The nonlinear Kneser problem for singular in phase variables second-order differential equations
Abstract
For the singular in phase variables differential equation u'' = f (t, u, u') ), sufficient conditions are found for the existence of a solution satisfying the conditions Phi(u) = c, u(t) > 0, u'(t) < 0 for t > 0, where Phi : C([0, a]; R+) to R+ is a continuous nondecreasing functional, c > 0, and a > 0.
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Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 http://www.boundaryvalueproblems.com/content/2014/1/147 R E S E A R C H Open Access The nonlinear Kneser problem for singular in phase variables second-order differential equations Nino Partsvania1,2* and Bedˇ rich P˚uža3 Dedicated to our dear teacher, Prof. Ivan Kiguradze *Correspondence: [email protected] 1A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6 Tamarashvili St., Tbilisi, 0177, Georgia 2International Black Sea University, 2 David Agmashenebeli Alley 13km, Tbilisi, 0131, Georgia Full list of author information is available at the end of the article Abstract For the singular in phase variables differential equation u =f(t,u,u), sufficient conditions are found for the existence of a solution satisfying the conditions ϕ(u)=c,u(t)>0, u(t)<0 fort>0, where ϕ:C([0,a];R+)→R+is a continuous nondecreasing functional, c>0,anda>0. MSC: 34B16; 34B40 Keywords: differential equation; second order; singular in phase variables; Kneser solution; Kneser problem; nonlinear 1 Statement of the problem and formulation of the main results Suppose D=(t,x,y):t>,x>,y< ,R+=[,+∞[, and f:D→R+is a continuous function. Consider the differential equation u =ft,u,u. (.) A continuous function u:R+→R+is said to be the Kneser solution of Eq. (.)ifitis twice continuously differentiable in the interval ],+∞[, and in this interval it satisfies the inequalities u(t)>, u(t)< and the differential equation (.). ©2014 Partsvania and P˚uža; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 2 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 In the present paper, we investigate the problem on the existence of a Kneser solution of Eq. (.) satisfying the condition ϕ(u)=c,(.) where ϕ:C([,a]; R+)→R+is a continuous, nondecreasing functional, a>,andc>. It is natural to name this problem the nonlinear Kneser problem since it was first studied by Kneser []inthecasewhereEq.(.) and condition (.)havetheforms u =f(t,u), (.) u() = c,(.) where f:R+×R+→R+is a continuous function. Particularly, in []itisprovedthatif fis a nondecreasing in the second argument function satisfying the local Lipschitz condition in this argument and f(t,)≡, then for an arbitrarily fixed c> , the differential equation (.) has a unique Kneser solution satisfying condition (.). years later since Kneser’s paper was published, in their study of the problem on the distribution of electrons in a heavy atom, Fermi []andThomas[] had to investigate the problem analogous to the Kneser one for the concrete second-order differential equation u =t– u .(.) In particular, they have proved that Eq. (.) has a unique solution satisfying the boundary conditions u() = , lim t→+∞u(t)=. (.) It is easy to see that a solution of problem (.), (.) is a Kneser solution of Eq. (.)and vice versa, a Kneser solution of that equation, satisfying the initial condition u() = , (.) is a solution of problem (.), (.). Therefore, problem (.), (.)isequivalenttothe Kneser problem for Eq. (.) with the initial condition (.). After the papers by Fermi and Thomas were published, many mathematicians have been interested in the Kneser-type problems, and such problems have been investigated in detail for a wide class of differential equations and systems. Most of the results on the solvability and unique solvability of the Kneser problem for second-order nonlinear differential equations, obtained until the beginning of s of the last century, are reflected in the monograph by Sansone []. From further investigations, first of all the paper by Hartman and Wintner [], where the Kneser problem for Eq. (.) was studied in the case when f:R+×R→R+is a continuous function, should be noted. Kiguradze [, ] studied the same problem in the case when the function f:],+∞[×R→Rhas a nonintegrable singularity in the first argument at the point t=.
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 3 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 The Kneser problem for singular in a time variable higher-order nonlinear differential equations first was studied by Kiguradze in [], where the optimal conditions are established for the solvability of the above-mentioned problem (see, [, Sect. ] as well). Analogous results were obtained by Kiguradze and Rachůnková []fortheKneserproblem with a nonlinear initial condition. Sufficient conditions for the solvability of the Kneser-type problems for nonlinear differential systems were obtained by Chanturia [], Coffman [], Hartman and Wintner [], Kiguradze and Rachůnková [], and Rachůnková [, ]. In all the above-mentioned works, differential equations and systems, not having singularities in phase variables, are considered. The Kneser problem for the differential equation with a singularity in one of the phase variables first was investigated by Kiguradze []. However, in this paper we consider not the general differential equation but the EmdenFowler type higher-order differential equation u(n)=p(t)u–λ. As for the general differential equation (.) with singularities in phase variables, for it the Kneser problem has been practically unstudied so far. The aim of the present paper is to fill this gap. In what follows it is assumed that the function fsatisfies the inequality g(t)≤xλ|y|μf(t,x,y)≤g(t)(.) in the domain D.Hereλand μare nonnegative constants, λ+μ>,andgi:],+∞[→R+ (i= ,) are continuous functions, not equal identically to zero in an arbitrary neighborhood of +∞,i.e., there exists a sequence of positive numbers (tk)+∞ k= such that lim k→+∞tk=+∞,gi(tk)> (i=,;k=,,...). (.) Consequently, Eq. (.) has singularities in phase variables since either lim x→f(tk,x,y)=+∞for y<(k=,,...) or lim y→f(tk,x,y)=+∞for x>(k=,,...). Throughout the paper, the following notation and definitions are used. ν=+λ+μ +μ. (.) C([,a]; R) is the Banach space of continuous functions u:[,a]→Rwith the norm uC=maxu(t):≤t≤a, C[,a]; R+=u∈C[, a];R:u(t)≥for≤t≤a.
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 4 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 Afunctionalϕ:C([,a];R+)→R+is said to be nondecreasing if for any u∈C([,a];R+) and u∈C([,a]; R+)theinequalityϕ(u+u)≥ϕ(u)holds. For any x∈R+,weputϕ(x)=ϕ(u), where u(t)≡x. AKnesersolutionuof Eq. (.) is called vanishing at infinity if limt→+∞u(t)=,andit is called remote from zero if limt→+∞u(t)>. Theorem . If Eq.(.)has a Kneser solution u,then +∞ t g(s)ds <+∞for t >, +∞ +∞ t g(s)ds +μ dt <+∞, (.) and u(t)>v(t;δ)for t ≥, (.) where v(t;δ)=δν+(+μ) +μν+∞ t+∞ s g(x)dx +μ ds ν , (.) δ=lim t→+∞u(t). (.) Corollary . If condition (.)holds and c<ϕv(·;), (.) then problem (.), (.)has no Kneser solution. Theorem . If +∞ t g(s)ds <+∞for t >, +∞ +∞ t g(s)ds +μ dt <+∞, (.) then for any positive number δEq.(.)has at least one Kneser solution satisfying equality (.). Theorem . If along with (.)the condition +∞ t g(s) vλ (s;)ds <+∞for t >, +∞ +∞ t g(s) vλ (s;)ds +μ dt <+∞(.) is satisfied,then Eq.(.)has at least one vanishing at infinity Kneser solution. According to Corollary ., for small cproblem (.), (.) has no Kneser solution. Thus we can expect the solvability of that problem only for large c. Suppose that condition (.) holds. Then obviously condition (.) is satisfied as well. We introduce the function v(t;δ)=δ++∞ t( + μ)+∞ s g(x) vλ (x;δ)dx +μ ds for t≥,δ> , (.)
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 5 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 and the number c=infϕv(·;δ):δ> . (.) Theorem . Let the function gsatisfy condition (.), and lim x→+∞ϕ(x)=+∞. (.) If,moreover, c>c, (.) then problem (.), (.)has at least one Kneser solution. Remark . In the case, where conditions (.)holdand c∈ϕv(·;),c, the question on the existence of a Kneser solution of problem (.), (.) remains open. Considernowthecasewhere g(t)≡g(t), =const≥, (.) i.e.,thecasewhereinequality(.)hastheform g(t)≤xλ|y|μf(t,x,y)≤g(t). From Theorems .,.,and. we immediately have the following corollary. Corollary . Let the function gsatisfy identity (.), and let the functional ϕsatisfy condition (.). Then the following assertions are equivalent: (i) the function gsatisfies conditions (.); (ii) Eq.(.)has at least one remote from zero Kneser solution; (iii) for any δ>,problem (.), (.)has at least one Kneser solution; (iv) for any sufficiently large c>,problem (.), (.)has at least one Kneser solution. The following statement is also valid. Corollary . Let the function gsatisfy identity (.), and the functional ϕsatisfy condition (.). Let,moreover,there exist numbers αand βsuch that lim inf t→tαg(t)>, lim sup t→tαg(t)<+∞, (.) lim inf t→+∞tβg(t)>, lim sup t→+∞tβg(t)<+∞. (.) Then the following assertions are equivalent: (i) α<+μ,β>+μ; (ii) Eq.(.)has at least one remote from zero Kneser solution;
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 6 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 (iii) Eq.(.)has at least one vanishing at infinity Kneser solution; (iv) for any δ>,problem (.), (.)has at least one Kneser solution; (v) for any sufficiently large c>,problem (.), (.)has at least one Kneser solution. Remark . In Theorem . and its corollaries it can be assumed, for example, that ϕ(u)=a ψu(s)dσ(s), where ψ:R+→R+is a continuous, nondecreasing function, and σ:[,a]→Ris a nondecreasing function such that lim x→+∞ψ(x)=+∞,σ(a)–σ() > . Remark . The above-formulated theorems and their corollaries cover the case, where the function fhas a nonintegrable singularity in a time variable at the point t= . Indeed, if conditions (.)hold,where<α<+μ,then t f(s,x,y)ds =+∞for (t,x,y)∈D. 2 Auxiliary propositions 2.1 Lemmas on aprioriestimates Consider the differential inequalities u(t) μu(t)≥g(t)u–λτ(t)(.) and g(t)u–λτ(t)≤u(t) μu(t)≤g(t)u–λτ(t). (.) Everywhere in this section it is assumed that λand μare nonnegative constants, τ:R+→ R+is a continuous function such that τ(t)≥tfor t∈R+,(.) and gi:],+∞[→R+(i= , ) are continuous functions, not equal identically to zero in an arbitrary neighborhood of +∞,i.e., there exists a sequence of positive numbers (tk)+∞ k= such that condition (.)issatisfied. A continuous function u:R+→],+∞[issaidtobethe Kneser solution of the differential inequality (.)(of the differential inequality (.)) if it is twice continuously differentiable in the interval ],+∞[ and in this interval along with the inequality u(t) < satisfies the differential inequality (.) (the differential inequality (.)). Lemma . If the differential inequality (.)has a Kneser solution u,then the function gsatisfies condition (.), and u admits estimate (.), where vis the function given by equality (.), and ν,δare numbers given by equalities (.)and (.).
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 7 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 Proof In view of (.)andthefactthatuis a Kneser solution, from (.)wefind u(t) +μ=(+μ)+∞ t uλτ(x)u(x) μu(x)u–λτ(x)dx >(+μ)u–λ(t)+∞ t uλτ(x)u(x) μu(x)dx ≥( + μ)u–λ(t)+∞ t g(x)dx, and, consequently, –u(t)u(t)λ +μ>( + μ)+∞ t g(x)dx +μ for t>. If we integrate this inequality from tto +∞,thenduetoequalities(.)and(.), we obtain u(t)ν>δν+(+μ) +μν+∞ t+∞ s g(x)dx +μ ds for t≥. Therefore condition (.)issatisfiedandthefunctionuadmits estimate (.). Let b∈],+∞[, and along with (.)lettheinequality τ(t)≤bfor ≤t≤b(.) be fulfilled. A continuous function u:[,b]→],+∞[issaidtobea Kneser solution of the differential inequality (.)(of the differential inequality (.)) in the interval [, b]if it is continuously differentiable in the interval ], b] and in this interval along with the inequality u(t) < satisfies the differential inequality (.) (the differential inequality (.)). The following lemma can be proved analogously to Lemma .. Lemma . Let inequality (.)be fulfilled and the differential inequality (.)in the interval [,b]have a Kneser solution u.Then b b s g(x)dx +μ ds <+∞,(.) and that solution admits the estimate u(t)>w(t;δ,b)for ≤t≤b,(.) where δ=u(b)and w(t;δ,b)=δν+(+μ) +μνb tb s g(x)dx +μ ds ν for ≤t≤b.(.) Lemma . Let along with (.)the condition b b s g(x)dx +μ ds <+∞(.)
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 8 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 be fulfilled,and let the differential inequality (.)in the interval [,b]have a Kneser solution u.Then this solution along with (.)admits the estimates u(t)≤w(t;δ,ε,b)for ≤t≤b,(.) ε+μ+w–λ(;δ,ε,b)b t g(s)ds +μ ≤–u(t) ≤–w(t;δ,ε,b)for <t<b,(.) where δ=u(b), ε=|u(b)|,and w(t;δ,ε,b)=δ+b tε+μ+(+μ)b s g(x)dx wλ (τ(x);δ,b) +μ ds (.) for ≤t≤b. Proof First note that the validity of condition (.) guarantees the validity of condition (.). On the other hand, by Lemma . the function uadmits estimate (.). By virtue of this estimate and the fact that uis a Kneser solution, (.)and(.)yield u(t) +μ=ε+μ+(+μ)b tu(x) μu(x)dx ≥ε+μ+b t g(x)u–λτ(x)dx ≥ε+μ+u–λ()b t g(x)dx for < t≤b,(.) u(t) +μ≤ε+μ+(+μ)b t g(x)u–λ(x)dx ≤ε+μ+(+μ)b t g(x) wλ (x;δ,b)dx =–w(t;δ,ε,b)+μ, and, consequently, –u(t)≤–w(t;δ,ε,b)for<t≤b. (.) Integration of this inequality from tto bresults in estimate (.). If along with (.) we take into account inequalities (.)and(.), then the validity of estimate (.) becomes evident. 2.2 Lemma on the solvability of a nonlinear Kneser problem on a finite interval Let b>a. Consider the problem on the existence of a Kneser solution of Eq. (.)inthe interval [, b] satisfying condition (.). If condition (.) holds, then for any δ>,ε>,andt∈[,b], we assume w(t;δ,ε,b)=δ+b tε+μ+(+μ)b s g(x)dx wλ (x;δ,b) +μ ds.(.)
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 9 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 Lemma . Let condition (.)be fulfilled and there exist numbers ε>,δ>,and δ∗>δsuch that ϕδ∗>c(.) and ϕw(·;δ,ε,b)<c.(.) Then problem (.), (.)has a Kneser solution u in the interval [, b]such that δ≤u(b)≤δ∗,u(b)=–ε.(.) Proof For arbitrarily fixed δ∈[δ,δ∗] and natural number k, we consider the Cauchy problem u(t)=u(t) –μft,uτk(t),uτk(t),(.) u(b)=δ,u(b)=–ε,(.) where f(t,x,y)=|y|μf(t,x,y), (.) τk(t)=⎧ ⎨ ⎩ t+b kfor ≤t≤b–b k, bfor b–b k<t≤b.(.) By virtue of conditions (.), (.) and equalities (.), (.), problem (.), (.)has auniquesolutionintheinterval[,b], which is a Kneser solution of the differential inequality (.) as well, where τ(x)≡τk(x). Denote this solution by uk(t;δ). It is clear that the function (t,δ)→uk(t;δ) is continuous on [,b]×[δ,δ∗], and on ],b[×[δ,δ∗]itsatisfies the inequalities uk(t;δ)>δ,u k(t;δ)≤–ε.(.) On the other hand, by Lemma .,on],b]×[δ,δ∗] this function admits the estimates uk(t;δ)≤wk(t;δ,ε,b)≤r,(.) –u k(t;δ)≤–w k(t;δ,ε,b)≤–w (t), (.) where wk(t;δ,ε,b)=δ+b tε+μ+(+μ)b s g(x)dx wλ (τk(x);δ,b) +μ ds, w(t)=δ∗+b tε+μ+(+μ)δ–λ b s g(x)dx +μ ds, r=w().
Partsvania and P˚uža Boundary Value Problems 2014, 2014:147 Page 16 of 17 http://www.boundaryvalueproblems.com/content/2014/1/147 Then due to condition (.), there exists a sequence of positive numbers (εk)+∞ k= satisfying (.) and the inequalities ϕw(·;δ,εk,bk)<c(k=,,...). (.) By Lemma . and conditions (.), (.), for any natural k,problem(.), (.)hasa Kneser solution ukin the interval [, bk]suchthat δ≤δk≤δ∗,u(bk)=–εk, where δk=u(bk). Without loss of generality, we can assume that the sequence (δk)+∞ k= is converging. Put δ=lim k→+∞δk. By Lemma .,thesequence(uk)+∞ k= contains a uniformly converging on every finite interval from R+subsequence (ukm)+∞ m= such that the function, defined by the equality u(t)= lim m→+∞ukm(t)fort≥, is a Kneser solution of problem (.), (.). On the other hand, if in the equality ϕ(ukm)=c we pass to the limit as m→+∞, then it becomes clear that usatisfies condition (.)as well. Thus uis a Kneser solution of problem (.), (.). To convince ourselves that Corollary . is valid, it suffices to note that if conditions (.)-(.) are fulfilled, then each of conditions (.), (.), (.)issatisfiediffα<+μ and β>+μ. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript. Author details 1A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6 Tamarashvili St., Tbilisi, 0177, Georgia. 2International Black Sea University, 2 David Agmashenebeli Alley 13km, Tbilisi, 0131, Georgia. 3Faculty of Business and Management, Brno University of Technology, Kolejní 2906/4, Brno, 612 00, Czech Republic. Acknowledgements For the first author this work is supported by the Shota Rustaveli National Science Foundation (Project # FR/317/5-101/12), and for the second author this work is supported by the Internal Grant Agency at Brno University of Technology (Project # FP-S-13-2148). Received: 11 February 2014 Accepted: 29 May 2014 References 1. Kneser, A: Untersuchung und asymptotische darstellung der integrale gewisser differentialgleichungen bei grossen reellen werthen der arguments, I. J. Reine Angew. Math. 116, 173-212 (1896) 2. Fermi, E: Un metodo statistico per la determinazione di alcune proprietà dell’atomo. Rend. R. Accad. Naz. Lincei 6, 602-607 (1927) 3. Thomas, LH: The calculation of atomic fields. Proc. Camb. Philos. Soc. 23, 542-548 (1927) 4. Sansone, G: Ordinary Differential Equations, vol. I. Izdat. Inostranno˘ ı Literatury, Moscow (1953) (trans. in Russian) 5. Hartman, P, Wintner, A: On the non-increasing solutions of y =f(x,y,y).Am.J.Math.73(2), 390-404 (1951) 6. Kiguradze, IT: On non-negative non-increasing solutions of non-linear second order differential equations. Ann. Mat. Pura Appl. 81, 169-192 (1969)
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