Abstract
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ− fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results.
Highlights
Quadratic integral equations have gained much attention and many authors studied the existence of solutions for several classes of nonlinear quadratic integral equations
Quadratic integral equations have been appeared in many useful application and problems of the real world
We proved the existence of the maximal and minimal solutions of the quadratic integral Equation (1)
Summary
Quadratic integral equations have gained much attention and many authors studied the existence of solutions for several classes of nonlinear quadratic integral equations (see e.g., [1,2,3,4,5,6,7,8,9,10,11]). Proved the existence of at least one positive nondecreasing continuous solution to the Equation (1) under the assumption that the functions f and g satisfy the conditions of the Carathèodory Theorem [14]. We proved the existence of the maximal and minimal solutions of the quadratic integral Equation (1). We shall generalize these results and obtain similar ones for the fractional quadratic φ− integral Equation (2), which in turn gives the existence as well as the existence of many key integral and functional equations that arise in nonlinear analysis and its applications. Iφα may be known as the fractional integral of the function f (t) with respect to φ(t), which is defined for any monotonic increasing function φ(t) ≥ 0, with a continuous derivative
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