This paper illustrates a convexity technique to be employed with functional fixed point theorems in proving the existence of solutions to boundary value problems

Avery, Richard I.

Eloe, Paul W.

Henderson, Johnny

English

University of Dayton

University of DaytoneCommonsMathematics Faculty Publications Department of Mathematics2013An example employing convexity in functionalfixed point argumentsRichard I. AveryDakota State UniversityPaul W. EloeUniversity of Dayton, peloe1@udayton.eduJohnny HendersonBaylor UniversityFollow this and additional works at: https://ecommons.udayton.edu/mth_fac_pubPart of the Mathematics CommonsThis Article is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion inMathematics Faculty Publications by an authorized administrator of eCommons. For more information, please contact frice1@udayton.edu,mschlangen1@udayton.edu.eCommons CitationAvery, Richard I.; Eloe, Paul W.; and Henderson, Johnny, "An example employing convexity in functional fixed point arguments"(2013). Mathematics Faculty Publications. 128.https://ecommons.udayton.edu/mth_fac_pub/128INTERNATIONALPUBLICATIONS USACommunications on Applied Nonlinear AnalysisVolume 20(2013), Number 1, 87–96An Example Employing Convexity in FunctionalFixed Point ArgumentsRichard I. AveryDakota State UniversityCollege of Arts and SciencesMadison, South Dakota 57042 USARich.Avery@dsu.eduPaul W. EloeUniversity of DaytonDepartment of MathematicsDayton, Ohio 45469-2316 USAPaul.Eloe@notes.udayton.eduJohnny HendersonBaylor UniversityDepartment of MathematicsWaco, Texas 76798-7328 USAJohnny Henderson@baylor.eduCommunicated by the Editors(Received November 3, 2011; Accepted December 10, 2012)AbstractThis paper illustrates a convexity technique to be employed with functional fixed pointtheorems in proving the existence of solutions to boundary value problems.Key words: Multiple fixed-point theorems, convexity, Leggett-Williams, expansion,compression.AMS Subject Classification: 47H110, 34B151 IntroductionFunctional arguments to prove the existence of solutions to boundary value problems origi-nated with the work of Leggett-Williams [13] when the authors utilized concave functionals8788 R.I. Avery, P.W. Eloe & J. Hendersonin conjunction with the norm. The five functionals fixed point theorem [3] generalized theLeggett-Williams fixed point theorem by providing flexibility in choosing a convex func-tional instead of using the norm when proving the existence of triple positive solutions.Axiomatic index theory has been used to create and extend many functional fixed pointtheorems. Functional fixed point theorems (including [1, 7, 8, 14]) can be traced back toLeggett and Williams [13] when they presented criteria which guaranteed the existence of afixed point for a completely continuous map that did not require the operator to be invariantwith regards to the concave functional boundary of a functional wedge. Avery, Henderson,and O’Regan [5], in a dual of the Leggett-Williams fixed point theorem, gave conditionswhich guaranteed the existence of a fixed point for a completely continuous map that didnot require the operator to be invariant with regards to the convex functional boundary ofa functional wedge.Although there is a lot more flexibility that can be utilized by using the functional fixedpoint theorems when proving existence of solutions to boundary value problems, bound-ing the nonlinear term on intervals by constants as was done in arguments utilizing Kras-nosel’skii’s fixed point theorem [11] and the original Leggett-Williams [13] fixed point theo-rem in the past is still the most prevalant technique. In this paper we provide an example ofan alternative technique using a functional involving the derivative as well as bounding thenonlinearity by functions defined in terms of the nonlinearity. We provide an example that isthe sum of an increasing and decreasing function which is not well suited to being boundedeither above or below on intervals by constants due to the concavity of the nonlinearity.2 PreliminariesIn this section we will state the definitions that are used in the remainder of the paper.Definition 2.1 Let E be a real Banach space. A nonempty closed convex set P ⊂ E iscalled a cone if for all x ∈ P and λ ≥ 0, λx ∈ P and if x,−x ∈ P then x = 0.Every cone P ⊂ E induces an ordering in E given by x ≤ y if and only if y − x ∈ P.Definition 2.2 An operator is called completely continuous if it is continuous and mapsbounded sets into precompact sets.Definition 2.3 A map α is said to be a nonnegative continuous concave functional on acone P of a real Banach space E if α : P → [0,∞) is continuous andα(tx+ (1− t)y) ≥ tα(x) + (1− t)α(y)for all x, y ∈ P and t ∈ [0, 1]. Similarly we say the map β is a nonnegative continuousconvex functional on a cone P of a real Banach space E if β : P → [0,∞) is continuous andβ(tx+ (1− t)y) ≤ tβ(x) + (1 − t)β(y)for all x, y ∈ P and t ∈ [0, 1].Convexity in Functional Arguments 89Let ψ and δ be nonnegative continuous functionals on P ; then, for positive real numbers aand b, we define the following sets:P (ψ, b) = {x ∈ P : ψ(x) < b} and P (ψ, δ, a, b) = {x ∈ P : a < ψ(x) and δ(x) < b}.Theorem 2.4 Let P be a cone in a real Banach space E, α and ψ be nonnegative continuousconcave functionals on P , δ and β be nonnegative continuous convex functionals on P , andT : P → P be a completely continuous operator. If there exist nonnegative numbers a, b, cand d such that(A1) {x ∈ P : a < α(x) and δ(x) < b} 6= ∅;(A2) if x ∈ P with δ(x) = b and α(x) ≥ a, then δ(Tx) < b;(A3) if x ∈ P with δ(x) = b and α(Tx) < a, then δ(Tx) < b;(A4) {x ∈ P : c < ψ(x) and β(x) < d} 6= ∅;(A5) if x ∈ P with ψ(x) = c and β(x) ≤ d, then ψ(Tx) > c;(A6) if x ∈ P with ψ(x) = c and β(Tx) > d, then ψ(Tx) > c;and if(H1) P (δ, b) ⊂ P (ψ, c) with P (ψ, c) being bounded, then T has a fixed point x∗ in P (δ, ψ, b, c);(H2) P (ψ, c) ⊂ P (δ, b) with P (δ, b) being bounded, then T has a fixed point x∗ in P (ψ, δ, c, b).We say that T is LW-inward with respect to P (α, δ, a, b) if the conditions (A1), (A2),and (A3) are satisfied and P (δ, b) is bounded. Similarly we say that T is LW-outward withrespect to P (ψ, β, c, d) if the conditions (A4), (A5), and (A6) are satisfied and P (ψ, c) isbounded.3 Main Result - Using ConvexityIn this section we will illustrate the key techniques for verifying the existence of a positivesolution for a right focal boundary value problem using our main result. Right focal bound-ary value problems have received substantial study for many years. For an early paper, wemention the classical paper by Jackson [10], and for more recent studies, we cite [2] and[4], to name just a couple. For our purposes in this paper, under the expansion condition(H1) we apply the properties of a Green’s function, bound the nonlinearity by functionsderived from the nonlinearity, and demonstrate how to utilize a functional defined in termsof the derivative to obtain the necessary inequalities to apply the compression and expansionfixed point theorem of functional type, Theorem 2.4. To proceed, consider the second-ordernonlinear right focal boundary value problemx′′(t) + f(x(t)) = 0, t ∈ (0, 1), (3.1)x(0) = 0 = x′(1), (3.2)90 R.I. Avery, P.W. Eloe & J. Hendersonwhere f : R → [0,∞) is continuous. If x is continuous on [0, 1] and a fixed point of theoperator T defined byTx(t) :=∫ 10G(t, s)f(x(s))ds,whereG(t, s) = min{t, s}, (t, s) ∈ [0, 1]× [0, 1]is the Green’s function for the operator L defined byLx(t) := −x′′,with right-focal boundary conditionsx(0) = 0 = x′(1),then it is well known that x is a solution of the boundary value problem (3.1), (3.2).Throughout this section of the paper we will use the facts that G(t, s) is nonnegative, andfor each fixed s ∈ [0, 1], the Green’s function is nondecreasing in t, as well as a concavityproperty of the Green’s function which is given bymins∈[0,1]G(y, s)G(w, s)≥ ywfor 0 ≤ y < w ≤ 1. (3.3)Define the cone P ⊂ E = C1[0, 1] byP := {x ∈ E : x is nonnegative, nondecreasing, concave, and x(0) = 0} ,and for x ∈ P , define the concave functionals α and ψ on P byψ(x) := mint∈[12 ,1]x(t) = x(12)and α(x) := mint∈[14 ,1]x(t) = x(14),and define the convex functionals β and δ on P byδ(x) := maxt∈[0,1]x(t) = x(1) and β(x) := maxt∈[12 ,1]x′(t) = x′(12).Thus if x ∈ P , then by the concavity of x we have x (12) ≥ (12)x(1), sincex(12)− x(0)12 − 0≥ x(1)− x(0)1− 0 .Therefore, for all x ∈ P , we have (12)δ(x) ≤ ψ(x). (3.4)In the following theorem, we demonstrate how to apply condition (H1) of Theorem 2.4to prove the existence of at least one positive solution to (3.1), (3.2).Convexity in Functional Arguments 91Theorem 3.1 If m,M , b and c are positive real numbers with b < c, and f : [0, 2c]→ [0,∞)is a continuous function such that(i)∫ 114(s − 14 ) g(s) ds < b2 where g(s) := maxr∈[bs,b]f(r),(ii)∫ 140 s h(s) ds < m where h(s) := maxr∈[2bs,b]f(r) for s ∈ [0, 14],(iii)∫ 114s k(s) ds < b−m where k(s) := maxr∈[ b(2s+1)3 ,b]f(r) for s ∈ [14 , 1],(iv) f(3c2)> 4c+M where |f(x)− f(y)| ≤ M whenever x, y ∈ [c, 2c] and |x− y| ≤ c2 ,then the right focal problem (3.1), (3.2) has at least one positive solution x∗ ∈ P (δ, ψ, b, c).Proof: Let a = b2and d = 2c. For x ∈ P (δ, ψ, b, c), if t ∈ (0, 1), then by the properties ofthe Green’s function(Tx)′′(t) = −f(x(t))andTx(0) = 0 = (Tx)′(1).By the Arzela-Ascoli Theorem it is a standard exercise to show that T is a completelycontinuous operator using the properties of G and f , and for each x ∈ P (δ, ψ, b, c), Tx ∈ P .Therefore we have thatT : P (β, αµ, b, c)→ Pis continuous. Thus by Dugundji’s Theorem, there is a continuous extension, which we willagain denote by T , such that T : P → P (a proof can be found in [6]). We will now verifythat properties (i)− (v) imply that T is LW-inward with respect to P (α, δ, a, b) and that Tis LW-outward with respect to P (ψ, β, c, d).Claim 1: T is LW-inward with respect to P (α, δ, a, b).Define the function xL ∈ P byxL(t) ≡ 5a√t(2−√t)3.Thenα(xL) = xL(14)=5a4> a,andδ(xL) = xL(1) =5a3< 2a = b.Therefore, we have that{x ∈ P : a < α(x) and δ(x) < b} 6= ∅,and if x ∈ P (δ, b), then‖x‖ = δ(x) ≤ b.92 R.I. Avery, P.W. Eloe & J. HendersonThus P (δ, b) is bounded as well.Subclaim 1.1: δ(Tx) < b for all x ∈ P , with δ(x) = b and α(Tx) < a.Let x ∈ P with δ(x) = b and α(Tx) < a. By the concavity of x, we have x(t)−x(0)t−0 ≥x(1)−x(0)1−0 = b. Hence x(t) ≥ bt and since x is increasing with x(1) = b, we have bt ≤ x(t) ≤ b,for all t ∈ [0, 1]. Thusδ(Tx) =∫ 10G(1, s) f(x(s)) ds =∫ 10s f(x(s)) ds=∫ 140s f(x(s)) ds+∫ 114s f(x(s)) ds=∫ 140s f(x(s)) ds+∫ 114(14)f(x(s)) ds+∫ 114(s − 14)f(x(s)) ds=∫ 10G(14, s)f(x(s)) ds +∫ 114(s − 14)f(x(s)) ds= α(Tx) +∫ 114(s− 14)g(s) ds< a+ a = b,and we have verified that δ(Tx) < b.Subclaim 1.2: If x ∈ P with δ(x) = b and α(x) ≥ a, then δ(Tx) < b.Let x ∈ P with δ(x) = b and α(x) ≥ a. Thus by the concavity of x, and since x is increasingwith x(1) = b, we haveas14≤ x(s) ≤ b for s ∈[0,14],anda(s − 14)1− 14+ a ≤ x(s) ≤ b for s ∈[14, 1].Thusδ(Tx) =∫ 10G(1, s) f(x(s)) ds =∫ 10s f(x(s)) ds=∫ 140s f(x(s)) ds+∫ 114s f(x(s)) ds≤∫ 140s h(s) ds+∫ 114s k(s) ds< b,Convexity in Functional Arguments 93and we have verified that δ(Tx) < b.Therefore, we have verified that T is LW-inward with respect to I(α, δ, a, b).Claim 2: T is LW-outward with respect to O(ψ, β, c, d).Define the function xO ∈ P byxO(t) ≡ 3ct(2− t)2 .Thenψ(xO) = xO(12)=9c8> c,andβ(xO) = x′O(1) =3c2< 2c = d.Therefore, we have that{x ∈ P : c < ψ(x) and β(x) < d} 6= ∅,and P (ψ, c) is a bounded subset of the cone P , since if x ∈ P (ψ, c), then by (3.4) we havethat (12)‖x‖ =(12)δ(x) ≤ ψ(x),and so‖x‖ =≤ 2ψ(x) ≤ 2c.Subclaim 2.1: ψ(Tx) > c for all x ∈ P , with ψ(x) = c and β(x) ≤ d.Let x ∈ P with ψ(x) = c and β(x) ≤ d = 2c. Then by the concavity of x for t ∈ (12, 1], wehavex(t)− x (12)t− 12< x′(12)= d = 2c,and since x(12)= ψ(x) = c, we havec ≤ x(t) ≤ 2ctfor t ∈ [12 , 1]. Thus for each t ∈ [12 , 1] ,|x(t)− 3c2| ≤ c2.Therefore,94 R.I. Avery, P.W. Eloe & J. Hendersonψ(Tx) =∫ 10G(12, s)f(x(s)) ds≥∫ 112(12)f(x(s)) ds≥(12)∫ 112f(3c2)−M ds>(12)[(2c+M2)− M2]= c.Subclaim 2.2: ψ(Tx) > c for all x ∈ P , with ψ(x) = c and β(Tx) > d.Let x ∈ P with β(Tx) > d and ψ(x) = c. Thusβ(Tx) = (Tx)′(12)=∫ 112f(x(s)) ds > d = 2c,andψ(Tx) =∫ 10G(12, s) f(x(s)) ds=∫ 120s f(x(s)) ds+∫ 112(12)f(x(s)) ds≥(12)β(Tx) = c.Therefore we have verified that T is LW-outward with respect to O(ψ, β, c, d).Definexsubset(t) ≡∫ 10(b+ c) G(t, s)ds =(b+ c)t(2 − t)2.Then xsubset(t) ∈ P (δ, ψ, b, c), sinceψ(xsubset) = xsubset(12)=(b+ c)(12) (32)2=3(b+ c)8< candδ(xsubset) = xsubset(1) =b+ c2> b.Consequently we have that {x ∈ P : b < δ(x) and ψ(x) < c} 6= ∅, and if x ∈ P (δ, b), thenψ(x) = x(12)≤ x(1) = δ(x) ≤ b < c.Convexity in Functional Arguments 95Thus P (β, b) ⊂ P (ψ, c). Therefore, (H1) of Theorem 2.4 has been satisfied; thus theoperator T has at least one fixed point x∗ ∈ P (δ, ψ, b, c), which is a desired solution of (3.1),(3.2). 2When we have that f is the sum of an increasing and a decreasing function, then de-termining the maximum value of these monotonic functions on a given interval is trivial.In the next corollary we demonstrate that if more restrictive conditions defined using themonotonic functions are satisfied, then the result is even easier to verify and simpler tostate.Corollary 3.2 If b and c are positive real numbers with b < c, M is a positive number,f(x) = f1(x) + f2(x) is a continuous function with f1, f2 : [0, 2c]→ [0,∞), f1 is decreasing,and f2 is increasing such that(i)∫ 114(s − 14 ) (f1(bs) + f2(b)) ds < b2 ,(ii)∫ 140s (f1(2bs) + f2(b)) ds +∫ 114s(f1(b(2s+1)3)+ f2(b))ds < b,(iii) f(3c2)> 4c+M where |f(x)− f(y)| ≤ M whenever x, y ∈ [c, 2c] and |x− y| ≤ c2 ,then the right focal problem (3.1), (3.2) has at least one positive solution x∗ ∈ P (δ, ψ, b, c).The following example illustrates the preceding Corollary.Example: Letf(x) = e−x + x2.Then the boundary value problemx′′ + f(x) = 0, for t ∈ (0, 1),with right focal boundary conditionsx(0) = 0 = x′(1),has at least one positive solution x∗, which can be verified by Corollary 3.2 using b = 1,c = 10, and M = 50.References[1] D. R. Anderson and R. I. Avery, Fixed point theorem of cone expansion and compressionof functional type, J. Difference Equations Appl. 8 (2002), 1073–1083.[2] D. R. Anderson, R. I. Avery and J. Henderson, Functional expansion - compressionfixed point theorem of Leggett-Williams type, Electron. J. Differential Equations 2010(2010), Number 63, 1-9.[3] R. I. Avery, A generalization of the Leggett-Williams fixed point theorem, MSR Hot-Line 3, No.7 (1999), 9–14.96 R.I. Avery, P.W. Eloe & J. Henderson[4] R. I. Avery, J. Henderson and D. R. Anderson, Existence of a positive solution to aright focal boundary value problem, Electron. J. Qual. Theory Differ. Equ. 2010, No.5, 6 pp. (electronic).[5] R. I. Avery, J. Henderson and D. O’Regan, Dual of the compression-expansion fixedpoint theorems, Fixed Point Theory Appl. 2007 (2007), Article ID 90715, 11 pages.[6] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.[7] D. Guo, A new fixed point theorem, Acta Math. Sinica 24 (1981), 444–450.[8] D. Guo, Some fixed point theorems on cone maps, Kexeu Tongbao 29 (1984), 575–578.[9] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, AcademicPress, San Diego, 1988.[10] L. K. Jackson, Boundary value problems for Lipschitz equations. Differential Equations(Proc. Eighth Fall Conf., Oklahoma State Univ., Stillwater, Okla., 1979), pp. 31–50,Academic Press, New York-London-Toronto, Ont., 1980.[11] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, P. Noordhoff, Gronin-gen, The Netherlands, 1964.[12] M. K. Kwong, The topological nature of Krasnosel’skii’s cone fixed point theorem,Nonlinear Anal. 69 (2008), 891–897.[13] R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operatorson ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673–688.[14] J. Sun and G. Zhang, A generalization of the cone expansion and compression fixedpoint theorem and applications, Nonlinear Anal. 67 (2007), 579-586.

An example employing convexity in functional fixed point arguments

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An example employing convexity in functional fixed point arguments

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