/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 2, 2019, 654-668 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global A differential Game Related to Terrorism: Stackelberg Differential Game of E-differentiable and E-convex function Abd El-Monem A. Megahed Basic Science Department, Faculty of Computes and Informatics, Suez Canal University,Ismalia, Egypt Abstract. In this work, the Stackelberg differential game of E-differentiable and E-convex func- tion is studied in order to fight the terrorism taking into account the government’s procedures such as education quality, better job opportunity, social justice, religious awareness and security arrangements. We consider Stackelberg differential game. Firstly, the government is the leader and the terrorist organization is the follower. Secondly, the terrorist organization is the leader and the government is a follower. Furthermore, we apply the necessary conditions of the Stackelberg differential game for these cases to obtain the optimal strategy of this problem. 2010 Mathematics Subject Classifications: 49N70, 49N90, 91A23 Key Words and Phrases: Game theory, Terrorism, stackelberg differential game, Government Activities, E-differentiable 1. Introduction The government’s tasks are very important for counter-terror, the government must be applied some subject of mathematics to get methods for combating terrorism, particularly Operations Research. Counter-terror measures range comes from security arrangements and the government’s procedures to freezing assets of a terrorist organization or even to invade their territories for assassinating the terrorists. Since any action against terrorists must be taken into account their reactions. In this paper, the Stackelberg approach is used for studying the interaction strategies of the government and the terrorist organization. Therefore, we use the concept of E-differentiable and E-convex functions to transform a non-differentiable and non-convex function to E-differentiable and E-convex function [1, 2]. In [3] it is imposed that the success of combating terrorism depends on public opinion, while in [4] the efficiency of ”water” and ”fire” strategies are compared. Hsia[5, 6] introduced the fuzzy differential game to guard a territory movable and not movable, a Nash-collative DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3375 Email address: a megahed15@yahoo.com (A. Megahed) http://www.ejpam.com 654 c© 2019 EJPAM All rights reserved. A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 655 differential game is presented in [7]. A min-max fuzzy differential game with fuzzy on the objective and control, and the large-scale differential game are discussed in [8–11]. The terrorism infighting by Stackelberg and Nash strategies are introduced in [12], the interactions strategics between R&D, defense and preemption is presented in [13], a min- max differential game approach is studied to fight the terrorism [14, 15]. The policies of anti-terrorism and economy of terrorism are introduced in [16, 17]. The study of terror support and recruitment (defense and peace economic) is presented in [18]. Megahed [19] discussed that governments must be made some procedures for fighting the terrorism such as unemployment, justice social, religious awareness, improving the education with considering the security measurements The organization’s power is measured by the terrorist’s activities, the organization’s resources such as weapons, financial capital, and technological expertise. The power of terror organizations changes with the time, the recruitment of terrorists is through existing terrorists. The decreasing rate of terrorists is affected by their own action and the anti- terrorist actions of the government through of the education quality, increase the chances of labor, social justice, religious awareness and security arrangements. The objectives of the government drive his utility from the loss of the terrorist resources and their activities but incur costs for combating terror and dis-utility from the terror organizations, the later tries to maximize its power both by its size and its terrorist attacks. In this work, we study how to help the governments to counterterrorism, the Stackel- berg differential game plays the main role to combat the terrorism. 2. Problem Formulation The differential game with the state variable z(t) which describe the resource of the terror organization (TO). It may also include weapons, financial capital, the network of supporters, etc. and another state variable M(t) which describe the government’s activities, the education quality, increasing the chances of labor, social justice, religious awareness and security arrangements, t ∈ [0,∞) is the time. The two players are the government and (TO) with non-negative strategy v1(t), v2(t) respectively. The stock of resources of (TO) grows according to the growth of a linear function g(z) i.e., g(z) = rz, r > 0, and the government’s procedures grow according to the function A(M) = µM , where µ > 0 is the growth rate of the government’s procedures. Carrying out attacks make a reduction the growth of the resource stock as it affects negatively the number of terrorists (e.g. suicide bombing or caught and killed terrorists). Furthermore, weapons and financial means, it may even include a reduction of the network of supporters. However, the growth reduction of resources stocks does not only depend on the strength of attacks v2(t) but it is also influenced by fighting the terrorists v1(t). This effectiveness of the control variables of the two players on the growth of the resource may be denoted as ”harvest function h(v1, v2). As a consequence, the dynamic equations of the resources stocks and government’s tasks z(t),M(t) respectively can be written as ż = rz(t)− h(v1(t), v2(t)), z(0) = z0 > 0 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 656 Ṁ = µM − av1 + bv2, M(0) =M0 > 0 where h(v1, v2)is non-differentiable and not convex, z0 denotes the initial stock of ter- rorist’s resources,M0 is the initial government’s procedures and a, b are positive constants. Moreover, we assume that z(t) ≥ 0, M(t) ≥ 0 for all t ≥ 0 (1) Now, we define an operator E : Rn → Rn such that (h ◦ E)(v1, v2) is differentiable and convex. Then ẏ = rz(t)− (h ◦ E)(v1, v2) Since the increase rate of counter-terror measure and attacks leads to a reduction of growth, we assume that the partial derivatives are greater than zero: (h ◦ E)v1(v1, v2) > 0, (h ◦ E)v2(v1, v2) > 0. The counter-terror measures exhibit marginally decreasing efficiency (h ◦ E)v1v2 < 0. Moreover the increasing rate of attacks induces disproportional higher losses of resources i.e. (h ◦ E)v2v2 > 0. Finally, the instruments reinforce each other, i.e. (h ◦ E)v1v2 > 0, which makes economical sense. This positive interaction means that the minor efficiency of fighting terrorism increases with the strength of terrorist’s attacks. In addition to, the Inada conditions of the economy are assumed to be fulfilled lim v1→0 (h ◦ E)v1(v1, v2) = ∞, lim v1→∞ (h ◦ E)v1(v1, v2) = 0 (2) lim v2→0 (h ◦ E)v2(v1v2) = 0, lim v2→∞ (h ◦ E)v2(v1, v2) = ∞ (3) This gives that the optimal strategies are nonnegative v1(t) ≥ 0, and v2(t) ≥ 0, t > 0 Player 1 (the government) derives its benefit from the loss of the terrorist resources and their activitiesM(t) besides showing the dis-utility of these terrorist organization but incur costs for combating terror. For simplicity, all these terms must be linear. Thus, the objective of the government is max v1t) { J1 = ∫ ∞ 0 e−η1t [ω(h ◦ E)(v1(t), v2(t)) + qM(t)− cz(t)− kv2(t)− αv1(t)] dt } (4) where ω, c, k, q and α are positive constants. The second player (TO) derives its benefit from the resource stock z(t) and the terrorist actions at intensity v2(t). Then the objective of (To) problem is max v2(t) { J2 = ∫ ∞ 0 e−η2t [σz(t) + βv2(t)− γM(t)] dt } (5) where σ, β and γ are positive constants. The decreasing rates ηi, i = 1, 2 are assumed to be greater than the growth and activity rates, r, µ ηi > r, ηi > µ for i = 1, 2 (6) In this paper, we will derive the Stackelberg equilibria. The solution procedures rely on Pontryagin’s maximum [11] A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 657 3. The stackelberg differential game The Stackelberg approach is a hierarchical solution concept, since one of the players, the leader, has a stronger position in the decision process. This solution is the result of sequential decisions: the leader announces his strategy firstly so that the other players, the followers, can only react to the leader strategy. In this paper, we consider the government plays the leader’s role and the terrorist’s organizations (TO) play the follower’s role, and another case is vice versa. 3.1. Stackelberg game with the government is the leader and (TO) is the follower The government chooses the rate of counter-terror measures before (TO) decides on the rate of attacks. Consider the following problem, where the government and the terrorist organizations are is the leader and the follower respectively maxv1(t) J1 = ∫ ∞ 0 e−η1t [ω(h ◦ E)(v1(t), v2(t)) + qM(t)− cz(t)− kv2(t)− αv1(t)] dt maxv2(t) J2 = ∫ ∞ 0 e−η2t [σz(t) + βv2(t)− γM(t)] dt y· = rz(t)− (h ◦ E)(v1(t), v2(t)), z(0) = z0 > 0 , z(t) ≥ 0 M · = µM(t)− av1 + bv2, M(0) =M0 > 0 ,M(t) ≥ 0                (7) 3.1.1. The follower Problem Firstly, we find the optimization of the follower (To) which consider the action of a leader is given maxv2(t) J2 = ∫ ∞ 0 e−η2t [σz(t) + βv2(t)− γ M(t)] dt y· = rz(t)− (h ◦ E)(v1(t), v2(t)), z(0) = z0 } (8) The Hamiltonian function of the follower (To), H2, is defined by H2(z(t), v1(t), v2(t), λ1(t)) = σz(t) + βv2(t)− γ M(t) + λ1(t)(rz(t)− (h ◦E)(v1(t), v2(t))) (9) From the necessary conditions ∂H2 ∂v2 = β − λ1(h ◦ E)v2 = 0 and consider the harvest function h(v1, v2) = v 1 τ 1 v 1 δ 2 , where τ, δ are positive integers Consider the following operator E(v1, v2) = (v1 nτ , v2 mδ), m,n are positive integer, then (h ◦ E)(v1, v2) = vn1 v m 2 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 658 and (h ◦ E)v2 = mvn1 v m−1 2 = β λ1 and thus v∗2(v1) = ( β mλ1 ) 1 m−1 v n 1−m 1 (10) and the harvest function (h ◦ E)(v1, v ∗ 2) = ( β mλ1 ) m m−1 v −n m−1 1 (11) The adjoint variable λ1 satisfy the following differential equation λ̇1 = λ1η2 − ∂H2 ∂z = λ1(η2 − r)− σ (12) Remark 1. Sincem > 1 (10) and (11) implies to any increase of combating measures leads to a more cautious behavior of the terrorists as well as a lower harvest of the terrorists’ resources. 3.1.2. The leader Problem (The government) To obtain the optimal strategy v1 for the government, the government must be taken into account the optimal strategy of the follower which is represented by an additional costate equation. maxv1(t) J1 = ∫ ∞ 0 e−η1t [ ω ( β mλ1 ) m m−1 v −n m−1 1 + qM(t)− cz(t)− k ( β mλ1 ) 1 m−1 v n 1−m 1 − αv1(t) ] dt ż = rz(t)− ( β mλ1 ) m m−1 v −n m−1 1 , Ṁ = µM(t)− av1 + b ( β mλ1 ) 1 m−1 v n 1−m 1 , λ̇1 = λ1(η2 − r)− σ                  (13) The Hamiltonian function of the leader H1 = ( (ω − ψ1) ( β mλ1 ) m m−1 + (bψ2 − k) ( β mλ1 ) 1 m−1 ) v −n m−1 1 + (q + µψ2)M(t) −(α+ aψ2)v1 + (rψ1 − c)z(t) + ψ3(λ1(η2 − r)− σ) from the necessary conditions, ∂H1 ∂v1 = n m− 1 [ (ψ1 − ω) ( β mλ1 ) m m−1 + (k + bψ2) ( β mλ1 ) 1 m−1 ] v 1−n−m m−1 1 − (α+ aψ2) = 0 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 659 v1 = [ (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 ] m−1 1−n−m ( β mλ1 ) m m+n−1 (14) The adjoint variables satisfy the following differential equations ψ̇1 = η1ψ1 − ∂H1 ∂y = (η1 − r)ψ1 + c ψ̇2 = η1ψ2 − ∂H1 ∂M = (η1 − µ)ψ2 − q ψ̇3 = η1ψ3 − ∂H1 ∂λ1 = (η1 − η2 + r)ψ3 + m λ1(m−1) [ ω − ψ1 + (bψ2−k)λ1 β ] ( β mλ1 ) m m−1 v −m m−1+ 1            (15) Remark 2. The adjoint variable ψ3 of the leader with respect to the adjoint variable λ1 of the follower has no influence on the optimization of the leader Proposition 1. A feasible solution of the game with the government is the leader and the (TO) is the follower is exists if and only if k < bψ2 + β(ω − ψ1) mλ1 (16) Proof. . The optimal solution of the leader and follower is feasible if and only if (α+ aψ2)m(m− 1)λ1 nβ(ψ1 − ω) + nm(k + bψ2)λ1 > 0 then, nβ(ω − ψ1) + nm(bψ2 − k)λ1 > 0 and thus k < bψ2 + β(ω − ψ1) mλ1 The optimal strategies are given by v1 = [ (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 ] m−1 1−n−m ( β mλ1 ) m m+n−1 (17) v2 = [ (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ] n n+m−1 ( β mλ1 ) 1−n n+m−1 (18) and the harvest function with the operator E is (h ◦ E)(v1, v2) = [ (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ] n n+m−1 ( β mλ1 ) n n+m−1 (19) Proposition 2. The values of the steady state for the inventory resources and the gov- ernments’s procedures are given by z∞ = 1 r [ (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ] n n+m−1 ( β mλ1 ) n n+m−1 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 660 M∞ = 1 µ [ a ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 ) m−1 1−n−m ( β mλ1 ) m m+n−1 ] − 1 µ [ b ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ) n n+m−1 ( β mλ1 ) 1−n n+m−1 ] Proof. . The solution of the differential equation ˙z(t) = rz(t)− (h ◦ E)(v1, v2) is z(t) e−rt = (h ◦ E)(v1, v2) r e−rt + c1 where c1 is the constant, for t→ ∞, then c1 = 0, and z∞ = 1 r [ (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ] n n+m−1 ( β mλ1 ) n n+m−1 Also, the solution of the differential equation Ṁ = µM + bv2 − av1 is Me−µt = 1 µ e−µt(av1 − bv2) + c0(constant) For t→ ∞ then c0 = 0 and M∞ = 1 µ [ a ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 ) m−1 1−n−m ( β mλ1 ) m m+n−1 ] − 1 µ [ b ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ) n n+m−1 ( β mλ1 ) 1−n n+m−1 ] Proposition 3. (i) The government as the leader is more active but the (TO) as the follower is more cautiously if and only if aβ mλ1 > b ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 )( β mλ1 ) (ii) The government as the leader is more cautiously and the (TO) as the follower is more aggressively if and only if aβ mλ1 < b ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 )( β mλ1 ) Proof. A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 661 (i) The government as the leader is more active if and only if M(t) > 0, from Proposi- tion 2 we have a ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 ) m−1 1−n−m ( β mλ1 ) m m+n−1 −b ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k) ) n n+m−1 ( β mλ1 ) 1−n n+m−1 > 0 Then aβ mλ1 > b ( (α+ aψ2)(m− 1)β nβ(ω − ψ1) + nm(bψ2 − k)λ1 )( β mλ1 ) (ii) The proof of 2 is similar to 1 with less than sign instead of greater than sign Lemma 1. The objectives of the government and (TO) are given by J1 = 1 η1 [ ω ( β mλ1 ) m m−1 − k ( β mλ1 ) 1 m−1 ] v −n m−1 1 + q η1−µ ( M0 − av1−bv2 η1µ ) + q µη1 (av1 − bv2)− c η1−r ( y0 − h◦E(v1,v2) r ) + (h◦E)(v1,v2) rη1 J2 = σ η2 − r ( z0 − h ◦ E(v1, v2) r ) + σh ◦ E(v1, v2) η2 − γ η2 − µ ( M0 − av1 − bv2 µ ) − γ(av1 − bv2) µη2 + βv2 η2 Proof. . The solution of the differential equation Ṁ(t) = µM − av1 + bv2 is Me−µ(t) = 1 µ (av1 − bv2)e −µt + c4(constant) For t→ 0 then c4 =M0 − av1−bv2 µ ,and thus M(t) = ( M0 − av1 − bv2 µ ) eµt + av1 − bv2 µ (20) similarly the solution of the differential equation ż(t) = rz(t)− h(v1, v2) z(t) = ( z0 − h(v1, v2) r ) ert + h(v1.v2) r (21) from (20) and (21) in the objectives of the leader and follower,and by integration we have J1 = 1 η1 [ ω ( β mλ1 ) m m−1 − k ( β mλ1 ) 1 m−1 ] v −n m−1 1 + q η1−µ ( M0 − av1−bv2 η1µ ) + q µη1 (av1 − bv2)− c η1−r ( y0 − h◦E(v1,v2) r ) + (h◦E)(v1,v2) rη1 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 662 J2 = σ η2 − r ( z0 − h ◦ E(v1, v2) r ) + σh ◦ E(v1, v2) η2 − γ η2 − µ ( M0 − av1 − bv2 µ ) − γ(av1 − bv2) µη2 + βv2 η2 where v1, v2 and h(v1.v2) are defined in (17), (18), and (19) Remark 3. As shown in (20), (21) and according to the condition (1), we must be assume that, M0 > av1−bv2 µ , and z0 > h◦E(v1.v2) r 3.2. The Stackelberg with the (TO) is the leader and the government is the follower In this case, the (TO) attacks the government before the government makes counter- terror measures. Consider the following problem, where the(TO) is the leader and gov- ernment is the follower. 3.2.1. The follower problem (The government problem) Firstly, we consider the optimization of the follower ( government) which consider the action of leader is given maxv1(t) J1 = ∫ ∞ 0 e−η1t [ω(h ◦ E)(v1(t), v2(t)) + qM(t)− cz(t)− kv2(t)− αv1(t)] dt ż = rz(t)− (h ◦ E)(v1(t), v2(t)), z(0) = z0 > 0 , z(t) ≥ 0 Ṁ = µM(t)− av1 + bv2, M(0) =M0 > 0 ,M(t) ≥ 0    (22) The Hamiltonian function of the follower H1 is defined by H1 = ω(h ◦ E)(v1.v2) + qM − cz(t)− kv2 − αv1 + λ1(ry − h) + λ2(µM − av1 + bv2) From the necessary conditions ∂H1 ∂v1 = ω(h ◦ E)v1 − α− λ1(h ◦ E)v1 − λ2a = 0 (h ◦ E)v1 = α+ λ2a ω − λ1 and consider the harvest function h(v1, v2) = v 1 τ 1 v 1 δ 2 , where τ, δ are positive integers Consider the following operator E(v1, v2) = (v1 nτ , v2 mδ), m,n are positive integers, then (h ◦ E)(v1, v2) = vn1 v m 2 Then (h ◦ E)v1 = nvn−1 1 vm2 = α+ λ2a ω − λ1 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 663 and thus v∗1(v2) = ( α+ λ2a n(ω − λ1) ) 1 n−1 v m 1−n 2 (23) and the harvest function (h ◦ E)(v∗1(v2), v2) = ( α+ λ2a n(ω − λ1) ) n n−1 v m 1−n 2 (24) Remark 4. : The follower (government) reacts with a higher strength of counter-terror measures in case (TO) intensifies its rate attacks, as shown in (23) which implies to a higher harvest (i) If m 1−n = 1 see Fig.1(An increasing of counter-terror measures and the terrorists is more cautious), (ii) If m 1−n < 1 see Fig.2 (The (TO) is more aggressively when the government increases their counter-terror measures), (iii) If m 1−n > 1 see Fig.3 (The government is more aggressively with any incrossing of (TO) attacks). The adjoint variables λ1, λ2 satisfy the following differential equations λ̇1 = η1λ1 − ∂H1 ∂y = (η1 − r)λ1 + c λ̇2 = η1λ2 − ∂H1 ∂M = (η1 − µ)λ2 − q 3.2.2. The leader problem (TO) To obtain on the optimal strategy v2 for the (TO), it must be taken into account the optimal strategy of the follower which is represented by an additional costate equation maxv2(t) J2 = ∫ ∞ 0 e−η2t [σz(t) + βv2(t)− γ M(t)] dt ż = rz(t)− ( α+λ2a n(ω−λ1) ) n n−1 v m 1−n 2 , λ̇1 = (η1 − r)λ1 + c λ̇2 = (η1 − µ)λ2 − q            (25) The Hamiltonian function of the leader H2 = σz(t) + βv2(t)− γ M(t) + Ψ1 ( rz(t)− ( α+ λ2a n(ω − λ1) ) n n−1 v m 1−n 2 ) (26) + Ψ2((η1 − r)λ1 + c) + Ψ3((η1 − µ)λ2 − q) A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 664 From the necessary conditions ∂H2 ∂v2 = β −Ψ1 m 1− n ( α+ λ2a n(ω − λ1) ) n n−1 v m+n−1 1−n 2 = 0 and thus v2 = ( β(1− n) Ψ1m ) 1−n n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 (27) The adjoint variables satisfy the differential equations Ψ̇1 = η1Ψ1 − ∂H2 ∂y = (η1 − r) psi1 − σ Ψ̇2 = η1Ψ2 − ∂H2 ∂λ1 = rΨ2 + ψ1n (n−1)(ω−λ1) ( α+aλ2 n(ω−λ1) ) n 1−n v m 1−n 2 Ψ̇3 = η1ψ3 − ∂H2 ∂λ2 = µΨ3 + aΨ1 (n−1)(ω−λ1) ( α+aλ2 n(ω−λ1) ) 1 1−n v m 1−n 2              (28) Proposition 4. The optimal strategies, the harvest function and the steady state values of state variables z(t) and M(t) with the (TO) is the leader and the government is follower are given by v1 = ( β(1− n) Ψ1m ) m n+m−1 ( α+ aλ2 n(ω − λ1) ) 1−m n+m−1 v2 = ( β(1− n) Ψ1m ) 1−n n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 (h ◦ E)(v1, v2) = ( β(1− n) Ψ1m ) m n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 z∞ = 1 r ( β(1− n) Ψ1m ) m n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 M∞ = 1 µ [ a ( β(1− n) Ψ1m ) m n+m−1 ( α+ aλ2 n(ω − λ1) ) 1−n n+m−1 − b ( β(1− n) Ψ1m ) 1−n n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 ] Proof. from (23), (24), and (27) we get on v1, v2, and (h ◦ E)(v1, v2) The solution of the differential equation Ṁ(t) = µM − av1 + bv2 is Me−µ(t) = 1 µ (av1 − bv2)e −µt + c5(constant) For t→ ∞ then c4 = 0, then M(t) = av1−bv2 µ Similarly the solution of the differential equation ż = rz(t)− ( α+λ2a n(ω−λ1) ) n n−1 v m 1−n 2 is z(t) = 1 r ( α+ λ2a m(ω − λ1) ) n n−1 v m 1−n 2 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 665 From (23),(24) and (27), we have M∞ = 1 µ [ a ( β(1− n) Ψ1m ) δ n+m−1 ( α+ aλ2 n(ω − λ1) ) 1−m n+m−1 − b ( β(1− n) Ψ1m ) 1−n n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 ] z∞ = 1 r ( β(1− n) Ψ1m ) m n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 Lemma 2. The objectives functional of the leader (TO), J2, and the follower, J1, are given by J1 = ω(h ◦ E)(v1, v2) η1 + q η1 − µ ( M0 − av1 − bv2 µ ) + av1 − bv2 µη1 − c η1 − r ( z0 − (h ◦ E)(v1, v2) r ) − c (h ◦ E)(v1, v2) rη1 J2 = σ η2 − r ( z0 − (h ◦ E)(v1, v2) r ) + σ(h ◦ E)(v1, v2) rη2 + βv2 η2 − γ η2 − µ ( M0 − av1 − bv2 µ ) − γ(av1 − bv2) µη2 Proof. . From (20) and (21) in the follower’s objective, J1, and the leader’s objective, J2, then J1 = ∫ ∞ 0 e−η1t [ ω(h ◦ E)(v1, v2) + q ( M0 − av1 − bv2 µ ) eµt + q(av1 − bv2) µ ] − c e−η1t [( z0 − (h ◦ E)(v1, v2) r ) ert − c(h ◦ E)(v1, v2) r − kv2 − αv1 ] dt = ω(h ◦ E)(v1, v2) η1 + q η1 − µ ( M0 − av1 − bv2 µ ) + q(av1 − bv2) µη1 − c η1 − r ( z0 − (h ◦ E)(v1, v2) r ) − c(h ◦ E)(v1, v2) rη1 − kv2 η1 − αv1 η1 and J2 = ∫ ∞ 0 e−η2t [ σ ( z0 − (h ◦ E)(v1, v2) r ) ert + σ(h ◦ E)(v1, v2) r ] − e−η2t [ γ ( M0 − av1 − bv2 µ ) eµt + γ(av1 − bv2) µ + βv2 ] dt = σ η2 − r ( z0 − (h ◦ E)(v1, v2) r + ) + σ(h ◦ E)(v1, v2) rη2 − γ η2 − µ ( M0 − av1 − bv2 µ ) − γ(av1 − bv2) µη2 + βv2 η2 A. Megahed / Eur. J. Pure Appl. Math, 12 (2) (2019), 654-668 666 Proposition 5. (i) The government as the follower is more cautiously and the (TO) as leader is more aggressively if and only if a ( β(1− n) Psi1m ) < b ( α+ aλ2 n(ω − λ1) ) (ii) The government as the follower is more active and the (TO) as leader is more cautiously if and only if a ( β(1− n) Psi1m ) > b ( α+ aλ2 n(ω − λ1) ) Proof. . (i) The government as the follower is more cautiously and the (TO) as leader is more aggressively if M(t) < 0, Since M(t) = av1−bv2 µ then, av1 < bv2. From the values v1 and v2 of Proposition 4, we find that a ( β(1− n) Ψ1m ) m n+m−1 ( α+ aλ2 n(ω − λ1) ) 1−m n+m−1 < b ( β(1− n) Ψ1δ ) 1−n n+m−1 ( α+ aλ2 n(ω − λ1) ) n n+m−1 and thus a ( β(1− n) Ψ1m ) < b ( α+ aλ2 n(ω − λ1) ) (ii) The proof of 2 is similar to 1 with greater than sign instead of less than sign Acknowledgements My highly grateful to my Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt. REFERENCES 667 References [1] Youness, E. A. : E-convex sets, E-convex functions and E-convex programming, J. Optim. Theory Appl. 102(3)(1999) 439-450 [2] Megahed et al.,Optimality conditions of E-convex programming for an E-differentiable function, Journal of Inequalities and Applications (2013) 2013:246 [3] JP. Caulkins, G. Feichtinger, D.Grass, G.Tragler, Optimal control of terrorism and global reputation: A case study with novel threshold behavior, Oper. Res. Lett. 3(2009) 387-391. [4] JP. Caulkins, G.Feichtinger, D.Grass, G.Tragler, Optimizing Counterterror opera- tions: Should one fight with ”Fire ”or ”water”?Comput. Oper. Res. 35, (2008)1874- 1855 [5] K.H.Hsia, and J. G.Hsie, A first approach to fuzzy differential Game problem: guard- ing Territory, Fuzzy set.Syst. , No.55, (1993), pp. 157-167. [6] I.C.Hung, K.H.Hsia, and L.W.Chen, Fuzzy Differential Game of Guarding a Movable Territory, Inform. Sciences No.91, (1993)pp.113-131. [7] E.Youness, J.B.Hughes, and El-kholy, Parametric Nash Collative Differential Games, Math. Comput.Model.,Vol.26, No.2, (1997), pp. 97-105. [8] E.Youness and A.A.Megahed, A study on Fuzzy Differential Game, Le Matematche, Vol. VI, Fasc.1, (2001), pp.97-107. [9] E.Youness, A.A.Megahed, A Study on Large Scale Continuous Differential Games, Bull. Cul. Math. Soc.,94(5), (2002) 359-368. [10] S .Hegazy, A.A Megahed,E.Youness and A. Elbanna , Min-Max Zero-Sum two Persons Fuzzy Continuous Differential Games, IJAM, Vol.21, No.1, (2008), pp. 1-16. [11] A.A. Megahed, S. Hegazy, Min-Max Zero two Persons Continuous Differential Game with Fuzzy Control, AJCEM 2:, 2 March (2013) p.86-98 [12] A.J. Nova,G.Feichtinger,G.Leitmann, A differential game related to terrorism: Nash and Stackelberg strategies, J. Optim. Theory Appl. 144, (2010), 533-555 [13] Abhra Roy, Jomon Aliyas Paul, Terrorism deterrence in a two country frame- work: strategic interactions between R&D, defense and pre-emption, Ann.Oper. Res., V.211,Issue 1 (December 2013), pp.399-432. [14] Abd El-Monem.A. Megahed, A differential game related to terrorism: Min-Max Zero- Sum two persons differential game, NCA, published at 28-11-2016 ,pp.1-6 [15] Abd El-Monem.A. Megahed (2017) The development of A differential Game Related to Terrorism:Min-Max Differential Game,JEMS,25, Issue 3, pp. 308-312 REFERENCES 668 [16] Enders, W. and T. Sandler (2012) Political Economy of Terrorism, Cambridge Uni- versity Press. Cambridge. [17] Enders, W. and T. Sandler (1993) The effectiveness of anti-terrorism policies: A vectorautoregression-intervention analysis, American Political Science Review 87, 829-844. [18] Faria, J.R. and D. Arce (2005) Terror support and recruitment, Defence and Peace Economics 16, 263-273. [19] Abd El-Monem A. Megahed, The Stackelberg diferential game forcounter-terrorism, Quality & Quantity published at 19-3-2018 pp.1-14