2015, No.84, 1–10; doi: 10.14232/ejqtde.2015.1.84 http://www.math.u-szeged.hu/ejqtde/
A note on the existence of positive solutions of singular initial-value problem for second order
differential equations
Afgan Aslanov
BMathematics and Computing Department, Beykent University, Ayaza ˘ga, ¸Si¸sli, Istanbul, Turkey Received 16 September 2015, appeared 27 November 2015
Communicated by Ivan Kiguradze
Abstract. We are interested in the existence of positive solutions to initial-value prob- lems for second-order nonlinear singular differential equations. Existence of solutions is proven under conditions which are directly applicable and considerably weaker than previously known conditions.
Keywords: second order equations, existence, Emden–Fowler equation.
2010 Mathematics Subject Classification: 34A12.
1 Introduction
In recent years, the studies of singular initial value problems for second order differential equation have attracted the attention of many mathematicians and physicists (see for example, [1–22])
Agarwal and O’Regan [1] established the existence theorems for the positive solution of the problems
(py0)0+pqg(y) =0, t∈[0,T) y(0) =a>0,
tlim→0+p(t)y0(t) =0
(1.1)
and
(py0)0+pqg(y) =0, t∈[0,T) y(0) =a>0,
y0(0) =0,
(1.2)
where 0< T≤∞, p≥0, q≥0 andg:[0,∞)→[0,∞).
BEmail: [email protected]
Theorem 1.1([1]). Suppose the following conditions are satisfied
p∈ C[0,T)∩C1(0,T) with p>0on(0,T) (1.3) q∈ L1p[0,t∗] for any t∗∈ (0,T)with q>0on(0,T), (1.4) where L1r[0,a]is the space of functions u(t)withRa
0 |u(t)|r(t)dt<∞,
Z t∗
0
1 p(s)
Z s
0 p(x)q(x)dxds<∞ for any t∗ ∈(0,T) (1.5) and
g:[0,∞)→[0,∞)is continuous, nondecreasing on[0,∞)and g(u)>0for u>0. (1.6) Let
H(z) =
Z a
z
dx
g(x) for0<z≤ a and assume
Z t∗
0
1 p(s)
Z s
0 p(x)q(x)τ(x)dxds<a for any t∗ ∈ (0,T), (1.7) here
τ(x) = g
H−1 Z x
0
1 p(w)
Z w
0 p(z)q(z)dzdw
.
Then(1.1) has a solution y ∈ C[0,T)with py0 ∈ C[0,T), (py0)0 ∈ L1pq(0,T)and0 < y(t) ≤ a for t∈ [0,T).In addition if either
p(0)6=0 (1.8)
or
p(0) =0 and lim
t→0+
p(t)q(t)
p0(t) =0 (1.9)
holds, then y is a solution of (1.2).
The condition (1.7) makes this theorem difficult for application. We try to establish a more general and applicable condition instead of (1.6) and (1.7).
2 Main results
Theorem 2.1. Suppose(1.3)–(1.5)hold. In addition we assume Z t∗
0
1 p(s)
Z s
0
p(x)q(x)g(a)dxds< a (2.1) for any t∗ ∈ (0,T0).Then
a) (1.1) has a solution y ∈ C[0,T0)with py0 ∈ C[0,T0), (py0)0 ∈ L1pq(0,T0)and0 < y(t)≤ a for t∈[0,T0).
b) If RT1
T0
1 p(s)
Rs
0 p(x)q(x)g(T0)ds < y(T0),and conditions(1.3)–(1.6)satisfied then the solution can be extended to the interval[0,T1).
By monotonicity ofg(x), it follows from (1.7) that Z t∗
0
1 p(s)
Z s
0 p(x)q(x)τ(x)dxds≤
Z t∗
0
1 p(s)
Z s
0 p(x)q(x)g(a)dxds,
and therefore the condition (2.1) is stronger than condition (1.7) in the Theorem 1.1. But the statement b) of the Theorem 2.1 allows to extend the condition to the new intervals [T0,T1), [T1,T2), . . . and therefore this theorem can be considered as a generalization of the Theo- rem1.1.
For the existence of the inverse of the functionH(z), the Theorem1.1 proposes the condi- tion q> 0 on (0,T)and g(u) > 0 for u > 0. Since we shall not deal with the function H(z), we shall prove more general theorem.
Theorem 2.2. Suppose the following conditions are satisfied
p∈ C[0,T0)∩C1(0,T0) with p>0on(0,T0) (2.2) q∈ L1p[0,t∗] for any t∗ ∈(0,T0) with q≥0on(0,T0), (2.3) where L1r[0,a]is the space of functions u(t)withRa
0 |u(t)|r(t)dt< ∞,
Z t∗
0
1 p(s)
Z s
0 p(x)q(x)dxds<∞ for any t∗ ∈(0,T0), (2.4) g:[0,∞)→[0,∞)is nondecreasing on[0,∞)with g(u)≥0for u>0 (2.5) and
Z t∗
0
1 p(s)
Z s
0 p(x)q(x)g(a)dx<a for any t∗ ∈(0,T0).Then
a) (1.1) has a solution y ∈ C[0,T0)with py0 ∈ C[0,T0),(py0)0 ∈ L1pq(0,T0)and0 < y(t) ≤ a for t ∈[0,T0).
b) If RT1
T0
1 p(s)
Rs
0 p(x)q(x)g(T0)ds < y(T0), and conditions (2.2)–(2.5) satisfied then solution can be extended to the interval[0,T1).
In addition if either(1.8)and(1.9)holds, then y is a solution of (1.2).
Proof of Theorem2.2. Let us take y0(t) ≡ a, and define y1(t), y2(t), . . . from the recurrence relations
yn(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(yn−1(x))dxds, n=1, 2, . . . (2.6) For the sequence{yn(t)}we obtain
y1(t)≤y0(t) =a (2.7)
y2(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y1(x))dxds≥a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y0(x))dxds=y1(t), y3(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y2(x))dxds≤a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y1(x))dxds=y2(t), y3(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y2(x))dxds≥a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(a)dxds= y1(t),
y4(t) = a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y3(x))dxds≥ a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y2(x))dxds=y3(t), y4(t) = a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y3(x))dxds≤ a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y1(x))dxds=y2(t), ...
That is, {y2n(t)} and {y2n+1(t)} are monotonically nonincreasing and nondecresing se- quences, consecutively. Let us show that these sequences are equicontinuous. Indeed we have
|yn(t)−yn(r)|=
Z t
r
1 p(s)
Z s
0 p(x)q(x)g(yn−1(x))dxds≤ M Z t
r
1 p(s)
Z s
0 p(x)q(x)dxds, where
M=max{g(u): 0≤ u≤a}
and it follows from (2.4) that the right-hand side can be taken < ε for |t−r| < δ, regardless of the choice oft andr: the function ϕ(t) = Rt
0 1 p(s)
Rs
0 p(x)q(x)dxdsis (uniformly) continuous on [0,t∗]for any t∗ < T0. That is, the bounded and equicontinuous sequences {y2n(t)}and {y2n+1(t)}both have a limit. Denote by
nlim→∞y2n(t)≡u(t),
nlim→∞y2n+1(t)≡v(t).
Clearly we haveu(t)≥v(t). Now Lebesgue’s dominated theorem guarantees that u(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(v(x))dxds and v(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(u(x))dxds.
(2.8)
If u(t) = v(t) we have that the function u(t) is the solution of the problem (1.1), indeed it follows from
u(t) =a−
Z t
0
1 p(s)
Z s
0
p(x)q(x)g(u(x))dxds that
u0(t) =− 1 p(t)
Z t
0 p(x)q(x)g(u(x))dx, pu0 = −
Z t
0 p(x)q(x)g(u(x))dx, pu00
= −pqg(u).
So we supposeu(t)6= v(t)and consider the operatorN :C[0,T0)→C[0,T0)defined by Ny(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y(x))dxds. (2.9) Next let
K ={y∈C[0,T0):v(t)≤y(t)≤u(t)fort∈[0,T0)}.
Clearly K is closed, convex, bounded subset of C[0,T0) and N : K → K. Let us show that N : K → K is continuous and compact. Continuity follows from Lebesgue’s dominated con- vergence theorem: if yn(t) → y(t), then Nyn(t) → Ny(t). To show that N is completely continuous lety(t)∈K,t∗< T0, then
|Ny(t)−Ny(r)| ≤M
Z t
r
1 p(x)
Z x
0 p(z)q(z)dzds
fort,r∈ [0,t∗], that is Ncompletely continuous on[0,T0).
The Schauder–Tychonoff theorem guarantees that N has a fixed point w ∈ K, i.e. w is a solution of (1.1).
Now ifw(T0)>0, andRT1 T0
1 p(s)
Rs
0 p(x)q(x)g(T0)ds<w(T0) =bwe take y0(t) =
(w, if 0≤t ≤T0 b, if T0≤t <T1 yn(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(yn−1(x))dxds, n=1, 2, . . .
(2.10)
and in like manner we obtain the solutionwof the problem (1.2) on the interval[0,T1). Clearly we obtain for this solution
w(T0) =b= w(T0) =a−
Z T0
0
1 p(s)
Z s
0 p(x)q(x)g(w(x))dxds and
w0(T0+) = lim
t→0+
w(T0+t)−w(T0) t
=− lim
t→0+
RT0+t T0
1 p(s)
Rs
0 p(x)q(x)g(w(x))dxds
t .
Using L’Hôpital’s rule we obtain w0(T0+) = − lim
t→0+
1 p(T0+t)
Z T0+t
0 p(x)q(x)g(w(x))dx
=− 1 p(T0)
Z T0
0
p(x)q(x)g(w(x))dx=w0(T0−) and thereforew0 ∈ C[0,T1). It is also clear that pw0 is differentiable and
pw00
=−pqg(w) for all t∈[0,T1).
If (1.8) or (1.9) holds we easily havew0(0) =0 and thereforewis the solution of (1.2).
Now we will prove the stronger result which generalizes the Theorems1.1,2.1and2.2.
Consider the problem
(py0)0+p(t)h(t,y) =0, t∈[0,T) y(0) =a>0,
tlim→0+p(t)y0(t) =0
(2.11)
or
(py0)0+p(t)h(t,y) =0, t∈ [0,T) y(0) =a>0,
y0(0) =0
(2.12)
and some preliminary problem
(pz0)0+p(t)ϕ(t) =0, t∈ [0,T) z(0) =a >0,
tlim→0+p(t)z0(t) =0
(2.13)
or
(pz0)0+p(t)ϕ(t) =0, t∈ [0,T) z(0) =a >0,
z0(0) =0.
(2.14)
Theorem 2.3. Suppose that p∈C[0,T0)∩C1(0,T0)with p>0on(0,T0)and p is integrable, k(t)≥ ϕ(t)−h(t,y)≥0, t∈(0,T0) (2.15) where p(t)k(t)and p(t)ϕ(t)are integrable with
Z t∗
0
1 p(s)
Z s
0 p(t)k(x)dxds< ∞ and Z t∗
0
1 p(s)
Z s
0 p(t)ϕ(x)dxds<∞ for any t∗ ∈ (0,T0),
(2.16)
ϕ(t)∈ C(0,T0)and such that the problem(2.13) has nonnegative solution z(t),for each t ∈ (0,T0), h(t,·) is continuous, for each fixed y, h(·,y) is measurable on [0,T0], then the problem (2.11) has nonnegative solution on[0,T0).
Note 2.4. Theorem2.3is the generalization of Theorem2.2in the following sense: since g(y) in the Theorem2.2 is nonincreasing we have that g(a) ≥ g(y)and therefore for the solution of the problem (1.1) we have
y(t) =a−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(y)dxds
=a−
Z t
0
1 p(s)
Z s
0 p(x)q(x) [g(y)−g(a)]dxds−
Z t
0
1 p(s)
Z s
0 p(x)q(x)g(a)dxds, that is
y(t) =z(t) +
Z t
0
1 p(s)
Z s
0
[g(a)−g(y)]p(x)q(x)dxds,
wherez(t)is the solution of the problem (2.13) withϕ(t) = g(a). Thus Theorem2.2is a special case of Theorem2.3 with ϕ(t) =g(a)andk(t) =g(a)−g(y).
Note 2.5. Theorem2.3shows that the nondecreasing condition ofg(y)in the statement of the Theorem2.2 can be omitted and therefore the scope of problems can be seriously extended.
Proof of Theorem2.3. It follows from the condition (2.16) that the problem (2.13) has nonnega- tive solution:
z= a−
Z t
0
1 p(s)
Z s
0 p(x)ϕ(x))dxds (2.17) on some interval[0,T0).
We will show that the problem (2.11) is equivalent to the (integral) equation:
y(t) =z(t) +
Z t
0
1 p(s)
Z s
0 p(x) [ϕ(x)−h(x,y)]dxds. (2.18) Let us calculate the derivativesy0(t)and(py0(t))0 from (2.18) by using the Leibniz rule:
y0(t) =z0(t) + 1 p(t)
Z t
0 p(x) [ϕ(x)−h(x,y(x))]dx, p(t)y0(t) = p(t)z0(t) +
Z t
0 p(x) [ϕ(x)−h(x,y(x))]dx, py0(t)0 = pz0(t)0+p(t)ϕ(t)−p(t)h(t,y(t)),
and since (pz0)0+p(t)ϕ(t) = 0 we obtain(py0(t))0+p(t)h(t,y(t)) =0. That is, the equation (2.18) is equivalent to the problem (2.11). Let us consider the recurrence relations
y0(t) =z(t), y1(t) =z(t) +
Z t
0
1 p(s)
Z s
0
p(x) [ϕ(x)−h(x,y0)]dxds, . . . yn(t) =z(t) +
Z t
0
1 p(s)
Z s
0 p(x) [ϕ(x)−h(x,yn−1)]dxds, . . .
(2.19)
We have
|yn(t)−z(t)| ≤
Z t
0
1 p(s)
Z s
0 p(x) [ϕ(x)−h(x,yn−1)]dxds
≤
Z t
0
1 p(s)
Z s
0 p(x)k(x)dxds
and
|yn(t)−z(t)−(yn(r)−z(r))| (2.20)
=
Z t
r
1 p(s)
Z s
0 p(x) [ϕ(x)−h(x,yn−1)]dxds
≤
Z t
r
1 p(s)
Z s
0 p(x)k(x)dxds.
Thus, the sequence {yn(t)−z(t)} is uniformly bounded and equicontinuous on [0,t∗] for any t∗ < T0 and therefore by Ascoli–Arzelà lemma, there exists a continuous w(t)such that ynk(t)−z(t) → w(t)uniformly on [0,t∗]. Without loss of generality, sayyn(t)−z(t)→ w(t) or yn(t)→z(t) +w(t)≡y(t). Then we obtain
y(t) =z(t) + lim
n→∞ Z t
0
1 p(s)
Z s
0 p(x) [ϕ(x)−h(x,yn)]dxds
=z(t) +
Z t
0
1 p(s)
Z s
0 p(x) [ϕ(x)−h(x,y(x))]dxds
(2.21)
using the Lebesgue dominated convergence theorem. Thus y(t) ≥ 0 is the solution of the problem (2.11).
Example 2.6. The problem
(t−1/3y0)0+t−1/3 1
2
t2,5 t3+ (y−1)2
=0, t∈ h0, √31
4
i
y(0) =1,
tlim→0+p(t)y0(t) =0 has a positive solutiony=−t3/2+1. Forh(t,y)we have
h(t,y) = 1 2
t2,5
t3+ (y−1)2 ≤ 1 2√
t ≡ ϕ(t), and
z(t) =−2t3/2+1≥0 is the solution of the problem
(t−1/3z0)0+t−1/3ϕ(t) =0, z(0) =1,
tlim→0+p(t)z0(t) =0.
For the iteration sequenceynin (2.19) we obtain (by using standard Latex tools) y0(t) =−2t3/2+1,
y1(t) =−2t3/2+1+
Z t
0 s1/3 Z s
0 x−1/3 1
2x1/2 − 1 2
x2.5 x3+4x3
dxds
=−2t3/2+1+ 8
5t3/2 =1− 2 5t32, y2(t) =1− 50
29t32, y3(t) =1−1682
3341t32, y4(t) =1−1. 595 6t32, y5(t) =1−0.564 03t32, . . . ,
y17(t) =1−0.711 85t32, . . .
Conflict of interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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