On the monotonicity property for a certain class of second order differential equations

Bounded and Zero-Convergent Solutions of a Class of Stieltjes Integro-Differential Equations

“…This answered an earlier problem posed in [7] and extended the results for the linear case. Corollary 3 improves further the result of [1] by relaxing assumptions on the function /. Whereas the authors of [ 1 ] used a fixed point theorem for operators defined by Schauder's linearization device, we are able to obtain a more general result by a more or less standard application of the SchauderTychonov Fixed Point Theorem.…”

“…From the above, it follows that all proper solutions of ( 1 ) are nonoscillatory and, as in [1], can be divided into the following two classes: A = {x, a proper solution of (1) : 1tx > bx : x{t)x\t) > 0 for t > tx} , B = {x, a proper solution of (1) : x(t)x (t) < 0 for t > bx}.…”

“…In a recent paper [1] Cecchi, Marini, and Villari considered the nonlinear equation (3) (P(t)x'(t))' = q(t)f(x(t)) and gave conditions under which there exist solutions that are convergent to zero and, at the same time, solutions that have nonzero limit at infinity. This answered an earlier problem posed in [7] and extended the results for the linear case.…”

“…Thus, the solutions in class A are either eventually increasing and positive or eventually decreasing and negative, and those in class B are either nonincreasing and positive or nondecreasing and negative. The asymptotic behavior of solutions of equation (3) that belong to class (A) was studied in [6] and [8], whereas the work in [ 1 ] was devoted to a study of the asymptotic behavior of solutions in class (B). In this work, as indicated earlier, we examine the asymptotic behavior of solutions of (1) in classes (A) and (B), in § §2 and 3, respectively.…”

Existence of periodic solutions of fourth-order nonlinear difference equations

“…By using various methods and techniques, such as Schauder fixed point theorem, the cone theoretic fixed point theorem, the method of upper and lower solutions, coincidence degree theory, a series of existence results of nontrivial solutions for differential equations have been obtained in [4,6,8,20]. Critical point theory is also an important tool to deal with problems on differential equations [9,11,12,24,29,33].…”