Yongkun Li

Multiple positive solutions for first-order impulsive integral boundary value problems on time scales

In this paper, we first present a class of first-order nonlinear impulsive integral boundary value problems on time scales. Then, using the well-known Guo-Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem, some criteria for the existence of at least one, two, and three positive solutions are established for the problem under consideration, respectively. Finally, examples are presented to illustrate the main results.MSC: 34B10; 34B37; 34N05.

Multiple positive solutions for second-order -Laplacian dynamic equations with integral boundary conditions.

We are concerned with the following second-order -Laplacian dynamic equations on time scales , , with integral boundary conditions , . By using Legget-Williams fixed point theorem, some criteria for the existence of at least three positive solutions are established. An example is presented to illustrate the main result.

Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a -Laplacian conformable fractional differential equation boundary value problem on time scale , , , where denotes the conformable fractional derivative of of order at , is the forward jump operator, , and . By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects

AbstractIn this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effectsn {uΔ2(t)+A(σ(t))u(σ(t))+∇F(σ(t),u(σ(t)))=0,Δ-a.e.xa0t∈[0,T]Tκ;u(0)−u(T)=uΔ(0)−uΔ(T)=0,(ui)Δ(tj+)−(ui)Δ(tj−)=Iij(ui(tj)),i=1,2,…,N,j=1,2,…,p,n where t0=0<t1<t2<⋯<tp<tp+1=T, tj∈[0,T]T (j=0,1,2,…,p+1), u(t)=(u1(t),u2(t),…,uN(t))∈RN, A(t)=[dlm(t)] is a symmetric N×N matrix-valued function defined on [0,T]T with dlm∈L∞([0,T]T,R) for all l,m=1,2,…,N, Iij:R→R (i=1,2,…,N, j=1,2,…,p) are continuous and F:[0,T]T×RN→R. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.MSC:34B37, 34N05.

Multiple positive solutions for a fourth-order integral boundary value problem on time scales

In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnoselskill. To demonstrate the applications of our results, an example is also given in the article.

Extremal Solutions of Periodic Boundary Value Problems for First-Order Impulsive Integrodifferential Equations of Mixed-Type on Time Scales

We consider the existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive integrodifferential equations of mixed-type on time scales by establishing a comparison result and using the monotone iterative technique.

Three Solutions for Forced Duffing-Type Equations with Damping Term

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. , .