Investigation of a Class of Implicit Anti-Periodic Boundary Value Problems

This research is devoted to studying a class of implicit fractional order differential equations ($\mathrm{FODEs}$) under anti-periodic boundary conditions ($\mathrm{APBCs}$). With the help of classical fixed point theory due to $\mathrm{Schauder}$ and $\mathrm{Banach}$, we derive some adequate results about the existence of at least one solution. Moreover, this manuscript discusses the concept of stability results including Ulam-Hyers (HU) stability, generalized Hyers-Ulam (GHU) stability, Hyers-Ulam Rassias (HUR) stability, and generalized Hyers-Ulam- Rassias (GHUR)stability. Finally, we give three examples to illustrate our results.

Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations

This paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the ϕ-Laplacian equation ( ϕ ( u ′ ) ) ′ + a ( t ) g ( u ) = 0 , (\phi \left(u^{\prime} ))^{\prime} +a\left(t)g\left(u)=0, where ϕ is a homeomorphism with ϕ(0) = 0, a(t) is a stepwise indefinite weight and g(u) is a continuous function. When dealing with the p-Laplacian differential operator ϕ(s) = ∣s∣ p−2 s with p > 1, and the nonlinear term g(u) = u γ with γ ∈ R \gamma \in {\mathbb{R}} , we prove the existence of a unique positive solution when γ ∈ ]− ∞ \infty , (1 − 2p)/(p − 1)] ∪ ]p − 1, + ∞ \infty [.

Optimal estimation of unknown data of periodic boundary value problems for first order linear impulsive systems of ordinary differential equations from indirect noisy observations of their solutions

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.