This paper examines a three point, nonlocal boundary value problem for a second order ordinary differential equation. We make use of a generalization of the fixed point theorem of compression and expansion of functional type to obtain the existence of positive solutions

Ehrke, J.

Electronic Journal of Qualitative Theory of Differential Equations

Repository of the Academy's Library

Electronic Journal of Qualitative Theory of Differential Equations2009, No. 55, 1-8; http://www.math.u-szeged.hu/ejqtde/POSITIVE SOLUTIONS OF A SECOND-ORDER THREE-POINTBOUNDARY VALUE PROBLEM VIA FUNCTIONALCOMPRESSION-EXPANSIONJOHN EHRKEAbstract. This paper examines a three point, nonlocal boundary value problem for asecond order ordinary differential equation. We make use of a generalization of the fixedpoint theorem of compression and expansion of functional type to obtain the existence ofpositive solutions.1. IntroductionThe literature concerning nonlocal boundary value problems is extensive. Moreover, appli-cations of fixed point theorems to positive solutions of such has sparked much interest, see[9, 10, 11, 12]. Many of the results involving nonlocal boundary value problems are summa-rized in [9]. A more general summary of positive solutions results for a variety of differentialequation types is available in [1]. Our interest lies in the nature of nonlocal problems asbeing the limiting problem of more classical focal problems. For example, we will considerthe second order, three-point BVPx′′(t) + f(x(t)) = 0, t ∈ (0, 1) (1.1)with nonlocal conditionsx(0) = 0 and x(η) = x(1) (1.2)where, η ∈ (0, 1) is fixed. With the BVP (1.1),(1.2) we associate the right focal BVP (1.1)with boundary conditions x(0) = 0, x′(1) = 0. We say that this is the limiting problem for(1.1),(1.2) in the sense that the “limit” occurs when we think of η → 1−. Such problemsand even characterizations are not new. For example, in [9] it is mentioned that for suchproblems often times numerical and experimental computation of x′(1) is more difficultthan that ofx(η)− x(1)η − 1. It is for this reason that we are keen to investigate problemsof this type. We will make use of some newer results concerning the Green’s function ofsuch problems in guaranteeing the existence of positive solutions for (1.1), (1.2). This articledifferentiates itself from these results in the application of a functional expansion-compressionargument to obtain positive solutions, due [5]. Where much of the current literature usesnorm based, Krasnosel’skii fixed point results to obtain positive solutions for the cases whenf is superlinear or sublinear, we will only assume that f : R → [0,∞) is continuous and willEJQTDE, 2009 No. 55, p. 1be using a functional approach highlighted in [2, 3, 5]. Remarks concerning a larger class ofproblems will be given at the end.2. PreliminariesOur results are based on the functional compression-expansion fixed point theorem, due [5].We will list some preliminary material to be used later. We begin by defining a set of proper-ties to which our functionals must adhere. Throughout this paper we assume 0 < η ≤ q < 1are fixed.Definition 1. A nonempty closed convex set P ⊂ E, real Banach space, is called a cone ifit satisfies each of following conditions:(i) ∀x ∈ P , λ ≥ 0, λx ∈ P ,(ii) ∀x ∈ P , −x ∈ P , implies x = 0.Definition 2. A map α is said to be a nonnegative, continuous, concave functional on acone P ⊂ E if α : P → [0,∞) is continuous and satisfies the concave relationshipα(tx+ (1− t)y) ≥ tα(x) + (1− t)α(y),∀x, y ∈ P and t ∈ [0, 1].Definition 3. A map γ is said to be a nonnegative, continuous, convex functional on a coneP ⊂ E if γ : P → [0,∞) is continuous and satisfies the convex relationshipγ(tx+ (1− t)y) ≤ tγ(x) + (1− t)γ(y),∀x, y ∈ P and t ∈ [0, 1].Definition 4. A map ψ is said to be a sublinear functional ifψ(tx) ≤ tψ(x),∀x ∈ P , t ∈ [0, 1].Property A1. Let P be a cone in a real Banach space E and Ω be a bounded open subsetof E with 0 ∈ Ω. Then a continuous functional β : P → [0,∞) is said to satisfy propertyA1 if one of the following conditions hold:(i) β is convex, β(0) = 0, β(x) 6= 0 if x 6= 0 and infx∈P∩∂Ωβ(x) > 0,(ii) β is sublinear, β(0) = 0, β(x) 6= 0 if x 6= 0, and infx∈P∩∂Ωβ(x) > 0,EJQTDE, 2009 No. 55, p. 2(iii) β is concave and unbounded.Property A2. Let P be a cone in a real Banach space E and Ω be a bounded open subsetof E with 0 ∈ Ω. Then a continuous functional β : P → [0,∞) is said to satisfy propertyA2 if one of the following conditions hold:(i) β is convex, β(0) = 0 and β(x) 6= 0, if x 6= 0,(ii) β is sublinear, β(0) = 0 and β(x) 6= 0 if x 6= 0,(iii) β(x+ y) ≥ β(x) + β(y) for all, x, y ∈ P , β(0) = 0, β(x) 6= 0 if x 6= 0.Theorem 1. Let Ω1 and Ω2 be two bounded open sets in a Banach space E such that 0 ∈ Ω1and Ω1 ⊆ Ω2 and P is a cone in E. Suppose A : P ∩(Ω2−Ω1)→ P is completely continuous,α and γ are non-negative continuous functionals on P , and one of the two conditions:(C1) α satisfies Property A1 with α(Ax) ≥ α(x), for all x ∈ P ∩ ∂Ω1, and γ satisfiesProperty A2 with γ(Ax) ≤ γ(x), for all x ∈ P ∩ ∂Ω2(C2) α satisfies Property A2 with α(Ax) ≤ α(x), for all x ∈ P ∩ ∂Ω1, and γ satisfiesProperty A1 with γ(Ax) ≥ γ(x), for all x ∈ P ∩ ∂Ω2,is satisfied. Then A has at least one fixed point in P ∩ (Ω2 − Ω1).We will obtain positive solutions for (1.1), (1.2) by looking for a fixed point of a completelycontinuous operator which satisfies certain properties. A solution to (1.1), (1.2) is given byx(t) =∫ 10G∗(t, s)f(x(s)) ds (2.1)where G∗(t, s) is the Green’s function for the operator Lx := −x′′ = 0 satisfying boundaryconditions (1.2). Recent results in the literature, though are characterizing Green’s functionsfor nonlocal problems as being part local and part nonlocal. For example, throughout thispaper we will think of G∗(t, s) as being given byG∗(t, s) = G(t, s) +11− ηG(η, s) (2.2)where G(t, s) is the Green’s function for the problem with local conditions x(0) = x(1) = 0,given byG(t, s) :={t(1− s) t ≤ ss(1− t) s ≤ t(2.3)EJQTDE, 2009 No. 55, p. 3Some useful properties of the Green’s function in this case are:(G1)∫ 10G(t, s) ds =t(1− t)2for 0 ≤ t, s ≤ 1(G2) G∗(t, s) ≥ 0 for 0 ≤ t, s ≤ 1(G3) G∗(t, s) ≤ G∗(s, s) for 0 ≤ t, s ≤ 1(G4) G∗(t, s) ≤2− η1− ηfor 0 ≤ t, s ≤ 1The construction of Green’s functions for multi-point problems, using two-point BVP Green’sfunctions is considered in various places in the literature. The benefits of this approach aretwo-fold. We can easily make use of the positivity of the Green’s function in this case andresults using these Green’s function methods are more easily generalizable to higher order(or larger multiple-point) results.3. Main ResultWe will make use of the generalization of the fixed point theorem of compression and expan-sion of functional type to obtain positive solutions, but first we define the parameters underwhich we will satisfy the theorem.Let E = C[0, 1] endowed with the maximum norm be our Banach space. We define our coneP as follows:P :=x is non-decreasing on [0, q]x ∈ E x is non-increasing on [q, 1]x is non-negative on [0, 1]x(0) = 0. (3.1)The functionals which will satisfy the requirements of the theorem are given byα(x) = mint∈[η,1]x(t) = x(η) = x(1) (3.2)andγ(x) = maxt∈[0,1]x(t) = ‖x‖ = x(q) (3.3)which satisfy properties A1(iii) and A2(i), respectively. We will consider the setsΩ1 = {x : γ(x) < r}, Ω2 = {x : α(x) < R} (3.4)EJQTDE, 2009 No. 55, p. 4as well. We are now ready to impose growth conditions on f(x) to apply the theorem.Theorem 2. Suppose there exists positive numbers r and R such that 0 < r < ηR, andsuppose that f satisfies the following growth conditions:(i) f(x) ≥2Rη(2− η)for all x ∈ [0, R].(ii) f(x) ≤2rq(2− q)for all x ∈ [r, R].Then the second order BVP, (1.1), (1.2) has at least one positive solution in P ∩ (Ω2 −Ω1).Proof. We will verify each of the hypotheses of the functional compression-expansion theoremto obtain the result.Claim 1. A : P → P is a completely continuous operator.Standard arguments show that A is completely continuous. Let x ∈ P . Then by theproperties of G∗(t, s), Ax(t) ≥ 0, (Ax)′′(t) = −f(x(t)) ≤ 0. Moreover, there exists aq ∈ [0, 1] such that by the properties of∂G∗(t, s)∂t={1− s, t ≤ s−s, s ≤ tAx is non-decreasing on [0, q] and non-increasing on [q, 1]. Hence, Ax ∈ P giving A : P → P .Claim 2. If x ∈ P ∩ ∂Ω1, then γ(Ax) ≤ γ(x).We employ properties of the Green’s function as well as our growth conditions on f toresolve this claim:EJQTDE, 2009 No. 55, p. 5γ(Ax) = Ax(q)=∫ 10G∗(q, s)f(x(s)) ds=∫ 10G(q, s)f(x(s)) ds+11− η∫ 10G(η, s)f(x(s)) ds≤(2rq(2− q))[∫ 10G(q, s) ds+11− η∫ 10G(η, s) ds]=(2rq(2− q))(q(1− q)2+11− η·η(1− η)2)≤(2rq(2− q))(q(1− q)2+q2)=(2rq(2− q))(q(2− q)2)= r = γ(x)Claim 3. If x ∈ P ∩ ∂Ω2, then α(Ax) ≥ α(x).As above we use the Green’s function properties and growth conditions on f to resolvethis claim:α(Ax) = Ax(η)=∫ 10G∗(n, s)f(x(s)) ds=∫ 10G(η, s)f(x(s)) ds+11− η∫ 10G(η, s)f(x(s)) ds=(2− η1− η)∫ 10G(η, s)f(x(s)) ds≥(2− η1− η)·(2Rη(2− η))∫ 10G(η, s) ds= R = α(x)EJQTDE, 2009 No. 55, p. 6Having satisfied the hypotheses for the functional compression-expansion theorem for appro-priately conditioned functionals, α and γ, we can conclude that the operator A has a fixedpoint in Ω2 − Ω1.4. RemarksSimilar results have been proved using fixed point theorems based on the Krasnosel’skiiresult. Echoing the sentiment expressed in [2, 3, 5] these arguments greatly simplify theresults of [9, 12] as well as others in establishing existence results for nonlocal boundaryvalue problems. Let us consider for a moment the right focal “limit” problem (1.1) withboundary conditions x(0) = x′(1) = 0. In [3], this problem is considered where the growthconditions for the nonlinearity look like(i) f(x) ≥ 2R, x ∈ [0, R](ii) f(x) ≤rη(1− η), x ∈ [r, R].Notice that as η → 1− the growth conditions, (i) and (ii), coincide. Likewise when weexamine the BVP considered in [5] with boundary conditions x(0) = x(1) = 0 we note thatthis problem is related to our result for when η → 0+ and q → 1/2. In this case, it isinteresting to note the growth conditions for the nonlinearity look like(i) f(x) ≥ 16R, x ∈ [R, 2R](ii) f(x) ≤ 8r, x ∈ [0, r].As above it is interesting to note that in our result above if you let q → 1/2 then growthcondition (ii) can be replaced by f(x) ≤ 83r which is very similar to what is obtained in[5]. The motivation for the work in this regard is a unifying theorem that simultaneouslygives existence results for all three problem types. As seen in this paper these ideas are notwithout some foundation. The additional freedom in choosing functionals provided by theresult of [5] makes this possible and deserves more attention in the future.References[1] Agarwal, R.P., O’Regan, D., Wong, P.; Positive Solutions of Differential, Difference, and Integral Equa-tions, Kluwer Acad. Publ., Boston, 1999.[2] Anderson, D., Avery R.; Fixed point theorem of cone expansion and compression of functional type,Journal of Difference Equations and Applications, 8 (2002), 1073-1083.[3] Anderson, D., Avery R., Krueger, R.; An extension of the fixed point theorem of cone expansion andcompression of functional type, Comm. on Appl. Nonl. Anal., 12 (2003), 83-94.EJQTDE, 2009 No. 55, p. 7[4] Anderson, D., Avery R.; A Fourth-Order, Four-Point Right Focal Boundary Value Problem, RockyMountain Journal of Mathematics, 36 (2006), 367-380.[5] Avery, R., Henderson, J., O’Regan, D.; Functional Compression-Expansion Fixed Point Theorem, Elec-tronic Journal of Differential Equations, 22 (2008), 1-12.[6] Avery, R., Henderson, J., O’Regan, D.; Four functionals fixed point theorem, Mathematical and Com-puter Modelling, 48 (2008), 1081-1089.[7] Guo, D., Lakshmikantham, V.; Nonlinear problems in abstract cones, Academic Press, San Diego, 1988.[8] Leggett, R., Williams, L.; Multiple Positive Fixed Points of Nonlinear Operators on Ordered BanachSpaces, Indiana University Mathematics Journal, 28 (1979), 673-688.[9] Ma, R.; A Survey On Nonlocal Boundary Value Problems, Applied Mathematics E-Notes, 7 (2007),257-279.[10] Ma, R.; Existence of positive solutions for superlinear semipositone m-point boundary value problems,Proc. Edinburgh Math. Soc., 46 (2003), 279-292.[11] Ma, R., Thompson, B.; Positive solutions of nonlinear three point boundary value problem, J. Math.Anal. Appl., 279 (2003), 1216-1227.[12] Sun, J., Wei, J.; Positive Solutions to Systems of Nonlinear Second-Order Three-Point Boundary ValueProblems, Applied Mathematics E-Notes, 9 (2009), 55-62.(Received July 14, 2009)Abilene Christian University, Abilene, TX, USA.E-mail address : john.ehrke@acu.eduEJQTDE, 2009 No. 55, p. 8

Positive solutions of a second-order three-point boundary value problem via functional compression-expansion

John Ehrke

Crossref

Electronic journal of qualitative theory of differential equations

J. Ehrke

Directory of Open Access Journals

Ehrke John

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2009, No. 55, 1-8; http://www.math.u-szeged.hu/ejqtde/POSITIVE SOLUTIONS OF A SECOND-ORDER THREE-POINTBOUNDARY VALUE PROBLEM VIA FUNCTIONALCOMPRESSION-EXPANSIONJOHN EHRKEAbstract. This paper examines a three point, nonlocal boundary value problem for asecond order ordinary differential equation. We make use of a generalization of the fixedpoint theorem of compression and expansion of functional type to obtain the existence ofpositive solutions.1. IntroductionThe literature concerning nonlocal boundary value problems is extensive. Moreover, appli-cations of fixed point theorems to positive solutions of such has sparked much interest, see[9, 10, 11, 12]. Many of the results involving nonlocal boundary value problems are summa-rized in [9]. A more general summary of positive solutions results for a variety of differentialequation types is available in [1]. Our interest lies in the nature of nonlocal problems asbeing the limiting problem of more classical focal problems. For example, we will considerthe second order, three-point BVPx′′(t) + f(x(t)) = 0, t ∈ (0, 1) (1.1)with nonlocal conditionsx(0) = 0 and x(η) = x(1) (1.2)where, η ∈ (0, 1) is fixed. With the BVP (1.1),(1.2) we associate the right focal BVP (1.1)with boundary conditions x(0) = 0, x′(1) = 0. We say that this is the limiting problem for(1.1),(1.2) in the sense that the “limit” occurs when we think of η → 1−. Such problemsand even characterizations are not new. For example, in [9] it is mentioned that for suchproblems often times numerical and experimental computation of x′(1) is more difficultthan that ofx(η) − x(1)η − 1. It is for this reason that we are keen to investigate problemsof this type. We will make use of some newer results concerning the Green’s function ofsuch problems in guaranteeing the existence of positive solutions for (1.1), (1.2). This articledifferentiates itself from these results in the application of a functional expansion-compressionargument to obtain positive solutions, due [5]. Where much of the current literature usesnorm based, Krasnosel’skii fixed point results to obtain positive solutions for the cases whenf is superlinear or sublinear, we will only assume that f : R → [0,∞) is continuous and willEJQTDE, 2009 No. 55, p. 1be using a functional approach highlighted in [2, 3, 5]. Remarks concerning a larger class ofproblems will be given at the end.2. PreliminariesOur results are based on the functional compression-expansion fixed point theorem, due [5].We will list some preliminary material to be used later. We begin by defining a set of proper-ties to which our functionals must adhere. Throughout this paper we assume 0 < η ≤ q < 1are fixed.Definition 1. A nonempty closed convex set P ⊂ E, real Banach space, is called a cone ifit satisfies each of following conditions:(i) ∀x ∈ P , λ ≥ 0, λx ∈ P ,(ii) ∀x ∈ P , −x ∈ P , implies x = 0.Definition 2. A map α is said to be a nonnegative, continuous, concave functional on acone P ⊂ E if α : P → [0,∞) is continuous and satisfies the concave relationshipα(tx+ (1 − t)y) ≥ tα(x) + (1 − t)α(y),∀x, y ∈ P and t ∈ [0, 1].Definition 3. A map γ is said to be a nonnegative, continuous, convex functional on a coneP ⊂ E if γ : P → [0,∞) is continuous and satisfies the convex relationshipγ(tx+ (1 − t)y) ≤ tγ(x) + (1 − t)γ(y),∀x, y ∈ P and t ∈ [0, 1].Definition 4. A map ψ is said to be a sublinear functional ifψ(tx) ≤ tψ(x),∀x ∈ P , t ∈ [0, 1].Property A1. Let P be a cone in a real Banach space E and Ω be a bounded open subsetof E with 0 ∈ Ω. Then a continuous functional β : P → [0,∞) is said to satisfy propertyA1 if one of the following conditions hold:(i) β is convex, β(0) = 0, β(x) 6= 0 if x 6= 0 and infx∈P∩∂Ωβ(x) > 0,(ii) β is sublinear, β(0) = 0, β(x) 6= 0 if x 6= 0, and infx∈P∩∂Ωβ(x) > 0,EJQTDE, 2009 No. 55, p. 2(iii) β is concave and unbounded.Property A2. Let P be a cone in a real Banach space E and Ω be a bounded open subsetof E with 0 ∈ Ω. Then a continuous functional β : P → [0,∞) is said to satisfy propertyA2 if one of the following conditions hold:(i) β is convex, β(0) = 0 and β(x) 6= 0, if x 6= 0,(ii) β is sublinear, β(0) = 0 and β(x) 6= 0 if x 6= 0,(iii) β(x+ y) ≥ β(x) + β(y) for all, x, y ∈ P , β(0) = 0, β(x) 6= 0 if x 6= 0.Theorem 1. Let Ω1 and Ω2 be two bounded open sets in a Banach space E such that 0 ∈ Ω1and Ω1 ⊆ Ω2 and P is a cone in E. Suppose A : P ∩(Ω2−Ω1) → P is completely continuous,α and γ are non-negative continuous functionals on P , and one of the two conditions:(C1) α satisfies Property A1 with α(Ax) ≥ α(x), for all x ∈ P ∩ ∂Ω1, and γ satisfiesProperty A2 with γ(Ax) ≤ γ(x), for all x ∈ P ∩ ∂Ω2(C2) α satisfies Property A2 with α(Ax) ≤ α(x), for all x ∈ P ∩ ∂Ω1, and γ satisfiesProperty A1 with γ(Ax) ≥ γ(x), for all x ∈ P ∩ ∂Ω2,is satisfied. Then A has at least one fixed point in P ∩ (Ω2 − Ω1).We will obtain positive solutions for (1.1), (1.2) by looking for a fixed point of a completelycontinuous operator which satisfies certain properties. A solution to (1.1), (1.2) is given byx(t) =∫ 10G∗(t, s)f(x(s)) ds (2.1)where G∗(t, s) is the Green’s function for the operator Lx := −x′′ = 0 satisfying boundaryconditions (1.2). Recent results in the literature, though are characterizing Green’s functionsfor nonlocal problems as being part local and part nonlocal. For example, throughout thispaper we will think of G∗(t, s) as being given byG∗(t, s) = G(t, s) +11 − ηG(η, s) (2.2)where G(t, s) is the Green’s function for the problem with local conditions x(0) = x(1) = 0,given byG(t, s) :={t(1 − s) t ≤ ss(1 − t) s ≤ t(2.3)EJQTDE, 2009 No. 55, p. 3Some useful properties of the Green’s function in this case are:(G1)∫ 10G(t, s) ds =t(1 − t)2for 0 ≤ t, s ≤ 1(G2) G∗(t, s) ≥ 0 for 0 ≤ t, s ≤ 1(G3) G∗(t, s) ≤ G∗(s, s) for 0 ≤ t, s ≤ 1(G4) G∗(t, s) ≤2 − η1 − ηfor 0 ≤ t, s ≤ 1The construction of Green’s functions for multi-point problems, using two-point BVP Green’sfunctions is considered in various places in the literature. The benefits of this approach aretwo-fold. We can easily make use of the positivity of the Green’s function in this case andresults using these Green’s function methods are more easily generalizable to higher order(or larger multiple-point) results.3. Main ResultWe will make use of the generalization of the fixed point theorem of compression and expan-sion of functional type to obtain positive solutions, but first we define the parameters underwhich we will satisfy the theorem.Let E = C[0, 1] endowed with the maximum norm be our Banach space. We define our coneP as follows:P :=x is non-decreasing on [0, q]x ∈ E x is non-increasing on [q, 1]x is non-negative on [0, 1]x(0) = 0.(3.1)The functionals which will satisfy the requirements of the theorem are given byα(x) = mint∈[η,1]x(t) = x(η) = x(1) (3.2)andγ(x) = maxt∈[0,1]x(t) = ‖x‖ = x(q) (3.3)which satisfy properties A1(iii) and A2(i), respectively. We will consider the setsΩ1 = {x : γ(x) < r}, Ω2 = {x : α(x) < R} (3.4)EJQTDE, 2009 No. 55, p. 4as well. We are now ready to impose growth conditions on f(x) to apply the theorem.Theorem 2. Suppose there exists positive numbers r and R such that 0 < r < ηR, andsuppose that f satisfies the following growth conditions:(i) f(x) ≥2Rη(2 − η)for all x ∈ [0, R].(ii) f(x) ≤2rq(2 − q)for all x ∈ [r, R].Then the second order BVP, (1.1), (1.2) has at least one positive solution in P ∩ (Ω2 −Ω1).Proof. We will verify each of the hypotheses of the functional compression-expansion theoremto obtain the result.Claim 1. A : P → P is a completely continuous operator.Standard arguments show that A is completely continuous. Let x ∈ P . Then by theproperties of G∗(t, s), Ax(t) ≥ 0, (Ax)′′(t) = −f(x(t)) ≤ 0. Moreover, there exists aq ∈ [0, 1] such that by the properties of∂G∗(t, s)∂t={1 − s, t ≤ s−s, s ≤ tAx is non-decreasing on [0, q] and non-increasing on [q, 1]. Hence, Ax ∈ P giving A : P → P .Claim 2. If x ∈ P ∩ ∂Ω1, then γ(Ax) ≤ γ(x).We employ properties of the Green’s function as well as our growth conditions on f toresolve this claim:EJQTDE, 2009 No. 55, p. 5γ(Ax) = Ax(q)=∫ 10G∗(q, s)f(x(s)) ds=∫ 10G(q, s)f(x(s)) ds+11 − η∫ 10G(η, s)f(x(s)) ds≤(2rq(2 − q)) [∫ 10G(q, s) ds+11 − η∫ 10G(η, s) ds]=(2rq(2 − q)) (q(1 − q)2+11 − η·η(1 − η)2)≤(2rq(2 − q)) (q(1 − q)2+q2)=(2rq(2 − q)) (q(2 − q)2)= r = γ(x)Claim 3. If x ∈ P ∩ ∂Ω2, then α(Ax) ≥ α(x).As above we use the Green’s function properties and growth conditions on f to resolvethis claim:α(Ax) = Ax(η)=∫ 10G∗(n, s)f(x(s)) ds=∫ 10G(η, s)f(x(s)) ds+11 − η∫ 10G(η, s)f(x(s)) ds=(2 − η1 − η)∫ 10G(η, s)f(x(s)) ds≥(2 − η1 − η)·(2Rη(2 − η))∫ 10G(η, s) ds= R = α(x)EJQTDE, 2009 No. 55, p. 6Having satisfied the hypotheses for the functional compression-expansion theorem for appro-priately conditioned functionals, α and γ, we can conclude that the operator A has a fixedpoint in Ω2 − Ω1.4. RemarksSimilar results have been proved using fixed point theorems based on the Krasnosel’skiiresult. Echoing the sentiment expressed in [2, 3, 5] these arguments greatly simplify theresults of [9, 12] as well as others in establishing existence results for nonlocal boundaryvalue problems. Let us consider for a moment the right focal “limit” problem (1.1) withboundary conditions x(0) = x′(1) = 0. In [3], this problem is considered where the growthconditions for the nonlinearity look like(i) f(x) ≥ 2R, x ∈ [0, R](ii) f(x) ≤rη(1 − η), x ∈ [r, R].Notice that as η → 1− the growth conditions, (i) and (ii), coincide. Likewise when weexamine the BVP considered in [5] with boundary conditions x(0) = x(1) = 0 we note thatthis problem is related to our result for when η → 0+ and q → 1/2. In this case, it isinteresting to note the growth conditions for the nonlinearity look like(i) f(x) ≥ 16R, x ∈ [R, 2R](ii) f(x) ≤ 8r, x ∈ [0, r].As above it is interesting to note that in our result above if you let q → 1/2 then growthcondition (ii) can be replaced by f(x) ≤ 83r which is very similar to what is obtained in[5]. The motivation for the work in this regard is a unifying theorem that simultaneouslygives existence results for all three problem types. As seen in this paper these ideas are notwithout some foundation. The additional freedom in choosing functionals provided by theresult of [5] makes this possible and deserves more attention in the future.References[1] Agarwal, R.P., O’Regan, D., Wong, P.; Positive Solutions of Differential, Difference, and Integral Equa-tions, Kluwer Acad. Publ., Boston, 1999.[2] Anderson, D., Avery R.; Fixed point theorem of cone expansion and compression of functional type,Journal of Difference Equations and Applications, 8 (2002), 1073-1083.[3] Anderson, D., Avery R., Krueger, R.; An extension of the fixed point theorem of cone expansion andcompression of functional type, Comm. on Appl. Nonl. Anal., 12 (2003), 83-94.EJQTDE, 2009 No. 55, p. 7[4] Anderson, D., Avery R.; A Fourth-Order, Four-Point Right Focal Boundary Value Problem, RockyMountain Journal of Mathematics, 36 (2006), 367-380.[5] Avery, R., Henderson, J., O’Regan, D.; Functional Compression-Expansion Fixed Point Theorem, Elec-tronic Journal of Differential Equations, 22 (2008), 1-12.[6] Avery, R., Henderson, J., O’Regan, D.; Four functionals fixed point theorem, Mathematical and Com-puter Modelling, 48 (2008), 1081-1089.[7] Guo, D., Lakshmikantham, V.; Nonlinear problems in abstract cones, Academic Press, San Diego, 1988.[8] Leggett, R., Williams, L.; Multiple Positive Fixed Points of Nonlinear Operators on Ordered BanachSpaces, Indiana University Mathematics Journal, 28 (1979), 673-688.[9] Ma, R.; A Survey On Nonlocal Boundary Value Problems, Applied Mathematics E-Notes, 7 (2007),257-279.[10] Ma, R.; Existence of positive solutions for superlinear semipositone m-point boundary value problems,Proc. Edinburgh Math. Soc., 46 (2003), 279-292.[11] Ma, R., Thompson, B.; Positive solutions of nonlinear three point boundary value problem, J. Math.Anal. Appl., 279 (2003), 1216-1227.[12] Sun, J., Wei, J.; Positive Solutions to Systems of Nonlinear Second-Order Three-Point Boundary ValueProblems, Applied Mathematics E-Notes, 9 (2009), 55-62.(Received July 14, 2009)Abilene Christian University, Abilene, TX, USA.E-mail address : john.ehrke@acu.eduEJQTDE, 2009 No. 55, p. 8

Positive solutions of a second-order three-point boundary value problem via functional compressions-expansion

http://acta.bibl.u-szeged.hu/42205/1/ejqtde_2009_055_001-008.pdf

Positive solutions of a second-order three-point boundary value problem via functional compression-expansion

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library

Crossref

Directory of Open Access Journals

University of Szeged