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We study the existence and multiplicity of positive solutions to the periodic problem u ′′ = p ⁢ ( t ) ⁢ u - q ⁢ ( t , u ) ⁢ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\quad u^{\prime}(0)=u^{\prime}(\omega), where p , f ∈ L ⁢ ( [ 0 , ω ] ) p,f\in L([0,\omega]) and q : [ 0 , ω ] × R → R q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R} is a Carathéodory function. By using the method of lower and upper functions, we show some properties of the solution set of the considered problem and, in particular, the existence of a minimal positive solution.

In this paper, we consider the existence of analytic invariant curves of an iterative equation $$f(f(x)) = x{{\rm{e}}^{a - x - f(x)}}$$ which arises from Ricker-type second-order equation. By reducing the equation with the Schroder transformation to an auxiliary equation, the author discusses not only that the parameter at resonance, i.e., at a root of the unity, but also the parameter near resonance under the Brjuno condition.

A second order nonlinear differential equation is considered. An algorithm for reconstructing an input from inaccurate measurements of the solution at discrete times is designed. The algorithm based on the constructions of feedback control theory and theory of ill-posed problems is stable with respect to informational noises and computational errors.

In this paper, under the generalized assumptions of the memory kernel function, using the idea introduced by Lasiecka and Wang (Intrinsic decay rate estimates for semilinear abstract second order equations with memory. Springer INdAM Series 10, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Chap. 14, Springer, Switzerland, 2014) we prove the general decay result to an initial boundary value problem of a wave equation with variable density. Our general assumptions of the memory kernel function generalize the assumptions in the literature (Alabau-Boussouira et al. in J Funct Anal 254(5):1342–1372, 2008; Messaoudi and Khulaifi in Appl Math Lett 66:16–22, 2017).

This work aims to study the removal of lead (II) from aqueous solutions with Ca-Alginate (CA) and Coffee-Calcium-Alginate (CCA). The coffee biomass were successfully prepared as the material to be encapsulated in calcium alginate. The characterization of the synthesized CA and CCA was performed using fourier transform infrared and scanning electron microscopy. The method used was batch study. Various factors which affected adsorption efficiency of lead (II) ions by CA and CCA, such as pH, agitation time, and adsorbent dose were investigated for determination of optimum experimental conditions. The result showed that CA and CCA had significant effects on adsorption of lead (ІІ) ions at pH = 4, agitation time of more than 120 min, and the adsorbent dose was 0.05 gram. Moreover, the Langmuir isotherm model showed that maximum adsorption capacity (qm) was 163.66 mg/g and 176.99 mg/g respectively for CA and CCA. The Langmuir isotherm was better described the adsorption equilibrium. Both of the adsorbent fitted to pseudo second order equations. These results demonstrated that CA and CCA show great potential to remove Pb(ІІ) ions from aqueous solutions.

We consider the following second order equation u¨(t)+γu˙(t)+(I−T)u(t)+∇ϕ(u(t))=0, where T:H→H is quasi-nonexpansive and Lipschitz continuous on bounded sets and ϕ:H→R is a continuously differentia...

We consider the singular initial value problem for the second order differential equation. We interested in the existence of positive solutions and proved an easily applicable theorem on the existence of positive solutions to initial-value problems for second-order nonlinear singular differential equations. The previously established result on the existence of positive solution by Agarwal and O'Regan is not easy for the applications due to very complex definition of a new function and their properties in the statement of their theorem. We used the Lebesgue dominated convergence theorem and the Schauder--Tychonoff theorem in the proof of the main result. The main result can be easily applied for the singular and regular type of problems.

In this study, mitigation of Cu(II) from aqueous solutions was evaluated using the sandal Santalum Album as biosorbent. The removal capacity of the Cu(II) was studied in batch experiments including such as initial metal concentration (25-400 mg/L), biosorbent dose (0.05-0.2 g), effects of contact time (0-1440 min), kinetics and pre-treatment of the biosorbent. The maximum removal of Cu(II) was obtained at the optimum pH of 5. The maximum equilibrium uptake for Cu(II) was Ba(OH)2-SA(248.8 mg/g-1) at 400 ppm. Kinetics studies explained that, mitigation of the Cu(II) was very fast from 0 to 120 min and achieved equilibrium at 240 min. The equilibrium was well explained by Frendlich isotherm. Kinetics was well fitted to pseudo second order equation with high R2 values.

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second-order equations with real constant coefficients in the layer RTn+1=Rn×(0,T), where n≥ 1 a...

In the paper explicit functional continuous Runge–Kutta and Runge–Kutta–Nystrom methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the special form are constructed with the reuse of the last stage of the step. The order conditions for Runge–Kutta–Nystrom methods are derived. Methods of orders three, four and five which require less computations than the known methods are presented. Numerical solution of the test problems confirm the convergence order of the new methods and their lower computational cost is performed.

Let D ( t 0 , e ) be the splitting distance of the stable and unstable manifold of a time-periodic second order equation. We expand D ( t 0 , e ) as a formal power series in e as D ( t 0 , e ) = E 0 ( t 0 ) + e E 1 ( t 0 ) + ⋯ + e n E n ( t 0 ) + ⋯ . In this paper we derive an explicit integral formula for E 1 ( t 0 ) . We also evaluate E 1 ( t 0 ) to prove the existence of homoclinic tangles for an equation to which the Poincare/Melnikov method fails to apply.

In this paper, we study the well-posedness of a linear inverse problem for a mixed-type second-order multidimensional equation of the first kind. We prove the unique solvability of this problem in a certain function class with the help of “e-regularization”, a priori estimates, and successive approximation methods.

Maxilon Blue 5G dye and Turkey green clay minerals used in adsorption experiments are of great importance recently due to their biological and physicochemical properties. Therefore, kinetic parameters of the adsorption of maxilon blue 5G dye on green clay were investigated. Studies were performed under adsorption conditions such as pH (5-11), maxilon blue 5G dye concentration (0.5.10-6 - 2.10-5 M) and temperature (298-328 K). Turkey green clay and maxilon blue 5G dye were characterized by BET, SEM and XRF. The most favorable conditions for the dye adsorption of clay mineral, which is the support substance, were found to be pH 11 and the temperature was 328 K. The kinetic datas obtained from experimental studies were investigated on three different kinetic models such as pseudo-first-order, pseudo-second-order and intraparticle diffusion and it was determined that the adsorption event was carried out more compatible with the second order equation. Thermodynamic functions such as activation enegy (Ea), enthalpy (ΔH), Gibbs free mechanism (ΔG) and entropy (ΔS) were calculated. According to the results obtained from the experimental datas, the interaction between green clay and maxilon blue 5G are a physical interaction and experimental processes of adsorption are endothermic. These results indicated that green clay can be used as an adsorbent for the adsorption of the maxilon blue 5G (MB-5G).

We consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x ″ + f ( t , x ) = 0 . The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems. We show how a rotation number approach, together with the Poincare–Birkhoff theorem and the phase-plane analysis of the spiral properties, allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity. These systems may also be resonant. In particular, our approach can be used to deal with the problems without both global existence of the Cauchy problems associated to the equation and the sign assumption on f. The typical example is a partially superlinear second order equation. Our main result generalizes some classical results of Jacobowitz and Hartman, among others.

In this paper we prove the unique solvability and smoothness of the solution of a nonlocal boundary-value problem for a multidimensional mixed type second-order equation of the second kind in Sobolev space Wℓ2(Q), (2≤ℓ is an integer). First, we have studied the unique solvability of the problems in the space W22(Q). Solution uniqueness for a nonlocal boundary-value problem for a mixed-type equation of the second kind is proved by the methods of a priori estimates.Further, to prove the solution existence in the space W22(Q), the Fourier method is used. In other words, the problem under consideration is reduced to the study of unique solvability of a nonlocal boundary value problem for an infinite number of systems of second-order equations of mixed type of the second kind. For the unique solvability of the problems obtained, the ``ε-regularization'' method is used, i.e, the unique solvability of a nonlocal boundary-value problem for an infinite number of systems of composite-type equations with a small parameter was studied by the methods of functional analysis. The necessary a priori estimates were obtained for the problems under consideration. Basing on these estimates and using the theorem on weak compactness as well as the limit transition, solutions for an infinite number of systems of second-order equations of mixed type of the second kind are obtained. Then, using Steklov-Parseval equality for solving an infinite number of systems of second-order equations of mixed type of the second kind, the unique solvability of original problem was obtained. At the end of the paper, the smoothness of the problem's solution is studied.

We proved a new theorem on the existence of positive solutions to initial-value problems for second-order nonlinear singular differential equations. The existence of solutions is proven under considerably weaker than previously known conditions.

We present an overview of recent results of the author related to the Aleksandrov type estimates with power of summability of the free term $$d_{0}

Various results and techniques, such as Bohl-Perron theorem, a priori solution estimates, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equations $$ \dot{x_i}(t)=A_i(t)x_i(h_i(t)) +\sum_{j=1}^n \sum_{k=1}^{m_{ij}} B_{ij}^k(t)x_j(h_{ij}^k(t)) + \sum_{j=1}^n\int\limits_{g_{ij}(t)}^t K_{ij}(t,s)x_j(s)ds,~i=1,\dots,n. $$ Here $x_i$ are unknown vector functions, $A_i, B_{ij}^k, K_{ij}$ are matrix functions, $h_i,h_{ij}^k, g_{ij}$ are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations., 14 pages

In this paper, we consider a parabolic second order equation with changing direction of time. Under certain conditions, the existence of regular solutions in a suitable weighted Sobolev space is proven by regularization. The uniqueness of a generalized solution is also established as a consequence.

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form$$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ωwhere the coefficients$$a_{ij}$$aijbelong to the Central (that is, Local) Sarason class CVMO andfis assumed to be in some Modified Local Generalized Morrey Spaces$$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$LM~{x0}p,φ. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.

International audience; This article proposes a new control law for an embedded DC distributed network supplied by a supercapacitor module (as a supplementary source) and a battery module (as the main generator) for transportation applications. A novel control algorithm based on the nonlinear differential flatness approach is studied and implemented in the laboratory. Using the differential flatness theory, straightforward solutions to nonlinear system stability problems and energy management have been developed. To evaluate the performance of the studied control technique, a hardware power electronics system is designed and implemented with a fully digital calculation (real-time system) realized with a MicroLabBox dSPACE platform (dual-core processor and FPGA). Obtained test bench results with a small scale prototype platform (a supercapacitor module of 160 V, 6 F and a battery module of 120 V, 40 Ah) corroborate the excellent control structure during drive cycles: steady-state and dynamics.

Con el propósito de determinar el efecto y la dosis óptima de potasio en la producción de minitubérculos del clon mexicano de papa 99-39, se establecieron experimentos en julio y septiembre de 2015, donde se probaron concentraciones de 0 (testigo absoluto), 150, 250 (testigo comercial), 350, 450 y 550 mg L-1 en solución nutritiva en sistema hidropónico. Los resultados indicaron que la mayor altura de planta e índice de área foliar se encontraron en los tratamientos de mayor concentración de K. La dosis de 350 mg L-1 de K produjo 19% más tubérculos de diámetro igual o mayor de 15 mm por planta que el testigo comercial y 194% más que el testigo absoluto. La ecuación de segundo orden (R2= 0.94) permitió estimar que la dosis óptima para el clon de papa 99-39 fue de 400 mg L-1 de potasio, con lo que se puede producir hasta 20.67 minitubérculos por planta. Hubo significancia estadística en la interacción fecha x dosis para producción de tubérculos por planta y para el índice de cosecha, con mayor producción en la siembra de julio que en la época de otoño invierno; contrariamente, se encontró que el índice de cosecha fue mayor en la época de otoño- invierno. Concentraciones elevadas de potasio durante todo el ciclo de cultivo promovieron la producción de un mayor número de tubérculos.

We investigate strongly nonlinear differential equations of the type $$\bigl(\Phi \bigl(k(t) u'(t) \bigr) \bigr)'= f \bigl(t,u(t),u'(t) \bigr), \quad\text{a.e. on } [0,T], $$ where Φ is a strictly increasing homeomorphism and the nonnegative function k may vanish on a set of measure zero. By using the upper and lower solutions method, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions involving the function k. Our existence results require a weak form of a Wintner–Nagumo growth condition.

We develop a new method for studying inverse problems of mathematical physics by using systems of equations of ray expansion of solutions to parabolic and hyperbolic equations with variable coefficients.

Some method is proposed for studying the identification problems for second-order equations of evolutionary type, in particular, parabolic.We give the new representations of solutions and coefficients of such type of equations using integral transformations.

We obtain some new Gronwall type inequalities where, instead of linear growth assumptions, we allow quadratic (or more) growth provided some additional conditions are satisfied. Applications are made to both local and nonlocal boundary value problems for some second order ordinary differential equations which have quadratic growth in the derivative terms.

We prove exact rate of decay for solutions to a class of second order ordinary differential equations with degenerate potentials, in particular, for potential functions that grow as different powers in different directions in a neigborhood of zero. As a tool we derive some decay estimates for scalar second order equations with non-autonomous damping.

An optimal control problem with second order equation is considered. For this problem Pontryagin maximum principle is obtained. Also non-triviality condition is proved.

This paper deals with the periodic solutions of second order equations with asymptotical non-resonance. Using the point of view that the force is a perturbation, we can think that, asymptotically, the solutions of forced non-autonomous equation behave as those of the autonomous equation. Then, under a sharp integral condition, we prove that the periodic solution of non-autonomous equation can be estimated by using time map of autonomous equation. The existence of periodic solutions is thus proved via qualitative analysis and topological degree theory. The main result in this paper generalize a existence result obtained by Capietto, Mawhin and Zanolin.