Zdeněk Opluštil

On oscillations of solutions to second-order linear delay differential equations

Abstract. New oscillation criteria are proved for second-order two-term linear delay differential equations with locally integrable coefficients and measurable delays. In particular, a suitable a priori lower bound of non-oscillatory solutions is found, which plays a crucial role in the proofs of the results obtained.

Fredholm alternative for the second-order singular Dirichlet problem

AbstractConsider the singular Dirichlet problem u″=p(t)u+q(t);u(a)=0,u(b)=0, where p,q:]a,b[→R are locally Lebesgue integrable functions. It is proved that if ∫ab(s−a)(b−s)[p(s)]−ds<+∞, then the Fredholm alternative remains true.MSC:34B05.

Oscillatory criteria for two-dimensional system of difference equations

Abstract Some oscillation criteria are established for two-dimensional systems of first order linear difference equations.

On a Nonlocal Boundary Value Problem for First Order Nonlinear Functional Differential Equations

A nonlocal boundary value problem for nonlinear functional equations is studied. New effective conditions are found for solvability a unique solvability of considered problem. Obtained results are concretized for differential equation with deviating argument.

On non-oscillation for certain system of non-linear ordinary differential equations

Abstract We consider the following two-dimensional system of non-linear equations: u ′ = g ⁢ ( t ) ⁢ | v | 1 α ⁢ sgn ⁡ v , v ′ = - p ⁢ ( t ) ⁢ | u | α ⁢ sgn ⁡ u , u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u, where α > 0 {\alpha>0} , and g : [ 0 , + ∞ [ → [ 0 , + ∞ [ {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and p : [ 0 , + ∞ [ → ℝ {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on [ 0 , + ∞ ] {[0,+\infty]} . We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

Fredholm’s third theorem for second-order singular Dirichlet problem

AbstractConsider the singular Dirichlet problem u″=p(t)u+q(t);u(a)=0,u(b)=0, where p,q:]a,b[→R are locally Lebesgue integrable functions. It is proved that if ∫ab(s−a)(b−s)[p(s)]−ds<+∞and∫ab(s−a)(b−s)|q(s)|ds<+∞, then Fredholm’s third theorem remains true.MSC:34B05.

Solvability of a nonlocal boundary value problem for linear functional differential equations

AbstractIn the paper, the problem on the existence and uniqueness of a solution to the nonlocal problem u′(t)=ℓ(u)(t)+q(t),u(a)=h(u)+c is considered, where ℓ:C([a,b];R)→L([a,b];R) and h:C([a,b];R)→R are linear bounded operators, q∈L([a,b];R), and c∈R.MSC:34K06, 34K10.

Solvability of a linear non-local boundary value problem for nonlinear functional differential equations

Abstract New sufficient conditions are established for the solvability as well as unique solvability of a linear non-local boundary value problem for nonlinear functional differential equations.

On constant sign solutions (nonpositive) of certain functional differential inequality

New sufficient conditions are found guaranteeing that every solution to the problem u′(t)⩾l(u)(t), u(a)⩾h(u) is nonpositive, where l:C([a,b];R)→L([a,b];R) is a linear bounded operator and h:C([a,b];R)→R is a linear bounded functional. Obtained results are concretized for the differential inequalities with argument deviations.