It is well known that the primary tools in the proofs of existence and, sometimes, uniqueness of solution of elliptic boundary value problems of type (1) are (W^{2,2} -a priori bounds.
Consequently, we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type Neumann, Robin, or Wentzell, may be uniformly continuous (up to the boundary).
The many interests in (1.1) may stem from the fact that boundary value problems of type (1.1) model various dynamic systems with m degrees of freedom in which m states are observed at m times; see Meyer [1].
The recurrence sequences determined by equalities (5.1), (5.2) and (5.3), (5.4) arise in a natural way when boundary value problems of type (4.1), (4.1) and (4.3), (4.4) are considered.
In this section, the main results for a consistent boundary value problem of types (1) and (2) are analytically presented.
In this paper, motivated by the aforementioned works, we consider a more general class of boundary value problems of Caputo type fractional differential equations and non-separated type multi-point and multi-strip boundary conditions.
In this paper we investigate vector-valued parabolic initial boundary value problems of relaxation type.
Note that: If (r(t)equiv0) for all (tin[1,T]), then the problem (1.2) is reduced to the Sturm-Liouville fractional boundary value problem of Hadamard type of the form left { textstylebegin{array}{l} D^{beta} (p(t)D^{alpha}x t))=g t, x t)),quad 1< t< T, x(1)=-x(T),quad quad D^{alpha}x(1)=-D^{alpha}x(T).
If (r(t)equiv0) for all (tin[1,T]), then the problem (1.2) is reduced to the Sturm-Liouville fractional boundary value problem of Hadamard type of the form left { textstylebegin{array}{l} D^{beta} (p(t)D^{alpha}x t))=g t, x t)),quad 1< t< T, x(1)=-x(T),quad quad D^{alpha}x(1)=-D^{alpha}x(T).
In recent years, the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types.
