A note on the existence of positive solutions of singular initial-value problem for second order differential equations

. We are interested in the existence of positive solutions to initial-value problems 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.


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, ) Agarwal  where 0 < T ≤ ∞, p ≥ 0, q ≥ 0 and g : [0, ∞) → [0, ∞).
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).

b) If
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 [T 0 , T 1 ), [T 1 , T 2 ), . . . and therefore this theorem can be considered as a generalization of the Theorem 1.1.
For the existence of the inverse of the function H(z), the Theorem 1.1 proposes the condition 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 In addition if either (1.8) and (1.9) holds, then y is a solution of (1.2).
Proof of Theorem 2.2. Let us take y 0 (t) ≡ a, and define y 1 (t), y 2 (t), . . . from the recurrence relations For the sequence {y n (t)} we obtain That is, {y 2n (t)} and {y 2n+1 (t)} are monotonically nonincreasing and nondecresing sequences, consecutively. Let us show that these sequences are equicontinuous. Indeed we have |y n (t) − y n (r)| =     p(x)q(x)g(y(x))dxds. (2.9) Clearly K is closed, convex, bounded subset of C[0, T 0 ) and N : K → K. Let us show that N : K → K is continuous and compact. Continuity follows from Lebesgue's dominated convergence theorem: if y n (t) → y(t), then Ny n (t) → Ny(t). To show that N is completely continuous let y(t) ∈ K, t * < T 0 , then that is N completely continuous on [0, T 0 ). The Schauder-Tychonoff theorem guarantees that N has a fixed point w ∈ K, i.e. w is a solution of (1.1). Now if w(T 0 ) > 0, and Using L'Hôpital's rule we obtain and therefore w ∈ C[0, T 1 ). It is also clear that pw is differentiable and for all t ∈ [0, T 1 ).
If (1.8) or (1.9) holds we easily have w (0) = 0 and therefore w is the solution of (1.2). Now we will prove the stronger result which generalizes the Theorems 1. and some preliminary problem (2.14) where p(t)k(t) and p(t)ϕ(t) are integrable with t * 0 1 p(s) s 0 p(t)k(x)dxds < ∞ and t * 0 1 p(s) s 0 p(t)ϕ(x)dxds < ∞ for any t * ∈ (0, T 0 ), (2.16) ϕ(t) ∈ C(0, T 0 ) and such that the problem (2.13) has nonnegative solution z(t), for each t ∈ (0, T 0 ), h(t, ·) is continuous, for each fixed y, h(·, y) is measurable on [0, T 0 ], then the problem (2.11) has nonnegative solution on [0, T 0 ). Note 2.4. Theorem 2.3 is the generalization of Theorem 2.2 in the following sense: since g(y) in the Theorem 2.2 is nonincreasing we have that g(a) ≥ g(y) and therefore for the solution of the problem (1.1) we have where z(t) is the solution of the problem (2.13) with ϕ(t) = g(a). Thus Theorem 2.2 is a special case of Theorem 2.3 with ϕ(t) = g(a) and k(t) = g(a) − g(y). Note 2.5. Theorem 2.3 shows that the nondecreasing condition of g(y) in the statement of the Theorem 2.2 can be omitted and therefore the scope of problems can be seriously extended.
Proof of Theorem 2.3. It follows from the condition (2.16) that the problem (2.13) has nonnegative solution: on some interval [0, T 0 ). We will show that the problem (2.11) is equivalent to the (integral) equation: (2.18) Let us calculate the derivatives y (t) and (py (t)) from (2.18) by using the Leibniz rule: and since (pz ) + p(t)ϕ(t) = 0 we obtain (py (t)) + 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 y 0 (t) = z(t),  Thus, the sequence {y n (t) − z(t)} is uniformly bounded and equicontinuous on [0, t * ] for any t * < T 0 and therefore by Ascoli-Arzelà lemma, there exists a continuous w(t) such that y n k (t) − z(t) → w(t) uniformly on [0, t * ]. Without loss of generality, say y n (t) − z(t) → w(t) or y n (t) → z(t) + w(t) ≡ y(t). Then we obtain y(t) = z(t) + lim