for F∈ϝ.We now present the following definitions:Definition 14.
Letα,η:X×X→R+, θ∈Ω, T,S:X→P(X), and F:R4→R+.(1)
A pair(T,S) is called a multivalued (α,η)-(θ,ϝ) rational contraction pair if, for any x,y∈X with α(x,y)≥η(x,y) and H(Tx,Sy)>0, the following condition holds: (17)θHTx,Sy≤θMT,Sx,y)FNT,Sx,ywhere(18)MT,Sx,y=maxdx,y,dx,Tx,dy,Sy,12dx,Sy,dy,Sy1+dx,Tx1+dx,yand(19)NT,Sx,y=dx,Tx,dy,Sy,dx,Sy,dy,Tx.(2)
A pair(S,T) is called a multivalued α,η-θ,ϝ rational contraction if, for any x,y∈X with α(x,y)≥η(x,y) and H(Sx,Ty)>0, the following condition holds: (20)θHSx,Ty≤θMS,Tx,y)FNS,Tx,ywhere(21)MS,Tx,y=maxdx,y,dx,Sx,dy,Ty,12dx,Ty,dy,Ty1+dx,Sx1+dx,yand(22)NS,Tx,y=dx,Sx,dy,Ty,dx,Ty,dy,Sx.(3)
A mappingT:X→P(X) is called a multivalued (α,η)-(θ,ϝ) rational contraction if, for any x,y∈X with α(x,y)≥η(x,y) and H(Tx,Ty)>0, the following condition holds: (23)αx,yθHTx,Ty≤θMx,yFNx,ywhere(24)Mx,y=maxdx,y,dx,Tx,dy,Ty,12dx,Ty,dy,Ty1+dx,Tx1+dx,yand(25)Nx,y=dx,Tx,dy,Ty,dx,Ty,dy,Tx.Remark 15.
(1)
Ifα,η:X×X→R+ are defined as α=η(x,y)=1 for all x,y∈X in Definition 14, then the pairs of mappings (T,S) and (S,T) are multivalued (θ,ϝ) generalized rational contractions. (2)
Ifα,η:X×X→R+ are defined as α(x,y)=η(x,y)=1 for all x,y∈X in Definition 14, then T is a multivalued (θ,ϝ) generalized rational contraction mapping.Theorem 16.
LetT,S:X→CB(X). Suppose that the pairs (T,S) and (S,T) are multivalued (α,η)-(θ,ϝ) rational contractions such that(C1)
the pairs(T,S) and (S,T) are triangular (α,η)-closed;(C2)
there existx0∈X and x1∈Tx0 with α(x0,x1)≥η(x0,x1);(C3)
(T,S) and (S,T) are (α,η)-continuous.
Then there existsx~∈X such that x~∈Tx~∩Sx~.Proof.
IfMT,S(x,y)=MS,T(x,y)=0, for some x,y∈X, then we have our conclusion. Assume that MT,S, MS,T>0 for all x,y∈X. By assumption there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥η(x0,x1). If x0=x1 or d(x1,S1)=0 then the result follows. Assume that x0≠x1 and x1∉Sx1 then (26)0<dx1,Sx1≤HTx0,Sx1.This implies (27)1<θdx1,Sx1≤θHTx0,Sx1.Since α(x0,x1)≥η(x0,x1) and (T,S) is multivalued (α,η)-(θ,ϝ) rational contraction, we obtain that(28)1<θdx1,Sx1≤HTx0,Sx1≤θMT,Sx0,x1FNT,Sx0,x1where(29)NT,Sx0,x1=dx0,Tx0,dx1,Sx1,dx0,Sx1,dx1,Tx0≤dx0,x1,dx1,Sx1,dx0,Sx1,dx1,x1=dx0,x1,dx1,Sx1,dx0,Sx1,0.Since d(x0,x1).d(x1,Sx1).d(x0,Sx1).0=0, by (A∗) there exists c∈[0,1) such that (30)Fdx0,x1,dx1,Sx1,dx0,Sx1,0=c.Therefore, from (29) and using the fact that F is nondecreasing we obtain(31)FNT,Sx0,x1≤Fdx0,x1,dx1,Sx1,dx0,Sx1,0=c.Also, (32)MT,Sx0,x1=max⁡dx0,x1,dx0,Tx0,dx1,Sx1,12dx0,Sx1,dx1,Sx11+dx0,Tx01+dx0,x1≤maxdx0,x1,dx0,x1,dx1,Sx1,12dx0,Sx1,dx1,Sx11+dx0,x11+dx0,x1=maxdx0,x1,dx0,x1,dx1,Sx1,12dx0,Sx1,dx1,Sx11+dx0,x11+dx0,x1, since (33)12dx0,Sx1≤12dx0,x1+12dx1,Sx1≤maxdx0,x1,dx1,Sx1. Thus,(34)MT,Sx0,x1≤maxdx0,x1,dx1,Sx1. Replacing (31) and (34) in (28) we get (35)θdx1,Sx1≤θmaxdx0,x1,dx1,Sx1c. If max{d(x0,x1),d(x1,Sx1))}=d(x1,Sx1), then (36)θdx1,Sx1≤dx1,Sx1)c, a contradiction. Hence max{d(x0,x1),d(x1,Sx1))}=d(x0,x1). Therefore,(37)1≤θdx1,Sx1≤θdx0,x1c.By (θ′3) we have (38)θdx1,Sx1=infy∈Sx1⁡θdx1,y. Thus, there exists x2∈Sx1 such that (39)θdx1,Sx1=θdx1,x2. Then from (37) we have(40)θdx1,x2≤θdx0,x1c.
Sincex1∈Tx0, x2∈Sx1, α(x0,x1)≥η(x0,x1), and (T,S) is (α,η)-closed, we have α(x1,x2)≥η(x1,x2). If x1=x2 or d(x2,Tx2)=0 then the result follows. Suppose that x1≠x2 and x2∉Tx2. Thus (41)1<θdx2,Tx2≤θHSx1,Tx2. As α(x1,x2)≥η(x1,x2), we have(42)1<θdx2,Tx2≤θHSx1,Tx2≤θMS,Tx1,x2FNT,Sx1,x2 where(43)NT,Sx1,x2=dx1,Tx1,dx2,Sx2,dx1,Sx2,dx2,Tx1≤dx1,x2,dx2,Sx2,dx1,Sx2,dx2,x2=dx1,x2,dx2,Sx2,dx1,Sx2,0.Since d(x1,x2).d(x2,Sx2).d(x1,Sx2).0=0, by (A∗) there exists c∈[0,1) such that (44)Fdx1,x2,dx2,Sx2,dx1,Sx2,0=c Therefore, from (43) and using the fact that F is nondecreasing we have(45)FNT,Sx0,x1≤Fdx1,x2,dx2,Sx2,dx1,Sx2,0=c.Further, (46)MS,Tx1,x2=maxdx1,x2,dx1,Sx1,dx2,Tx2,12dx1,Tx2,dx2,Tx21+dx1,Sx11+dx1,x2≤maxdx1,x2,dx1,x2,dx2,Tx2,12dx1,Tx2,dx2,Tx21+dx1,x21+dx1,x2=maxdx1,x2,dx1,x2,dx2,Tx2,12dx1,Tx2,dx2,Tx21+dx1,x21+dx1,x2. Since (47)12dx1,Tx2≤12dx1,x2+12dx2,Tx2≤maxdx1,x2,dx2,Tx2, we get(48)MS,Tx1,x2≤maxdx1,x2,dx2,Tx2.Replacing (45) and (48) in (42) we have (49)θdx2,Tx2≤θmaxdx1,x2,dx2,Tx2c. If max{d(x1,x2),d(x2,Tx2))}=d(x2,Tx2), then (50)θdx2,Tx2≤θdx2,Tx2c, a contradiction. Hence max{d(x1,x2),d(x2,Tx2))}=d(x1,x2). Therefore,(51)1≤dx2,Tx2≤θdx1,x2)c.By (θ′3) we have (52)θdx2,Tx2=infy∈Tx2⁡θdx2,y. Thus, there exists x3∈Tx2 such that (53)θdx2,Tx2=dx2,x3. Therefore, from (51) we have (54)θdx2,x3≤θdx1,x2c. Furthermore, from (40) we have (55)θdx2,x3≤θdx1,x2c≤θdx0,x1c2. Proceeding in the same manner, we obtain a sequence {xi} in X such that x2i≠x2i+1, x2i∉Tx2i, x2i+1∉Sx2i+1, x2i+1∈Tx2i, and x2i+2∈Sx2i+1 with α(x2i,x2i+1)≥η(x2i,x2i+1) and it satisfies(56)θdx2i+1,x2i+2≤θdx0,x1c2i+1,for each i∈N0. As x2i+1∈Tx2i, x2i+2∈Sx2i+1, and α(x2i,x2i+1)≥η(x2i,x2i+1), we have α(x2i+1,x2i+2)≥η(x2i+1,x2i+2). Then, (57)0<dx2i+2,Tx2i+2≤HSx2i+1,Tx2i+2. Therefore,(58)θdx2i+2,Tx2i+2≤θHSx2i+1,Tx2i+2≤αx2i+1,x2i+2θHSx2i+1,Tx2i+2≤θMS,Tx2i+1,x2i+2FNS,Tx2i+1,x2i+2. Thus,(59)NS,Tx2i+1,x2i+2dx2i+1,Sx2i+1,dx2i+2,Tx2i+2,dx2i+1,Tx2i+2,dx2i+2,Sx2i+1≤dx2i+1,x2i+2,dx2i+2,Tx2i+2,dx2i+1,Tx2i+2,dx2i+2,x2i+2=dx2i+1,x2i+2,dx2i+2,Tx2i+2,dx2i+1,Tx2i+2,0. Since d(x2i+1,x2i+2).d(x2i+2,Tx2i+2).d(x2i+1,Tx2i+2).0=0 by (A∗) there exists c∈[0,1) such that F(d(x2i+1,x2i+2),d(x2i+2,Tx2i+2),d(x2i+1,Tx2i+2),0)=c. Thus, from definition of F and (59) we obtain(60)FNS,Tx2i+1,x2i+2≤c.Also, (61)12dx2i+1,Tx2i+2≤12dx2i+1,x2i+2+12dx2i+2,Tx2i+2≤maxdx2i+1,x2i+2,dx2i+2,Tx2i+2. Then(62)MS,Tx2i+1,x2i+2=maxdx2i+1,x2i+2,dx2i+1,Sx2i+1,dx2i+2,Tx2i+2,12dx2i+1,Tx2i+2dx2i+2,Tx2i+21+dx2i+1,Sx2i+11+dx2i+1,x2i+2≤maxdx2i+1,x2i+2,dx2i+1,x2i+2,dx2i+2,Tx2i+212dx2i+1,Tx2i+2,dx2i+2,Tx2i+21+dx2i+1,x2i+21+dx2i+1,x2i+2≤maxdx2i+1,x2i+2,dx2i+2,Tx2i+2. Therefore, replacing (60) and (62) in (58) we obtain (63)θdx2i+2,Tx2i+2≤θmax⁡dx2i+1,x2i+2,dx2i+2,Tx2i+2)c. If max{d(x2i+1,x2i+2),d(x2i+2,Tx2i+2)}=d(x2i+2,Tx2i+2) then (64)θdx2i+2,Tx2i+2≤θdx2i+2,Tx2i+2c a contradiction. Further, (65)θdx2i+2,Tx2i+2≤θ(dx2i+1,x2i+2c. Again, using (θ′3) we have (66)θdx2i+2,Tx2i+2=infy∈Tx2i+2⁡θdx2i+2,y. Therefore, there exists x2i+3∈Tx2i+2 such that (67)θdx2i+2,Tx2i+2=θdx2i+2,x2i+3. Thus, we have (68)θdx2i+2,x2i+3≤θdx2i+1,x2i+2c. From (56), we have (69)θdx2i+2,x2i+3≤θ(dx2i+1,x2i+2c≤θ(dx0,x1c2i+2. Hence, we have a sequence {xn} in X and α(xn,xn+1)≥η(xn,xn+1) such that (70)θdxn+1,xn+2≤θdx0,x1cn+1, for all n∈N0. On taking limit as n→∞, we obtain (71)1<limn→∞⁡θdxn+1,xn+2≤1 which implies that limn→∞θ(d(xn+1,xn+2))=1. Then by (θ2) we obtain(72)limn→∞⁡dxn+1,xn+2=0.Now, we show that {xn} is a Cauchy sequence. If {xn} is not Cauchy, then there exist ϵ>0 and m(i)>n(i)>i for all i∈N0 such that (73)dxni,xmi≥ϵanddxni,xmi-1<ϵ. Thus, (74)ϵ≤dxni,xmi≤dxni,xmk-1+dxmi-1,xmi<ϵ+dxmi-1,xmi. Therefore, from the above inequality and (72), we obtain(75)limk→∞⁡dxni,xmi=ϵ.Also,(76)dxni,xmi≤dxni,xni+1+dxni+1,xmiand(77)dxni+1,xmi≤dxni+1,xni+dxni,xmi.On taking limit as i→∞ in (76) and using (75) (77), we have (78)ϵ≤limi→∞⁡dxni+1,xmi≤ϵ. Therefore,(79)limi→∞⁡dxni+1,xmi=ϵ.Similarly, we obtain that(80)limi→∞⁡dxni+1,xmi+1=limk→∞⁡dxni,xmk+1=limi→∞⁡dxni+1,Txmi=ϵ.By Lemma 10, we have (81)αxni,xmi≥ηxni,xmi. If d(xn(i)+1,Txm(i))=0, we have (82)dxni,xmi≤dxni,xni+1+dxni+1,Txmi+dTxmi,xmi≤dxni,xni+1+dxni+1,Txmi+dxmi+1,xmi≤dxni,xni+1+dxmi+1,xmi. Taking limit i→∞ we have (83)limi→∞⁡dxni,xmi=0 a contradiction to our assumption. Thus, assume that d(xn(i)+1,Txm(i))>0. Therefore, we have(84)θdxni+1,Txmi≤θdSxni,Txmi≤θMS,Txni,xmiFNS,Txni,xmiwhere(85)NS,Txni,xmi=dxni,Sxni,dxmi,Txmi,dxni,Txmi,dxmi,Sxni≤dxni,xni+1,dxmi,xmi+1,dxni,xmi+1,dxmi,xni+1. Taking limit i→∞ in (85),(86)limi→∞⁡NS,Txni,xmi≤limi→∞⁡dxni,xni+1,limi→∞⁡dxmi,xmk+1,limi→∞⁡dxni,xmi+1,limi→∞dxmi,xni+1=0,0,0,0. Now, by (A∗) there exists c∈[0,1) such that F(0,0,0,0)=c. Thus, using the continuity of F and (86), (87)Flimi→∞⁡NS,Txni,xmi≤c. Moreover,(88)MS,Txni,xmi=maxdxni,xmi,dxni,Sxni,dxmi,Txmi,12dxni,Txmi,dxmi,Txmi1+dxni,Sxni1+dxni,xmi≤maxdxni,xmi,dxni,xni+1,dxmi,xmi+1,12dxni,xmk+1,dxmi,xmi+1)(1+dxni,xni+11+dxni,xmi. Taking limits as i→∞ and using (72) and (80) we obtain that(89)limi→∞⁡MS,Txni,xmi≤ϵ.Thus, using (84) and the continuity of θ we have (90)θlimi→∞⁡dxnk+1,Txmi)≤θlimi→∞⁡MS,Txni,xmi)Flimi→∞NS,Txni,xmi. From (89), we obtain (91)θϵ≤θϵc<θϵ a contradiction. Hence, {xn} is a Cauchy. Since X is a complete metric space, there exists x~∈X such that limn→∞d(xn,x~)=0. As the pair (T,S) is (α,η)-continuous, we have limn→∞H(Tx2n,Sx~)=0.
Note that(92)dx~,Sx~≤dx~,x2n+1+dx2n+1,Sx~≤dx~,x2n+1+HTx2n,Sx~. On taking limit as n→∞ on both sides of the above inequality, we obtain that d(x~,Sx~)=0 and hence x~∈Sx~. As (S,T) is (α,η)-continuous, we have limn→∞H(Sx2n+1,Tx~)=0. Also, (93)dx~,Tx~≤dx~,x2n+2+dx2n+2,Tx~≤dx~,x2n+2+HSx2n+1,Tx~. On taking limit as n→∞ on both sides of the above inequality, we obtain that d(x~,Tx~)=0 and hence x~∈Tx~. Thus there exists x~ such that x~∈Tx~∩Sx~.We may omit the(α,η)-continuity condition in the above theorem by condition (H).If{xn} is a sequence in X with α(xn,xn+1)≥η(xn,xn+1) for all n∈N and limn→∞d(xn,x)=0 for some x∈X, then α(xn,x)≥η(xn,x) for all n∈N.Theorem 17.
LetT,S:X→CB(X). Suppose that the pairs (T,S) and (S,T) are multivalued (α,η)-(θ,ϝ) rational contractions such that(C1)
(T,S) and (S,T) are triangular (α,η)-closed mapping;(C2)
there existx0∈X and x1∈Tx0 with α(x0,x1)≥η(x0,x1);(C3)
the pair(α,η) satisfies condition (H).
Then there existsx~∈X such that x~∈Tx~∩Sx~.Proof.
As in Theorem16, we obtain a Cauchy sequence {xn} in the complete metric space X with limn→∞d(x2n,x~)=0 where x~∈X and α(xn,xn+1)≥η(xn,xn+1). As the pair (α,η) satisfies condition (H), α(xn,x~)≥η(xn,x~) for all n∈N0. We need to show that x~ is the common fixed point. Suppose on the contrary that x~∉Sx~.
From condition (H), we obtainα(x2n,x~)≥η(x2n,x~) as α(x2n,x2n+1)≥η(x2n,x2n+1) and limn→∞d(x2n,x~)=0. If d(x2n+1,Sx~)=0 then (94)dx~,Sx~≤dx~,x2n+1+dx2n+1,Sx~≤dx~,x2n+1 Taking limit k→∞ in the above equation we obtain (95)dx~,Sx~=0 a contradiction to our assumption. Thus, we assume d(x2n+1,Sx~)>0. Then, (96)0<dx2n+1,Sx~≤HT2n,Sx~. Further,(97)θdx2n+1,Sx~≤θHT2n,Sx~≤αx2n,x~θHT2n,Sx~≤θMT,Sx2n,x~F(NT,Sx2n,x~ where (98)NT,Sx2n,x~=dx2n,Tx2n,dx~,Sx~,dx~,Tx2n,dx2n,Sx~≤dx2n,x2n+1,dx~,Sx~,dx~,x2n+1,dx2n,Sx~. Taking n→∞ in the above inequality we have (99)limn→∞⁡NT,Sx2n,x~≤limn→∞⁡dx2n,x2n+1,limn→∞⁡dx~,Sx~,limn→∞⁡dx~,x2n+1),limn→∞dx2n,Sx~=0,dx~,Sx~,0,limn→∞⁡dx2n,Sx~. Thus, as previously shown, by (A∗) and continuity of F we have (100)limn→∞⁡FNT,Sx2n,x~≤c. Also, (101)MT,Sx2n,x~=maxdx2n,x~,dx2n,Tx2n,dx~,Sx~,12dx2n,Sx~,dx~,Sx~1+dx2n,Tx2n1+dx2n,x~≤maxdx2n,x~,dx2n,x2n+1,dx~,Sx~,12dx2n,Sx~,dx~,Sx~1+dx2n,x2n+11+dx2n,x~. Hence (102)limn→∞MT,Sx2n,x~≤dx~,Sx~. Moreover, from (97) we have (103)θdx2n+1,Sx~≤θMT,Sx2n,x~FNT,Sx2n,x~ On taking limit as n→∞ in the above inequality and using (θ3′), we obtain that (104)θlimn→∞dx2n+1,Sx~≤θlimn→∞⁡MT,Sx2n,x~limn→∞⁡FNT,S(x2n,x~≤θdx~,Sx~c<θdx~,Sx~. It follows from (θ1) that (105)limn→∞⁡dx2n+1,Sx~<dx~,Sx~. That is, (106)dx~,Sx~<dx~,Sx~, a contradiction. Hence, d(x~,Sx~)=0. Similarly, we can show d(x~,Tx~)=0. Hence, x~∈Sx~∩Tx~Example 18.
LetX=R+ and (107)dx,y=maxx,y,x≠y0,x=y.Define the mappings S,T:X→CB(X) by (108)Sx=x-1,x+1,x>10,x16,0≤x≤1.and (109)Tx=1,x+1,x>10,x8,0≤x≤1.Define α,η:X×X and H:R4→R+ by (110)αx,y=1,0≤x,y≤10,eitherxory∉0,1,(111)ηx,y=x2forallx∈X and (112)Fs1,s2,s3,s4=18. It is evident that both the pairs (T,S) and (S,T) are triangular α-η-closed. Now, we show that the pair (T,S) is a (α,η)-(θ,ϝ) rational contraction for θ(t)=et. That is, we need to show that (113)HTx,SyMT,Sx,y≤FNT,Sx,y for all x,y∈X.
Letx,y∈[0,1]. Without any loss of generality, we may assume that y≤x. Thus, (114)HTx,SyMT,Sx,y=x/8x=x/8x=18. Thus, the pair T,S is a α,η-θ,ϝ. Similarly, S,T is α,η-θ,ϝ rational for F as defined above. If x0=1, T1=0,1/8 then α1,0=1=η1,0. For any sequence αxn,xn+1≥ηxn,xn+1 for all n∈N, we have {xn}∈[0,1] and {xn} converges to some x∈[0,1]. Thus α(xn,x)≥η(xn,x). All the conditions of Theorem 17 are satisfied and x~=0 is the common fixed point of S and T.Corollary 19.
LetT,S:X→CB(X). Suppose that the pairs (T,S) and (S,T) are continuous multivalued (θ,ϝ) rational contractions, then there exists x~∈X such that x~∈Tx~∩Sx~.Proof.
Defineα(x,y)=η(x,y)=1 for all x,y∈X. Then the result follows from Theorem 16.Theorem 20.
LetT:X→CB(X) be a multivalued (α,η)-(θ,ϝ) rational contraction mapping such that(C1)
T is a triangular (α,η)-closed mapping;(C2)
there existx0∈X and x1∈Tx0 with α(x0,x1)≥η(x0,x1);(C3)
T is (α,η)-continuous.
Then there existsx~∈X such that x~∈Tx~.Proof.
By choosingT=S in Theorem 16 the result follows.Theorem 21.
LetT:X→CB(X) be a multivalued (α,η)-(θ,ϝ) rational contraction mapping such that(C1)
T is a triangular (α,η)-closed mapping;(C2)
there existx0∈X and x1∈Tx0 with α(x0,x1)≥η(x0,x1);(C3)
the pair(α,η) satisfies condition (H).
Then there existsx~∈X such that x~∈Tx~.Proof.
By choosingT=S in Theorem 17 the result follows.Theorem 22.
LetT:X→CB(X) be a multivalued mapping. Suppose that T satisfies the following conditions:(C1)
For anyx,y∈X such that α(x,y)≥η(x,y) and 0≤c<1, we have (115)HTx,Ty≤cMx,ywhere(116)Mx,y=maxdx,y,dx,Tx,dy,Ty,12dx,Ty,dy,Ty1+dx,Tx1+dx,y;(C2)
T is a triangular (α,η)-closed mapping;(C3)
there existx0∈X and x1∈Tx0 with α(x0,x1)≥η(x0,x1);(C4)
T is (α,η)-continuous or (α,η) satisfies condition (H).
Then there existsx~∈X such that x~∈Tx~.Proof.
Takingθ(t)=et and F(s1,s2,s3,s4)=c where c∈[0,1) in Theorems 20 and 21, the result follows.Example 23.
LetX=0,∞ and (117)dx,y=x+y,x≠y0,x=y Define the mapping T:X→CB(X) by (118)Tx=x6,x9;0≤x≤12xx+1;x>1 Define α,η:X×X→R+ as(119)αx,y=1;0≤x,y≤118;otherwise and (120)ηx,y=14forallx,y∈X.
Clearly,T is a triangular (α,η)-closed mapping. If x0=1 and x1=1/6, then α(x0,x1)=1>1/4=η(x,y). Note that (C(3)) is also satisfied. Let x,y∈X, then α(x,y)≥1 if 0≤x,y≤1. Assume that x≠y, then (121)eHTx,Ty=emaxδTx,Ty,δTy,Tx=emaxx/6+y/9,x/9+y/6≤ex+y/4≤eMx,y/4=eMx,y)1/2.Therefore, T is a multivalued (α,η)-(θ,ϝ) rational contraction with θ(t)=et and F(s1,s2,s3,s4)=1/2. Thus, all the conditions of Theorem 21 are satisfied and x~=0 is a fixed point of T.Corollary 24.
LetT,S:X→CB(X). Suppose that, for any x,y∈X such that α(x,y)≥η(x,y), and 0≤c<1, the pairs (T,S) and (S,T) satisfy (122)θHTx,Sy≤θdx,yc whenever (H(Tx,Sy))>0 and (123)θHSx,Ty≤θdx,ycwhenever (H(Sx,Ty))>0. If the following conditions also hold:(C1)
(T,S) and (S,T) are triangular (α,η)-closed mappings;(C2)
there existx0∈X and x1∈Tx0 with α(x0,x1)≥η(x0,x1);(C3)
(T,S) and (S,T) are (α,η)-continuous.
Then there existsx~∈X such that x~∈Tx~∩Sx~.Now, we apply the results for the existence of common fixed points of single valued mappings on a complete metric space.Definition 25.
Letf,g:X→X be two mappings on a nonempty set X and α,η:X×X→R+. A pair (f,g) is called (α,η) -admissible if for any x,y∈X, with α(x,y)≥η(x,y), we have α(fx,gy)≥η(fx,gy).Denote the set of fixed points off and g by F(f) and F(g), respectively.Theorem 26.
Letf,g:X→X. Suppose that the pairs (f,g) and (g,f) are (α,η)-(θ,ϝ) rational contractions such that(C1)
the pairs(f,g) and (g,f) are triangular (α,η)-admissible mappings;(C2)
there existsx0∈X such that α(x0,fx0)≥η(x0,fx0);(C3)
(f,g) and (g,f) are (α,η)-continuous.
ThenF(f)∩F(g)≠ϕ.Proof.
DefineT,S:X→CB(X) as Tx={fx} and Sx={gx}. Then Theorem 16 implies the result.Theorem 27.
Letf,g:X→X. Suppose that the pairs (f,g) and (g,f) are (α,η)-(θ,ϝ) rational contractions such that(C1)
(f,g) and (g,f) are triangular (α,η)-admissible mapping;(C2)
there existsx0∈X such that α(x0,fx0)≥η(x0,fx0);(C3)
the pair(α,η) satisfies condition (H).
ThenF(f)∩F(g)≠ϕ.Proof.
DefineT,S:X→CB(X) as Tx={fx} and Sx={gx}. Then Theorem 17 implies the result.Example 28.
LetX=[0,∞) and (124)dx,y=x+y,x≠y0,x=yDefine f,g:X→X by (125)fx=x10,0≤x≤11-x2+110,x>1andgx=x10,0≤x≤12x-1210,x>1 Define α,η:X×X→[0,∞) by (126)αx,y=1,0≤x,y≤1110,x>1 and (127)ηx,y=1forallx,y∈X. Clearly, (f,g) and (g,f) are triangular (α,η)-admissible mappings.
DefineF:R4→R+ as (128)Fs1,s2,s3,s4=12. Note that pairs (f,g) and (g,f) are (α,η)-continuous. Also, (f,g) and (g,f) are (α,η)-(θ,ϝ) rational contractions for θ(t)=t. If x0=0 and fx0=0, then α(0,0)=1=η(0,0). Thus all the conditions of Theorem 26 are satisfied. Thus, f and g have a common fixed point x~=0 in X.Corollary 29.
Letf:X→X be an (α,η)-continuous (α,η)-(θ,ϝ) rational contraction. Then f has a fixed point in X if there exists x0∈X such that α(x0,fx0)≥η(x0,fx0) and f is triangular (α,η)-admissible. Furthermore, the fixed point is unique if α(x,y)≥η(x,x).Corollary 30.
Letf:X→X be a triangular (α,η)-admissible (α,η)-(θ,ϝ) rational contraction. Then f has a fixed point in X provided that there exists x0∈X such that α(x0,fx0)≥η(x0,fx0) and (α,η) satisfies condition (H). Furthermore, the fixed point is unique if α(x,y)≥η(x,x).Example 31.
LetX=0,1/4,1/2. Note that(129)dx,y=x+y2,x≠y0,x=y defines the metric on X. Define T:X→X by (130)Tx=x2,x=0,120,x=14.Define α,η:X×X→R+ and (131)αx,y=coshx+y4,ifx,y∈0,0,14,14,12,12,0,14,0,1214,otherwise, and (132)ηx,y=1forallx,y∈X. Clearly, T is (α,η)-continuous. Define F:R4→R+ as (133)Fs1,s2,s3,s4=910e4min⁡s1,s2,s3,s4. Let θ(t)=1+t. We only consider the case where (x,y)=(0,1/2); all other cases are trivial. Note that (134)θdTx,Ty=1+dTx,Ty=1+18≤1+389/10≤1+Mx,y9/10≤θMx,y)HNx,y.
Thus,T is an (α,η)-(θ,ϝ) rational contraction. Also, T is triangular (α,η)-admissible. Let x0=0, then α(x0,Tx0)≥1=η(x0,Tx0). All the conditions of Corollary 29 are satisfied and x~=0 is a fixed point of T.


## 3. Application to Nonlinear Elastic Beam Equations

We study the existence of solutions of fourth-order two-point boundary value problem given by(135)μ4t=ft,μt,t∈0,1,μ0=μ1=μ′0=μ′1=0,which represents the bending of an elastic beam clamped at both ends. The boundary value problem in (135) can be written as [17](136)μt=∫01Gt,sfs,μsdsfort∈0,1,where the Green function associated with the given boundary value problem is given by (137)Gt,s=16t21-s2s-t+21-ts,0≤t≤s≤1s21-t2t-s+21-st,0≤s≤t≤1 where supt∈I∫01G(t,s)=1/384 (see [18]).LetX=(C[0,1],[0,∞)) be the space of all continuous functions defined on [0,1]. The metric on X is given by(138)dx,y=maxt∈0,1⁡xt-ytfor all t∈[0,1]. Note that the space X=(C([0,1]),d) is complete metric space.Theorem 32.
Suppose that the following hypotheses are satisfied:(1)
f:[0,1]×[0,∞)→[0,∞) is continuous;(2)
f(s,.):[0,∞)→[0,∞) is nondecreasing for each s∈[0,1];(3)
forμ(s)≤ν(s) for μ,ν∈X, we have (139)fs,μs-fs,νs≤Mμs,νs,for anys∈[0,1], where (140)Mμs,νs=maxdμs,νs,dμs,fs,μs,dνs,fs,νs,12dμs,fs,νs,dνs,fs,νs1+dμs,fs,μs1+dμs,νs;(4)
there existsμ0∈X such that, for all t∈[0,1], (141)μ0t≤∫01Gt,sfs,μ0sds.
Then problem (135) has a solution in X.Proof.
Letμ∈X. Define the operator T:X→X by(142)Tμt=∫01Gt,sfs,μsdsfort∈0,1.Clearly, T is continuous. Define α,η:X×X→[0,∞) as (143)αμ,ν=1,μt≤νt,t∈0,10,otherwise. and (144)ημ,ν=1forallμ,ν∈X. Clearly, (α,η) is triangular. Also, since f(s,.) is nondecreasing, then for any μ,ν∈X such that μ(t)≤ν(t) for all t∈[0,1] we obtain (145)Tμt=∫01Gt,sfs,μsds≤∫01Gt,sfs,νsds=Tνt. Hence, α(Tμ,Tν)≥1. Since α(μ,ν)≥1, T is (α,η)-admissible. Now, for all μ,ν∈X such that μ(s)≤ν(s) for all s∈[0,1] we have (146)Tμt-Tνt=∫01Gt,sfs,μsds-∫01Gt,sfs,νsds≤∫01Gt,sfs,μs-fs,νsds≤∫01Gt,sMμs,νsds=Mμs,νs∫01Gt,sds where t∈[0,1]. Using the fact that supt∈[0,1]∫01G(t,s)ds=1/384 we have (147)dTμ,Tν≤1384Mμ,ν. Now, (148)dTμ,TνedTμ,Tν≤1384Mμ,νe1/384Mμ,ν≤1384Mμ,νeMμ,ν
Now, passing through exponential we obtain(149)edTμ,TνedTμ,Tν≤e1/384Mμ,νeMμ,ν≤eMμ,νeMμ,ν1/384. Thus, T satisfies (150)θdTμ,Tν≤θ)Mμ,ν)FNμ,ν with θ(t)=etet and F(N(μ,ν))=1/384<1. Since all the conditions of Corollary 29 are satisfied, then problem (135) has a solution in X.


## 4. An Application to First-Order Periodic Problem

In this section, we establish the necessary conditions for existence of a fixed point of a mapping in the setting of a partially ordered metric space. Throughout this section, we assume that(X,d,≼) is a partially ordered metric space.Definition 33.
A sequence{xn}⊂X is called ≼-preserving if xn≼xn+1 for all n∈N0.Definition 34.
A mappingT:X→P(X) is called ≼-closed if, for any x,y∈X with x≼u, u≼v for all u∈Tx and v∈Ty.Definition 35.
A mappingT:X→CB(X) is called ≼-continuous at a point x∈X if, for any sequence {xn} in X, xn≼xn+1 and limn→∞⁡d(xn,x)=0 implies that limn→∞⁡H(Txn,Tx)=0 for all n∈N0. We say that the mapping T is ≼-continuous on X if T≼ -continuous at every x∈X.Corollary 36.
LetT:X→CB(X) be ≼-(θ,ϝ) rational contraction such that T is a ≼-closed mapping and T is ≼-continuous. If there exists x0 such that x0≼Tx0, then there exists x~∈X such that x~∈Tx~.Proof.
Defineα,η:X×X→R+ by α(x,y)=1 whenever x≼y and α(x,y)=0 whenever x⋠y and η(x,y)=1 whenever x≼y. Thus, the result follows from Theorem 21.Example 37.
LetX={0,1,2}. Define a metric d on X by (151)d0,1=d1,0=7,d0,2=d2,0=3,d1,2=d2,1=4Define x≼y by (152)≼≔0,0,1,1,2,2,0,1,0,2. Note that X is a partially ordered metric space. Define the mapping T:X→CB(X) by T0=T2={0} and T1={2}. It can easily be shown that T is ≼-closed, ≼-continuous, and ≼-(θ,ϝ) rational contraction for θ(t)=et and c=9/10. If x0=0 and x1=1∈Tx0, then we have x0≼x1. Note that all the conditions of Corollary 36 are satisfied and {0,2} is the set of fixed points of T.Corollary 38.
Supposef:X→X is a ≼-(θ,ϝ) rational contraction. If f is ≼-closed and ≼-continuous and there exists x0∈X such that x0≼fx0, then f has a fixed point in X.Corollary 39.
Supposef:X→X is a ≼-closed and ≼-continuous mapping that satisfies (153)θdTx,Ty≤θdx,yHNx,y for any x,y∈X with x≼y. Then f has a fixed point in X.We now apply the Corollary38 in proving the existence of solution of the first-order periodic problem.LetX=(C[0,W],R) be the space of all continuous functions defined on [0,W]. The metric on X is given by(154)dx,y=maxt∈0,W⁡xt-yt.Define the partial order on X by (155)x≼yifandonlyifxt≤yt for all t∈[0,W]. Note that the space X=(C([0,W]),d) is partially ordered complete metric space. The following first-order periodic problem is given by(156)u′t=ft,ut,t∈0,Wμ0=μW,where W>0 and f:[0,W]×R→R is a continuous function. The problem in (156) can be written as(157)μ′t+λμt=ft,μt+λμt,t∈0,Wandλ>0μ0=μW.Problem (157) is equivalent to (158)μt=∫0WGt,sfs,μs+λμsds where G:[0,W]×[0,W]→R is defined by (159)Gt,s=eλW+s-teλW-1,0≤s≤t≤Weλs-teλW-1,0≤t≤s≤W. Note that (160)∫0WGt,sds=1λ.The following definition will be used in our theorem.Definition 40.
A lower solution for (156) is a function β∈C([0,W],R) differentiable on [0,W] such that (161)β′t≤ft,βt,forallt∈0,W,β0≤βW.Theorem 41.
Suppose that following conditions hold:(1)
f:[0,W]×R→R is a continuous function;(2)
f(s,.):R→R is a nondecreasing function for each s∈[0,W];(3)
there existsk∈R+ where k/λ<1 such that (162)fs,μs+λμs-fs,νs+λvs≤kMμs,vswhere(163)Mμs,νs=maxdμs,νs,dμs,fs,μs,dνs,fs,νs,12dμs,fs,νs,dνs,fs,νs1+dμs,fs,μs1+dμs,νsholds for allμ≼ν and s∈[0,W];(4)
there exists a lower solution of problem (156).
Then problem (156) has a solution in X.Proof.
Letμ∈X. Define the integral operator T:X→X as (164)Tμt=∫0WGt,sfs,μs+λμsdsfort∈0,W. Clearly, T is continuous. For any μ,ν∈X such that μ≼ν we obtain (165)dTμ,Tν=supt∈0,WTμt-Tνt≤supt∈0,W∫0WGt,sfs,μs-λμs-fs,νs-λνsds≤supt∈0,W∫0WGt,skMμs,νsds≤kMμ,νsupt∈0,W∫0WGt,sds≤kMμ,νλ. Since k/λ<1 we obtain (166)dTμ,TνedTμ,Tν≤kMμ,νλekMμ,ν/λ≤kMμ,νλeMμ,ν.Passing through exponential, we have (167)edTμ,TνedTμ,Tν≤ekMμ,ν/λeMμ,ν=eMμ,νeMμ,νk/λ. Setting k/λ=c, we obtain that (168)edTμ,TνedTμ,Tν≤eMμ,ν)eMμ,νc. Thus, (169)θdTμ,Tν≤θMμ,νFNμ,ν for θ(t)=etet, F(s1,s2,s3,s4)=c where c<1 and μ≼ν. Since f(s,.) is nondecreasing then for any μ,ν∈X such that μ(t)≤ν(t) for all t∈[0,W], we have (170)Tμt=∫0TGt,sfs,μs+λμsds≤∫0WGt,sfs,νs+λνsds=Tνt. Therefore, T is ≼-closed. If β∈X is a lower solution of (156), then by simple calculations we have (171)βt≤∫0WGt,sfs,βs+λβsds≤Tβt. Hence, by the Corollary 38, problem (156) has a solution in X.Remark 42.
Our results generalize, extend, and refine several results in the literature.(1)
Our results dealing with single valued mappings can be viewed as an extension and generalization of Banach fixed point theorem [1]. It is worth mentioning that the results in [15] are not a generalization of the Banach fixed point theorem.(2)
Theorems20 and 21 extend Nadler’s theorem [5], Bianchini’s Theorem [19], and Hancer’s theorem [6].(3)
Corollaries29 and 30 generalize Theorems 2.3 and 2.4 in [4] and refine Theorems 2.5 and 2.7 in [15].



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