Second order linear differential equations with almost periodic solutions

In this paper, a characterization of almost periodic strongly continuous Sine operator function is given, and in a Hilbert space, it is shown that the almost periodicity of a Sine operator function implies that of the corresponding Cosine operator function.     

First order functional differential equations with periodic boundary conditions

In this article, the authors prove an existence theorem for periodic boundary value problems for first order functional differential equations (FDE) in Banach algebras under mixed generalized Lipschitz and Carathéodory conditions. The existence of extremal positive solutions is also proved under certain monotonicity conditions. An example illustrating the results is included.     

Advanced differential equations with nonlinear boundary conditions

We investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions. Sufficient conditions when the problem has extremal solutions or a unique solution are formulated. Linear advanced differential inequalities are also discussed.     

First order differential equations with piecewise constant arguments and nonlinear boundary value conditions

This paper is devoted to the study of differential equations with piecewise constant arguments coupled with nonlinear boundary value conditions. Under suitable assumptions on the data of the equation, by means of the method of (weakly coupled) lower and upper solutions, we derive the existence of extremal solutions and extremal quasi-solutions. Moreover some results are given concerning the uniqueness of solutions. Furthermore, we deduce some maximum principles related to the linear equation which allow us to develop the monotone iterative method. Some illustrative examples are also presented.     

On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions

We prove the existence of positive solutions of second-order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions. Obtained results yield positive periodic solutions of the equation on the whole real axis, provided that the coefficients are periodic.     

Two point boundary value problems associated with a system of first order nonlinear impulse differential equations

We show assumptions of boundedness and Lipschitz conditions that guarantee the existence of a unique solution to a nonlinear differential boundary value problem. The result utilizes a variation of parameters of formula due to V. Lakshmikantham, et. al. and depends upon a contraction mapping.     

Second order differential operators with Feller-Wentzell type boundary conditions

In this paper, we study a basic generation problem concerning the second order differential operator in the space C[0,1] of complex continuous functions equipped with Feller-Wentzell type boundary conditions, which originates from the work of Feller [W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952) 468-519]. We prove successfully that the operator, under suitable assumptions, generates a strongly continuous cosine function on C[0,1] (or on a subspace of C[0,1]), by means of an operator matrix analysis combined with perturbation, approximation, and similarity techniques.     

On a system of second order differential equations with periodic impulse coefficients

A thorough investigation of the systemd~2y(x):dx~2 p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤xx_0>0) -η, x_0≤x<2π(η>0)p(x)=p(x 2π),-∞相似文献   

Fourth order nonselfadjoint differential equations with clamped-free boundary conditions

Summary A class of nonselfadjoint fourth order differential equations is investigated in this paper by a pair of equations of the second order. Special attention is given to establishing conditions for the existence of solutions subject to two point boundary conditions, and is achieved basically through various characterizations, comparison theorems and related eigenvalue problems. Entrata in Redazione il 1^o giugno 1977. The first author is supported by the NRC of Canada under Grant number A3105.     

Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions

This paper is concerned with the existence of solutions for systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. By establishing new comparison results and applying the monotone iterative technique, we obtain the sufficient conditions for the existence of extremal system of solutions.     

Mild and strong solutions to few types of fractional order nonlinear equations with periodic boundary conditions

In this paper, the two fractional periodic boundary value problems $$_0^C D_{0 + }^\alpha u\left( t \right) - \lambda u\left( t \right) = f\left( {t,u\left( t \right)} \right), u\left( 0 \right) = u\left( 1 \right), 0 < \alpha < 1,$$ and $$_0^C D_{0 + }^\beta u\left( t \right) - \lambda u\left( t \right) = f\left( {t,u\left( t \right)} \right), u\left( 0 \right) = u\left( 1 \right),u'\left( 0 \right) = 0 1 < \beta < 2,$$ will be studied where ₀ ^C D _t ^α is the ordinary Caputo fractional derivative and λ ∈ ? ?{0}. Under some suitable assumptions on the function f, the existence of at least one mild solution will be proved. Under some conditions, the uniqueness of this mild solution will be proved to both problems. Finally, these mild solutions will be strong solutions under certain conditions.     

Variational approach to impulsive differential equations with periodic boundary conditions

In this paper, we obtain some new existence results of solutions of impulsive differential equations with periodic boundary conditions. The main tool that we use is critical point theory. Our results generalize some existing results on periodic solutions for second order ordinary differential equations even when the impulses are absent.     

A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation $-(1/w)({\mathit{pu}}^{\prime }{)}^{\prime }+\mu u=\mathit{Fu}$, where F is a nonlinear operator, with nonlinear boundary conditions. Under appropriate assumptions on $p,w,F$ and the boundary conditions, the existence of solutions is established. If the problem has a lower solution and an upper solution, then we use a quasilinearization method to obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.     

Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions

The paper deals with the impulsive nonlinear boundary value problem

     

PERTURBATION RESULTS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

Two perturbation results on the linear differential function f" ∏(z)A(z)f = 0 areobtained, where ∏(z) and A(z) are periodic entire functions with period 2πi and σe(∏) ＜σe(A).     

Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations

This paper is concerned with the solvability of periodic boundary value problems for nonlinear impulsive functional differential equations

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Existence of periodic solutions of nonlinear systems of differential equations with impulse effect

Sufficient conditions for the existence of periodic solutions of nonlinear systems of differential equations with impulses in a canonic domain have been found in the paper.     

Strongly nonlinear second order differential inclusions with generalized boundary conditions

In this paper, we study second order differential inclusions in with a maximal monotone term and generalized boundary conditions. The nonlinear differential operator need not be necessary homogeneous and incorporates as a special case the one-dimensional p-Laplacian. The generalized boundary conditions incorporate as special cases well-known problems such as the Dirichlet (Picard), Neumann and periodic problems. As application to the proven results we obtain existence theorems for both “convex” and “nonconvex” problems when the maximal monotone term A is defined everywhere and when not (case of variational inequalities).     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.