Existence and Uniqueness Results of Coupled Fractional-Order Differential Systems Involving Riemann–Liouville Derivative in the Space <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>W</mi><mrow><msup><mi>a</mi><mo>+</mo></msup></mrow><mrow><msub><mi>γ</mi><mn>1</mn></msub><mo>,</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>W</mi><mrow><msup><mi>a</mi><mo>+</mo></msup></mrow><mrow><msub><mi>γ</mi><mn>2</mn></msub><mo>,</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with Perov’s Fixed Point Theorem

Abstract

This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces E=Wa+γ1,1(a,b)×Wa+γ2,1(a,b). Our strategy is to endow the space E with a vector-valued norm and apply the Perov fixed point theorem. An example is given to show the usefulness of our main results.

Keywords

• coupled system
• fractional differential equations
• Riemann–Liouville derivative
• generalized Banach space
• fixed point theorems
• convergent to zero matrix