summary:In this note we consider the boundary value problem $y''=f(x,y,y')$ $\,(x\in [0,X];X>0)$, $y(0)=0$, $y(X)=a>0$; where $f$ is a real function which may be singular at $y=0$. We prove an existence theorem of positive solutions to the previous problem, under different hypotheses of Theorem 2 of L.E. Bobisud [J. Math. Anal. Appl. 173 (1993), 69–83], that extends and improves Theorem 3.2 of D. O'Regan [J. Differential Equations 84 (1990), 228–251]

Bonanno, Gabriele

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Commentationes Mathematicae Universitatis CarolinaeGabriele BonannoAn existence theorem of positive solutions to a singular nonlinear boundaryvalue problemCommentationes Mathematicae Universitatis Carolinae, Vol. 36 (1995), No. 4, 609--614Persistent URL: http://dml.cz/dmlcz/118790Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1995Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 36,4 (1995)609–614 609An existence theorem of positive solutions toa singular nonlinear boundary value problemGabriele BonannoAbstract. In this note we consider the boundary value problem y′′ = f(x, y, y′) (x ∈[0, X];X > 0), y(0) = 0, y(X) = a > 0; where f is a real function which may be singularat y = 0. We prove an existence theorem of positive solutions to the previous problem,under different hypotheses of Theorem 2 of L.E. Bobisud [J. Math. Anal. Appl. 173(1993), 69–83], that extends and improves Theorem 3.2 of D. O’Regan [J. DifferentialEquations 84 (1990), 228–251].Keywords: ordinary differential equations, singular boundary value problem, positivesolutionsClassification: 34B15Let f be a real function defined on [0, X ]× (0,∞) × (−∞,∞); L1([0, X ]) thespace of all (equivalence classes of) measurable functions ψ : [0, X ]→ R such that‖ψ‖L1([0,X]) =∫ X0 |ψ(x)| dx < ∞; W 2,1([0, X ]) the space of all u ∈ C1([0, X ])such that u′ is absolutely continuous in [0, X ] and u′′ ∈ L1([0, X ]).Consider the problem(P)y′′ = f(x, y, y′)y(0) = 0y(X) = a > 0 .A function u : [0, X ] → [0,∞) is said to be a generalized solution to (P) ifu ∈ W 2,1([0, X ]), u(0) = 0, u(X) = a and, for almost every x ∈ [0, X ], onehas u′′(x) = f(x, u(x), u′(x)). When the function f is continuous in [0, X ] ×(0,∞)× (−∞,∞), any generalized solution to problem (P) is a classical one, thatis u ∈ C1([0, X ]) ∩ C2((0, X ]) and u′′(x) = f(x, u(x), u′(x)) for every x ∈ (0, X ].Positive solutions to singular nonlinear boundary value problems appear ina variety of applications. Consequently, they have been studied by many authors(see, for instance, [2], [4] and the references given there). In particular, amongthe latest contributions, there are the following two theorems.Theorem A ([2, Theorem 2]). Let X ≥ 1 be fixed. Assume the following hy-potheses.(H1) f ∈ C([0, X ] × (0,∞) × (−∞,∞)) and f(x, y, z) is locally Lipschitz in yand z on [0, X ]× (0,∞)× (−∞,∞).610 G.Bonanno(H2) zf(x, y, z) ≤ 0 on [0, X ]× (0,∞)× (−∞,∞).(H3) There exist a nonnegative function f1 continuous on [0, 1], a nonnegative,nonincreasing function g1 continuous on (0, a], and a function h1 positiveand continuous on (a,∞] such that(i) f(x, y, z) ≥ −f1(x)g1(y)h1(z)z on [0, X ]× (0, a]× [a,∞),(ii) f1(s)g1(aX s) ∈ L1([0, 1]),(iii)∫ ∞a dv/vh1(v) >∫ 10 f1(s)g1(aX s) dshold.(H4) PutH(z) =∫ za1h1(v)dv; and M1 = H−1(∫ a0g1(u) du),there exist a constant k > M1 and a measurable function F on [0, X ]satisfying(i) |f(x, y, z)| ≤ F (x) for 0 ≤ x ≤ X , aX x ≤ y ≤ k, and |z| ≤ k,(ii)∫ X0 F (x) dx <∞.Then, the problem (P) has at least one solution u ∈ C1([0, X ]) ∩C2((0, X ]) such that u(x) > 0 for every x ∈ (0, X ].Theorem B ([4, Theorem 3.2 and subsequent remark]). Consider the problem(P0)y′′ +Ψ(x)h(x, y) = 0 0 < x < 1y(0) = 0y(1) = a > 0 .where h and Ψ satisfy(K1)(i) h is continuous on [0, 1]× (0,∞);(ii) limy→0+ h(x, y) =∞ for each x ∈ [0, 1];(iii) 0 < h(x, y) ≤ g(y) on [0, 1], where g is continuous and nonincreasingon (0,∞).(iv) In addition 1/Ψ ∈ C([0, 1]) with Ψ > 0 on (0, 1).(K2) There exist p > 1, q > 1 with1p +1q = 1 together with∫ 10 Ψp(z) dz < ∞and∫ 10 gq(u) du <∞.(K3) For each constant M > 0 there exists η(x) continuous and positive on[0, 1] such that h(x, y) ≥ η(x) on [0, 1]× (0,M ].Then, the problem (P0) has at least one solution u ∈ C([0, 1])∩C2((0, 1)) suchthat u(x) > 0 for every x ∈ (0, 1].An existence theorem of positive solutions to a singular nonlinear boundary value problem 611The purpose of this note is to establish Theorem 1 below. We remark thatour result extends and improve Theorem B (see Remark 3) and is independent ofTheorem A. In particular, contrary to (H1), we assume that f is continuous in yand z. Moreover, the condition f(x, y, 0) ≡ 0, which is implied by (H2), does notfollow from our assumptions.Let r > 0, X > 0 and x ∈ [0, X ]. Here and in the sequel, W (r, x) stands forthe set{(y, z) ∈ (0,∞)× (−∞,∞) : aX x ≤ y ≤ a+Xr; |z| ≤ aX + 2r}. Let nowf be a real function defined on [0, X ]× (0,∞)× (−∞,∞). For every x ∈ [0, X ],we putMr(x) = sup(y,z)∈W (r,x)|f(x, y, z)| and mr(x) = sup(y,z)∈W (r,x)f(x, y, z).Theorem 1. Let f be a real function defined in [0, X ] × (0,∞) × (−∞,∞).Assume that(a) the function (y, z)→ f(x, y, z) is continuous for almost every x ∈ [0, X ];(b) the function x → f(x, y, z) is measurable for every (y, z) ∈ (0,∞) ×(−∞,∞);(c) there exists r > 0 such that the function Mr belongs to L1([0, X ]) andone has‖Mr‖L1([0,X]) ≤ r;(d) for almost every x ∈ [0, X ], one hasmr(x) < 0.Then, the problem (P) has at least one generalized solution u ∈ W 2,1([0, X ])such that u(x) > 0 for every x ∈ (0, X ].Proof: Consider the setK ={v ∈ L1([0, X ]) : −mr(x) ≤ v(x) ≤Mr(x) a.e. in [0, X ]}.Of course, K is nonempty and convex. By the Dunford-Pettis theorem (see,for instance, [3, Theorem 1, p. 101]), it is also weakly compact. For every v ∈L1([0, X ]) and every x ∈ [0, X ], we put(1)φ1(v)(x) =aXx+X − xX∫ x0sv(s) ds+xX∫ Xx(X − s)v(s) ds;φ2(v)(x) =aX− 1X∫ X0sv(s) ds+∫ Xxv(s) ds;Obviously, one has φ1(v)(0) = 0, φ1(v)(X) = a, [φ1(v)]′ = φ2(v); [φ1(v)]′′ =[φ2(v)]′ = −v; φ1(v) ∈ W 2,1([0, X ]), moreover, if v(x) > 0 for almost x ∈ [0, X ],therefore φ1(x) > 0 for every x ∈ (0, X ]. We now putG(v)(x) = −f(x, φ1(v)(x), φ2(v)(x))612 G.Bonannofor every v ∈ L1([0, X ]) and for every x ∈ (0, X ].Let us prove that G(K) ⊆ K. To this end, fix v ∈ K and observe that, by (1)and (c), one hasaXx ≤ φ1(v)(x) ≤ a+∫ x0Xv(s) ds+∫ XxXv(s) ds≤ a+X ‖Mr‖L1([0,X]) ≤ a+Xr;|φ2(v)(x)| ≤aX+1X∫ X0Xv(s) ds+∫ X0v(s) ds≤ aX+ 2 ‖Mr‖L1([0,X]) ≤aX+ 2r.Therefore, (φ1(v)(x), φ2(v)(x)) ∈ W (r, x) for every x ∈ (0, X ]. Hence, for almostevery x ∈ [0, X ], one has:−mr(x) ≤ −f(x, φ1(v)(x), φ2(v)(x)) ≤Mr(x).This implies that G(v) ∈ K.Now, let us prove that the operator G is weakly sequentially continuous. Letv ∈ K and let {vn} be a sequence in K weakly converging to v in L1([0, X ]).From (1) it follows that, for every x ∈ [0, X ], limn→∞ φ1(vn)(x) = φ1(v)(x);limn→∞ φ2(vn)(x) = φ2(v)(x). Therefore, by (a), the sequence {G(vn)} convergesalmost everywhere in [0, X ] to G(v). Bearing in mind that for almost everyx ∈ [0, X ] and every n ∈ N one has|G(vn)(x)| ≤Mr(x),the Lebesgue Dominated Convergence theorem yields limn→∞G(vn) = G(v) inL1([0, X ]). So, {G(vn)} converges weakly to G(v) in L1([0, X ]).We now have proved that the function G : K → K verifies all that assumptionsof Theorem 1 of [1]. Then, there is v ∈ K such that v = G(v). The functionu(x) = φ1(v)(x), x ∈ [0, X ], satisfies our conclusion. Remark 1. This theorem ensures the existence of positive solutions even iff(x, y, z) is not locally Lipschitz in y and z. For example, the problem(P1)y′′ = −(sen y)1/3|y′|1/3 − xy−1/2|y′|1/2 − x3y(0) = 0y(1) = a > 0 ,owing to Theorem 1, has at least one positive solution u ∈ C1([0, X ])∩C2((0, X ]).Indeed, taking into account that∫ X0sup(y,z)∈W (r,x)|f(x, y, z)| dx ≤( aX+ 2r)1/3X +23X2√a( aX+ 2r)1/2+X44An existence theorem of positive solutions to a singular nonlinear boundary value problem 613andlimr→∞r − X44(aX + 2r)1/3X + 23X2√a(aX + 2r)1/2=∞,there exists r > 0 such that ‖Mr‖L1([0,X]) < r. Hence, it is easily seen that allthe assumptions of Theorem 1 hold.We cannot apply Theorem A to the problem (P1), even because f(x, y, 0) =x3 6≡ 0.We also observe that assumption (H3) and (H4) of Theorem A and assumption(c) of Theorem 1 are mutually independent.Remark 2. We explicitly observe that in Theorem 1 f may be singular at someset Ω ⊆ [0, X ], with |Ω| = 0 (|Ω| denotes the Lebesgue measure of Ω). Particularly,if f ∈ C((0, X)×(0,∞)×(−∞,∞)) and the assumptions (c) and (d) of Theorem 1hold, then there exists at least one function u ∈ C1([0, X ]) ∩ C2((0, X)) suchthat u(0) = 0, u(X)=a and, for every x ∈ (0, X), u′′(x) = f(x, u(x), u′(x)) andu(x) > 0.Remark 3. Theorem 1 extends and improves Theorem B. Indeed, the assump-tions of Theorem B, even without the condition limy→0+ h(x, y) =∞, imply theones of Theorem 1. Let us prove this. Of course, from (i) and (iv) of (K1), (a) and(b) follow; (c) is verified by choosing r = ‖Ψ‖Lp([0,1])(1a)1/q ‖g‖Lq([0,a]), since, by(iii) of (K1), (K2) and Hölder inequality, one has∫ 10supaXx≤y≤a+Xr|Ψ(x)h(x, y)| dx ≤≤∫ 10Ψ(x)g( aXx)dx ≤ ‖Ψ‖Lp([0,1])(1a)1/q‖g‖Lq([0,a]);(d) follows from (iv) of (K1) and (K3), since in (0, a+Xr] one has Ψ(x)h(x, y) ≥Ψ(x)η(x) > 0, therefore−mr(x) = infaXx≤y≤a+XrΨ(x)h(x, y) ≥ Ψ(x)η(x) > 0for every x ∈ (0, 1). Hence, our claim is proved.Now, consider the problem(P2)y′′ + x[∣∣∣sen 1y∣∣∣1/2+ y1/2 + x]= 0y(0) = 0y(1) = a > 0 .614 G.BonannoOwing to Theorem 1, the problem (P2) has at least one positive solution u ∈C1([0, X ]) ∩ C2((0, X ]). Indeed, taking into account that∫ X0supaXx≤y≤a+Xr|f(x, y)| dx ≤ X22+X22(a+Xr)1/2 +X33andlimr→∞r −(X22 +X33)X22 (a+Xr)1/2=∞,there exists r > 0 such that ‖Mr‖L1([0,X]) < r. Hence, it is easily seen that allhypotheses of Theorem 1 hold and our claim is proved.In the previous example the condition limy→0+ h(x, y) =∞ is not satisfied andmoreover there is no function g(y), nonincreasing in (0,∞), such that h(x, y) ≤g(y), as it is required by Theorem B.References[1] Arino O., Gautier S., Penot J.P., A fixed point theorem for sequentially continuous map-pings with application to ordinary differential equations, Funkcial. Ekv. 27 (1984), 273–279.[2] Bobisud L.E., Existence of positive solutions to some nonlinear singular boundary valueproblems on finite and infinite intervals, J. Math. Anal. Appl. 173 (1993), 69–83.[3] Diestel J., Uhl J.J., Vector Measures, Math. Survey, no. 15, Amer. Soc., 1977.[4] O’Regan D., Existence of positive solutions to some singular and nonsingular second orderboundary value problems, J. Differential Equations 84 (1990), 228–251.Dipartimento di Ingegneria Elettronica e Matematica Applicata, Università diReggio Calabria, via E. Cuzzocrea 48, 89128 Reggio Calabria, Italy(Received November 15, 1994)

An existence theorem of positive solutions to a singular nonlinear boundary value problem

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An existence theorem of positive solutions to a singular nonlinear boundary value problem

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