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Research article

Mechanistic modeling of alarm signaling in seed-harvester ants


  • Received: 05 January 2024 Revised: 28 February 2024 Accepted: 05 March 2024 Published: 27 March 2024
  • Ant colonies demonstrate a finely tuned alarm response to potential threats, offering a uniquely manageable empirical setting for exploring adaptive information diffusion within groups. To effectively address potential dangers, a social group must swiftly communicate the threat throughout the collective while conserving energy in the event that the threat is unfounded. Through a combination of modeling, simulation, and empirical observations of alarm spread and damping patterns, we identified the behavioral rules governing this adaptive response. Experimental trials involving alarmed ant workers (Pogonomyrmex californicus) released into a tranquil group of nestmates revealed a consistent pattern of rapid alarm propagation followed by a comparatively extended decay period [1]. The experiments in [1] showed that individual ants exhibiting alarm behavior increased their movement speed, with variations in response to alarm stimuli, particularly during the peak of the reaction. We used the data in [1] to investigate whether these observed characteristics alone could account for the swift mobility increase and gradual decay of alarm excitement. Our self-propelled particle model incorporated a switch-like mechanism for ants' response to alarm signals and individual variations in the intensity of speed increased after encountering these signals. This study aligned with the established hypothesis that individual ants possess cognitive abilities to process and disseminate information, contributing to collective cognition within the colony (see [2] and the references therein). The elements examined in this research support this hypothesis by reproducing statistical features of the empirical speed distribution across various parameter values.

    Citation: Michael R. Lin, Xiaohui Guo, Asma Azizi, Jennifer H. Fewell, Fabio Milner. Mechanistic modeling of alarm signaling in seed-harvester ants[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5536-5555. doi: 10.3934/mbe.2024244

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  • Ant colonies demonstrate a finely tuned alarm response to potential threats, offering a uniquely manageable empirical setting for exploring adaptive information diffusion within groups. To effectively address potential dangers, a social group must swiftly communicate the threat throughout the collective while conserving energy in the event that the threat is unfounded. Through a combination of modeling, simulation, and empirical observations of alarm spread and damping patterns, we identified the behavioral rules governing this adaptive response. Experimental trials involving alarmed ant workers (Pogonomyrmex californicus) released into a tranquil group of nestmates revealed a consistent pattern of rapid alarm propagation followed by a comparatively extended decay period [1]. The experiments in [1] showed that individual ants exhibiting alarm behavior increased their movement speed, with variations in response to alarm stimuli, particularly during the peak of the reaction. We used the data in [1] to investigate whether these observed characteristics alone could account for the swift mobility increase and gradual decay of alarm excitement. Our self-propelled particle model incorporated a switch-like mechanism for ants' response to alarm signals and individual variations in the intensity of speed increased after encountering these signals. This study aligned with the established hypothesis that individual ants possess cognitive abilities to process and disseminate information, contributing to collective cognition within the colony (see [2] and the references therein). The elements examined in this research support this hypothesis by reproducing statistical features of the empirical speed distribution across various parameter values.



    The new era of fixed point theory associated with metrics is now ineluctably associated with medical biological sciences, abstract terminology, space analysis and epidemiological data mining through engineering. This were often persisted by extending metric fixed point theory to a profusion of literature from computational engineering, fluid mechanics, and medical science. Fixed point theory has made a brief appearance as its own literature in the analysis of metric spaces, as keep referring to many other mathematical groups. Popular uses of metric fixed point theory involve defining and/or generalizing the various metric spaces and the notion of contractions. These extensions are also rendered with the intended consequence of a deeper comprehension of the geometric properties of Banach spaces, set theory, and non-expensive mappings.

    A qualitative principle that concerns seeking conditions on the set M structure and choosing a mapping on M to get a fixed point is generally referred to as a fixed point theorem. The fixed point theory framework falls from the larger field of nonlinear functional analysis. Many of the natural sciences and engineering physical questions are usually developed in the form of numerical and analytical equations. Fixed-point assumptions find potential advantages in proving the existence of the solutions of some differential and integral equations which occur in the analysis of heat and mass transfer problems, chemical and electro-chemical processes, fluid dynamics, molecular physics and in many other fields. In 2006, Mustafa and Sims [17] introduced the concept of G-metric space as a generalization of metric space.

    It is calculated that investigators get their fresh outputs from engineering mathematics and/or its applications from 60%. For example, non-linear integral equations: it has been commonly utilized both in engineering and technology streams of all kinds. These are also appealing to researchers because of the simplicity of using non-linear integral equations and/or their implementations for approximation/numerical/data analysis strategies. In bio-medical sciences, evolution, database technology and computational systems, the steady flow of non-linear integral equations will create fresh avenues in broad directions. Non-linear integral equations are gradually becoming methods for different aspects of hydrodynamics, cognitive science, respectively (see for example [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]). The impetus of this work is to prove coupled coincidence point theorems for two mappings via rational type contractions satisfying mixed g-monotone property which are the generalizations of theorems of Chouhan and Richa Sharma [5] and extensions of some other existed results.

    The basic definitions and propositions which are used to derive our main results are given below and also note that G-metric and g-monotone property are denoted by ϑmetric and B-monotone property respectively through out this paper.

    Definition 1.1. ([17]) Let M be a set wihch is nonempty and ϑ:M×M×MR such that

    (ϑ1) ϑ(a,b,c)0 for all a,b,cM with ϑ(a,b,c)=0 if a=b=c;

    (ϑ2) ϑ(a,a,b)>0 for all a,bM with ab;

    (ϑ3) ϑ(a,a,b)ϑ(a,b,c) for all a,b,cM with cb;

    (ϑ4) ϑ(a,b,c)=ϑ(a,c,b)=ϑ(b,a,c)=ϑ(c,a,b)=ϑ(b,c,a)=ϑ(c,b,a) for all a,b,cM;

    (ϑ5) ϑ(a,b,c)ϑ(a,d,d)+ϑ(d,b,c) for all a,b,c,dM.

    Then the pair (M,ϑ) is called a ϑ-metric space with ϑ-metric ϑ on M. Axioms (ϑ4) and (ϑ5) are referred to as the symmetry and the rectangle inequality (of ϑ) respectively.

    If (M,) is partially ordered set in the definition of ϑ-metric space, then (M,ϑ,) is partially ordered ϑ-metric space.

    Given a ϑ-metric space (M,ϑ), define

    ρϑ(a,b)=ϑ(a,b,b)+ϑ(a,a,b) for all a,b,cM. (1.1)

    Then it is seen in [17] that ϑ is a metric on M, and that the family of all ϑ-balls {Bϑ(a,r):aM,r>0} is the base topology, called the ϑ-metric topology τ(ϑ) on M, where Bϑ(a,r)={bM:ϑ(a,b,b)<r}. Further, it was shown that the ϑ-metric topology coincides with the metric topology induced by the metric ϑ, which allows us to readily transform many concepts from metric spaces into ϑ-metric space.

    Definition 1.2. ([17]) If a sequence aκ n=1 in (M,ϑ) converges to an element pM in the ϑ-metric topology τ(ϑ), then aκ n=1 is called ϑ-convergent with limit p.

    Proposition 1.1. ([17]) If (M,ϑ) is a ϑ-metric space, then following are equivalent.

    (ⅰ) aκ κ=1 is ϑ-convergent to a;

    (ⅱ) ϑ(aκ,aκ,a)0 as κ;

    (ⅲ) ϑ(aκ,a,a)0 as κ;

    (ⅳ) ϑ(aκ,aη,a)0 as κ,η.

    Definition 1.3. ([17]) A sequence aκ in (M,ϑ)is called ϑ-Cauchy if for every σ>0 there exists a positive integer N such that ϑ(aκ,aη,aζ)<σ for all κ,η,ζN.

    Proposition 1.2. ([18]) If (M,ϑ) is a ϑ-metric space, then aκ is ϑ-Cauchy if and only if for every σ>0, there exists a positive integer N such that ϑ(aκ,aη,aη)<σ for all η,κN.

    Proposition 1.3. ([17]) Every ϑ-convergent sequence in (M,ϑ) is ϑ-Cauchy.

    Definition 1.4. ([17]) If every ϑ-Cauchy sequence in M converges in M, then (M,ϑ) is called ϑ-complete.

    Proposition 1.4. ([17]) If (M,ϑ)is a ϑ-metric space and B is a self-map on M, then B is ϑ-continuous at a point aM iff the sequenceBaκ converges to Ba whenever aκ converges to a.

    Proposition 1.5. ([17]) The ϑ-metric ϑ(a,b,c) is continuous jointly in all the variables a,b and c.

    Proposition 1.6. ([17]) If (M,ϑ) is a ϑ-metric space, then

    (ⅰ) if ϑ(a,b,c)=0 then a=b=c;

    (ⅱ) ϑ(a,b,c)ϑ(a,a,b)+ϑ(a,a,c);

    (ⅲ) ϑ(a,b,b)2ϑ(b,a,a);

    (ⅳ) ϑ(a,b,c)ϑ(a,x,c)+ϑ(x,b,c) for all a,b,c,xM.

    Definition 1.5. ([6]) Let (M,ϑ) be a ϑ-metric space and X:M×MM be a mapping on M×M. Then X is called continuous if the sequence X(an,bn) converge to X(a,b) whenever the sequences an and bn are converge to a and b respectively.

    Definition 1.6. ([11]) Let M be a set which is nonempty and X:M×MM, B:MM be two mappings. Then X is said to be commute with B if X(Ba,Bb)=B(X(a,b)).

    Definition 1.7. ([11]) Let M be a set which is nonempty set and X:M×MM, B:MM be two mappings. Then X is said to have mixed B-monotone property if

    a1,a2M,Ba1Ba2X(a1,b)X(a2,b)andb1,b2M,Bb1Bb2X(b1,a)X(b2,a)for all a,bM.

    If B is an identity mapping in the above deinition, then X has mixed monotone property.

    Definition 1.8. ([2]) If M is a set which is nonempty and X:M×MM is a mapping such thatX(a,b)=a and X(b,a)=b, then (a,b)M×M is called coupled fixed point of X.

    Definition 1.9. ([17]) If M is a set which is nonempty set and X:M×MM, B:MM are two mappings such that X(a,b)=Ba and X(b,a)=Bb, then (a,b)M×M is called coupled coincidence point of X and B.

    If B is an identity mapping in the above definition, then (a,b) is called coupled fixed point of a mapping X.

    Theorem 2.1. Let (M,ϑ,) be a partially ordered complete ϑ-metric space and X:M×MM, B:MM be two continuous mappings such that X has mixed B-monotone property and

    (i) there exist α,β,γ[0,1) and L0 with 8α+β+γ<1 such that

    ϑ(X(x,y),X(u,v),X(w,z))αϑ(B,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+βϑ(Bx,Bu,Bw),+γϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)),      ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)),      ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))}for allx,y,u,v,w,zMwithBxBuBw and ByBvBz; (2.1)

    (ii) X(M×M)B(M);

    (iii) B commutes with X.

    If there exist x0,y0M such that Bx0X(x0,y0) and By0X(y0,x0), then X and B have a coupled coincidence point in M×M.

    Proof. Let x0 and y0 be any two elements in M such that Bx0X(x0,y0) and By0X(y0,x0). Since X(M×M)B(M), we construct two sequences xκκ=1 and yκκ=1 in M as follows:

    Bxκ+1=X(xκ,yκ) and Byκ+1=X(yκ,xκ) for κN.

    Since X has mixed B-monotone property, we have

    Bxκ=X(xκ1,yκ1)X(xκ,yκ1)X(xκ,yκ)=Bxκ+1 and Byκ+1=X(yκ,xκ)X(yκ1,xκ)X(yκ1,xκ1)=Byκ.

    Now using (2.1) with x=xκ, y=yκ, u=w=xκ1 and v=z=yκ1, we get

    ϑ(Bxκ+1,Bxκ,Bxκ)=ϑ(X(xκ,yκ),X(xκ1,yκ1),X(xκ1,yκ1))αϑ(Bxκ,X(xκ,yκ),X(xκ,yκ))ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1))ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1))[ϑ(Bxκ,Bxκ1,Bxκ1)]2+βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)  +Lmin{ϑ(Bxκ,X(xκ1,yκ1),X(xκ1,yκ1)),ϑ(Bxκ1,X(xκ,yκ),X(xκ1,yκ1)),ϑ(Bxκ1,X(xκ,yκ),X(xκ1,yκ1)),ϑ(Bxκ,X(xκ,yκ),X(yκ,xκ)),ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1)),ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1))}=αϑ(Bxκ,Bxκ+1,Bxκ+1)ϑ(Bxκ1,Bxκ,Bxκ)ϑ(Bxκ1,Bxκ,Bxκ)[ϑ(Bxκ,Bxκ1,Bxκ1)]2+βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)+Lmin{ϑ(Bxκ,Bxκ,Bxκ),ϑ(Bxκ1,Bxκ+1,Bxκ),ϑ(Bxκ1,Bxκ+1,Bxκ)),ϑ(Bxκ,Bxκ+1,Bxκ+1),ϑ(Bxκ1,Bxκ,Bxκ),ϑ(Bxκ1,Bxκ,Bxκ)}8αϑ(Bxκ+1,Bxκ,,Bxκ)ϑ(Bxκ,Bxκ1,Bxκ1)ϑ(Bxκ,Bxκ1,Bxκ1)[ϑ(Bxκ,Bxκ1,Bxκ1)]2+βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)

    so that

    ϑ(Bxκ+1,Bxκ,Bxκ)1(18α)[βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)] (2.2)

    Similarly,

    ϑ(Byκ+1,Byκ,Byκ)1(18α)[βϑ(Byκ,Byκ1,Byκ1)+γϑ(Bxκ,Bxκ1,Bxκ1)] (2.3)

    Adding (2.2) and (2.3), we have

    ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ)β+γ18α[ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)]. (2.4)

    Let Δκ=ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ) and c=β+γ18α

    where 0c<1 inview of choice of α,β and γ.

    Now the inequality (2.4) becomes as follows

    Δκc.Δκ1 for κN

    Consequently ΔκcΔκ1c2Δκ2...cκΔ0

    If Δ0=0, we have ϑ(Bx1,Bx0,Bx0)+ϑ(By1,By0,By0)=0

    or ϑ(X(x0,y0),Bx0,Bx0)+ϑ(X(y0,x0),By0,By0)=0 which implies that X(x0,y0)=Bx0 and X(y0,x0)=By0.

    That is, (x0,y0) is a coupled coincidence point of X and B.

    Suppose that Δ0>0.

    Now by applying rectangle inequality of ϑ-metric repeatedly and using inequality ΔκcκΔ0, we have

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxη,Bxκ+1,Bxκ+1)]+[ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byη,Byκ+1,Byκ+1)][ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+ϑ(Bxη,Bxκ+2,Bxκ+2)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+ϑ(Byη,Byκ+2,Byκ+2)][ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)++ϑ(Bxη,Bxη1,Bxη1)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)++ϑ(Byη,Byη1,Byη1)]=Δκ+Δκ+1+Δκ+2++Δη1[cκ+cκ+1++cη1]Δ0cκ11cΔ0 for η>κ

    or

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)cκ11cΔ0 for η>κ. (2.5)

    Since 0c<1, cκ0 as κ.

    Now applying limit as κ with η>κ in the inequality (2.5), we have

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)0 which follows that xκκ=1 and yκκ=1 are cauchy sequences in M.

    Since (M,ϑ) is a partially ordered complete ϑ-metric space, there exist p,qM such that xκp and yκq.

    Now we prove that (p,q) is a coupled coincidence point of X and B.

    Since X commutes with B, we have

    X(Bxκ,Byκ)=B(X(xκ,yκ))=B(Bxκ+1)   andX(Byκ,Bxκ)=B(X(yκ,xκ))=B(Byκ+1)

    Since X and B are continuous, we have

    limκX(Bxκ,Byκ)=limκB(Bxκ+1)=Bp   andlimκX(Byκ,Bxκ)=limκB(Byκ+1)=Bq

    Snice ϑ is continuous in all its variables, we have

    ϑ(X(p,q),Bp,Bp)=ϑ(limκX(Bxκ,Byκ),Bp,Bp))=ϑ(Bp,Bp,Bp)

    so that

    ϑ(X(p,q),Bp,Bp)=0

    which implies that X(p,q)=Bp.

    Similarly, it can be proved that X(q,p)=Bq.

    Hence (p,q) is a coupled coincidence point of X and B.

    Remark 2.1. (ⅰ.) If we assume B is an identity mapping and γ=0 in the Theorem 2.1, then we get Theorem 3.1 in the results of the Chouhan and Richa Sharma [5];

    (ⅱ.) If we take, α=0 and β=α, γ=β in the Theorem 2.1, we get Theorem 3.1 in the results of Chandok et al. [3];

    (ⅲ.) By taking γ=0 and L=0, we get Theorem 2.1 in the results of Chakrabarti [4].

    That is Theorem 2.1 is generalization and extension of above three results.

    The following is the example to illusrative Theorem 2.1.

    Example 2.1. Let M=[0,1], with ϑ-metric ϑ(x,y,z)={0,   x=y=z,max{x,y,z},otherwise.

    Define partial order on X as xy for any x,yM. Then(M,ϑ,) is a partially ordered ϑ-complete.

    Define X:M×MM by X(x,y)={x3(y2+2), xy0    otherwise

    and B:MM by Bx=x4.

    Then clearlyX, B are continuous and X satisfies mixed B-monotone property. We show that X satisfies the inequality (2.1) with α=β=γ=132 so that 08α+β+γ<1 and for any L0.

    Let x,y,z,u,v,wM be such that xuw and yvz.

    We discuss four cases:

    Case (ⅰ): If xy,uv and wz then we have X(x,y)=x3(y2+2), X(u,v)=u3(v2+2) and X(w,z)=w3(z2+2)

    ϑ(X(x,y),X(u,v),X(w,z))=max{x3(y2+2),u3(v2+2),w3(z2+2)}=x3(y2+2)132[16wuxx2+x4+u4]=132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)),      ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)),      ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z)}

    Case (ⅱ): If xy,uv and w<z then we have X(x,y)=x3(y2+2), X(u,v)=u3(v2+2) and X(w,z)=0

    ϑ(X(x,y),X(u,v),X(w,z))=max{x3(y2+2),u3(v2+2),0}=x3(y2+2)132[16wuxx2+x4+u4]132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)),      ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)),      ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))}

    Case (ⅲ): If xy,u<v and w<z, then we have X(x,y)=x3(y2+2), X(u,v)=0 and X(w,z)=0

    ϑ(X(x,y),X(u,v),X(w,z))=max{x3(y2+2),0,0}=x3(y2+2)132[16wuxx2+x4+u4]132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)),      ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)),      ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))}

    Case (iv): If x<y,u<v and w<z, then X(x,y)=x3(y2+2), X(u,v)=0 and X(w,z)=0, it follows that

    ϑ(X(x,y),X(u,v),X(w,z))132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)),   ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)),   ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))}

    In similar manner, the cases x<y,uv,wz; x<y,u<v,wz and all others can be handled.

    Thus X and B satify all the conditions of Theorem 2.1 and also note that (0,0) is a coupled coincidence point of X and B.

    Theorem 2.2. Let (M,ϑ,) be a partially ordered complete ϑ-metric space and X:M×MM, B:MM be two continuous mappings such that X has mixed B-monotone property and

    (i) there exist α,β,γ[0,1) and L0 with 2α+β+γ<1 such that

    ϑ(X(x,y),X(u,v),X(w,z))αϑ(Bx,X(x,y),X(y,x))[1+ϑ(Bu,X(u,v),X(u,v))][1+ϑ(Bw,X(w,z),X(w,z))][1+2ϑ(Bx,Bu,Bw)]2+βϑ(Bx,Bu,Bw),+γϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)),      ϑ(Bx,X(x,y),X(x,y),ϑ(Bu,X(u,v),X(u,v))}for allx,y,u,v,w,zMwithBxBuBw and ByBvBz; (2.6)

    (ii) X(M×M)g(M);

    (iii) B commutes with X.

    If there exist x0,y0M such that Bx0X(x0,y0) and By0X(y0,x0), then X and B have a coupled coincidence point in M×M.

    Proof. Let x0 and y0 be any two elements in M such that Bx0X(x0,y0) and By0X(y0,x0). Since X(M×M)B(M), we constuct two sequences xκκ=1 and yκκ=1 in M as follows:

    Bxκ+1=X(xκ,yκ) and Byκ+1=X(yκ,xκ) for κN.

    Since X has mixed B-monotone property, we have

    Bxκ=X(xκ1,yκ1)X(xκ,yκ1)X(xκ,yκ)=Bxκ+1 and Byκ+1=X(yκ,xκ)X(yκ1,xκ)X(yκ1,xκ1)=Byκ.

    Now using (2.1) with x=xκ, y=yκ, u=w=xκ1 and v=z=yκ1, we get

    ϑ(Bxκ+1,Bxκ,Bxκ)=ϑ(X(xκ,yκ),X(xκ1,yκ1),X(xκ1,yκ1))αϑ(Bxκ,X(xκ,yκ),X(xκ,yκ))[1+ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1))][1+ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1))][1+2ϑ(Bxκ,Bxκ1,Bxκ1)]2+βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)  +Lmin{ϑ(Bxκ,X(xκ1,yκ1),X(xκ1,yκ1)),ϑ(Bxκ1,X(xκ,yκ),X(xκ1,yκ1)),ϑ(Bxκ,X(xκ,yκ),X(yκ,xκ)),ϑ(Bxκ1,X(xκ1,yκ1),X(xκ1,yκ1))}=αϑ(Bxκ,Bxκ+1,Bxκ+1)[1+ϑ(Bxκ1,Bxκ,Bxκ)][1+ϑ(Bxκ1,Bxκ,Bxκ)][1+2ϑ(Bxκ,Bxκ1,Bxκ1)]2+βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)+Lmin{ϑ(Bxκ,Bxκ,Bxκ),ϑ(Bxκ1,Bxκ+1,Bxκ),ϑ(Bxκ,Bxκ+1,Bxκ+1),ϑ(Bxκ1,Bxκ,Bxκ)}α2ϑ(Bxκ+1,Bxκ,Bxκ)[1+2ϑ(Bxκ,Bxκ1,Bxκ1)][1+2ϑ(Bxκ,Bxκ1,Bxκ1)][1+2ϑ(Bxκ,Bxκ1,Bxκ1)]2+βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)

    so that

    ϑ(Bxκ+1,Bxκ,Bxκ)1(12α)[βϑ(Bxκ,Bxκ1,Bxκ1)+γϑ(Byκ,Byκ1,Byκ1)] (2.7)

    Similarly,

    ϑ(Byκ+1,Byκ,Byκ)1(12α)[βϑ(Byκ,Byκ1,Byκ1)+γϑ(Bxκ,Bxκ1,Bxκ1)] (2.8)

    Adding (2.7) and (2.8), we have

    ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ)β+γ12α[ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)]. (2.9)

    Let Δκ=ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ) and c=β+γ12α where 0c<1 inview of choice of α,β and γ.

    Now the inequality (2.9) becomes as follows

    Δκc.Δκ1 for nN.

    Consequently ΔκcΔκ1c2Δκ2...cκΔ0.

    If Δ0=0, we have ϑ(Bx1,Bx0,Bx0)+ϑ(By1,By0,By0)=0

    or ϑ(X(x0,y0),Bx0,Bx0)+ϑ(X(y0,x0),By0,By0)=0 which implies that X(x0,y0)=Bx0 and X(y0,x0)=By0.

    That is, (x0,y0) is a coupled coincidence point of X and B.

    Suppose that Δ0>0.

    Now using repeated application of rectangle inequality of ϑ-metric and inequality ΔκcκΔ0, we have

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxη,Bxκ+1,Bxκ+1)]+[ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byη,Byκ+1,Byκ+1)][ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+ϑ(Bxη,Bxκ+2,Bxκ+2)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+ϑ(Byη,Byκ+2,Byκ+2)][ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)++ϑ(Bxη,Bxη1,Bxη1)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)++ϑ(Byη,Byη1,Byη1)]=Δκ+Δκ+1+Δκ+2++Δη1[cκ+cκ+1++cη1]Δ0cκ11cΔ0 for η>κ

    or

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)cκ11cΔ0 for η>κ. (2.10)

    Since 0c<1, cκ0 as κ.

    Now applying limit as κ with η>κ in the inequality (2.10), we have

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)0 which follows that xκκ=1 and yκκ=1 are ϑcauchy sequences in M.

    Since (M,ϑ) is a partially ordered ϑ-complete, there exist a,bM such that xκa and yκb.

    Now we prove that (a,b) is a coupled coincidence point of X and B.

    Since X commutes with B, we have

    X(Bxκ,Byκ)=B(X(xκ,yκ))=B(Bxκ+1)   andX(Byκ,Bxκ)=B(X(yκ,xκ))=B(Byκ+1)

    Since X and B are continuous, we have

    limκX(Bxκ,Byκ)=limκB(Bxκ+1)=Ba   andlimκX(Byκ,Bxκ)=limκB(Byκ+1)=Bb

    Since ϑ is continuous in all its variables, we have

    ϑ(X(a,b),Ba,Bb)=ϑ(limκX(Bxκ,Byκ),Ba,Ba)=ϑ(Ba,Ba,Ba)

    so that

    ϑ(X(a,b),Bb,Bb)=0

    which implies that X(a,b)=Ba.

    Similarly, it can be verified that X(b,a)=Bb.

    Thus(a,b) is a coupled coincidence point of X and B in M×M.

    Remark 2.2. If we take B is an identity mapping and γ=0 in the Theorem 2.2, we get Theorem 3.1 in the results of the Chouhan and Richa Sharma[5].

    Theorem 2.3. Let (M,ϑ,) be a partially ordered complete ϑ-metric space and X:M×MM, B:MM be two continuous mappings such that X has mixed B-monotone property and

    (i) there exist non negative real numbers Ψ1,Ψ2,Ψ3,Ψ4,Ψ5,Ψ6,Ψ7,Ψ8 and Ψ9 with 0Ψ1+Ψ2+Ψ3+Ψ4+Ψ5+Ψ6+Ψ7+Ψ8+Ψ9<1 such that

    ϑ(X(x,y),X(u,v),X(w,z))Ψ1ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)2+Ψ2ϑ(X(x,y),X(u,v),X(w,z))ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ3ϑ(X(x,y),X(u,v),X(w,z))ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)+Ψ4ϑ(Bx,Bx,X(x,y))ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ5ϑ(Bx,Bx,X(x,y))ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)+Ψ6ϑ(Bu,Bu,X(u,v))ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ7ϑ(Bu,Bu,X(u,v))ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)+Ψ8ϑ(Bw,Bw,X(w,z))ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ9ϑ(Bw,Bw,X(w,z))ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)for allx,y,u,v,w,zMwithBxBuBw and ByBvBz; (2.11)

    (ii) X(M×M)B(M);

    (iii) B commutes with X.

    If there exist x0,y0M such that Bx0X(x0,y0) and By0X(y0,x0), then X and B have a coupled coincidence point in M×M.

    Proof. Let x0 and y0 be any two elements in M such that Bx0X(x0,y0) and By0X(y0,x0). Since X(M×M)g(M), we constuct two sequences xκκ=1 and yκκ=1 in M as follows:

    Bxκ+1=X(xκ,yκ) and Byκ+1=X(yκ,xκ) for κN.

    Since X has mixed B-monotone property, we have

    Bxκ=X(xκ1,yκ1)X(xκ,yκ1)X(xκ,yκ)=Bxκ+1 and Byκ+1=X(yκ,xκ)X(yκ1,xκ)X(yκ1,xκ1)=Byκ.

    Now using (2.11) with x=xκ, y=yκ, u=w=xκ1 and v=z=yκ1, we get

    ϑ(Bxκ+1,Bxκ,Bxκ)=ϑ(X(xκ,yκ),X(xκ1,yκ1),X(xκ1,yκ1))Ψ1ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)2+Ψ2ϑ(X(xκ,yκ),X(xκ1,yκ1),X(xκ1,yκ1))ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ3ϑ(X(xκ,yκ),X(xκ1,yκ1),X(xκ1,yκ1))ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)+Ψ4ϑ(Bxκ,Bxκ,X(xκ,yκ))ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ5ϑ(Bxκ,Bxκ,X(xκ,yκ))ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)+Ψ6ϑ(Bxκ1,Bxκ1,X(xκ1,yκ1))ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ7ϑ(Bxκ1,Bxκ1,X(xκ1,yκ1))ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)+Ψ8ϑ(Bxκ1,Bxκ1,X(xκ1,yκ1))ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ9ϑ(Bxκ1,Bxκ1,X(xκ1,yκ1))ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)
    =Ψ1ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)2+Ψ2ϑ(Bxκ+1,Bxκ,Bxκ)ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byn1),+Ψ3ϑ(Bxκ+1,Bxκ,Bxκ)ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)+Ψ4ϑ(Bxκ,Bxκ,Bxκ+1)ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ5ϑ(Bxκ,Bxκ,Bxκ+1)ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)+Ψ6ϑ(Bxκ1,Bxκ1,Bxκ)ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ7ϑ(Bxκ1,Bxκ1,Bxκ)ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)+Ψ8ϑ(Bxκ1,Bxκ1,Bxκ)ϑ(Bxκ,Bxκ1,Bxκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1),+Ψ9ϑ(Bxκ1,Bxκ1,Bxκ)ϑ(Byκ,Byκ1,Byκ1)1+ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)
    Ψ1ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)2+Ψ2ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ3ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ4ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ5ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ6ϑ(Bxκ1,Bxκ1,Bxκ)+Ψ7ϑ(Bxκ1,Bxκ1,Bxκ)+Ψ8ϑ(Bxκ1,Bxκ1,Bxκ)+Ψ9ϑ(Bxκ1,Bxκ1,Bxκ)

    so that

    ϑ(Bxκ+1,Bxκ,Bxκ)Ψ12+Ψ6+Ψ7+Ψ8+Ψ9[1(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Bxκ,Bxκ1,Bxκ1)+Ψ12[1(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Byκ,Byκ1,Byκ1) (2.12)

    Similarly,

    ϑ(Byκ+1,Byκ,Byκ)Ψ12+Ψ6+Ψ7+Ψ8+Ψ9[1(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Byκ,Byκ1,Byκ1)+Ψ12[1(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Bxκ,Bxκ1,Bxκ1) (2.13)

    Adding (2.12) and (2.13), we have

    ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ)Ψ1+Ψ6+Ψ7+Ψ8+Ψ91(Ψ2+Ψ3+Ψ4+Ψ5)[ϑ(Bxκ,Bxκ1,Bxκ1)+ϑ(Byκ,Byκ1,Byκ1)]. (2.14)

    Let Δκ=ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ) and c=Ψ1+Ψ6+Ψ7+Ψ8+Ψ91(Ψ2+Ψ3+Ψ4+Ψ5) where 0c<1 inview of choice of Ψ1,Ψ2,Ψ3,Ψ4,Ψ5,Ψ6,Ψ7,Ψ8 and Ψ9.

    Now the inequality (2.14) becomes as follows

    Δκc.Δκ1 for nN,

    Consequently ΔκcΔκ1c2Δκ2...cκΔ0.

    If Δ0=0, we have ϑ(Bx1,Bx0,Bx0)+ϑ(By1,By0,By0)=0,

    or ϑ(X(x0,y0),Bx0,Bx0)+ϑ(X(y0,x0),By0,By0)=0 which implies that X(x0,y0)=Bx0 and X(y0,x0)=By0.

    That is, (x0,y0) is a coupled coincidence point of X and B.

    Suppose that Δ0>0.

    Now using rectangle inequality of ϑ-metric repeatedly and inequality ΔκcκΔ0, we have

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxη,Bxκ+1,Bxκ+1)]+[ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byη,Byκ+1,Byκ+1)][ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+ϑ(Bxη,Bxκ+2,Bxκ+2)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+ϑ(Byη,Byκ+2,Byκ+2)][ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)++ϑ(Bxη,Bxη1,Bxη1)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)++ϑ(Byη,Byη1,Byη1)]=Δκ+Δκ+1+Δκ+2++Δη1[cκ+cκ+1++cη1]Δ0cκ11cΔ0 for η>κ

    or

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)cκ11cΔ0 for η>κ. (2.15)

    Since 0c<1, cκ0 as κ.

    Now applying limit as κ with η>κ in the inequality (2.15), we have

    ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)0 which follows that xκκ=1 and yκκ=1 are ϑcauchy sequences in M.

    Since (M,ϑ) is a partially ordered complete ϑ-metric space, there exist r,sM such that xκr and yκs.

    Now we prove that (r,s) is a coupled coincidence point of X and B.

    Since X commutes with B, we have

    X(Bxκ,Byκ)=B(X(xκ,yκ))=B(Bxκ+1)   andX(Byκ,Bxκ)=B(X(yκ,xκ))=B(Byκ+1)

    Since X and B are continuous, we have

    limκX(Bxκ,Byκ)=limκB(Bxκ+1)=Br   andlimκX(Byκ,Bxκ)=limκB(Byκ+1)=Bs

    Since ϑ is continuous in all its variables, we have

    ϑ(X(r,s),Br,Bs)=ϑ(limκX(Bxκ,Byκ),Br,Br)=ϑ(Br,Br,Br),

    so that

    ϑ(X(r,s),Br,Bs)=0

    which implies that X(r,s)=Br.

    Similarly, it can be proved that X(s,r)=Bs.

    Hence (s,r) is a coupled coincidence point of X and B.

    Taking α=0, L=0 and B is an identity mapping in Theorem 2.1, we get

    Corollary 2.1. Let (M,ϑ,) be a partially ordered complete ϑ-metric space and X:M×MM be a continuous mapping such that X has mixed monotone property and there exist β,γ[0,1) with β+γ<1 such that

    ϑ(X(x,y),X(u,v),X(w,z))βϑ(x,u,w)+γϑ(y,v,z). (2.16)

    If there exist x0,y0M such that x0X(x0,y0) and y0X(y0,x0), then X has a coupled fixed point in M×M.

    Consider the following system of nonlinear integral equations:

    f(s)=q(s)+a0λ(s,t)[X1(t,f(t))+X2(t,g(t))]dt,g(s)=q(s)+a0λ(s,t)[X1(t,g(t))+X2(t,f(t))]dt,s[0,L],L>0. (3.1)

    Let M=C([0,L],R) be the class of all real valued continuous functions on [0,L].

    Define

    ϑ(a,b,c)=sup{|a(s)b(s)|/s[0,L]}x+sup{|b(s)c(s)|/s[0,L]}+sup{|c(s)a(s)|/s[0,L]}

    and the partial ordered relation on M as

    aba(s)b(s) for all a,bM and s[0,L]. (3.2)

    Then (M,ϑ,) is a partially ordered complete ϑ-metric space. We make the following assumptions:

    (a) The mappings X1:[0,L]×RR, X2:[0,L]×RR, q:[0,L]R and λ:[0,L]×R[0,) are continuous;

    (b) there exist c>0 and β,γ[0,1) with β+γ<1 such that

    0X1(s,b)X1(s,a)cβ(ba)0X2(s,a)X2(s,b)cγ(ba)

    for all a,bR with ba and s[0,L];

    (c) csup{L0λ(s,t)dt:s[0,L]}<1;

    (d) there exists u0 and v0 in M such that

    u0(s)q(s)+L0λ(s,t)[X1(t,u0(t))+X2(t,v0(t))]dt,v0(s)q(s)+L0λ(s,t)[X1(t,v0(t))+X2(t,u0(t))]dt.

    Then the system (3.1) has a solution in M×M where M=C([0,L],R). To achieve this, define X:M×MM as

    X(f,g)(s)=q(s)+L0λ(s,t)[X1(t,f(t))+X2(t,g(t))]dt for all f,gM and s[0,L]. (3.3)

    Using condition (b), it can be shown that X has mixed monotone property. Now for x,y,u,v,w,zM with xuw, yvz,

    ϑ(X(x,y),X(u,v),X(w,z))=sup{|X(x,y)(s)X(u,v)(s)|/s[0,L]}+sup{|X(u,v)(s)X(w,z)(s)|/s[0,L]}+sup{|X(w,z)(s)X(x,y)(s)|/s[0,L]}=sup{|L0λ(s,t)[X1(t,x(t))+X2(t,y(t))]dtL0λ(s,t)[X1(t,u(t))+X2(t,v(t))]dt|/s[0,L]}+sup{|L0λ(s,t)[X1(t,u(t))+X2(t,v(t))]dtL0λ(s,t)[X1(t,w(t))+X2(t,z(t))]dt|/s[0,L]}+sup{|L0λ(s,t)[X1(t,w(t))+X2(t,z(t))]dtL0λ(s,t)[X1(t,x(t))+X2(t,y(t))]dt|/s[0,L]}
    sup{|L0[X1(t,x(t))X1(t,u(t))]||λ(s,t)|dt/s[0,L]}+sup|L0[X2(t,y(t))X2(t,v(t))]||λ(s,t)|dt/s[0,L]}+sup{|L0[X1(t,u(t))X1(t,w(t))]||λ(s,t)|dt/s[0,L]}+sup|L0[X2(t,v(t))X2(t,z(t))]||λ(s,t)|dt/s[0,L]}+sup{|L0[X1(t,w(t))X1(t,x(t))]||λ(s,t)|dt/s[0,L]}+sup|L0[X2(t,z(t))X2(t,y(t))]||λ(s,t)|dt/s[0,L]}cβsup{L0|x(t)u(t)||λ(s,t)|dt/s[0,L]}+cγsup{L0|y(t)v(t)||λ(s,t)|dt/s[0,L]}+cβsup{L0|u(t)w(t)||λ(s,t)|dt/s[0,L]}+cγsup{L0|v(t)z(t)||λ(s,t)|dt/s[0,L]}+cβsup{L0|w(t)x(t)||λ(s,t)|dt/s[0,L]}+cγsup{L0|z(t)y(t)||λ(s,t)|dt/s[0,L]}
    β[sup{|x(s)u(s)|/s[0,L]}+sup{|u(s)w(s)|/s[0,L]}+sup{|w(s)x(s)|/s[0,L]}]c sup{L0|λ(s,t)|dt/s[0,L]}+γ[sup{|y(s)v(s)|/s[0,L]}+sup{|v(s)z(s)|/s[0,L]}+sup{|z(s)y(s)|/s[0,L]}]c sup{L0|λ(s,t)|dt/s[0,L]}β[sup{|x(s)u(s)|/s[0,L]}+sup{|u(s)w(s)|/s[0,L]}+sup{|w(s)x(s)|/s[0,L]]+γsup{|y(s)v(s)|/s[0,L]}+sup{|v(s)z(s)|/s[0,L]}+sup{|z(s)y(s)|/s[0,L]]=βϑ(x,u,w)+γϑ(y,v,z)

    So that

    ϑ(X(x,y),X(u,v),X(w,z))βϑ(x,u,w)+γϑ(y,v,z)

    Hence all the conditions of Corollary 2.1 are satisfied. Therefore, X has a coupled fixed point in M×M. In other words, the system (3.1) of nonlinear integral equations has a solution in M×M.

    The aforesaid application is illustrated by the following example:

    Example 3.1. Let M=C([0,1],R), Now consider the integral equation in M as

    X(f,g)(s)=s3+74+10t224(s+3)[f(t)+2g(t)+3]dt. (3.4)

    Then clearly the above equation is in the form of following equation:

    X(f,g)(s)=q(s)+L0λ(s,t)[X1(t,f(t))+X2(t,g(t))]dt for all f,gM and s[0,L],

    where q(s)=s3+74, λ(s,t)=t224(s+3), X1(t,s)=s, X2(t,s)=2s+3 and L=1.

    That is, (3.4) is a special case of (3.3) in Theorem 3.1.

    Here it is easy to verify that the functions q(s),λ(s,t),X1(t,s) and X2(t,s) are continuous. Moreover, there exist c=9,  β=13 and γ=12 with β+γ<1 such that

    0X1(s,b)X1(s,a)cβ(ba)0X2(s,a)X2(s,b)cγ(ba)

    for all a,bR with ba and s[0,1].

    and

    csup{L0λ(s,t)dt:s[0,L]}=9sup{10t224(s+3)dt:s[0,1]}.=9sup{172(s+3):s[0,1]}<1.

    Thus the conditions (a), (b) and (c) of Theorem 3.1 are satisfied.

    Now consider u0(s)=1 and v0(s)=1. Then we have

    q(s)+10λ(s,t)[X1(t,v0(t))+X2(t,u0(t))]dt=s3+74+10t224(s+3)[1+24]dt=s3+74+148(s+3)1

    That is, v0X(v0,u0). Similarly, it can be shown that u0X(u0,v0).

    Thus all the conditions of Theorem 3.1 are satisfied. It follows that the integral Eq (3.4) has a solution in M×M with M=C([0,1],R).

    Some coupled coincidence point theorems for two mappings established using rational type contractions in the setting of partially ordered Gmetric spaces. By considering Gmetric space, we propose a fairly simple solution for a system of nonlinear integral equations by using fixed point technique. Moreover, supporting example (exact solution) is provided to strengthen our obtained results.

    The third and fourth authors 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 interests.



    [1] X. Guo, M. R. Lin, A. Azizi, L. P. Saldyt, Y. Kang, T. P. Pavlic, et al., Decoding alarm signal propagation of seed-harvester ants using automated movement tracking and supervised machine learning, Proc. R. Soc. B, 289 (2022), 20212176. https://doi.org/10.1098/rspb.2021.2176 doi: 10.1098/rspb.2021.2176
    [2] O. Feinerman, A. Korman, Individual versus collective cognition in social insects, J. Exp. Biol., 220 (2017), 73–82. https://doi.org/10.1242/jeb.143891 doi: 10.1242/jeb.143891
    [3] B. Doerr, M. Fouz, T. Friedrich, Why rumors spread fast in social networks, Commun. ACM, 55 (2012), 70–75. https://doi.org/10.1145/2184319.2184338 doi: 10.1145/2184319.2184338
    [4] L. Bonnasse-Gahot, H. Berestycki, M. Depuiset, M. B. Gordon, S. Roché, N. Rodriguez, et al., Epidemiological modelling of the 2005 french riots: A spreading wave and the role of contagion, Sci. Rep., 8 (2018), 107. https://doi.org/10.1038/s41598-017-18093-4 doi: 10.1038/s41598-017-18093-4
    [5] D. A. Sprague, T. House, Evidence for complex contagion models of social contagion from observational data, PLoS One, 12 (2017), 1–12. https://doi.org/10.1371/journal.pone.0180802 doi: 10.1371/journal.pone.0180802
    [6] C. E. Coltart, B. Lindsey, I. Ghinai, A. M. Johnson, D. L. Heymann, The ebola outbreak, 2013–2016: Old lessons for new epidemics, Phil. Trans. R. Soc. B, 372 (2017), 20160297. https://doi.org/10.1098/rstb.2016.0297 doi: 10.1098/rstb.2016.0297
    [7] B. Hölldobler, E. O. Wilson, The Ants, Harvard University Press, 1990.
    [8] F. E. Regnier, E. O. Wilson, The alarm-defence system of the ant acanthomyops claviger, J. Insect Physiol., 14 (1968), 955–970. https://doi.org/10.1016/0022-1910(68)90006-1 doi: 10.1016/0022-1910(68)90006-1
    [9] W. H. Bossert, E. O. Wilson, The analysis of olfactory communication among animals, J. Theor. Biol., 5 (1963), 443–469. https://doi.org/10.1016/0022-5193(63)90089-4 doi: 10.1016/0022-5193(63)90089-4
    [10] E. Frehland, B. Kleutsch, H. Markl, Modelling a two-dimensional random alarm process, BioSystems, 18 (1985), 197–208. https://doi.org/10.1016/0303-2647(85)90071-1 doi: 10.1016/0303-2647(85)90071-1
    [11] D. J. McGurk, J. Frost, E. J. Eisenbraun, K. Vick, W. A. Drew, J. Young, Volatile compounds in ants: Identification of 4-methyl-3-heptanone from Pogonomyrmex ants, J. Insect Physiol., 12 (1966), 1435–1441. https://doi.org/10.1016/0022-1910(66)90157-0 doi: 10.1016/0022-1910(66)90157-0
    [12] J. B. Xavier, J. M. Monk, S. Poudel, C. J. Norsigian, A. V. Sastry, C. Liao, et al., Mathematical models to study the biology of pathogens and the infectious diseases they cause, Iscience, 25 (2022), 104079. https://doi.org/10.1016/j.isci.2022.104079 doi: 10.1016/j.isci.2022.104079
    [13] P. Törnberg, Echo chambers and viral misinformation: Modeling fake news as complex contagion, PLoS One, 13 (2018), 1–21. https://doi.org/10.1371/journal.pone.0203958 doi: 10.1371/journal.pone.0203958
    [14] G. I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Springer, 2013. https://doi.org/10.1007/978-94-015-8798-3
    [15] L. G. de Pillis, A. E. Radunskaya, A mathematical model of immune response to tumor invasion, in Computational Fluid and Solid Mechanics 2003, Elsevier, (2003), 1661–1668. https://doi.org/10.1016/B978-008044046-0.50404-8
    [16] L. G. de Pillis, A. Eladdadi, A. E. Radunskaya, Modeling cancer-immune responses to therapy, J. Pharmacokinet. Pharmacodyn., 41 (2014), 461–478. https://doi.org/10.1007/s10928-014-9386-9 doi: 10.1007/s10928-014-9386-9
    [17] A. M. Smith, Validated models of immune response to virus infection, Curr. Opin. Syst. Biol., 12 (2018), 46–52. https://doi.org/10.1016/j.coisb.2018.10.005 doi: 10.1016/j.coisb.2018.10.005
    [18] J. M. Conway, R. M. Ribeiro, Modeling the immune response to hive infection, Curr. Opin. Syst. Biol., 12 (2018), 61–69. https://doi.org/10.1016/j.coisb.2018.10.006 doi: 10.1016/j.coisb.2018.10.006
    [19] S. Legewie, N. Blüthgen, H. Herzel, Mathematical modeling identifies inhibitors of apoptosis as mediators of positive feedback and bistability, PLoS Comput. Biol., 2 (2006), e120. https://doi.org/10.1371/journal.pcbi.0020120 doi: 10.1371/journal.pcbi.0020120
    [20] T. Fasciano, H. Nguyen, A. Dornhaus, M. C. Shin, Tracking multiple ants in a colony, in 2013 IEEE Workshop on Applications of Computer Vision (WACV), IEEE, (2013), 534–540. https://doi.org/10.1109/WACV.2013.6475065
    [21] C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, in SIGGRAPH '87: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, ACM, (1987), 25–34. https://doi.org/10.1145/37401.37406
    [22] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226–1229. https://doi.org/10.1103/PhysRevLett.75.1226 doi: 10.1103/PhysRevLett.75.1226
    [23] H. Chaté, F. Ginelli, G. Grégoire, F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113. https://doi.org/10.1103/PhysRevE.77.046113 doi: 10.1103/PhysRevE.77.046113
    [24] I. D. Couzin, J. Krause, R. James, G. D. Ruxton, N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theor. Biol., 218 (2002), 1–11. https://doi.org/10.1006/jtbi.2002.3065 doi: 10.1006/jtbi.2002.3065
    [25] H. S. Fisher, L. Giomi, H. E. Hoekstra, L. Mahadevan, The dynamics of sperm cooperation in a competitive environment, Proc. R. Soc. B, 281 (2014), 20140296. https://doi.org/10.1098/rspb.2014.0296 doi: 10.1098/rspb.2014.0296
    [26] H. Hildenbrandt, C. Carere, C. K. Hemelrijk, Self-organized aerial displays of thousands of starlings: A model, Behav. Ecol., 21 (2010), 1349–1359. https://doi.org/10.1093/beheco/arq149 doi: 10.1093/beheco/arq149
    [27] J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, Convergence properties of the nelder–mead simplex method in low dimensions, SIAM J. Optim., 9 (1998), 112–147. https://doi.org/10.1137/S1052623496303470 doi: 10.1137/S1052623496303470
    [28] A. Lipp, H. Wolf, F. Lehmann, Walking on inclines: Energetics of locomotion in the ant Camponotus, J. Exp. Biol., 208 (2005), 707–719. https://doi.org/10.1242/jeb.01434 doi: 10.1242/jeb.01434
    [29] N. C. Holt, G. N. Askew, Locomotion on a slope in leaf-cutter ants: Metabolic energy use, behavioural adaptations and the implications for route selection on hilly terrain, J. Exp. Biol., 215 (2012), 2545–2550. https://doi.org/10.1242/jeb.057695 doi: 10.1242/jeb.057695
    [30] M. J. Greene, D. M. Gordon, Interaction rate informs harvester ant task decisions, Behav. Ecol., 18 (2007), 451–455. https://doi.org/10.1093/beheco/arl105 doi: 10.1093/beheco/arl105
    [31] D. M. Gordon, N. J. Mehdiabadi, Encounter rate and task allocation in harvester ants, Behav. Ecol. Sociobiol., 45 (1999), 370–377. https://doi.org/10.1007/s002650050573 doi: 10.1007/s002650050573
    [32] D. M. Gordon, The regulation of foraging activity in red harvester ant colonies, Am. Nat., 159 (2002), 509–518. https://doi.org/10.1086/339461 doi: 10.1086/339461
    [33] N. Razin, J. Eckmann, O. Feinerman, Desert ants achieve reliable recruitment across noisy interactions, J. R. Soc. Interface, 10 (2013), 20130079. https://doi.org/10.1098/rsif.2013.0079
    [34] S. C. Pratt, Behavioral mechanisms of collective nest-site choice by the ant Temnothorax curvispinosus, Insect. Soc., 52 (2005), 383–392. https://doi.org/10.1007/s00040-005-0823-z doi: 10.1007/s00040-005-0823-z
    [35] S. C. Pratt, Quorum sensing by encounter rates in the ant Temnothorax albipennis, Behav. Ecol., 16 (2005), 488–496. https://doi.org/10.1093/beheco/ari020 doi: 10.1093/beheco/ari020
    [36] A. Dornhaus, Specialization does not predict individual efficiency in an ant, PLoS Biol., 6 (2008), e285. https://doi.org/10.1371/journal.pbio.0060285 doi: 10.1371/journal.pbio.0060285
    [37] S. N. Beshers, J. H. Fewell, Models of division of labor in social insects, Annu. Rev. Entomol., 46 (2001), 413–440. https://doi.org/10.1146/annurev.ento.46.1.413 doi: 10.1146/annurev.ento.46.1.413
    [38] D. Charbonneau, C. Poff, H. Nguyen, M. C. Shin, K. Kierstead, A. Dornhaus, Who are the "lazy" ants? The function of inactivity in social insects and a possible role of constraint: Inactive ants are corpulent and may be young and/or selfish, Integr. Comp. Biol., 57 (2017), 649–667. https://doi.org/10.1093/icb/icx029 doi: 10.1093/icb/icx029
    [39] A. Bernadou, J. Busch, J. Heinze, Diversity in identity: Behavioral flexibility, dominance, and age polyethism in a clonal ant, Behav. Ecol. Sociobiol., 69 (2015), 1365–1375. https://doi.org/10.1007/s00265-015-1950-9 doi: 10.1007/s00265-015-1950-9
    [40] E. J. H. Robinson, T. O. Richardson, A. B. Sendova-Franks, O. Feinerman, N. R. Franks, Radio tagging reveals the roles of corpulence, experience and social information in ant decision making, Behav. Ecol. Sociobiol., 63 (2009), 627–636. https://doi.org/10.1007/s00265-008-0696-z doi: 10.1007/s00265-008-0696-z
    [41] H. G. Tanner, A. Jadbabaie, G. J. Pappas, Stable flocking of mobile agents, part I: Fixed topology, in 42nd IEEE International Conference on Decision and Control, IEEE, (2003), 2010–2015. https://doi.org/10.1109/CDC.2003.1272910
    [42] A. Kolpas, M. Busch, H. Li, I. D. Couzin, L. Petzold, J. Moehlis, How the spatial position of individuals affects their influence on swarms: A numerical comparison of two popular swarm dynamics models, PloS One, 8 (2013), e58525. https://doi.org/10.1371/journal.pone.0058525 doi: 10.1371/journal.pone.0058525
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