# ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS

Save this PDF as:

Size: px
Start display at page:

Download "ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS"

## Transcription

1 ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL 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 two-point boundary value problem, two-dimensional 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 multi-point 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 two-point 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 two-dimensional discontinuous differential systems. Russian) Trudy Inst. Prikl. Mat. im. I. N. Vekua 8198), B. L. Shekhter, On singular boundary value problems for two-dimensional 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

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation

7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity

### TRIPLE FIXED POINTS IN ORDERED METRIC SPACES

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197-207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

### BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam

Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Pacific Journal of Mathematics

Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

### CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### 0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup

456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### ON A DIFFERENTIAL EQUATION WITH LEFT AND RIGHT FRACTIONAL DERIVATIVES. Abstract

ON A DIFFERENTIAL EQUATION WITH LEFT AND RIGHT FRACTIONAL DERIVATIVES Teodor M. Atanackovic and Bogoljub Stankovic 2 Abstract We treat the fractional order differential equation that contains the left

### t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

### Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

### Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

### ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE. 1. Introduction

SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Singuläre Phänomene und Skalierung in mathematischen Modellen

RHEINISCHE FRIDRICH-WILHELM-UNIVERSITÄT BONN Diffusion Problem With Boundary Conditions of Hysteresis Type N.D.Botkin, K.-H.Hoffmann, A.M.Meirmanov, V.N.Starovoitov no. Sonderforschungsbereich 6 Singuläre

### Quasi Contraction and Fixed Points

Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa-00168, 6 Pages doi:10.5899/2012/jnaa-00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady

### WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION?

WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION? DAVID M MASON 1 Statistics Program, University of Delaware Newark, DE 19717 email: davidm@udeledu JOEL ZINN 2 Department of Mathematics,

### THE ROBUSTNESS AGAINST DEPENDENCE OF NONPARAMETRIC TESTS FOR THE TWO-SAMPLE LOCATION PROBLEM

APPLICATIOES MATHEMATICAE,4 (1995), pp. 469 476 P. GRZEGORZEWSKI (Warszawa) THE ROBUSTESS AGAIST DEPEDECE OF OPARAMETRIC TESTS FOR THE TWO-SAMPLE LOCATIO PROBLEM Abstract. onparametric tests for the two-sample

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS

FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure

### Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend

### A new continuous dependence result for impulsive retarded functional differential equations

CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Practice Problems Solutions

Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

### DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction

DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER DMITRY KHAVINSON, ERIK LUNDBERG, HERMANN RENDER. Introduction A function u is said to be harmonic if u := n j= 2 u = 0. Given a domain

### Existence and multiplicity of solutions for a Neumann-type p(x)-laplacian equation with nonsmooth potential. 1 Introduction

Electronic Journal of Qualitative Theory of Differential Equations 20, No. 7, -0; http://www.math.u-szeged.hu/ejqtde/ Existence and multiplicity of solutions for a Neumann-type p(x)-laplacian equation

### Minimizing the Number of Machines in a Unit-Time Scheduling Problem

Minimizing the Number of Machines in a Unit-Time Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.bas-net.by Frank

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

### Some Problems of Second-Order Rational Difference Equations with Quadratic Terms

International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 1, pp. 11 21 (2014) http://campus.mst.edu/ijde Some Problems of Second-Order Rational Difference Equations with Quadratic

### Computing divisors and common multiples of quasi-linear ordinary differential equations

Computing divisors and common multiples of quasi-linear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univ-lille1.fr

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS

GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS GUSTAVO A. FERNÁNDEZ-ALCOBER AND ALEXANDER MORETÓ Abstract. We study the finite groups G for which the set cd(g) of irreducible complex

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

### Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### The inflation of attractors and their discretization: the autonomous case

The inflation of attractors and their discretization: the autonomous case P. E. Kloeden Fachbereich Mathematik, Johann Wolfgang Goethe Universität, D-654 Frankfurt am Main, Germany E-mail: kloeden@math.uni-frankfurt.de

### Limits and convergence.

Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

### Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES. 1. Introduction

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES Abstract. In this article the notion of acceleration convergence of double sequences

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =

### TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh

Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp -. TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*

### 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.

Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### Numerical methods for American options

Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

### Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008

Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 1 Basic Theorems of Complex Analysis 1 1.1 The Complex Plane........................

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### ( ) ( ) [2],[3],[4],[5],[6],[7]

( ) ( ) Lebesgue Takagi, - []., [2],[3],[4],[5],[6],[7].,. [] M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math., (984), 83 99. [2] T. Sekiguchi and Y. Shiota, A

### The Fourier Series of a Periodic Function

1 Chapter 1 he Fourier Series of a Periodic Function 1.1 Introduction Notation 1.1. We use the letter with a double meaning: a) [, 1) b) In the notations L p (), C(), C n () and C () we use the letter

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

### On the Fractional Derivatives at Extreme Points

Electronic Journal of Qualitative Theory of Differential Equations 212, No. 55, 1-5; http://www.math.u-szeged.hu/ejqtde/ On the Fractional Derivatives at Extreme Points Mohammed Al-Refai Department of

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### University of Maryland Fraternity & Sorority Life Spring 2015 Academic Report

University of Maryland Fraternity & Sorority Life Academic Report Academic and Population Statistics Population: # of Students: # of New Members: Avg. Size: Avg. GPA: % of the Undergraduate Population

### GREEN S FUNCTION AND MAXIMUM PRINCIPLE FOR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH IMPULSES

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Number 2, Summer 2000 GREEN S FUNCTION AND MAXIMUM PRINCIPLE FOR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH IMPULSES ALBERTO CABADA, EDUARDO LIZ

### KATO S INEQUALITY UP TO THE BOUNDARY. Contents 1. Introduction 1 2. Properties of functions in X 4 3. Proof of Theorem 1.1 6 4.

KATO S INEQUALITY UP TO THE BOUNDARY HAÏM BREZIS(1),(2) AND AUGUSTO C. PONCE (3) Abstract. We show that if u is a finite measure in then, under suitable assumptions on u near, u + is also a finite measure

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### Lecture notes on Ordinary Differential Equations. Plan of lectures

1 Lecture notes on Ordinary Differential Equations Annual Foundation School, IIT Kanpur, Dec.3-28, 2007. by Dept. of Mathematics, IIT Bombay, Mumbai-76. e-mail: sivaji.ganesh@gmail.com References Plan

### ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### A Formalization of the Turing Test Evgeny Chutchev

A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### 1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1

Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between

### The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

### Some remarks on two-asset options pricing and stochastic dependence of asset prices

Some remarks on two-asset options pricing and stochastic dependence of asset prices G. Rapuch & T. Roncalli Groupe de Recherche Opérationnelle, Crédit Lyonnais, France July 16, 001 Abstract In this short

### Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization