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Research article

Common fixed point results for couples (f,g) and (S,T) satisfy strong common limit range property

  • Received: 01 January 2020 Accepted: 26 March 2020 Published: 08 April 2020
  • MSC : 47H10, 54H25

  • In this manuscript, we introduce strong common limit range property for couples (f,g) and (S,T) and by means of this new concept we establish common fixed point results for hybrid pair via (F,φ)-contraction and rational type contraction conditions. Further, we give some examples to support and illustrate our result. Using the established results existence of solution to the system of integral and differential equations are also discussed. We provide example where the main theorem is applicable but relevant classic result in literature fail to have a common fixed point.

    Citation: Muhammad Shoaib, Muhammad Sarwar, Thabet Abdeljawad. Common fixed point results for couples (f,g) and (S,T) satisfy strong common limit range property[J]. AIMS Mathematics, 2020, 5(4): 3480-3494. doi: 10.3934/math.2020226

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  • In this manuscript, we introduce strong common limit range property for couples (f,g) and (S,T) and by means of this new concept we establish common fixed point results for hybrid pair via (F,φ)-contraction and rational type contraction conditions. Further, we give some examples to support and illustrate our result. Using the established results existence of solution to the system of integral and differential equations are also discussed. We provide example where the main theorem is applicable but relevant classic result in literature fail to have a common fixed point.


    In the area of fixed point theory the study of common fixed point for couple of mapping is a new research area (see [1,7] and the references therein). For hybrid pair the (E. A) property was introduced by Kamran [6] and established coincidence and fixed points results via hybrid strict contractive conditions. To obtain common fixed point results for hybrid type contractive condition, Liu et al. [8] extended this property to common (E. A) property for hybrid pairs of (set-valued and single) mappings. For mapping of single-valued Sintunavarat and Kumam [13] put together the notion of the common limit range (CLR) property and showed its dominance on the property (E. A). For the mapping of hybrid pair the common limit range property defined by Imdad et al. [5] for set-valued fixed point mapping in the semi-metric (symmetric) spaces.

    Afrah [2] generalized this property for single hybrid pair of mapping and then the same author in [3] extended the CLRg property for two hybrid pairs of of mapping. In fuzzy metric spaces, by means of CLRg property, Rold and Sintunavarat [14] established common fixed point results. In 2016, Shoaib and Sarwar [11] studied applications to two functional equations by using set-valued fixed point theorems for a pair of maps and made use of (CLR) property via generalized contractions. For generalized hybrid (F,φ)-contractions Nashine et al.[10] established common fixed point results and used the common limit range property. For more detail see([4,12,15,16,17,18]).

    Motivated by above, using strong common limit range property, we derived set-valued common fixed point results in metric space for couples of maps. Using this property, established hybrid common fixed point results. Also explain the property by giving examples.

    In the whole paper CB(˜N) shows the class of all bounded and closed subsets of ˜N, R+ is the set of positive real numbers and R the set of real numbers respectively.

    Definition 1.1. [9] Maps f:˜N˜N, S:˜NCB(˜N) are said to be occasionally S-weakly commuting if there exists ξ˜N such that fξSξ and ffξfSξ.

    Definition 1.2. [13] If for a sequence {ξn} in ˜N, limnfξn=limngξn=fu for some u˜N. Then, f,g:˜N˜N are said to have the common limit range property of f with respect to g (shortly, the (CLRf)-property w.r.t to g).

    The following two definitions can be found in [3].

    Definition 1.3. If for a sequence {ξn} in ˜N and Ω1CB(˜N), limnfξn=fuΩ1=limnSξn for some u˜N. Then, S:˜NCB(˜N), f:˜N˜N, over metric space (˜N,d) are said to have the common limit range property of f with respect to S (shortly, (CLRf)-property w.r.t to S).

    Definition 1.4. If for sequences {ξn},{ζn} in ˜N and Ω1,Ω2CB(˜N), limnSξn=Ω1, limnTζn=Ω2, limnfξn=limngζn=fuΩ1Ω2, for some u˜N. Then, S,T:˜NCB(˜N), f,g:˜N˜N on metric space (˜N,d) are said to have the common limit in the range of f with respect to S (shortly, (CLRf)-property w.r.t S).

    The below definition is new and it is a modification of Definition 1.3 for couples of functions.

    Definition 1.5. Assume f,g:˜N˜N and S,T:˜NCB(˜N) are functions defined on metric space (˜N,d). Then the couple (f,g) and the couple (S,T) are said to fulfil the common limit in the range of f with respect to (S,T) via g (shortly, (CLRf)-property with respect to (S,T) via g) if there exist sequences {ξn} and {ζn} in ˜N and Ω1,Ω2CB(˜N) such that, for some u˜N we have limnSξn=Ω1, limnTζn=Ω2 and limnfξn=limngζn=fuΩ1Ω2.

    Remark 1.6. Clearly, if f=g and S=T in Definition 1.4 then we reobtain Definition 1.3.

    The below definition announces the so-called the strong common limit range property.

    Definition 1.7. If the couples (f,g) and (S,T) satisfy the (CLRf)-property with respect to (S,T) via g and the (CLRg) property with respect to (S,T) via f then we say the couples (f,g) and (S,T) fulfil the strong common limit range property.

    Remark 1.8. f and S fulfil strong common limit range property if there exists {ξn} and {ζn} in ˜N and Ω1,Ω2CB(˜N) such that, for some u˜N we have limnSξn=Ω1, limnSζn=Ω2 and limnfξn=limnfζn=fuΩ1Ω2.

    Remark 1.9. Couples (f,g) and (S,T) said to be not strong common limit range property. if limnfξn and limngζn exist but not equal to fu or their does not exist {ξn}, {ζn} such that limnfξn=limngζn=fu.

    Example 1.10. Let ˜N=[0,) with the usual metric. Define f,g:˜N˜N and S,T:˜NטNCB(˜N) by f(x)=1+x, g(x)=x2, S(x)=[1,2+2x], T(x)=[1,2+3x4], x˜N.

    Consider the sequences {ςn}={1+1n}, {ζn}={2+1n},

    Clearly limnS(ςn)=[1,4]=Ω1,

    limnT(ζn)=[1,72]=Ω2, limnf(ςn)=limng(ζn)=2=f(1)=g(2)Ω1Ω2.

    Therefore couples the (f,g) and (S,T) fulfil the strong common limit range property.

    Example 1.11. Let ˜N=[0,10) with the usual metric. Define f,g:˜N˜N and S,T:˜NטNCB(˜N) by f(x)=1, g(x)=2, S(x)=[2x,2+2x], T(x)=[x,2+3x4], x˜N.

    For any choice of {ςn} and {ζn}, the couples the (f,g) and (S,T) does not hold the strong common limit range property.

    Now, we provide definitions defined for set-valued mappings in a metric space (˜N,d). Defined the function H:CB(˜N)×CB(˜N)R+ for ˜N1,˜N2CB(˜N) by

    H(˜N1,˜N2)=max{supς1˜N1d(ς1,˜N2),supζ1˜N2d(ζ1,˜N1)},

    where

    d(ξ1,˜N1)=inf{d(ξ1,ζ1):ζ1˜N1},
    D(˜N1,˜N2)=inf{d(ς1,ζ1):ς1˜N1,ζ1˜N2}.

    and

    δ(˜N1,˜N2)=sup{d(ς1,ζ1):ς1˜N1,ζ1˜N2}

    Lemma 1.12. [19] Let (˜N,d) be a metric space. For any ˜N1,˜N2CB(˜N). We have

    d(ξ,˜N2)H(˜N1,˜N2), for all ξ˜N1.

    Definition 1.13. [20] Let η:˜N˜N and λ:˜NCB(˜N) be a mapping. Then η are known occasionally weakly λ- commuting there exist x in ˜N such that ηηxληx for ηxλx.

    Theorem 1.14. [3] Let f,g:˜N˜N and S,T:˜NCB(˜N) be a maps on metric space (˜N,d) satisfying the following condition.

    (a)The pairs (S,f) and (T,g) have common limit range property (CLRf),

    (b)for all x,y˜N

    Hp(Sx,Ty))φ(Δ(x,y))), (1.1)

    where,

    Δ(x,y)=max{dp(fx,gy),dp(fx,Sx)dp(gy,Ty)1+dp(fx,gy),dp(fx,Ty)dp(gy,Sx)1+dp(fx,gy)}.

    Here, p1, and φ:[0,)[0,) is continuous monotone increasing function such that φ(0)=0 and φ(t)<t for all t>0. If f(˜N) and g(˜N) are closed subsets of ˜N, then we have the following:

    (A1) (f,S) have coincidence point.

    (A2) (g,T) have coincidence point.

    (A3) f and S has a common fixed point, if ffv=fv and f and S are weakly compatible at v;

    (A4) g and T has a common fixed point, if ffv=fv and g and T are weakly compatible at v;

    (A5) if (A3) and (A4) holds. Then g, f, T and S have a common fixed point.

    Definition 1.15. [17] Let Fs denotes the class of all mapping F1:R+R, with the below conditions

    (1) F1 is strictly increasing and continuous;

    (2) limnαn=0 if and only if limnF1(αn)=;

    (3) For {αn}R+, limnαn=0, there exists q(0,1), such that limα0+(αn)qF1(αn)=0.

    Thoroughly in this section Φ denote the below class

    Φ={φ:R+R+,uppersemicontineous,increasingsuchthatlimκ1τ+1φ(κ1)<φ(τ1),φ(τ1)<τ1,forallτ1>0}.

    Theorem 1.16. Let f,g:˜N˜N and S,T:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose that (S,g) and (T,f) have strong common limit range property. Furthermore assume that

    τ+F(Hp(Tx,Sy))F(φ(Δ(x,y))), (1.2)

    where, Hp(Tx,Sy)>0 and

    Δ(x,y)=α[dp(fx,gy)]+β[dp(fx,Sy)dp(gy,Tx)1+dp(fx,gy)]+γ[dp(fx,Tx)+dp(gy,Sy)]+σ[dp(fx,Sy)]+η[dp(gy,Tx)].

    Here, τR+, α+β+γ+σ+η<1, p1, FFs and φΦ. Then the below assumption holds.

    (A1) (f,T) have coincidence point.

    (A2) (g,S) have coincidence point.

    (A3) T and f has a common fixed point, if ffv=fv and f is occasionally T-weakly commuting at v;

    (A4) g and S has a common fixed point, if ggu=gu and g is occasionally S-weakly commuting at w;

    (A5) if (A3) and (A4) holds. Then g, f, T and S have a common fixed point.

    Proof. Since f,g:˜N˜N and T,S:˜NCB(˜N) have strong common limit range property, therefore there exists a sequence {ξn} and {ζn} in ˜N and Ω1,Ω2CB(˜N) such that,

    limnTξn=Ω1,limnSζn=Ω2andlimnfξn=limngζn=fu=gvΩ1Ω2.

    for some u,v˜N.

    Now, we show gvSv, suppose gvSv. then putting x=ξn,y=v in inequality (1.2), we have

    τ+F(Hp(Tξn,Sv))F(φ(Δ(ξn,v))), (1.3)

    where,

    Δ(ξn,v)=α[dp(fξn,gv)]+β[dp(fξn,Sζn)dp(gv,Tξn)1+dp(fξn,gv)]+γ[dp(fξn,Tξn)+dp(gv,Sv)]+σ[dp(fξn,Sv)]+η[dp(gv,Tξn)].

    By taking limit to Δ, we have

    limnΔ(ξn,v)=α[dp(fu,gv)]+β[dp(fu,Ω2)dp(gv,Ω1)1+dp(fu,gv)]+γ[dp(fu,Ω1)+dp(gv,Sv)]+σ[dp(fu,Sv)]+η[dp(fu,Ω1)].
    limnΔ(ξn,v)=γ[dp(gv,Sv)]+σ[dp(fu,Sv)]+η[dp(fu,Ω1].
    limnΔ(ξn,w)=(γ+σ)[dp(gv,Sv)]. (1.4)

    Apply limit over (1.3) and by using (1.4), we have

    τ+F(Hp(Ω1,Sv))F(φ(α((γ+σ)(dp(gv,Sv))))).

    Which implies that

    F(Hp(Ω1,Sv))F(φ(α((γ+σ)(dp(gv,Sv))))).

    Using definitions of F and φ, we have

    Hp(Ω1,Sv)α((γ+σ)(dp(gv,Sv))).

    But α<1 and using Lemma 1.12

    dp(gv,Sv)Hp(Ω1,Sv)α(dp(gv,Sv))<dp(gv,Sv). (1.5)

    Which is contradiction. Hence, gvSv.

    Again from (1.2), we have

    τ+F(Hp(Tu,Sζn))F(φ(Δ(u,ζn))), (1.6)

    where,

    Δ(u,ζn)=α[dp(fu,gζn)]+β[dp(fu,Sζn)dp(gζn,Tu)1+dp(fu,gζn)]+γ[dp(fu,Tu)+dp(gζn,Sζn)]+σ[dp(fu,Sζn)]+η[dp(gζn,Tu)].

    By taking limit to Δ, we have

    limnΔ(u,ζn)=α[dp(fu,fu)]+β[dp(fu,Ω2)dp(gu,Tu)1+dp(fu,gu)]+γ[dp(fu,Tu)+dp(fu,Ω2)]+σ[dp(fu,Ω2)]+η[dp(fu,Tu)].
    limnΔ(u,ζn)=(γ+η)dp(fu,Tu). (1.7)

    By taking limit to (1.6) and using (1.7), we have

    τ+F(Hp((Tu,Ω2))F(φ(α((γ+η)dp(fu,Tu)))).

    Which implies that

    F(Hp((Tu,Ω2))F(φ(α((γ+η)dp(fu,Tu)))).

    Using definitions of F and φ, we have

    Hp((Tu,Ω2))α((γ+η)dp(fu,Tu)).

    But α<1 and using Lemma 1.12

    dp(fu,Tu)Hp(Tu,Ω2)α((γ+η)dp(fu,Tu))<dp(fu,Tu).

    Which is contradiction. Hence, fuTu. Since ffv=fv and fvTfv, therefore γ=fγTγ.

    Similarly γ=gγSγ. (A5) hold obviously.

    Example 1.17. Let ˜N=[0,) is a metric(w.r.t) the usual metric. S,T:˜NCB(˜N), f,g:˜N˜N and φ:R+R+ define by S(x)=[1,5+2βx], T(x)=[1,5+βx], f(x)=4x, g(x)=2x x˜N and φ(t)=βt, 0<β<1.

    Consider the sequences {ξn}={1+1n}, {ζn}={2+1n},

    Now, limnS(ξn)=[1,5+2β] limnT(ζn)=[1,5+2β],limng(ζn)=limnf(ξn)=4=f(1)=g(2)[1,5+β][1,5+2β].

    Therefore couples (f,g) and (S,T) satisfy the strong common limit range. Now

    H(Sx,Ty)=H([1,5+2βx],[1,5+βy])
    =max{d([1,5+2βx],[1,5+βy]),d([1,5+βx],[1,5+2βy])},
    =max{|βx2βy|,0},
    =β2d(gx,fu)
    =12.φ(d(gx,fu))
    12.φ(Δ(x,y))
    e16.φ(Δ(x,y)).

    By taking natural logarithm on both sides we conclude by Theorem 1.20 that C(S,f) and C(T,g). where C(S,f) represent coincidence point of S and f.

    Example 1.18. Let ˜N=(11,11) with the usual metric. Define S,T:˜NטNCB(˜N), f,g:˜N˜N, φ:R+R+ and F:R+R by S(x)=[6,2+βx4], T(x)=[6,2+β6x], f(x)=x2, g(x)=x3 x˜N, φ(t)=βt, 0<β<1 and F(t)=ln(t).

    Consider the sequences {ξn}={2+1n}, {ζn}={3+1n},

    Now, limnS(ξn)=[6,2+β2] limnT(ζn)=[6,2+β2],limng(ζn)=limnf(ξn)=1=f(2)=g(3)[6,2+β2][6,2+β2].

    Therefore couples (f,g) and (S,T) satisfy the strong common limit range. Now

    Now,

    H(Sx,Ty)=H([5,1+αx4],[5,1+αy6])
    =max{d([5,1+αx4],[5,1+αy6]),d([5,1+αx6],[5,1+αy4])},
    =max{|αx6αy4|,0},
    =α2d(gx,fy)
    =12φ(d(gx,fy))
    12φ(Δ(x,y))
    e16φ(Δ(x,y)).

    Taking logarithm on both sides and p=1, we conclude that all the other condition of Theorem 1.16 are satisfied. Therefore (f,g) and (S,T) have coincidence point point.

    Remark 1.19. From above examples it is clear that

    ● Theorem 1.14 is not applicable to Example 1.18 because f(˜N)=(112,112) nor g(˜N)=(113,113) are closed.

    Theorem 1.20. Let f,g:˜N˜N and S,T:˜NCB(˜N) are mapping on metric space (˜N,d). Furthermore assume that (S,g) and (T,f) have strong common limit range property and

    τ+F(Hp(Tx,Sy)F(φ(Δ(x,y))), (1.8)

    where, Hp(Tx,Sy)>0 and

    Δ(x,y)=max{dp(fx,Sy),dp(gy,Tx),dp(fx,gy),dp(fx,Sy)+dp(gy,Tx)2,dp(fx,Sy)dp(gy,Tx)1+dp(fx,gy),dp(gy,Tx)dp(fx,Sy)1+dp(fx,gy),dp(fx,Sy)dp(gy,Tx)1+Dp(Sx,Tx)}.

    Here, τR+, p1, FFs and φΦ. Then the below condition holds.

    (A1) (f,T) have coincidence point.

    (A2) (g,S) have coincidence point.

    (A3) T and f has a common fixed point, if ffv=fv and f is occasionally T-weakly commuting at v;

    (A4) S and g has a common fixed point, if ggw=gw and g is occasionally S-weakly commuting at w;

    (A5) if (A3) and (A4) holds. Then f, S, g and T have a common fixed point.

    Proof. Since f,g:˜N˜N and T,S:˜NCB(˜N) have strong (CLR)-property, therefore there exists a sequence {ξn} and {ζn} in ˜N and Ω1,Ω2CB(˜N) such that,

    limnTξn=Ω1,limnSζn=Ω2andlimnfξn=limngζn=fu=gvΩ1Ω2,

    for some u,v˜N. Now, we show gvSv, suppose gvSv, by putting x=ξn,y=v in inequality (1.22), we have

    τ+F(Hp(Tξn,Sv)F(φ(Δ(ξn,v))), (1.9)
    Δ(ξn,v)=max{dp(fξn,Sv),dp(gv,Tξn),dp(fξn,gv),dp(fξn,Sv)+dp(gv,Tξn)2,dp(fξn,Sv)dp(gv,Tξn)1+dp(fξn,gv),dp(gv,Tξn)dp(fξn,Sv)1+dp(fξn,gv),dp(fξn,Sv)dp(gv,Tξn)1+Dp(Sξn,Tξn)}.

    Taking limit to Δ, we have

    limnΔ(ξn,v)=max{dp(gv,Sv),dp(gv,Ω1),dp(gv,gv),dp(fv,Sv)+dp(gv,Ω1)2,dp(gv,Sv)dp(gv,Ω1)1+dp(gv,gv),dp(gv,Ω1)dp(gv,Sv)1+dp(gv,gv),dp(gv,Sv)dp(gv,Ω1)1+Dp(Ω2,Ω1)}.
    limnΔ(ξn,v)=max{dp(gv,Sv),dp(gv,Sv)2}.
    limnΔ(ξn,v)=dp(gv,Sv). (1.10)

    By taking limit over (1.9), and using (1.10), we have

    τ+F(Hp(Ω1,Sv))F(φ(α(dp(gv,Sv)))).

    Which implies that

    F(Hp(Ω1,Sv))F(φ(α(dp(gv,Sv)))).

    Using definitions of F and φ, we have

    Hp(Ω1,Sv)α(dp(gv,Sv)).

    But α<1 and using Lemma 1.12

    dp(gv,Sv)Hp(Ω1,Sv)α(dp(gv,Sv))<dp(gv,Sv). (1.11)

    which is contradiction. Hence, gvSv.

    We show fuTu, suppose fuTu. then putting x=u,y=ζn in inequality (1.22), we have

    τ+F(Hp(Tu,Sζn)F(φ(Δ(u,ζn))), (1.12)
    Δ(v,ζn)=max{dp(fu,Sζn),dp(gζn,Tu),dp(fu,gζn),dp(fu,Sζn)+dp(gζn,Tu)2,dp(fu,Sζn)dp(gζn,Tu)1+dp(fu,gζn),dp(gζn,Tu)dp(fu,Sζn)1+dp(fu,gζn),dp(fu,Sζn)dp(gζn,Tu)1+Dp(Su,Tu)}.

    By taking limit over Δ, we have

    limnΔ(u,ζn)=max{dp(fu,Ω2),dp(fu,Tu),dp(fu,fu),dp(fu,Ω2)+dp(fu,Tu)2,dp(fu,Ω2)dp(fu,Tu)1+dp(fu,fu),dp(fu,Tu)dp(fu,Ω2)1+dp(fu,fu),dp(fu,Ω2)dp(fu,Tu)1+Dp(Su,Tu)}.
    limnΔ(u,ζn)=max{dp(fu,Tu),dp(fu,Tu)2}.
    limnΔ(u,ζn)=dp(fu,Tu) (1.13)

    By taking limit over (1.12) and using (1.13), we have

    τ+F(Hp(Tu,Ω2)F(φ(αdp(fu,Tu))).

    Which implies that

    F(Hp(Tu,Ω2))F(φ(αdp(fu,Tu))).

    Using definitions of F and φ, we have

    Hp(Tu,Ω2)αdp(fu,Tu).

    But α<1 and using Lemma 1.12

    dp(fu,Tu)Hp(Tu,Ω2)αdp(fu,Tu)<dp(fu,Tu).

    Which is contradiction. Hence, fuTu.

    Succeeding the parallel line of Theorem 1.16, we can achieved that S,T,f and g have common coupled fixed point.

    If S=T and f=g in Theorem 1.16, by using Remark 1.6, we have

    Corollary 1.21. Let f:˜N˜N and S:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose (S,f) have strong common limit range property. Furthermore

    τ+F(Hp(Sx,Sy))F(φ(Δ(x,y))),

    where, H(Sx,Sy)>0 and

    Δ(x,y)=α[dp(fx,fy)]+β[dp(fx,Sy)dp(fy,Sx)1+dp(fx,fy)]+γ[dp(fx,Sx)+dp(fy,Sy)]+σ[dp(fx,Sy)]+η[dp(fy,Sx)].

    Here, τR+, α+β+γ+σ+η<1, p1, FFs and φΦ. Then the below condition holds.

    (A1) (f,S) have coincidence point.

    (A2) f and S has a common fixed point, if ffv=fv and f is occasionally S-weakly commuting at v; Then f, S have a common fixed point.

    If f=g and S=T in Theorem 1.20 by using Remark 1.6, we have

    Corollary 1.22. Let f:˜N˜N and S:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose (S,f) have strong common limit range property. Furthermore

    τ+F(Hp(Sx,Sy)F(φ(Δ(x,y))),

    where, H(Sx,Sy)>0 and

    Δ(x,y)=max{dp(fx,Sy),dp(fy,Sx),dp(fx,fy),dp(fx,Sy)+dp(fy,Sx)2,dp(fx,Sy)dp(fy,Sx)1+dp(fx,fy),dp(fy,Sx)dp(fx,Sy)1+dp(fx,fy),dp(fx,Sy)dp(fy,Sx)}.

    Here, τR+, p1, FFs and φΦ. Then the below assumption holds.

    (A1) (f,S) have coincidence point.

    (A2) f and S has a common fixed point, if ffv=fv and f is occasionally S-weakly commuting at v; Then f, S have a common fixed point.

    Theorem 1.23. Let f,g:˜N˜N and S,T:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose that (S,g) and (T,f) have strong common limit range property. Moreover assume that

    Hp(Tx,Sy)αdp(fx,Tx)dp(fx,Sy)+dp(gy,Sy)dp(gy,Tx)1+dp(fx,Sy)+dp(gy,Tx), (1.14)

    where, 0<α<1. Then

    (A1) (f,T) have coincidence point.

    (A2) (g,S) have coincidence point.

    (A3) f and T has a common fixed point, if ffv=fv and f is occasionally T-weakly commuting at v;

    (A4) g and S has a common fixed point, if ggu=gu and g is occasionally S-weakly commuting at w;

    (A5) if (A3) and (A4) holds. Then f, g, S and T have a common fixed point.

    Proof. Since f,g:˜N˜N and T,S:˜NCB(˜N) have strong (CLR)-property, therefore there exists a sequence {ξn} and {ζn} in ˜N and Ω1,Ω2CB(˜N) such that,

    limnTξn=Ω1,limnSζn=Ω2andlimnfξn=limngζn=fu=gvΩ1Ω2,

    for some u,v˜N. Now, we show gvSv, suppose gvSv, by putting x=ξn,y=u in inequality (1.14), we have

    Hp(Tξn,Sv)αdp(fξn,Tξn)dp(fξn,Sv)+dp(gv,Sv)dp(gv,Tξn)1+dp(fξn,Sv)+dp(gv,Tξn), (1.15)

    Applying limit, we have

    Hp(Ω1,Sv)=0.

    By using Lemma 1.12

    dp(gv,Sv)Hp(Ω1,Sv)=0. (1.16)

    which is possible if gvSv.

    We show fuTu, suppose fuTu. then putting x=u,y=ζn in inequality (1.19), we have

    Hp(Tu,Sζn)αdp(fu,Tu)dp(fu,Sζn)+dp(gζn,Sζn)dp(gζn,Tu)1+dp(fu,Sζn)+dp(gζn,Tu), (1.17)

    Taking limit, we have

    Hp(Tu,Ω2)=0.

    Using Lemma 1.12, we have

    Hp(fu,Tu)Hp(Tu,Ω2). (1.18)

    Which is possible only if fuTu.

    After succeeding the similar lines of Theorem 1.16 we can obtained that S,T,f and g have common coupled fixed point.

    Theorem 1.24. Let f,g:˜N˜N and S,T:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose that (S,g) and (T,f) have strong common limit range property. Moreover assume that

    Hp(Tx,Sy){αdp(fx,Tx)dp(fx,Sy)+dp(gy,Sy)dp(gy,Tx)dp(fx,Sy)+dp(gy,Tx),ifΔ0,0,ifΔ=0.

    Where Δ=dp(fx,Sy)+dp(gy,Tx), 0<α<1. Then

    (A1) (f,T) have coincidence point.

    (A2) (g,S) have coincidence point.

    (A3) T and f has a common fixed point, if ffv=fv and f is occasionally T-weakly commuting at v;

    (A4) S and g has a common fixed point, if ggu=gu and g is occasionally S-weakly commuting at w;

    (A5) if (A3) and (A4) holds. Then g, f, S and T have a common fixed point.

    Succeeding the steps of Theorem 1.23 we can obtained that S,T,f and g have common coupled fixed point.

    If f=g, T=S in Theorem 1.24 by using Remark 1.6, we get

    Corollary 1.25. Let f:˜N˜N and S:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose that (S,f) have strong common limit range property. Furthermore

    Hp(Sx,Sy){αdp(fx,Sx)dp(fx,Sy)+dp(gy,Sy)dp(fy,Sx)dp(fx,Sy)+dp(fy,Sx),ifΔ0,0,ifΔ=0.

    Where Δ=dp(fx,Sy)+dp(gy,Tx), 0<α<1. Then

    (A1) (f,S) have coincidence point.

    (A2) S and f has a common fixed point, if ffv=fv and f is occasionally S-weakly commuting at v. Then f, S have a common fixed point.

    If T=S, f=g in Theorem 1.23. From Remark 1.6, we have

    Corollary 1.26. Let f:˜N˜N and S:˜NCB(˜N) be a maps on metric space (˜N,d). Suppose (S,f) have strong common limit range property. Furthermore

    Hp(Sx,Sy)αdp(fx,Sx)dp(fx,Sy)+dp(fy,Sy)dp(fy,Sx)1+dp(fx,Sy)+dp(fy,Sx), (1.19)

    where, 0<α<1. Then

    (A1) (f,S) have coincidence point.

    (A2) f and S has a common fixed point, if ffv=fv and f is occasionally S-weakly commuting at v. Then S, f have a common fixed point.

    Now, we study solutions of 2nd kind general nonlinear system of Fredholm integral equations given by

    {x(t)=ϕ(t)+qpQ1(t,s,x(s))ds,t[p,q],y(t)=ϕ(t)+qpQ2(t,s,y(s))ds,t[p,q],. (2.1)

    Let ˜N=C[p,q] be the set of all continuous function defined on [p,q]. Define

    d:˜NטNR+, by

    d(x,y)=xy.

    Where x=sup{|x(t)|:t[p,q]}. Then (˜N,d) is a complete d metric space on ˜N. We give the following theorem.

    Theorem 2.1. Assume that the following assumptions hold

    (A1) Qj:[p,q]×[p,q]×R+R+, for j=1,2 and ϕ:R+R+ is continuous;

    (A2) there exist a continuous function G:[p,q]×[p,q][0,) such that,

    |Q1(t,s,u)Q2(t,s,v)|G(t,s)γ(|uv|),

    for each t,s[p,q], 0<γ<1,

    (A3) supt,s[p,q]qp|G(t,s)|dseτ for τ>0.

    Then the system of integral equations 2.1 has a common solution in C([p,q]).

    Proof. Define S,T:C([p,q])C([p,q]) by,

    Sx(t)=ϕ(t)+qpQ1(t,s,x(s))ds,t[p,q].
    Ty(t)=ϕ(t)+qpQ2(t,s,y(s))ds,t[p,q].

    Now we have,

    d(Sx(t),Ty(t))=supt[p,q]|Sx(t)Ty(t)|supt[p,q]qp|Q1(t,s,x(s))Q2(t,s,y(s))|dssupt[p,q]qpG(t,s)γ(|x(s)y(s)|)dssupt[p,q]γ(|x(t)y(t)|)supt[p,q]qpG(t,s)dssupt[p,q]γ(|x(t)y(t)|)eτ=γ(x(t)y(t))eτ=γ(d(x(t),y(t)))eτ.

    By taking natural log to both side, we have

    τ+F(Hp(Sx,Ty)F(φ(Δ(x,y))).

    Define f(x)=g(x)=x, F(t)=ln(t), φ(t)=γt, and p=1 then by Theorem 1.20 the system (2.1) has a common solution in ˜N.

    With the help of Theorem 1.20, one can also solve the following coupled system of nonlinear fractional ordered differential equations given by

    {cDβu(t)+^g1(v(t))=0, 1<β2, t[0,1],cDβv(t)+^g2(w(t))=0, 1<β2,u(0)=v(0)=a, u(1)=v(1)=b,where a,b are constant. (2.2)

    Where ^g1,^g2:[0,1]×[0,)[0,). Then the equivalent system of integral equations corresponding to (2.2) is given by

    {u(t)=ϕ(t)+10G(t,s)^g1(v(s)ds, t[0,1],v(t)=ϕ(t)+10G(t,s)^g2(w(s)ds, t[0,1], (2.3)

    Where G(t,s) is the Green's function

    G(t,s)={(ts)β1t(1s)β1Γ(β), 0st1,t(1s)β1Γ(β),0ts1,

    and continuous on [0,1]×[0,1]. Moreover supt[0,1]10|G(t,s)|ds1. Further, using Q(t,s,x(s))=G(t,s)^g1(v(s) etc. Then the system (2.3)become

    {x(t)=ϕ(t)+10Q1(t,s,x(s))ds, t[0,1],y(t)=ϕ(t)+10Q2(t,s,y(s))ds, t[0,1]. (2.4)

    Clearly by Theorem 1.20 the System(2.4) has a solution, which is the corresponding solution of the system of nonlinear fractional differential equation(2.2).

    In this work, we introduced strong common limit range property for couples (f,g) and (S,T) to relaxed the conditions of completeness (closedness), the containment of the range of the mappings, convexity of the underline space and continuity of the mappings and by means of this new concept we established common fixed point results for hybrid pair via (F,φ)-contraction and rational type contraction conditions. Further, using the established results existence of solution to the system of integral and differential equations are also discussed. We provided example where the main theorem is applicable but relevant classic result in literature fail to have a common fixed point.

    The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

    The authors declare that they have no competing interest.



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