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.