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Review

Efficacy and safety of non-invasive brain stimulation techniques for the treatment of nicotine addiction: A systematic review of randomized controlled trials

  • Received: 22 May 2024 Revised: 17 June 2024 Accepted: 25 June 2024 Published: 01 July 2024
  • Non-Invasive Brain Stimulation (NIBS) techniques seem to be effective in treating tobacco use disorder. We aimed to analyze what kinds of protocols are used to treat nicotine addiction in term of cessation and/or reduction and to evaluate the long-term effects of NIBS techniques.We searched PubMed, Scopus, and Web of Science for papers published, with combinations of the following search terms: “Non-invasive brain stimulation OR TMS OR transcranial magnetic stimulation OR tDCS OR transcranial direct current stimulation OR transcranial electrical stimulation OR TES AND Nicotine addiction”.We conducted a preliminary search, which revealed papers on the topic. Articles were included in the review according to the following inclusion criteria: English language, publication in peer reviewed journals, articles about studies performed on non-invasive brain stimulations techniques, and RCT studies. Studies involving clinical populations with organic or psychiatric diseases were excluded. We found 280 articles. Of these, at the first screening and conducted by title and abstract, 63 studies were excluded after duplicates were removed (118). After the second screening conducted by full-text examination, 45 articles were excluded. Ten studies met the inclusion criteria and were included in the review.The clinical benefits of NIBS, including the fast onset and minor side effects, showed that this kind of treatment could be helpful in patients with a long history of smoking in terms of cessation and abstinence rates.

    Citation: Fiammetta Iannuzzo, Silvia Crudo, Gianpaolo Antonio Basile, Fortunato Battaglia, Carmenrita Infortuna, Maria Rosaria Anna Muscatello, Antonio Bruno. Efficacy and safety of non-invasive brain stimulation techniques for the treatment of nicotine addiction: A systematic review of randomized controlled trials[J]. AIMS Neuroscience, 2024, 11(3): 212-225. doi: 10.3934/Neuroscience.2024014

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  • Non-Invasive Brain Stimulation (NIBS) techniques seem to be effective in treating tobacco use disorder. We aimed to analyze what kinds of protocols are used to treat nicotine addiction in term of cessation and/or reduction and to evaluate the long-term effects of NIBS techniques.We searched PubMed, Scopus, and Web of Science for papers published, with combinations of the following search terms: “Non-invasive brain stimulation OR TMS OR transcranial magnetic stimulation OR tDCS OR transcranial direct current stimulation OR transcranial electrical stimulation OR TES AND Nicotine addiction”.We conducted a preliminary search, which revealed papers on the topic. Articles were included in the review according to the following inclusion criteria: English language, publication in peer reviewed journals, articles about studies performed on non-invasive brain stimulations techniques, and RCT studies. Studies involving clinical populations with organic or psychiatric diseases were excluded. We found 280 articles. Of these, at the first screening and conducted by title and abstract, 63 studies were excluded after duplicates were removed (118). After the second screening conducted by full-text examination, 45 articles were excluded. Ten studies met the inclusion criteria and were included in the review.The clinical benefits of NIBS, including the fast onset and minor side effects, showed that this kind of treatment could be helpful in patients with a long history of smoking in terms of cessation and abstinence rates.


    Segre [1] made a pioneering attempt in the development of special algebra. He conceptualized the commutative generalization of complex numbers, bicomplex numbers, tricomplex numbers, etc. as elements of an infinite set of algebras. Subsequently, in the 1930s, researchers contributed in this area [2,3,4]. The next fifty years failed to witness any advancement in this field. Later, Price [5] developed the bicomplex algebra and function theory. Recent works in this subject [6,7] find some significant applications in different fields of mathematical sciences as well as other branches of science and technology. An impressive body of work has been developed by a number of researchers. Among these works, an important work on elementary functions of bicomplex numbers has been done by Luna-Elizaarrarˊas et al. [8]. Choi et al. [9] proved some common fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Jebril [10] proved some common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces. In 2017, Dhivya and Marudai [11] introduced the concept of a complex partial metric space, suggested a plan to expand the results and proved some common fixed point theorems under a rational expression contraction condition. In 2019, Mani and Mishra [12] proved coupled fixed point theorems on a complex partial metric space using different types of contractive conditions. In 2021, Gunaseelan et al. [13] proved common fixed point theorems on a complex partial metric space. In 2021, Beg et al.[14] proved fixed point theorems on a bicomplex valued metric space. In 2021, Zhaohui et al. [15] proved common fixed theorems on a bicomplex partial metric space. In this paper, we prove coupled fixed point theorems on a bicomplex partial metric space. An example is provided to verify the effectiveness and applicability of our main results. An application of these results to Fredholm integral equations and nonlinear integral equations is given.

    Throughout this paper, we denote the set of real, complex and bicomplex numbers, respectively, as C0, C1 and C2. Segre [1] defined the complex number as follows:

    z=ϑ1+ϑ2i1,

    where ϑ1,ϑ2C0, i21=1. We denote the set ofcomplex numbers C1 as:

    C1={z:z=ϑ1+ϑ2i1,ϑ1,ϑ2C0}.

    Let zC1; then, |z|=(ϑ21+ϑ22)12. The norm ||.|| of an element in C1 is the positive real valued function ||.||:C1C+0 defined by

    ||z||=(ϑ21+ϑ22)12.

    Segre [1] defined the bicomplex number as follows:

    ς=ϑ1+ϑ2i1+ϑ3i2+ϑ4i1i2,

    where ϑ1,ϑ2,ϑ3,ϑ4C0, and independent units i1,i2 are such that i21=i22=1 and i1i2=i2i1. We denote the set of bicomplex numbers C2 as:

    C2={ς:ς=ϑ1+ϑ2i1+ϑ3i2+ϑ4i1i2,ϑ1,ϑ2,ϑ3,ϑ4C0},

    i.e.,

    C2={ς:ς=z1+i2z2,z1,z2C1},

    where z1=ϑ1+ϑ2i1C1 and z2=ϑ3+ϑ4i1C1. If ς=z1+i2z2 and η=ω1+i2ω2 are any two bicomplex numbers, then the sum is ς±η=(z1+i2z2)±(ω1+i2ω2)=z1±ω1+i2(z2±ω2), and the product is ς.η=(z1+i2z2)(ω1+i2ω2)=(z1ω1z2ω2)+i2(z1ω2+z2ω1).

    There are four idempotent elements in C2: They are 0,1,e1=1+i1i22,e2=1i1i22 of which e1 and e2 are nontrivial, such that e1+e2=1 and e1e2=0. Every bicomplex number z1+i2z2 can be uniquely expressed as the combination of e1 and e2, namely

    ς=z1+i2z2=(z1i1z2)e1+(z1+i1z2)e2.

    This representation of ς is known as the idempotent representation of a bicomplex number, and the complex coefficients ς1=(z1i1z2) and ς2=(z1+i1z2) are known as the idempotent components of the bicomplex number ς.

    An element ς=z1+i2z2C2 is said to be invertible if there exists another element η in C2 such that ςη=1, and η is said to be inverse (multiplicative) of ς. Consequently, ς is said to be the inverse(multiplicative) of η. An element which has an inverse in C2 is said to be a non-singular element of C2, and an element which does not have an inverse in C2 is said to be a singular element of C2.

    An element ς=z1+i2z2C2 is non-singular if and only if ||z21+z22||0 and singular if and only if ||z21+z22||=0. When it exists, the inverse of ς is as follows.

    ς1=η=z1i2z2z21+z22.

    Zero is the only element in C0 which does not have a multiplicative inverse, and in C1, 0=0+i10 is the only element which does not have a multiplicative inverse. We denote the set of singular elements of C0 and C1 by O0 and O1, respectively. However, there is more than one element in C2 which does not have a multiplicative inverse: for example, e1 and e2. We denote this set by O2, and clearly O0={0}=O1O2.

    A bicomplex number ς=ϑ1+ϑ2i1+ϑ3i2+ϑ4i1i2C2 is said to be degenerated (or singular) if the matrix

    (ϑ1ϑ2ϑ3ϑ4)

    is degenerated (or singular). The norm ||.|| of an element in C2 is the positive real valued function ||.||:C2C+0 defined by

    ||ς||=||z1+i2z2||={||z21||+||z22||}12=[|z1i1z2|2+|z1+i1z2|22]12=(ϑ21+ϑ22+ϑ23+ϑ24)12,

    where ς=ϑ1+ϑ2i1+ϑ3i2+ϑ4i1i2=z1+i2z2C2.

    The linear space C2 with respect to a defined norm is a normed linear space, and C2 is complete. Therefore, C2 is a Banach space. If ς,ηC2, then ||ςη||2||ς||||η|| holds instead of ||ςη||||ς||||η||, and therefore C2 is not a Banach algebra. For any two bicomplex numbers ς,ηC2, we can verify the following:

    1. ςi2η||ς||||η||,

    2. ||ς+η||||ς||+||η||,

    3. ||ϑς||=|ϑ|||ς||, where ϑ is a real number,

    4. ||ςη||2||ς||||η||, and the equality holds only when at least one of ς and η is degenerated,

    5. ||ς1||=||ς||1 if ς is a degenerated bicomplex number with 0ς,

    6. ||ςη||=||ς||||η||, if η is a degenerated bicomplex number.

    The partial order relation i2 on C2 is defined as follows. Let C2 be the set of bicomplex numbers and ς=z1+i2z2, η=ω1+i2ω2C2. Then, ςi2η if and only if z1ω1 and z2ω2, i.e., ςi2η if one of the following conditions is satisfied:

    1. z1=ω1, z2=ω2,

    2. z1ω1, z2=ω2,

    3. z1=ω1, z2ω2,

    4. z1ω1, z2ω2.

    In particular, we can write ςi2η if ςi2η and ςη, i.e., one of 2, 3 and 4 is satisfied, and we will write ςi2η if only 4 is satisfied.

    Now, let us recall some basic concepts and notations, which will be used in the sequel.

    Definition 2.1. [15] A bicomplex partial metric on a non-void set U is a function ρbcpms:U×UC+2, where C+2={ς:ς=ϑ1+ϑ2i1+ϑ3i2+ϑ4i1i2,ϑ1,ϑ2,ϑ3,ϑ4C+0} and C+0={ϑ1C0|ϑ10} such that for all φ,ζ,zU:

    1. 0i2ρbcpms(φ,φ)i2ρbcpms(φ,ζ) (small self-distances),

    2. ρbcpms(φ,ζ)=ρbcpms(ζ,φ) (symmetry),

    3. ρbcpms(φ,φ)=ρbcpms(φ,ζ)=ρbcpms(ζ,ζ) if and only if φ=ζ (equality),

    4. ρbcpms(φ,ζ)i2ρbcpms(φ,z)+ρbcpms(z,ζ)ρbcpms(z,z) (triangularity) .

    A bicomplex partial metric space is a pair (U,ρbcpms) such that U is a non-void set and ρbcpms is a bicomplex partial metric on U.

    Example 2.2. Let U=[0,) be endowed with bicomplex partial metric space ρbcpms:U×UC+2 with ρbcpms(φ,ζ)=max{φ,ζ}ei2θ, where ei2θ=cosθ+i2sinθ, for all φ,ζU and 0θπ2. Obviously, (U,ρbcpms) is a bicomplex partial metric space.

    Definition 2.3. [15] A bicomplex partial metric space U is said to be a T0 space if for any pair of distinct points of U, there exists at least one open set which contains one of them but not the other.

    Theorem 2.4. [15] Let (U,ρbcpms) be a bicomplex partial metric space; then, (U,ρbcpms) is T0.

    Definition 2.5. [15] Let (U,ρbcpms) be a bicomplex partial metric space. A sequence {φτ} in U is said to be convergent and converges to φU if for every 0i2ϵC+2 there exists NN such that φτBρbcpms(φ,ϵ)={ωU:ρbcpms(φ,ω)<ϵ+ρbcpms(φ,φ)} for all τN, and it is denoted by limτφτ=φ.

    Lemma 2.6. [15] Let (U,ρbcpms) be a bicomplex partial metric space. A sequence {φτ}U is converges to φU iff ρbcpms(φ,φ)=limτρbcpms(φ,φτ).

    Definition 2.7. [15] Let (U,ρbcpms) be a bicomplex partial metric space. A sequence {φτ} in U is said to be a Cauchy sequence in (U,ρbcpms) if for any ϵ>0 there exist ϑC+2 and NN such that ||ρbcpms(φτ,φυ)ϑ||<ϵ for all τ,υN.

    Definition 2.8. [15] Let (U,ρbcpms) be a bicomplex partial metric space. Let {φτ} be any sequence in U. Then,

    1. If every Cauchy sequence in U is convergent in U, then (U,ρbcpms) is said to be a complete bicomplex partial metric space.

    2. A mapping S:UU is said to be continuous at φ0U if for every ϵ>0, there exists δ>0 such that S(Bρbcpms(φ0,δ))Bρbcpms(S(φ0,ϵ)).

    Lemma 2.9. [15] Let (U,ρbcpms) be a bicomplex partial metric space and {φτ} be a sequence in U. Then, {φτ} is a Cauchy sequence in U iff limτ,υρbcpms(φτ,φυ)=ρbcpms(φ,φ).

    Definition 2.10. Let (U,ρbcpms) be a bicomplex partial metric space. Then, an element (φ,ζ)U×U is said to be a coupled fixed point of the mapping S:U×UU if S(φ,ζ)=φ and S(ζ,φ)=ζ.

    Theorem 2.11. [15] Let (U,ρbcpms) be a complete bicomplex partial metric space and S,T:UU be two continuous mappings such that

    ρbcpms(Sφ,Tζ)i2lmax{ρbcpms(φ,ζ),ρbcpms(φ,Sφ),ρbcpms(ζ,Tζ),12(ρbcpms(φ,Tζ)+ρbcpms(ζ,Sφ))},

    for all φ,ζU, where 0l<1. Then, the pair (S,T) has a unique common fixed point, and ρbcpms(φ,φ)=0.

    Inspired by Theorem 2.11, here we prove coupled fixed point theorems on a bicomplex partial metric space with an application.

    Theorem 3.1. Let (U,ρbcpms) be a complete bicomplex partial metric space. Suppose that the mapping S:U×UU satisfies the following contractive condition:

    ρbcpms(S(φ,ζ),S(ν,μ))i2λρbcpms(S(φ,ζ),φ)+lρbcpms(S(ν,μ),ν),

    for all φ,ζ,ν,μU, where λ,l are nonnegative constants with λ+l<1. Then, S has a unique coupled fixed point.

    Proof. Choose ν0,μ0U and set ν1=S(ν0,μ0) and μ1=S(μ0,ν0). Continuing this process, set ντ+1=S(ντ,μτ) and μτ+1=S(μτ,ντ). Then,

    ρbcpms(ντ,ντ+1)=ρbcpms(S(ντ1,μτ1),S(ντ,μτ))i2λρbcpms(S(ντ1,μτ1),ντ1)+lρbcpms(S(ντ,μτ),ντ)=λρbcpms(ντ,ντ1)+lρbcpms(ντ+1,ντ)ρbcpms(ντ,ντ+1)i2λ1lρbcpms(ντ,ντ1),

    which implies that

    ||ρbcpms(ντ,ντ+1)||z||ρbcpms(ντ1,ντ)|| (3.1)

    where z=λ1l<1. Similarly, one can prove that

    ||ρbcpms(μτ,μτ+1)||z||ρbcpms(μτ1,μτ)||. (3.2)

    From (3.1) and (3.2), we get

    ||ρbcpms(ντ,ντ+1)||+||ρbcpms(μτ,μτ+1)||z(||ρbcpms(ντ1,ντ)||+||ρbcpms(μτ1,μτ)||),

    where z<1.

    Also,

    ||ρbcpms(ντ+1,ντ+2)||z||ρbcpms(ντ,ντ+1)|| (3.3)
    ||ρbcpms(μτ+1,μτ+2)||z||ρbcpms(μτ,μτ+1)||. (3.4)

    From (3.3) and (3.4), we get

    ||ρbcpms(ντ+1,ντ+2)||+||ρbcpms(μτ+1,μτ+2)||z(||ρbcpms(ντ,ντ+1)||+||ρbcpms(μτ,μτ+1)||).

    Repeating this way, we get

    ||ρbcpms(ντ,ντ+1)||+||ρbcpms(μτ,μτ+1)||z(||ρbcpms(μτ1,μτ)||+||ρbcpms(ντ1,ντ)||)z2(||ρbcpms(μτ2,μτ1)||+||ρbcpms(ντ2,ντ1)||)zτ(||ρbcpms(μ0,μ1)||+||ρbcpms(ν0,ν1)||).

    Now, if ||ρbcpms(ντ,ντ+1)||+||ρbcpms(μτ,μτ+1)||=γτ, then

    γτzγτ1zτγ0. (3.5)

    If γ0=0, then ||ρbcpms(ν0,ν1)||+||ρbcpms(μ0,μ1)||=0. Hence, ν0=ν1=S(ν0,μ0) and μ0=μ1=S(μ0,μ0), which implies that (ν0,μ0) is a coupled fixed point of S. Let γ0>0. For each τυ, we have

    ρbcpms(ντ,νυ)i2ρbcpms(ντ,ντ1)+ρbcpms(ντ1,ντ2)ρbcpms(ντ1,ντ1)+ρbcpms(ντ2,ντ3)+ρbcpms(ντ3,ντ4)ρbcpms(ντ3,ντ3)++ρbcpms(νυ+2,νυ+1)+ρbcpms(νυ+1,νυ)ρbcpms(νυ+1,νυ+1)i2ρbcpms(ντ,ντ1)+ρbcpms(ντ1,ντ2)++ρbcpms(νυ+1,νυ),

    which implies that

    ||ρbcpms(ντ,νυ)||||ρbcpms(ντ,ντ1)||+||ρbcpms(ντ1,ντ2)||++||ρbcpms(νυ+1,νυ)||.

    Similarly, one can prove that

    ||ρbcpms(μτ,μυ)||||ρbcpms(μτ,μτ1)||+||ρbcpms(μτ1,μτ2)||++||ρbcpms(μυ+1,μυ)||.

    Thus,

    ||ρbcpms(ντ,νυ)||+||ρbcpms(μτ,μυ)||γτ1+γτ2+γτ3++γυ(zτ1+zτ2++zυ)γ0zυ1zγ00asυ,

    which implies that {ντ} and {μτ} are Cauchy sequences in (U,ρbcpms). Since the bicomplex partial metric space (U,ρbcpms) is complete, there exist ν,μU such that {ντ}ν and {μτ}μ as τ, and

    ρbcpms(ν,ν)=limτρbcpms(ν,ντ)=limτ,υρbcpms(ντ,νυ)=0,ρbcpms(μ,μ)=limτρbcpms(μ,μτ)=limτ,υρbcpms(μτ,μυ)=0.

    We now show that ν=S(ν,μ). We suppose on the contrary that νS(ν,μ) and μS(μ,ν), so that 0i2ρbcpms(ν,S(ν,μ))=l1 and 0i2ρbcpms(μ,S(μ,ν))=l2. Then,

    l1=ρbcpms(ν,S(ν,μ))i2ρbcpms(ν,ντ+1)+ρbcpms(ντ+1,S(ν,μ))=ρbcpms(ν,ντ+1)+ρbcpms(S(ντ,μτ),S(ν,μ))i2ρbcpms(ν,ντ+1)+λρbcpms(ντ1,ντ)+lρbcpms(S(ν,μ),ν)i211lρbcpms(ν,ντ+1)+λ1lρbcpms(ντ1,ντ),

    which implies that

    ||l1||11l||ρbcpms(ν,ντ+1)||+λ1l||ρbcpms(ντ1,ντ)||.

    As τ, ||l1||0. This is a contradiction, and therefore ||ρbcpms(ν,S(ν,μ))||=0 implies ν=S(ν,μ). Similarly, we can prove that μ=S(μ,ν). Thus (ν,μ) is a coupled fixed point of S. Now, if (g,h) is another coupled fixed point of S, then

    ρbcpms(ν,g)=ρbcpms(S(ν,μ),S(g,h))i2λρbcpms(S(ν,μ),ν)+lρbcpms(S(g,h),g)=λρbcpms(ν,ν)+lρbcpms(g,g)=0.

    Thus, we have g=ν. Similarly, we get h=μ. Therefore S has a unique coupled fixed point.

    Corollary 3.2. Let (U,ρbcpms) be a complete bicomplex partial metric space. Suppose that the mapping S:U×UU satisfies the following contractive condition:

    ρbcpms(S(φ,ζ),S(ν,μ))i2λ(ρbcpms(S(φ,ζ),φ)+ρbcpms(S(ν,μ),ν)), (3.6)

    for all φ,ζ,ν,μU, where 0λ<12. Then, S has a unique coupled fixed point.

    Theorem 3.3. Let (U,ρbcpms) be a complete complex partial metric space. Suppose that the mapping S:U×UU satisfies the following contractive condition:

    ρbcpms(S(φ,ζ),S(ν,μ))i2λρbcpms(φ,ν)+lρbcpms(ζ,μ),

    for all φ,ζ,ν,μU, where λ,l are nonnegative constants with λ+l<1. Then, S has a unique coupled fixed point.

    Proof. Choose ν0,μ0U and set ν1=S(ν0,μ0) and μ1=S(μ0,ν0). Continuing this process, set ντ+1=S(ντ,μτ) and μτ+1=S(μτ,ντ). Then,

    ρbcpms(ντ,ντ+1)=ρbcpms(S(ντ1,μτ1),S(ντ,μτ))i2λρbcpms(ντ1,ντ)+lρbcpms(μτ1,μτ),

    which implies that

    ||ρbcpms(ντ,ντ+1)||λ||ρbcpms(ντ1,ντ)||+l||ρbcpms(μτ1,μτ)||. (3.7)

    Similarly, one can prove that

    ||ρbcpms(μτ,μτ+1)||λ||ρbcpms(μτ1,μτ)||+l||ρbcpms(ντ1,ντ)||. (3.8)

    From (3.7) and (3.8), we get

    ||ρbcpms(ντ,ντ+1)||+||ρbcpms(μτ,μτ+1)||(λ+l)(||ρbcpms(μτ1,μτ)||+||ρbcpms(ντ1,ντ)||)=α(||ρbcpms(μτ1,μτ)||+||ρbcpms(ντ1,ντ)||),

    where α=λ+l<1. Also,

    ||ρbcpms(ντ+1,ντ+2)||λ||ρbcpms(ντ,ντ+1)||+l||ρbcpms(μτ,μτ+1)|| (3.9)
    ||ρbcpms(μτ+1,μτ+2)||λ||ρbcpms(μτ,μτ+1)||+l||ρbcpms(ντ,ντ+1)||. (3.10)

    From (3.9) and (3.10), we get

    ||ρbcpms(ντ+1,ντ+2)||+||ρbcpms(μτ+1,μτ+2)||(λ+l)(||ρbcpms(μτ,μτ+1)||+||ρbcpms(ντ,ντ+1)||)=α(||ρbcpms(μτ,μτ+1)||+||ρbcpms(ντ,ντ+1)||).

    Repeating this way, we get

    ||ρbcpms(ντ,νn+1)||+||ρbcpms(μτ,μτ+1)||α(||ρbcpms(μτ1,μτ)||+||ρbcpms(ντ1,ντ)||)α2(||ρbcpms(μτ2,μτ1)||+||ρbcpms(ντ2,ντ1)||)ατ(||ρbcpms(μ0,μ1)||+||ρbcpms(ν0,ν1)||).

    Now, if ||ρbcpms(ντ,ντ+1)||+||ρbcpms(μτ,μτ+1)||=γτ, then

    γταγτ1ατγ0. (3.11)

    If γ0=0, then ||ρbcpms(ν0,ν1)||+||ρbcpms(μ0,μ1)||=0. Hence, ν0=ν1=S(ν0,μ0) and μ0=μ1=S(μ0,ν0), which implies that (ν0,μ0) is a coupled fixed point of S. Let γ0>0. For each τυ, we have

    ρbcpms(ντ,νυ)i2ρbcpms(ντ,ντ1)+ρbcpms(ντ1,ντ2)ρbcpms(ντ1,ντ1)+ρbcpms(ντ2,ντ3)+ρbcpms(ντ3,ντ4)ρbcpms(ντ3,ντ3)++ρbcpms(νυ+2,νυ+1)+ρbcpms(νυ+1,νυ)ρbcpms(νυ+1,νυ+1)i2ρbcpms(ντ,ντ1)+ρbcpms(ντ1,ντ2)++ρbcpms(νυ+1,νυ),

    which implies that

    ||ρbcpms(ντ,νυ)||||ρbcpms(ντ,ντ1)||+||ρbcpms(ντ1,ντ2)||++||ρbcpms(νυ+1,νυ)||.

    Similarly, one can prove that

    ||ρbcpms(μτ,μυ)||||ρbcpms(μτ,μτ1)||+||ρbcpms(μτ1,μτ2)||++||ρbcpms(μυ+1,μυ)||.

    Thus,

    ||ρbcpms(ντ,νυ)||+||ρbcpms(μτ,μυ)||γτ1+γτ2+γτ3++γυ(ατ1+ατ2++αυ)γ0αυ1αγ0asτ,

    which implies that {ντ} and {μτ} are Cauchy sequences in (U,ρbcpms). Since the bicomplex partial metric space (U,ρbcpms) is complete, there exist ν,μU such that {ντ}ν and {μτ}μ as τ, and

    ρbcpms(ν,ν)=limτρbcpms(ν,ντ)=limτ,υρbcpms(ντ,νυ)=0,ρbcpms(μ,μ)=limτρbcpms(μ,μτ)=limτ,υρbcpms(μτ,μυ)=0.

    Therefore,

    ρbcpms(S(ν,μ),ν)ρbcpms(S(ν,μ),ντ+1)+ρbcpms(ντ+1,ν)ρbcpms(ντ+1,ντ+1),ρbcpms(S(ν,μ)),S(ντ,μτ)+ρbcpms(ντ+1,ν)λρbcpms(ντ,ν)+lρbcpms(μτ,μ)+ρbcpms(ντ+1,ν).

    As τ, from (3.6) and (3.12) we obtain ρbcpms(S(ν,μ),ν)=0. Therefore S(ν,μ)=ν. Similarly, we can prove S(μ,ν)=μ, which implies that (ν,μ) is a coupled fixed point of S. Now, if (g1,h1) is another coupled fixed point of S, then

    ρbcpms(g1,ν)=ρbcpms(S(g1,h1),S(ν,μ))i2λρbcpms(g1,ν)+lρbcpms(h1,μ),ρbcpms(h1,μ)=ρbcpms(S(h1,g1),S(μ,ν))i2λρbcpms(h1,μ)+lρbcpms(g1,ν),

    which implies that

    ||ρbcpms(g1,ν)||λ||ρbcpms(g1,ν)||+l||ρbcpms(h1,μ)||, (3.12)
    ||ρbcpms(h1,μ)||λ||ρbcpms(h1,μ)||+l||ρbcpms(g1,ν)||. (3.13)

    From (3.12) and (3.13), we get

    ||ρbcpms(g1,ν)||+||ρbcpms(h1,μ)||(λ+l)[||ρbcpms(g1,ν)||+||ρbcpms(h1,μ)||].

    Since λ+l<1, this implies that ||ρbcpms(g1,ν)||+||ρbcpms(h1,μ)||=0. Therefore, ν=g1 and μ=h1. Thus, S has a unique coupled fixed point.

    Corollary 3.4. Let (U,ρbcpms) be a complete bicomplex partial metric space. Suppose that the mapping S:U×UU satisfies the following contractive condition:

    ρbcpms(S(φ,ζ),S(ν,μ))i2λ(ρbcpms(φ,ν)+ρbcpms(ζ,μ)), (3.14)

    for all φ,ζ,ν,μU, where 0λ<12. Then, S has a unique coupled fixed point.

    Example 3.5. Let U=[0,) and define the bicomplex partial metric ρbcpms:U×UC+2 defined by

    ρbcpms(φ,ζ)=max{φ,ζ}ei2θ,0θπ2.

    We define a partial order in C+2 as φζ iff φζ. Clearly, (U,ρbcpms) is a complete bicomplex partial metric space.

    Consider the mapping S:U×UU defined by

    S(φ,ζ)=φ+ζ4φ,ζU.

    Now,

    ρbcpms(S(φ,ζ),S(ν,μ))=ρbcpms(φ+ζ4,ν+μ4)=14max{φ+ζ,ν+μ}ei2θi214[max{φ,ν}+max{ζ,μ}]ei2θ=14[ρbcpms(φ,ν)+ρbcpms(ζ,μ)]=λ(ρbcpms(φ,ν)+ρbcpms(ζ,μ)),

    for all φ,ζ,ν,μU, where 0λ=14<12. Therefore, all the conditions of Corollary 3.4 are satisfied, then the mapping S has a unique coupled fixed point (0,0) in U.

    As an application of Theorem 3.3, we find an existence and uniqueness result for a type of the following system of nonlinear integral equations:

    φ(μ)=M0κ(μ,p)[G1(p,φ(p))+G2(p,ζ(p))]dp+δ(μ),ζ(μ)=M0κ(μ,p)[G1(p,ζ(p))+G2(p,φ(p))]dp+δ(μ),μ,[0,M],M1. (4.1)

    Let U=C([0,M],R) be the class of all real valued continuous functions on [0,M]. We define a partial order in C+2 as xy iff xy. Define S:U×UU by

    S(φ,ζ)(μ)=M0κ(μ,p)[G1(p,φ(p))+G2(p,ζ(p))]dp+δ(μ).

    Obviously, (φ(μ),ζ(μ)) is a solution of system of nonlinear integral equations (4.1) iff (φ(μ),ζ(μ)) is a coupled fixed point of S. Define ρbcpms:U×UC2 by

    ρbcpms(φ,ζ)=(|φζ|+1)ei2θ,

    for all φ,ζU, where 0θπ2. Now, we state and prove our result as follows.

    Theorem 4.1. Suppose the following:

    1. The mappings G1:[0,M]×RR, G2:[0,M]×RR, δ:[0,M]R and κ:[0,M]×R[0,) are continuous.

    2. There exists η>0, and λ,l are nonnegative constants with λ+l<1, such that

    |G1(p,φ(p))G1(p,ζ(p))|i2ηλ(|φζ|+1)12,|G2(p,ζ(p))G2(p,φ(p))|i2ηl(|ζφ|+1)12.

    3. M0η|κ(μ,p)|dpi21.

    Then, the integral equation (4.1) has a unique solution in U.

    Proof. Consider

    ρbcpms(S(φ,ζ),S(ν,Φ))=(|S(φ,ζ)S(ν,Φ)|+1)ei2θ=(|M0κ(μ,p)[G1(p,φ(p))+G2(p,ζ(p))]dp+δ(μ)(M0κ(μ,p)[G1(p,ν(p))+G2(p,Φ(p))]dp+δ(μ))|+1)ei2θ=(|M0κ(μ,p)[G1(p,φ(p))G1(p,ν(p))+G2(p,ζ(p))G2(p,Φ(p))]dp|+1)ei2θi2(M0|κ(μ,p)|[|G1(p,φ(p))G1(p,ν(p))|+|G2(p,ζ(p))G2(p,Φ(p))|]dp+1)ei2θi2(M0|κ(μ,p)|dp(ηλ(|φν|+1)12+ηl(|ζΦ|+1)12)+1)ei2θ=(M0η|κ(μ,p)|dp(λ(|φν|+1)+l(|ζΦ|+1)))ei2θi2(λ(|φν|+1)+l(|ζΦ|+1))ei2θ=λρbcpms(φ,ν)+lρbcpms(ζ,Φ)

    for all φ,ζ,ν,ΦU. Hence, all the hypotheses of Theorem 3.3 are verified, and consequently, the integral equation (4.1) has a unique solution.

    Example 4.2. Let U=C([0,1],R). Now, consider the integral equation in U as

    φ(μ)=10μp23(μ+5)[11+φ(p)+12+ζ(p)]dp+6μ25ζ(μ)=10μp23(μ+5)[11+ζ(p)+12+φ(p)]dp+6μ25. (4.2)

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

    φ(μ)=M0κ(μ,p)[G1(p,φ(p))+G2(p,ζ(p))]dp+δ(μ),ζ(μ)=M0κ(μ,p)[G1(p,ζ(p))+G2(p,φ(p))]dp+δ(μ),μ,[0,M], (4.3)

    where δ(μ)=6μ25, κ(μ,p)=μp23(μ+5), G1(p,μ)=11+μ, G2(p,μ)=12+μ and M=1. That is, (4.2) is a special case of (4.1) in Theorem 4.1. Here, it is easy to verify that the functions δ(μ), κ(μ,p), G1(p,μ) and G2(p,μ) are continuous. Moreover, there exist η=10, λ=13 and l=14 with λ+l<1 such that

    |G1(p,φ)G1(p,ζ)|ηλ(|φζ|+1)12,|G2(p,ζ)G2(p,φ)|ηl(|ζφ|+1)12

    and M0η|κ(μ,p)|dp=10ημp23(μ+5)dp=μη23(μ+5)<1. Therefore, all the conditions of Theorem 3.3 are satisfied. Hence, system (4.2) has a unique solution (φ,ζ) in U×U.

    As an application of Corollary 3.4, we find an existence and uniqueness result for a type of the following system of Fredholm integral equations:

    φ(μ)=EG(μ,p,φ(p),ζ(p))dp+δ(μ),μ,pE,ζ(μ)=EG(μ,p,ζ(p),φ(p))dp+δ(μ),μ,pE, (4.4)

    where E is a measurable, G:E×E×R×RR, and δL(E). Let U=L(E). We define a partial order in C+2 as xy iff xy. Define S:U×UU by

    S(φ,ζ)(μ)=EG(μ,p,φ(p),ζ(p))dp+δ(μ).

    Obviously, (φ(μ),ζ(μ)) is a solution of the system of Fredholm integral equations (4.4) iff (φ(μ),ζ(μ)) is a coupled fixed point of S. Define ρbcpms:U×UC2 by

    ρbcpms(φ,ζ)=(|φζ|+1)ei2θ,

    for all φ,ζU, where 0θπ2. Now, we state and prove our result as follows.

    Theorem 4.3. Suppose the following:

    1. There exists a continuous function κ:E×ER such that

    |G(μ,p,φ(p),ζ(p))G(μ,p,ν(p),Φ(p))|i2|κ(μ,p)|(|φ(p)ν(p)|+|ζ(p)Φ(p)|2),

    for all φ,ζ,ν,ΦU, μ,pE.

    2. E|κ(μ,p)|dpi214i21.

    Then, the integral equation (4.4) has a unique solution in U.

    Proof. Consider

    ρbcpms(S(φ,ζ),S(ν,Φ))=(|S(φ,ζ)S(ν,Φ)|+1)ei2θ=(|EG(μ,p,φ(p),ζ(p))dp+δ(μ)(EG(μ,p,ν(p),Φ(p))dp+δ(μ))|+1)ei2θ=(|E(G(μ,p,φ(p),ζ(p))G(μ,p,ν(p),Φ(p)))dp|+1)ei2θi2(E|G(μ,p,φ(p),ζ(p))G(μ,p,ν(p),Φ(p))|dp+1)ei2θi2(E|κ(μ,p)|(|φ(p)ν(p)|+|ζ(p)Φ(p)|2)dp+1)ei2θi2(E|κ(μ,p)|dp(|φ(p)ν(p)|+|ζ(p)Φ(p)|2)+1)ei2θi214(|φ(p)ν(p)|+|ζ(p)Φ(p)|2+4)ei2θi214(ρbcpms(φ,ν)+ρbcpms(ζ,Φ))=λ(ρbcpms(φ,ν)+ρbcpms(ζ,Φ)),

    for all φ,ζ,ν,ΦU, where 0λ=14<12. Hence, all the hypotheses of Corollary 3.4 are verified, and consequently, the integral equation (4.4) has a unique solution.

    In this paper, we proved coupled fixed point theorems on a bicomplex partial metric space. An illustrative example and an application on a bicomplex partial metric space were given.

    The authors declare no conflict of interest.


    Acknowledgments



    None.

    Conflict of interest



    Authors declare no conflict of interests.

    Author Contributions



    Fiammetta Iannuzzo: conceptualization, investigation, data curation, formal analysis, and writing – original draft.
    Silvia Crudo: investigation, data curation, formal analysis.
    Gianpaolo Antonio Basile: investigation, data curation, formal analysis.
    Fortunato Battaglia: validation, data curation, formal analysis.
    Carmenrita Infortuna: validation, data curation, formal analysis.
    Maria Rosaria Anna Muscatello: supervision, and writing – review & editing
    Antonio Bruno: conceptualisation, methodology, supervision, writing – original draft, and writing – review & editing.

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