The standard methods of applying iterative techniques do not apply when the nonlinear term is neither monotonic (corresponding to an increasing or decreasing operator) nor Lipschitz (corresponding to a condensing operator). However, by applying the Layered Compression–Expansion Theorem in conjunction with an alternative inversion technique, we show how one can apply monotonicity techniques to a right focal boundary value problem

Anderson, Douglas R.

Avery, Richard I.

Henderson, Johnny

Avery Richard I.

Anderson Douglas R.

Henderson Johnny

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2019, No. 51, 1–7; https://doi.org/10.14232/ejqtde.2019.1.51 www.math.u-szeged.hu/ejqtde/Alternative iterative techniqueRichard I. AveryB 1, Douglas R. Anderson2 and Johnny Henderson31Dakota State University, College of Arts and Sciences, Madison, South Dakota 57042 USA2Concordia College, Department of Mathematics, Moorhead, MN 56562 USA3Baylor University, Department of Mathematics, Waco, Texas 76798 USAReceived 19 February 2019, appeared 5 August 2019Communicated by Paul EloeAbstract. The standard methods of applying iterative techniques do not apply whenthe nonlinear term is neither monotonic (corresponding to an increasing or decreasingoperator) nor Lipschitz (corresponding to a condensing operator). However, by apply-ing the Layered Compression–Expansion Theorem in conjunction with an alternativeinversion technique, we show how one can apply monotonicity techniques to a rightfocal boundary value problem.Keywords: iteration, alternate inversion, layered, sum of operators, right focal problem.2010 Mathematics Subject Classification: 47H10, 34B15.1 IntroductionIn the Layered Compression–Expansion Fixed Point Theorem [1] it was shown that, if theoperator T = R + S and one could find r∗ and s∗ such thatR(r∗ + s∗) = r∗ and S(r∗ + s∗) = s∗,then x∗ = r∗ + s∗ is a fixed point of T, sinceTx∗ = Rx∗ + Sx∗ = R(r∗ + s∗) + S(r∗ + s∗) = r∗ + s∗ = x∗.To find a solution of a right focal boundary value problem, we will use this fixed point result,in conjunction with a variation of an alternative inversion technique presented by Avery–Peterson [2, 3] and Burton–Zhang [4, 5] which converts fixed point problems of the formx(t) =∫ 10G(t, s) f (x(s)) dsinto fixed point problems of the formu(t) = f(∫ 10G(t, s) u(s) ds).BCorresponding author. Email: rich.avery@dsu.edu2 R. I. Avery, D. R. Anderson and J. HendersonIn particular, we will apply these modern techniques to the boundary value problemx′′(t) + f+(x(t)) + f−(x(t)) = 0, t ∈ (0, 1),x(0) = 0 = x′(1),where f+ : R→ [0, ∞) is a strictly increasing, differentiable function and f− : R→ [0, ∞) is astrictly decreasing, differentiable function to arrive at a fixed point problem of the formu(t) = f+(∫ 10G(t, s)(u(s) + f−(f−1+ (u(s))))ds)which one can solve by iteration under suitable conditions. In our main results section weshow how to convert the differential equation into a fixed point problem of this form andconclude with an example.2 Main resultsLet f+ : R → [0, ∞) be a strictly increasing, differentiable function and f− : R → [0, ∞)be a strictly decreasing, differentiable function and consider the right focal boundary valueproblemx′′(t) + f+(x(t)) + f−(x(t)) = 0, t ∈ (0, 1), (2.1)x(0) = 0 = x′(1). (2.2)If we let x = r + s then the right focal boundary value problem (2.1), (2.2) becomes theequivalent boundary value problemr′′(t) + f+(r(t) + s(t)) + s′′(t) + f−(r(t) + s(t)) = 0, t ∈ (0, 1), (2.3)r(0) + s(0) = 0 = r′(1) + s′(1). (2.4)Any solution of the system of boundary value problemsr′′(t) + f+(r(t) + s(t)) = 0, s′′(t) + f−(r(t) + s(t)) = 0, t ∈ (0, 1), (2.5)r(0) = 0 = r′(1), s(0) = 0 = s′(1) (2.6)corresponds to a solution of (2.3), (2.4) which corresponds to a solution of the original focalboundary value problem (2.1), (2.2). If we letu(t) = −r′′(t) and v(t) = −s′′(t),thenr(t) =∫ 10G(t, τ)u(τ)dτ (2.7)ands(t) =∫ 10G(t, τ)v(τ)dτ; (2.8)here,G(t, τ) = min{t, τ}, (t, τ) ∈ [0, 1]× [0, 1], (2.9)Alternative iterative technique 3is the Green’s function for the right focal boundary value problem. Substituting for r, r′′, s ands′′ in (2.5), (2.6) we arrive at the equivalent fixed point formulation given byu(t) = f+(∫ 10G(t, τ)(u(τ) + v(τ))dτ), t ∈ [0, 1] (2.10)v(t) = f−(∫ 10G(t, τ)(u(τ) + v(τ))dτ), t ∈ [0, 1]. (2.11)Since the functions f− and f+ are assumed to be strictly monotonic, the system (2.10), (2.11)is equivalent tou(t) = f+(∫ 10G(t, τ)(u(τ) + f−( f−1+ (u(τ)))dτ), t ∈ [0, 1] (2.12)v(t) = f−( f−1+ (u(t)), t ∈ [0, 1]. (2.13)Let E = C[0, 1], P be the subset of nonnegative elements of E, and for u ∈ E define‖u‖ = maxt∈[0,1]|u(t)|.Thus P is a cone in the Banach space E and the norm is a monotonic norm on P. The proofof our main result hinges on the existence of an equivalent norm which is monotone on Pwhich is a normality condition on our cone. See Theorem 1.1.1 of Nonlinear problems in abstractcones [6] for a thorough discussion of normality as well as Theorem 2.1 for its application inmonotone iterative techniques.Let u ∈ P, and define the operatorA : E→ Eby(Au)(t) = f+(∫ 10G(t, τ)(u(τ) + f−( f−1+ (u(τ)))dτ), t ∈ [0, 1]. (2.14)We are now ready to state and prove our main result which provides a new perspective tothe monotone iterative technique presented in Chapter 2 of the Monotone iterative techniquesfor nonlinear differential equations [7] monograph due to the form of the nonlinear term ( f =f+ + f−).Theorem 2.1. Let G and A be given by (2.9) and (2.14), respectively. Suppose d > 0 withf+(d + f−(f−1+ (d))2)≤ d,and (f−(f−1+))′(z) ≤ −1for all z ∈ (0, d). Also assume thatf+ : [0, d]→ [0, ∞)is a strictly increasing, differentiable function andf− : [0, d]→ [0, ∞)4 R. I. Avery, D. R. Anderson and J. Hendersonis a strictly decreasing, differentiable function. Then the sequence un defined recursively byu0(t) ≡ d for all t ∈ [0, 1]andun+1 = Aun, for n ≥ 0converges to a fixed point u∗ of the operator A and for v∗ = f−( f−1+ (u∗)) we have thatx∗ = r∗ + s∗,wherer∗(t) =∫ 10G(t, τ)u∗(τ)dτ and s∗(t) =∫ 10G(t, τ)v∗(τ)dτ,is a solution of (2.1), (2.2).Proof. Let d > 0 and suppose thatf+(d + f−(f−1+ (d))2)≤ d,(f−(f−1+))′(z) ≤ −1,for all z ∈ (0, d) withf+ : [0, d]→ [0, ∞)being a strictly increasing, differentiable function andf− : [0, d]→ [0, ∞)being a strictly decreasing, differentiable function. It is a standard exercise by the Arzelà–Ascoli Theorem to show that A in (2.14) is a completely continuous operator, using the prop-erties of G in (2.9) and the continuity of f+, f− and f−1+ . LetPd = {x ∈ P : ‖x‖ ≤ d}.Claim: un is a decreasing sequence and un ∈ Pd.We proceed by induction. Since f+ is an increasing function, for all t ∈ [0, 1], we have thatu1(t) = Au0(t)= f+(∫ 10G(t, τ)(d + f−( f−1+ (d))dτ)= f+(t(2− t)(d + f−(f−1+ (d)))2)≤ f+(d + f−(f−1+ (d))2)≤ d = u0(t).Alternative iterative technique 5Hence u1 ≤ u0, and since the norm is monotonic we have that ‖u1‖ ≤ ‖u0‖ = d; consequentlyu1 ∈ Pd.For induction purposes, assume that um ≤ um−1 and um ∈ Pd for all m ≤ n. For every τ ∈ [0, 1]since un(τ), un−1(τ) ∈ [0, d], by the Mean Value Theorem there is a z ∈ (0, d) such thatf−(f−1+ (un(τ)))− f−(f−1+ (un−1(τ)))=(f−(f−1+))′(z)(un(τ)− un−1(τ)).Since (f−(f−1+))′(z) ≤ −1for all z ∈ (0, d), we have thatf−(f−1+ (un(τ)))− f−(f−1+ (un−1(τ)))≤−1(un(τ)−un−1(τ)) = un−1(τ)−un(τ).It follows thatun(τ) + f−(f−1+ (un(τ)))≤ un−1(τ) + f−(f−1+ (un−1(τ))),and therefore ∫ 10G(t, τ)(un(τ) + f−(f−1+ (un(τ))))dτ≤∫ 10G(t, τ)(un−1(τ) + f−(f−1+ (un−1(τ))))dτ.Since f+ is an increasing function, we havef+(∫ 10G(t, τ)(un(τ) + f−(f−1+ (un(τ))))dτ)≤ f+(∫ 10G(t, τ)(un−1(τ) + f−(f−1+ (un−1(τ))))dτ);that is,un+1 ≤ un,and by the monotonicity of the norm we have that‖un+1‖ ≤ ‖un‖ ≤ d.Hence un+1 ∈ Pd. Therefore we have that{un}∞n=1is a monotonic sequence in the bounded, closed subset Pd of the cone P.Thus, since A in (2.14) is completely continuous withun+1 = A(un),there is a subsequence {unk}∞k=1 withunk → u∗ ∈ Pd.6 R. I. Avery, D. R. Anderson and J. HendersonThe norm is monotonic, thus we have thatun → u∗ ∈ Pd,since for any k ∈N, when j ≥ nk we have‖uj − u∗‖ ≤ ‖unk − u∗‖ since uj − u∗ ≤ unk − u∗.Therefore u∗ is a fixed point of A, and forv∗ = f−( f−1+ (u∗))we have a solution (u∗, v∗) for (2.10), (2.11). Hence, we have, using (2.7), (2.8), thatx∗ = r∗ + s∗,wherer∗(t) =∫ 10G(t, τ)u∗(τ)dτ and s∗(t) =∫ 10G(t, τ)v∗(τ)dτ,is a solution of our original focal boundary value problem (2.1), (2.2).Example 2.2. The function f (x) = x2 +√8e√8−√x hasf+(x) =x2which is a strictly increasing, differentiable function andf−(x) =√8e√8−√xwhich is a strictly decreasing, differentiable function on (0, 4). Alsof−1+ (x) = 2x,and thus (for d = 4)f+(4 + f−(f−1+ (4))2)=(12)(4 +√8e√8−√82)≤ 4and (f−(f−1+))′(x) =(−√8e√8−√2x√2x)< −1for all x ∈ (0, 4). Therefore, the sequence un, defined recursively byu0(t) ≡ 4 for all t ∈ [0, 1]andun+1 = Aun, for n ≥ 0,converges to a fixed point u∗ of the operator A and forv∗ = f−( f−1+ (u∗)) =√8e√8−√2u∗ ,we have thatx∗ = r∗ + s∗,wherer∗(t) =∫ 10G(t, τ)u∗(τ)dτ and s(t) =∫ 10G(t, τ)v∗(τ)dτ,is a solution of (2.1), (2.2).Alternative iterative technique 7Remark 2.3. While other methods can be applied easily and are straightforward to obtainexistence of solutions for Example 2.2, we chose it as an example illustrating the iteration ofthe paper, but for which standard iteration techniques do not apply. Our goal has been toexpand the application of iteration for which computers can play a critical role.References[1] D. Anderson, R. Avery, J. Henderson, Layered compression–expansion fixed pointtheorem, Results in Fixed Point Theory and Applications 2018, Article ID 201825, 10 pp.https://doi.org/10.30697/rfpta-2018-25[2] R. I. Avery, D. O’Regan, J. Henderson, Dual of the compression–expansion fixed pointtheorems, Fixed Point Theory Appl. 2007, Article ID 90715, 11 pp. MR2358026[3] R. I. Avery, A. C. Peterson, Multiple positive solutions of a discrete second order conju-gate problem, PanAmer. Math. J. 8(1998), No. 3, 1–12. MR1642636[4] T. A. Burton, Fixed points, differential equations, and proper mappings, Semin. FixedPoint Theory Cluj-Napoca 3(2002), 19–32. MR1929745[5] T. A. Burton, B. Zhang, Periodicity in delay equations by direct fixed point mapping,Differential Equations Dynam. Systems 6(1998), 413–424. MR1790185[6] D. J. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reportsin Mathematics in Science and Engineering, Vol. 5, Academic Press, Boston, MA, 1988.MR959889[7] G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone iterative techniques for nonlin-ear differential equations, Monographs, Advanced Texts and Surveys in Pure and AppliedMathematics, Vol. 27, Pitman, Boston, MA, USA, 1985. 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Alternative iterative technique

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Alternative iterative technique

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