GREEN S FUNCTIONAL FOR SECOND-ORDER LINEAR DIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS

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1 Electronic Journal of Differential Equations, Vol. 1 (1), No. 11, pp ISSN: URL: or ftp ejde.math.txstate.edu GREEN S FUNCTIONAL FOR SECOND-ORDER LINEAR DIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS KAMİL ORUÇOĞLU, KEMAL ÖZEN Abstract. In this work, we present a new constructive technique which is based on Green s functional concept. According to this technique, a linear completely nonhomogeneous nonlocal problem for a second-order ordinary differential equation is reduced to one and only one integral equation in order to identify the Green s solution. The coefficients of the equation are assumed to be generally variable nonsmooth functions satisfying some general properties such as p-integrability and boundedness. A system of three integro-algebraic equations called the special adjoint system is obtained for this problem. A solution of this special adjoint system is Green s functional which enables us to determine the Green s function and the Green s solution for the problem. Some illustrative applications and comparisons are provided with some known results. 1. Introduction Green functions of linear boundary-value problems for ordinary differential equations with smooth coefficients have been investigated in detail in several studies [1, 13, 16, 17, 18]. In this work, a linear nonlocal problem is studied for a secondorder differential equation. The coefficients of the equation are assumed to be generally nonsmooth functions satisfying some general properties such as p-integrability and boundedness. The operator of this equation, in general, does not have a formal adjoint operator, or any extension of the traditional type for this operator exists only on a space of distributions [7, 16]. In addition, the considered problem does not have a meaningful traditional type adjoint problem, even for simple cases of a differential equation and nonlocal conditions. Due to these facts, some serious difficulties arise in the application of the classical methods for such a problem. As can be seen from [1], similar difficulties arise even for classical type boundary value problems if the coefficients of the differential equation are, for example, continuous nonsmooth functions. For this reason, a Green s functional approach is introduced for the investigation of the considered problem. This approach is based on [1,, 3, 4, 14] and on some methods of functional analysis. The main idea of this approach is related to the usage of a new concept of the adjoint problem named adjoint system. Such an adjoint system includes three integro-algebraic equations Mathematics Subject Classification. 34A3, 34B5, 34B1, 34B7, 45A5. Key words and phrases. Green s function; nonlocal boundary conditions; nonsmooth coefficient; adjoint problem. c 1 Texas State University - San Marcos. Submitted February 3, 1. Published July 19, 1. 1

2 K. ORUÇOĞLU, K. ÖZEN EJDE-1/11 with an unknown element (f (ξ), f 1, f ) in which f (ξ) is a function, and f j for j =, 1 are real numbers. One of these equations is an integral equation with respect to f (ξ) and generally includes f j as parameters. The other two equation can be considered as a system of algebraic equations with respect to f and f 1, and they may include some integral functionals defined on f (ξ). The form of the adjoint system depends on the operators of the equation and the conditions. The role of the adjoint system is similar to that of the adjoint operator equation in the general theory of the linear operator equations in Banach spaces [5, 8, 1]. The integral representation of the solution is obtained by a concept of the Green functional which is introduced as a special solution f(x) = (f (ξ, x), f 1 (x), f (x)) of the corresponding adjoint system having a special free term depending on x as a parameter. The first component f (ξ, x) of Green functional f(x) is corresponded to Green s function for the problem. The other two components f j (x) for j =, 1 correspond to the unit effects of the conditions. If the homogeneous problem has a nontrivial solution, then the Green functional does not exist. In summary, this approach is principally different from the classical methods used for constructing Green functions[17].. Statement of the problem Let R be the set of real numbers. Let G = (, x 1 ) be a bounded open interval in R. Let L p (G) with 1 p < be the space of p-integrable functions on G. Let L (G) be the space of measurable and essentially bounded functions on G, and let (G) with 1 p be the space of all functions u = u(x) L p (G) having derivatives d k u/dx k L p (G), where k = 1,. The norm on the space W p () (G) is defined as u () W p (G) = dk u dx k L p(g). W () p k= We consider the second-order boundary value problem (V u)(x) u (x) + A 1 (x)u (x) + A (x)u(x) = z (x), x G, (.1) subject to the nonlocal boundary conditions V 1 u a 1 u( ) + b 1 u ( ) + V u a u( ) + b u ( ) + g 1 (ξ)u (ξ)dξ = z 1, g (ξ)u (ξ)dξ = z, (.) which are more general conditions than the ones in [4]. We investigate a solution to the problem in the space W p = W p () (G). Furthermore, we assume that the following conditions are satisfied: A i L p (G) and g i L q (G) for i =, 1 are given functions; a i, b i for i =, 1 are given real numbers; z L p (G) is a given function and z i for i =, 1 are given real numbers. Problem (.1)-(.) is a linear completely nonhomogeneous problem which can be considered as an operator equation: V u = z, (.3) with the linear operator V = (V, V 1, V ) and z = (z (x), z 1, z ).

3 EJDE-1/11 GREEN S FUNCTIONAL FOR A NONLOCAL PROBLEM 3 The assumptions considered above guarantee that V is bounded from W p to the Banach space E p L p (G) R R consisting of element z = (z (x), z 1, z ) with z Ep = z Lp(G) + z 1 + z, 1 p. If, for a given z E p, the problem (.1)-(.) has a unique solution u W p with u Wp c z Ep, then this problem is called a well-posed problem, where c is a constant independent of z. Problem (.1)-(.) is well-posed if and only if V : W p E p is a (linear) homeomorphism. 3. Adjoint space of the solution space Problem (.1)-(.) is investigated by means of a new concept of the adjoint problem. This concept is introduced in the papers [, 3] by the adjoint operator V of V. Some isomorphic decompositions of the space W p of solutions and its adjoint space Wp are employed. Any function u W p can be represented as u(x) = u(α) + u (α)(x α) + x α (x ξ)u (ξ)dξ (3.1) where α is a given point in G which is the set of closure points for G. Furthermore, the trace or value operators D u = u(γ), D 1 u = u (γ) are bounded and surjective from W p onto R for a given point γ of G. In addition, the values u(α), u (α) and the derivative u (x) are unrelated elements of the function u W p such that for any real numbers ν, ν 1 and any function ν L p (G), there exists one and only one u W p such that u(α) = ν, u (α) = ν 1 and u (x) = ν (x). Therefore, there exists a linear homeomorphism between W p and E p. In other words, the space W p has the isomorphic decomposition W p = L p (G) R R. Theorem 3.1 ([4]). If 1 p <, then any linear bounded functional F Wp can be represented as F (u) = u (x)ϕ (x)dx + u ( )ϕ 1 + u( )ϕ (3.) with a unique element ϕ = (ϕ (x), ϕ 1, ϕ ) E q where p + q = pq. Any linear bounded functional F W can be represented as F (u) = u (x)dϕ + u ( )ϕ 1 + u( )ϕ (3.3) with a unique element ϕ = (ϕ (e), ϕ 1, ϕ ) Ê1 = (BA(, µ)) R R where µ is the Lebesgue measure on R, is σ-algebra of the µ-measurable subsets e G and BA(, µ) is the space of all bounded additive functions ϕ (e) defined on with ϕ (e) = when µ(e) = [8]. The inverse is also valid; that is, if ϕ E q, then (3.) is bounded on W p for 1 p < and p + q = pq. If ϕ Ê1, then (3.3) is bounded on W. Proof. [4] The operator Nu (u (x), u ( ), u( )) : W p E p is bounded and has a bounded inverse N 1 represented by x u(x) = (N 1 h)(x) (x ξ)h (ξ)dξ + h 1 (x ) + h, h = (h (x), h 1, h ) E p. (3.4)

4 4 K. ORUÇOĞLU, K. ÖZEN EJDE-1/11 The kernel ker N of N is trivial and the image Im N of N is equal to E p. Hence, there exists a bounded adjoint operator N : Ep Wp with ker N = {} and Im N = Wp. In other words, for a given F Wp there exists a unique ψ Ep such that F = N ψ or F (u) = ψ(nu), u W p. (3.5) If 1 p <, then Ep = E q in the sense of an isomorphism [8]. Therefore, the functional ψ can be represented by ψ(h) = ϕ (x)h (x)dx + ϕ 1 h 1 + ϕ h, h E p, (3.6) with a unique element ϕ = (ϕ (x), ϕ 1, ϕ ) E q. By expressions (3.5) and (3.6), any F Wp can uniquely be represented by (3.). For a given ϕ E q, the functional F represented by (3.) is bounded on W p. Hence, (3.) is a general form for the functional F Wp. The proof is complete due to that the case p = can also be shown [4]. Theorem 3.1 guarantees that W p = E q for all 1 p <, and W = E = Ê1. The space E 1 can also be considered as a subspace of the space Ê1 (see [3, 4]). 4. Adjoint operator and adjoint system of the integro-algebraic equations Investigating an explicit form for the adjoint operator V of V is taken into consideration in this section. To this end, any f = (f (x), f 1, f ) E q is taken as a linear bounded functional on E p and also f(v u) f (x)(v u)(x)dx + f 1 (V 1 u) + f (V u), u W p, (4.1) can be assumed. By substituting expressions (.1) and (.), and expression (3.1) (for α = ) of u W p into (4.1), we have f(v u) f (x)[u (x) + A 1 (x){u ( ) + + A (x){u( ) + u ( )(x ) + + f 1 {a 1 u( ) + b 1 u ( ) + + f {a u( ) + b u ( ) + After some calculations, we can obtain f(v u) x x1 f (x)(v u)(x)dx + x x u (ξ)dξ} (x ξ)u (ξ)dξ}]dx g 1 (ξ)u (ξ)dξ} g (ξ)u (ξ)dξ}. 1 f i (V i u) i= = (w f)(ξ)u (ξ)dξ + (w 1 f)u ( ) + (w f)u( ) (wf)(u), f E q, u W p, 1 p (4.) (4.3)

5 EJDE-1/11 GREEN S FUNCTIONAL FOR A NONLOCAL PROBLEM 5 where (w f)(ξ) = f (ξ) + f 1 g 1 (ξ) + f g (ξ) + w 1 f = b 1 f 1 + b f + w f = a 1 f 1 + a f + x x1 ξ f (s){a 1 (s) + A (s)(s ξ)}ds, f (s){a 1 (s) + A (s)(s )}ds f (s)a (s)ds. (4.4) The operators w, w 1, w are linear and bounded from the space E q of the triples f = (f (x), f 1, f ) into the spaces L q (G), R, R respectively. Therefore, the operator w = (w, w 1, w ) : E q E q represented by wf = (w f, w 1 f, w f) is linear and bounded. By (4.3) and Theorem 3.1, the operator w is an adjoint operator for the operator V when 1 p <, in other words, V = w. When p =, w : E 1 E 1 is bounded; in this case, the operator w is the restriction of the adjoint operator V : E W of V onto E 1 E. Equation (.3) can be transformed into the following equivalent equation V Sh = z, (4.5) with an unknown h = (h, h 1, h ) E p by the transformation u = Sh where S = N 1. If u = Sh, then u (x) = h (x), u ( ) = h 1, u( ) = h. Hence, equation (4.3) can be rewritten as 1 f(v Sh) f (x)(v Sh)(x)dx + f i (V i Sh) i= = (w f)(ξ)h (ξ)dξ + (w 1 f)h 1 + (w f)h (wf)(h), f E q, h E p, 1 p. (4.6) Therefore, one of the operators V S and w becomes an adjoint operator for the other one. Consequently, the equation wf = ϕ, (4.7) with an unknown function f = (f (x), f 1, f ) E q and a given function ϕ = (ϕ (x), ϕ 1, ϕ ) E q can be considered as an adjoint equation of (4.5)(or of (.3)) for all 1 p. Equation (4.7) can be written in explicit form as the system of equations (w f)(ξ) = ϕ (ξ), ξ G, w 1 f = ϕ 1, w f = ϕ. (4.8) By the expressions (4.4), the first equation in (4.8) is an integral equation for f (ξ) and includes f 1 and f as parameters; on the other hand, the other equations in (4.8) constitute a system of two algebraic equations for the unknowns f 1 and f and they include some integral functionals defined on f (ξ). In other words, (4.8) is a system of three integro-algebraic equations. This system called the adjoint system for (4.5)(or (.3)) is constructed by using (4.3) which is actually a formula of integration by parts in a nonclassical form. The traditional type of an adjoint problem is defined by the classical Green s formula of integration by parts [17], therefore, has a sense only for some restricted class of problems.

6 6 K. ORUÇOĞLU, K. ÖZEN EJDE-1/11 5. Solvability conditions for completely nonhomogeneous problem The operator Q = w I q is considered where I q is the identity operator on E q ; i.e., I q f = f for all f E q. This operator can also be defined as Q = (Q, Q 1, Q ) with (Q f)(ξ) = (w f)(ξ) f (ξ), ξ G, (5.1) Q i f = w i f f i, i =, 1. By the expressions (4.4) and the conditions imposed on A i and g i for i =, 1, Q m : E q L q (G) is a compact operator, and also Q i : E q R for i =, 1 are compact operators where 1 < p <. That is, Q : E q E q is a compact operator, and therefore has a compact adjoint operator Q : E p E p. Since w = Q + I q and V S = Q + I p, where I p = Iq, (4.5) and (4.7) are canonical Fredholm type equations, and S is a right regularizer of (.3) [1]. Consequently, we have the following theorem. Theorem 5.1 ([4]). If 1 < p <, then V u = has either only the trivial solution or a finite number of linearly independent solutions in W p : (1) If V u = has only the trivial solution in W p, then also wf = has only the trivial solution in E q. Then, the operators V : W p E p and w : E q E q become linear homeomorphisms. () If V u = has m linearly independent solutions u 1, u,..., u m in W p, then wf = also has m linearly independent solutions f 1 = (f 1 (x), f 1 1, f 1 ),..., f m = (f m (x), f m 1, f m ) in E q. In this case, (.3) and (4.7) have solutions u W p and f E q for given z E p and ϕ E q if and only if the conditions and f i (ξ)z (ξ)dξ + f i 1 z 1 + f i z =, i = 1,,..., m, (5.) ϕ (ξ)u i (ξ)dξ + ϕ 1 u i( ) + ϕ u i ( ) =, i = 1,,..., m, (5.3) are satisfied, respectively. 6. Green s Functional Consider the following equation given in the form of a functional identity (wf)(u) = u(x), u W p, (6.1) where f = (f (ξ), f 1, f ) E q is an unknown triple and x G is a parameter. Definition. [4] Suppose that f(x) = (f (ξ, x), f 1 (x), f (x)) E q is a triple with parameter x G. If f = f(x) is a solution of (6.1) for a given x G, then f(x) is called a Green s functional of V (or of (.3)). Due to that the operator I Wp,C of the imbedding of W p into the space C(G) of continuous functions on G is bounded, the linear functional θ(x) defined by θ(x)(u) = u(x) is bounded on W p for a given x G. On the other hand, (wf)(u) = (V f)(u). Thus, (6.1) can also be written as [, 3] (V f) = θ(x).

7 EJDE-1/11 GREEN S FUNCTIONAL FOR A NONLOCAL PROBLEM 7 In other words, (6.1) can be considered as a special case of the adjoint equation V f = ψ for some ψ = θ(x). By substituting α = into (3.1) and using (4.3), we can rewrite (6.1) as (w f)(ξ)u (ξ)dξ + (w 1 f)u ( ) + (w f)u( ) x (6.) = (x ξ)u (ξ)dξ + u ( )(x ) + u( ), f E q, u W p. The elements u (ξ) L p (G), u ( ) R and u( ) R of the function u W p are unrelated. Then, we can construct the system (w f)(ξ) = (x ξ)h(x ξ), ξ G, (w 1 f) = (x ), (w f) = 1, (6.3) where H(x ξ) is a Heaviside function on R. Equation (6.1) is equivalent to the system (6.3) which is a special case for the adjoint system (4.8) when ϕ (ξ) = (x ξ)h(x ξ), ϕ 1 = x and ϕ = 1. Therefore, f(x) is a Green s functional if and only if f(x) is a solution of system (6.3) for an arbitrary x G. For a solution u W p of (.3) and a Green s functional f(x), we can rewrite (4.3) as = f (ξ, x)z (ξ)dξ + f 1 (x)z 1 + f (x)z (x ξ)h(x ξ)u (ξ)dξ + u ( )(x ) + u( ). (6.4) The right hand side of (6.4) is equal to u(x). Therefore, we can state the following theorem. Theorem 6.1 ([4]). If (.3) has at least one Green s functional f(x), then any solution u W p of (.3) can be represented by u(x) = Additionally, V u = has only the trivial solution. f (ξ, x)z (ξ)dξ + f 1 (x)z 1 + f (x)z. (6.5) Since one of the operators V : W p E p and w : E q E q is a homeomorphism, so is the other, and, there exists a unique Green s functional, where 1 p. Necessary and sufficient conditions for the existence of a Green s functional can be given in the following theorem for 1 < p <. Theorem 6. ([4]). If there exists a Green s functional, then it is unique. Additionally, a Green s functional exists if and only if V u = has only the trivial solution. From Theorems 5.1, 6.1, 6. can be easily shown. Remark. If V u = has a nontrivial solution, then a Green s functional corresponding to V u = z does not exist due to Theorem 6.1. In this case, V u = z usually has no solution unless z is of a specific type. Therefore, a representation of the existing solution of V u = z are constructed by a concept of the generalized Green s functional [3, 4].

8 8 K. ORUÇOĞLU, K. ÖZEN EJDE-1/11 It must be noted that the proposed Green s functional approach can also be employed some classes of nonlinear equations involving linear nonlocal conditions to transform into the corresponding integral equations and then solve them. The corresponding integral equations will naturally become of nonlinear type. These nonlinear integral equations can be solved approximately even if they can not be solved exactly. 7. Some applications In this section, some applications to such problems involving nonlocal boundary conditions are implemented in order to emphasize the preferability of the presented approach. Example 7.1. First, we seek for the Green s solution to the following problem, which has been considered in [15]: u (x) = f(x), x G = (, 1), (7.1) u() = γ u (ξ ), u(1) = γ 1 u (ξ 1 ), (7.) where f(x) L p (G), ξ, ξ 1 G and γ, γ 1 R. We can rewrite this problem as Thus, we have (V u)(x) u (x) = f(x) = z (x), x G = (, 1), V 1 u u(1) γ 1 u (ξ 1 ) = = z 1, V u u() γ u (ξ ) = = z. a 1 = 1, b 1 = 1 γ 1, g 1 (ξ) = 1 ξ γ 1 H(ξ 1 ξ), a = 1, b = γ, g (ξ) = γ H(ξ ξ), and A i (x) = z i = for i =, 1, where H(ξ 1 ξ) and H(ξ ξ) are Heaviside functions on R. Consequently, the special adjoint system (6.3) corresponding to this problem can be constructed in the form f (ξ) + f 1 {1 ξ γ 1 H(ξ 1 ξ)} f γ H(ξ ξ) = (x ξ)h(x ξ), (7.3) f 1 (1 γ 1 ) f γ = x, (7.4) f 1 + f = 1, (7.5) where ξ (, 1). We firstly determine f 1 and f with using only (7.4) and (7.5) under the condition 1 = 1 γ 1 + γ in order to solve (7.3)-(7.5). Thus, we have f 1 = x + γ, f = 1 γ 1 x. 1 1 After substituting f 1 and f into equation (7.3), f (ξ) becomes f (ξ) = (x ξ)h(x ξ) + (1 γ 1 x) 1 γ H(ξ ξ) (x + γ ) 1 {1 ξ γ 1 H(ξ 1 ξ)}. Thus, the Green s functional f(x) = (f (ξ, x), f 1 (x), f (x)) for the problem has been determined. The first component f (ξ, x) = f (ξ) is the Green s function for the problem. After substituting ξ = s for notational compatibility, f (ξ, x) is equal

9 EJDE-1/11 GREEN S FUNCTIONAL FOR A NONLOCAL PROBLEM 9 to the Green s function constructed in [15] for the problem. By (6.5) in Theorem 6.1, the representation of the existing solution for the problem can be given as u(x) = [(x ξ)h(x ξ) + (1 γ 1 x) 1 γ H(ξ ξ) (x + γ ) {1 ξ γ 1 H(ξ 1 ξ)}]f(ξ)dξ. 1 Example 7.. Next, we construct the Green s solution to the following problem, which has been considered in [15]: u (x) = f(x), x G = (, 1), (7.6) u() = γ (1 + t)u(t)dt, u(1) = γ 1 u(t)dt, (7.7) where f(x) L p (G) and γ, γ 1 R. We can rewrite this problem as Then, we have (V u)(x) u (x) = f(x) = z (x), x G = (, 1), V 1 u u(1) γ 1 u(t)dt = = z 1, V u u() γ (1 + t)u(t)dt = = z. a 1 = 1 γ 1, b 1 = 1 γ 1, g 1(ξ) = 1 ξ γ 1 ( 1 ξ + ξ ), a = 1 3 γ, b = 5 6 γ, g (ξ) = γ ( ξ3 6 + ξ 3 ξ ), and A i (x) = z i = for i =, 1. Consequently, the special adjoint system (6.3) corresponding to this problem is of the form f (ξ) + f 1 {1 ξ γ 1 ( 1 ξ + ξ )} f γ ( ξ3 6 + ξ 3 ξ + 5 ) = (x ξ)h(x ξ), 6 (7.8) (1 γ 1 )f γ f = x, (7.9) (1 γ 1 )f 1 + (1 3 γ )f = 1, (7.1) where ξ (, 1) and, H(x ξ) is Heaviside function on R. We firstly determine f 1 and f with using only (7.9) and (7.1) under the condition = (1 γ1 )(1 3 γ ) γ (1 γ 1 ) in order to solve (7.8)-(7.1). Hence, we have f 1 = (1 3 γ )x γ, f = 1 γ1 x(1 γ 1 ). After substituting f 1 and f into equation (7.8), f (ξ) becomes γ1 [1 f (ξ) = (x ξ)h(x ξ) + x(1 γ 1)] γ ( ξ3 6 + ξ 3 ξ )

10 1 K. ORUÇOĞLU, K. ÖZEN EJDE-1/11 [(1 3 γ )x γ ] {1 ξ γ 1 ( 1 ξ + ξ )}. Thus, the Green s functional f(x) = (f (ξ, x), f 1 (x), f (x)) for the problem has been determined. The first component f (ξ, x) = f (ξ) is the Green s function for the problem. After substituting ξ = s for notational compatibility, f (ξ, x) is equal to the Green s function constructed in [15] for the problem. By (6.5) in Theorem 6.1, the representation of the existing solution for the problem can be given as u(x) = γ1 [1 [(x ξ)h(x ξ) + x(1 γ 1)] [(1 3 γ )x γ ] {1 ξ γ 1 ( 1 ξ + ξ )}]f(ξ)dξ. γ ( ξ3 6 + ξ 3 ξ ) Example 7.3. Finally, we consider the following problem to reduce an integral equation by using the Green s functional concept: u (x) + λu(x) = f(x), x G = (, 1), (7.11) u() = α u(β ) + α 1 u (β 1 ) + γ u(t)dt, (7.1) u(1) = α 1 u(β 1 ) + α 11 u (β 11 ) + γ 1 u(t)dt, (7.13) where f(x) L p (G), α, α 1, α 1, α 11, γ, γ 1 R, β, β 1, β 1, β 11 G and, λ is a constant. We can rewrite this problem as Then, we have (V u)(x) u (x) + λu(x) = f(x) = z (x), x G = (, 1), V 1 u u(1) α 1 u(β 1 ) α 11 u (β 11 ) γ 1 u(t)dt = = z 1, V u u() α u(β ) α 1 u (β 1 ) γ u(t)dt = = z. a 1 = 1 α 1 γ 1, b 1 = 1 α 1 β 1 α 11 γ 1, a = 1 α γ, b = β α 1 γ, (1 ξ) g 1 (ξ) = 1 ξ α 1 (β 1 ξ)h(β 1 ξ) α 11 H(β 11 ξ) γ 1, (1 ξ) g (ξ) = α (β ξ)h(β ξ) α 1 H(β 1 ξ) γ, and A (x) = λ, A 1 (x) = z 1 = z = and, H( ) is Heaviside function on R.

11 EJDE-1/11 GREEN S FUNCTIONAL FOR A NONLOCAL PROBLEM 11 Consequently, the special adjoint system (6.3) corresponding to this problem is of the form f (ξ) + ξ f (s)λ(s ξ)ds (1 ξ) + f 1 {1 ξ α 1 (β 1 ξ)h(β 1 ξ) α 11 H(β 11 ξ) γ 1 } (1 ξ) + f { α (β ξ)h(β ξ) α 1 H(β 1 ξ) γ } = (x ξ)h(x ξ), (1 α 1 β 1 α 11 γ 1 )f 1 + ( β α 1 γ )f + (1 α 1 γ 1 )f 1 + (1 α γ )f + (7.14) f (s)λsds = x, (7.15) f (s)λds = 1, (7.16) where ξ (, 1). We denote f (s)λsds and f (s)λds by E and F respectively, and then determine f 1 and f with using only (7.15) and (7.16) under the condition 3 = b 1 a b a 1 = (1 α 1 β 1 α 11 γ1 )(1 α γ ) ( β α 1 γ )(1 α 1 γ 1 ) in order to solve (7.14)-(7.16). As a result, we have f 1 = a (x E) b (1 F ) 3, f = b 1(1 F ) a 1 (x E) 3. (7.17) After substituting f 1 and f into (7.14), we have f (ξ) + ξ f (s)λ(s ξ)ds + { a (x E) b (1 F ) 3 } (1 ξ) {1 ξ α 1 (β 1 ξ)h(β 1 ξ) α 11 H(β 11 ξ) γ 1 } + { b 1(1 F ) a 1 (x E) } 3 (1 ξ) { α (β ξ)h(β ξ) α 1 H(β 1 ξ) γ } = (x ξ)h(x ξ). (7.18) As can be seen from the denotations E and F, (7.18) is an integral equation for f (ξ). After f (ξ) is determined by solving this integral equation, and then f 1 (x) and f (x) by (7.17); the Green s functional f(x) = (f (ξ, x), f 1 (x), f (x)) for the problem will have been constructed. The first component f (ξ, x) = f (ξ) will become the Green s function for the problem. Consequently, it must be noticed that the Green s function is constructed under the condition 3 in addition to the solvability conditions for the problem. References [1] Akhiev, S. S.; Representations of the solutions of some linear operator equations, Soviet Math. Dokl., 1() (198), [] Akhiev, S. S.; Fundamental solutions of functional differential equations and their representations, Soviet Math. Dokl., 9() (1984), [3] Akhiev, S. S. and Oruçoğlu, K.; Fundamental Solutions of Some Linear Operator Equations and Applications, Acta Applicandae Mathematicae, 71 (), 1-3.

12 1 K. ORUÇOĞLU, K. ÖZEN EJDE-1/11 [4] Akhiev, S. S.; Green and Generalized Green s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations, Acta Applicandae Mathematicae, 95 (7), [5] Brown, A. L. and Page, A.; Elements of Functional Analysis, New York, 197. [6] Denche, M. and Kourta, A.; Boundary Value Problem for Second-Order Differential Operators with Mixed Nonlocal Boundary Conditions, Journal of Inequalities in Pure and Applied Mathematics, 5() (4), [7] Hörmander, L.; Linear Partial Differential Operators, Springer-Verlag, New York, [8] Kantorovich, L. V. and Akilov, G. P.; Functional Analysis (nd ed, translated by Howard L. Silcock), Pergamon Press, New York, 198. [9] Karakostas, G. L. and Tsamatos, P. C.; Sufficient Conditions for the Existence of Nonnegative Solutions of A Nonlocal Boundary Value Problem, Applied Mathematics Letters, 15(4) (), [1] Krein, S. G.; Linear Equations in Banach Space, Nauka, Moscow, 1971 (in Russian). [11] Liu, X., Jiang, W. and Guo, Y.; Multi-Point Boundary Value Problems for Higher Order Differential Equations, Applied Mathematics E-Notes, 4 (4), [1] Ma, R.; Existence Theorems for A Second Order m-point Boundary Value Problem, Journal of Mathematical Analysis and Applications, 11() (1997), [13] Naimark, M. A.; Linear Differential operators, Nauka, Moscow, 1969 (in Russian). [14] Orucoglu, K.; A New Green Function Concept for Fourth-Order Differential Equations, Electronic Journal of Differential Equations, 8 (5), 1-1. [15] Roman, S. and Stikonas, A.; Green s Functions for Stationary Problems with Nonlocal Boundary Conditions, Lithuanian Mathematical Journal, 49() (9), 19-. [16] Shilov, G. E.; Mathematical Analysis: Second special course, Nauka, Moscow, 1965; English transl., Generalized Functions and Partial Differential Equations, Gordon and Breach, New York, [17] Stakgold, I.; Green s Functions and Boundary Value Problems, Wiley-Interscience Publications, New York, [18] Tikhonov, A. N., Vasil eva, A. B. and Sveshnikov, A. G.; Differential Equations, Nauka, Moscow, 198 (in Russian). Kamil Oruçoğlu Istanbul Technical University, Department of Mathematics, Istanbul, 34469, Turkey address: koruc@itu.edu.tr Kemal Özen Istanbul Technical University, Department of Mathematics, Istanbul, 34469, Turkey, Namık Kemal University, Department of Mathematics, Tekirdağ, 593, Turkey address: ozenke@itu.edu.tr