Abstract
In this paper we consider a nonlinear boundary value problem generated by a fourth order differential equation on the semi-infinite interval in which the lim-4 case holds for fourth order differential expression at infinity. Using the well-known Banach and Schauder fixed point theorems we prove the existence and uniqueness theorems for the nonlinear boundary value problem.
Highlights
1 Introduction In the literature a kind of first order nonlinear boundary value problems is of the form y = A(x)y + F(x, y), ( )
Where A is a n×n matrix defined on some interval I ⊂ R, F is a n× vector which is continuous on I × S, S ⊂ Rn, T is a bounded linear operator defined on the space of bounded and continuous Rn-valued functions on I and r is a n × vector in Rn
Similar nonlinear boundary value problem has been studied by Agarwal et al [ ] on a time scale [, ∞)T = [, ∞) ∩ T as y = A(x)y + F(x, y), ( )
Summary
Where A is a n×n matrix defined on some interval I ⊂ R, F is a n× vector which is continuous on I × S, S ⊂ Rn, T is a bounded linear operator defined on the space of bounded and continuous Rn-valued functions on I and r is a n × vector in Rn. Existence and uniqueness theorems of the solutions of the problem ( ), ( ) have been obtained in many papers. Second order nonlinear boundary value problems have been investigated by many authors. We should note that there are several works in the field of the existence and uniqueness theorems of the second order nonlinear boundary value problems.
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