On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients
aa r X i v : . [ m a t h . D S ] J un On Nonoscillation of Mixed Advanced-Delay DifferentialEquations with Positive and Negative Coefficients
Leonid Berezansky Department of Mathematics, Ben-Gurion University of the Negev,Beer-Sheva 84105, IsraelElena Braverman Department of Mathematics and Statistics, University of Calgary,2500 University Drive N.W., Calgary, AB T2N 1N4, CanadaandSandra Pinelas Universidade dos A¸cores, Departamento de Matem´atica,R. M˜ae de Deus, 9500-321 Ponta Delgada, Portugal
Abstract
For a mixed (advanced–delay) differential equation with variable delays and coeffi-cients ˙ x ( t ) ± a ( t ) x ( g ( t )) ∓ b ( t ) x ( h ( t )) = 0 , t ≥ t where a ( t ) ≥ , b ( t ) ≥ , g ( t ) ≤ t, h ( t ) ≥ t explicit nonoscillation conditions are obtained. AMS(MOS) subject classification.
Keywords: mixed differential equations, delay equations, advanced equations, positivesolutions
Differential equations with delayed in advanced arguments occur in many applied problems,see [1, 2, 3, 4, 5, 6], especially in mathematical economics. There are natural delays in impact Partially supported by Israeli Ministry of Absorption Partially supported by the NSERC Research Grant x ( t ) + δ a ( t ) x ( g ( t )) + δ b ( t ) x ( h ( t )) = 0 , t ≥ t (1)with variable coefficients a ( t ) ≥ , b ( t ) ≥
0, one delayed ( g ( t ) ≤ t ) and one advanced( h ( t ) ≥ t ) arguments. To the best of our knowledge, oscillation of such equations has notbeen studied before, except partial cases of autonomous equations [20]-[24], equations of thesecond or higher order [25]-[27] and equations with constant delays [28]. In [29] nonoscillationonly of equation (1) and higher order equations was considered, where δ and δ have thesame sign. In [30] the author considers a differential equation with a deviating argumentwithout the assumption that it is either a delay or an advanced equation. Hence the results of[30] can be applied to MDE (1). The results of the present paper and of [30] are independent.We consider equation (1) under the following conditions:(a1) a ( t ) , b ( t ) , g ( t ) , h ( t ) are Lebesgue measurable locally essentially bounded functions, a ( t ) ≥ , b ( t ) ≥ g ( t ) ≤ t, h ( t ) ≥ t, lim t →∞ g ( t ) = ∞ .For equation (1) we can consider the same initial value problem as for delay equations: x ( t ) = ϕ ( t ) , t < t , x ( t ) = x . (2) Definition.
An absolutely continuous on each interval [ t , b ] function x : IR → IR is called asolution of problem (1)-(2), if it satisfies equation (1) for almost all t ∈ [ t , ∞ ) and equalities(2) for t ≤ t .In the present paper we will not discuss existence and uniqueness conditions for a solutionof the problem (1)-(2), instead as mentioned before we will only discuss asymptotic propertiesof the solutions. Definition.
Solution x ( t ) , t ≤ t < ∞ , of a differential equation or inequality is called nonoscillatory if there exists T such that x ( t ) = 0 for any t ≥ T and oscillatory otherwise.2n [31] for equation (1) the cases when the delay and the advanced term have the samesign ( δ = δ = 1 or δ = δ = −
1) were investigated and the following main results wereobtained.
Theorem A
Suppose for the equation ˙ x ( t ) + a ( t ) x ( g ( t )) + b ( t ) x ( h ( t )) = 0 (3) (a1)-(a2) hold, functions a ( t ) , b ( t ) , g ( t ) , h ( t ) are equicontinuous on [0 , ∞ ) and lim sup t →∞ [ t − g ( t )] < ∞ , lim sup t →∞ [ h ( t ) − t )] < ∞ . (4) If the delay equation ˙ x ( t ) + a ( t ) x ( g ( t )) + b ( t ) x ( t ) = 0 has a nonoscillatory solution then equation (3) also has a nonoscillatory solution.In particular, if lim sup t →∞ Z tg ( t ) a ( s ) exp (Z sg ( s ) b ( τ ) dτ ) ds < e then equation (3) has a nonoscillatory solution. Theorem B
Suppose for the equation ˙ x ( t ) − a ( t ) x ( g ( t )) − b ( t ) x ( h ( t )) = 0 (5) (a1)-(a2) hold, functions a ( t ) , b ( t ) , g ( t ) , h ( t ) are equicontinuous on [0 , ∞ ) and condition (4)holds. If the advanced equation ˙ x ( t ) − a ( t ) x ( t ) − b ( t ) x ( h ( t )) = 0 has a nonoscillatory solution then equation (5) also has a nonoscillatory solution.In particular, if lim t →∞ sup Z h ( t ) t b ( s ) exp (Z h ( s ) s a ( τ ) dτ ) ds < e then equation (5) has a nonoscillatory solution. In the present paper we consider the two remaining cases where coefficients have differentsigns: δ = 1 , δ = − δ = − , δ = 1. For these cases we obtain some rather naturalexplicit nonoscillation conditions for equation (1): in Section 2 for the former case, in Section3 for the latter one. Finally, Section 4 involves discussion of the results and outlines someopen problems on MDE.It is important to emphasize that in applications of ADE and MDE, for example in eco-nomics, it is interesting to obtain not only positive solutions but also positive monotonesolutions, which keep the trend. Most of the theorems in this paper present sufficient con-ditions when solutions of this kind exist. Some of the results also present explicit estimatesfor positive solutions.We note that the methods applied in the present paper are different from [31]. The criteria(like Theorem 2) and non-explicit results of the general form (Theorem 3) are supplementedby corollaries which provide easily verified sufficient conditions.3 Positive Delay Term, Negative Advanced Term
In this section we consider the case δ = 1 , δ = − x ( t ) + a ( t ) x ( g ( t )) − b ( t ) x ( h ( t )) = 0 , t ≥ t . (6) Theorem 1
Suppose (a1)-(a2) hold and a ( t ) ≥ b ( t ) . Then the following conditions areequivalent:1. Differential inequality ˙ x ( t ) + a ( t ) x ( g ( t )) − b ( t ) x ( h ( t )) ≤ , t ≥ t , (7) has an eventually nonincreasing positive solution.2. Integral inequality u ( t ) ≥ a ( t ) exp (Z tg ( t ) u ( s ) ds ) − b ( t ) exp ( − Z h ( t ) t u ( s ) ds ) , t ≥ t , (8) has a nonnegative locally integrable solution for some t ≥ t , where we assume u ( t ) = 0 for t < t .3. Differential equation (6) has an eventually positive nonincreasing solution. Proof. ⇒ x be a solution of (7) such that x ( t ) > , ˙ x ( t ) ≤ , t ≥ t . For some t ≥ t we have g ( t ) ≥ t for t ≥ t . Denote u ( t ) = − ˙ x ( t ) /x ( t ) , t ≥ t , u ( t ) = 0, t < t .Then x ( t ) = x ( t ) exp (cid:26) − Z tt u ( s ) ds (cid:27) , t ≥ t . (9)After substituting (9) into (7) and carrying the exponent out of the brackets we obtain − exp (cid:26) − Z tt u ( s ) ds (cid:27) x ( t ) " u ( t ) − a ( t ) exp (Z tg ( t ) u ( s ) ds ) + b ( t ) exp ( − Z h ( t ) t u ( s ) ds ) ≤ . Hence (8) holds.2) ⇒ u ( t ) ≥ , t ≥ t is a solution of inequality (8). Consider the followingsequence u n +1 ( t ) = a ( t ) exp (Z tg ( t ) u n ( s ) ds ) − b ( t ) exp ( − Z h ( t ) t u n ( s ) ds ) , n ≥ . (10)Since u n ( t ) ≥ a ( t ) − b ( t ) ≥ u ≥ u then by induction0 ≤ u n +1 ( t ) ≤ u n ( t ) ≤ . . . ≤ u ( t ) . Hence there exists a pointwise limit u ( t ) = lim n →∞ u n ( t ) . u ( t ) = a ( t ) exp (Z tg ( t ) u ( s ) ds ) − b ( t ) exp ( − Z h ( t ) t u ( s ) ds ) . Then x ( t ) denoted by (9) is a nonnegative nonincreasing solution of equation (6).Implication 3) ⇒
1) is evident. ⊓⊔ For comparison consider now the following MDE˙ x ( t ) + a ( t ) x ( g ( t )) − b ( t ) x ( h ( t )) = 0 , t ≥ t . (11) Corollary 1.1
Suppose (a1)-(a2) hold for a , b , h , g , a , b , h , g and b ( t ) ≤ b ( t ) ≤ a ( t ) ≤ a ( t ) , g ( t ) ≥ g ( t ) , h ( t ) ≤ h ( t ) . (12)If equation (11) has an eventually positive solution with an eventually nonpositive derivativethen the same is valid for equation (6). Proof.
Suppose (11) has an eventually positive solution with an eventually nonpositivederivative. By Theorem 1 the integral inequality u ( t ) ≥ a ( t ) exp (Z tg ( t ) u ( s ) ds ) − b ( t ) exp ( − Z h ( t ) t u ( s ) ds ) , t ≥ t , has a nonnegative locally integrable solution u ( t ) for some t . Inequalities (12) imply that u ( t ) also satisfies (8). Thus by Theorem 1 equation (6) has an eventually positive solutionwith an eventually nonpositive derivative. ⊓⊔ Corollary 1.2
Suppose (a1)-(a2) hold, for t sufficiently large a ( t ) ≥ b ( t ) and b ( t ) ≥ a ( t ) " exp (Z tg ( t ) a ( s ) ds ) − exp (Z h ( t ) t a ( s ) ds ) . Then there exists an eventually positive solution with an eventually nonpositive derivativeof equation (6).
Proof . It is easy to see that u ( t ) = a ( t ) is a nonnegative solution of inequality (8). ⊓⊔ Corollary 1.3
Suppose (a1)-(a2) hold and there exist a > , b > , τ > , σ > b ≤ b ( t ) ≤ a ( t ) ≤ a, g ( t ) ≥ t − τ, h ( t ) ≤ t + σ. If there exists a solution λ > − λ + ae λτ − be − λσ = 0 (13)then equation (6) has an eventually positive solution with an eventually nonpositive deriva-tive. Proof.
The function x ( t ) = e − λt is a positive solution of the equation˙ x ( t ) + ax ( t − τ ) − bx ( t + σ ) = 0 , t ≥ . (14)5y Corollary 1.1 equation (6) has a nonoscillatory solution. ⊓⊔ Corollary 1.4
Suppose (a1)-(a2) hold, a ( t ) ≥ b ( t ) and there exists a nonoscillatory solutionof the delay equation ˙ x ( t ) + a ( t ) x ( g ( t )) = 0 . (15)Then there exists an eventually positive solution with an eventually nonpositive derivativeof equation (6). Proof.
Theorem 1 in [32] implies that there exists a nonnegative solution u ( t ) of the in-equality u ( t ) ≥ a ( t ) exp (Z tg ( t ) u ( s ) ds ) . Hence u ( t ) is also a nonnegative solution of inequality (8), then by Theorem 1 equation (6)has a nonoscillatory solution. ⊓⊔ Remark.
Equation (15) has a nonoscillatory solution [8] if for t sufficiently large Z tg ( t ) a ( s ) ds ≤ e . Corollary 1.5
Suppose (a1)-(a2) hold, a ( t ) ≥ b ( t ), the integral inequality (8) has a non-negative solution for t ≥ t and R ∞ [ a ( s ) − b ( s )] ds = ∞ . Then there exists an eventuallypositive solution x ( t ) with an eventually nonpositive derivative of equation (6) such thatlim t →∞ x ( t ) = 0. Proof.
By the assumption of the theorem there exists a nonnegative solution u ( t ) of equa-tion (8), which obviously satisfies u ( t ) ≥ a ( t ) − b ( t ). Then the function x ( t ) defined by (9)is a solution of (6). For this solution we have 0 < x ( t ) ≤ x ( t ) exp (cid:26) − Z t [ a ( s ) − b ( s )] ds (cid:27) .Hence lim t →∞ x ( t ) = 0. ⊓⊔ Corollaries 1.4 and 1.5 imply the following statement.
Corollary 1.6
Suppose (a1)-(a2) hold, a ( t ) ≥ b ( t ), for t sufficiently large R tg ( t ) a ( s ) ds ≤ e and R ∞ [ a ( s ) − b ( s )] ds = ∞ . Then the equation˙ x ( t ) + a ( t ) x ( t ) − b ( t ) x ( h ( t )) = 0has an eventually positive solution x ( t ) with an eventually nonpositive derivative such thatlim t →∞ x ( t ) = 0. Example 1.
Consider the equation˙ x ( t ) + 1 . x ( t − . − . x ( t + 0 .
3) = 0 . (16)Then u ( t ) ≡ . e . − . e − . ≈ . < . · . . > e ≈ . x ( t ) + 1 . x ( t − .
3) = 0 (17)6nd ˙ x ( t ) − . x ( t + 0 .
3) = 0 (18)are oscillatory [9]. The characteristic equation − λ + 1 . e . λ − . e − . λ = 0 (19)has three real roots: λ ≈ − . λ ≈ . λ ≈ . e − λ t , e − λ t and e − λ t are three nonoscillatory solutions of (16), the first one is unbounded on [0 , ∞ ) and the othertwo are bounded and have a negative derivative. Example 2.
Consider the equation˙ x ( t ) + (1 .
375 + 0 .
025 sin t ) x ( t − . − (1 .
325 + 0 .
025 cos t ) x ( t + 0 .
3) = 0 . (20)Since 1 . ≤ .
325 + 0 .
025 cos t ≤ . ≤ .
375 + 0 .
025 sin t ≤ . x ( t ) with an eventually nonpositive derivative. Moreover, since the integral of a continuousnonnegative periodic function R ∞ (0 . .
025 sin t − .
025 cos t ) dt diverges, then by Corollary1.5 this solution satisfies lim t →∞ x ( t ) = 0. Theorem 2
Suppose (a1)-(a2) hold and b ( t ) ≥ a ( t ) . Then the following conditions areequivalent:1. Differential inequality ˙ x ( t ) + a ( t ) x ( g ( t )) − b ( t ) x ( h ( t )) ≥ , t ≥ t , (21) has an eventually positive solution with an eventually nonnegative derivative.2. Integral inequality u ( t ) ≥ b ( t ) exp (Z h ( t ) t u ( s ) ds ) − a ( t ) exp ( − Z tg ( t ) u ( s ) ds ) , t ≥ t , (22) u ( t ) = 0 , t < t , has a nonnegative locally integrable solution for some t ≥ t .3. Differential equation (6) has an eventually positive solution with an eventually non-negative derivative. Proof. ⇒ . Let x be a solution of (21) such that x ( t ) > , ˙ x ( t ) ≥ , t ≥ t . For some t ≥ t we have g ( t ) ≥ t for t ≥ t . Denote u ( t ) = ˙ x ( t ) /x ( t ) , t ≥ t . Then x ( t ) = x ( t ) exp (cid:26)Z tt u ( s ) ds (cid:27) , t ≥ t . (23)After substituting (23) into (21) and carrying the exponent out of the brackets we obtainexp (cid:26)Z tt u ( s ) ds (cid:27) x ( t ) " u ( t ) − b ( t ) exp (Z h ( t ) t u ( s ) ds ) + a ( t ) exp ( − Z tg ( t ) u ( s ) ds ) ≥ . ⇒ . Let u ( t ) ≥ u n +1 ( t ) = b ( t ) exp (Z h ( t ) t u n ( s ) ds ) − a ( t ) exp ( − Z tg ( t ) u n ( s ) ds ) , n ≥ . (24)Inequalities u n ( t ) ≥ b ( t ) − a ( t ) and u ≥ u imply0 ≤ u n +1 ( t ) ≤ u n ( t ) ≤ . . . ≤ u ( t ) . Hence there exists a pointwise limit u ( t ) = lim n →∞ u n ( t ) . By the Lebesgue convergence theorem and (24) we have u ( t ) = b ( t ) exp (Z h ( t ) t u ( s ) ds ) − a ( t ) exp ( − Z tg ( t ) u ( s ) ds ) . (25)Then x ( t ) defined by (23) is a positive solution of equation (6) with a nonnegative derivative.Implication 3) ⇒
1) is evident. ⊓⊔ The proofs of the following corollaries can be given very similar to those given for Theorem1 and hence we omit most of them.
Corollary 2.1
Suppose (a1)-(a2) hold for equations (6) and (11), a ( t ) ≤ a ( t ) ≤ b ( t ) ≤ b ( t ) , g ( t ) ≥ g ( t ) , h ( t ) ≤ h ( t ) . If equation (11) has an eventually positive solution with an eventually nonnegative derivativethen so does equation (6).
Corollary 2.2
Suppose (a1)-(a2) hold, for t sufficiently large b ( t ) ≥ a ( t ) and a ( t ) ≥ b ( t ) " exp (Z h ( t ) t b ( s ) ds ) − exp (Z tg ( t ) b ( s ) ds ) . Then there exists an eventually positive solution with an eventually nonnegative derivativeof equation (6).
Proof . It is easy to see that u ( t ) = b ( t ) is a nonnegative solution of inequality (22). ⊓⊔ Corollary 2.3
Suppose (a1)-(a2) hold and there exist a > , b > , τ > , σ > a ≤ a ( t ) ≤ b ( t ) ≤ b, g ( t ) ≥ t − τ, h ( t ) ≤ t + σ. If there exists a positive solution λ > λ + ae − λτ − be λσ = 0 (26)8hen there exists an eventually positive solution with an eventually nonnegative derivativeof equation (6). Corollary 2.4
Suppose (a1)-(a2) hold, b ( t ) ≥ a ( t ) and there exists a nonoscillatory solutionof the advanced equation ˙ x ( t ) − b ( t ) x ( h ( t )) = 0 . (27)Then there exists an eventually positive solution with an eventually nonnegative derivativeof equation (6). Remark.
If for t sufficiently large Z h ( t ) t b ( s ) ds ≤ e , then [9] there exists a nonoscillatory solution of (27). Corollary 2.5
Suppose (a1)-(a2) hold, b ( t ) ≥ a ( t ), the integral inequality (22) has a nonneg-ative solution for t ≥ t and R ∞ [ b ( s ) − a ( s )] ds = ∞ . Then there exists an eventually positivesolution with an eventually nonnegative derivative x ( t ) of (6) such that lim t →∞ x ( t ) = ∞ . In this section we consider the following scalar MDE (1) with δ = − , δ = 1:˙ x ( t ) − a ( t ) x ( g ( t )) + b ( t ) x ( h ( t )) = 0 , t ≥ t , (28) Theorem 3
Suppose that a ( t ) , b ( t ) are continuous bounded on [0 , ∞ ) functions, g ( t ) and h ( t ) are equicontinuous on [0 , ∞ ) functions, there exist positive numbers a , a , b , b , τ , σ , t such that a ≤ a ( t ) ≤ a , b ≤ b ( t ) ≤ b , t − g ( t ) ≤ τ, h ( t ) − t ≤ σ, t ≥ t , (29) and the following algebraic system ( a e yτ − b e − yσ ≤ x,b e xσ − a e − xτ ≤ y (30) has a solution x = d > , y = d > .Then equation (28) has a nonoscillatory solution. Proof.
Define the following operator in the space C [ t , ∞ ) of all bounded continuous on[ t , ∞ ) functions with the usual sup-norm( Au )( t ) = a ( t ) exp ( − Z tg ( t ) u ( s ) ds ) − b ( t ) exp (Z h ( t ) t u ( s ) ds ) , t ≥ t , ( t ) = 0 , t ≤ t . Let x = d , y = d be a positive solution of system (30). Then the inequality − d ≤ u ( t ) ≤ d implies − d ≤ ( Au )( t ) ≤ d . It means that A ( S ) ⊂ S , where S = { u ( t ) : − d ≤ u ( t ) ≤ d } . Now we will prove that A ( S ) is a compact set in the space C [ t , ∞ ). Denote the integraloperators ( Hu )( t ) := Z tg ( t ) u ( s ) ds, ( Ru )( t ) := Z h ( t ) t u ( s ) ds. We have for u ∈ S | ( Hu )( t ) | ≤ max { d , d } τ, | ( Ru )( t ) | ≤ max { d , d } σ. Hence the sets H ( S ) and R ( S ) are bounded in the space C [ t , ∞ ).Let u ∈ S . Then | ( Hu )( τ ) − ( Hu )( τ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z g ( τ ) g ( τ ) | u ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z τ τ | u ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ max { d , d } ( | g ( τ ) − g ( τ ) | + | τ − τ | )and, similarly, | ( Ru )( τ ) − ( Ru )( τ ) | ≤ max { d , d } ( | h ( τ ) − h ( τ ) | + | τ − τ | ) . Since g , h are equicontinuous in [ t , ∞ ), then functions in H ( S ) and R ( S ) are also equicon-tinuous. Then the sets H ( S ) and R ( S ) are compact, consequently, A ( S ) is also a compactset.Schauder’s fix point theorem implies that there exists a solution u ∈ S of operatorequation u = Au . Therefore the function x ( t ) = x ( t ) exp (cid:26)Z tt u ( s ) ds (cid:27) , t ≥ t , is a positivesolution of equation (28). ⊓⊔ Corollary 3.1
Suppose that a ( t ), b ( t ) are continuous bounded on [0 , ∞ ) functions, g ( t ) and h ( t ) are equicontinuous on [0 , ∞ ) and there exist positive numbers a , a , b , b , τ , σ suchthat (29) is satisfied and at least one of the following conditions holds:1) b < a , < a − b < τ + σ ln a b , a < b , < b − a < τ + σ ln b a . Then equation (28) has a nonoscillatory solution.
Proof.
It suffices to prove that system (30) has a positive solution. Suppose condition 1)holds. For the second condition the proof is similar.10et us define the functions f ( x ) = b e xσ − a e − xτ and g ( y ) = a e yτ − b e − yσ which areboth monotone increasing. We have f (0) < f ( x ) = 0, where x = 1 τ + σ ln a b . Since g ( y ) is a monotone function then there exists the monotone increasing inverse function g − ( x ), for which we have g − ( x ) = 0, where x = a − b > h ( x ) = g − ( x ) − f ( x ). Condition 1) implies x < x , then f ( x ) < h ( x ) > g ( y ) = a e yτ − b e − yσ = x we have a e yτ ≤ x + b for y ≥
0. Then g − ( x ) ≤ τ ln x + b a and h ( x ) ≤ τ ln x + b a − b e xσ + a for x large enough. Hence lim x →∞ h ( x ) = −∞ . Since h is continuous, then there exists x > x > h ( x ) = 0. It means that f ( x ) = g − ( x ). Therefore x , y = f ( x ) is a solution of the system (30). Hence equation(28) has a nonoscillatory solution. ⊓⊔ If the conditions of Corollary 3.1 do not hold we can apply numerical methods to provethat system (30) has a positive solution.
Example 3.
Consider the equation˙ x ( t ) − (1 . . t ) x ( t − . − . t )+(1 . . t ) x ( t +0 . . t ) = 0 , t ≥ . (31)Then a = 1 . a = 1 . b = 1 . b = 1 . τ = 0 . σ = 0 . x = 2, y = 3, since a e yτ − b e − yσ = 1 . e . − . e − . ≈ . < x = 2 ,b e xσ − a e − xτ ≤ y = 1 . e . − . e − . ≈ . < y = 3 . Hence equation (31) has a nonoscillatory solution.Fig. 1 illustrates the domain of values ( x, y ) satisfying the system of inequalities (30) forequation (31) which is between the two curves.
Example 4.
Consider the equation with constant coefficients and variable advance anddelay ˙ x ( t ) − ax ( g ( t )) + bx ( h ( t )) = 0 , t ≥ , (32)where t ≥ g ( t ) ≥ t − . t ≤ h ( t ) ≤ t + 0 .
3, like in Example 3. Thus in (30) we have τ = 0 . δ = 0 .
3. All values below the curve in Fig. 2 are such that the system of inequalities (30)has a positive solution and hence equation (32) has a nonoscillatory solution.For comparison, we also included the line0 . a + 0 . b < e . (33)11 y x1.8exp(0.3x)-1.2exp(-0.2x)1.6exp(0.2y)-1.4exp(-0.3y) Figure 1: The domain of values ( x, y ) satisfying the system of inequalities (30) for equation(31) is between the curves. The chosen value x = 2, y = 3 inside the domain is also markedon the graph. Remark.
The autonomous equation˙ x ( t ) − ax ( t − τ ) + bx ( t + σ ) = 0 , a > , b > , τ > , σ > , (34)always has a positive solution e λt , where λ is a solution of the characteristic equation f ( λ ) = λ − ae − τλ + be σλ = 0 . (35)Since lim λ →±∞ f ( λ ) = ±∞ , then there is always a real λ satisfying (35). There is a positivesolution satisfying lim t →∞ x ( t ) = ∞ if b < a and a positive solution satisfying lim t →∞ x ( t ) = 0if b > a . In this paper we presented results for equations with variable arguments and coefficients,one delay and one advanced term in the case when coefficients have any of four possible signcombinations; the results for coefficients of different signs are new. If the delayed term ispositive, we not only claim the existence of a positive solution but present sufficient conditionsunder which its asymptotics can be deduced (i.e., a nonincreasing positive solutions whichtends to zero or a nondecreasing solution which tends to infinity).Below we present some open problems and topics for research and discussion.1. If we consider the characteristic equation of the autonomous equation (14) it is easy toprove that equation (14) has a positive solution for any positive coefficients a > , b > b abounds where (30) has a positive solutionline (33) Figure 2: The domain of values a, b such that the system of inequalities (30) for equation(32) has a positive solution is under the curve.Prove or disprove:If conditions (a1)-(a2) hold, a ≤ a ( t ) ≤ a , b ≤ b ( t ) ≤ b , t − g ( t ) ≤ τ , h ( t ) − t ≤ σ, then equation (6) has an eventually positive solution.Consider the same problem for equation (28).2. Prove or disprove:If a ( t ) ≥ b ( t ) ≥ x ( t ) + [ a ( t ) − b ( t )] x ( t ) = 0 (36)tend to zero as t → ∞ . If b ( t ) ≥ a ( t ) ≥ t → ∞ then allsolutions of (36) tend to infinity.3. For equation (6) obtain sufficient conditions under which there exists a positive solutionwhich tends to some d = 0. Is convergence of Z ∞ a ( s ) ds and Z ∞ b ( s ) ds sufficient,together with the other conditions of Theorem 1? Consider the same problem for(28). We remark that for first order delay neutral equations with positive and negativecoefficients in [33, 34] sufficient conditions were found under which all solutions eitheroscillate or tend to zero, and for the second order neutral delay equation such conditionscan be found in the recent paper [35]. It would be interesting to find conditions whichguarantee the existence of a positive solution (which tends to zero) for these equations.For some sufficient conditions for linear models without a neutral part see [36].13. Find sufficient conditions when equation (28) has a positive nonincreasing solution x ( t ) such that lim t →∞ x ( t ) = 0 or find sufficient conditions when equation (28) has apositive solution x ( t ) with a nonnegative derivative such that lim t →∞ x ( t ) = ∞ .5. Obtain sufficient conditions when the equation with one term which can be both ad-vanced and delayed and an oscillating coefficient˙ x ( t ) + a ( t ) x ( g ( t )) = 0 (37)has a positive solution. For instance, if a ( t )[ t − g ( t )] is either positive or negative forany t , then (37) can be rewritten in the form (6) or (28), thus some conditions can bededuced from the results of the present paper.6. Prove or disprove:Suppose a ( t ) ≥ , b ( t ) ≥ , Z ∞ [ a ( t ) + b ( t )] dt = ∞ .If equation (3) has a positive solution then this equation is asymptotically stable.If equation (5) has a positive solution then the absolute value of any nontrivial solutiontends to infinity.7. Instead of initial condition (2) for MDE we can formulate the “final” conditions x ( t ) = ϕ ( t ) , t > t , x ( t ) = x , (38)studying nonoscillation for t < T ≤ t and asymptotics for t → −∞ . How would theresults of the present paper change if obtained for (6),(38) or (28),(38)? References [1] R. Frish and H. Holme, The characteristic solutions of mixed difference and differentialequation occurring in economic dynamics,
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