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In this paper, we shall derive explicit error estimates in L ∞ norm between a given function f ( x ) ∈ PC ( n ) [ a , b ], 4 ≤ n ≤ 6 and its quintic Lidstone-spline interpolate. The results obtained are then used to establish precise error bounds for the approximated and biquintic Lidstone-spline interpolates. We also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results.

We consider the following boundary value problems:(−1)n−py(n) = a(t)h(y), n ⩽ 2, t ∈(0,1),y(i)(0) = 0, 0 ⩾ i ⩾ p − 1, y(i)(1) = 0, 0 ⩾ i ⩾ n − p − 1,and(−1)n−pΔny = F(k,y,Δy,…,Δn−1y), n ⩽ 2, 0 ⩾ k ⩾ m,Δiy(0), 0 ⩾ i ⩾ p − 1, Δiy(m+p+1)=0, 0 ⩾ i ⩾ n − p − 1,where 1 ≤ p ≤ n − 1 is fixed. By employing fixed-point theorems for operators on a cone, existence criteria are developed for multiple (at least three) positive solutions of the boundary value problems. As an application, we also establish the existence of radial solutions of certain partial difference equations. Several examples are included to dwell upon the importance of the results obtained.

We offer sufficient conditions for the oscillation of all solutions of the partial difference equations y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+P(m,n,y(m−k,n−l)) = Q (m,n,y(m − k,n −l)) , and y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+ ∑ i=1 τ P i (m,n,y(m − k i , n − l i )) = ∑ i=1 τ Q i (m,n,y(m − k i , n − l i )) . Several examples, which dwell upon the importance of our results, are also included.

We consider the following boundary value problem: (−1) n−p Δ n y=λF(k,y,Δy,…,Δ n−1 y), n≪2, 0≤k≤m, Δ i y(0)=0, 0≤i≤p−1; Δ i y(m+n−i)=0, 0≤i≤n−p−1, where 1 ≤ p ≤ n − 1 is fixed and λ > 0. A characterization of the values of λ is carried out so that the boundary value problem has a positive solution. Next, for λ = 1, criteria are developed for the existence of two positive solutions of the boundary value problem. In addition, for particular cases we also offer upper and lower bounds for these positive solutions. Several examples are included to dwell upon the importance of the results obtained.

We shall provide existence criteria for double positive solutions of the (n,p) boundary value problem δ n y+F(k,y,δy,…,δ n-2 y)=G(k,y,δy…,δ n−1 y) , n−1⩽k⩽N , δ i y(0)=0 , 0⩽i⩽n−2 , δ p y(N+n−p)=0 , where n ≥ 2 and 0 ≤ p ≤ n − 1 is fixed. Upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when n = 2. Several examples are included to dwell upon the importance of the results obtained.

We consider the following differential equation on a time scale T, y^@D^@D(t)+P(t,y(@s(t)))=0,t@?[a,b]@?T, together with focal boundary conditions, y(a)=0,y^@D(@s(b))=0, where a,b @e T and a < @s (b). By using two different fixed-point theorems, criteria are established for the existence of triple positive solutions of the boundary value problem. We also include some examples to illustrate the results obtained.

We define Lidstone polynomials, provide their explicit representations, give their relations with Bernoulli polynomials, obtain best possible error estimates in Lidstone interpolation, and establish several best possible/sharp inequalities which compare favourably with the known results. Next, we use these results to study even order differential equations together with Lidstone boundary conditions.

We consider the following boundary value problem, (-1)^n^-^1y^@D^^^n(t)=(-1)^p^+^1F(t,y(@s^n^-^1(t))),[email protected]?[a,b]@?T, y^@D^^^n(a)=0,[email protected][email protected]?p-1, y^@D^^^n(@s(b))=0,[email protected][email protected]?n-1, where n >= 2, 1 @? p @? n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.

For the half-linear difference equation Δ[|Δy(k)|α−1Δy(k)]=∑i=1npi(k)|y(gi(k))|α−1y(gi(k)),k≥a where α > 0, we shall offer sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions. Several examples which dwell upon the importance of our results are also included.

We consider the following boundary value problems: (−1)n−py(n) = a(t)h(y), n ⩽ 2, t ∈(0,1), y(i)(0) = 0, 0 ⩾ i ⩾ p − 1, y(i)(1) = 0, 0 ⩾ i ⩾ n − p − 1, and (−1)n−pΔny = F(k,y,Δy,…,Δn−1y), n ⩽ 2, 0 ⩾ k ⩾ m, Δiy(0), 0 ⩾ i ⩾ p − 1, Δiy(m+p+1)=0, 0 ⩾ i ⩾ n − p − 1, where 1 ≤ p ≤ n − 1 is fixed. By employing fixed-point theorems for operators on a cone, existence criteria are developed for multiple (at least three) positive solutions of the boundary value problems. As an application, we also establish the existence of radial solutions of certain partial difference equations. Several examples are included to dwell upon the importance of the results obtained.