In this work, we discuss the existence, uniqueness and uniform of the solution of stability non-local problem (1 - 3).
Now we complete the proof by proving the equivalence between the integral equation (4) and the non-local problem (1 - 3).
El-Sayed et al. [12] proved the existence of a unique uniformly stable solution of the non-local problem D α x i ( t ) = ∑ j = 1 n a i j ( t ) x j ( t ) + ∑ j = 1 n b i j ( t ) x j ( t - r j ) + h i ( t ), t > 0, x ( t ) = Φ ( t ) f o r t < 0, lim t → 0 - Φ ( t ) = O a n d I β x ( t ) | t = 0 = O, β ∈ ( 0, 1 ).
Recently, a lot of attention has been focused on the study of fractional and non-local problems; see [2 6].
The resulting equations are composed of a system of fourth order PDEs coupled with a non-linear, non-local optimization problem to determine the conjugate fields.
Wang et al. [26] investigated the existence theory and proved some conditions for uniqueness and derived some data dependency results of solutions using topological degree technique by considering some classes of non-local Cauchy problems including BVPs and impulsive Cauchy problems (ICPs) to FDEs.
"It can be a very local problem.
This is no local problem.
This is not just a local problem.
And, of course, well-posed solvability of the Cauchy problem and non-local boundary value problems for operator-differential equations as well as related spectral problems (see, e.g., Shkalikov [13], Gorbachuk and Gorbachuk [14], Agarwal et al. [15]) are also of great interest.
In [16], Hao et al. gave a very nice approximation for this problem by using a non-local boundary value problem method.
