Existence of solutions for a third order differential equation with integral boundary conditions

• Published in: Computers & mathematics with applications (1987) Vol. 53; no. 1; pp. 144 - 154
• Main Authors: Wang, Youyu, Ge, Weigao
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 2007
• Subjects: Boundary conditions; Boundary value problems; Differential equations; Existence; Integral boundary conditions; Integrals; Leray–Schauder degree; Mathematical analysis; Mathematical models; Nagumo condition; Upper and lower solutions; Leray–Schauder degree; Nagumo condition; Upper and lower solutions; Integral boundary conditions; Existence
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2007.01.002

In this paper, we consider the following third order differential equation ( ϕ ( u ″ ) ) ′ + f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) = 0 , 0 < t < 1 , subject to the following integral boundary conditions { u ( 0 ) = 0 , u ′ ( 0 ) − k 1 u ″ ( 0 ) = ∫ 0 1 h 1 ( u ( s ) ) d s , u ′ ( 1 ) + k 2 u ″ ( 1 ) = ∫ 0 1 h 2 ( u ( s ) ) d s , where f : [ 0 , 1 ] × R 3 → R and h i : R → R are continuous and k 1 , k 2 ≥ 0 , ϕ ( u ) is a continuous and strictly increasing function with ϕ ( 0 ) = 0 , ϕ ( R ) = R , where R = ( − ∞ , + ∞ ) . The existence result to the above boundary value problem is obtained by applying the method of upper and lower solutions and Leray–Schauder degree theory.