Then, we give a different proof for the existence and uniqueness of the solutions for linear coupled boundary value conditions for a differential system, i.e., Lemma 2.5.
There is large bibliography on papers related with lower and upper solutions with nonlinear boundary value conditions for first- and higher-order equations.
Remark 2. We remark that the methods used in this paper are also applicable for the cases of the other boundary value conditions, for example, Dirichlet boundary value conditions.
Up to now, coupled boundary value conditions for a fractional differential system with generalized p-Laplacian like system (1) have seldom been considered when (lambda_{1}), (lambda_{2}) are different.
It is a well-known fact that the boundary value problems with Sturm-Liouville boundary value conditions for integral order differential equations have important physical and applied background and have been studied in many literatures, while BVPs (1.3) and (1.4) are the nonlocal boundary value problems, which are not able to substitute BVP (1.5).
We will devote the paper to considering the existence of a solution of coupled integral boundary value conditions for the second-order ordinary differential system (ODS for short) { − x ″ ( t ) = f 1 ( t, x ( t ), y ( t ) ), t ∈ ( 0, 1 ), − y ″ ( t ) = f 2 ( t, x ( t ), y ( t ) ), t ∈ ( 0, 1 ), x ( 0 ) = y ( 0 ) = 0, x ( 1 ) = ∫ 0 1 y ( t ) d A ( t ), y ( 1 ) = ∫ 0 1 x ( t ) d B ( t ), (1.1).
In this article, we consider the following nonlinear impulsive fractional differentialequation with generalized periodic boundary value conditions (for short BVPs (1.1)): { D t q c u ( t ) = f ( t, u ( t ) ), t ∈ J ′ = J ∖ { t 1, …, t m }, J = [ 0, 1 ], Δ u ( t k ) = I k ( u ( t k ) ), Δ u ′ ( t k ) = J k ( u ( t k ) ), k = 1, …, m, a u ( 0 ) − b u ( 1 ) = 0, a u ′ ( 0 ) − b u ′ ( 1 ) = 0, (1.1).
By the way, the author has been interested in the boundary value condition of a degenerate parabolic equation for some time, one may refer to [9].
The variational method and critical point theory are employed to investigate the existence of solutions for 2 th-order difference equation for with boundary value condition by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional.
Then ({u_{n}(x)}) converges strongly to a point (p_{0}(x) in (L_{1}+L_{3})^{-1}0 ), which is the common solution of the Laplacian equation with Signorini boundary value condition (E) and the following variational inequality: for (forall z in (L_{1}+L_{3})^{-1}0), bigllangle (T-eta f p_{0}, p_{0}-z bigrrangle leq 0. (23).
