ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS


 Kristin Palmer
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1 ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute of the Georgian Academy of Sciences Tbilisi, Georgia 1991 Mathematics Subject Classification. 34K1, 34K15 Key words and phrases. Nonlinear twopoint boundary value problem, twodimensional differential system with advanced arguments, nonnegative solution 1. Statement of the Problem and Formulation of the Existence and Uniqueness Theorems Consider the differential system with the boundary conditions u i t) = f i t, u1 τ i1 t)), u 2 τ i2 t)) ) i = 1, 2) 1.1) ϕ u 1 ), u 2 ) ) =, u 1 t) = u 1 a), u 2 t) = for t a, 1.2) where f i : [, a] R 2 R i = 1, 2) satisfy the local Carathéodory conditions, while ϕ : R 2 R and τ ik : [, a] [, + [ i, k = 1, 2) are continuous functions. We are interested in the case, where f i t,, ) =, f i t, x, y) for t a, x, y i = 1, 2) 1.3) and the function ϕ satisfies one of the following two conditions: and ϕ, ) <, ϕx, y) > for x > r, y 1.4) ϕ, ) <, ϕx, y) > for x, y, x + y > r, 1.5) kig ninopa EJQTDE, 1999 No. 5, p. 1
2 where r is a positive constant. Put τik t) = τ ik t) for τ ik t) a a for τ ik t) > a i, k = 1, 2). 1.6) Under a solution of the problem 1.1), 1.2) is understood an absolutely continuous vector function u 1, u 2 ) : [, a] R 2 satisfying almost everywhere on [, a] the differential system and also the boundary conditions u it) = f i t, u1 τ i1t)), u 2 τ i2t)) ) i = 1, 2) 1.1 ) ϕ u 1 ), u 2 ) ) =, u 2 a) =. 1.2 ) A solution u 1, u 2 ) of the problem 1.1), 1.2) is called nonnegative if u 1 t), u 2 t) for t a. If 1.3) holds, then it is obvious that each component of a nonnegative solution of the problem 1.1), 1.2) is a nonincreasing function. For the case, where τ ik t) t i, k = 1, 2), the boundary value problems of the type 1.1), 1.2) have been investigated by quite a number of authors see, e.g., [4 8, 14 21] and the references therein). In this paper the case, where τ ik t) t for t a i, k = 1, 2), 1.7) is considered and the optimal, in a certain sense, sufficient conditions are established for the existence and uniqueness of a solution of the problem 1.1), 1.2). Some of these results see, e.g., Corollaries 1.1 and 1.4) are specific for advanced differential systems and have no analogues for the system u it) = f i t, u1 t), u 2 t) ) i = 1, 2). Theorems proven below and also their corollaries make the previous wellknown results [1 3, 9 12] on the solvability and unique solvability of the boundary value problems for the differential systems with deviated arguments more complete. Along with 1.1) we will consider its perturbation u it) = f i t, u1 τ i1 t) + ε), u 2 τ i2 t) + ε) ) i = 1, 2), 1.1 ε ) where ε >. As it will be proved below 1, for every ε > the problem 1.1 ε ), 1.2) has at least one nonnegative solution provided the conditions 1.3), 1.4), and 1.7) are fulfilled. 1 See Lemma 2.4. EJQTDE, 1999 No. 5, p. 2
3 Theorem 1.1 below reduces the question of the solvability of the problem 1.1), 1.2) to obtaining uniform a priori estimates of second components of solutions of the problem 1.1 ε ), 1.2) with respect to the parameter ε. Such estimates can be derived in rather general situations and therefore from Theorem 1.1 the effective and optimal in a sense conditions are obtained for the solvability of the problem 1.1), 1.2) see Corollaries and Theorem 1.2). Theorem 1.1. Let the conditions 1.3), 1.4), and 1.7) be fulfilled and let there exist positive numbers ε and ρ such that for any ε ], ε ] the second component of an arbitrary nonnegative solution u 1, u 2 ) of the problem 1.1 ε ), 1.2) admits the estimate u 2 ) ρ. 1.8) Then the problem 1.1), 1.2) has at least one nonnegative solution. Corollary 1.1. Let the conditions 1.3), 1.4) be fulfilled and let τ 1i t) t i = 1, 2), τ 21 t) t, τ 22 t) > t for t a. 1.9) Then the problem 1.1), 1.2) has at least one nonnegative solution. Remark 1.1. The condition 1.9) in Corollary 1.1 is essential and it cannot be replaced by the condition 1.7). To convince ourselves that this is so, consider the boundary value problem where u 1 t) =, u 2 t) = u 1 t) + u 2 t) ) λ, 1.1) u 1 ) = 1, u 2 a) =, 1.11) λ ) a It is seen that for that problem all the conditions of Corollary 1.1, except 1.9), are fulfilled. Instead of 1.9) there takes place the condition 1.7). Nevertheless, the problem 1.1), 1.11) has no solution. Indeed, should this problem have a solution u 1, u 2 ), the function u 2 would be positive on [, a[ and a a = 1 + u2 s) ) λ du2 s) = 1 λ u2 ) ) 1 λ 1 < λ 1 λ 1. But the latter inequality contradicts 1.12). EJQTDE, 1999 No. 5, p. 3
4 Corollary 1.2. Let the conditions 1.3), 1.4), and 1.7) be fulfilled and let there exist y > such that f 2 t, x, y) ht)ωy) for t a, x r, y y, 1.13) where h : [, a] [, + [ is a summable function and ω : [y, + [ ], + [ is a nondecreasing continuous function satisfying the condition + y dy = ) ωy) Then the problem 1.1), 1.2) has at least one nonnegative solution. Remark 1.2. As is shown above, if a = 1/ε and λ = 1+ε, then the problem 1.1), 1.11) has no solution. This fact shows that the condition 1.14) in Corollary 1.2 cannot be replaced by the condition no matter how small ε > is. + y yε dy ωy) = + Corollary 1.3. Let the conditions 1.3), 1.4), and 1.7) be fulfilled and let there exist numbers a i ], a] i = 1, 2) and y > such that and τ 12 t) a 2 for t a 1, 1.15) f 1 t, x, y) δt, y) for t a 1, x r, y y, 1.16) f 2 t, x, y) [ ht) + f 1 t, x, y) ] ωy) for t a 2, x r, y y, 1.17) where δ : [, a 1 ] [y, + [ [, + [ is a summable in the first and nondecreasing in the second argument function, while h : [, a 2 ] [, + [ and ω : [y, + [ ], + [ are summable and nondecreasing continuous functions, respectively. Let, moreover, τ 1i t) τ 2i t) i = 1, 2), a 1 lim δt, y) dt > r, 1.18) y + and ω satisfy the condition 1.14). Then the problem 1.1), 1.2) has at least one nonnegative solution. EJQTDE, 1999 No. 5, p. 4
5 Remark 1.3. The condition 1.15) in Corollary 1.3 cannot be replaced by the condition no matter how small ε > is. differential system u 1 t) = u 2a), τ 12 t) a 2 + ε for t a ) As an example verifying this fact, consider the u 2 t) = gt) u 1 t) + u 2 t) ) 1+ 1 ε with the boundary conditions 1.11), where ε ], a[ and for t a ε gt) = 1 for a ε < t a. 1.2) It is seen that for this problem the conditions 1.16) 1.18) hold and instead of 1.15) the condition 1.19) is fulfilled, where τ 12 t) a, a 1 = a, a 2 = a ε, δt, y) y, ht), ωy) 1. But the problem 1.2), 1.11) has no solution. Indeed, if we assume that 1.2), 1.11) has a solution u 1, u 2 ), then we will obtain the contradiction, i.e., a ε = 1 + u2 s) ) 1 ε 1 du 2 s) = ε ε 1 + u 2 a ε) ) 1 ε < ε. a ε In contrast to Corollary 1.3, Corollaries 1.4 and 1.5 below catch the effect of an advanced argument τ 22. Corollary 1.4. Let the conditions 1.3), 1.4), and 1.7) be fulfilled and let for some a i ], a] i = 1, 2) and y > the inequalities 1.15), 1.16), and τ 22 t) a 2 for t a 1.21) hold, where δ : [, a 1 ] [y, + [ [, + [ is a summable in the first and nondecreasing in the second argument function satisfying the condition 1.18). Then the problem 1.1), 1.2) has at least one nonnegative solution. Remark 1.4. It is obvious from the example 1.2), 1.11) that it is impossible in Corollary 1.4 to replace the condition 1.21) by the condition no matter how small ε > is. τ 22 t) a 2 ε for t a Corollary 1.5. Let the conditions 1.3), 1.4), and 1.7) hold and let, moreover, for some a ], 1] ], a 1/α ] the inequalities τ 12 t) t α for t a, 1.22) f 1 t, x, y) lt β y λ 1 for t a, x r, y, 1.23) EJQTDE, 1999 No. 5, p. 5
6 and f 2 t, x, y) ht)1 + y) 1+λ 2 for t a, x r, y 1.24) be fulfilled, where < α 1, β > 1, l >, λ 1 >, λ 2 and h : [, a ] [, + [ is a measurable function satisfying the condition a [τ 22 t)] 1+β)λ 2 αλ 1 ht) dt < ) Then the problem 1.1), 1.2) has at least one nonnegative solution. Remark 1.5. The condition 1.25) in Corollary 1.5 cannot be replaced by the condition a [τ 22 t)] 1+β)λ 2 αλ +ε 1 ht) dt < ) no matter how small ε > is. As an example, consider the differential system u 1t) = u 2 t), u 2t) = γt λ 1 δ u 1 t) + u 2 t) ) 1+λ with the boundary conditions 1.11), where < δ < ε < λ, γ > λ δ λ λ δ ) λη δ λ 1.27) 1.28) and η = min{a, 1}. For the system 1.27) the conditions 1.22) 1.24) hold and instead of 1.25) there takes place the condition 1.26), where τ 12 t) τ 22 t) t, a = η, α = 1, β =, l = 1, λ 1 = 1, λ 2 = λ, r = 1, and ht) = 2 1+λ γt λ 1 δ. Show that the problem 1.27), 1.11) has no solution. Assume the contrary that this problem has a solution u 1, u 2 ). Then u 1 t) > u 1 η) >, u 2 t) > for t < η, and η u 2 t) dt = 1 u 1 η), u 1 η) + u 2 t) ) 1 λ u 2 t) > γt λ 1 δ for < t < η. The integration of the latter inequality from to t yields and hence u1 η) + u 2 t) ) λ u1 η) + u 2 ) ) λ > λγ λ δ tλ δ for < t < η ) 1 λ δ λ u 1 η) + u 2 t) < t 1+ δ λ for < t < η. λγ EJQTDE, 1999 No. 5, p. 6
7 Integrating this inequality from to η, we obtain ηu 1 η) + 1 u 1 η) < λ ) 1 λ δ λ η δ λ. δ λγ However, since < u 1 η) < 1, we have ηu 1 η) + 1 u 1 η) > η. Thus which contradicts 1.28). λ δ ) 1 λ δ λ η δ λ > η, λγ Theorem 1.2. If the conditions 1.3), 1.5), and 1.7) are fulfilled, then the problem 1.1), 1.2) has at least one nonnegative solution. The uniqueness of a solution of the problem 1.1), 1.2) is closely connected with the uniqueness of a solution of the system 1.1) with the Cauchy conditions The following theorem is valid. u 1 t) = c, u 2 t) = for t a. 1.29) Theorem 1.3. Let the condition 1.7) be fulfilled and for any c R the Cauchy problem 1.1), 1.29) have no more than one solution. Let, moreover, the functions f i i = 1, 2) not increase in the last two arguments, while the function ϕ increase in the first argument and not decrease in the second argument. Then the problem 1.1), 1.2) has no more than one solution. Theorem 1.4. Let 1.7) be fulfilled and for any c R the Cauchy problem 1.1), 1.29) have no more than one solution. Let, moreover, the function f 1 not increase in the last two arguments, f 2 decrease in the second argument and not increase in the third argument, while the function ϕ be such that ϕx, y) < ϕx, y) for x < x, y < y. 1.3) Then the problem 1.1), 1.2) has no more than one solution. Remark 1.6. For the uniqueness of a solution of the problem 1.1), 1.29) it is sufficient that either the functions f i i = 1, 2) satisfy in the last two arguments the local Lipschitz condition or the functions τ ik i, k = 1, 2) satisfy the inequalities τ ik t) > t for t a i, k = 1, 2). As an example, consider the boundary value problem 2 u i t) = p ik t) uk τ ik t)) λ ik sgnu k τ ik t))) i = 1, 2), k=1 u 1 ) + αu 2 ) = β, u 1 t) = u 1 a), u 2 t) = for t a, EJQTDE, 1999 No. 5, p. 7
8 where λ ik > i, k = 1, 2), α, β >, p ik : [, a] [, + [ i, k = 1, 2) are summable functions and τ ik : [, a] [, + [ i, k = 1, 2) are continuous functions satisfying the inequalities 1.7). By virtue of Corollaries 1.1, 1.3 and Theorems 1.2, 1.3 this problem has the unique nonnegative solution if, besides the above given, one of the following three conditions is fulfilled: i) τ ik t) > t for t a i, k = 1, 2); ii) α >, λ ik 1 i, k = 1, 2); iii) α =, λ ik 1 i, k = 1, 2), λ λ 12 and there exist a i ], a] i = 1, 2) such that τ 12 t) a 2 for t a 1, p 12 t) > for < t < a 2, and { } p22 t) vrai max p 12 t) : < t < a 2 < Some Auxiliary Statements 2.1. Lemmas on properties of solutions of an auxiliary Cauchy problem. Lemma 2.1. If the conditions 1.3) and τ ik t) > t for t a i, k = 1, 2) 2.1) are fulfilled, then for any nonnegative c the problem 1.1), 1.29) has the unique solution u 1, c), u 2, c)). Moreover, the functions t, c) u i t, c) i = 1, 2) are continuous on the set [, a] [, + [ and satisfy on the same set the inequalities u 1 t, c) c, u 2 t, c). 2.2) Proof. In view of 2.1) there exists a natural number m such that τ ik t) t > a m for t a i, k = 1, 2). Put t j = ja m j =,..., m). Then τ ik t) > t j for t j 1 t t j i, k = 1, 2; j = 1,..., m). 2.3) Suppose that for some c the problem 1.1), 1.29) has a solution u 1,c), u 2,c)). Then on account of u i t, c) = u ij t, c) for t j t t j+1 i = 1, 2; j = m 1,..., ), EJQTDE, 1999 No. 5, p. 8
9 where t j+1 t u 1m t, c) = c, u 2m t, c) = for t a, 2.4) u ij t, c) = u i j+1 t j+1, c) f i s, u1 j+1 τ i1 s), c), u 2 j+1 τ i2 s), c) ) ds for t j t t j+1, u ij t, c) = u i j+1 t, c) for t > t j+1 i = 1, 2; j = m 1,..., ), 2.5) we conclude that if the problem 1.1), 1.29) has a solution, then this solution is unique since the functions u ij i = 1, 2; j = m,..., ) are defined uniquely. Let us now suppose that u ij : [t j, + [ [, + [ R i = 1, 2; j = m 1,..., ) are the functions given by 2.4) and 2.5). Then proceeding from the conditions 1.3) and 2.3) we prove by the induction that these functions are continuous, u 1j t, c) c, u 2j t, c) for t t j, c j = m 1,..., ), and for any j {,..., m 1} and c the restriction of the vector function u 1j, c), u 2j, c)) on [t j, a] is a solution of the problem 1.1), 1.29) on [t j, a]. Consequently, the vector function u 1, c), u 2, c)) with the components u i t, c) = u i t, c) for t a i = 1, 2) is the unique solution of the problem 1.1), 1.29). Moreover, u i : [, a] [, + [ R i = 1, 2) are continuous and satisfy 2.2). Lemma 2.2. Let 1.7) be fulfilled and the functions f i i = 1, 2) be nonincreasing in the last two arguments. Let, moreover, there exist c 1 and c 2 > c 1 such that for c {c 1, c 2 } the problem 1.1), 1.29) has the unique solution u 1, c), u 2, c)). Then u 1 t, c 2 ) c 2 c 1 + u 1 t, c 1 ), u 2 t, c 2 ) u 2 t, c 1 ) for t a. 2.6) Proof. For each j {1, 2} the vector function u 1, c j ), u 2, c j )) is a solution of the differential system 1.1 ) under the conditions Using the transformation u 1 a, c j ) = c j, u 2 a, c j ) =. s = a t, v i s) = u i t) i = 1, 2) we can rewrite the system 1.1 ) in the form v i s) = f i s, v1 ζ i1 s)), v 2 ζ i2 s)) ) i = 1, 2), 2.7) where f i s, x, y) = f i a s, x, y), ζ ik s) = a τ ika s) i, k = 1, 2). EJQTDE, 1999 No. 5, p. 9
10 On the basis of 1.6) and 1.7) we have ζ ik s) s for s a i, k = 1, 2). 2.8) According to one of the conditions of the lemma, for each j {1, 2} the system 2.7) under the initial conditions v 1 ) = c j, v 2 ) = has the unique solution v 1, c j ), v 2, c j )), and On the other hand, v i s, c j ) = u i a s, c j ) i = 1, 2). c 1 < c 2, the functions f i i = 1, 2) do not decrease in the last two arguments, and the functions ζ ik i, k = 1, 2) satisfy 2.8). By virtue of Corollary 1.9 from [13] the above conditions guarantee the validity of the inequalities It is obvious that Consequently, v i s, c 1 ) v i s, c 2 ) for s a i = 1, 2). v i s, c 1) v i s, c 2) for s a i = 1, 2). u i t, c 1 ) u i t, c 2 ), u i t, c 1) u i t, c 2) for t a i = 1, 2). Hence the inequalities 2.6) follow immediately. For every natural m consider the Cauchy problem u i t) = f i t, u1 τ i1m t)), u 2 τ i2m t)) ) i = 1, 2), 2.9) u 1 t) = c m, u 2 t) = for t a, 2.1) where τ ikm : [, a] [, + [ i, k = 1, 2) are continuous functions. The following lemma holds. Lemma 2.3. Let lim τ ikmt) = τ ik t) uniformly on [, a] i, k = 1, 2), 2.11) m + lim c m = c 2.12) m + and let there exist positive number ρ such that for every natural m the problem 2.9), 2.1) has a solution u 1m, u 2m ) satisfying the inequality u 1m t) + u 2m t) ρ for t a. 2.13) EJQTDE, 1999 No. 5, p. 1
11 Then from the sequences u im ) + m=1 i = 1, 2) we can choose uniformly converging subsequences u imj ) + j=1 i = 1, 2) such that the vector function u 1, u 2 ), where u i t) = lim j + u im j t) for t a i = 1, 2), 2.14) is a solution of the problem 1.1), 1.29). Proof. According to 2.13), for every m we have where u 1m t) + u 2m t) f t) for t a, { f 2 } t) = max f i t, x, y) : x + y ρ i=1 and, as is evident, f is summable on [, a]. Therefore the sequences u im ) + m=1 i = 1, 2) are uniformly bounded and equicontinuous on [, a]. By virtue of the Arzela Ascoli lemma, these sequences contain subsequences u imj ) + j=1 i = 1, 2) converging uniformly on [, a]. Let u 1 and u 2 be functions given by 2.14). If in the equalities u 1mj t) = c mj a u 2mj t) = t a t f 1 s, u1mj τ 11mj s)), u 2mj τ 12mj s)) ) ds, f 2 s, u1mj τ 21mj s)), u 2mj τ 22mj s)) ) ds for t a we pass to the limit as j +, then by virtue of 2.1) 2.12) and the Lebesgue theorem concerning the passage to the limit under the integral sign we find that u 1 t) = c u 2 t) = a t a t f 1 s, u1 τ 11 s)), u 2 τ 12 s)) ) ds, f 2 s, u1 τ 21 s)), u 2 τ 22 s)) ) ds for t a and the vector function u 1, u 2 ) satisfies 1.29). Consequently, u 1, u 2 ) is a solution of the problem 1.1), 1.29) Lemma on the solvability of the problem 1.1), 1.2). Lemma 2.4. If the conditions 1.3), 1.4), and 2.1) are fulfilled, then the problem 1.1), 1.2) has at least one nonnegative solution. EJQTDE, 1999 No. 5, p. 11
12 Proof. By Lemma 2.1, for every nonnegative c the problem 1.1), 1.29) has the unique solution u 1, c), u 2, c)), while the functions t, c) u i t, c) i = 1, 2) are continuous on the set [, a] [, + [ and satisfy on the same set the inequalities 2.2). Hence, according to 1.3), it becomes evident that and Put u i t, ) = for t a i = 1, 2) 2.15) u 1, c) c, u 2, c) for c >. 2.16) ϕ c) = ϕ u 1, c), u 2, c) ). It is clear that ϕ : [, + [ R is a continuous function. On the other hand, in view of 1.4), 2.15), and 2.16) we have Thus there exists c ], r] such that Consequently, ϕ ) <, ϕ r). ϕ c ) =. ϕ u 1, c ), u 2, c ) ) = and thus the vector function u 1, c ), u 2, c )) is a nonnegative solution of the problem 1.1), 1.2) Lemmas on a priori estimates. First of all consider the system of differential inequalities with the initial condition u 1 t) δt, u 2a )), u 2 t) [ ht) + u 1 t) ] ωu 2 t)) 2.17) u 1 ) r, 2.18) where δ : [, a ] [, + [ [, + [ is a continuous in the first and nondecreasing in the second argument function, h : [, a ] [, + [ is a summable function and ω : [, + [ ], + [ is a nondecreasing continuous function. A vector function u 1, u 2 ) with the nonnegative components u i : [, a ] [, + [ i = 1, 2) is said to be a nonnegative solution of the problem 2.17), 2.18) if the functions u 1 and u 2 are absolutely continuous, the function u 1 satisfies the inequality 2.18), and the system of differential inequalities 2.17) holds almost everywhere on [, a ]. EJQTDE, 1999 No. 5, p. 12
13 and Lemma 2.5. Let a lim δs, y) ds > r 2.19) y + + dy = ) ωy) Then there exists a positive number ρ such that the second component of an arbitrary nonnegative solution u 1, u 2 ) of the problem 2.17), 2.18) admits the estimate u 2 t) ρ for t a. 2.21) Proof. By virtue of 2.19) and 2.2) there exist positive numbers ρ and ρ 1 such that and a δs, y) ds > r for y > ρ ) ρ ρ 1 dy ωy) = r + a hs) ds. 2.23) Let u 1, u 2 ) be an arbitrary nonnegative solution of the problem 2.17), 2.18). Then a a u 1 s) ds = u 1 s) ds = u 1) u 1 a ) r, which, owing to 2.17), results in u 2 t) u 2 a ) a r u 1s) ds dy a ωy) = t r + u 2 s) ds ωu 2 s)) Taking into account 2.22), from 2.24) we get a a a δs, u 2 a )) ds, 2.24) hs) ds + a u 1 s) ds hs) ds for t a. 2.25) u 2 a ) ρ 1. EJQTDE, 1999 No. 5, p. 13
14 On the basis of the last inequality and 2.23), from 2.25) we find the estimate 2.21). Finally, on the segment [, a ] consider the system of differential inequalities where u 1 t) ltβ u λ 1 2 tα ), ht) 1 + u 2 τt)) ) 1+λ 2 u 2t), 2.26) a ], 1], 2.27) < α 1, β > 1, l >, λ 1 >, λ 2, while h : [, a ] [, + [ and τ : [, a ] [, a ] are measurable functions. Lemma 2.6. Let τt) t for t a and a [τt)] 1+β)λ 2 αλ 1 ht) dt < ) Then there exists a positive number ρ such that the second component of an arbitrary nonnegative solution u 1, u 2 ) of the problem 2.26), 2.18) admits the estimate and Proof. Put ρ = h t) = [ r1 + β) l u 2 ) < ρ. [ r1 + β) l ] 1 λ 1 1+β αλ a 1 ] 1 λ 1 [τt)] 1+β αλ 1 ) λ2ht) ) a exp ) h s) ds. Then by 2.28), ρ < +. Let u 1, u 2 ) be an arbitrary nonnegative solution of the problem 2.26), 2.18). Then Thus r u 1 t) u λ 1 2 tα )l t t t u 1s) ds l s β ds = s β u λ 1 2 s α ) ds l 1 + β t1+β u λ 1 2 tα ) for < t a. ) 1 r1 + β) u 2 t α λ 1 ) t 1+β λ 1 for < t a, l EJQTDE, 1999 No. 5, p. 14
15 whence by virtue of 2.27) we get ) 1 r1 + β) λ 1 u 2 t) t 1+β αλ 1 l for < t a. According to the latter estimate and the inequalities 2.26), we have and ) 1 r1 + β) λ 1 1+β αλ u 2 a ) a 1, l u 2 τt)) u 2 t), 1 + u2 t) ) ht) [ 1 + u2 τt)) ] λ 2 [ 1 + u2 τt)) ] Thus h t) 1 + u 2 t) ) for < t a. 1 + u 2 t) 1 + u 2 a ) ) a ) exp h s) ds ρ for t a. t 3. Proofs of the Existence and Uniqueness Theorems Proof of Theorem 1.1. Let τ ikm t) = τ ik t) + ε i, k = 1, 2; m = 1, 2,... ). m Then by virtue of Lemma 2.4, for every natural m the system 2.9) has a nonnegative solution u 1m, u 2m ) satisfying the boundary conditions ϕ u 1m ), u 2m ) ) =, u 1m t) = u 1m a), u 2m t) = for t a. 3.1) By the condition of the theorem, u 2m ) ρ m = 1, 2,... ). 3.2) On the other hand, taking into account 1.4) and 3.1), we find u 1m ) r m = 1, 2,... ). 3.3) In view of 1.3) and the fact that u 1m and u 2m are nonnegative, we can conclude that these functions are nonincreasing. Thus it becomes clear from 3.2) and 3.3) that for every natural m the estimate 2.13), where ρ = r + ρ, is valid. It is also obvious that u 1m, u 2m ) is a solution of the problem 2.9), 2.1), where c m = u 1m a). Without loss of generality it can be assumed that the sequence c m ) + m=1 is convergent. Denote by c the limit of that sequence. According to Lemma 2.3, we can choose from u im ) + m=1 i = 1, 2) uniformly converging subsequences u imj ) + j=1 i = 1, 2), and the vector function u 1, u 2 ) whose EJQTDE, 1999 No. 5, p. 15
16 components are given by 2.14) is a solution of the problem 1.1), 1.29). On the other hand, if in the equality ϕ u 1mj ), u 2mj ) ) = we pass to the limit as j +, then, taking into account the fact that ϕ is continuous, we obtain ϕ u 1 ), u 2 ) ) =. Therefore u 1, u 2 ) is a nonnegative solution of the problem 1.1), 1.2). Proof of Corollary 1.1. Choose a natural number m so that τ 22 t) > t + a m for t a. Put t j = ja m, τt) = t j+1 for t j < t t j+1 j =,..., m 1). 3.4) Then τ 22 t) > τt) for t a. 3.5) Introduce the function and the numbers ht, y) = max { f 2 t, x, z) : x r, z y } 3.6) ρ m =, ρ j = ρ j+1 + t j+1 t j hs, ρ j+1 ) ds j = m 1,..., ). 3.7) Let ε ], 1] be an arbitrarily fixed number and u 1, u 2 ) be a nonnegative solution of the problem 1.1 ε ), 1.2). By Theorem 1.1, to prove Corollary 1.1 it is sufficient to show that u 2 admits the estimate 1.8). By virtue of 1.3), the functions u 1 and u 2 are nonincreasing. From this fact, on account of 1.4) and 3.5) it follows that and u 1 t) u 1 ) r for t a 3.8) u 2 τ 22 t) + ε) u 2 τt)) for t a. 3.9) In view of 3.6), 3.8), and 3.9), from 1.1 ε ) we get u 2t) ht, u 2 τt))) for t a, EJQTDE, 1999 No. 5, p. 16
17 whence with regard for 3.4) and since u 2 t m ) =, we find that Therefore the estimate 1.8) is valid. u 2 t) ρ j for t j t a j = m 1,..., ). Proof of Corollary 1.2. Without loss of generality it can be assumed below that If now we suppose ht) > 1 ωy ) max { f 2 t, x, y) : x r, y y }. 3.1) then owing to 1.13), we will have ωy) = ωy ) for y y, f 2 t, x, y) ht)ωy) for t a, x r, y. 3.11) On the other hand, by virtue of 1.14) there exists ρ > such that ρ dy ωy) = a hs) ds. 3.12) Let u 1, u 2 ) be a nonnegative solution of the problem 1.1 ε ), 1.2) for some ε ], 1]. By Theorem 1.1, to prove Corollary 1.2 it suffices to show that u 2 admits the estimate 1.8). By 1.3), the functions u 1 and u 2 are nonincreasing. Hence, taking into account 1.4) and 1.7), we get 3.8) and u 2 τ 22 t) + ε) u 2 t) for t a. 3.13) According to 3.8), 3.11), and 3.13), for almost all t [, a] we have u 2t) ht)ωu 2 t)) as long as ω is a nondecreasing function. Moreover, u 2 a) =. Thus u 2 ) dy ωy) = a u 2s) a ωu 2 s)) ds Hence by virtue of 3.12) we obtain the estimate 1.8). hs) ds. Proof of Corollary 1.3. Without loss of generality we assume that the function h satisfies the inequality 3.1) on [, a 2 ]. If we now suppose a = a 2, and ωy) = ωy ) for y y, δt, y) = for t a 1, y y δt, y) = for a 1 < t a, y, EJQTDE, 1999 No. 5, p. 17
18 then in view of 1.15) 1.17) we get τ 12 t) a for t a 1, f 1 t, x, y) δt, y) for t a, x r, y, f 2 t, x, y) [ ht) + f 1 t, x, y) ] ωy) for t a, x r, y. 3.14) On the other hand, by 1.18) and 1.14) the functions δ and ω satisfy the conditions 2.19) and 2.2), respectively. Let ρ be the positive constant appearing in Lemma 2.5. Put 1 for y ρ ψy) = 2 y for ρ < y < 2ρ, 3.15) ρ for y 2ρ and consider the differential system f 2 t, x, y) = ψy)f 2 t, x, y) 3.16) u 1 t) = f 1 t, u1 τ 11 t)), u 2 τ 12 t)) ), u 2 t) = f 2 t, u1 τ 21 t)), u 2 τ 22 t)) ). 3.17) In view of 3.15) and 3.16) f 2 t, x, y) h t) for t a, x r, y, 3.18) where h t) = max { f 2 t, x, y) : x r, y 2ρ } 3.19) and h is summable on [, a]. By virtue of Corollary 1.2 the condition 3.18) guarantees the existence of a nonnegative solution u 1, u 2 ) of the problem 3.17), 1.2). By the conditions 1.3) and 1.4), the functions u 1 and u 2 are nonincreasing and u 1 satisfies the inequalities 3.8). If along with this fact we take into account the conditions 1.7), 3.14) 3.16), then we will see that the restriction of u 1, u 2 ) on [, a ] is a solution of the problem 2.17), 2.18). Hence if we take into consideration how ρ is, then we will get the estimate 2.21). Consequently, u 2 τ 22 t)) ρ for t a. 3.2) According to this estimate, from 3.15) 3.17) follows that u 1, u 2 ) is a solution of the system 1.1). Proof of Corollary 1.4. By virtue of 1.18) we can find ρ 1 y so that a 1 δs, ρ 1 ) ds > r. 3.21) EJQTDE, 1999 No. 5, p. 18
19 and Put ht) = max { f 2 t, x, y) : x r, y ρ 1 } ρ = ρ 1 + a ) hs) ds. 3.23) Let ψ, f 2 and h be the functions defined by 3.15), 3.16) and 3.19), respectively. Then f 2 satisfies the condition 3.18). By this condition and Corollary 1.2 the problem 3.17), 1.2) has a nonnegative solution u 1, u 2 ). By virtue of 1.3) and 1.4) the functions u 1 and u 2 do not increase and u 1 admits the estimate 3.8). Let us now show that u 2 a 2 ) < ρ ) Assume the contrary that u 2 a 2 ) ρ 1. Then in view of 1.15) and 1.16) we get and u 2 τ 12 t)) u 2 a 2 ) ρ 1 for t a 1 u 1 t) δt, ρ 1) for t a 1. Integrating the latter inequality from to a 1 and taking into account 3.8), we obtain r a 1 δs, ρ 1 ) ds. But this contradicts 3.21). The contradiction obtained proves that the estimate 3.24) is valid. Hence by virtue of 1.21) we get u 2 τ 22 t)) < ρ 1 for t a. According to this estimate and the conditions 3.8), 3.22), we have u 2 t) ht) for t a. If along with the above inequality we take into account 3.23) and 3.24), then we will get u 2 ) u 2 a 2 ) + a 2 hs) ds < ρ. Consequently, the estimate 3.2) is valid. In view of this estimate, from 3.15) 3.17) follows that u 1, u 2 ) is a solution of the system 1.1). Proof of Corollary 1.5. Introduce the function τt) = min { a, τ 22 t) }. EJQTDE, 1999 No. 5, p. 19
20 Then by virtue of 1.25), the condition 2.28) is fulfilled. Let ρ be the positive constant appearing in Lemma 2.6 and let ψ and f 2 be the functions defined by 3.15) and 3.16). By the conditions 1.3), 1.4) and Corollary 1.2, the problem 3.17), 1.2) has a nonnegative solution u 1, u 2 ), the functions u 1 and u 2 do not increase and u 1 admits the estimate 3.8). If along with this fact we take into account the conditions 1.7), 1.22) 1.24), then it will become evident that the restriction of u 1, u 2 ) on [, a ] is a solution of the problem 2.26), 2.18). Hence if we take into consideration how ρ is, then we obtain the estimate 3.2). According to this estimate, from 3.15) 3.17) follows that u 1, u 2 ) is a solution of the system 1.1). Proof of Theorem 1.2. First of all note that 1.4) follows from 1.5). Let us now suppose that u 1, u 2 ) is an arbitrary nonnegative solution of the problem 1.1 ε ), 1.2) for some ε ], 1]. Then according to 1.5) we have u 1 ) + u 2 ) r. Therefore the estimate 1.8), where ρ = r, is valid. From the above reasoning it is clear that all the conditions of Theorem 1.1 are fulfilled, which guarantees the existence of at least one nonnegative solution of the problem 1.1), 1.2). Proof of Theorem 1.3. Assume the contrary that the problem 1.1), 1.2) has two different solutions u 1, u 2 ) and u 1, u 2 ). Suppose u 1 a) = c, u 1 a) = c. 3.25) Then u 1, u 2 ) is a solution of the problem 1.1), 1.29) with c = c, while u 1, u 2 ) is a solution of the same problem with c = c. Thus c c, since according to one of the conditions of the theorem the problem 1.1), 1.29) has no more than one solution for an arbitrarily fixed c. Without loss of generality it can be assumed that Then by virtue of Lemma 2.2 we have c > c. u 1 t) c c + u 1 t) > u 1 t), u 2 t) u 2 t) for t a. 3.26) Consequently, and u 1 ) > u 1 ), u 2 ) u 2 ) ϕ u 1 ), u 2 ) ) > ϕ u 1 ), u 2 ) ), 3.27) EJQTDE, 1999 No. 5, p. 2
21 since ϕ is an increasing in the first argument and nondecreasing in the second argument function. But the latter inequality contradicts the equalities ϕ u 1 ), u 2 ) ) =, ϕ u 1 ), u 2 ) ) =. 3.28) The contradiction obtained proves the validity of the theorem. Proof of Theorem 1.4. Assume the contrary that the problem 1.1), 1.2) has two different solutions u 1, u 2 ) and u 1, u 2 ). Then, as is shown when proving Theorem 1.3, c c, where c and c are the numbers defined by 3.25). For the sake of definiteness we assume that c > c. Then by virtue of Lemma 2.2 the inequalities 3.26) are fulfilled. Thus u 2 t) = f 2 t, u1 τ 21 t)), u 2 τ 22 t)) ) < < f 2 t, u1 τ 21 t)), u 2 τ 22 t)) ) = u 2 t) for t a, 3.29) since f 2 is a function decreasing in the second and nonincreasing in the third argument. From 3.26) and 3.29) we have u 1 ) > u 1 ), u 2 ) > u 2 ). Hence by virtue of 1.3) we obtain the inequality 3.27), which contradicts 3.28). The contradiction obtained proves the validity of the theorem. Acknowledgement This work was supported by INTAS Grant No References 1. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the theory of functional differential equations. Russian) Nauka, Moscow, S. R. Bernfeld and V. Lakshmikantham, An introduction to nonlinear boundary value problems. Academic Press, Inc., New York and London, Sh. Gelashvili and I. Kiguradze, On multipoint boundary value problems for systems of functional differential and difference equations. Mem. Differential Equations Math. Phys ), I. Kiguradze, On a boundary value problem for a system of two differential equations. Russian) Proc. Tbilisi Univ. 1137) A1971), I. Kiguradze, Boundary value problems for systems of ordinary differential equations. J. Soviet Math ), No. 2, I. Kiguradze, Initial and boundary value problems for systems of ordinary differential equations, I. Russian) Metsniereba, Tbilisi, EJQTDE, 1999 No. 5, p. 21
22 7. I. Kiguradze and N. R. Lezhava, On some nonlinear twopoint boundary value problems. Russian) Differentsial nye Uravneniya 11974), No. 12, I. Kiguradze and B. Půža, On some boundary value problems for a system of ordinary differential equations. Russian) Differentsial nye Uravneniya ), No. 12, I. Kiguradze and B. Půža, On boundary value problems for systems of linear functional differential equations. Czechoslovak Math. J ), No. 2, I. Kiguradze and B. Půža, Conti Opial type theorems for systems of functional differential equations. Russian) Differentsial nye Uravneniya ), No. 2, I. Kiguradze and B. Půža, On the solvability of boundary value problems for systems of nonlinear differential equations with deviating arguments. Mem. Differential Equations Math. Phys ), I. Kiguradze and B. Půža, On the solvability of nonlinear boundary value problems for functional differential equations. Georgian Math. J ), No. 3, I. Kiguradze and Z. Sokhadze, On singular functional differential inequalities. Georgian Math. J ), No. 3, G. N. Milshtein, On a boundary value problem for a system of two differential equations. Russian) Differentsial nye Uravneniya 11965), No. 12, A. I. Perov, On a boundary value problem for a system of two differential equations. Russian) Dokl. AN SSSR ), No. 3, B. L. Shekhter, On a boundary value problem for twodimensional discontinuous differential systems. Russian) Trudy Inst. Prikl. Mat. im. I. N. Vekua 8198), B. L. Shekhter, On singular boundary value problems for twodimensional differential systems. Arch. Math ), No. 1, N. I. Vasil ev, Some boundary value problems for a system of two differential equations of first order, I. Russian) Latv. Mat. Ezhegodnik 51969), N. I. Vasil ev, Some boundary value problems for a system of two differential equations of first order, II. Russian) Latv. Mat. Ezhegodnik 61969), N. I. Vasil ev and Yu. A. Klokov, The foundations of the theory of boundary value problems of ordinary differential equations. Russian) Zinatne, Riga, P. Waltman, Existence and uniqueness of solutions of boundary value problem for two dimensional systems of nonlinear differential equations. Trans. Amer. Math. Soc ), No. 1, EJQTDE, 1999 No. 5, p. 22
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