Positive solutions for singular boundary value problem with p-Laplacian.Abstract In this paper, five functionals fixed point theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. is extended and applied to singular SINGULAR, construction. In grammar the singular is used to express only one, not plural. Johnson. 2. In law, the singular frequently includes the plural. boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. with p-Laplacian. The existence of at least three positive solutions is obtained. Keywords Keywords are the words that are used to reveal the internal structure of an author's reasoning. While they are used primarily for rhetoric, they are also used in a strictly grammatical sense for structural composition, reasoning, and comprehension. and Phrases: Positive solutions, Cone, Fixed point theorem. 1. Introduction Nonlinear A system in which the output is not a uniform relationship to the input. nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. multi-point boundary value problems have been studied extensively in the literature (see [?], [?], and [?], and the references cited therein). Singular differential boundary value problems arise in many nonlinear complex phenomena in the science, engineering and technology and have been studied extensively (see [1-4, 8, 9]). Agarwal et al. [?] consider the following one-dimensional one-di·men·sion·al adj. 1. Having or existing in one dimension only. 2. Lacking depth; superficial. one-dimensional Adjective 1. having one dimension 2. singular equation [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]. The existence of one positive solution have been obtained by using upper-lower solution method. Xian Xu [?] consider the following boundary value problems [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], give some existence results for at least one positive solution by using Leray-Schauder continuation continuation - continuation passing style theorem. Motivated mo·ti·vate tr.v. mo·ti·vat·ed, mo·ti·vat·ing, mo·ti·vates To provide with an incentive; move to action; impel. mo by the results mentioned, we study the existence of three positive solutions for the following BVP BVP bovine viral papillomatosis. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1) where [[PHI phi n. Symbol  The 21st letter of the Greek alphabet.PHI, n See health information, protected. ].sub.p] v [colon colon, in anatomy colon, in anatomy: see intestine. colon, in punctuation colon, in writing: see punctuation. colon Segment that makes up most of the large intestine. , equals] |v|[.sup.p-2]v, p > 1, 0 [less than or equal to] [alpha] < 1, f(t, u) : [0, 1] x (0, [infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]) [right arrow] [0, [infinity]), f may have singularity (1) See technology singularity. (2) (Singularity) An experimental operating system from Microsoft for the x86 platform written almost entirely in C#, a .NET managed code language. Released in 2007, Singularity is a non-Windows research project. at u = 0. e(t) is a nonnegative non·neg·a·tive adj. Of, relating to, or being a quantity that is either positive or zero. Adj. 1. nonnegative - either positive or zero measurable function In mathematics, measurable functions are well-behaved functions between measurable spaces. Functions studied in analysis that are not measurable are generally considered pathological. defined on (0, 1), and e(t) is not identically zero In mathematics, identically zero is a term used to describe a function which is equal to the zero function and not merely zero at a particular point in its domain. on any compact subinterval of (0, 1). Furthermore e(t) satisfies 0 < [[integral].sub.0.sup.1] e(t)dt < + [infinity]. When [alpha] = 0, (1.1) becomes the Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. . As far as we know, there were not any papers to consider singular 3-point boundary value problem with p-laplacian operator by means of five functionals fixed point theorem[19]. The main reason is that A : [bar.[P([gamma], c)]] [right arrow] [bar.[P([gamma], c)]] is difficult to verified ver·i·fy tr.v. ver·i·fied, ver·i·fy·ing, ver·i·fies 1. To prove the truth of by presentation of evidence or testimony; substantiate. 2. . To implying the theorem we improve the condition of five functionals fixed point theorem such that we only need to verify (1) To prove the correctness of data. (2) In data entry operations, to compare the keystrokes of a second operator with the data entered by the first operator to ensure that the data were typed in accurately. See validate. A : [partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential ]P([gamma], c) [right arrow] [bar.[P([gamma], c)]]. At least three positive solutions for singular p-laplacian equation(1.1) are obtained. By a positive solution of problem (1.1), we mean a function y [member of] [C.sup.1][0, 1], [[PHI].sub.p] (y') [member of] [C.sup.1](0, 1] satisfying the boundary value problem (1.1) and that y(t) > 0 for t [member of] (0, 1]. The paper is organized as follows. Section 2 present some background definitions and five functionals fixed point theorem. Furthermore, we extend the five functionals fixed point theorem. The main result about the existence of solutions to BVP (1.1) are given in Section 3. Section 4 present the proof of existence theorem In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s) ..', or more generally 'for all x, y, ... there exist(s) ...'. . 2. Background Knowledge and Results Let [gamma], [beta], [theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ] be nonnegative continuous convex Convex Curved, as in the shape of the outside of a circle. Usually referring to the price/required yield relationship for option-free bonds. functionals on K and let [alpha], [psi PSI - Portable Scheme Interpreter ] be nonnegative continuous concave Concave Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex. functionals on K. Then for nonnegative numbers h, a, b, d, and c, we define the following convex sets In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent : P([gamma], c) = {x [member of] K | [gamma](x) < c}, P([gamma], [alpha], a, c) = {x [member of] K | a [less than or equal to] [alpha](x), [gamma](x) [less than or equal to] c}, Q ([gamma], [beta], d, c) = {x [member of] K | [beta](x) [less than or equal to] d, [gamma](x) [less than or equal to] c}, P([gamma], [theta], [alpha], a, b, c) = {x [member of] K | a [less than or equal to] [alpha](x), [theta](x) [less than or equal to] b, [gamma](x) [less than or equal to] c}, Q([gamma], [beta], [psi], h, d, c) = {x [member of] K | h [less than or equal to] [psi](x), [beta](x) [less than or equal to] d, [gamma](x) [less than or equal to] c}. Theorem 2.1[19]. Let K be a cone in a real Banach space (mathematics) Banach space - A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. E. Let [alpha] and [psi] be nonnegative continuous concave functionals on K and [gamma], [beta], and [theta] are nonnegative continuous convex functionals on K such that for some positive numbers c and m, [alpha](x) [less than or equal to] [beta](x) and ||x|| [less than or equal to] m[gamma](x) for all x [member of] [bar.[P([gamma], c)]]. Suppose further that A : [bar.[P([gamma], c)]] [right arrow] [bar.[P([gamma], c)]] is completely continuous and there exist h, d, a, b [greater than or equal to] 0 with 0 < d < a such that each of the following is satisfied: ([A.sub.1]){x [member of] P([gamma], [theta], [alpha], a, b, c)|[alpha](x) > a} [not equal to] [null A character that is all 0 bits. Also written as "NUL," it is the first character in the ASCII and EBCDIC data codes. In hex, it displays and prints as 00; in decimal, it may appear as a single zero in a chart of codes, but displays and prints as a blank space. ] and [alpha](Ax) > a for x [member of] P([gamma], [theta], [alpha], a, b, c), ([A.sub.2]){x [member of] Q([gamma], [beta], [psi], h, d, c)|[beta](x) < d} [not equal to] [null] and [beta](Ax) < d for x [member of] Q([alpha], [beta], [psi], h, d, c), ([A.sub.3])[alpha](Ax) > a provided x [member of] P([gamma], [alpha], a, c) with [theta](Ax) > b, ([A.sub.4])[beta](Ax) < d provided x [member of] Q([gamma], [beta], d, c) with [psi](Ax) < h. Then A has at least three fixed points [x.sub.1], [x.sub.2], [x.sub.3] [member of] [bar.[P([gamma], c)]] such that [beta]([x.sub.1]) < d, a < [alpha]([x.sub.2]), and d < [beta]([x.sub.3]) with [alpha]([x.sub.3]) < a. Theorem 2.2. Let K be a cone in a real Banach space E. Let [alpha] and [psi] be nonnegative continuous concave functionals on K and [gamma], [beta], and [theta] are nonnegative continuous convex functionals on K such that for some positive numbers c and m, [alpha](x) [less than or equal to] [beta](x) and ||x|| [less than or equal to] m[gamma](x) for all x [member of] [bar.[P([gamma], c)]]. Suppose further that A : [bar.[P([gamma], c)]] [right arrow] K, is completely continuous, A[|.sub.[partial derivative]P([gamma],c)] : [partial derivative]P([gamma], c) [right arrow] [bar.[P([gamma], c)]] and there exist h, d, a, b [greater than or equal to] 0 with 0 < d < a such that each of the following is satisfied: ([C.sub.1]){x [member of] P([gamma], [theta], [alpha], a, b, c)|[alpha](x) > a} [not equal to] [null] and [alpha]([THETA] [o] Ax) > a for x [member of] P([gamma], [theta], [alpha], a, b, c), ([C.sub.2]){x [member of] Q([gamma], [beta], [psi], h, d, c)|[beta](x) < d} [not equal to] [null] and [beta]([THETA] [o] Ax) < d for x [member of] Q([gamma], [beta], [psi], h, d, c), ([C.sub.3])[alpha]([THETA] [o] Ax) > a provided x [member of] P([gamma], [alpha], a, c) with [theta]([THETA] [o] Ax) > b, ([C.sub.4])[beta]([THETA] [o] Ax) < d provided x [member of] Q([gamma], [beta], d, c) with [psi]([THETA] [o] Ax) < h, where [THETA] : K [right arrow] [bar.[P([gamma], c)]] is a contraction contraction, in physics contraction, in physics: see expansion. contraction, in grammar contraction, in writing: see abbreviation. contraction - reduction operator such that [THETA]u = u for u [member of] [bar.[P([gamma], c)]]; [THETA]u [member of] [partial derivative]P([gamma], c) for u [??] [bar.[P([gamma], c)]]. Then A has at least three fixed points [x.sub.1], [x.sub.2], [x.sub.3] [member of] [bar.[P([gamma], c)]] such that [beta]([x.sub.1]) < d,a < [alpha]([x.sub.2]), and d < [beta]([x.sub.3]) with [alpha]([x.sub.3]) < a. Proof. By the definitions of A and [THETA], [THETA] [o] A : [bar.[P([gamma], c)]] [right arrow] [bar.[P([gamma], c)]] is completely continuous. By Theorem 2.1 and ([C.sub.1]) - ([C.sub.4]), [THETA] [o] A has at least three fixed points, i.e., ([THETA] [o] A)[x.sub.i] = [x.sub.i], i = 1, 2, 3. We claim A[x.sub.i] = [x.sub.i]. In fact, if A[x.sub.i] [member of] [bar.[P([gamma], c)]], then ([THETA] [o] A)[x.sub.i] = A[x.sub.i] = [x.sub.i]; if A[x.sub.i] [??] [bar.[P([gamma], c)]], then [x.sub.i] = ([THETA] [o] A)[x.sub.i] [member of] [partial derivative]P([gamma], c). So A[x.sub.i] [member of] [bar.[P([gamma], c)]], a contradiction CONTRADICTION. The incompatibility, contrariety, and evident opposition of two ideas, which are the subject of one and the same proposition. 2. In general, when a party accused of a crime contradicts himself, it is presumed he does so because he is guilty for . Thus A[x.sub.i] [member of] [bar.[P([gamma], c)]] and A[x.sub.i] = [x.sub.i], i = 1, 2, 3. 3. Main Result In this paper we will use the following conditions (H1) the nonlinear term f(t, y) satisfies f(t, y) [less than or equal to] g(y) + h(y) for t [member of] [0, 1] with f continuous on (0, [infinity]), g > 0 continuous, non-increasing on (0, [infinity]) and h continuous on [0, [infinity]), (H2) there exists an [epsilon] > 0 such that f(t, u) is non-increasing in u [less than or equal to] [epsilon] for all t [member of] [0, 1], (H3) [[integral].sub.0.sup.1] e(s)g(ms)ds < [infinity], [for all] m > 0, (H4) for each constant r > 0, there exists a function [[phi].sub.r] continuous on [0, 1] and positive (0, 1) such that f(t, u) [greater than or equal to] [[phi].sub.r](t) on [0, 1] x (0, r]. The main result of this paper is the following Theorem 3.1. Suppose (H1)-(H4) hold. If there exist 0 < [t.sub.1] < [t.sub.2] [less than or equal to] 1/2, 0 < [t.sub.3] [less than or equal to] 1/2, 0 < d < a < [[t.sub.2]/[t.sub.1]]a [less than or equal to] c, such that f(t, u) [greater than or equal to] max{[[[[PHI].sub.p](2a/[t.sub.1])]/[[integral].sub.[t.sub.1].sup.[[[t.sub.1] + [t.sub.2]]/2] e([theta])d[theta]], [[[[PHI].sub.p](2a/[1 - [t.sub.2]])]/[[integral].sub.[[[t.sub.1] + [t.sub.2]]/2].sup.[t.sub.2]] e([theta])d[theta]]]}, (t, u) [member of] [0, 1] x [[t.sub.1]a, [[t.sub.2]/[t.sub.1]]a + 1], (2) [[integral].sub.0.sup.1] e([theta]) (g([alpha][eta](1 - [eta])d[t.sub.3][theta]) + [max.[u[member of][0,d+1]]] h(u)) d[theta] [less than or equal to] [[PHI].sub.p](d), (3) [[integral].sub.0.sup.1] e([theta]) (g([alpha][eta](1 - [eta])c[theta]) + [max.[u[member of][0,c+1]]] h(u)) d[theta] < [[PHI].sub.p](c). (4) Then problem (1.1) has at least three positive solutions [u.sub.i] [member of] [C.sup.1] [0, 1], [[PHI].sub.p][u'.sub.i] [member of] [C.sup.1](0, 1], with [u.sub.i] [member of] [bar.[P([gamma], c)]], i = 1, 2, 3 such that [max.[t[member of][0,1]]] [u.sub.1](t) [less than or equal to] d, [[u.sub.2]([t.sub.1]) + [u.sub.2]([t.sub.2])]/2 [greater than or equal to] a, and [max.[t[member of][0,1]]] [u.sub.3](t) [greater than or equal to] d, with [[u.sub.3]([t.sub.1]) + [u.sub.3]([t.sub.2])]/2 [less than or equal to] a. 4. Related Lemmas This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. and Proof of Theorem 3.1 To prove the main result, we need some lemmas as follows: Lemma lemma (lĕm`ə): see theorem. (logic) lemma - A result already proved, which is needed in the proof of some further result. 4.1. Suppose y : [0, 1] [right arrow] (0, [infinity]) is continuous, 0 [less than or equal to] [alpha] < 1. Then boundary value problem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5) has a unique solution u [member of] [C.sup.1] [0, 1], [[PHI].sub.p]u' [member of] [C.sup.1] [0, 1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6) where [[integral].sub.0.sup.[[sigma].sub.y]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [[integral].sub.[[sigma].sub.y].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [[integral].sub.[[sigma].sub.y].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [[alpha]/[1 - [alpha]]] [[integral].sub.[eta].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds. Proof. Firstly, integrating BVP (4.1) from 0 to s, then integrating again from 0 to t in s, the solution of BVP (4.1) may be represented as : u(t) = [[integral].sub.0.sup.t] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds (7) where [[sigma].sub.y] satisfies [[integral].sub.[eta].sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds + (1 - [alpha]) [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = 0, i.e. [[integral].sub.0.sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds (8) Set F([sigma]) = [[integral].sub.0.sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.[sigma]] y([theta])d[theta]) ds - [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[sigma]] y([theta])d[theta]) ds = (1 - [alpha]) [[integral].sub.0.sup.[eta]] [[phi].sub.q] ([[integral].sub.s.sup.[sigma]] y([theta])d[theta]) ds + [[integral].sub.[eta].sup.1] [[phi].sub.q] ([[integral].sub.s.sup.[sigma]] y([theta])d[theta]) ds. Clearly F([sigma]) is continuous and increasing with respect to [sigma] and F([[sigma].sub.1]) < 0 for [[sigma].sub.1] = 0; F([[sigma].sub.2]) > 0 for [[sigma].sub.2] = 1. (9) So there exists a unique [[sigma].sub.y] [member of] [0, 1] satisfying F([[sigma].sub.y]) = 0. Thus there is a unique solution u to BVP(4.1). If [eta] [less than or equal to] [[sigma].sub.y], by (4.3) (4.4) and t [greater than or equal to] [[sigma].sub.y] we have u(t) = - [[integral].sub.t.sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [[integral].sub.t.sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds. If [eta] [greater than or equal to] [[sigma].sub.y], by (4.3) (4.4) and t [greater than or equal to] [[sigma].sub.y] we have u(t) = - [[integral].sub.t.sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [[integral].sub.t.sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [alpha] [[integral].sub.0.sup.[[sigma].sub.y]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds + [alpha] [[integral].sub.[[sigma].sub.y].sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds. (10) Then [[integral].sub.0.sup.[[sigma].sub.y]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = u([[sigma].sub.y]) = [[integral].sub.[[sigma].sub.y].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [alpha] [[integral].sub.0.sup.[[sigma].sub.y]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds + [alpha] [[integral].sub.[[sigma].sub.y].sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds. i.e., [[integral].sub.0.sup.[[sigma].sub.y]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [1/[1 - [alpha]]] ([[integral].sub.[[sigma].sub.y].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [alpha] [[integral].sub.[[sigma].sub.y].sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds). So (4.6) becomes u(t) = [[integral].sub.t.sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [[alpha]/[1 - [alpha]]] ([[integral].sub.[[sigma].sub.y].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [alpha] [[integral].sub.[[sigma].sub.y].sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds + [alpha] [[integral].sub.[[sigma].sub.y].sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.y]] y([theta])d[theta]) ds = [[integral].sub.t.sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds + [[alpha]/[1 - [alpha]]] [[integral].sub.[eta].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.y].sup.s] y([theta])d[theta]) ds. Thus (4.2) holds. Let E = C[0, 1], ||u|| = [sup.[t[member of][0,1]]] |u(t)| for u [member of] E, K = {u [member of] E : u(t) is nonnegative, concave valued on [0, 1], u(0) = 0, u(1) = [alpha]u([eta])}. Finally on K we define the nonnegative continuous concave functionals [alpha], [psi] and nonnegative continuous convex functionals [beta], [theta], [gamma] by [gamma](u) = [max.[t[member of][0,1]]] u(t), [psi](u) = [min.[t[member of][[t.sub.3],1-[t.sub.3]]]] u(t), [beta](u) = [max.[t[member of][0,1]]] u(t), [alpha](u) = [u([t.sub.1]) + u([t.sub.2])]/2, [theta](u) = [max.[t[member of][[t.sub.1],[t.sub.2]]]] u(t), for 0 < [t.sub.1] < [t.sub.2] [less than or equal to] 1/2, 0 < [t.sub.3] [less than or equal to] 1/2. It is obvious that for each u [member of] P, [alpha](u) [less than or equal to] [beta](u). Lemma 4.2 Let u [member of] K. Then u(t) [greater than or equal to] [alpha][eta](1 - [eta])||u||t, [for all] t [member of] [0, 1]. Proof. Without loss of generality Without loss of generality (abbreviated to WLOG or WOLOG and less commonly stated as without any loss of generality) is a frequently used expression in mathematics. , suppose u([xi]) = u(1). By the concavity con·cav·i·ty n. A hollow or depression that is curved like the inner surface of a sphere. concavity, n 1. the condition of being concave. n 2. of u we have u(t) [greater than or equal to] u([xi])t = u(1)t, t [member of] [0, [xi]]. In addition, u(t) [greater than or equal to] u(1) [greater than or equal to] u(1)t for t [member of] [[xi], 1]. By u [member of] P, we have u(1) = [alpha]u([eta]) [greater than or equal to] [alpha][eta](1 - [eta])||u||. So u(t) [greater than or equal to] [alpha][eta](1 - [eta])||u||t, t [member of] [0, 1]. Consider boundary value problem [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define the operator T by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCTI]. (12) Lemma 4.3. Suppose the conditions in Theorem 3.1 hold, then T has at least three fixed points, [u.sub.in] [member of] [bar.[P([gamma], c)]], i = 1, 2, 3 satisfying [max.[t[member of][0,1]]] [u.sub.1n](t) < d, [[u.sub.2n]([t.sub.1]) + [u.sub.2n]([t.sub.2])]/2 > a, and [max.[t[member of][0,1]]] [u.sub.3n](t) > d, [[u.sub.3n]([t.sub.1]) + [u.sub.3n]([t.sub.2])]/2 < a. Proof. We will finish the proof applying Theorem 2.2. First we show T : [partial derivative]P([gamma], c) [right arrow] [bar.[P([gamma], c)]]. For any u [member of] [partial derivative]P([gamma], c), we have c [greater than or equal to] u(t) [greater than or equal to] [alpha][eta](1 - [eta])||u||t = [alpha][eta](1 - [eta])ct, t [member of] [0, 1], by Lemma 4.2. So [u]* = u. By Lemma 4.1, (H1), (H3) and (3.3) we have [gamma](Tu) = [max.[t[member of][0,1]]] Tu(t) = (Tu)([[sigma].sub.u]) = [[integral].sub.0.sup.[[sigma].sub.u]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds [less than or equal to] [[PHI].sub.q] ([[integral].sub.0.sup.1] e([theta])f([theta], u([theta]) + [1/n])d[theta]) [less than or equal to] [[PHI].sub.q] ([[integral].sub.0.sup.1] e([theta]) (g([alpha][eta](1 - [eta])c[theta]) + [max.[u[member of][0,c+1]]] h(u)) d[theta]) < c So T : [partial derivative]P([gamma], c) [right arrow] [bar.[P([gamma], c)]]. In addition, standard argument shows that T : [bar.[P([gamma], c)]] [right arrow] K is completely continuous. Following we will show ([C.sub.1]) - ([C.sub.4]) hold. Therefore T has at least three fixed points. Let b = [[t.sub.2]/[t.sub.1]]a, h = d[t.sub.3], {u [member of] P([gamma], [theta], [alpha], a, [[t.sub.2]/[t.sub.1]]a, c)|[alpha](u) > a} [not equal to] [null] {u [member of] Q([gamma], [beta], [psi], d[t.sub.3], d, c)|[beta](u) < d} [not equal to] [null]. In the following claims, we verify the remaining conditions of Theorem 2.2. Let ([THETA]u)(t) = min{u(t), c}. So [THETA] : K [right arrow] [bar.[P([gamma], c)]] is a contraction operator. Claim 1. If u [member of] P([gamma], [theta], [alpha], a, [[t.sub.2]/[t.sub.1]]a, c), then [alpha]([THETA] [o] Tu) > a. By u [member of] P([gamma], [theta], [alpha], a, [[t.sub.2]/[t.sub.1]]a, c), we claim u(t) [member of] [[t.sub.1]a, [[t.sub.2]/[t.sub.1]]a] holds for t [member of] [[t.sub.1], [t.sub.2]]. In fact, by u [member of] K we have [min.[t[member of][[t.sub.1],[t.sub.2]]]] u(t) = u([t.sub.1]) [greater than or equal to] [t.sub.1]||u|| [greater than or equal to] [t.sub.1][alpha](u) [greater than or equal to] [t.sub.1]a, for [t.sub.1] < [t.sub.2] [less than or equal to] [[sigma].sub.u]; [min.[t[member of][[t.sub.1],[t.sub.2]]]] u(t) = u([t.sub.2]) [greater than or equal to] (1 - [t.sub.2])||u|| [greater than or equal to] [t.sub.1][alpha](u) [greater than or equal to] [t.sub.1]a for [[sigma].sub.u] [less than or equal to] [t.sub.1] < [t.sub.2]; [min.[t[member of][[t.sub.1],[t.sub.2]]]] u(t) = min{u([t.sub.1]), u([t.sub.2])} [greater than or equal to] [t.sub.1]a for [t.sub.1] [less than or equal to] [[sigma].sub.u] [less than or equal to] [t.sub.2]. When [t.sub.1] [less than or equal to] [t.sub.2] [less than or equal to] [[sigma].sub.u], it follows from (3.1) that [alpha]([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + ([THETA] [o] T)u([t.sub.2])]/2 > ([THETA] [o] T)u([t.sub.1]) = min{[[integral].sub.0.sup.[t.sub.1]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds, c} [greater than or equal to] min{[[integral].sub.0.sup.[t.sub.1]] [[PHI].sub.q] ([[integral].sub.[t.sub.1].sup.[t.sub.2]] e([theta])f([theta], u([theta]) + [1/n])d[theta]) ds, c} [greater than or equal to] min{[t.sub.1][[PHI].sub.q] ([[integral].sub.[t.sub.1].sup.[t.sub.2]] e([theta]) [[[[PHI].sub.p](2a/[t.sub.1])]/[[[integral].sub.[t.sub.1].sup.[[[t.sub.1] + [t.sub.2]]/2]] e(s)ds]]d[theta]), c} > a. When [[sigma].sub.u] [less than or equal to] [t.sub.1] [less than or equal to] [t.sub.2], similar to the above process we obtain [alpha]([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + ([THETA] [o] T)u([t.sub.2])]/2 > ([THETA] [o] T)u([t.sub.2]) > a. When [t.sub.1] [less than or equal to] [[sigma].sub.u] [less than or equal to] [t.sub.2], it follows from (3.1) that [alpha]([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + ([THETA] [o] T)u([t.sub.2])]/2 > [1/2] min{[[integral].sub.0.sup.[t.sub.1]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u]* + [1/n])d[theta]) ds, c} + min{[[integral].sub.[t.sub.2].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u]* + [1/n])d[theta])ds, c} > [1/2] min{[t.sub.1] [[PHI].sub.q] ([[integral].sub.[t.sub.1].sup.[[[t.sub.1] + [t.sub.2]]/2]] e([theta])f([theta], u([theta]) + [1/n])d[theta]), c} or [1/2] min((1 - [t.sub.2])[[PHI].sub.q] ([[integral].sub.[[[t.sub.1] + [t.sub.2]]/2].sup.[t.sub.2]] e([theta])f([theta], u([theta]) + [1/n])d[theta]), c) > a. Claim 2. If u [member of] Q([gamma], [beta], [psi], d[t.sub.3], d, c), then [beta]([THETA] [o] Tu) < a. For u [member of] Q([gamma], [beta], [psi], d[t.sub.3], d, c), we have [beta](u) = [max.[t[member of][0,1]]] u(t) [less than or equal to] d, [psi](u) = [min.[t[member of][[t.sub.3],1 - [t.sub.3]]]] u(t) [greater than or equal to] d[t.sub.3]. It follows from Lemma 4.2 that u(t) [greater than or equal to] [alpha][eta](1 - [eta])||u||t [greater than or equal to] [alpha][eta](1 - [eta])d[t.sub.3]t, t [member of] [0, 1]. So [u]* = u. By (H1) (H3) (3.2) we have [beta] ([THETA] [o] Tu) = [max.[t[member of][0,1]]] ([THETA] [o] T)u(t) = min{[max.[t[member of][0,1]]] Tu(t), c} = min{Tu([[sigma].sub.u]), c} [less than or equal to] min{[[PHI].sub.q] ([[integral].sub.0.sup.1] e([theta])(g([alpha][eta](1 - [eta])d[t.sub.3][theta]) + [max.[u[member of][0,d+1]]] h(u))d[theta]), c} < d. Claim 3. If u [member of] P([gamma], [alpha], a, c) with [theta]([THETA] [o] Tu) > [[t.sub.2]/[t.sub.1]]a, then ([THETA] [o] Tu) > a. For [eta] [less than or equal to] [[sigma].sub.u], [[sigma].sub.u] [less than or equal to] [t.sub.1] we have [alpha]([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + ([THETA] [o] T)u([t.sub.2])]/2 > ([THETA] [o] T)u([t.sub.2]) = min{[[integral].sub.[t.sub.2].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds, c} [greater than or equal to] min{[[1 - [t.sub.2]]/[1 - [t.sub.1]]] [[integral].sub.[t.sub.1].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds, c} = [[1 - [t.sub.2]]/[1 - [t.sub.1]]] [max.[t[member of][[t.sub.1],[t.sub.2]]]] [THETA] [o] Tu(t) = [[1 - [t.sub.2]]/[1 - [t.sub.1]]] [theta]([THETA] [o] Tu) > [[t.sub.1]/[t.sub.2]][theta]([THETA] [o] Tu) > a. For [eta] [less than or equal to] [[sigma].sub.u], [t.sub.2] [less than or equal to] [[sigma].sub.u] we have [alpha] ([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + ([THETA] [o] T)u([t.sub.2])]/2 > ([THETA] [o] T)u([t.sub.1]) = min{[[integral].sub.0.sup.[t.sub.1]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds, c} [greater than or equal to] min{[[t.sub.1]/[t.sub.2]] [[integral].sub.0.sup.[t.sub.2]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds, c} = [[t.sub.1]/[t.sub.2]] [min.[t[member of][[t.sub.1],[t.sub.2]]]] [THETA] [o] Tu(t) = [[t.sub.1]/[t.sub.2]] [theta]([THETA] [o] Tu) > a. For [eta] [less than or equal to] [[sigma].sub.u], [t.sub.1] [less than or equal to] [[sigma].sub.u] [less than or equal to] [t.sub.2], similar to the above process we have [alpha] ([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + ([THETA] [o] T)u([t.sub.2])]/2 > a. For [eta] [greater than or equal to] [[sigma].sub.u], [[sigma].sub.u] [less than or equal to] [t.sub.1] we have [alpha] ([THETA] [o] Tu) = [([THETA] [o] T)u([t.sub.1]) + [THETA] [o] T)u([t.sub.2])]/2 > ([THETA] [o] T)u([t.sub.2]) = min {[[integral].sub.[t.sub.2].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds + [[alpha]/[1 - [alpha]]] [[integral].sub.[eta].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds, c} [greater than or equal to] min {[[1 - [t.sub.2]]/[1 - [t.sub.1]]] [[integral].sub.[t.sub.1].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds + [[alpha]/[1 - [alpha]]] [[integral].sub.[eta].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds, c} [greater than or equal to] [[1 - [t.sub.2]]/[1 - [t.sub.1]]] [max.[t[member of][[t.sub.1],[t.sub.2]]]] [THETA] [o] Tu(t) = [[1 - [t.sub.2]]/[1 - [t.sub.1]]] [theta]([THETA] Tu) > [[t.sub.1]/[t.sub.2]][theta]([THETA] [o] Tu) > a. For the case [eta] [greater than or equal to] [[sigma].sub.u], [t.sub.2] [less than or equal to] [[sigma].sub.u] and [eta] [greater than or equal to] [[sigma].sub.u], [t.sub.1] [less than or equal to] [[sigma].sub.u] [less than or equal to] [t.sub.2]. The proof is similar to [eta] [less than or equal to] [[sigma].sub.u], [t.sub.2] [less than or equal to] [[sigma].sub.u] and [eta] [less than or equal to] [[sigma].sub.u], [t.sub.1] [less than or equal to] [[sigma].sub.u] [less than or equal to] [t.sub.2], respectively. So we omit o·mit tr.v. o·mit·ted, o·mit·ting, o·mits 1. To fail to include or mention; leave out: omit a word. 2. a. To pass over; neglect. b. it here. Claim 4. If u [member of] Q([gamma], [beta], d, c) with [psi] ([THETA] [o] Tu) < d[t.sub.3], then [beta]([THETA] [o] Tu) < d. For u [member of] Q([gamma], [beta], d, c), u(t) [member of] [0, d], t [member of] [0, 1]. So [u]* = u. For [eta] [less than or equal to] [[sigma].sub.u], 1 - [t.sub.3] [less than or equal to] [[sigma].sub.u] we have [beta]([THETA] [o] Tu) = ([THETA] [o] Tu)([[sigma].sub.u]) = min {[[integral].sub.0.sup.[[sigma].sub.u]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds, c} [less than or equal to] min {[[[sigma].sub.u]/[t.sub.3]] [[integral].sub.0.sup.[t.sub.3]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds, c} [less than or equal to] [1/[t.sub.3]] min{[min.[t[member of][[t.sub.3],1-[t.sub.3]]]] Tu(t), c} = [1/[t.sub.3]] [THETA]([psi](Tu)) = [1/[t.sub.3]][psi]([THETA] [o] Tu) < d. For [eta] [less than or equal to] [[sigma].sub.u], [[sigma].sub.u] [less than or equal to] [t.sub.3] [less than or equal to] 1 - [t.sub.3] we have [beta]([THETA] [o] Tu) = ([THETA] [o] Tu)([[sigma].sub.u]) = min{[[integral].sub.[[sigma].sub.u].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds, c} [less than or equal to] min{[[1 - [[sigma].sub.u]]/[t.sub.3]] [[integral].sub.[1-[t.sub.3]].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u].sup.s] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta]) ds, c} = [1/[t.sub.3]] min{[min.[t[member of][[t.sub.3],1-[t.sub.3]]]] Tu(t), c} = [1/[t.sub.3]][psi]([THETA] [o] Tu) < d. For [eta] [less than or equal to] [[sigma].sub.u], [t.sub.3] [less than or equal to] [[sigma].sub.u] [less than or equal to] 1 - [t.sub.3], we have the following two inequlities hold. [[integral].sub.0.sup.[[sigma].sub.u]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds [less than or equal to] [[[sigma].sub.u]/[t.sub.3]] [[integral].sub.0.sup.[t.sub.3]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], u([theta]) + [1/n])d[theta])ds [less than or equal to] [Tu([t.sub.3])]/[t.sub.3] and [[integral].sub.0.sup.[[sigma].sub.u]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds [less than or equal to] [1/[t.sub.3]] Tu(1 - [t.sub.3]). Then [beta]([THETA] [o] [T.sub.1]u) = ([THETA] [o] [T.sub.1]u)([[sigma].sub.u]) = min{[[integral].sub.0.sup.[[sigma].sub.u]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u]] e([theta])f([theta], [u([theta])]* + [1/n])d[theta])ds, c} [less than or equal to] min{[1/[t.sub.3]]Tu([t.sub.3]), [1/[t.sub.3]]Tu(1 - [t.sub.3]), c} [less than or equal to] [1/[t.sub.3]] min{Tu([t.sub.3]), Tu(1 - [t.sub.3]), c} = [1/[t.sub.3]] min{[psi](Tu), c} = [1/[t.sub.3]][psi]([THETA] [o] Tu) < d. For [[sigma].sub.u] [less than or equal to] [eta], 1 - [t.sub.3] [less than or equal to] [[sigma].sub.u], [[sigma].sub.u] [less than or equal to] [eta], [[sigma].sub.u] [less than or equal to] [t.sub.3] and [[sigma].sub.u] [less than or equal to] [eta], [t.sub.3] [less than or equal to] [[sigma].sub.u] [less than or equal to] 1 - [t.sub.3], we can show [beta]([THETA] [o] Tu) < d similarly. Therefore the hypotheses of Theorem 2.2 are satisfied and there exists three fixed points [u.sub.1n], [u.sub.2n], [u.sub.3n] [member of] [bar.[P([gamma], c)]] such that [beta]([u.sub.1n]) = [max.[t[member of][0,1]]] [u.sub.1n](t) < d, [alpha]([u.sub.2n]) = [[u.sub.2n]([t.sub.1]) + [u.sub.2n]([t.sub.2])]/2 > a. (13) [beta]([u.sub.3n]) = [max.[t[member of][0,1]]] [u.sub.3n](t) > d, [alpha]([u.sub.3n]) = [[u.sub.3n]([t.sub.1]) + [u.sub.3n]([t.sub.2])]/2 < a. (14) Lemma 4.5. Set S = {[u.sub.n]|[u.sub.n] is a fixed point of operator T}, A = {[[sigma].sub.u.sub.n]|[[sigma].sub.u.sub.n] [member of] (0, 1), [u'.sub.n]([[sigma].sub.u.sub.n]) = 0, [u.sub.n] is a fixed point of operator T}. Then 0 < in f A [less than or equal to] supA < 1. Proof. For any [u.sub.n] [member of] S, by the conclusion of Lemma 4.1, for [eta] [less than or equal to] [[sigma].sub.u.sub.n], [[sigma].sub.u.sub.n] satisfies [[integral].sub.0.sup.[[sigma].sub.u.sub.n]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u.sub.n]] e([theta])f([theta], [u]* + [1/n])d[theta]) ds = [[integral].sub.[[sigma].sub.u.sub.n].sup.1] [[PHI].sub.q] ([[integral].sub.[[sigma].sub.u.sub.n].sup.s] e([theta])f([theta], [u]* + [1/n])d[theta])ds + [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u.sub.n]] e([theta])f([theta], [u]* + [1/n])d[theta])ds. (15) If [[sigma].sub.u.sub.n] [right arrow] 1, taking limits on both sides of (4.12), we have [[integral].sub.0.sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.1] e([theta])f([theta], [u]* + [1/n])d[theta]) ds = [alpha] [[integral].sub.0.sup.[eta]] [[PHI].sub.q] ([[integral].sub.s.sup.[[sigma].sub.u.sub.n]] e([theta])f([theta], [u]* + [1/n])d[theta]) ds < [[integral].sub.0.sup.1] [[PHI].sub.q] ([[integral].sub.s.sup.1] e([theta])f([theta], [u]* + [1/n])d[theta]) ds, a contradiction. For [[sigma].sub.u.sub.n] [less than or equal to] [eta], similarly we get a contradiction. The proof is completed. Lemma 4.6. Let [u.sub.n] be a fixed point of operator T. Suppose ([H.sub.4]) holds, then there exists m > 0 (independent of n) such that [u.sub.n](t) [greater than or equal to] mt. Proof. Noticing [u.sub.n](0) = 0, u"(0) [less than or equal to] 0, t [member of] [0, 1], we have [u.sub.n](t) [greater than or equal to] [u.sub.n](1)t. So it is sufficient to prove there exists m (independent of m) such that [u.sub.n](1) [greater than or equal to] m. [u.sub.n](1) satisfied [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So there exists m (independent of n) such that [u.sub.n](1) [greater than or equal to] m. The proof is completed. Proof of Theorem 3.1. The proof is achieved in three steps. Step 1. Let the operator T be defined by (4.8) for every n [member of] N. It follows from Lemma 4.4 that T has at least three fixed points [u.sub.in], i = 1, 2, 3 such that [u.sub.in] [member of] [bar.[P([gamma], c)]], i.e. 0 [less than or equal to] [u.sub.in](t) [less than or equal to] c on [0, 1] and satisfied [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16) Step 2. Let S, A be defined as Lemma 4.5, we will show there exists an infinite (mathematics) infinite - 1. Bigger than any natural number. There are various formal set definitions in set theory: a set X is infinite if (i) There is a bijection between X and a proper subset of X. (ii) There is an injection from the set N of natural numbers to X. subset A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. [N.sup.+] of S such that {[u.sub.in]}[.sub.n[member of][N.sup.+]] is a relative compact set of C[0, 1]. From (4.10), (4.11), Lemma 4.2 and Lemma 4.6 we obtain d > [u.sub.1n](t) [greater than or equal to] mt, c [greater than or equal to] [u.sub.3n](t) [greater than or equal to] [alpha][eta](1 - [eta])||[u.sub.3n]||t > [alpha][eta](1 - [eta])dt, c [greater than or equal to] [u.sub.2n](t) [greater than or equal to] [alpha][eta](1 - [eta])||[u.sub.2n]||t [greater than or equal to] [alpha][eta](1 - [eta])[alpha]([u.sub.2n])t > [alpha][eta](1 - [eta])at. Thus {[u.sub.in]}[.sub.[N.sup.+]], i = 1, 2, 3, are uniformly bounded. Without loss of generality, we suppose m't [less than or equal to] [u.sub.in](t) [less than or equal to] c, t [member of] [0, 1], i = 1, 2, 3. Standard argument shows that {[u.sub.in]}[.sub.n[member of][N.sup.+]], i = 1, 2, 3, are equi-continuous family on [0, 1]. The Arzela-Ascoli theorem guarantees the existence of the subsequence sub·se·quence n. 1. Something that is subsequent; a sequel. 2. The fact or quality of being subsequent. 3. Mathematics A sequence that is contained in another sequence. Noun 1. [N.sup.+] of S and a function [u.sub.i] [member of] C[0, 1] with [u.sub.in] converging con·verge v. con·verged, con·verg·ing, con·verg·es v.intr. 1. a. To tend toward or approach an intersecting point: lines that converge. b. uniformly on [0, 1] to [u.sub.i] as n [right arrow] [infinity] through [N.sup.+]. Also [u.sub.i](0) = 0, [u.sub.i](1) = [alpha][u.sub.i]([eta]), i = 1, 2, 3. [u.sub.i](t) > 0 for t [member of] (0, 1]. In addition, mt [less than or equal to] [u.sub.1](t) [less than or equal to] d, [alpha][eta](1 - [eta])at [less than or equal to] [u.sub.2](t) [less than or equal to] c, [alpha][eta](1 - [eta])dt [less than or equal to] [u.sub.3](t) [less than or equal to] c, and [[u.sub.2]([t.sub.1]) + [u.sub.2]([t.sub.2])]/2 [greater than or equal to] a, [max.[t[member of][0,1]]] [u.sub.3](t) [greater than or equal to] d, [[u.sub.3]([t.sub.1]) + [u.sub.3]([t.sub.2])]/2 [less than or equal to] a. Step 3. For any [u.sub.in] [member of] {[u.sub.in]}[.sub.n[member of][N.sub.1]] we have [u.sub.in](t) = [u.sub.in]([xi]) + [[integral].sub.[xi].sup.t] [[PHI].sub.q] ([[PHI].sub.q][u'.sub.in]([xi]) - [[integral].sub.[xi].sup.s] e([theta])f([theta], [u.sub.in]([theta]) + [1/n])d[theta])ds (17) for t [member of] (0, 1)\{[xi]} where [xi] = 1/2, [[integral].sub.[xi].sup.t] [[PHI].sub.q] ([[PHI].sub.p][u'.sub.in]([xi]) - [[integral].sub.[xi].sup.s] e([theta])f([theta], [u.sub.in]([theta]) + [1/n])d[theta]) ds = [u.sub.in](t) - [u.sub.in]([xi]). Fixed t [member of] (0, 1)\{[xi]}. Without loss of generality, suppose t = [t.sub.0]. Let [u.sub.in] [right arrow] [u.sub.i] uniformly on [[xi], t] [union] [t, [xi]]. Following we show [lim lim abbr. Mathematics limit .[n[right arrow][infinity]]] [u'.sub.in]([xi]) = [u'.sub.i]([xi]). [u.sub.in]([t.sub.0]) - [u.sub.in]([xi]) - [u.sub.i]([t.sub.0]) + [u.sub.i]([xi]) = [[integral].sub.[xi].sup.[t.sub.0]] [[PHI].sub.q] ([[PHI].sub.p][u'.sub.in]([xi]) - [[integral].sub.[xi].sup.s] e([theta])f([theta], [u.sub.in]([theta]) + [1/n])d[theta]) - [[PHI].sub.q] ([[PHI].sub.p][u'.sub.i]([xi]) - [[integral].sub.[xi].sup.s] e([theta])f([theta], [u.sub.i]([theta]))d[theta]) ds. By the mean value theorem In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the "average" derivative of the section. for integrals then implies (logic) implies - (=> or a thin right arrow) A binary Boolean function and logical connective. A => B is true unless A is true and B is false. The truth table is A B | A => B ----+------- F F | T F T | T T F | F T T | T It is surprising at first that A => that there exists [[eta].sub.in] [member of] [0, 1] with [u.sub.in]([t.sub.0]) - [u.sub.in]([xi]) - [u.sub.i]([t.sub.0]) + [u.sub.i]([xi]) = [[PHI].sub.q] ([[PHI].sub.p][u'.sub.in]([xi]) - [[integral].sub.[xi].sup.[[eta].sub.in]] e([theta])f([theta], [u.sub.in]([theta]) + [1/n])d[theta]) - [[PHI].sub.q] ([[PHI].sub.p][u'.sub.i]([xi]) - [[PHI].sub.[xi].sup.[[eta].sub.in]] e([theta])f([theta], [u.sub.i]([theta]))d[theta]) and now let [u.sub.in] [right allow] [u.sub.i] uniformly on [[xi], [t.sub.0]] [union] [[t.sub.0], [xi]], we have [lim.[n[right arrow][infinity]]] [u'.sub.in]([xi]) = [u'.sub.i]([xi]). So we have [u.sub.i](t) = [u.sub.i]([xi]) + [[integral].sub.[xi].sup.t] [[PHI].sub.q] ([[PHI].sub.p][u'.sub.i]([xi]) - [[integral].sub.[xi].sup.s] e([theta])f([theta], [u.sub.i]([theta]))d[theta]) ds, t [member of] (0, 1)\{[xi]}. By direct computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. , we have for t [member of] (0, 1)\{[xi]}, ([[PHI].sub.p][u'.sub.i])' + e(t)f(t, [u.sub.i](t)) = 0 (4.15) If we take [xi] = 1/4 in (4.16), in a similar way above we see that (4.15) also holds for t = 1/2. Obviously [u.sub.i](0) = 0, [u.sub.i](1) = [alpha][u.sub.i]([eta]). So [u.sub.i](t), i = 1, 2, 3 are three positive solutions of BVP (1.1). References [1] R. P. Agarwal and D. O'Regan O'Regan can refer to:
Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. 130 (1996), 333-355. [2] R. P. Agarwal and D. O,Regan Regan young girl gruesomely infested with the devil. [Am. Lit.: The Exorcist] See : Possession , Second-order initial value problems of singular type, J. Math. Anal anal (a´n'l) relating to the anus. a·nal adj. 1. Of, relating to, or near the anus. 2. . Appl. 229 (1999), 441-451. [3] R. P. Agarwal, H. Lu and D. O,Regan, Eigenvalues eigenvalues statistical term meaning latent root. and the one-dimensional p-Laplacian, J. Math. Anal. Appl. 266 (2002), 383-400. [4] R. P. Agarwal, Haishen Lu and D. O,Regan, Existence theorems for the one-dimensional p-Laplacian equation with sign changing nonlinearities, Appl. Math. Comput. 143 (2003), 15-38. [5] C. P. Gupta Gupta (g  p`tə), Indian dynasty, A.D. c.320–c.550, whose empire at its height encompassed much of N India. Ancient Indian culture reached a high point during this period. , A note on a second order three-point boundary
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[6] R. Ma, Positive solutions of a nonlinear m-point boundary value problems, J. Math. Anal. Appl. 256 (2001), 556-567. [7] J. Wang (Wang Laboratories, Inc., Lowell, MA) A computer services and network integration company. Wang was one of the major early contributors to the computing industry from its founder's invention that made core memory possible, to leadership in desktop calculators and word processors. and W. Gao, A singular boundary value problem for the one-dimensional p-Laplacian, J. Math. Anal. Appl. 201 (1996), 851-866. [8] Xian Xu, Positive solutions for singular m-point boundary value problems with parameter (1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. , J. Math. Anal. Appl. 291 (2004), 352-367. [9] Q. Yao Yao Various Bantu-speaking peoples inhabiting southern Tanzania, northern Mozambique, and southern Malawi. In the colonial era the Yao were prominent as slave traders. They were never completely united but lived as small groups ruled by chiefs. and H. Lu, Positive solutions of one-dimensional singular p-Laplace equation, Acta acta (ăk`tə), official texts of ancient Rome, written or carved on stone or metal. Usually acta were texts made public, although publication was sometimes restricted. Acta were first posted or carved for general reading c.131 B.C. Math. Sinica 41 (1998), 1255-1264. Ti-An Yu ([dagger]) and Wei-Gao Ge ([double dagger double dagger n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ]) Department of Applied Mathematics, Beijing Institute of Technology Beijing Institute of Technology (BIT,北京理工大学) is a university located in Beijing, People's Republic of China. History Founded in 1940 as Yan'an Academy of Natural Science. Beijing Beijing (bā-jĭng) or Peking (pē-kĭng, pā–), city (1994 est. urban pop. 6,093,300; 1994 est. total pop. 7,240,700), capital of the People's Republic of China. It is in central Hebei prov. 100081, People's Republic People's Republic n. A political organization founded and controlled by a national Communist party. of China Received May 12, 2005, Accepted September September: see month. 9, 2005. * 2000 Mathematics Subject Classification. Primary 34B16; Secondary 34B10. ([dagger]) E-mail: tianyu2992@163.com ([double dagger]) E-mail: gew@bit.edu See .edu. (networking) edu - ("education") The top-level domain for educational establishments in the USA (and some other countries). E.g. "mit.edu". The UK equivalent is "ac.uk". .cn |
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