Taking in Theorem 3.1, we can obtain the existence of three distinct solutions for the Duffing-type equation without perturbation (1.2) as following.
Using the variational principle of Ricceri and a local mountain pass lemma, we study the existence of three distinct solutions for the following resonant Duffing-type equations with damping and perturbed term, a.e., and without perturbed term, a.e.,.
Due to Lemma 2.7 with (Gamma=lambda H ), we conclude that for each compact interval ([a_{0},b_{0} ]subset( hat{ell}, +infty)), there exists a number (tau > 0) such that, for each (thetain[0,tau]), the problem (N) has at least three distinct solutions for every (lambdain[a_{0},b_{0} ]).
Then, there exists a constant ℓ̂ with the following property: for each compact interval ([a_{0},b_{0} ]subset( hat{ell}, +infty)), there exists a number (tau > 0) such that, for each (thetain[0,tau]), the problem (N) has at least three distinct solutions for every (lambdain[a_{0},b_{0} ]).
SQL LIMS, for example, has distinct solutions for pharmaceutical laboratories, chemical, nourishing, forensic and water analysis laboratories.
This pressure apparently triggered independent evolution of several distinct solutions for the set of relatively simple reactions that are required for ROS detoxification.
In contrast to earlier studies, Chen and Chen [10] offered a distinct solution for image authentication in 2012.
They do support, however a distinct solution for in-group Tetrapulmonata, with Pedipalpi instead of Labellata, and only one study [ 5] recovered Poecilophysidea with moderate support.
In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices.
If y1 x) and y2 x) are two distinct solutions of the equation, then any combination ay1 x) + by2 x will also be a solution, called the general solution, for any constants a and b.
Given that distinct solutions are required for each of these problems, Newton and colleagues have stressed the need for a consensus on definitions and an end to interchangeable use of these terms [ 14].
