Electron. J. Differential Equations, Vol. 2016 (2016), No. 251, pp. 1-17.

Positive solutions for a second-order Phi-Laplacian equations with limiting nonlocal boundary conditions

George L. Karakostas, Konstantina G. Palaska, Panagiotis Ch. Tsamatos

Abstract:
Motivated, mainly, by the works of Fewster-Young and Tisdell [9,10] and Orpel [30], as well as the papers by Karakostas [21,22,23], we give sufficient conditions to guarantee the existence of (nontrivial) solutions of the second-order Phi-Laplacian equation
$$ \frac{1}{p(t)}\frac{d}{dt}[p(t)\Phi(u'(t))]+(Fu)(t)=0,\quad\text{a.e. } t\in[0,1]=:I, $$
which satisfy the nonlocal boundary value conditions of the limiting Sturm-Liouville form
$$ \lim_{t\to 0}[p(t)\Phi(u'(t))]=\int_0^1u(s)d\eta(s),\quad \lim_{t\to 1}[p(t)\Phi(u'(t))]=-\int_0^1u(s)d\zeta(s). $$
Here $\Phi$ is an increasing homeomorphism of the real line onto itself and F is an operator acting on the function u and on its first derivative with the characteristic property that $u\to p(Fu)$ is a $C^0$-type, or $C^1$-type Caratheodory operator, a meaning introduced here. Examples are given to illustrate both cases.

Submitted April 19, 2016. Published September 20, 2016.
Math Subject Classifications: 34B18, 34B10.
Key Words: Positive solution; Sturm-Liouville equation; Phi-Laplacian; Schauder's fixed point theorem.

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George L. Karakostas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: gkarako@uoi.gr, gkarako@hotmail.com
Konstantina G. Palaska
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: cpalaska@cc.uoi.gr
Panagiotis Ch. Tsamatos
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: ptsamato@cc.uoi.gr

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