Electron. J. Diff. Equ., Vol. 2015 (2015), No. 08, pp. 1-14.

Existence and uniqueness for superlinear second-order differential equations on the half-line

Imed Bachar, Habib Maagli

Abstract:
We prove the existence and uniqueness, and study the global behavior of a positive continuous solution to the superlinear second-order differential equation
$$\displaylines{ \frac{1}{A(t)}(A(t)u'(t))'=u(t)g(t,u(t)),\quad t\in (0,\infty ), \cr u(0)=a,\quad \lim_{t\to\infty} \frac{u(t)}{\rho (t)}=b, }$$
where a,b are nonnegative constants such that a+b>0, A is a continuous function on $[0,\infty)$, positive and continuously differentiable on $(0,\infty )$ such that 1/A is integrable on [0,1] and $\int_0^{\infty }1/A(t)\,dt=\infty $. Here $\rho (t)=\int_0^t 1/A(s)\,ds$, for $t\geq 0$ and g(t,s) is a nonnegative continuous function satisfying suitable integrability condition. Our Approach is based on estimates of the Green's function and a perturbation argument. Finally two illustrative examples are given.

Submitted October 11, 2014. Published January 5, 2015.
Math Subject Classifications: 34B15, 34B18, 34B27.
Key Words: Second order differential equation; boundary value problem; half-line; Green's function; positive solution.

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Imed Bachar
King Saud University, College of Science
Mathematics Department, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: abachar@ksu.edu.sa
Habib Mâagli
King Abdulaziz University, College of Sciences and Arts
Rabigh Campus, Department of Mathematics P.O. Box 344
Rabigh 21911, Saudi Arabia
email: habib.maagli@fst.rnu.tn, abobaker@kau.edu.sa

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