Abstract
In this article, we study a class of nonlinear fractional differential equation for the existence and uniqueness of a positive solution and the Hyers–Ulam-type stability. To proceed this work, we utilize the tools of fixed point theory and nonlinear analysis to investigate the concern theory. We convert fractional differential equation into an integral alternative form with the help of the Greens function. Using the desired function, we studied the existence of a positive solution and uniqueness for proposed class of fractional differential equation. In next section of this work, the author presents stability analysis for considered problem and developed the conditions for Ulam’s type stabilities. Furthermore, we also provided two examples to illustrate our main work.
Highlights
Fractional calculus is known as the generalization of traditional calculus
One of the important aspects of aforementioned field that attended the attention of large number of researchers has existence of the solution for boundary value problems (BVPs) of fractional differential equations (FDEs)
The area devoted to study boundary value problems via topological degree theory for classical order differential equations has been well studied by researches and published number of articles and books
Summary
Fractional calculus is known as the generalization of traditional calculus. In the last few decades, the aforesaid field attended more attention of researchers due to its variety of applications in diverse field of social science and physical science, like physics, chemistry, economics and mechanics. One of the important aspects of aforementioned field that attended the attention of large number of researchers has existence of the solution for boundary value problems (BVPs) of fractional differential equations (FDEs). Ali and Khan [27] study the following BVPs of FDEs with non-local boundary conditions involving fractional integral which is given by cD v(t) = f (t, v(t)), t ∈ J = [0, 1], T v(0) = g(v), v(1) = Γ(q) ∫ (t − s)q−1v(s)ds, where cD represents Caputo fractional derivatives and g(v) is non-local function, f ∶ J × R → R is continuous function Another important aspect of concerned theory, which attracted the attention of researchers, has the area devoted of stability analysis of BVPs of FDEs. There are various types of stabilities present in the literature of fractional calculus. In order to justify the desired results, we provide two examples in last section of the work
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