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Positive solutions to nonlinear singular second order boundary value problems

100%
Bonanno G.
Annales Polonici Mathematici
|
1996
|
tom 64
|
nr 3
237-251
EN
Existence theorems of positive solutions to a class of singular second order boundary value problems of the form y'' + f(x,y,y') = 0, 0 < x < 1, are established. It is not required that the function (x,y,z) → f(x,y,z) be nonincreasing in y and/or z, as is generally assumed. However, when (x,y,z) → f(x,y,z) is nonincreasing in y and z, some of O'Regan's results [J. Differential Equations 84 (1990), 228-251] are improved. The proofs of the main theorems are based on a fixed point theorem for weakly sequentially continuous operators.
2
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Existence of solutions for a multivalued boundary value problem with non-convex and unbounded right-hand side

63%
Averna D., Bonanno G.
Annales Polonici Mathematici
|
1999
|
tom 71
|
nr 3
253-271
EN
Let $F:[a,b] × ℝ^n × ℝ^n → 2^{ℝ^n}$ be a multifunction with possibly non-convex and unbounded values. The main result of this paper (Theorem 1) asserts that, given the multivalued boundary value problem ($P_F$)    {u'' ∈ F(t,u,u'),                    u(a) = u(b) = ϑ_{ℝ^n}, if an appropriate restriction of the multifunction F has non-empty and closed values and satisfies the lower Scorza Dragoni property and a weak integrable boundedness type condition, then we can substitute the problem ($P_F$) with another one ($P_G$), with a suitable convex right-hand side G, such that every generalized solution of ($P_G$) is also a generalized solution of ($P_F$) (see also Remark 1 and Corollary 1). As a consequence of our results, in conjunction with those in [13] and [18], some existence theorems for multivalued boundary value problems are then presented (see Theorem 2, Corollary 2 and Theorem 3). Finally, some applications are given to the existence of generalized solutions for two implicit boundary value problems (Theorems 4-6).
3
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Infinitely many solutions for a mixed boundary value problem

63%
Bonanno G., Tornatore E.
Annales Polonici Mathematici
|
2010
|
tom 99
|
nr 3
285-293
EN
The existence of infinitely many solutions for a mixed boundary value problem is established. The approach is based on variational methods.
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