Abstract
In this paper, we investigate a class of nonlinear two-term fractional differential equations involving two fractional orders deltain(1,2] and tauin (0,delta) with integral boundary value conditions. By the Schauder fixed point theorem we obtain the existence of positive solutions based on the method of upper and lower solutions. Then we obtain the uniqueness result by the Banach contraction mapping principle. Examples are given to illustrate our main results.
Highlights
Fractional differential equations are an important tool to describe many processes and phenomena of science and engineering [1,2,3]
The theory of lower and upper solutions is known to be an effective method for proving the existence of solutions to fractional differential equations
To the best of our knowledge, no paper has considered the existence of positive solutions for nonlinear fractional differential equations with integral boundary conditions
Summary
Fractional differential equations are an important tool to describe many processes and phenomena of science and engineering [1,2,3]. We study positive solutions for the integral boundary value problems. Staněk [29], applying the Schauder fixed point theorem, considered the existence, multiplicity, and uniqueness of solutions to the periodic boundary value problem. To the best of our knowledge, no paper has considered the existence of positive solutions for nonlinear fractional differential equations with integral boundary conditions. We prove the existence of positive solutions to the boundary value problem (1.1)–(1.2) by the Schauder fixed point theorem and the method of upper and lower solutions. To use the fixed point theorem, according to Lemma 2.6, we define the operator T as. The functions x and x are called a pair of upper and lower solutions of the boundary value problem (1.1)–(1.2). The boundary value problem (1.1)–(1.2) has at least one positive solution x ∈ X, and x(t) ≤ x(t) ≤ x(t), t ∈ [0, 1]
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