Introduction

The searching problem for the lost targets that move with random movement is one of the most important contemporary problems that the mathematicians seek to solve it in different ways to reduce the cost of the search. One of these random movements is the Brownian motion. In an earlier work, this problem has been discussed on the real line, plan, and space by studying different kinds of search plans such as the linear search plan as in El-Hadidy et al [1, 2]., El-Rayes et al. [3], and Corwin [4]. They presented the finiteness of the search strategy which was used by founding the conditions that make E(τ) (the first interviewing time expected value for one of the searchers and the target where τ denote to this time) finite. In addition, they showed the existence of this optimal search plan and El-Hadidy [1] found it.

One of the famous search techniques is the coordinated search technique. Reyniers [5, 6] was the first one who investigated this technique on the line. This technique considered two searchers starting together at the origin of the real line with unit speed. These searchers aimed to find the expected time for detecting a hidden target after one of them return to the origin to tell the other. In addition, they aimed to minimize this expected time. But, the returning time increases this expected time. Here, we will treat this defect by not repeating the searching process on the searched parts.

In recent works, El-Hadidy and Abou-Gabal [7] presented a coordinated search technique to find a random Walk target, which moves randomly on the real line. Rather than showing the existence of the optimal search strategy that minimizes this first interviewing time, El-Hadidy and Abou-Gabal [7] found the conditions that make E(τ) finite. El-Hadidy et al. [8,9,10] also studied this technique on the plan and space when the target is hidden at any random point. They found the expected value of the time to detect the target by one of the searchers. Besides that, they found the optimal search plan that minimizes this expected value. For different kinds of search plans on the line, plan, and space, the reader can see, El-Hadidy et al. [11,12,13,14,15,16,17,18,19,20], Kagan and Ben-Gal [21], Guerrier and Holcman [22], Palyulin et al. [23], Radmard and Croft [24], Stone et al. [25], and Jia et al. [26].

In this work, we use the technique which was presented in Reyniers [5, 6] and El-Hadidy and Abou-Gabal [7] but without repeating the searching process on the searched parts of n disjoint real lines. This technique is the generalization of the technique which was used in El-Hadidy and Alzulaibani [27]. We use n cooperative searchers to seek a Brownian target which moves in one of the n disjoint cylinders (real lines). This idea is supported through modern communication methods. Each line has two searchers starting at the same time from the origin. One of these searchers goes towards +∞ to search the right part of the real line and the other goes towards to −∞ to search the left part of the real line. This will reduce E(τ) (cost). Thus, we aim to show the existence of a finite search plan and compute this expected value.

The organization of this paper appears as follows: the “Problem formulation” section describes our search problem. We show the existence and finiteness of our search plan in the “Existence of the finite search plan” section. The computation of E(τ) appears in “Computational expected value for τ” section. Finally, we discuss the results and future works which were obtained in the conclusion part.

Problem formulation

We always seek for reducing the cost to detect the Brownian moving target on one of the n real lines. To do this, we use n cooperative searchers as in El-Hadidy et al. [1, 2, 20, 28]. But, the technique which was used in El-Hadidy et al. [1, 2, 20, 28] wasted a lot of costs because the searching process is repeated in the same part more than once. Our model does not waste this cost because the searchers do not repeat the searching process by using the connection technology methods. Besides that, we calculate the approximated expected value of the Brownian target detection.

The searching framework

The space of search

The search space is the n axis of n cylinders (n real lines Li, i = 1, 2, ..., n), see Fig. 1.

Fig. 1
figure 1

The search space (n disjoint cylinders)

Full size image

The target

Our target moves with a Brownian motion {B(t), t ≥ 0}, where the target’s initial position is unknown, but its probability distribution is known at time 0.

The means of search

All the searchers Sj, j = 1, 2, ..., 2n start the searching process from the origin. Each line Li, i = 1, 2, ..., n needs two unit speed searchers S2j − 1  and  S2j, where the searcher S2j − 1 searches in the right part and S2j in the left part of the real line. After searching successively common distances, they do not return to the origin of Li. The searching process on Li would conduct as follows: begin at H0i = 0 and go to the right (left) as far as \( {H}_{1i}^{\left(+\right)}\left({H}_{2i}^{\left(-\right)}\right) \). Then, the first one of S2j − 1  and  S2j which meets the target will tell the other by using the wireless communication methods. If the target is not found, then do the above step again at the point \( {H}_{1i}^{\left(+\right)}\left({H}_{2i}^{\left(-\right)}\right) \) to search the right (left) part of \( {H}_{1i}^{\left(+\right)}\left({H}_{2i}^{\left(-\right)}\right) \) as far as \( \left[{H}_{3i}^{\left(+\right)}-{H}_{1i}^{\left(+\right)}\right]\left(\left[|{H}_{4i}^{\left(-\right)}-{H}_{2i}^{\left(-\right)}|\right]\right) \) and so on, see Fig. 2.

Fig 2
figure 2

The search plan without returning to the origin of S2j − 1  and  S2j for finding a Brownian target on Li

Full size image

Existence of the finite search plan

In this section, we aim to get the conditions that make the search plan be finite. On the line Li, i = 1, 2, ..., n the two searchers S2j − 1  and  S2j are coordinating their search and following the search paths \( {\varphi}_{2j-1}:\kern0.48em {R}^{+}\to R\kern0.36em \mathrm{and}\kern0.5em {\overline{\varphi}}_{2j}:\kern0.48em {R}^{+}\to R, \) respectively, to meet the target, where

$$ \mid \varphi {}_{2j-1}\left({t}_1\right)-\varphi {}_{2j-1}\left({t}_2\right)\mid \le {v}^{\left(+\right)}\mid {t}_1-{t}_2\mid \&\mid {\overline{\varphi}}_{2j}\left({t}_1\right)-{\overline{\varphi}}_{2j}\left({t}_2\right)\mid \le {v}^{\left(-\right)}\mid {t}_1-{t}_2\mid \forall \kern0.36em {t}_1,{t}_2\in {R}^{+}, $$
(1)

(i.e., the search paths should satisfy the Lipschitz condition), and v(+) and v(−) are the velocities of the searchers in the right and left part, respectively. Assuming that θi(+), θi(−), λi(+), and λi(−) are positive integer numbers greater than one. For any line, Li, i = 1, 2, ..., n define the sequences \( {\displaystyle \begin{array}{l}\\ {}{\left\{{G}_{ki}^{\left(+\right)}\right\}}_{k\ge 1,i=1,2,...,n},{\left\{{G}_{ki}^{\left(-\right)}\right\}}_{k\ge 1,i=1,2,...,n},{\left\{{H}_{ki}^{\left(+\right)}\right\}}_{k\ge 1,i=1,2,...,n}\end{array}} \) and \( {\left\{{H}_{ki}^{\left(-\right)}\right\}}_{k\ge 1,i=1,2,...,n} \) (see El-Rayes et al. [3]), respectively, to obtain the distances which the searchers should do them as a function of θi(+), θi(−), λi(+), and λi(−). The main idea of this model is all searchers do not return to the origin as in Fig. 1 that is reducing E(τ). Thus, we can define \( {G}_{ik}^{\left(+\right)}={\lambda}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^k-1\right),{G}_{ik}^{\left(-\right)}={\lambda}_{i\left(-\right)}\left({\theta}_{i\left(-\right)}^k-1\right). \) This leads to the traveled distances at the time step k, k = 1, 2, ... are given by \( {H}_{ki}^{\left(+\right)}={C_i}^{\left(+\right)}\left({G}_{ki}^{\left(+\right)}+2\right),{H}_{ki}^{\left(-\right)}={C_i}^{\left(-\right)}\left({G}_{ki}^{\left(-\right)}+2\right), \) in the right and left part of Li, i = 1, 2, ..., n, respectively, where \( {C_i}^{\left(+\right)}=\frac{{\varpi_i}^{\left(+\right)}\left({\theta}_{i\left(+\right)}-1\right)}{\left({\theta}_{i\left(+\right)}+1\right)},{C_i}^{\left(-\right)}=\frac{{\varpi_i}^{\left(-\right)}\left({\theta}_{i\left(-\right)}-1\right)}{\left({\theta}_{i\left(-\right)}+1\right)}, \)and ϖi(+), ϖi(−) are rational numbers. Consequently, the search path in the right part of Li, i = 1, 2, ..., n can be defined as follows: for any t ∈ R+, if \( {G}_{\left(2k-3\right)i}^{\left(+\right)}\le t\le {G}_{\left(2k-1\right)i}^{\left(+\right)} \), we have

$$ {\varphi}_i(t)=\left({H}_{\left(2k-1\right)i}^{\left(+\right)}-{H}_{\left(2k-3\right)i}^{\left(+\right)}\right)+\left(t-{G}_{\left(2k-1\right)i}^{\left(+\right)}\right) $$
(2)

And, in the left part for any t ∈ R+, if \( {G}_{\left(2k-2\right)i}^{\left(-\right)}\le t\le {G}_{(2k)i}^{\left(-\right)} \), we have

$$ {\overline{\varphi}}_i(t)=-\left[\left({H}_{(2k)i}^{\left(-\right)}-{H}_{\left(2k-2\right)i}^{\left(-\right)}\right)+\left(t-{G}_{(2k)i}^{\left(-\right)}\right)\;\right] $$
(3)

Let the notations \( \psi \left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)=B\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)-{C_i}^{\left(+\right)}\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right),\tilde{\psi}\left({G}_{(2k)i}^{\left(-\right)}\right)=B\left({G}_{(2k)i}^{\left(-\right)}\right)+{C_i}^{\left(-\right)}\left({G}_{(2k)i}^{\left(-\right)}\right),i=1,2,...,n;k=1,2,... \) are held. Then,

$$ \tau =\operatorname{inf}\ \left\{t:\mathrm{either}\ \mathrm{one}\ \mathrm{of}\ {\varphi}_i(t)={Z}_{0i}+B(t)\kern0.24em \mathrm{or}\kern0.5em {\overline{\varphi}}_i(t)={Z}_{0i}+B(t)\right\},i=1,2,...,n\Big\}, $$
(4)

where Z0i is a random variable which represents the target’s initial position on Li, i = 1, 2, ..., n and independent with {B(t), t > 0}. Also, let the search plan be represented by \( \left({\varphi}_1,{\varphi}_2,...,{\varphi}_n,{\overline{\varphi}}_1,{\overline{\varphi}}_2,...,{\overline{\varphi}}_n\right)\in {\varPhi}_0 \), where \( {\varPhi}_0=\left\{\left(\varPhi, \overline{\varPhi}\right):{\varphi}_i\in \varPhi, {\overline{\varphi}}_i\in \overline{\varPhi}\kern0.3em \forall i=1,2,...,n\right\} \).

For t > 0, we have an infinite number of possible outcomes, thus the probability density function which represents the target’s position can be described as a calendar for the continuity of the relative frequencies of the data in a given interval. Consequently, the probability of any certain value of the target’s position at time step k is equal to zero. Logically, the first interviewing time event and the target’s position zi ∈ Zi on any line Li, i = 1, 2, ..., n at time step k on [0, ∞] or [−∞, 0] have a known probability value greater than zero. This leads us to study our problem in a new space (probability space). Let this probability space be (Ω, Σ, γ) where Ω is the sample space of all expected meeting points, Σ is the σ− algebra which represents the collection of all mutually exclusive events that show the target’s position at any time step k, and γ is the probability measure which used as a measure of an integrator factor into our probability space. Now, we will study the existence of the finite search plan as in the following theorems.

Theorem 1: The combination of the search plans \( \left({\varphi}_1,{\varphi}_2,...,{\varphi}_n,{\overline{\varphi}}_1,{\overline{\varphi}}_2,...,{\overline{\varphi}}_n\right)\in {\varPhi}_0 \) is finite if the following \( \underset{0}{\overset{\infty }{\int }}\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}P\left(\psi \left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>-{z}_i\right){\gamma}_i\left(d{z}_i\right) \) and \( \underset{-\infty }{\overset{0}{\int }}\sum \limits_{k=2}^{\infty }{\theta}_{\left(-\right)}^{2i-2}P\left(\tilde{\psi}\left({G}_{\left(2k-2\right)}^{\left(-\right)}\right)<-{z}_i\right){\gamma}_i\left(d{z}_i\right) \) are finite for all i = 1, 2, ..., n, where \( \sum \limits_{i=1}^n{\gamma}_i=1. \)

Proof Since the target may be met at one of n lines, then \( {\tau}_{\varphi_i},i=1,2,...,n \) “the first interviewing time between one of the searchers S2j − 1, j = 1, 2, ..., 2n in the right parts of all lines Li, i = 1, 2, ..., n and the Brownian target” and \( {\tau}_{{\overline{\varphi}}_i},i=1,2,...,n \) “the first interviewing time between one of the searchers S2j, j = 1, 2, ..., 2n in the left parts and the Brownian target” of all lines Li, i = 1, 2, ..., n are mutually exhaustive events. Then, for any k ≥ 0, we have

$$ {\displaystyle \begin{array}{c}P\left(\tau >t\right)=P\left({\tau}_{\varphi_1}>t\kern0.3em \mathrm{or}\kern0.3em {\tau}_{\varphi_2}>t\ \mathrm{or}\ ...\kern0.3em \mathrm{or}\kern0.3em {\tau}_{\varphi_n}>t\kern0.3em \mathrm{or}\kern0.3em {\tau}_{{\overline{\varphi}}_1}>t\ \mathrm{or}\ {\tau}_{{\overline{\varphi}}_2}>t\kern0.3em \mathrm{or}\kern0.3em ...\kern0.3em \mathrm{or}\kern0.3em {\tau}_{{\overline{\varphi}}_n}>t\right)\\ {}=\underset{i=1}{\overset{n}{\cup }}\left(P\left({\tau}_{\varphi_i}>t\right)+P\left({\tau}_{{\overline{\varphi}}_i}>t\right)\right)\\ {}=\sum \limits_{i=1}^n\left(P\left({\tau}_{\varphi_i}>t\right)+P\left({\tau}_{{\overline{\varphi}}_i}>t\right)\right).\end{array}} $$

Then, we obtain

$$ {\displaystyle \begin{array}{c}E\;\left(\tau \right)=\underset{0}{\overset{\infty }{\int }}P\;\left(\tau >t\right) dt\le \sum \limits_{1=1}^n\kern0.1em \sum \limits_{k=1}^{\infty}\kern0.1em \left[\underset{G_{\left(2k-3\right)i}^{\left(+\right)}}{\overset{G_{\left(2k-1\right)i}^{\left(+\right)}}{\int }}P\left({\tau}_{\varphi_i}>t\right)\; dt+\underset{G_{\left(2k-2\right)i}^{\left(-\right)}}{\overset{G_{\left(2k-1\right)i}^{\left(-\right)}}{\int }}P\left({\tau}_{{\overline{\varphi}}_i}>t\right)\; dt\right]\\ {}\le \sum \limits_{i=1}^n\sum \limits_{k=1}^{\infty}\kern0.1em \left[\left({G}_{\left(2k-1\right)i}^{\left(+\right)}-{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\;P\;\left({\tau}_{\varphi_i}>{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\right)+\left({G}_{(2k)i}^{\left(-\right)}-{G}_{\left(2k-2\right)i}^{\left(-\right)}\right)\;P\;\left({\tau}_{{\overline{\varphi}}_i}>{G}_{\left(2k-2\right)i}^{\left(-\right)}\right)\;\Big]\\ {}\le \sum \limits_{i=1}^n\sum \limits_{k=1}^{\infty}\kern0.1em \left[{\lambda}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^{2k-1}-{\theta}_{i\left(+\right)}^{2k-3}\right)\;P\;\left({\tau}_{\varphi_i}>{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)+{\lambda}_{i\left(-\right)}\left({\theta}_{i\left(-\right)}^{2k}-{\theta}_{i\left(-\right)}^{2k-2}\right)\;P\;\left({\tau}_{{\overline{\varphi}}_i}>{G}_{\left(2k-2\right)i}^{\left(-\right)}\;\right)\right]\\ {}=\sum \limits_{i=1}^n\Big({\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}P\;\left({\tau}_{\varphi_i}>0\right)+{\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}P\;\left({\tau}_{\varphi_i}>{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\\ {}+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}P\;\left({\tau}_{{\overline{\varphi}}_i}>0\right)+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}^2\left({\theta}_{i\left(-\right)}^2-1\right)\;\sum \limits_{k=2}^{\infty }{\theta}_{\left(-\right)}^{2k-2}P\;\left({\tau}_{{\overline{\varphi}}_i}>{G}_{\left(2k-2\right)i}^{\left(-\right)}\right)\Big).\end{array}} $$
(5)

Also, at time step k on the line Li, i = 1, 2, ..., n, we have

$$ P\left({\tau}_{\varphi_i}>{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\le \underset{0}{\overset{\infty }{\int }}P\left({Z}_{0i}+B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>{H}_{\left(2k-3\right)i}^{\left(+\right)}|{Z}_{0i}={z}_i\right){\gamma}_i\left(d{z}_i\right),k\ge 1. $$

By using the above notation \( \psi \left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)=B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)-{C_i}^{\left(+\right)}\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right), \) we can get \( \psi \left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>-{z}_i \) (El-Hadidy et al. [1, 2]), then \( P\left({\tau}_{\varphi_i}>{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\le \underset{0}{\overset{\infty }{\int }}P\left(\psi \left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>-{z}_i\right){\gamma}_i\left(d{z}_i\right), \) and by using the another notation \( \tilde{\psi}\left({G}_{(2k)i}^{\left(-\right)}\right)=B\left({G}_{(2k)i}^{\left(-\right)}\right)+{C_i}^{\left(-\right)}\left({G}_{(2k)i}^{\left(-\right)}\right), \) we get \( P\left({\tau}_{{\overline{\varphi}}_i}>{G}_{\left(2k-2\right)}^{\left(-\right)}\right)\le \underset{-\infty }{\overset{0}{\int }}P\left(\tilde{\psi}\left({G}_{\left(2k-2\right)i}^{\left(-\right)}\right)<-{z}_i\right){\gamma}_i\left(d{z}_i\right). \) Consequently,

$$ E\left(\tau \right)=\sum \limits_{i=1}^n\left[{g}_i+{\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^2-1\right)\underset{0}{\overset{\infty }{\int }}{M}_i\left({z}_i\right){\gamma}_i\left(d{z}_i\right)+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}^2\left({\theta}_{i\left(-\right)}^2-1\right)\;\underset{-\infty }{\overset{0}{\int }}{L}_i\left({z}_i\right){\gamma}_i\left(d{z}_i\right)\right], $$

where gi = (λi(+)θi(+) + λi(−)θi(−))P (τ > 0), P (τ > 0) is the knowing initial probability of τ and \( {M}_i\left({z}_i\right)=\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}P\left(\psi \left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>-{z}_i\right) \) and \( {L}_i\left({z}_i\right)=\sum \limits_{k=2}^{\infty }{\theta}_{i\left(-\right)}^{2k-2}P\left(\tilde{\psi}\left({G}_{\left(2k-2\right)i}^{\left(-\right)}\right)<-{z}_i\right). \) Thus, E(τ) is finite if \( \underset{0}{\overset{\infty }{\int }}{M}_i\left({z}_i\right){\gamma}_i\left(d{z}_i\right) \) and \( \underset{-\infty }{\overset{0}{\int }}{L}_i(z){\gamma}_i\left(d{z}_i\right) \) are finite. The prove is completed.■

This model is the first investigation of the cooperative search technique by using 2n searchers which reduces E(τ). But the above conditions in Theorem 1 are not sufficient to get the finite expected search time. Thus, we want to get more conditions to make this model more applicable and effective.

Theorem 2: At any time step k, the chosen search plan \( {\varPhi}_0=\left\{\left(\varPhi, \overline{\varPhi}\right):{\varphi}_i\in \varPhi, {\overline{\varphi}}_i\in \overline{\varPhi}\forall i=1,2,...,n\right\} \) should satisfy \( \left[{M}_1\left({z}_{k1}\right),{M}_2\left({z}_{k2}\right),...,{M}_n\left({z}_{kn}\right)\right]\le \left[{\tilde{M}}_1\left(|{z}_{k1}|\right),{\tilde{M}}_2\left(|{z}_{k2}|\right),...,{\tilde{M}}_n\left(|{z}_{kn}|\right)\right] \) and \( \left[{L}_1\left({z}_{k1}\right),{L}_2\left({z}_{k2}\right),...,{L}_n\left({z}_{kn}\right)\right]\le \left[{\tilde{L}}_1\left(|{z}_{k1}|\right),{\tilde{L}}_2\left(|{z}_{k2}|\right),...,{\tilde{L}}_n\left(|{z}_{kn}|\right)\right] \) where \( \left[{\tilde{M}}_1\left(|{z}_{k1}|\right),{\tilde{M}}_2\left(|{z}_{k2}|\right),...,{\tilde{M}}_n\left(|{z}_{kn}|\right)\right] \)and \( \left[{\tilde{L}}_1\left(|{z}_{k1}|\right),{\tilde{L}}_2\left(|{z}_{k2}|\right),...,{\tilde{L}}_n\left(|{z}_{kn}|\right)\right] \) are vectors of linear functions.

Proof At time step k, if zki ≤ 0 on the line Li, i = 1, 2, ..., n, then [M1(zk1), M2(zk2), ..., Mn(zkn)] ≤ [M1(0), M2(0), ..., Mn(0)] but for zki > 0, we get \( {M}_i(0)=\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}P\left(\psi \left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)>0\right). \) Consequently,

$$ \left[{M}_1\left({z}_{k1}\right),{M}_2\left({z}_{k2}\right),...,{M}_n\left({z}_{kn}\right)\right]=\left[{M}_1(0)+\sum \limits_{k=2}^{\infty }{\theta}_{1\left(+\right)}^{2k-2}P\left(\psi \left({G}_{\left(2k-1\right)1}^{\left(+\right)}\right)\le 0\right),{M}_2(0)+\sum \limits_{k=2}^{\infty }{\theta}_{2\left(+\right)}^{2k-2}P\left(\psi \left({G}_{\left(2k-1\right)2}^{\left(+\right)}\right)\le 0\right),...,{M}_n(0)+\sum \limits_{k=2}^{\infty }{\theta}_{n\left(+\right)}^{2k-2}P\left(\psi \left({G}_{\left(2k-1\right)n}^{\left(+\right)}\right)\le 0\right)\right]. $$
(6)

If the target starts its motion at time step k on the real line Li, i = 1, 2, ..., n from the random point zki with drift μ and variance σ2, then on the right part of Liand for \( t\ge {G}_{\left(2k-1\right)i}^{\left(+\right)}>0, \) we have

$$ P\left(B(t)\ge {\alpha}_it\right)\le P\left(B\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)\ge {\alpha}_i{G}_{\left(2k-1\right)i}^{\left(+\right)}\right)=P\left(\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}Z+\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}\ge {\alpha}_i{G}_{\left(2k-1\right)i}^{\left(+\right)}\right), $$

where αi, i = 1, 2, ..., n are constants. This leads to

$$ p\left(Z\ge \frac{\left({\alpha}_i-\mu \right){G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\right)=\underset{k}{\overset{\infty }{\int }}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{z^2}{2}} dz=\underset{0}{\overset{\infty }{\int }}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\left(x+{\kappa}_i\right)}^2}{2}} dx\le \frac{1}{2}{\varepsilon}^{G_{\left(2k-1\right)i}^{\left(+\right)}}, $$
(7)

where \( {\kappa}_i=\frac{\left({\alpha}_i-\mu \right){G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}},i=1,2,...,n \). Also, in the left part of Li, we can get

$$ P\left(Z<\frac{\left({\alpha}_i-\mu \right){G}_{\left(2k-2\right)i}^{\left(-\right)}}{\sigma \sqrt{G_{\left(2k-2\right)i}^{\left(-\right)}}}\right)\le 1-\frac{1}{2}{\varepsilon}^{G_{\left(2k-2\right)i}^{\left(-\right)}}. $$
(8)

From (7) and (8) in (6), we obtain

\( \left[{M}_1(0),{M}_2(0),...,{M}_n(0)\right]<\left[\sum \limits_{k=2}^{\infty }{\theta}_{1\left(+\right)}^{2k-2}{\varepsilon}_1^{G_{\left(2k-1\right)1}^{\left(+\right)}},\sum \limits_{k=2}^{\infty }{\theta}_{2\left(+\right)}^{2k-2}{\varepsilon}_2^{G_{\left(2k-1\right)2}^{\left(+\right)}},...,\sum \limits_{k=2}^{\infty }{\theta}_{n\left(+\right)}^{2k-2}{\varepsilon}_n^{G_{\left(2k-1\right)n}^{\left(+\right)}}\right],\kern1.00em 0<{\varepsilon}_i<1,i=1,2,...,n. \)For any two random positions z2i ≤ z1i of the target on the right part of Li, we get\( P\left({z}_{2i}\le B(t)\le {z}_{1i}\right)\le P\left({z}_{2i}\le B\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)\le {z}_{1i}\right) \) is non-increasing with time \( {G}_{\left(2k-1\right)i}^{\left(+\right)}, \) where \( t\ge {G}_{\left(2k-1\right)i}^{\left(+\right)}\ge \max \left(\frac{z_{1i}}{\mu },\frac{z_{2i}}{\mu}\right), \)

because

$$ {\displaystyle \begin{array}{c}P\left({z}_{2i}\le B\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)\le {z}_{1i}\right)=P\left({z}_{2i}\le \sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}\right)Z+\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}\le {z}_{1i}\Big)\\ {}=P\left(\frac{z_{2i}-\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\le Z\le \frac{z_{1i}-\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\right).\end{array}} $$

If μ < 0, then \( \frac{z_{2i}-\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\ge 0,\frac{d}{d{G}_{\left(2k-1\right)i}^{\left(+\right)}}\left(\frac{z_{2i}-\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\right)\ge 0,\frac{z_{1i}-\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\le 0 \) and \( \frac{d}{d{G}_{\left(2k-1\right)i}^{\left(+\right)}}\left(\frac{z_{1i}-\mu {G}_{\left(2k-1\right)i}^{\left(+\right)}}{\sigma \sqrt{G_{\left(2k-1\right)i}^{\left(+\right)}}}\right)\le 0 \). Thus, \( P\left({z}_{2i}\le B\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)\le {z}_{1i}\right) \) be non-increasing.

Assuming that \( \left[{\psi}_1(k),{\psi}_2(k),...,{\psi}_n(k)\right]=\left[\sum \limits_{\ell =1}^k{y}_{\ell 1},\sum \limits_{\ell =1}^k{y}_{\ell 2},...,\sum \limits_{\ell =1}^k{y}_{\mathit{\ell n}}\right], \) where {yℓi}ℓ ≥ 1, i = 1, 2, ..., n is a sequence of i.i.d.r.vs which represents the target position at time step k on the line Li, i = 1, 2, ..., n and yik~N(μ − αi, σ2). Consider the following: \( \left[{d}_{k1},{d}_{k2},...,{d}_{kn}\right]=\left[{G}_{\left(2k+1\right)1}^{\left(+\right)},{G}_{\left(2k+1\right)2}^{\left(+\right)},...,{G}_{\left(2k+1\right)n}^{\left(+\right)}\right]=\left[{\lambda}_{1\left(+\right)}\left({\theta}_{1\left(+\right)}^{2k+1}-1\right),{\lambda}_{2\left(+\right)}\left({\theta}_{2\left(+\right)}^{2k+1}-1\right),...,{\lambda}_{n\left(+\right)}\left({\theta}_{n\left(+\right)}^{2k+1}-1\right)\right] \) and \( \left[{U}_1\left(j,j+1\right),{U}_2\left(j,j+1\right),...,{U}_n\left(j,j+1\right)\right]=\left[\sum \limits_{k=1}^{\infty }P\left[-\left(j+1\right)<{\psi}_1(k)\le -j\right],\sum \limits_{k=1}^{\infty }P\left[-\left(j+1\right)<{\psi}_2(k)\le -j\right],...,\sum \limits_{k=1}^{\infty }P\left[-\left(j+1\right)<{\psi}_n(k)\le -j\right]\right] \). By choosing \( \left[{d}_{m1},{d}_{m2},...,{d}_{mn}\right]=\left[\max \left(0,\frac{z_{m1}}{\mu}\right),\max \left(0,\frac{z_{m2}}{\mu}\right),...,\max \left(0,\frac{z_{mn}}{\mu}\right)\right] \) and also for any line Li, i = 1, 2, ..., n, we choose \( {a}_i(k)=P\left[-{z}_{ki}<{\psi}_i(k)\le 0\right]=\sum \limits_{j=1}^{\mid {z}_{ki}\mid }P\left[-\left(j+1\right)<{\psi}_i(n)\le -j\right] \). Thus, if k > dmi for all k ≥ 1, i = 1, 2, ..., n and since \( P\left({z}_{2i}\le B\left({G}_{\left(2k-1\right)i}^{\left(+\right)}\right)\le {z}_{1i}\right) \) is non-increasing with time \( {G}_{\left(2k-1\right)i}^{\left(+\right)} \) at step k, then we have ai(k) is a non-increasing also. Consequently, \( \left[{M}_1\left({z}_{k1}\right),{M}_2\left({z}_{k2}\right),...,{M}_n\left({z}_{kn}\right)\right]-\left[{M}_1(0),{M}_2(0),...,{M}_n(0)\right]=\left[\sum \limits_{k=1}^m{\theta}_{1\left(+\right)}^{2k+1}{a}_1\left({d}_{k1}\right)+{\alpha}_1\sum \limits_{k=m+1}^{\infty}\left({d}_{k1}-{d}_{\left(k-1\right)1}\right){a}_1\left({d}_{k1}\right),\sum \limits_{k=1}^m{\theta}_{2\left(+\right)}^{2k+1}{a}_2\left({d}_{k2}\right)+{\alpha}_2\sum \limits_{k=m+1}^{\infty}\left({d}_{k2}-{d}_{\left(k-1\right)2}\right){a}_2\left({d}_{k2}\right),...,\sum \limits_{k=1}^m{\theta}_{n\left(+\right)}^{2k+1}{a}_n\left({d}_{kn}\right)+{\alpha}_n\sum \limits_{k=m+1}^{\infty}\left({d}_{kn}-{d}_{\left(k-1\right)n}\right){a}_n\left({d}_{kn}\right)\right]. \) From Lemma 1 in El-Hadidy et al [1], we have \( \left[{M}_1\left({z}_{k1}\right),{M}_2\left({z}_{k2}\right),...,{M}_n\left({z}_{kn}\right)\right]-\left[{M}_1(0),{M}_2(0),...,{M}_n(0)\right]\le \left[\sum \limits_{k=1}^m{\theta}_{1\left(+\right)}^{2k+1}+{\alpha}_1\sum \limits_{k={d}_m}^{\infty }{a}_1(k),\sum \limits_{k=1}^m{\theta}_{2\left(+\right)}^{2k+1}+{\alpha}_2\sum \limits_{k={d}_m}^{\infty }{a}_2(k),...\sum \limits_{k=1}^m{\theta}_{n\left(+\right)}^{2k+1}+{\alpha}_2\sum \limits_{k={d}_m}^{\infty }{a}_n(k)\right]\le \left[{d}_{m1}+{\lambda}_{1\left(+\right)}\left(1-{\theta}_{n\left(+\right)}\right)\sum \limits_{j=0}^{\mid {z}_{k1}\mid }{U}_1\left(j,j+1\right),{d}_{m2}+{\lambda}_{2\left(+\right)}\left(1-{\theta}_{2\left(+\right)}\right)+\sum \limits_{j=0}^{\mid {z}_{k2}\mid }{U}_2\left(j,j+1\right),...{d}_{mn}+{\lambda}_{n\left(+\right)}\left(1-{\theta}_{n\left(+\right)}+\sum \limits_{j=0}^{\mid {z}_{kn}\mid }{U}_n\left(j,j+1\right)\right)\right]. \)

Since \( \left[{\psi}_1(k),{\psi}_2(k),...,{\psi}_n(k)\right]=\left[\sum \limits_{\ell =1}^k{y}_{\ell 1},\sum \limits_{\ell =1}^k{y}_{\ell 2},...,\sum \limits_{\ell =1}^k{y}_{\mathit{\ell n}}\right], \) and {yℓi}ℓ ≥ 1,i=1,2,...,n, then [U1(j, j + 1), U2(j, j + 1), ..., Un(j, j + 1)] satisfies the renewal theorem as in Feller [29]. Hence, [U1(j, j + 1), U2(j, j + 1), ..., Un(j, j + 1)] is bounded ∀j by a constant. Hence, \( \left[{M}_1\left({z}_{k1}\right),{M}_2\left({z}_{k2}\right),...,{M}_n\left({z}_{kn}\right)\right]\le \left[{M}_1(0),{M}_2(0),...,{M}_n(0)\right]+\left[{M}_1,{M}_2,...,{M}_n\right]+\left[{M}_1\left(|{z}_{k1}|\right),{M}_2\left(|{z}_{k2}|\right),...,{M}_n\left(|{z}_{kn}|\right)\right]\left[{\tilde{M}}_1\left(|{z}_{k1}|\right),{\tilde{M}}_2\left(|{z}_{k2}|\right),...,{\tilde{M}}_n\left(|{z}_{kn}|\right)\right]. \)By the same method, we can show that \( \left[{L}_1\left({z}_{k1}\right),{L}_2\left({z}_{k2}\right),...,{L}_n\left({z}_{kn}\right)\right]\le \left[{\tilde{L}}_1\left(|{z}_{k1}|\right),{\tilde{L}}_2\left(|{z}_{k2}|\right),...,{\tilde{L}}_n\left(|{z}_{kn}|\right)\right] \) in the left parts of Li, i = 1, 2, ..., n.■

In addition, to the conditions in the previous theorems, we need to prove another important condition that is E ∣ Z0 ∣  < ∞. This condition confirms the existence of the finiteness.

Theorem 3: If \( \left({\varphi}_1,{\varphi}_2,...,{\varphi}_n,{\overline{\varphi}}_1,{\overline{\varphi}}_2,...,{\overline{\varphi}}_n\right)\in {\varPhi}_0 \) are a combination of finite search, then E ∣ Z0∣ is finite.

Proof It is clear that E(τ) < ∞ if (P(τ is finite) = 1. Thus, we have

$$ {\displaystyle \begin{array}{c}P\left(\tau\;\mathrm{is}\; finite\right)=P\left({\tau}_{\varphi_1}\;\mathrm{is}\; finite\kern0.17em or\;{\tau}_{\varphi_2}\;\mathrm{is}\; finite\kern0.17em or...\mathrm{or}\;{\tau}_{\varphi_n}\;\mathrm{is}\; finite\kern0.17em or\kern0.2em {\tau}_{{\overline{\varphi}}_1}\;\mathrm{is}\; finite\kern0.34em or\kern0.2em {\tau}_{{\overline{\varphi}}_2} is\kern0.17em finite\kern0.34em or...\kern0.2em \mathrm{or}\;{\tau}_{{\overline{\varphi}}_n}\;\mathrm{is}\; finite\right)\\ {}=\underset{i=1}{\overset{n}{\cup }}\left(P\left({\tau}_{\varphi_i}\; is\kern0.17em finite\right)+P\left({\tau}_{{\overline{\varphi}}_i}\;\mathrm{is}\; finite\right)\right)\\ {}=\sum \limits_{i=1}^n\left(P\left({\tau}_{\varphi_i}\;\mathrm{is}\; finite\right)+P\left({\tau}_{{\overline{\varphi}}_i}\;\mathrm{is}\; finite\right)\right).\end{array}} $$

But, we have only one of \( \Big(P\left({\tau}_{\varphi_i} is\kern0.17em finite\right)=1\;\mathrm{or}\;P\left({\tau}_{{\overline{\varphi}}_i}\;\mathrm{is}\kern0.17em \mathrm{finite}\right)=1 \) for all i = 1, 2, ..., n. If we suppose that \( P\left({\tau}_{\varphi_i}\;\mathrm{is}\; finite\right)+P\left({\tau}_{{\overline{\varphi}}_i}\;\mathrm{is}\; finite\right)=1 \), then we have \( {Z}_0={\varphi}_i\left({\tau}_{\varphi_i}\right)-B\left({\tau}_{\varphi_i}\right)+{\overline{\varphi}}_i\left({\tau}_{{\overline{\varphi}}_i}\right)-B\left({\tau}_{{\overline{\varphi}}_i}\right)={\varphi}_i\left({\tau}_{\varphi_i}\right)+{\overline{\varphi}}_i\left({\tau}_{{\overline{\varphi}}_i}\right)-\left(B\left({\tau}_{\varphi_i}\right)+B\left({\tau}_{{\overline{\varphi}}_i}\right)\right) \) with probability one and hence, \( \mid {Z}_0\mid \le \mid {\varphi}_i\left({\tau}_{\varphi_i}\right)+{\overline{\varphi}}_i\left({\tau}_{{\overline{\varphi}}_i}\right)\mid +\mid B\left({\tau}_{\varphi_i}\right)+B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid \le {\tau}_{\varphi_i}+{\tau}_{{\overline{\varphi}}_i}+\mid B\left({\tau}_{\varphi_i}\right)+B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid, E\mid {Z}_0\mid \le E\left({\tau}_{\varphi_i}\right)+E\left({\tau}_{{\overline{\varphi}}_i}\right)+E\mid B\left({\tau}_{\varphi_i}\right)\mid +E\mid B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid \). But, \( \mid B\left({\tau}_{\varphi_i}\right)\mid \le {\tau}_{\varphi_i} \) and \( \mid B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid \le {\tau}_{{\overline{\varphi}}_i} \), then \( E\mid B\left({\tau}_{\varphi_i}\right)+B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid \le E\left({\tau}_{\varphi_i}+{\tau}_{{\overline{\varphi}}_i}\right) \) leads to \( E\mid B\left({\tau}_{\varphi_i}\right)\mid +E\mid B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid \le E\left({\tau}_{\varphi_i}\right)+E\left({\tau}_{{\overline{\varphi}}_i}\right) \). If \( E\left({\tau}_{\varphi_i}\right)<\infty \) and \( E\left({\tau}_{{\overline{\varphi}}_i}\right)<\infty \), then \( E\mid B\left({\tau}_{\varphi_i}\right)\mid <\infty \),\( E\mid B\left({\tau}_{{\overline{\varphi}}_i}\right)\mid <\infty \), and E ∣ Z0∣ is finite. Also, if \( P\left({\tau}_{\varphi_{\hslash }}\;\mathrm{is}\; finite\right)+p\left({\tau}_{{\overline{\varphi}}_{\hslash }}\;\mathrm{is}\; finite\right) \) = 1 for all ℏ ≠ i, i = 1, 2, ..., n then \( {Z}_0={\varphi}_i\left({\tau}_{\varphi_i}\right)-B\left({\tau}_{\varphi_i}\right)+{\overline{\varphi}}_i\left({\tau}_{{\overline{\varphi}}_i}\right)-B\left({\tau}_{{\overline{\varphi}}_i}\right)={\varphi}_i\left({\tau}_{\varphi_i}\right)+{\overline{\varphi}}_i\left({\tau}_{{\overline{\varphi}}_i}\right)-\left(B\left({\tau}_{\varphi_i}\right)+B\left({\tau}_{{\overline{\varphi}}_i}\right)\right) \) with probability one; similarly, we get E ∣ Z0∣ is finite. ■

The direct result to the realization of the previous theorems confirming a finite search plan if E ∣ Z0∣ is finite.

Computational expected value for τ

From the Markovian property of the Brownian motion and its dependence on normal distribution, we can use this information to compute E(τ) which depends on the probability of meeting the target on the line Li, i = 1, 2, ..., n at the time step k. This probability is given by \( {G}_{\left(2k-1\right)i}^{\left(+\right)} \) which is depending on the probability of the meeting at the time \( {G}_{\left(2k-3\right)i}^{\left(+\right)} \).

Theorem 4: The approximated value of E(τ)is given by

$$ E\left(\tau \right)=\sum \limits_{i=1}^n\left({\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}+{\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}\left[1-\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\right]\; d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right){\gamma}_i\left(d{z}_i\right)+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}^2\left({\theta}_{i\left(-\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{\left(-\right)}^{2k-2}\underset{-\infty }{\overset{0}{\int }}\underset{-\infty }{\overset{H_{\left(2k-2\right)i}^{\left(-\right)}+{z}_i}{\int }}\varPhi \left({G}_{\left(2k-4\right)i}^{\left(-\right)}+{z}_i\right)\; d\varPhi \left({G}_{\left(2k-4\right)i}^{\left(-\right)}+{z}_i\right){\gamma}_i\left(d{z}_i\right)\right). $$
(9)

where \( \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\sim N\left(0,\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\right) \) and \( \left({G}_{\left(2k-4\right)i}^{\left(-\right)}+{z}_i\right)\sim N\left(0,\left({G}_{\left(2k-2\right)i}^{\left(-\right)}-{G}_{\left(2k-4\right)i}^{\left(-\right)}\right)\right) \) with the distribution functions \( \varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right) \) and \( \varPhi \left({G}_{\left(2k-4\right)i}^{\left(-\right)}+{z}_i\right) \), respectively, at time step k on the line Li, i = 1, 2, ..., n.

Proof From (5), we get

$$ E\left(\tau \right)=\sum \limits_{i=1}^n\left({\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}P\;\left({\tau}_{\varphi_i}>0\right)+{\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}P\left({\tau}_{\varphi_i}>{G}_{\left(2k-3\right)i}^{\left(+\right)}\right)+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}P\;\left({\tau}_{{\overline{\varphi}}_i}>0\right)+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}^2\left({\theta}_{i\left(-\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{\left(-\right)}^{2k-2}P\;\left({\tau}_{{\overline{\varphi}}_i}>{G}_{\left(2k-2\right)i}^{\left(-\right)}\right)\right). $$
(10)

Also, at the time step k on the right part of the line Li, i = 1, 2, ..., n, we have

$$ P\Big({\tau}_{\varphi_i}>\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\le \underset{0}{\overset{\infty }{\int }}P\left({Z}_{0i}+B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>\left({H}_{\left(2k-3\right)i}^{\left(+\right)}\right)|{Z}_{0i}={z}_i\right){\gamma}_i\left(d{z}_i\right). $$

Consequently, we can find that

$$ P\left({Z}_{0i}+B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>{H}_{\left(2k-3\right)i}^{\left(+\right)}|{Z}_{0i}={z}_i\right)=P\left(\right(B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)>\left({H}_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i\right)=1-P\Big(B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\le \left({H}_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i\right)\right). $$
(11)

Generally, for any times t, s, we have B(t) − B(s) is a normal random variable with mean 0 and variance t − s where s < t. Therefore, we can find that

$$ {\displaystyle \begin{array}{c}P\left(B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\le \left({H}_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i\right)\right)\le \underset{-\infty }{\overset{\infty }{\int }}P\Big(B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\le \left({H}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right),\left[B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)+B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)-B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\right]\\ {}\le \left({H}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\mid B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)=\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\Big)\times \frac{1}{\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\sqrt{2\pi }}\\ {}\times {e}^{-\frac{1}{2}{\left(\frac{G_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i}{\Big({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}}\right)}^2}d\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\\ {}=\underset{-\infty }{\overset{\infty }{\int }}P\left(\left[B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)+B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\right]-B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\le \left({H}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)|B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)=\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\right)\\ {}\times \frac{1}{\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{G_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i}{\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)}\right)}^2}d\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\end{array}} $$

(see Klebaner [30]). By considering that the first meeting is not done at the time \( t={G}_{\left(2k-5\right)i}^{\left(+\right)}\Big) \), then

$$ {\displaystyle \begin{array}{l}P\left(B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\le \left({H}_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i\right)\right)=\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}P\right(\left[\right(B\left({G}_{\left(2k-3\right)i}^{\left(+\right)}\right)\le \left({H}_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i\right)\Big]-\left[\left(B\left({G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\le \left({H}_{\left(2k-5\right)i}^{\left(+\right)}\right)-{z}_i\right)\right]\\ {}\kern1.50em \begin{array}{cccc}& & & \end{array}\begin{array}{cccc}& & & \end{array}\begin{array}{ccc}& & \end{array}\begin{array}{cc}& \end{array}\le -\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\Big)\times \frac{1}{\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\sqrt{2\pi }}\\ {}\begin{array}{cccc}& & & \end{array}\begin{array}{cccc}& & & \end{array}\begin{array}{cccc}& & & \end{array}\begin{array}{cccc}& & & \end{array}\begin{array}{cc}& \end{array}\times {e}^{-\frac{1}{2}{\left(\frac{G_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i}{\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)}\right)}^2}d\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\\ {}\begin{array}{cccc}\begin{array}{ccc}& & \end{array}& & & \end{array}\begin{array}{ccc}& & \end{array}=\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}\varPhi \Big(-\left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right) d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)=\underset{-\infty }{\overset{H_{2i-3}^{\left(+\right)}-z}{\int }}\left[1-\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\right] d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right),\end{array}} $$

where \( \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\sim N\left(0,\left({G}_{\left(2k-3\right)i}^{\left(+\right)}-{G}_{\left(2k-5\right)i}^{\left(+\right)}\right)\right) \) and \( \varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right) \) is the distribution function of this variable. By a similar manner, on the left part of line Li, i = 1, 2, ..., n, one can compute the probability at time step k by

$$ {\displaystyle \begin{array}{l}P\left({Z}_{0i}+B\left({G}_{\left(2k-2\right)i}^{\left(-\right)}\right)<{H}_{\left(2k-2\right)i}^{\left(-\right)}|{Z}_{0i}={z}_i\right)=P\left(B\left({G}_{\left(2k-2\right)i}^{\left(-\right)}\right)-\left({H}_{\left(2k-2\right)i}^{\left(-\right)}+{z}_i\right)<0\right)\\ {}\kern1.50em \begin{array}{cccc}& & & \end{array}\begin{array}{cccc}& & & \end{array}=\underset{-\infty }{\overset{H_{\left(2k-2\right)i}^{\left(-\right)}+{z}_i}{\int }}\varPhi \left({G}_{\left(2k-2\right)i}^{\left(-\right)}+{z}_i\right) d\varPhi \left({G}_{\left(2k-2\right)i}^{\left(-\right)}+{z}_i\right).\end{array}} $$

Since the target starts its motion on the line Li, i = 1, 2, ..., n from a random point zi, then we can assume that \( P\;\left({\tau}_{\varphi_i}>0\right)=P\;\left({\tau}_{{\overline{\varphi}}_i}>0\right)=1 \). In addition, the first interviewing time should be done on one real line by one searcher. Consequently, in (10), we can obtain the approximate value of E(τ) by

$$ E\left(\tau \right)=\sum \limits_{i=1}^n\left({\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}+{\lambda}_{i\left(+\right)}{\theta}_{i\left(+\right)}\left({\theta}_{i\left(+\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{i\left(+\right)}^{2k-2}\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}\left[1-\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\right]\; d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right){\gamma}_i\left(d{z}_i\right)+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}+{\lambda}_{i\left(-\right)}{\theta}_{i\left(-\right)}^2\left({\theta}_{i\left(-\right)}^2-1\right)\sum \limits_{k=2}^{\infty }{\theta}_{\left(-\right)}^{2k-2}\underset{-\infty }{\overset{0}{\int }}\underset{-\infty }{\overset{H_{\left(2k-2\right)i}^{\left(-\right)}+{z}_i}{\int }}\varPhi \left({G}_{\left(2k-4\right)i}^{\left(-\right)}+{z}_i\right)\; d\varPhi \left({G}_{\left(2k-4\right)i}^{\left(-\right)}+{z}_i\right){\gamma}_i\left(d{z}_i\right)\right).\blacksquare $$

If we have X~N(0, σ2), then \( \underset{-\infty }{\overset{0}{\int }} d\varPhi (x)=\frac{1}{2}. \) This leads to \( \frac{1}{2}<\underset{-\infty }{\overset{H_{2i-3}^{\left(+\right)}-z}{\int }} d\varPhi \left({G}_{2i-5}^{\left(+\right)}-z\right)<1 \) where \( \underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}\left[1-\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)\right] d\varPhi \left({G}_{2i-5}^{\left(+\right)}-{z}_i\right)=\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }} d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)-\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right) d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right) \) and \( {H}_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i>0 \). From the fact \( \underset{-\infty }{\overset{0}{\int }}\varPhi (x) d\varPhi (x)=\underset{0}{\overset{\frac{1}{2}}{\int }} xdx=\frac{1}{8}<\frac{1}{2} \), one can easily conclude that \( 0<\underset{-\infty }{\overset{H_{\left(2k-3\right)i}^{\left(+\right)}-{z}_i}{\int }}\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right) d\varPhi \left({G}_{\left(2k-5\right)i}^{\left(+\right)}-{z}_i\right)<\frac{1}{2} \). This also contributes to the assertiveness of the existence and finiteness of our search plan where the Brownian target is not evading the searchers.

Concluding remarks

  1. 1)

    The existence of a cooperative search plan with multiple searchers that used the coordinated linear search technique to find a Brownian target on one of n real line has been presented.

  2. 2)

    The random variable Z0 is used to give the target initial position.

  3. 3)

    We showed the finiteness of our continuous search plan by giving the conditions that make E(τ) is finite as in Theorem 1.

  4. 4)

    At any time step k, we showed that the chosen search plan \( {\varPhi}_0=\left\{\left(\varPhi, \overline{\varPhi}\right):{\varphi}_i\in \varPhi, {\overline{\varphi}}_i\in \overline{\varPhi}\kern0.3em \forall i=1,2,...,n\right\} \) should satisfy \( \left[{M}_1\left({z}_{k1}\right),{M}_2\left({z}_{k2}\right),...,{M}_n\left({z}_{kn}\right)\right]\le \left[{\tilde{M}}_1\left(|{z}_{k1}|\right),{\tilde{M}}_2\left(|{z}_{k2}|\right),...,{\tilde{M}}_n\left(|{z}_{kn}|\right)\right] \), and \( \left[{L}_1\left({z}_{k1}\right),{L}_2\left({z}_{k2}\right),...,{L}_n\left({z}_{kn}\right)\right]\le \left[{\tilde{L}}_1\left(|{z}_{k1}|\right),{\tilde{L}}_2\left(|{z}_{k2}|\right),...,{\tilde{L}}_n\left(|{z}_{kn}|\right)\right] \) where \( \left[{\tilde{M}}_1\left(|{z}_{k1}|\right),{\tilde{M}}_2\left(|{z}_{k2}|\right),...,{\tilde{M}}_n\left(|{z}_{kn}|\right)\right] \) and \( \left[{\tilde{L}}_1\left(|{z}_{k1}|\right),{\tilde{L}}_2\left(|{z}_{k2}|\right),...,{\tilde{L}}_n\left(|{z}_{kn}|\right)\right] \) are vectors of linear functions, see Theorem 2.

  5. 5)

    Besides proving that E ∣ Z0 ∣  < ∞ in Theorem 3, we compute the approximated value of E(τ) in Theorem 4.

  6. 6)

    In future research, one can study the finiteness of the expected value of the first meeting time between one of n-cooperative searchers and n-dimensional Brownian target and calculate this expected value.