Abstract
We study the existence of positive solutions for second-order differential equations with separated integral boundary conditions. The nonlinear part of the equation involves the derivative and may be singular for the second and third space variables. The result ensures existence of a positive solution when the parameters are in certain ranges. The proof depends on general properties of the associated Green’s function and the Krasnosel’skii–Guo fixed point theorem applied to a perturbed Hammerstein integral operator. Both numerical and analytical examples are constructed to illustrate applications of the theorem to a group of equations. The result generalizes previous work.
Highlights
We are interested in the following singular Boundary Value Problem (BVP) for second-order differential equations with non-local boundary conditions involving integrals:
In [14], existence of multiple positive solutions for nonlocal BVPs involving various integral conditions were obtained for the case that the nonlinear function f does not involve the first-order derivative
M − g1(s)h1(s)ds m − g2(s)h2(s)ds − g1(s)h2(s)ds g2(s)h1(s)ds = 0. This condition implies that BVP (2.1) has only the trivial solution:
Summary
We are interested in the following singular Boundary Value Problem (BVP) for second-order differential equations with non-local boundary conditions involving integrals:. In [14], existence of multiple positive solutions for nonlocal BVPs involving various integral conditions were obtained for the case that the nonlinear function f does not involve the first-order derivative. Results on non-existence of positive solutions for different types of nonlocal BVPs were discussed in [11].
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