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a b s t r a c tA Volterra system of difference equations of the formwhere n ∈ N, a s , b s , x s : N → R and K sp : N × N → R, s = 1, 2, . . . , r is studied. Sufficient conditions for the existence of asymptotically periodic solutions of this system are derived.Crown

Abstract. New explicit stability results are obtained for the following scalar linear difference equationand for some nonlinear Volterra difference equations.

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A linear Volterra difference equation of the formx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i),wherex:N0→R,a:N0→R,K:N0×N0→Randb:N0→R∖{0}isω-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on∏j=0ω-1b(j)is assumed. The results generalize some of the recent results.

Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problemẍ (t) = f (t, x (t)), x (0) = x (1) = 0 where f : [0, 1] × R → R is a jointly continuous function convex in x which does not need to satisfy any further growth conditions.

<abstract><p>In this work, some class of the fractional differential equations under fractional boundary conditions with the Katugampola derivative is considered. By proving the Lyapunov-type inequality, there are deduced the conditions for existence, and non-existence of the solutions to the considered boundary problem. Moreover, we present some examples to demonstrate the effectiveness and applications of the new results.</p></abstract>